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Using calculators to teach algebra : from numbers to operations to structure Murphy, Maureen 1996

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USING CALCULATORS TO TEACH ALGEBRA: FROM NUMBERS TO OPERATIONS TO STRUCTURE by MAUREEN THERESE MURPHY . S c . , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n THE FACULTY OF GRADUATE STUDIES Depar tmen t o f C u r r i c u l u m S t u d i e s We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRIT ISH COLUMBIA December 1996 ®Maureen T h e r e s e Murphy , 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of. this thesis for financial gain shall, not be allowed without my written permission. Department of ^ ^ w - L H t ^ - t l ^ t - ^ . - ^£^L^<_,tL^c^, The University of British Columbia Vancouver, Canada Date • .2-? . / f f j -DE-6 (2/88) Using Calculators to Teach Algebra i i Abstract The problems of teaching mathematics to unmathematical high school students are described and analyzed i n terms of d i f f e r e n c e s i n language, e s p e c i a l l y syntax, between arithmetic and algebra. There follows a d e t a i l e d d e s c r i p t i o n of a demonstrably successful a l t e r n a t i v e to s o - c a l l e d regular mathematics courses which depends e s s e n t i a l l y on the use of e l e c t r o n i c c a l c u l a t o r s . Over three years, 274 students i n 11 of the author's own mathematics classes, where algebra was taught i n t h i s a l t e r n a t i v e way, were observed f o r signs of mathematical i n s i g h t . Relevant anecdotes were recorded, analyzed, and categorized and are discussed so as to j u s t i f y the claims that, by using e l e c t r o n i c c a l c u l a t o r s , teachers can s h i f t the attention of students from numbers to mathematical operations and from operations to the o v e r a l l structure of algebraic expressions, and that students who are taught algebra i n t h i s way reveal spontaneously, both o r a l l y and i n wr i t i n g , that they have an understanding of algebra. F i n a l l y , i t i s suggested that the d i f f i c u l t i e s encountered by many students i n high school algebra could be reduced or eliminated i f unmathematical students were taught i n t h i s way alone and mathematical students were taught i n t h i s way f i r s t , before proceeding to s o - c a l l e d regular mathematics courses. i i i TABLE OF CONTENTS Abstract i i Table of Contents i i i L i s t of Tables v Li s t of Figures v i Chapter One Teaching Mathematics to Unmathematical Students 1 Observable Characteristics 1 Underlying Problems 2 Attempts at Solutions 4 Understanding from Doing 6 Successful Practice 7 Research Questions 9 Consequences 10 Chapter Two Mathematical Language in Arithmetic and Algebra 12 Same Words, Different Meanings 12 Variables 13 Operational Structure 14 Equals Signs 18 Solving Equations 19 Conclusion 23 Chapter Three Syntax of Mathematical Language 25 Syntax of Arithmetic 26 Syntax of the Reversal Method 30 Syntax of Conventional Algebra 31 Comparison of Conventional and Reversal Syntax 33 Dif f icul ty of Reification 35 Algebra Without Reification 37 Conclusion 40 Chapter Four Role of the Calculator 42 Responsibilities of Calculators and Students 42 Understanding from Doing 47 Conclusion 54 Chapter Five Research methods 57 Characteristics of Students Studied 57 Basic, "Star," and "Double-Star" Work 58 Classroom Act iv i t i es and Course Requirements 61 Quantitative Data About Students Studied 63 Data Collection: Attempts at Interviews 65 Reasons for Abandoning Interviews 72 Collecting Anecdotes 75 Recording, Transcribing, and Analyzing Anecdotes 79 Using Calculators to Teach Algebra i v G e n e r a l i z a b i l i t y and Limits 83 Chapter Six Expressing Mathematical Insight 85 Communication i n Calculator Language 85 Brackets 88 Non-Algebraic Calculator Language 95 Order of Operations 99 Use of Equals Sign 105 Differences Between Input and Execution 110 Conclusion 120 Chapter Seven Bridging the Gap Between Arithmetic and Algebra 122 Operations Versus Numbers 122 T r a n s i t i o n from Numbers to Letters 127 Nature of Variables 135 Structure of Expressions 138 Groups: S t a t i c Algebraic Expressions 149 Authority Behind "The Right Answer" 162 Conclusion 164 Chapter Eight Conclusion and Consequences 165 Conclusion 169 Consequences 177 Bibliography 173 Appendix 1 Mathematics 11A F i n a l Examination 175 Appendix 2 Students' Records i n Previous Mathematics Courses 179 Appendix 3 Use of the E l e c t r o n i c Calculator 183 Order of Operations 183 Two D i f f e r e n t Types of Calculator 191 Non-Algebraic Calculators 191 Algebraic Calculators 196 Appendix 4 Reversal Method of Solving Equations 200 Formulae 200 Reversing a Formula 201 Brackets 204 D i v i s i o n Lines as Brackets 204 Reversal of Powers 205 Groups 207 Solving for an Exponent 209 Multi-Step Solutions 210 Appendix 5 Two Methods of Poetic Description 215 Appendix 6 Outline of Mathematics 8A-11A 216 Using Calculators to Teach Algebra v LIST OF TABLES Table 1 Conventional vs. Reversal Syntax 33 Table 2 < Transformation of Letter Grades 60 Table 3 Actual Letter Grade Distribution 64 Table 4 Codification of Themes 80 Table 2 -A Symbols Used in Students' Records 179 Table 2-B Records of Students Taught By Terry Baker Between September 1991 and January 1 9 9 2 1 8 0 Table 2 -C Records of Students Taught By Terry Baker Between September 1 9 9 2 and January 1 9 9 3 181 Table 2-D Records of Students Taught by Maureen Murphy Between January 1 9 9 3 and June 1 9 9 3 1 8 2 Table 6 -A Contents of Mathematics 8 A - 1 1 A 2 1 7 Using Calculators to Teach Algebra v i LIST OF FIGURES Figure 1 D i s t r i b u t i o n of Letter Grades 65 Figure 2 D i s t r i b u t i o n of Anecdotes 82 Using Calculators to Teach Algebra 1 Chapter 1 Teaching Mathematics to Unmathematical Students The problem of how to teach mathematics to unmathematical students i s apparently never-ending. V a r i a t i o n i n a b i l i t i e s i s taken f o r granted i n e l e c t i v e school subjects such as a r t , music, and drama, to the point where no one i s considered u n i n t e l l i g e n t because he or she cannot draw, sing, or act. However, i t i s l e s s taken f o r granted i n mathematics, to the point where mathematics i s a made a compulsory subject i n B r i t i s h Columbia at l e a s t up to Grade 11; most educators, mathematical or not, think that a l l students could be successful i n the "regular" mathematics courses i f they were taught appropriately; and a great many unmathematical adults f e e l i t necessary to devise excuses for, or to conceal, t h e i r lack of success i n mathematics at school. Observable C h a r a c t e r i s t i c s I f i r s t encountered the phenomenon of unmathematical students i n 1974 as a beginning teacher of General Mathematics 10, whose intended " c l i e n t s " were students who had had l i t t l e or no success as determined by examinations i n the "regular" mathematics courses. I soon found that they had l i t t l e of what I, with a background of success i n mathematics up to fourth year u n i v e r s i t y honours, could recognize as "understanding" of mathematics. F i r s t , they were, i n general, poor at r e c a l l i n g number f a c t s . Second, they had d i f f i c u l t y determining on t h e i r own what Using Calculators to Teach Algebra 2 mathematical operations to perform on the numbers they were given in a problem so as to obtain the correct answer. Although they could sometimes determine when to add and when to subtract, they could almost never determine when to multiply and when to divide and, given the choice, they nearly always preferred to add rather than multiply, e.g. to determine the number of months in five years. Third, they could not apply the rules of algebra for simplifying or evaluating expressions or for solving equations. They seemed to have no understanding of what I saw as the structure of arithmetic or algebraic expressions or any of the formulae with which they were provided to help them solve problems. Underlying Problems It seemed to me that the problem was primarily one of language and communication. Attempting to teach mathematics to such unmathematical students was like speaking to people who did not understand my language, or rather people who understood the meaning of some of the words I used but not the syntax. To be sure, the students were a l l English-speaking: they could isolate in English what they were being told and what they were being asked when they were given a word problem. However, they could not translate the problem into mathematical language, which uses mathematical symbols and syntax, and they could not transform mathematical sentences according to the rules of algebra Using Calculators to Teach Algebra 3 so as to i s o l a t e one v a r i a b l e . To help students t r a n s l a t e a word problem into mathematical language, I attempted to devise rules based on what I c a l l e d "word clues," e.g. "and" i s a prompt to add or " l e s s " i s a prompt to subtract, but I found myself unable to do so because problems could be worded i n English i n so many d i f f e r e n t ways: a word that might s i g n a l "add" i n one problem could s i g n a l "subtract" i n another. The fa c t that mathematics i s a language i s suggested by the t i t l e of Pimm's book (1987) Speaking Mathematically. Mathematicians are seen by others p r i m a r i l y as people who communicate with each other about c e r t a i n kinds of things (such as numbers, geometrical shapes, functions, etc.) and therefore make only c e r t a i n kinds of statements (such as "y i s d i r e c t l y proportional to x" or " t h i s t r i a n g l e i s congruent to that one"). Unlike French or German, mathematics i s not an a l t e r n a t i v e language i n which to express common ideas, but a language whose purpose i s to communicate ideas which could not otherwise be expressed, expressed so exactly, or expressed so economically. Knowing mathematics involves aural comprehension, speech, reading, and writ i n g . I t includes knowledge of s p e l l i n g , pronunciation, syntax, and vocabulary (Pimm, 1987, p. 3). It also involves "communicative competence" - knowing how to use language appropriate to context. "A general p r i n c i p l e i n teaching any kind of communicative competence, spoken or written, i s that the speaking, l i s t e n i n g , w r i t i n g or reading should have U s i n g C a l c u l a t o r s t o T e a c h A l g e b r a 4 some g e n u i n e c o m m u n i c a t i v e p u r p o s e " ( S t u b b s , 1980, q u o t e d i n Pimm, 1987, p . 4 ) . A t t e m p t s a t S o l u t i o n s T h r o u g h o u t my t e a c h i n g c a r e e r I have c o n t i n u e d t o t e a c h s t u d e n t s who e x h i b i t s i m i l a r l a c k o f i n s i g h t . In 1979, I b e g a n t o t e a c h Consumer M a t h e m a t i c s 11, t h e n one o f two t e r m i n a l c o u r s e s i n m a t h e m a t i c s f o r " u n m a t h e m a t i c a l " s t u d e n t s ( r e p l a c e d some y e a r s l a t e r by M a t h e m a t i c s 11A) , where I e n c o u n t e r e d t h e same p r o b l e m s as i n G e n e r a l M a t h e m a t i c s 10. The c o n t e n t s o f t h e c o u r s e were markdown and s a l e p r i c e , t a x , s i m p l e i n t e r e s t , compound i n t e r e s t , mor tgages and a n n u i t i e s . C o m p u t a t i o n s i n t h e l a s t t h r e e t o p i c s were p e r f o r m e d w i t h t h e a i d o f s t a n d a r d i n t e r e s t and mor tgage t a b l e s . U n l i k e Grade 10 s t u d e n t s , t h e s e Grade 11 s t u d e n t s were n o t e x p e c t e d t o t a k e any f u r t h e r m a t h e m a t i c s c o u r s e s , s o I was n o t a c c o u n t a b l e t o any f u t u r e t e a c h e r . To s a t i s f y m y s e l f more t h a n my s t u d e n t s , I began t o m o d i f y t h e c u r r i c u l u m a c c o r d i n g t o t h r e e g u i d i n g c r i t e r i a : 1. I wou ld t e a c h no s u b j e c t i n w h i c h t h e s t u d e n t s c o u l d n o t , as a m a t t e r o f o b s e r v a b l e f a c t , remember how t o s o l v e t h e p r o b l e m s . 2 . I wou ld keep t o a minimum what I gave t h e s t u d e n t s ( f o r m u l a e , d a t a , t a b l e s , e t c . ) i n o r d e r f o r them t o s o l v e t h e p r o b l e m s . 3 . The c o u r s e would s a t i s f y me i n t e l l e c t u a l l y , and be r U s i n g C a l c u l a t o r s t o T e a c h A l g e b r a 5 r e c o g n i z a b l e by t h e s t u d e n t s , a s a m a t h e m a t i c s c o u r s e . T h e s e c r i t e r i a s p r a n g f r om my own p e r c e p t i o n s o f , and d e l i g h t i n , m a t h e m a t i c s a s a s u b j e c t 1. w h i c h d i d n o t r e q u i r e m e m o r i z a t i o n b e c a u s e i t s p r o p o s i t i o n s were s o o b v i o u s and s o l o g i c a l l y n e c e s s a r y . 2 . i n w h i c h , t h e r e f o r e , one c o u l d a l w a y s be s u r e o f s u c c e s s on exams . 3 . w h i c h c o u l d a l w a y s p r o v i d e t h e s t i m u l a t i o n o f c h a l l e n g e . I n a c c o r d a n c e w i t h t h e s e c r i t e r i a , I g r a d u a l l y i n t r o d u c e d t h r e e i n n o v a t i o n s . The f i r s t , i n an a t t e m p t t o s o l v e t h e p r o b l e m t h a t s o many o f my s t u d e n t s c o u l d n o t r e c a l l number f a c t s , was t o e n c o u r a g e them t o u s e e l e c t r o n i c c a l c u l a t o r s , w h i c h were j u s t b e c o m i n g common among h i g h s c h o o l s t u d e n t s . A t t h e same t i m e , c a l c u l a t o r s w o u l d make mor tgage and a n n u i t y t a b l e s o b s o l e t e . I a r g u e d ( i m p l i c i t l y ) t h a t i t was more e c o n o m i c a l ( i n t h e l o g i c a l s e n s e ) f o r s t u d e n t s t o c a r r y a r o u n d c a l c u l a t o r s t h a n books o f a d d i t i o n , m u l t i p l i c a t i o n , mo r tgage o r a n n u i t y t a b l e s . Howeve r , w i t h t h e a i d o f c a l c u l a t o r s , s t u d e n t s c o u l d a l l , i m m e d i a t e l y , e v a l u a t e a r i t h m e t i c e x p r e s s i o n s w h i c h c o n t a i n e d o n l y one o p e r a t i o n , e.g. 150 + 7 5 , s o I began a s k i n g them t o e v a l u a t e a r i t h m e t i c e x p r e s s i o n s c o n t a i n i n g more t h a n one o p e r a t i o n , e.g. 150 + 75 x 3 . I n t h e e a r l y 1 9 8 0 s , most o f t h e s t u d e n t s owned what I c a l l n o n - a l g e b r a i c c a l c u l a t o r s - c a l c u l a t o r s w h i c h e x e c u t e t h e o p e r a t i o n s o f a d d i n g , s u b t r a c t i n g , e t c . i n t h e o r d e r i n w h i c h t h e Using C a l c u l a t o r s to Teach Algebra 6 operat ion buttons are pressed. Therefore , for students to use such non-a lgebra ic c a l c u l a t o r s s u c c e s s f u l l y , I f i r s t had to make them f a m i l i a r with the conventional order of operat ions so that they would be able to make the t r a n s i t i o n from reading an a r i t h m e t i c express ion such as 150 + 75 x 3 to press ing the buttons 75 x 3 + 150 = on a c a l c u l a t o r . Second, s ince my students could not r e l i a b l y determine what operat ions to perform on the numbers i n a problem so as to answer the ques t ion , I began to give them the formulae or equations they needed to so lve problems. However, on the p r i n c i p l e of economy, I was u n w i l l i n g to g ive them every mathematical v a r i a t i o n on each formula. For example, while I was w i l l i n g to give them the formula I = P r t , I was not w i l l i n g to give them i n a d d i t i o n P = l / ( r t ) , r = l / ( P t ) , and t = l / ( P r ) . Accord ing ly , I made my t h i r d innovat ion: I modif ied , expanded, and began to teach a new method of s o l v i n g equations, the germ of which had been suggested to me by a co l league , which used the same language I had used to teach the use of c a l c u l a t o r s . Understanding from Doing I t w i l l be seen that these three innovations were e s s e n t i a l l y p r a c t i c a l : i f I found that my students could not, as a matter of observat ion , do what I was r e q u i r i n g them to do, I s u b s t i t u t e d something e l s e . My philosophy was to give the students whatever they needed ( a l b e i t a minimum) to be success fu l : i . e . to pass a Using Calculators to Teach Algebra 7 f i n a l exam on the problems I was requiring them to solve. My emphasis was on doing rather than on understanding. I argued that anything mathematical I could get my students to do with success, even with the aid of a calculator, was better than nothing. However, despite this motivation for the changes I made, I gradually began to see and encourage in my students signs of understanding and even creativity. Apparently, doing, understanding, and creating were linked in ways I had not suspected, as described by Pimm (1995). Successful Practice Once students had learned how to use calculators e f f i c i e n t l y and how to solve an equation for any variable i t contained, they could learn to solve word problems for which they were given the relevant formulae. For the rest of the course, therefore, I set problems in simple interest, compound interest, saving with regular deposits, and financing with regular payments. Appendix 1, comprising the students' f i n a l exam, shows the formulae and definitions with which they were provided. In each problem, students were required 1. to recognize the kind of problem (simple interest, compound interest, saving, or financing). 2. to extract the numerical data and to associate each datum with the appropriate variable. 3. to select the appropriate formula. Using C a l c u l a t o r s to Teach Algebra 8 4. to solve the formula for the unknown v a r i a b l e . 5. to evaluate the unknown v a r i a b l e . I have now taught approximately 1000 students t h i s Mathematics 11A course. The students wri te a one-quest ion qu iz every day, a ha l f -hour quiz every three days, four hour- long exams, and a f i n a l exam. Moreover, i n 1993, with a few days' n o t i c e , I was asked by my v i c e p r i n c i p a l and my department head to teach a s i m i l a r course to two c lasses of Grade 8 students who had been i d e n t i f i e d by t h e i r elementary school teachers as l i k e l y to have d i f f i c u l t y with the "regular" Mathematics 8 course . I began w r i t i n g and teaching a s i m i l a r course to these students, although with some modi f i ca t ions , necessary because of the students' immaturity. In 1994-95 I taught Mathematics 9A to some of these students as wel l as others who had not taken my Mathematics 8A course, and i n 1995-96 I taught Mathematics 10A to the same students as we l l as others who had taken only one or even ne i ther of my previous courses . In each case I wrote the course as I taught i t . I have now taught a t o t a l of approximately 1100 students i n my Mathematics 8A, 9A, 10A and 11A courses . Among those students I have found s i x who were, i n my p r o f e s s i o n a l op in ion , t r u l y unable to pass. A number of others have f a i l e d courses once, taken them again, and passed the second time, or even f a i l e d them twice . However, i n my opin ion and i n the opinions of the students themselves - opinions which they often assert p u b l i c l y i n c l a s s the second year to help b o l s t e r my a s s e r t i o n that my course i s Using C a l c u l a t o r s to Teach Algebra 9 passable - they f a i l e d because they d i d not at tend a l l the c las ses and/or d i d not work when they d i d at tend. Many of these students have had no, or almost no, success i n previous mathematics courses . Appendix 2 gives the h igh school records of three c lasses of these students , one taught Mathematics 11A by me, the others by another teacher. For example, one student who f i n i s h e d Mathematics 11A with 75% had passed no mathematics courses, modified or otherwise, s ince Grade 6. Two other teachers have now used my m a t e r i a l s . Appendix 2 gives the r e s u l t s obtained by one of them i n two c la s se s : they show that she had the same kind of success with the same k ind of students as I have had. Research Questions I have found that the observable, t e s tab le success of students i n my courses requires some emphasis, because again and again when I have of fered my mater ia l to other teachers , they have d e c l i n e d i t on the grounds that i t would be too d i f f i c u l t f or t h e i r students . However, even when they accept the fac t that unmathematical students can be and are success fu l i n these courses , teachers who have not taught from my mater ia l often express two reservat ions about i t . The f i r s t i s based on a fear that my students are merely repeat ing by r o t e , without anything that teachers would recognize Using C a l c u l a t o r s to Teach Algebra 10 as understanding, r u l e s which are so s t ruc tured as to make them f o o l - p r o o f . The second i s based on an impression that my mater ia l i s so d i f f e r e n t from the contents of the "regular" courses that i t cannot r e a l l y be d i g n i f i e d with the name of mathematics. Accord ing ly , t h i s thes i s addresses the fo l lowing quest ions: 1 . Is i t p o s s i b l e to s h i f t the a t t e n t i o n of students from numbers to operations by the use of e l e c t r o n i c c a l c u l a t o r s ? 2. Is i t p o s s i b l e to s h i f t the a t t e n t i o n of students from operations to the overall structure of a lgebra i c expressions and equations by the use of e l e c t r o n i c c a l c u l a t o r s ? 3. Do the responses of students who have been taught a lgebra i n c a l c u l a t o r language show that they understand algebra? I f the answers to these questions are "yes," then not only are my mathematics courses v i a b l e a l t e r n a t i v e s to the "regular" mathematics courses for unmathematical students , but a l s o , perhaps, they can serve as bridges between ar i thmet i c and a lgebra for students who w i l l be capable, before they leave h igh schoo l , of pass ing the "regular" courses . Consequences Moreover, my f indings w i l l have consequences for the study of how mathematics i s learned, taught, conceptual ized , and performed - a study which i n t e r e s t s some educators for i t s own sake, apart from i t s a p p l i c a t i o n to the design of mathematics courses . Accord ing ly , i n the re s t of t h i s t h e s i s , I Using C a l c u l a t o r s to Teach Algebra 11 1 . analyze how the language of a lgebra d i f f e r s from the language of a r i t h m e t i c . 2. recount how I teach the use of c a l c u l a t o r s , so as to make the reader f a m i l i a r with my language. 3. demonstrate how I use the same language i n teaching the "reversa l" method of s o l v i n g equations. 4. analyze the syntax of mathematical language of c a l c u l a t o r s , focuss ing on the d i f f erences between the language I use and the language used i n "regular" high school a lgebra courses . 5. analyze the funct ion and the author i ty of c a l c u l a t o r s as I use them. 6. descr ibe more f u l l y the students I teach. 7. report classroom observations i l l u s t r a t i n g the nature of the students' understanding of the mathematics I teach. 8 . argue that my mathematics courses are v i a b l e and a l together s a t i s f a c t o r y a l t e r n a t i v e s to the "regular" mathematics courses for unmathematical students. 9. argue that the success of my mathematics courses sheds l i g h t on the l e a r n in g and therefore on the teaching of mathematics. Using C a l c u l a t o r s to Teach Algebra 12 Chapter 2 Mathematical Language i n Ar i thmet ic and Algebra Pimm (1987) claimed that knowing mathematics involves a u r a l comprehension, speech, reading , and w r i t i n g . I t inc ludes knowledge of s p e l l i n g , pronunc ia t ion , syntax, and vocabulary . I now turn to a review of the mathematical language encountered by students i n high school , focus ing on the d i f f erences between the language of ar i thmet i c and the language of a lgebra and how these d i f f erences can become stumbling b locks for beginning a lgebra students . Same Words, D i f f e r e n t Meanings In an ana lys i s of the r o l e of language i n communication i n genera l , von G l a s e r s f e l d (1988) s a i d : . . . l anguage users must i n d i v i d u a l l y construct the meaning of words, phrases, sentences, and t e x t s . . . . the b a s i c elements out of which an i n d i v i d u a l ' s conceptual s t ruc tures are composed and the r e l a t i o n s by means of which they are he ld together cannot be t r a n s f e r r e d from one language user to a n o t h e r . . . (p. 10) Once we come to see t h i s e s s e n t i a l and inescapable s u b j e c t i v i t y of l i n g u i s t i c meaning, we can no longer maintain the preconceived not ion that words convey ideas or knowledge; nor can we be l i eve that a l i s t e n e r who apparently "understands" what we say must n e c e s s a r i l y have conceptual s t ruc tures that are i d e n t i c a l with ours . (p. 11) For example, Rotman (1991), i n a study of students at Lansing Community Col lege i n Michigan USA, found that " t y p i c a l a r i t h m e t i c courses do l i t t l e to prepare students to master algebra" (p. 1) and that students who were i n i t i a l l y unsuccessful i n a lgebra d i d Using C a l c u l a t o r s to Teach Algebra 13 not p r o f i t from the c o l l e g e ' s pre -a lgebra course, which comprised a review of a r i t h m e t i c . I t appears that although algebra i s perce ived by mathematicians to be a g e n e r a l i z a t i o n of a r i t h m e t i c , and although many of the symbols used i n a lgebra are the same as those used i n a r i t h m e t i c , d i f f erences i n the meanings of the language make the two subjects appear to many students to be almost u n r e l a t e d . S i m i l a r , although l e s s e r , misunderstandings occur when people from d i f f e r e n t Engl i sh-speaking countr ies t a l k to each other or when educated people t a l k to uneducated; they are "divided by a common language." V a r i a b l e s V a r i a b l e s represent for students perhaps the greatest d i f f e r e n c e between ar i thmet i c and a lgebra . As they graduate from a r i t h m e t i c to a lgebra , l e t t e r s replace numbers. Pimm (1995) points out that t h i s move "can provoke a sense of r e a l loss" (p. 90) on the part of students . Kieran (1989) d iscusses at some length the problems beginning a lgebra students have with v a r i a b l e s , p a r t l y because there are so many d i f f e r e n t kinds of v a r i a b l e . Fol lowing Kuchemann and C o l l i s (1975) , she l i s t s them as fo l lows: (a) L e t t e r evaluated: The l e t t e r i s assigned a numerical value from the outset . (b) L e t t e r not used: The l e t t e r i s ignored or i t s existence i s acknowledged without g i v i n g i t a meaning. (c) L e t t e r used as an object : The l e t t e r i s regarded as a shorthand for an object or as an object i n i t s Using Calculators to Teach Algebra 14 own r i g h t . (d) L e t t e r used as a s p e c i f i c unknown: The l e t t e r i s regarded as a s p e c i f i c but unknown number and can be operated on d i r e c t l y . (e) Letter used as a generalized number: The l e t t e r i s seen as representing, or at le a s t being able to take on, several values rather than j u s t one. (f) L e t t e r used as a va r i a b l e : The l e t t e r i s seen as representing a range of unspecified values, and a systematic r e l a t i o n s h i p i s seen to e x i s t between two such sets of values. (p. 42) Operational Structure Although l e t t e r s appear i n addition to numbers i n the move from arithmetic into algebra, students continue to see the f a m i l i a r operations signs +, - , x , /, and 4- (although the x sign, alone among the others, may now be omitted, perhaps f o r the f i r s t time). They also continue to see the same meaning attached to a small number or l e t t e r written somewhat above the l i n e - an exponent - i n d i c a t i n g the operation of " r a i s i n g to" a power, which has no conventional symbol of i t s own. The d i f f e r e n c e i s that i n algebra, unhappily f o r students, the operations can no longer be c a r r i e d out. I say "unhappily" because, according to Booth (1984), quoted by Kieran (1989), "Children consider mathematics to be an empirical subject which requires the production of numerical answers" (p. 39). Pimm (1987) notes that i n arithmetic, "merely w r i t i n g 6 + 9" i s "perceived as a powerful prompt to carry out the a d d i t i o n " (p. 167). As Pimm (1995) puts i t , "Operation symbols i n algebra are v i r t u a l and not actual as they seem to be i n arithmetic" (p. 88). Using Calculators to Teach Algebra 15 In the move from arithmetic to algebra, a t t e n t i o n s h i f t s away from the performance of the operations i n an expression - known as "getting the answer." What meaning, therefore, can beginning algebra students attach to the once f a m i l i a r mathematical operations: +, - , x , and exponents? For students who appear i n my mathematics classes f o r the f i r s t time, the answer seems to be "none." Yet an understanding of the operations i s key to an understanding of algebra. In c l a s s , Pimm pointed out that when geometrical objects formed on the computer screen by Geometer's Sketchpad are changed, they " r e t a i n the memory of how they were constructed." For. example, one can construct a l i n e segment, then i t s perpendicular b i s e c t o r . If one now "drags" one end of the o r i g i n a l l i n e segment, the perpendicular bisector moves as well so as to maintain i t s i d e n t i t y as the perpendicular b i s e c t o r of each of the successive new l i n e segments. In t h i s respect, Pimm pointed out, geometrical objects created with t h i s computer program are d i f f e r e n t from those created with p e n c i l and paper. In an analogous way, algebraic expressions can be sa i d to r e t a i n the memory of the operations that were i n some sense performed at t h e i r c reation and of the order i n which these operations were performed. Indeed, the language mathematicians use, and expect t h e i r students to use, to describe algebraic expressions bears witness to t h i s memory, as surnames bear witness to the occupations of our ancestors (Cooper, Taylor) and the names Using C a l c u l a t o r s to Teach Algebra 16 of towns to events i n t h e i r h i s t o r y , reasons for t h e i r ex is tence , e t c . (cf. C a r r y i n g - P l a c e i n Ontario or 100 M i l e House i n B r i t i s h Columbia). For example, mathematicians descr ibe a 2 - b 2 as the "difference" of two squares, implying that the squaring was done before the s u b t r a c t i o n . In exp la in ing that (a - 2 ) 2 + (b - 3 ) 2 cannot be fac tored , we are l i k e l y to c a l l i t the "sum" of two squares, implying that a d d i t i o n was the l a s t operat ion performed, that squaring immediately preceded it> and that the subtract ions can be ignored. In exp la in ing how to so lve (x + 2)(x - 3) = 0, we note that the l e f t s ide of the equation i s a "product," imply ing that m u l t i p l i c a t i o n was the l a s t operat ion performed. Moreover, u n t i l students know how to d i s t i n g u i s h between an a l g e b r a i c express ion which i s ( in t h i s sense) a product and one which i s a sum, they have not learned to "factor" or " s i m p l i f y . " I t appears, then, that although the ar i thmet i c operat ions +, - , x, and ra i s ing- to -a -power cannot be performed with l e t t e r s as they can with numbers, i t i s the operat ions that were performed " v i r t u a l l y " i n the cons truc t ion of an a l g e b r a i c express ion, together with the order i n which they were performed, that determines the character of the express ion. In f a c t , i t could be s a i d that the operat ions and t h e i r order are even more important i n algebra than i n a r i t h m e t i c . Given the ar i thmet i c expression 10 + 3 x 2, we need to know what + and x mean and the fac t that (by convention) the x i s performed before the +. When we perform these operations we "get the answer": 16. Using C a l c u l a t o r s to Teach Algebra 17 But 16 has a f l u i d rather than a determinate s t r u c t u r e . We can descr ibe i t equa l ly wel l as the product of 8 and 2, the sum of 9 and 7, the d i f f e r e n c e of 20 and 4, the square root of 256, the square of 4, the fourth power of 2, e t c . However, the a l g e b r a i c express ion m + be i s nothing more nor l ess nor other than the sum of m together with the product of b and c. I t has no other i d e n t i t y . I t cannot be fac tored . I t i s not a per fec t square. I t i s not a product or a quot ient . Only by invoking what would be looked on as "legal f i c t i o n " can we change i t s i d e n t i t y ; for example m + be = (m + be)2 :— = [V(m + be)] 2 m + be I t s i d e n t i t y i s completely determined by i t s s t r u c t u r e , and i t s s t r u c t u r e i s completely determined by the ( v i r t u a l ) operat ions which b u i l t i t and the order i n which they were performed. "The r e c o g n i t i o n and use of s tructure" (p. 33) i s inc luded by Kieran (1989) among the d i f f i c u l t i e s encountered by beginning a lgebra students . She def ines " a r i t h m e t i c / a l g e b r a i c s tructure" i n general as "the s t r u c t u r e of a system that i s comprised of a set of numbers/numerical v a r i a b l e s , some opera t ion( s ) , and the p r o p e r t i e s of the operat ion(s )" (p. 33). She says that s ince "much of school elementary a r i t h m e t i c i s or i en ted towards ' f i n d i n g the answer'" (p. 33), c h i l d r e n need not attend to the s t ruc ture of ar i thmet i c express ions . "However, i n a lgebra , they are requ ired to recognize and use the s t r u c t u r e that Using Calculators to Teach Algebra 18 they have been able to avoid in arithmetic" (p. 39). Equals Signs Besides the substitution of letters for numbers and the relegation of operations to the realm of the virtual, the transition from arithmetic to algebra also includes subtly new meanings for the = sign. Mathematicians interpret an = sign to mean that whatever i s written on the l e f t side of i t i s in some sense equivalent to whatever i s written on the right side. However, according to Herscovics and Kieran (1980), "students do not view 2 + 3 = 5 as an identity but rather interpret i t operationally, as indicated by their reading '2 and 3 make 5'" (p. 573). Behr et al. (1976, p. 9) suggest that "children do not view sentences like 2 + 3 = 3 + 2 as being sentences about number relationships. They do not see these as indicating the sameness of sets of objects. Indeed, i t appears that children consider these as 'do something' sentences." They also found that some students accepted 2 + 3 = 5 and 3 + 2 = 5 but not 2 + 3 = 3 + 2 . The students accepted 2 + 4 as meaningful, but as suggesting that something had to be done. They did not, for example, think of 2 + 4 as being a name for 6 (p. 2). Significantly, these researchers also found that there i s "no evidence to suggest that children change in their thinking about equality as they get older and progress to upper grades; in fact, Using Calculators to Teach Algebra 19 the evidence seems to be to the contrary" (p. 10). S i m i l a r l y , Mevarech and Yitschak (1983), quoted i n Kieran (1992), showed that c o l l e g e students often have a "poor" understanding of the meaning of the equals sign (p. 399). Kieran (1989) also points out the use of the = sign by students to s i g n i f y that what follows is' the r e s u l t of the c a l c u l a t i o n that precedes (p. 37). She quotes Vergnaud, Benhadj, and Dussouet (1979), who presented sixth-grade c h i l d r e n with the following problem: In an e x i s t i n g forest, 425 new trees were planted. A few years l a t e r , the 217 oldest trees were cut. The for e s t then contained 1063 trees. How many trees were there before the new trees were planted? (p. 37) Children t y p i c a l l y wrote 1063 + 217 = 1280 - 425 = 855. Kieran describes t h i s as "writing down the operations i n the order i n which they are c a r r i e d out and keeping a running t o t a l " (p. 37). Used i n t h i s way, the = sign could be described as "what you write before the answer" or "what you write to show that what follows i s the answer." Solving Equations With l e t t e r s instead of numbers, with the prompts suggested by the f a m i l i a r operation signs impossible to carry out but nevertheless responsible f o r the structure of expressions, and with a new meaning f or the = sign, i t i s not s u r p r i s i n g that students progressing from arithmetic to algebra have d i f f i c u l t y Using Calculators to Teach Algebra 20 solving equations, no matter which of the standard methods they are taught. (I use the words "formula" and "equation" interchangeably, since nearly a l l equations my students see are formulae; that i s , one side of the = sign is occupied by only one variable.) In an ar t i c l e significantly entitled "Early Learning: A Structural Perspective," Kieran (1989) notes that the structure of an equation "incorporates the characteristics of the structure of expressions..." (p. 34). In particular, she says, the surface structure of an equation i s no more than the surface structure of the two expressions and their equality. She goes on (p. 49) to describe Whitman's (1976) distinction between "formal" and "informal" equation-solving techniques as follows. The equation 69 - 96 = 37 7 - a could be solved "formally" as follows: 483 - 69a - 96 387 - 69a 128 - 69a 128 4 37(7 259 -259 --37a 32a 37 a) 37a 37a a or "informally" as follows: 69 minus what number gives 37? — 32 96 divided by what number i s 32? — 3 7 minus what number i s 3? — 4 Solution: a = 4 or by a combination of formal and informal techniques. Using C a l c u l a t o r s to Teach Algebra 21 F a c i l i t y with the formal technique c e r t a i n l y requ ires r e c o g n i t i o n and understanding of the s t r u c t u r e of the equat ion. However, f a c i l i t y with the informal technique o u t l i n e d above i m p l i c i t l y requires a knowledge of the order of the operat ions i n the o r i g i n a l equation, together (again i m p l i c i t l y ) with the knowledge that , a f t e r beginning with 37, operat ion 3 should be "undone" f i r s t , operat ion 2 second, and operat ion 1 l a s t . The informal technique, therefore , a l so requires r e c o g n i t i o n and understanding of the s t ruc ture of the equat ion. In the re s t of her 1989 paper (pp. 49-52), Kieran d iscusses the r e l a t i v e merits of "doing the same th ing to both s ides" and "transpos ing ," p o i n t i n g out while to teachers these may be v a r i a t i o n s of the same method, to , s tudents they appear q u i t e d i f f e r e n t . She a l so observes (1992) that " learning to operate on the s t r u c t u r e of an equation by performing the same operat ion on both s ides may be eas i er for students who already view equations as e n t i t i e s with symmetrical balance" (p. 401). Again , both of these methods r e q u i r e students to understand the s t r u c t u r e of the equation; that i s , the s t r u c t u r e of each s ide of the equation which i s comprised i n the operat ions and t h e i r order , together with the equals s i g n . Kieran (1992) r e f e r s to a l i m i t e d vers ion of the r e v e r s a l method as the "undoing" or "working backwards" method, i n which, she says, "to solve 2x + 4 = 18, the student takes the numerical r e s u l t on the r i g h t s ide and, proceeding i n a r i g h t - t o - l e f t order , Using C a l c u l a t o r s to Teach Algebra 22 undoes each operat ion as he/she comes to i t by r e p l a c i n g the g iven operat ion with i t s inverse; thus the student i s able to operate e x c l u s i v e l y with numbers and avoid d e a l i n g with the equivalence s t r u c t u r e of t h i s mathematical object" (p. 400). (I say " l imi ted" because "proceeding i n a r i g h t - t o - l e f t order" would not always produce a c o r r e c t s o l u t i o n and because the r e v e r s a l method i s not l i m i t e d to equations conta in ing only one v a r i a b l e . ) The r e v e r s a l method per se i s r e f e r r e d to i n Verse 28 i n the Ganita s e c t i on of Shukla's (1976) e d i t i o n of the Aryabhatiya, completed by the Hindu mathematician and astronomer Aryabhata i n AD 499: In the method of i n v e r s i o n m u l t i p l i e r s become d i v i s o r s and d i v i s o r s become m u l t i p l i e r s , a d d i t i v e becomes subtrac t ive and subtrac t ive becomes a d d i t i v e . Example: A number i s m u l t i p l i e d by 2; then increased by 1; then d i v i d e d by 5; then m u l t i p l i e d by 3; then diminished by 2; and then d i v i d e d by 7; the r e s u l t (thus obtained) i s 1. Say what i s the i n i t i a l number. S t a r t i n g from the l a s t number 1, i n the reverse order , i n v e r t i n g the operat ions , the r e s u l t i s 1 x 7, + 2, -r 3, x 5, - 1, T 2, i . e . 7. (p. 71) C l e a r l y , t h i s method involves r e c o g n i t i o n of the order of the o r i g i n a l operat ions , together with the knowledge that , a f t e r beginning with the o r i g i n a l r e s u l t , one must " invert" the operat ions i n the "reverse" order . But t h i s i s what c o n s t i t u t e s what we would c a l l the s t ruc ture of the equation that corresponds to the o r i g i n a l problem: 2x + 1 3 x - 2 1 - i S i m i l a r r u l e s for the i n v e r s i o n method occur i n other Hindu Using C a l c u l a t o r s to Teach Algebra 23 mathematical works AD 628-1356. An ancient Babylonian problem, quoted i n Fauvel (1987), a l so appears to be set up to be solved by the same method: I found a stone, (but) d i d not weigh i t ; (after) I weighed (out) 6 times ( i t s weight) , [added] 2 g i n , (and) added o n e - t h i r d of one-seventh m u l t i p l i e d by 24, I weighed ( i t ) : 1 ma-na. What was the o r i g i n ( a l weight) of the stone? The o r i g i n ( a l weight) of the stone was 4g g i n (p. 14). In t h i s problem, at f i r s t glance, the f ("one-third of one-seventh m u l t i p l i e d by 24") could be f of the o r i g i n a l weight of 8 8 the stone, y of a g i n , or y of a ma-na. However, from the answer s u p p l i e d , and from comparison with s i m i l a r problems, i t appears that i t i s 7 of the stone's weight increased by 2 gin. That i s , the stone's weight i s f i r s t m u l t i p l i e d by 6, then increased by 2 o g i n , and then m u l t i p l i e d by y to give 1 ma-na. The o r i g i n a l weight of the stone, therefore , i s c a l c u l a t e d by s t a r t i n g with 1 ma-na, d i v i d i n g by y, and subtrac t ing 2 g i n . I t appears that the implementation of any method of s o l v i n g an equation, whether "formal ," " in formal ," "doing the same th ing to both s i d e s , " "transposing," or r e v e r s a l , demands knowledge of the equat ion's s t r u c t u r e , i.e., the s t r u c t u r e of the expressions on the two s ides of the equals s ign and t h e i r e q u a l i t y . Conclus ion Although the i n t r o d u c t i o n of l e t t e r s i s the most v i s i b l e change as students move from ar i thmet ic to a lgebra , changes i n the meanings of operat ions and the equals s i gn , together with the Using C a l c u l a t o r s to Teach Algebra 24 necess i ty of understanding the s t ruc ture of a l g e b r a i c expressions and equations, are a l so sources of the d i f f i c u l t i e s so many students have when they move from ar i thmet i c to a lgebra . These changes and d i f f erences are analyzed fur ther i n the Chapter 3. However, to apprec iate f u l l y the d i f f erences and s i m i l a r i t i e s i n syntax among the languages of a r i t h m e t i c , the c a l c u l a t o r , my own algebra courses and the "regular" mathematics courses r e s p e c t i v e l y , the reader must f a m i l i a r i z e himself or h e r s e l f with the language I use i n my own courses i n teaching students how to use c a l c u l a t o r s (see Appendix 3) and how to solve equations by the r e v e r s a l method (see Appendix 4) . Both of these appendices are w r i t t e n as i f addressed to students, with explanatory notes to the more mathematical reader i n t e r p o l a t e d i n i t a l i c s . Using C a l c u l a t o r s to Teach Algebra 25 Chapter 3 Syntax of Mathematical Language Although algebra i s perceived by mathematicians to be a g e n e r a l i z a t i o n of a r i t h m e t i c , and although many of the symbols used i n a lgebra are the same as those used i n a r i t h m e t i c , the two subjects appear to many students to be completely unre la ted because of the d i f f erences of meaning analyzed i n Chapter 2. However, t h i s meaning cannot be d ivorced from the syntax of mathematical language, which I have not yet considered per se. The language of the c a l c u l a t o r , as i l l u s t r a t e d i n Appendix 3, i s e s s e n t i a l l y the language of a r i t h m e t i c , s ince the purpose of a c a l c u l a t o r i s to carry out ar i thmet i c operat ions . The language of the r e v e r s a l method of s o l v i n g equations, as i l l u s t r a t e d i n Appendix 4, i s s i m i l a r . In t h i s chapter, I w i l l f i r s t analyze the syntax of c a l c u l a t o r language and the syntax of the language used i n the r e v e r s a l method, i n order to show how s i m i l a r they a r e . Then I w i l l analyze the syntax of the language convent iona l ly used i n "regular" high school a lgebra courses i n order to show how d i f f e r e n t i t i s from the language of c a l c u l a t o r s and the r e v e r s a l method. F i n a l l y , I w i l l argue that the syntax of a r i t h m e t i c , c a l c u l a t o r s and the r e v e r s a l method should have equal s tatus with convent ional syntax and be used i n teaching students who have d i f f i c u l t y with conventional syntax. The Compact Edition of the Oxford English Dictionary (1980) Using C a l c u l a t o r s to Teach Algebra 26 def ines "syntax" as "the arrangement of words ( i n t h e i r appropriate forms) by which t h e i r connexion and r e l a t i o n i n a sentence are s h o w n . . . . the department of grammar which deals with the e s tab l i shed usages of grammatical cons truc t ion and the r u l e s deduced therefrom" (p. 3212). The fact that "syntax" i s an a t t r i b u t e of mathematical language i s a t tes ted to i n Pimm's Speaking Mathematically (1987), i n which the t i t l e of Chapter 7 i s "The syntax of mathematical forms." Syntax of Ar i thmet i c Confronted with the fo l lowing wr i t t en express ion, 3 ( 4 + 2 ) + 5 students who have passed my c a l c u l a t o r u n i t behave as fo l lows: reach for a c a l c u l a t o r , press the buttons (on an a l g e b r a i c c a l c u l a t o r ) 3 x ( 4 + 2 ) + 5 = o r ( o n a non-a lgebra ic c a l c u l a t o r ) 4 + 2 x 3 + 5 = , and write down the r e s u l t . That i s , they obey the "powerful prompt" to do something represented by the operat ion s igns (Pimm 1987, p . 167). They behave as i f I had s a i d to them, "Start with 4, add 2, m u l t i p l y by 3, add 5, press =, and t e l l me what you get ." Conversely, i f I t o l d them that I wanted them to "write a question" which, when handed to someone e l s e , would " t e l l that person" to s t a r t with 4, add 2, m u l t i p l y by 3, add 5, and report the answer, they would wri te ( 4 + 2 ) 3 + 5 (In p r a c t i c e , I do not have them perform the second of these two procedures u n t i l they have begun the u n i t on s o l v i n g equat ions . ) Using C a l c u l a t o r s to Teach Algebra 27 We may say, therefore , that one meaning of 3 ( 4 + 2 ) + 5 i s "s tart with 4 , add 2 , m u l t i p l y by three , add 5 , and ( i m p l i c i t i n the other i n s t r u c t i o n s ) report what you get ." Because they are us ing c a l c u l a t o r s , t h i s i s the meaning I , as a teacher, seek to v a l i d a t e among my students by the language I myself use, the language i l l u s t r a t e d i n Appendix 3 . Not ice that i n t h i s language, 2 , 3 , 4 , and 5 are nouns. However, the words used for the operat ions - "add" and "mult iply" - are verbs . Moreover, they are t r a n s i t i v e verbs , not i n t r a n s i t i v e ; that i s , they are capable of having d i r e c t o b j e c t s . We add, subtrac t , m u l t i p l y , and d i v i d e numbers and l e t t e r s . (In c o n t r a s t , "go" i s an i n t r a n s i t i v e verb; one cannot "go something.") T r a n s i t i v e verbs , but not i n t r a n s i t i v e verbs , can be used i n the a c t i v e or the pass ive v o i c e . For example, i f I say "I added four and two" or "Add four and two" (the l a t t e r i n the imperat ive , where the understood subject "you" i s omitted) , I am us ing the a c t i v e vo ice , while i f I say "Four and two were added" I am us ing the pass ive v o i c e . There can be a curious d i f f erence between the v e r b a l i z a t i o n of "add" and "subtract" on the one hand and that of "mult iply" and "divide" on the other . Consider how to v e r b a l i z e the f o l l o w i n g . 1 ) 4 + 2 2) 4 - 2 3) 4 x 2 4) 4 -r 2 One way, i f the operat ion s igns are read as verbs i n the Using C a l c u l a t o r s to Teach Algebra 28 imperat ive , i s to say, on the one hand, "Add two to four" and "Subtract two from f o u r , " but, on the other hand, "Mul t ip ly four by two" and "Divide four by two." I t i s not uncommon for students to say, "Add four by two" or "Subtract four by two," but t h i s i s not considered standard language. I t appears that the d i r e c t object of the verb i s the second number when we add or s u b t r a c t , but the f i r s t number when we m u l t i p l y or d i v i d e . One cou ld a l so say, "Add four and two" and "Mul t ip ly four and two," because a d d i t i o n and m u l t i p l i c a t i o n are commutative, but "Subtract four and two" and "Divide four and two" are.ambiguous. In p r a c t i c e , I ( i m p l i c i t l y ) take the symbols +, - , x , and •"-to mean "add," "subtract ," "mult iply by ," and "div ide by" r e s p e c t i v e l y . The students often subs t i tu te the words "plus ," "minus," and "times" for the f i r s t three r e s p e c t i v e l y , but t h e i r use of these words c l e a r l y shows that they consider them to be verbs (e.g. "Should I p lus t h i s next or times that?" or "So I have to minus t h i s by that .") As I note i n Appendix 4, I use the symbol y x to denote the operat ion of r a i s i n g to a power. With t h i s convention, I cou ld add the fo l lowing to the above l i s t 5) 4 y x 2 Then "2" would be the object of the operat ion s ign i n each of these numbered numerical phrases. So far I have t r a n s l a t e d in to E n g l i s h only an a r i t h m e t i c express ion . Consider now the fo l lowing ar i thmet i c equation: 3(4 + 2) + 5 = 23 Using C a l c u l a t o r s to Teach Algebra 29 The operat ions s igns are not now i n the imperative mood, because anything they enjoined on the reader has been c a r r i e d out . What i s wr i t t en i n the equation above could mean that whoever "did the question" (i.e., whoever c a r r i e d out the operat ions) s t a r t e d with 4, added 2, multiplied by 3, added 5, and got 23. In t h i s reading of the equation, the operations are verbs i n the i n d i c a t i v e mood, past tense. However, i t could a l so mean that when you s t a r t with 4, add 2, multiply by 3, and add 5, you get 23. In t h i s reading , the operat ions are verbs i n the i n d i c a t i v e mood, present tense. Moreover, i t could a l so mean that i f you s t a r t e d with 4, added 2, multiplied by 3, and added 5, you would get 23. In t h i s reading , the operations are verbs i n the c o n d i t i o n a l mood. Nevertheless , i n a l l three readings the operat ions remain verbs . Not ice that the = s ign i s read "got," "get," or "would get ." I a l so commonly read i t "end up wi th ." ("Get" i s probably more appropriate for a formula, which you se l ec t by i t s subject with the conscious purpose of c a l c u l a t i n g a value you want to know, as opposed to an equation, where you simply fol low the i n s t r u c t i o n s and are t o l d where you "end up.") I pass over brackets as a s y n t a c t i c a l element i n mathematical sentences and expressions because they have the same meaning i n algebra as they do i n ar i thmet i c (although brackets can have other meanings i n algebra as w e l l ) . I f I were to complete the s y n t a c t i c a l a n a l y s i s of a r i t h m e t i c , I would compare brackets to Using C a l c u l a t o r s to Teach Algebra 30 the commas around a n o n - r e s t r i c t i v e c lause . (Indeed, brackets may often be subs t i tu ted for these commas i n w r i t i n g ) . On the c a l c u l a t o r , the press ing of brackets around an a r i t h m e t i c express ion may be s a i d to t e l l the c a l c u l a t o r to wait a minute while the operator i n s e r t s and the c a l c u l a t o r evaluates a p a r e n t h e t i c a l express ion, jus t as the s l i g h t pause which i s the o r a l equivalent of a comma t e l l s the l i s t e n e r to wait a minute while the speaker i n s e r t s a p a r e n t h e t i c a l express ion . Syntax of the Reversal Method S y n t a c t i c a l l y , the language of the r e v e r s a l method of s o l v i n g equations I used i n Appendix 4 i s almost i d e n t i c a l to the language of a r i t h m e t i c . In f a c t , except for the f i r s t paragraph, the s ec t i on above, "Syntax of a r i t h m e t i c , " could be repeated here word for word simply by changing the numbers to l e t t e r s . For example, i n Appendix 4 I def ined a formula as "a set of mathematical i n s t r u c t i o n s for c a l c u l a t i n g the value of one v a r i a b l e from the values of other v a r i a b l e s . " As an example, I might e x p l a i n to my students that n = m + be i s a formula for c a l c u l a t i n g the value of n: i t says that i f you m u l t i p l y the value of b by the value of c and add the value of m, you w i l l get the value of n, the v a r i a b l e which i s the subject of the formula. A l t e r n a t i v e l y , i t says that i f you could m u l t i p l y b by c and add m, you would get n . The l e t t e r s have replaced numbers as the nouns. However, the Using C a l c u l a t o r s to Teach Algebra 31 operat ions , even though " v i r t u a l , " are s t i l l verbs . The only way i n which they d i f f e r from those of ar i thmet i c i s that they can no longer be i n the imperative mood or the past tense of the i n d i c a t i v e mood, for they cannot be c a r r i e d out . They can be i n the future tense: "If you do t h i s , you w i l l get t h a t , " but i t i s arguable that the most appropriate mood i s the c o n d i t i o n a l : "If you could do t h i s (or were to do t h i s ) , you would get t h a t . " Syntax of Conventional Algebra Consider t h i s sentence by Kieran (1989): " . . . the s t r u c t u r e of the express ion 3(x + 2) +5 inc ludes the surface s t r u c t u r e , that i s , the given ensemble of terms and operat ions - i n t h i s case, the m u l t i p l i c a t i o n of 3 by x + 2, fol lowed by the a d d i t i o n of 5 . . . " (p. 34). Notice that , i n t h i s sentence, as i n "reversa l" syntax, 3 and 5 are nouns. However, the s i m i l a r i t y ends there . Not ice that x + 2 i s a l so a noun. Moreover, the words used to descr ibe the operat ions - " m u l t i p l i c a t i o n " and "addit ion" - are nouns as w e l l . Even f u r t h e r , "the m u l t i p l i c a t i o n of 3 by x + 2, fol lowed by the a d d i t i o n of 5" i s a noun phrase - a noun with modi f i er s . Using the terminology of "sums" and "products ," one cou ld a l so say "the sum of 5 and 3(x + 2)" or "the sum of 5 with the product of 3 and (x + 2)" or (rather confus ingly) "the sum of 5 together with the product of 3 and the sum of x and 2." Contrast what I would say to my students: "We are being t o l d Using C a l c u l a t o r s to Teach Algebra 32 to s t a r t with x, then add 2, then m u l t i p l y by 3, and then add 5." Even when I d i scuss what I c a l l groups, where I come c l o s e s t to t r e a t i n g an a lgebra i c expression as a noun (see Appendix 4), I t a l k (as long as any i n d i v i d u a l student f inds i t necessary) about "the answer" to the express ion. For example, I would descr ibe the "3" i n the express ion 3(x + 2) + 5 as being m u l t i p l i e d by "the answer" you get when you add 2 to x (or would get i f you could add 2 to x ) . There i s an equal ly great d i f f erence i n the syntax of the = s ign between "regular" a lgebra i c methods and the r e v e r s a l method of s o l v i n g equations. Most commonly, mathematicians read the = s ign as " i s . " "Is" i s a copula verb, def ined by The Compact Edition of the Oxford English Dictionary (1980) as "that part of a p r o p o s i t i o n which connects the subject and the predicate" (p. 555). In mathematical sentences such as " 2 x 3 = 6," both the subject ("2 x 3") and the pred ica te ("6") are nouns. The predicate of a copula verb i s not l i k e the object of a t r a n s i t i v e verb: a copula verb does not "act on" i t s pred ica te as a t r a n s i t i v e verb "acts on" i t s objec t . For example, "I w i l l be a teacher" i s qu i t e d i f f e r e n t grammatically from "I w i l l h i t a teacher ." The copula verb i n d i c a t e s i n some sense an e q u a l i t y between the subject and the p r e d i c a t e . Nevertheless , the subject and the pred ica te of a copula verb are not interchangeable . For example, the sentence "A cow i s an animal" does not mean the same as "An animal i s a cow." Using C a l c u l a t o r s to Teach Algebra 33 S i m i l a r l y , Schoenfeld and A r c a v i (1988) point out tha t , i n a formal sense, the two statements "2 x 3 = 6" and "6 = 2 x 3" are equiva lent . But i n r e a l i t y , the f i r s t statement i s about m u l t i p l i c a t i o n , the second about f a c t o r i n g . "That i s , i n context , the equal s i gn i s not read as ' i s formal ly equivalent t o ' ; i t i s read as ' y i e l d s ' " (p. 424). Comparison of Conventional and Reversal Syntax The d i f f erences i n syntax between convent ional a lgebra and my courses , descr ibed above, are encapsulated i n the Table 1. Table 1 Conventional vs . Reversal Syntax Concept Conventional My courses adding, s u b t r a c t i n g , m u l t i p l y i n g , d i v i d i n g , exponent i a t i on conjunctions or " jo in ing words" denoting r e l a t i o n -ships between numbers or v a r i a b l e s verbs g i v i n g i n s t r u c t i o n s about act ions to be performed - "things to be done" a l g e b r a i c or numerical expressions mathematical "phrases" i n which the "nouns" are numbers and v a r i a b l e s and the "conjunctions" are operations sets of i n s t r u c t i o n s t e l l i n g you what to do with the values of v a r i a b l e s equals s i gn a s ign placed between two expressions to denote equa l i ty of value a s ign which separates what has to be done from what you end up with formulae or equations statements of the e q u a l i t y of two mathematical phrases sets of i n s t r u c t i o n s t e l l i n g you how to c a l c u l a t e the value of a v a r i a b l e from the values of other v a r i a b l e s Using C a l c u l a t o r s to Teach Algebra 34 Sfard (1991) uses the word "operat ional" to descr ibe the conception of an a lgebra i c expression as a set of i n s t r u c t i o n s and the word " s t r u c t u r a l " to descr ibe i t s conception as what I have c a l l e d a group: an e n t i t y , a noun, a "thing" i n i t s own r i g h t . She contras t s the two conceptions as fo l lows: Seeing a mathematical e n t i t y as an object means being capable of r e f e r r i n g to i t as i f i t were a r e a l th ing -a s t a t i c s t r u c t u r e , e x i s t i n g somewhere i n space and time. I t a l so means being able to recognize the idea "at a glance" and to manipulate i t as a whole, without going in to d e t a i l s . . . . In contras t , i n t e r p r e t i n g a not ion as a process impl ies regarding i t as a p o t e n t i a l ra ther than ac tua l e n t i t y , which comes in to existence upon request i n a sequence of a c t i o n s . Thus, whereas the s t r u c t u r a l conception i s s t a t i c . . . instantaneous, and i n t e g r a t i v e , the operat iona l i s dynamic, s equent ia l , and d e t a i l e d . In other words, there is a deep ontological gap between operational and structural conceptions, (p. 4) She a l so says that no information i s added i n a s h i f t from the opera t iona l approach to the s t r u c t u r a l approach; ins tead the computational processes are "caught in to a s t a t i c construct j u s t as water i s frozen i n a piece of ice" (p. 25). S fard uses the word reification (p. 14) for the t r a n s i t i o n from seeing expressions "operat ional ly" to seeing them " s t r u c t u r a l l y . " Reification i s not inc luded i n The Compact Edition of the Oxford English Dictionary (1980), but Sfard has constructed i t from the L a t i n words res, meaning "a thing" or "an objec t ," and facere, meaning "to make." To reify an express ion means to make i t in to a th ing ; i n grammatical terms, i t means to use a c o l l e c t i o n of nouns, pronouns, a d j e c t i v e s , verbs , and adverbs as a noun c lause i n a l a r g e r sentence. Fowler (1984) Using C a l c u l a t o r s to Teach Algebra 35 descr ibes a noun c lause as "subordinate words i n c l u d i n g a subject and p r e d i c a t e , but s y n t a c t i c a l l y equivalent to a noun" (p. 79). Kieran (1992) says that reification "seems to be a leap: a process s o l i d i f i e s i n t o an object , in to a s t a t i c s t r u c t u r e . The new e n t i t y i s detached from the process that produced i t " (p. 392). D i f f i c u l t y of Reification Sfard (1991) points out that "unlike mater ia l objects . . . advanced mathematical constructs are t o t a l l y i n a c c e s s i b l e to our senses - they can be seen only with our mind's e y e s . . . . Being capable of somehow "seeing" these i n v i s i b l e objects appears to be an e s s e n t i a l component of mathematical a b i l i t y ; lack of t h i s capac i ty may be one of the major reasons [why] mathematics appears p r a c t i c a l l y impermeable to so many 'well-formed minds'" (p. 3 ) . Kieran (1992) agrees with Sfard that there i s a need for a "lengthy t r a n s i t i o n per iod i n moving from operat iona l (procedural) to s t r u c t u r a l conceptions" (p. 395) and that the leap "is inherent ly so d i f f i c u l t that there may be students for whom the s t r u c t u r a l conception w i l l remain p r a c t i c a l l y out of reach whatever the teaching method" (p. 412). Thi s op in ion i s supported by the work of researchers who have found that students who have trouble with convent ional a lgebra can nevertheless be success fu l i n courses where they t rea t operat ions as "things to be done" rather than r e l a t i o n s h i p s among v a r i a b l e s . For example, Sutherland (1991) says that "resu l t s from our Using C a l c u l a t o r s to Teach Algebra 36 own s tudies of p u p i l s programming i n Logo have shown that p u p i l s as young as 10 years o l d can accept and use 'unclosed' a l g e b r a i c expressions i n a Logo programming context" and that "pupi ls a l so seem to accept these 'unclosed' a lgebra i c expressions i n a spreadsheet environment" (p. 40), both of which a p p l i c a t i o n s t r e a t operat ions as i n s t r u c t i o n s to the computer. Another example i s the New York C i t y Board of Educat ion ' s Computer Mathematics: An Introduction (1985), developed for h igh school students "who are not achiev ing success i n the t r a d i t i o n a l mathematics program." The course cons i s t s of 60 lessons which use computers and computer programming to help students understand perimeters of polygons, area , the order of ar i thmet i c operat ions , consumer mathematics, a lgebra i c equations, e t c . "The major focus of the program i s on problem-solving" with the teaching of computer programming "intended to provide a model of a problem-so lv ing process i n which a problem i s broken down i n t o i t s component parts" (New York C i t y Board of Education 1985). In the f i r s t lesson, students are t o l d that they have "just s imulated a computer" when they fol low the i n s t r u c t i o n s "Stand up, walk to the front of the room, wri te your f i r s t name on the chalkboard, say 'I love computers,' and r e t u r n to your seat" (p. D . This process i s genera l ized as "we had a p l a n ; you were given i n s t r u c t i o n s i n E n g l i s h (a language you understand); you memorized the i n s t r u c t i o n s ; when you heard the word RUN, you 'executed' Using C a l c u l a t o r s to Teach Algebra 37 (followed) the i n s t r u c t i o n s " (p. 2 ) . A f t e r the students type in to the computer the formula P = 3*S for the perimeter of an e q u i l a t e r a l t r i a n g l e , they are taught to c a l l the value of S the "input" and the value of P the "output" (p. 6 ) . Algebra Without Reification Sfard (1991) says, "The c a r e f u l ana lys i s of textbook d e f i n i t i o n s w i l l show that t r e a t i n g mathematical not ions as i f they r e f e r r e d to some abs trac t objects i s often not the only p o s s i b i l i t y . Although t h i s k ind of conception [the "s tructura l"] seems to p r e v a i l i n modern mathematics, there are accepted mathematical d e f i n i t i o n s which revea l qu i t e a d i f f e r e n t approach. Funct ion [for example] can be def ined not only as a set of ordered p a i r s , but a l so as a c e r t a i n computational process . . . Symmetry can be conceived as a s t a t i c property of geometric forms, but a l so as a k ind of transformation" (p. 4) . The l a t t e r type of d e s c r i p t i o n i s what she c a l l s the "operat iona l ." Sfard emphasizes that the operat iona l and the s t r u c t u r a l conceptions of the same mathematical not ion "are not mutually e x c l u s i v e . Although os tens ib ly incompatible (how can anything be a process and an object at the same t ime?) , they are i n fac t complementary. The term 'complementarity' i s used here i n much the same sense as i n phys ics , where e n t i t i e s at subatomic l e v e l must be regarded both as p a r t i c l e s and as waves to enable f u l l d e s c r i p t i o n and explanat ion of the observed phenomena." Using C a l c u l a t o r s to Teach Algebra 38 Sfard s tresses that "unlike 'conceptual ' and ' p r o c e d u r a l , ' or ' a l g o r i t h m i c ' and ' a b s t r a c t , ' the terms 'opera t iona l and ' s t r u c t u r a l ' r e f e r to inseparable , though d r a m a t i c a l l y d i f f e r e n t , facets of the same thing" (p. 9) . However, she admits that "there seems to be a consensus that the ' abs t rac t ' mathematics deserves the highest esteem" and that "algorithmic" mathematics "is somehow second-rate" (p. 9 ) . A f t e r o u t l i n i n g the h i s t o r i c a l development of the concepts of var ious kinds of numbers ( i n t e g r a l , r a t i o n a l , negat ive , imaginary) , through processes performed on other , a lready recognized, kinds of numbers (pp. 11-13), Sfard genera l i zes the development of each concept thus: 1. the preconceptual stage, at which mathematicians were ge t t ing used to c e r t a i n operat ions on the already known numbers . . . ; at t h i s p o i n t , the rout ine manipulations were treated as they were: as processes , and nothing e l se (there was no need for new objec t s , s ince a l l the computations were s t i l l r e s t r i c t e d to those procedures which produced the prev ious ly accepted numbers). 2. a long per iod of predominantly opera t iona l approach, dur ing which a new kind of number began to emerge out of the f a m i l i a r processes (what t r i g g e r e d t h i s s h i f t were c e r t a i n uncommon operat ions , p r e v i o u s l y regarded as t o t a l l y forbidden, but now accepted as u s e f u l , i f s trange); at t h i s stage, the j u s t introduced name of the new number served as a cryptogram for c e r t a i n operat ions rather than as a s i g n i f i e r of any "rea l" object . . . 3. the structural phase, when the number i n quest ion has eventual ly been recognized as a f u l l y f ledged mathematical objec t . From now on, d i f f e r e n t processes would be performed on t h i s new number, thus g i v i n g b i r t h to even more advanced kinds of numbers. She summarizes the h i s t o r i c a l development of mathematical concepts as fo l lows: Using C a l c u l a t o r s to Teach Algebra 39 Various processes had to be converted in to compact s t a t i c wholes to become the bas i c u n i t s of a new, higher l e v e l - t h e o r y . When we broaden our view and look at mathematics . . . as a whole, we come to r e a l i z e that i t i s a k ind of h i e r a r c h y , i n which what i s conceived pure ly o p e r a t i o n a l l y at one l e v e l should be conceived s t r u c t u r a l l y at a higher l e v e l . Such h ierarchy emerges i n a long sequence of " r e i f i c a t i o n s , " each one of them s t a r t i n g where the former ends, each one of them adding a new layer to the complex system of abstract no t ions , (p. 16) In what i s f ar from being a s i m p l i s t i c i d e n t i f i c a t i o n of mathematical h i s t o r y with i n d i v i d u a l development, S fard nevertheless suggests that "when a person gets acquainted with a new mathematical not ion , the operat iona l conception i s u s u a l l y the f i r s t to develop" (p. 16) and introduces the words interiorization, condensation, and reification to descr ibe the stages i n a person's concept-development which are analogous to the three h i s t o r i c a l stages above. This three-phase schema, she says, "is to be understood as a hierarchy, which impl ies that one stage cannot be reached before a l l the former steps have been taken" (p. 21). In a s ec t i on e n t i t l e d "Operational Approach: C e r t a i n l y Necessary, Sometimes a l so S u f f i c i e n t , " Sfard has t h i s to say: Apparent ly , a pure ly operat iona l way of looking at mathematics would be qu i te appropr ia te . Indeed, to an unprejudiced and i n s i g h t f u l person, the very not ion of "mathematical object" may appear superf luous: s ince processes seem to be the only r e a l concern of mathematics, why bother about these e l u s i v e , p h i l o s o p h i c a l l y problematic "things ," such as i n f i n i t e sets or "aggregates of ordered pairs"? T h e o r e t i c a l l y , i t would be poss ib le to do almost a l l the mathematics pure ly o p e r a t i o n a l l y ; we could proceed from elementary processes to h i g h e r - l e v e l processes and then to even more complex processes without ever r e f e r r i n g to any k ind of abstract objects (p. 2 3 ) . . . . Using C a l c u l a t o r s to Teach Algebra 4 0 Twentieth-century mathematics, however, seems to be so deeply permeated with the s t r u c t u r a l outlook that a modern mathematician has to be except iona l ly open-minded - indeed, not himself [sic] at a l l - to r e a l i z e that from a p h i l o s o p h i c a l (not psycho log i ca l ! ) po int of view he [sic] could do without "mathematical objects" (p. 24.) There i s no reason to turn process i n t o object unless we have some h i g h - l e v e l processes performed on t h i s s impler process , (p. 31) Davis and Hersh (1981) point out that our modern preoccupat ion with the s t r u c t u r a l i s h i s t o r i c a l l y " insu lar": The mathematics of Egypt, of Babylon, and of the ancient Orient was a l l of the a lgor i thmic type. D i a l e c t i c a l mathematics - s t r i c t l y l o g i c a l , deduct ive mathematics - o r i g i n a t e d with the Greeks. But i t d i d not d i s p l a c e the a l g o r i t h m i c . In E u c l i d , the r o l e of d i a l e c t i c i s to j u s t i f y a construct ions - i . e . , an a lgor i thm. I t i s only i n modern times that we f i n d mathematics with l i t t l e or no a lgor i thmic content , which we could c a l l purely d i a l e c t i c a l or e x i s t e n t i a l , (p. 182) Sfard (1991) admits that "at c e r t a i n stages of knowledge formation (or a c q u i s i t i o n ) the absence of a s t r u c t u r a l conception may hinder fur ther development" (p. 29) but s tresses that , both h i s t o r i c a l l y and i n d i v i d u a l l y , reification i s very d i f f i c u l t . Conclus ion The language of the c a l c u l a t o r i s the language of a r i t h m e t i c , i n which operations are c a r r i e d out and "answers" obtained. Thi s language can be transposed in to a lgebra with only minor changes. When i t i s , i t can be understood by students who have had l i t t l e or no success i n conventional courses . Moreover, unless students are going to progress to higher l e v e l s of mathematics ( in S f a r d ' s Using C a l c u l a t o r s to Teach Algebra 41 h i e r a r c h i c a l sense), there i s no v a l i d mathematical reason for i n s i s t i n g that they do mathematics the conventional way. Using C a l c u l a t o r s to Teach Algebra 42 Chapter 4 Role of the C a l c u l a t o r The use of c a l c u l a t o r s , even today, i s frequent ly condemned by parents who say they want t h e i r c h i l d r e n to l e a r n to th ink and by students who say they do not want to take the easy route but ra ther to understand what they are doing . I s t a r t e d us ing c a l c u l a t o r s for the p r a c t i c a l reason that my students could hot r e l y on t h e i r memory for number f a c t s , e i t h e r adding or m u l t i p l y i n g . I s t a r t e d asking students to do c a l c u l a t i o n s i n v o l v i n g more than one operat ion because c a l c u l a t i o n s i n v o l v i n g jus t one operat ion were too easy. However, I found that students could l e a r n a great dea l about the s t r u c t u r e of an ar i thmet ic expression by l e a r n i n g how to evaluate i t on a c a l c u l a t o r . In t h i s chapter I argue 1 . that the r e s p o n s i b i l i t i e s or tasks handed over by students to c a l c u l a t o r s are i r r e l e v a n t to the study of a lgebra . 2 . that understanding need not precede p r a c t i c e . R e s p o n s i b i l i t i e s of C a l c u l a t o r s and Students "Teachers can and do provide e l e c t r o n i c c a l c u l a t o r s , " says Pimm ( 1995 , p . 6 1 ) . "As wel l as ho ld ing number symbols, the e l e c t r o n i c c a l c u l a t o r a l so holds procedures over which I have no c o n t r o l . . . " (p. 6 2 ) . By au thor i z ing the use of c a l c u l a t o r s , he says, the teacher suggests "'you need not attend to t h i s : i t i s Using C a l c u l a t o r s to Teach Algebra 43 being taken care of for you'" (p. 61). C e r t a i n l y , i n my courses, students, with my permiss ion, turn over to t h e i r c a l c u l a t o r s r e s p o n s i b i l i t y for 1. the meaning of the numbers (how b i g or small they are , what a decimal po int means, and - with some c a l c u l a t o r s - rounding o f f a number to a g iven number of decimal p l a c e s ) . 2. knowing what you get when you add, subtrac t , m u l t i p l y , or d i v i d e two given numbers or r a i s e a number to a power. 3. remembering a number s tored i n i t s memory. 4. the execution of the operat ions . (If t h e i r c a l c u l a t o r s are a l g e b r a i c , students would a l so hand over to the c a l c u l a t o r r e s p o n s i b i l i t y for the order i n which the operat ions are executed i f I d i d not demand that they continue to number the operat ions to show t h e i r o r d e r . ) Th i s handing over of r e s p o n s i b i l i t i e s gives the c a l c u l a t o r a u t h o r i t y . I t means that the c a l c u l a t o r can be used to "prove" c e r t a i n fac t s i f they come into quest ion or d i spute . For example, the fac t that the c a l c u l a t o r d i s p l a y s 14904 when the buttons 36 x 414 = are pressed cons t i tu te s proof that 36 x 424 = 14904. Of course, the c a l c u l a t o r does not acquire t h i s au thor i ty untested: students check that i t gives 6 for 3 x 2 , for example. (They of ten perform such a check spontaneously when they obta in an answer which they th ink i s r i g h t but which they are t o l d i s wrong i n order to make sure that the c a l c u l a t o r i s working p r o p e r l y , s ince c a l c u l a t o r s w i l l g ive wrong answers when they break down and jus t before t h e i r b a t t e r i e s d i e . ) Using C a l c u l a t o r s to Teach Algebra 44 In a d d i t i o n , as I pointed out i n Appendix 4, I use c a l c u l a t o r s to j u s t i f y to students my statements about logarithms and the reverse of the y x operat ion . However, students r e t a i n r e s p o n s i b i l i t y for 1. press ing the correc t d i g i t buttons. 2. press ing the correc t operat ion buttons so that they operate on the c o r r e c t numbers, i . e . press ing the operat ions buttons i n the conventional order on an non-a lgebra ic c a l c u l a t o r or a f t e r an a l g e b r a i c c a l c u l a t o r r e g i s t e r s an e r r o r , and i n the w r i t t e n order on an a lgebra i c c a l c u l a t o r . 3. r e c a l l i n g a number from the c a l c u l a t o r ' s memory. As Pimm says, "As a user , my focus then becomes: have I g iven the machine the correc t things i n the c o r r e c t order to have i t do what I want?" (Pimm, 1995, p . 62). A f t e r d i s cus s ing the " r e l a t i v e absence of imagery assoc ia ted with a c a l c u l a t o r , " Pimm (1995) warns, "It i s an i n t e r e s t i n g and open quest ion whether t h i s . . . i s a p o t e n t i a l weakness - the mechanisms are opaque and therefore o f f e r very l i t t l e support - or a p o t e n t i a l s trength - l eav ing p u p i l s free to form t h e i r own imagery - with regard to us ing such devices to he lp ga in e i t h e r numerical f luency or understanding" (pp. 80-81). Pimm's statement that "the t h e o r e t i c a l j u s t i f i c a t i o n for 'why' the device [the c a l c u l a t o r ] works i s complex and opaque" (p. 83); h i s quest ion , "What mathematics has been embedded i n i t s c o n s t r u c t i o n and how e a s i l y a v a i l a b l e i s i t to a user?" (p. 83); and h i s warning that "This degree of opaci ty with regard to the Using C a l c u l a t o r s to Teach Algebra 45 method of funct ion ing a c t u a l l y cons t i tu te s a s h i f t away from even a c a l c u l a t i o n a l a i d toward a complete s u b s t i t u t i o n for c a l c u l a t i o n that i s r e s i s t a n t to understanding at any but a very deep l e v e l " (p. 84) would be important for my thes i s i f I were us ing c a l c u l a t o r s simply to c a l c u l a t e expressions conta in ing one operat ion (such as 12.6 x 13.2) . However, they are i r r e l e v a n t to my use of the c a l c u l a t o r , which has students focus on the order of the operat ions , not the numbers. (The fac t that i n t h e i r past courses my students have focused on the numbers and not the operations i s revealed by the i n i t i a l tendency of some to put the "ordering numbers" under the numbers i n the expression instead of under the operat ions . ) Indeed, for my purposes, the expressions 2 + 3 x 6 and 1.2 + 3.6 x 543 are equiva lent . As Pimm acknowledged i n c l a s s , even at t h i s point i n my course the numbers have a lready "dropped out ." Pimm (1995) a l so warned that " . . . i f the c a l c u l a t o r i s there , i t determines the sy l labus to an extent that may be f a r from des i rab le" (p. 84). There i s no denying that the c a l c u l a t o r has determined the sy l labus of my courses . As far as the a lgebra which succeeds the c a l c u l a t o r u n i t i s concerned, the use of the (non-algebraic) c a l c u l a t o r furnishes me with an excuse to have students order the operat ions i n an express ion i n a manner which reveals the s t r u c t u r e of the express ion . As far as the re s t of the course i s concerned, where students c a l c u l a t e i n t e r e s t and mortgage and annuity payments, Using C a l c u l a t o r s to Teach Algebra 46 e t c . , the c a l c u l a t o r i s e s s e n t i a l as a c a l c u l a t i n g dev ice: students must evaluate expressions conta in ing as many as twenty operat ions and obta in answers i n d o l l a r s and cents . (When my students reach the longest formulae, i n f a c t , they a c t u a l l y use programmable c a l c u l a t o r s , handing over to the c a l c u l a t o r s the fur ther r e s p o n s i b i l i t y of repeat ing given operat ions i n a g iven o r d e r . ) Pimm (1995) d i s t i n g u i s h e s between computational dev ices , " p r i m a r i l y concerned with automation and f luency of o p e r a t i o n , " and pedagogical devices , "about i l l u m i n a t i o n and understanding" (p. 84). "No c a l c u l a t i n g a i d i s p r i m a r i l y a pedagogic device -though they a l l provide untaught lessons neverthe less ," he warns (p. 63). However, there i s no quest ion i n my mind o r , as I s h a l l demonstrate i n Chapters 6 and 7, i n my students' minds, that when students use c a l c u l a t o r s i n the way I have descr ibed they are doing and t h i n k i n g mathematics. I observe, for example, how q u i c k l y t h e i r i n i t i a l j u b i l a t i o n at being allowed to use c a l c u l a t o r s disappears when they d i scover how much r e s p o n s i b i l i t y they r e t a i n . Used as I use i t , a c a l c u l a t o r may f a i r l y be c a l l e d a pedagogical device as wel l as a computational dev ice , a c a l c u l a t i n g a i d . Since the numbers have a lready , i n e f f e c t , "dropped out ," students are i n t e l l e c t u a l l y ready to s u b s t i t u t e l e t t e r s but r e t a i n the s t ruc ture impl ied by the operat ions - that i s , to begin algebra as a f a m i l i a r d i s c u s s i o n about a new k ind of Using C a l c u l a t o r s to Teach Algebra 47 objec t . Unless the c a l c u l a t o r "took care of" computation for them, students would f i n d i t impossible to focus on the order of the operat ions and the s t ruc ture of the express ion, any more than I can focus at the same time on the sound of each chord, the "shape" of each chorus, and the o v e r a l l s t ruc ture of Handel 's Messiah. Understanding from Doing In the context of h i s Chapter 4, e n t i t l e d "What Counts as a Number?" Pimm (1995) discusses the s imul tane i ty of and, at the same time, the tens ion between, c r e a t i n g mathematics, doing mathematics, and understanding mathematics. I designed my courses so that students could succeed i n doing something which both they and I would recognize as mathematics, but I concentrated on having the students do mathematics whether or not they understood i t , with very l i t t l e thought of developing what I would have c a l l e d i n s i g h t or understanding i n my students and with no thought of t h e i r c r e a t i n g i t . Accord ing ly , I introduced the c a l c u l a t o r as a computational a i d . I was qu i te w i l l i n g to make the "tradeoff between transparency . . . and f luency or e f f i c i e n c y . . . " (p. 63) descr ibed by Pimm (1995). I would have agreed that "any o v e r - r e l i a n c e on machines can lead to the atrophy of the human f a c u l t i e s involved" (p. 82), but I would have argued that anything mathematical I could get my students to do, even with the a i d of a c a l c u l a t o r , was be t t er than nothing . In my physics and sc ience courses , Using C a l c u l a t o r s to Teach Algebra 48 however, I shared with other teachers a "fear of apparent s o p h i s t i c a t i o n of performance unrooted i n understanding, and the perenn ia l de s i re . . . to be able to read comprehension from success fu l p r a c t i c e " (p. 77). S i m i l a r l y , I began to teach the r e v e r s a l method as an a lgor i thm for s o l v i n g equations - or , r a t h e r , an a lgor i thm for s o l v i n g equations for which I could make up f o o l - p r o o f r u l e s . Again , I would have agreed with Pimm (1995) that "there are two separate aspects of a lgori thms, namely the 'what to do' and the 'why t h i s i s what we do . . . ' " (p. 69), and I would have s a i d that I was concentrat ing e n t i r e l y on having students l e a r n "what to do ." Having thus introduced c a l c u l a t o r s and the r e v e r s a l method with the lowest (mathematically speaking) of motives, I have been b lessed beyond what I deserved. Besides t h e i r success at doing mathematics, I began to see and encourage s igns of understanding and even c r e a t i v i t y among my students , which I recount i n Chapters 6 and 7. What Pimm (1995) c a l l s the "confusion of the two funct ions" of c a l c u l a t o r s - the computational and the pedagogical - d i d not l ead , as he warns could happen, "to crossed in ten t ions and erroneous expectat ions , as wel l as missed opportun i t i e s" (p. 84). Today, I agree with Pimm that "working at such algorithms can r e s u l t i n meaning" (p. 61); that i n f a c t , " i t may wel l be that some l e v e l of p r a c t i c e i s requ ired before i t even makes sense to ask 'why?'" (p. 76); that " r e f l e c t i n g on the p r a c t i c e may thus Using C a l c u l a t o r s to Teach Algebra 49 provide some access to the 'embedded' mathematics" (p. 77); and that " a l l of these resources (phys i ca l , g e s t u r e - g r a p h i c a l , and l i n g u i s t i c ) contr ibute to the meaning" (p. 61). With V a l e r i e Walkerdine, whom he quotes on p. 75, I "draw a t t e n t i o n to the c r e a t i o n of meaning wi th in p r a c t i c e s . " A r i s i n g as i t does from a background of p r a c t i c e i n us ing c a l c u l a t o r s , any mathematical understanding or c r e a t i v i t y my students develop i s expressed i n terms of c a l c u l a t o r language, making i t (perhaps) d i f f i c u l t for anyone unaccustomed to that language to recognize i t . As Pimm (1995) says, "To the extent that understanding a r i s e s from r e f l e c t i o n on the p r a c t i c e , p a r t i c u l a r hard-won understandings may d i f f e r " (p. 86). I t appears that once students f i n d that they can do what they are being asked to do - once the threat of frequent and prolonged f a i l u r e i s removed - some of the i n s i g h t that I missed i n them when I f i r s t began to teach does, i n f a c t , emerge. The evidence i s that students frequent ly and spontaneously v e r b a l i z e t h e i r own observations about a l g e b r a i c expressions and equations. Since the language they use i s , l i k e mine, der ived from the language of c a l c u l a t o r s , I suspect that by us ing c a l c u l a t o r s as I do I have, unwi t t ing ly , bypassed a number of roadblocks for these students . The i n s i g h t s v e r b a l i z e d by my students, who continue to th ink of a l g e b r a i c expressions as b u i l d i n g i n s t r u c t i o n s or , at most, as e n t i t i e s someone has a c t u a l l y b u i l t , may be compared to what a man sees when he i s b u i l d i n g a house or looking at a house he has Using C a l c u l a t o r s to Teach Algebra 50 a c t u a l l y helped to b u i l d or has seen others b u i l d , as opposed to what someone sees who i s f a m i l i a r with b u i l d i n g i n genera l , but who looks at the house for the f i r s t time a f t e r i t i s f i n i s h e d . I t may be that we have a b ias i n favor of understanding before doing because understanding (on the part of somebody, i f not ourselves) precedes doing i n the l o g i c a l sense of "precedes." For example, no one could r e p a i r a car engine unless somebody had already understood car engines. However, r e f l e c t i n g on the contr ibut ions of the p h y s i c a l , the ges ture -graph ica l and the l i n g u i s t i c to the meaning of numbers, Pimm (1995) says, "It i s . . . far from obvious whether one should come before another" (p. 62). On page 9 of the Introduct ion , Pimm (1995) quotes Diderot : " A l l e z en avant, l a f o i [comprehension] vous v i e n d r a , " which can be t r a n s l a t e d as "Keep moving ahead; f a i t h [understanding] w i l l come" (p. 9). Thi s i n j u n c t i o n br ings to mind the fo l lowing: 1. Augustine of Hippo ( t rans la ted i n R e t t i g , 1993) wrote, "Understanding i s the recompense of f a i t h . Therefore , seek not to understand so that you may b e l i e v e , but be l i eve that you may understand" (p. 18). 2. Lewis (1952) wrote, "When you are not f e e l i n g p a r t i c u l a r l y f r i e n d l y but know you ought to be, the best th ing you can do, very o f ten , i s to put on a f r i e n d l y manner and behave as i f you were a n i c e r person than you a c t u a l l y are . And i n a few minutes, as we have a l l no t i ced , you w i l l r e a l l y be f e e l i n g f r i e n d l i e r than you Using C a l c u l a t o r s to Teach Algebra 51 were. Very of ten the only way to get a q u a l i t y i n r e a l i t y i s to s t a r t behaving as i f you had i t already" (p. 158). 3. Sfard (1991) quotes Jourdain (1956): '"When l o g i c a l l y -minded men objected' to the 'absurd' not ions of negative and imaginary (complex) numbers, 'mathematicians simply ignored them and s a i d , "Go on; f a i t h w i l l come to you . '" Those who could see the inner beauty of the idea thought that the new numbers, 'though apparent ly un in terpre tab le and even s e l f - c o n t r a d i c t o r y , must have l o g i c . So they [the numbers] were used with a f a i t h that was almost f i rm and was j u s t i f i e d much l a t e r ' " (pp. 29-30). 4. In the disagreement between observations and the c l a s s i c a l wave theory of blackbody r a d i a t i o n , Planck introduced the very u n c l a s s i c a l E « f i n order to b r i n g theory i n t o agreement with observat ion . However, as G i a n c o l i (1980) put i t : "Planck himsel f was not comfortable with i t and reported that he proposed i t only to b r i n g theory in to accord with experiment; and he s a i d he hoped that a be t t er explanat ion would soon come f o r t h . A l a s , i t d i d not , and Planck's quantum hypothesis has come to be accepted as a fac t of nature" (p. 650). In other words, the p r a c t i c e became the new understanding. 5. When my niece K a e t l i n was two, I showed her a photograph of our s t a f f when I took her to my school one day. " A c t u a l l y , " I s a i d , "this p i c t u r e was taken l a s t year ." She r e p l i e d , "So some of them must have r e t i r e d by now." I was used to such language from her; her mother had always spoken to her i n adul t terms. A few minutes l a t e r , I introduced her to one of the teachers , Using C a l c u l a t o r s to Teach Algebra 52 say ing , "Mr. H a r r i s i s going to r e t i r e t h i s year ." When H a r r i s mouthed to me over K a e t l i n ' s head, "She won't understand t h a t , " I asked K a e t l i n , "Do you know what ' r e t i r e ' means?" She answered ( in the tone of voice she would have used i f I had asked, "Do you know what I want for Christmas?"), "No?" "Then why d i d you say that some of those teachers must have r e t i r e d ? " I asked. She r e p l i e d (not deprecat ing ly , but i n the tone of one who i s e x p l a i n i n g something that should be obvious) , "Well , I 've heard the word." I suspect that a great many researchers formulate t h e i r quest ions a f t e r they see what kinds of answers are forthcoming. I know that a great many o u t l i n e s i n high school and u n i v e r s i t y w r i t i n g courses are put together, or at l ea s t cons iderably modif ied, a f t e r the essay has been w r i t t e n . There i s a profound t r u t h under ly ing the remark made by the l i t t l e g i r l when she was t o l d to th ink before she spoke: "How do I know what I th ink t i l l I see what I say?" I t i s f ar from obvious to me as a phys ics teacher, for example, whether students should f i r s t use the concept of "momentum" or l earn i t s d e f i n i t i o n . At l ea s t once a year, one of my students w i l l ask me what momentum is. The l a s t t ime, I asked i n r e t u r n , "Do you know what time i s?" The student s a i d she d i d . "What?" I asked. " T e l l me." When she had seen that she cou ld not, I pointed out to her that although she knew how to t a l k about time, t e l l what time i t was, and even so lve kinematics problems i n v o l v i n g time, she could not def ine i t . I a l so pointed out that Using C a l c u l a t o r s to Teach Algebra 53 she had learned what she d i d know about time through us ing the concept and observing others use i t and suggested that she could acquire the same "fee l ing" for momentum by us ing the word, l i s t e n i n g to me use the word, and s o l v i n g the assigned problems on the subjec t . Newton def ined the momentum of an object as a quant i ty measured by m u l t i p l y i n g the o b j e c t ' s mass by i t s v e l o c i t y . I t i s arguable that one reason for h i s i d e n t i f i c a t i o n of t h i s q u a n t i t y , measured i n t h i s way, i s that momentum i s a conserved quant i ty i n an i s o l a t e d system. Did momentum and i t s conservat ion s p r i n g f u l l y armed from Newton's head? Or was there a f r u i t f u l i n t e r p l a y of d e f i n i t i o n and proof s i m i l a r to the one descr ibed by Lakatos (1976)? I f the former, can we induce the same phenomenon i n our students? I f the l a t t e r , what consequences, i f any, does the i n t e r p l a y have for teachers of the subject? Should teachers s t a r t with the d e f i n i t i o n of momentum, which i s easy to understand but which seems to have no reason or purpose? Should they s t a r t with i t s conservat ion , before students have a sense of what i t i s that i s conserved? Or should they attempt to pa int a p i c t u r e l i k e an a r t i s t whose every brush stroke i s "informed" by h i s sense of what the p i c t u r e w i l l become? (According to The Compact Edition of the Oxford English Dictionary (1980), the t r a n s i t i v e verb "inform" - apart from the "informing" of a person i n the sense of pass ing on informat ion -means "to give form to; put in to form or shape; to g ive formative p r i n c i p l e or determinative character to; hence to stamp, impress, Using C a l c u l a t o r s to Teach Algebra 54 imbue, or impregnate with some s p e c i f i c q u a l i t y or a t t r i b u t e ; e s p e c i a l l y to impart some pervading, a c t i v e , or v i t a l q u a l i t y to ; to imbue with a s p i r i t ; to f i l l or a f f ec t (the mind or heart) with a f e e l i n g , thought, e t c . ; to i n s p i r e , animate" (p. 1431).) Conclus ion Although the i n t r o d u c t i o n of l e t t e r s i s the most v i s i b l e change as students move from ar i thmet ic to a lgebra , changes i n the meanings of operations and the equals s i gn , together with the necess i ty of understanding the s t ruc ture of a l g e b r a i c expressions and equations, are a l so sources of the d i f f i c u l t i e s so many students have when they move from ar i thmet ic to a lgebra . The language of the c a l c u l a t o r , which i s the language of a r i t h m e t i c , can be transposed in to algebra with only minor changes. When i t i s , i t can be understood by students who have had l i t t l e or no success i n "regular" courses . Moreover, unless students are going to progress to higher l e v e l s of mathematics ( i n S f a r d ' s h i e r a r c h i c a l sense), there i s no v a l i d mathematical reason for i n s i s t i n g that they do mathematics the convent ional way. Nevertheless , two reservat ions are commonly expressed about t h i s a l t e r n a t i v e way of teaching a lgebra: f i r s t , that students proceed by r o t e , without anything that teachers would recognize as understanding; and second, that i t does not deserve to be d i g n i f i e d with the name of mathematics. Accord ing ly , the observations and anecdotes recounted and analyzed i n Chapters 6 and 7 are d i r e c t e d towards demonstrating: Using C a l c u l a t o r s to Teach Algebra 55 1. that i t i s pos s ib l e to s h i f t the a t t e n t i o n of students from numbers to operations by the use of e l e c t r o n i c c a l c u l a t o r s . 2. that i t i s pos s ib l e to s h i f t the a t t e n t i o n of students from operat ions to the o v e r a l l s t ruc ture of a l g e b r a i c expressions and equations by the use of e l e c t r o n i c c a l c u l a t o r s . 3. that the responses of students who have been taught a lgebra i n c a l c u l a t o r language demonstrate that they understand a lgebra . I t fol lows not only that my mathematics courses are v i a b l e a l t e r n a t i v e s to the "regular" mathematics courses for unmathematical students, but a l so that they can serve as bridges between ar i thmet i c and algebra for students who w i l l be capable, before they leave high school , of passing the "regular" courses . In teaching a lgebra , which so many students f i n d so d i f f i c u l t , we should be w i l l i n g to use any method that has a p o s s i b i l i t y of success, even when i t s p h i l o s o p h i c a l j u s t i f i c a t i o n i s d o u b t f u l . Otherwise we are l i k e the nineteenth century captains who pred ic ted that steamships would never rep lace s a i l i n g ships because they would be the r u i n of good seamanship. We should be more l i k e Shakespeare than M i l t o n , as descr ibed i n Lewis' essay Variation in Shakespeare and Others ( 1969 ) . Here Lewis t e l l s of the famous problem presented to Samuel Johnson by Boswell: whether Shakespeare or M i l t o n had drawn the more admirable p i c t u r e of a man, as revealed i n Hamlet's d e s c r i p t i o n of h i s father i n Shakespeare's Hamlet and M i l t o n ' s d e s c r i p t i o n of Adam i n Paradise Lost (see Appendix 5 . ) Using C a l c u l a t o r s to Teach Algebra 56 Lewis says: The two passages i l l u s t r a t e two r a d i c a l l y d i f f e r e n t methods of poe t i c d e s c r i p t i o n . M i l t o n keeps h i s eye on the objec t , and b u i l d s up h i s p i c t u r e i n what seems a n a t u r a l order . I t i s d i s t i n g u i s h e d from a prose catalogue l a r g e l y by the verse , and by the e x q u i s i t e choice not of the r a r e s t words but of the words which w i l l seem the most nobly obvious when once they have been c h o s e n . . . . Shakespeare's method i s wholly d i f f e r e n t . Where M i l t o n marches s t e a d i l y forward, Shakespeare behaves rather l i k e a swallow. He dart s at the subject and glances away; and then he i s back again before your eyes can fol low him. I t i s as i f he kept on having t r i e s at i t , and being d i s s a t i s f i e d . He dart s image a f t e r image at you and s t i l l seems to th ink that he has not done enough. He br ings up a whole l i g h t a r t i l l e r y of mythology, and gets t i r e d of each p iece almost before he has f i r e d i t . He wants to see the object from a dozen d i f f e r e n t angles; i f the u n d i g n i f i e d word i s pardonable, he nibbles, l i k e a man [sic] t r y i n g a tough b i s c u i t now from t h i s s ide and now from tha t ." (pp. 74-75) Using C a l c u l a t o r s to Teach Algebra 57 Chapter 5 Research Methods Beginning i n September 1993 and ending i n June 1996, I conducted a study of my own students i n Mathematics 8A, 9A, 10A, and 11 A, with the o v e r r i d i n g purpose of documenting my students' i n s i g h t i n t o mathematics. My intent i n t h i s chapter i s twofold: 1. to descr ibe the c h a r a c t e r i s t i c s of the students I s tud ied , my classroom a c t i v i t i e s and course requirements, and the degree of success my students had i n my courses, so as to he lp the reader judge the relevance of my study to other students . 2. to descr ibe and j u s t i f y my research methods. C h a r a c t e r i s t i c s of Students Studied Students who take my courses have t y p i c a l l y had l i t t l e or no success i n previous mathematics courses, as documented by t h e i r school records . They and (usual ly) t h e i r parents have despaired of t h e i r ever being success fu l i n "regular" courses and of thus meeting the p r e r e q u i s i t e s for going to u n i v e r s i t y . Appendix 2, which contains the high school records of the students who took Mathematics 11A from me between January and June 1993 as we l l as those of two Mathematics 11A c lasses taught e a r l i e r by another teacher us ing my mater ia l s , documents t h i s lack of success . Students i n Mathematics 8A, who have not yet completed any Using C a l c u l a t o r s to Teach Algebra 58 h igh school mathematics courses, take the course because they have been recommended to do so by t h e i r elementary school teachers . A l l of these students t y p i c a l l y absent themselves from somewhere between four and f i v e times as many c lasses as students i n my physics c la s se s , who are more academic. Many of t h e i r absences c o n s t i t u t e ou tr igh t truancy, as the students would admit, but many others are accounted f o r , often i n advance, with excuses such as, "I had to help my parents move," "I had to go to ICBC," "I have to take my d r i v e r ' s t e s t ," "I have to go to the doctor , the d e n t i s t , e t c ." - a c t i v i t i e s which academic students and t h e i r parents would schedule outs ide school hours, not cons ider ing them s u f f i c i e n t l y important to make the students miss a c l a s s . They are a l so t y p i c a l l y somewhat unpunctual , prone to leave the classroom for long "bathroom breaks ," u n w i l l i n g to do homework, and d i sorganized , often appearing i n c l a s s without books, c a l c u l a t o r s or w r i t i n g u t e n s i l s . The d i f f erences between grades can be summarized by saying that the higher the grade l e v e l , the more q u i c k l y the students can l e a r n the mater ia l and the more complicated the problems they can so lve , as d e t a i l e d below. Bas i c , "Star ," and "Double-Star" Work Before I can descr ibe a t y p i c a l c l a s s and the requirements of my courses , I must exp la in what I mean by "star" and "double-star" work. I t i s common p r a c t i c e for teachers to present the same Using C a l c u l a t o r s to Teach Algebra 59 mater ia l to a l l the students i n a c l a s s , to examine them a l l on the same m a t e r i a l , and then to ass ign f i n a l l e t t e r grades according to the fo l lowing scheme, which i s standard i n B r i t i s h Columbia schools: 86% i A s 100% 73% < B < 86% 67% s C+ < 73% 60% < C < 67% 50% £ C - < 60% 40% £ D < 50% 0% £ E < 40% However, I found when I f i r s t s t a r t e d teaching - i n a l l my courses , not jus t "a l ternat ive" mathematics courses - that I could not mark the extra volume of very d i f f i c u l t work of which my "A" students were capable out of the 14% allowed between 86% and 100%, nor the ex tra volume of d i f f i c u l t work of which my "B" students were capable out of the 13% allowed between 73% and 86%. Nor, I found, was i t reasonable to requ ire my other students even to attempt the d i f f i c u l t work of which the "B" and "A" students were capable . Instead, I s t a r t e d des ignat ing a l l problems - on worksheets, quizzes , and exams - b a s i c , "s tar ," or "double s tar" and a l lowing students to omit "star" and "double s tar" problems i f they wished. However, I requ ired students to work on "star" problems i f they wanted (and were capable of achieving) a B and to work on both "star" and "double s tar" problems i f they wanted (and were capable of achieving) an A. With t h i s scheme, I ass ign l e t t e r grades according to Table 2, which shows, as an example, how I ass ign l e t t e r grades on the f i n a l exam i n Mathematics 11A, which has a p o s s i b l e 108 marks Using C a l c u l a t o r s to Teach Algebra 60 awarded for bas i c problems, a pos s ib l e 60 for "star" problems, and a p o s s i b l e 92 for "double s tar" problems. (This exam i s inc luded i n Appendix 1, which a l so shows how I inform students of the lower cuto f f for each l e t t e r grade.) The r i g h t s ide of Table 2 shows the convent ional l e t t e r - g r a d e scheme. In order to report marks to parents , I transform my marks (on the l e f t ) to standard marks (on the r i g h t ) by a continuous s er i e s of l i n e a r transformations which transform 100% to 100%, 0 to 0, and the lower cuto f f for each l e t t e r grade to the corresponding standard c u t o f f . Table 2 Transformation of L e t t e r Grades 108 + 60* + 92** = 260 .85(260) = 221.0 108 + 60* = 168 ,85(168) = 142.8 C+ 108 .85(108) = 91 .8 .70(108) = 75.6 .50(108) = 54.0 .25(108) = 27.0 0 C -C+ -100% - 86% - 73% - 67% - 60% - 50% - 40% Using C a l c u l a t o r s to Teach Algebra 61 Classroom A c t i v i t i e s and Course Requirements An hour- long c l a s s i n any of my mathematics courses t y p i c a l l y begins with my "going over" the problems on the "pract ice quiz" given at the end of the previous day. A "pract ice quiz" takes about f i v e or ten minutes and t y p i c a l l y comprises three problems taken from the day's worksheet, with only the l e t t e r s or the numbers changed: one b a s i c , one "star" and one "double s t a r . " The students do whichever of these problems they choose, but only one. The number of marks awarded for each problem r e f l e c t s i t s d i f f i c u l t y . Usua l ly I ask a student who I know made a mistake on the "pract ice quiz" to t e l l me, out loud, how to so lve the problem, thus g i v i n g me the opportunity to exp la in the nature of the e r r o r to the whole c l a s s . A l t e r n a t i v e l y , i n Grade 8, for two c las ses dur ing one semester, I had students who had solved the problems c o r r e c t l y "go over" them for the c l a s s , but I abandoned t h i s p r a c t i c e for var ious reasons, d iscussed l a t e r i n t h i s chapter . This i n t r o d u c t i o n serves to remind the students of what they were doing the previous day and what they w i l l be asked to do today. Then I t e l l them which worksheet they w i l l be working on today and what k ind of assessment w i l l fo l low: another "pract ice quiz" (which serves as a "rewrite" of the f i r s t one, with the mark recorded being the higher of the two), or , every three or four days, a q u i z , which t y p i c a l l y takes between twenty and t h i r t y minutes to complete and t y p i c a l l y contains f i v e b a s i c problems, two "star" problems and two "double s tar" problems. Students may Using C a l c u l a t o r s to Teach Algebra 62 "rewrite" a quiz (another vers ion) on a pre-announced day a f t e r schoo l . Again, the mark recorded i s the higher of the two. For the h a l f an hour i n the middle of the c l a s s , students t y p i c a l l y work on the day's worksheet by themselves or with f r i e n d s . Each worksheet has three vers ions , which take three days; each has an answer key suppl ied and students are encouraged to ask quest ions of me and/or the teaching aide ( i f there i s one) whenever t h e i r answer d i f f e r s from the one i n the key. I ass ign no homework, mostly because the students need he lp so f requent ly . I have found that i f I r e q u i r e them to do homework, the consc ient ious ones attempt to do the work but get "stuck" or , worse, do i t wrong, while the unconscient ious ones simply c la im that they t r i e d and got "stuck." At the end of the u n i t the students wri te an hour- long exam on the u n i t , which they may "rewri te ," i f they wish, on a p r e -announced day a f t e r school . As always, the mark recorded i s the higher of the two. At the end of the course, there i s a two-hour-long f i n a l exam. The students' f i n a l marks are c a l c u l a t e d by weighting the var ious components of my assessment as fo l lows: "pract ice quizzes" 20%, quizzes 20%, u n i t exams 40%, and the f i n a l exam 20%. I g ive no marks for s o - c a l l e d "compassionate reasons," arguing that the students' current marks are posted every day and that i f they want to improve t h e i r marks, they can always get extra he lp and do "rewrites ." Anyone who f i n i s h e s the course with Using C a l c u l a t o r s to Teach Algebra 63 l e s s than 49.5% i s requ ired to repeat i t . The contents of the f i v e courses - Mathematics 8A, 9A, 10A, and 11A - are s i m i l a r . The d i f ferences r e f l e c t the fac t that the higher the g r a d e - l e v e l , the more q u i c k l y the students can l e a r n the mater ia l and the more complicated the problems they can so lve . For example, I can teach Grade 11 students how to so lve equations, i n c l u d i n g groups and mul t i - s t ep s o l u t i o n s , i n f i v e three -hour- long worksheets, even i f they have not taken any of my previous courses . However, to teach Grade 8 students to so lve equations with far fewer operat ions , no groups and only one step takes e ight three -hour- long worksheets. Appendix 6 contains an o u t l i n e of the f i v e courses . Quant i ta t ive Data About Students Studied Table 3 shows the number of c lasses I taught i n each course between September 1993 and June 1996, the number of students i n each c l a s s , and the f i n a l d i s t r i b u t i o n of l e t t e r grades i n each c l a s s . Using C a l c u l a t o r s to Teach Algebra 64 Table 3 Actua l L e t t e r Grade D i s t r i b u t i o n Course Dates A B C+ C C - D E T o t a l Math. 8A Sep 93 -- Jan 94 3 11 9 1 0 1 1 26 Sep 93 -- Jan 94 0 4 4 5 2 0 0 15 Sep 94 -- Jan 95 0 8 6 3 2 1 1 21 Sep 94 -- Jan 95 2 3 6 4 0 2 1 18 Math. 9A Sep 94 -- Jan 95 0 8 10 4 3 1 1 27 Sep 95 -- Jan 96 0 3 7 1 3 4 0 18 Math. 10A Sep 95 -- Jan 96 1 10 10 3 4 5 0 33 Math. 11A Jan 93 -- Jun 93 0 6 8 4 6 6 0 30 Sep 93 -- Jan 94 1 3 12 4 • 5 4 3 32 Jan 95 -- Jun 95 2 8 9 3 3 1 1 27 Jan 96 -- Jun 96 0 1 7 4 5 6 4 27 T o t a l Sep 93 -- Jun 96 9 65 88 36 33 31 12 274 The graph i n Figure 1 shows the d i s t r i b u t i o n of these l e t t e r grades. Using C a l c u l a t o r s to Teach Algebra 65 Figure 1 90 -N 80 • u m b 70 • e r 60 • o f 50 • s 40 • t u d 30 • e n 20 • t s 10 -D i s t r i b u t i o n of L e t t e r Grades B C+ C C-L e t t e r grade D Considered i n conjunct ion with the above d e s c r i p t i o n of my course requirements, Table 3 and Figure 1 are evidence that I am ne i ther unduly l en ient nor unduly harsh i n eva luat ing students' achievement and thus support the c la im I made i n Chapter 1 that my students can and do succeed i n these courses, observably and t e s t a b l y . Data C o l l e c t i o n : Attempts at Interviews I now turn to my attempts to e l i c i t from students responses that would revea l t h e i r i n s i g h t in to the mathematics I was teaching them. By 1993, I had been c o l l e c t i n g very in formal ly for about fourteen years comments made by Mathematics 11A students which Using C a l c u l a t o r s to Teach Algebra 66 had s u r p r i s e d me by the i n s i g h t they revea led . In September 1993, a f t e r I had decided to begin a formal case study, I began making notes on them as they occurred . However, I a l so envisaged in terv iewing se l ec ted students and v ideotaping or audiotaping the in terv iews , perhaps asking the students to solve a problem and d i s c u s s i n g i t with them as they d i d so - an idea I eventua l ly abandoned, for reasons given below. Accord ing ly , on March 28 1994, I interviewed two students who had s u c c e s s f u l l y completed Mathematics 8A i n the f i r s t Mathematics 8A course I had ever taught. They had had t h e i r l a s t c l a s s January 27 1994 and t h e i r f i n a l exam January 28 1994. Both had worked at the A - l e v e l (that i s , so lved the "double-star" problems) throughout the course but for var ious reasons had f i n i s h e d with l e t t e r grades of B (one with 83% and the other with 81%). I interviewed them for about h a l f an hour and videotaped the in terv iew. To begin, I asked them to solve the fo l lowing equation together at the chalkboard: (This was a "double-star" quest ion . ) They so lved i t c o r r e c t l y , the f i r s t time, by w r i t i n g the fo l lowing on the board: 3 . 2 + (x - 1 . 2 ) 4 / 3 1 3 . 2 5 = 2 . 5 St. x 2 . 5 - 1 . 2 y x 4 / 3 + 3 . 2 y x 3 / 4 + 1 .2 x 1 3 . 2 5 - 3 . 2 e n d -=- 1 3 . 2 5 2 . 5 x Using C a l c u l a t o r s to Teach Algebra 6 7 (2.5x 13.25 - 3 . 2 ) 3 / 4 +1.2 = x= 13.99456778 Check: 3.2 + (13.99456778 - 1 .2) 4 / 3 _ 13.25 " ' 5 There was some d i s c u s s i o n about where the brackets should go i n the r e v e r s a l column and the fac t that they had to c a l c u l a t e the exponents 3/4 and, l a t e r , 4/3, and s tore the r e s u l t s i n the c a l c u l a t o r ' s memory before performing the other c a l c u l a t i o n s . (Both had used non-a lgebra ic c a l c u l a t o r s . ) The interview s a t i s f i e d me that the students had r e t a i n e d what they had learned . Although they had seen the equation I gave them on an exam at the end of January, ne i ther of them appeared to recognize i t . They were not, therefore , r e c a l l i n g how to do that i n d i v i d u a l quest ion , but rather remembering and apply ing general r u l e s . Indeed, they s a i d as much to each other as they so lved the problem. They apologized for stumbling s l i g h t l y and reminded me that they had not seen "this s tuf f" s ince the end of January, but they both agreed that " i t comes back to you." Having been t o l d that I wanted to interview them to convince my col leagues that my students r e a l l y d i d understand what they had learned i n Mathematics 8A, one of them turned to the camera and s a i d g r a t u i t o u s l y : "It looks d i f f i c u l t , but i t ' s not that d i f f i c u l t r e a l l y . There 's a l o t of steps, but there ' s r u l e s that t e l l you what to do . . . l i k e , there ' s r u l e s about bracket s ." Thi s student a l so s a i d that he l i k e d t h i s course "because Using C a l c u l a t o r s to Teach Algebra 68 i t ' s d i f f e r e n t every day. Last year Math was the same th ing day a f t e r day a f t e r day. It got b o r i n g . " when I asked whether that was not a l so true for Mathematics 8A, he r e p l i e d , "Oh, no. Every quest ion had something new i n i t , l i k e a new way to catch you . . . l i k e brackets i n d i f f e r e n t p laces ." (That, i n f a c t , was t rue : with f i v e operations i n each equation, one of them a power, a great many v a r i a t i o n s had been poss ib l e and I had explored them a l l . ) Questioned dur ing t h e i r s o l v i n g of the equation about the reason for arranging the operations i n the f i r s t column i n the order they chose, they responded with the r u l e s g i v i n g the convent ional order of operat ions: "You have to do what's i n brackets before anything else"; "You have to do the power before anything e l se - unless i t ' s i n brackets ." They j u s t i f i e d adding 3.2 before d i v i d i n g by 13.25 as fo l lows: "It has to be the top d i v i d e d by the bottom." They d i d not mention e x p l i c i t l y the "bracketing" e f f ec t of the long d i v i s i o n l i n e . Nor d i d they r e f e r to the fac t that the order of operat ions was merely conventional and not a matter of l o g i c . When asked why they had s tar ted with x, they s a i d f i r s t that i t was because that was what they were s o l v i n g f o r . Pressed harder , one of them expla ined: "You see, we're going to reverse a l l t h i s [po int ing to the operations i n the f i r s t column] and then x w i l l be down here [point ing to the bottom of the r e v e r s a l column] and then a l l t h i s s t u f f [po int ing to the reversed Using C a l c u l a t o r s to Teach Algebra 69 operat ions above x] w i l l t e l l us how to get x. Then, i f we do i t c o r r e c t l y , we can put x back i n here [point ing to the x i n the o r i g i n a l equat ion] ." Asked how he would know whether he had done i t c o r r e c t l y , he expla ined , "It w i l l work o u t , t o t h i s [po int ing to the 2.5 i n the o r i g i n a l equat ion] ." I was a l so in teres t ed to know how wel l they remembered something they had learned i n the f i r s t u n i t of Mathematics 8A, on c a l c u l a t o r s , but not seen again i n the u n i t on equat ion - so lv ing because they had not been asked to solve equations that would take more than one step, viz. the fac t that "you can add backwards but not s u b t r a c t . " I asked, "I see that you d i d the plus 3.2 backwards. What would you have done i f i t had been a minus instead?" I expected them to exp la in that "you can ' t do a minus backwards," but instead one of them s a i d , "The minus would have been over here [po int ing to the general area i n s i d e the brackets i n the o r i g i n a l equat ion] ." I am not sure I understand t h i s answer; the only way I can f i n d meaning for i t i s to assume that he had r e a l i z e d that any minus s igns i n equations would always come a f t e r the x and not before - a precaut ion I had had to take i n a l l the equations I had given them so that they would never be faced with a two-step s o l u t i o n . A f t e r the videotaped interview, I questioned the more voca l of the two students again, hoping to have him e l u c i d a t e why, i n Using C a l c u l a t o r s to Teach Algebra 70 c o n s t r u c t i n g the r e v e r s a l column, he had reversed both the operat ions and t h e i r order . This interview was audiotaped. "Because you have to reverse i t , " he s a i d . "That's how you reverse . I f you don't do i t that way, i t won't work . . . i t won't check . . . you won't get the r i g h t answer when you check." He a l so s a i d , "Because x has to come at the end . . . because x i s what you don't know." Pressed s t i l l f u r t h e r , and apparent ly t h i n k i n g that he was l e t t i n g me down, he s a i d , "I guess I don' t know. I guess I missed that b i t . S o r r y . " The fo l lowing week, on A p r i l 15 1994, I interviewed three other students who had a l so completed Mathematics 8A at the end of January and recorded the interview on audiotape. Two of the students I had requested interviews with , one of whom had achieved 94% and the other 86%. The other, who had achieved 73%, came along because she was a f r i e n d of one of the others . I asked them to solve the same equation I had given the f i r s t two students . The l ess able of the two "A" students so lved i t almost by h e r s e l f because the a t t e n t i o n of the other "A" student kept wandering (the interview took place i n a c a f e t e r i a f u l l of Physics 11 students r e w r i t i n g an exam). At f i r s t she solved i t wrong, d i v i n g by 13.25 before adding 3.2, but f i n i s h i n g the s o l u t i o n , c a l c u l a t i n g the value of x, s u b s t i t u t i n g i t back in to the o r i g i n a l equation, and f i n d i n g that i t d i d not "check" before saying, "Something's wrong. Oh, f- , what d i d I do [ looking back over her work]. Oh, f , I'm supposed to do the add f i r s t . " Using C a l c u l a t o r s to Teach Algebra 71 She then s c r i b b l e d out what she had done and, looking at my tape recorder , s a i d , " Y o u ' l l erase that , won't you?" before s o l v i n g the equation again c o r r e c t l y . When I asked her why her f i r s t s o l u t i o n had not been c o r r e c t , she s a i d , "Because I made a mistake." Asked to descr ibe her mistake, she s a i d , "I d i d things i n the wrong o r d e r . " Asked why that made a d i f f e r e n c e , she r e p l i e d , "Because then you don't get the r i g h t answer." I continued to press her, asking why i t was so important to do things i n the r i g h t order , and she s a i d , "Because i f you don ' t , i t won't check." I abandoned that l i n e of quest ioning and asked ins tead , "Can you e x p l a i n to me why t h i s whole method works?" She answered, "Because i t g ives you x." I asked why you should s t a r t with x and she s a i d , "Because then y o u ' l l end up with x." In answer to my quest ions why you reverse the operations and why you reverse t h e i r order , she s a i d twice, "Because that gives you the r i g h t answer." At t h i s p o i n t , she sa id p e t t i s h l y , "I don' t know." I s a i d , "Does t h i s method of f i n d i n g x make sense to you?" She s a i d , "Oh, yes ," i n a tone of great assurance. "But you jus t can ' t e x p l a i n i t ? " "Yeah," she s a i d . Then the t h i r d student - the one who had not been i n v i t e d -broke i n , "Why do you want to know t h i s , anyway?" I expla ined , as I had already explained to the other two, that some teachers thought that my students d i d not r e a l l y understand what they were doing but were merely proceeding by rote and she s a i d i n d i g n a n t l y , Using C a l c u l a t o r s to Teach Algebra 72 "Of course we do!" "Yes, but how can I prove i t ? " I asked. "Maybe you j u s t memorized what to do." "Memorized a l l that?" she s a i d i n bewilderment. "Anyway," I s a i d , "they would l i k e to know whether you can e x p l a i n how t h i s method of s o l v i n g equations works." She looked at me wonderingly and asked, "Why would anybody want to know that?" Reasons for Abandoning Interviews I have inc luded a d e s c r i p t i o n of the above interv iew i n order to e x p l a i n why I abandoned t h i s l i n e of research . I n i t i a l l y , I was disappointed with a l l the students' responses. They seemed to be f a m i l i a r with what to do and they could j u s t i f y what they were doing i n an u n f a m i l i a r example by general r u l e s , but they could not exp la in why the r u l e s "worked." T h e i r p r i n c i p a l j u s t i f i c a t i o n for the r u l e s seemed to be that they l ed to the "correct" value for x, namely the value that would "check" or s a t i s f y the o r i g i n a l equation - what seemed to me a u t i l i t a r i a n j u s t i f i c a t i o n . Then I began to wonder what I myself would have answered had I been asked the questions I had asked my students . Why does s o l v i n g for x involve revers ing the operations? Because i f you s t a r t with x and add 2, you have to subtract 2 to get back to x. Why? Because you won't get back to x i f you don' t - t r y i t and see! Why does s o l v i n g for x involve r e v e r s i n g the order of the operat ions? That , I found, i s even more d i f f i c u l t to answer. I e x p l a i n i t to students, i n p r a c t i c e , by g i v i n g them an example to Using C a l c u l a t o r s to Teach Algebra 73 show them that i t "works." The expressed r e a c t i o n of most students i s (eventual ly , i f not immediately), "Well , of course!" I had always had the subconscious idea that my j u s t i f i c a t i o n for t h i s procedure was u t i l i t a r i a n only . But now I began to wonder what other j u s t i f i c a t i o n there could be. Is i t , i n f a c t , p o s s i b l e to exp la in why performing the opposite operat ions i n the opposite order takes you back from the number you ended with to the number you s t a r t e d with? How would mathematicians "prove" i t ? Is i t t rue , as I began to suspect, that they would merely invoke mathematical laws that are i n r e a l i t y no more than names for what my students "see"? I began to th ink be t ter of my teaching techniques. I explained i n Appendix 4 that I j u s t i f y saying that y ^ / x i s the reverse of y x as fo l lows: I f (on a c a l c u l a t o r ) you press 2 y x 5, you get 32. I f you now press y x ( 1 -f 5 ) =, you get back to 2. What other k ind of explanat ion i s there? Mathematicians would say that ( y x ) ^ / x = y because the exponents should be m u l t i p l i e d . But what i s t h i s r u l e but an attempt to maintain the laws of exponents? I t i s pos s ib l e to exp la in the fac t that (2 5 ) 2 = 1024 = 2 1 0 , where the exponents are in tegers , i n terms of m u l t i p l i c a t i o n , but what i s the comparable meaning of a f r a c t i o n a l exponent? T r y i n g to give meaning to a f r a c t i o n a l exponent i n terms of m u l t i p l i c a t i o n i s l i k e t r y i n g to give meaning to d i v i s i o n i n terms of s u b t r a c t i o n . When I f i r s t s t a r t e d teaching, I looked for "ins ight" and deplored the fac t that so many students operated by r u l e . Was I Using C a l c u l a t o r s to Teach Algebra 74 now demanding that students who could "see" should a l so "prove" what they could see according to mathematical ru l e s? Thi s i s not to denigrate conventional mathematical proof . But as Balacheff (1988) showed, proof , e s p e c i a l l y proof which s a t i s f i e s students , can take many forms. Once students "see" that something i s t rue , perhaps i t should be acknowledged that for them there i s no more b a s i c reason for i t s t r u t h than that i t can be seen to be t r u e . C e r t a i n l y t h i s convention e x i s t s i n E u c l i d i a n proof , which i n h igh school i s seldom pushed back even as f ar as the axioms. C e r t a i n l y teachers are far more s a t i s f i e d with a genuine " J see!" from a student than from the most g l i b enunciat ion of the a p p l i c a b l e p r i n c i p l e . I r e a l i z e d that i n looking for evidence of mathematical i n s i g h t and understanding among my students, I might be more success fu l i f I waited for spontaneous utterances by the students - anecdotes such as those I had c o l l e c t e d in formal ly over the years . That meant, of course, that I would be r e s t r i c t e d to s tudying my own students , s ince students i n the few c lasses of Mathematics 11A taught by other teachers us ing my mater ia l s would not be l i k e l y to t a l k to me spontaneously about mathematics. On the other hand, any spontaneous utterances by my own students would be l i k e l y to take place i n my own c lasses (or at l ea s t i n my own classroom), i n the context of the teacher-student r e l a t i o n s h i p , with the r e s u l t that I cou ld , i f I wished, e x p l o i t the occasions by asking quest ions , whether the remark was d i r e c t e d Using C a l c u l a t o r s to Teach Algebra 75 to me, to a teaching a ide , or to another student, without g i v i n g the occas ion the appearance of a formal interview and thus g i v i n g r i s e to the sor t s of problems I have descr ibed with in terv iews . C o l l e c t i n g Anecdotes Accord ing ly , I abandoned a l l ideas of formal interviews and continued to be a l e r t for spontaneous utterances by my students which revealed mathematical i n s i g h t or understanding, e i t h e r because they were o r i g i n a l to the students or because, i f they were statements they had heard me make, they e n t a i l e d a s i g n i f i c a n t rewording. Beginning i n September 1994, I made a speech along the fo l lowing l i n e s to a l l my mathematics c lasses very s h o r t l y a f t e r we had begun the courses and they had begun to f i n d that they cou ld , i n f a c t , do what they were being asked to do: "I am doing my master's degree r i g h t now. I'm t r y i n g to prove that my students r e a l l y understand what they 're doing and don't j u s t fo l low the r u l e s b l i n d l y . " (At t h i s point there would u s u a l l y be some s igns of ind ignat ion among the s tudents . ) "I know you understand what you're doing, but I have to prove i t to other people . One of the ways I know you understand i s that q u i t e of ten you say things about mathematics that are much be t t er than the way I s a i d them. You cou ldn ' t do that i f you d i d n ' t understand what you were t a l k i n g about. When that happens, i f you don' t mind, I w i l l make a few notes on what you s a i d so that I w i l l remember i t . " Using Calculators to Teach Algebra 76 (In practice, with so many students waiting for my help at any given moment in a class, I always carry around with me a distinctively coloured piece of paper on which I or they write their names in a l i s t ; f i r s t , so that I w i l l help them in the order in which they asked, and second, so that they cannot use the fact that they are "waiting for me" as an excuse to stop working -they can continue with the next problem in the assurance that I wi l l get to them in turn. I also use this paper i f I need to write so as to demonstrate something to students I am helping, rather than using their notebooks. Accordingly, students very soon became used to seeing me writing on this paper during a class and within two days - at most - stopped trying to see "whether I had written down" anything they had said.) The students appeared to forget what I had said almost immediately - none of them ever referred to i t again unless I brought up the subject - so I am confident that I can discount any possibility of self-consciousness or desire to impress the teacher in the utterances I recorded. Moreover, I argue, the students could not know what I would consider worth recording. I expected that one particularly f r u i t f u l source of anecdotes would be lessons given by individual students to the rest of the class. Accordingly, in two Grade 8 classes between September 1994 and January 1995, I designated as "teachers" each day three students who had scored 100% on the previous day's practice quiz, one for each question, and asked them to explain to the rest of the class from the front of the room, using the overhead Using C a l c u l a t o r s to Teach Algebra 77 p r o j e c t o r , how to do that quest ion c o r r e c t l y . The students loved the idea and v i e d with each other a l l semester long for the p r i v i l e g e of being the "teacher." However, I eventua l ly gave up the p r a c t i c e for a number of reasons: 1 . I t took more time than I wanted to spend. On average, the students took h a l f an hour to do what I could do i n twenty minutes, p a r t l y because they wrote so slowly but a l so because of (what I considered to be) i r r e l e v a n t comments and quest ions from the other students . 2. The "teachers" c a r r i e d t h e i r own c o r r e c t l y done quizzes up to the front of the room and simply copied out what they had done, ins tead of l e t t i n g the other students see them i n the process of s o l v i n g the problem. 3. I began to see that the goal of the "teachers" was to imi ta te me as c l o s e l y as pos s ib l e - asking the quest ions I would ask i n my words and my tone of voice and us ing the same co lours of ink as I d i d to wri te the quest ion , number the operat ions , draw i n the bracket boxes, e t c . - ins tead of expounding the mathematics i n t h e i r own words. 4. The re s t of the students, instead of cha l l eng ing the "teachers" to exp la in the mathematics, made comments l i k e , "You're w r i t i n g i t too s m a l l , " "You're standing i n front of the p r o j e c t o r , " "You're us ing the wrong c o l o u r , " e t c . Most of the chal lenges to "teachers" to exp la in what they were doing came from me. I a l so expected that i t would be use fu l to audiotape what Using Calculators to Teach Algebra 78 students s a i d and I c a r r i e d a small audiotape recorder i n my b r i e f c a s e from September 1993 u n t i l June 1994. However, I found i t impractical to carry the tape recorder i n my hand at a l l times and p s y c h o l o g i c a l l y impossible (for me) to introduce a tape recorder into an informal, data-rich exchange between a student and me or between two students. In addition, I expected that i t might be re v e a l i n g to ask students i n a questionnaire to write what they thought about the course they had just completed. In f a c t , I d i d give such a questionnaire to the Mathematics 11A students I taught between January and June 1993, but t h e i r s p e c i f i c comments were a l l about how much they had enjoyed working "at t h e i r own l e v e l " ( r e f e r r i n g to the freedom they enjoyed to omit " s t a r " and "double-star" questions). Moreover, I found, t h e i r comments d i d not d i s t i n g u i s h between my a b i l i t y as a teacher and the nature of the course I was teaching. Although my students d i d not keep r e f e r r i n g to my study throughput the course, I nevertheless received the impression that they were "on my side" i n the project, as the following story demonstrates. On August 16 1996, as I was d r i v i n g home at about 10:00 p.m., the car i n front of me stopped to l e t o f f a passenger, who turned out to be a student to whom I had taught Mathematics 10A from September 1995 to January 1996 and Mathematics 11A immediately afterward, from January 1996 to June 1996. He recognized my car and came over to my d r i v e r ' s window. He asked me how my summer was going, and I r e p l i e d that I was Using C a l c u l a t o r s to Teach Algebra 79 engaged i n w r i t i n g my t h e s i s . He answered, "Oh, yes . . . don' t forget to give me c r e d i t for anything you put i n that I s a i d . " I laughed. By that time, the d r i v e r of the c a r , who had stopped on seeing h i s passenger approach the car behind, had backed up so that h i s car was p a r a l l e l to mine on my l e f t . Through h i s open passenger window, he c a l l e d , i n a pleased tone of v o i c e , "Oh, h i , Miss Murphy!" (I had taught him Mathematics 10A at the same time as the passenger.) The passenger then s a i d to the d r i v e r , "I'm j u s t t e l l i n g her to put my name i n her t h e s i s , " upon which the d r i v e r s a i d , "Yeah . . . i s there anything about me i n i t ? Are you sure you know how to s p e l l my name?" Recording, T r a n s c r i b i n g , and Analyz ing Anecdotes Between September 1993 and June 1996, I stayed a l e r t for spontaneous utterances by my students which revealed mathematical i n s i g h t or understanding, e i t h e r because they were o r i g i n a l to the students or because they e n t a i l e d a s i g n i f i c a n t rewording of what I had s a i d . Whenever such utterances occurred, I immediately j o t t e d down notes about what had been s a i d and/or done, i n c l u d i n g the students' names, t h e i r grade l e v e l s and the l e t t e r - g r a d e l e v e l s at which they h a b i t u a l l y worked. (This was not n e c e s s a r i l y the same as t h e i r f i n a l l e t t e r grade, as lack of attendance, l ack of a t t e n t i o n to the lesson, lack of regu lar p r a c t i c e i n c l a s s , miss ing exams, e tc . would often make t h e i r f i n a l mark lower than i t would otherwise have been.) Before the end of each c l a s s , Using C a l c u l a t o r s to Teach Algebra 80 u s u a l l y while the students were w r i t i n g the "pract ice q u i z , " I ampl i f i ed these notes i n w r i t i n g from memory. I thus c o l l e c t e d very f u l l notes of 77 anecdotes. On typ ing them out dur ing June and J u l y 1996, I observed, f i r s t , c e r t a i n r e p e t i t i o n s , and, second, twelve d i s t i n g u i s h a b l e themes. I coded the themes as shown i n Table 4 and ordered them according to i n c r e a s i n g depth of mathematical i n s i g h t , as shown i n Table 4. Table 4 C o d i f i c a t i o n of Themes No. Theme Code Number of Anecdotes 1 C a l c u l a t o r language LANG 2 2 Brackets BRAC 4 3 P e c u l i a r i t i e s of non-a lgebra i c c a l c u l a t o r s NONAL 2 4 Order of operations ORDER 3 5 Equals s ign EQUAL 7 6 Discrepancies between input and execution EXE 5 7 Importance of operations as opposed to numbers OPER 5 8 T r a n s i t i o n from numbers to l e t t e r s LET 7 9 Nature of v a r i a b l e s VAR 3 10 Structure of expressions STRUC 7 11 Groups GROUP 7 12 Author i ty AUTH 1 T o t a l 53 Using Calculators to Teach Algebra 81 By eliminating the repetitions among the anecdotes, I reduced the number to 53. Using the above codes, I labelled each anecdote with the single most appropriate code, although many anecdotes could clearly be placed in more than one category. Finally, I ordered the anecdotes within each category in increasing level of insight. These anecdotes are recounted and analyzed in this order in Chapters 6 and 7. With each anecdote i s included, in brackets after the name, the course the student was taking and the letter grade level at which that student habitually worked so that the reader can judge qualitatively how common this kind of insight might be. An insight expressed by a "C" student, for example, might be expected to be common among "B" and "A" students as well, while an insight expressed by an "A" student might be expected to be comparatively rare. The anecdotes I eventually recorded came from 36 different combinations of students and grades (because 17 of the 53 came from students who had already supplied another anecdote while they were in the same grade). I include the graph in Figure 2, which details the distribution of grade levels and "working" letter grades among the 36 student-grade combinations, in order to give an overall picture of the students from whom the anecdotes came and thus make possible a more reliable assessment of the relevance of my thesis to other mathematics students. Figure 2 shows that while eight anecdotes came from "A" students in Grade 8, the rest came f a i r l y indiscriminately from students working at a l l levels Using Calculators to Teach Algebra 82 i n a l l grades. Figure 2 D i s t r i b u t i o n of Anecdotes N 8 u m b 7 e r 6-o f 5-t * u d 3 e n 2 t ^ s 1-B C+ L e t t e r g r a d e s G r a d e 11 G r a d e 10 G r a d e 9 | | § G r a d e 8 • In recounting these anecdotes, I have divided them in t o two chapters. Chapter 6 includes the f i r s t s i x categories i n Table 4: c a l c u l a t o r language, brackets, p e c u l i a r i t i e s of non-algebraic c a l c u l a t o r s , order of operations, equals signs, and discrepancies between input and execution. Chapter 7 includes the other s i x : importance of operations as opposed to numbers, t r a n s i t i o n from numbers to l e t t e r s , nature of variabl e s , structure of expressions, groups, and authority. Chapter 6 can thus be s a i d to deal mainly with students' i n s i g h t s into the mathematical language of arithmetic and c a l c u l a t o r s , while Chapter 7 deals with t h e i r i n s i g h t s into algebra and the t r a n s i t i o n from arithmetic. Using C a l c u l a t o r s to Teach Algebra 83 G e n e r a l i z a b i l i t v and L i m i t s Fol lowing S f a r d ' s (1991) ana lys i s of reification and the comparison i n Chapter 3 of my courses with "regular" courses , i t appears that my case study of unmathematical students l e a r n i n g at the boundary between ar i thmet i c and algebra has consequences f o r : 1. research in to how mathematics i s learned, taught, conceptual ized , and performed - a t o p i c which i n t e r e s t s some educators for i t s own sake, apart from i t s a p p l i c a t i o n to teaching . 2. the design of high school mathematics courses for the unmathematical, who can l earn algebra without reification. 3. the design of high school mathematics course for the mathematical, who might be able to use a lgebra without reification to br idge the gap between ar i thmet i c and "regu lar ," or reified, a lgebra . The students I s tudied were those assigned to my Mathematics 8A-11A c lasses by school counse l lors and adminis trators according to the advice of t h e i r previous mathematics teachers , which i n turn was based on the students' poor performance i n previous mathematics courses . Since Mathematics 8A-11A courses i n general are not accepted by co l l eges and u n i v e r s i t i e s as p r e r e q u i s i t e s for entry , and s ince i t i s we l l known that those who take these courses are very seldom able to switch succes s fu l l y to the "regular" courses before they leave h igh school , students who take these courses , as we l l as t h e i r parents , have abandoned hope of academic post-secondary Using C a l c u l a t o r s to Teach Algebra 84 educat ion . I observed a l l my students without conscious d i s c r i m i n a t i o n , although more of my a t t e n t i o n was claimed by the more voca l and ex trover ted . My d e c i s i o n to record or not to record any mathematical exchange was based only on my percept ion of i t as i n s i g h t f u l or o r i g i n a l . The s o - c a l l e d Hawthorne e f f ec t appeared not to be a f a c t o r i n my research , as students appeared to forget very soon that I was observing them and could not expect to know what I would f i n d i n s i g h t f u l or o r i g i n a l . The e f f ec t s of the s e t t i n g can a l so be l a r g e l y d iscounted, as the s tudy's venue was my own classroom, with no one present other than those who would have been present i n any case, and I made no conscious d i f f e r e n c e i n my teaching because I was conducting a study; the exchanges I recorded were those that occur n a t u r a l l y i n a classroom and were s i m i l a r to those I had not i ced for many years before I began recording them. Although the courses I taught had been w r i t t e n so that students could be succes s fu l , I evaluated t h e i r success by means of o b j e c t i v e t es t s according to pre - se t standards and they d i d not pass the course unless they achieved at l ea s t 50%. Two other teachers have each taught my Mathematics 11A course (but not my Mathematics 8A-10A courses) to a number of other c lasses for a number of years , us ing my worksheets, qu izzes , and exams. Each has found the students' f i n a l marks to be s i m i l a r to mine (see Appendix 2). Using C a l c u l a t o r s to Teach Algebra 85 Chapter 6 Express ing Mathematical Ins ight The anecdotes inc luded i n t h i s chapter f a l l i n t o the f i r s t h a l f of my twelve categor ies : c a l c u l a t o r language, brackets , p e c u l i a r i t i e s of non-a lgebraic c a l c u l a t o r s , order of operat ions , equals s igns , and d i screpanc ies between input and execut ion. In genera l , they demonstrate that unmathematical students can become f luent i n c a l c u l a t o r language, which i s a d i a l e c t of a r i t h m e t i c language, and use i t to express i n s i g h t i n t o a r i t h m e t i c and even a lgebra . The brackets a f t e r each student 's name at the beginning of each anecdote inc ludes the course the student was tak ing at the time the inc ident occurred and the l e v e l at which the student h a b i t u a l l y worked, e.g. (Ma 9A - B ) . I have t r i e d to i n d i c a t e wherever I or a student emphasized a spoken word by the use of bo ld i t a l i c s thus: emphasized. Communication i n C a l c u l a t o r Language By far the most common r e a c t i o n among students to the news that they are going to be allowed to use c a l c u l a t o r s i s j u b i l a t i o n . Occas iona l ly , a student w i l l announce that he or she i s not going to use a c a l c u l a t o r because "I want to l e a r n to th ink for myself ." However, both j u b i l a t i o n and renunc ia t ion soon become tempered as students r e a l i z e that they have to t e l l c a l c u l a t o r s exact ly what to do. Within a few days of the Using C a l c u l a t o r s to Teach Algebra 86 beginning of a course, they see that c a l c u l a t o r s are as necessary i n t h e i r mathematics c l a s s as gym s t r i p i s i n a p h y s i c a l education c l a s s . Before long, they are speaking the language of c a l c u l a t o r s and a c t u a l l y v e r b a l i z i n g what i s wr i t t en on t h e i r c a l c u l a t o r s ' keyboards. In f a c t , some students, l i k e A l i s o n (Ma 9A - B) a c t u a l l y communicate i n w r i t i n g with me as the teacher by means of c a l c u l a t o r language. Asked to solve 7x - 6 = 3.95 for x, she c a l c u l a t e d that x = 1.421428571 and then wrote (ch) 7 x 1 .421 428571 - 6 = 3.949999997 (mode 7,2 = 3.95) "(ch)" means "check." In the "check" she s u b s t i t u t e d her value for x in to the o r i g i n a l equation to see whether i t "checked"; that i s , made the l e f t s ide of the equation equal to 3.95. She found that i t d i d not; that i t came ins tead to 3.94999997. "Mode 7,2" are the buttons she pushed on her c a l c u l a t o r (a Casio) to round o f f the d i s p l a y to two decimal p lace s . By "mode 7,2 = 3.95," she meant that when she rounded o f f 3.94999997 to two decimal p laces , she got 3.95, as she had expected, and so her value for x d i d , i n f a c t , "check." The comma and the = s ign were not buttons she had to press; the comma simply separated the 7 from the 2 and the = s ign merely i n d i c a t e d that 3.95 was the r e s u l t of press ing the buttons "mode 7 2" - a use for the = s ign which i s extremely common, as I w i l l show l a t e r i n t h i s chapter . The fac t that c a l c u l a t o r language thus enabled A l i s o n to Using C a l c u l a t o r s to Teach Algebra 87 express very economically what she wanted to say i s good evidence that c a l c u l a t o r language deserves to be d i g n i f i e d as "mathematical language." However, students do not use the language of c a l c u l a t o r s merely to communicate with other people; they a l so use i t to communicate with c a l c u l a t o r s themselves. C h r i s t i n e (Ma 11A - C+) showed that she considered h e r s e l f to be communicating with her c a l c u l a t o r when she began us ing my formula for simple i n t e r e s t , viz. Prt/u 100 where P represents the p r i n c i p a l , r the i n t e r e s t r a t e , t the time, u the number of time u n i t s per year (1 for a time given i n years , 12 for months, 52 for weeks, and 365 for days) . Christine: Last year we learned an easier formula for this. I: Did you? What was it? Christine: I think it was [writing] I = Prt I: I put 100 into my formula because the r is always a percentage. That way you don't have to remember to move the decimal point. Christine: OK. But why do we need u? I: Suppose you leave your money in the bank for 5 years. What are you going to put into the calculator for t? Christine: That's not what I asked you. I: I know, but u is connected to t . . . [I repeated the Using C a l c u l a t o r s to Teach Algebra 88 question.] Christine: Five. I: OK. Now suppose Kelly leaves her money in the bank for 5 months. What is she going to put into the calculator for t? Christine: Five ... Oh, but that would give her the same answer ... I see ... you can't put the words on to the calculator ... but you have to tell the calculator what you mean ... Oh, OK. In other words, you have to. t e l l the c a l c u l a t o r what you mean through numbers, because that i s the only language i t understands. Brackets The l a s t two examples i l l u s t r a t e students' t r a n s l a t i n g from E n g l i s h to c a l c u l a t o r language, but the reverse can happen as w e l l . Bob (Ma 10A - A) was encouraging Jack (Ma 10A - C-) to t r y double - s tar quest ions . Accord ing ly , given the values of the v a r i a b l e s (which inc luded p = 10 and q = 1.5), Jack was i n the process of eva luat ing the fo l lowing expression on h i s (a lgebraic) c a l c u l a t o r , more or l ess by fo l lowing Bob's orders . 1 Q + ma pq He had jus t punched the large r ight -hand square bracket . Jack: [Reciting as he punched the buttons] y to the x, one, divided by ... Bob: No; you need brackets around here [he drew brackets around the entire exponent]. Using C a l c u l a t o r s to Teach Algebra 89 Jack repeated the part he had done c o r r e c t l y the previous t ime. Jack: . . . [emphatically - he was going to get it right this time] bracket y to the x, one, divided by . . . Bob: No! No! No! You have to punch the y to the x before the bracket! Jack: Why? Bob: 'Cause the y to the x is in here. He l a i d h i s p e n c i l po int exact ly where the dot i s below. Bob: You have to tell the calculator that you're going to do a y to the x, right? But then you have to tell it to wait while you work out the exponent, 'cause the exponent has to be worked out. Jack repeated the part he had done c o r r e c t l y a l l a long . Jack: . . . y to the x, bracket, one, divided by, ten, times, Bob: No, you need brackets here as well [drawing in brackets around pq]. Jack: [Exasperatedly] What do we need those for? Bob: 'Cause you've got to tell the calculator to wait again. It has to do this [laying his pencil point between the p and the q] before it divides. Jack: [Pushing his paper away] Forget it, man. Q + ma 1 + — Using C a l c u l a t o r s to Teach Algebra 90 Notice how Bob t r a n s l a t e d from c a l c u l a t o r language ("brackets") to E n g l i s h ("wait") and back again: Brackets mean the c a l c u l a t o r has to wait and i f you want the c a l c u l a t o r to wait you have to press brackets . Notice a l so how necessary the c a l c u l a t o r a p p e l l a t i o n "y to the x" i s to Bob's exp lanat ion . In t h i s exchange, Bob was emphasizing communication with the c a l c u l a t o r . However, students can a l so used c a l c u l a t o r language to descr ibe mathematical communication with other people , as Tom r e a l i z e d i n what fo l lows . Obviously mimicking my habi t s as a teacher, Bahman (Ma 9A - B) was exp la in ing to the c l a s s how to do yes terday's p r a c t i c e qu iz problem, viz. to solve the fo l lowing equation for y. y + b — — - p = z He had e l i c i t e d from another student that he should wr i te + b + p T a x a - p - b z y and was now asking Abdul (Ma 9A - C+) whether any brackets should be i n s e r t e d i n the second column. Bahman: Do we need any brackets? Abdul: Yes, between the plus and the times. Bahman: Can you tell us why? Abdul: [In a sing-song tone] 'Cause you always need brackets if an add or a subtract comes right before a multiply or a divide. Bahman: [Putting in the brackets, and no longer mimicking me] Using C a l c u l a t o r s to Teach Algebra 91 But why? Abdul: I don't know. Bahman: Can anybody tell me? Scott (Ma 9A - C): 'Cause an algebraic calculator needs brackets to make it do a times before a minus. Bahman: [Patronisingly] Very good. I: But what if someone has a non-algebraic calculator? Bahman: Huh? I: Non-algebraic calculators don't need brackets. Does that mean that, say, Kathy doesn't need to put in those brackets? Bahman: No ... Kathy needs to put in the brackets. I: Why? Tom (Ma 9A - C): [After a pause] So anyone else doing the problem knows that she wants them to do the times before the minus. I: Right! In other words, brackets aren't just to tell calculators what to do. They're to tell people what to do. Tom: They're to tell people who know the rules what to do. I: What rules? Tom: The order of operations. Here, even before my f i n a l summing-up, Tom was t a l k i n g about communication with people by means of c a l c u l a t o r language and not merely communication with a c a l c u l a t o r , s ince he r e a l i z e d that even with her c a l c u l a t o r , Kathy would have to wri te i n the brackets . Of a s l i g h t l y higher order i s the c a l c u l a t o r language used by Using C a l c u l a t o r s to Teach Algebra 92 Bob (Ma 9A - A) to demonstrate to me that , contrary to my i n s t r u c t i o n s , i t was not necessary to punch in to the c a l c u l a t o r the ( implied) brackets around the express ion above a long d i v i s i o n l i n e unless that expression contained a + or a - . Asked to so lve for x, he had wr i t t en x 1 8 x 3 x 6 - i - 6 - r 3 1 8 x Bob: You know how you can prove you don't need brackets around 18 times 6? I: How? Bob: You don't need a bracket here, right? He pointed to the space between "x 6" and 3" i n the second column. J ; Right. Bob: So when I write this out normally . . . He wrote 1 8 x 6 3 . . . that means I don't need brackets here, right? [sketching in brackets around "18 x 6. " I: Right. Bob: [triumphantly] So that proves I don't have to punch them in on my calculator! About ten minutes l a t e r he addressed the same quest ion aga in . Using C a l c u l a t o r s to Teach Algebra 9 3 Bob: You know that question where I proved you don't need brackets? I: Yes? Bob: There's another way to prove it. I: Is there? How? Bob: If I don't need brackets here . . . He pointed to the space between " x 6 " and "T 3 " i n the second column of h i s s o l u t i o n for x. . . . then this [pointing to the - immediately below the space] doesn't have to be a long line, right? I: Right. Bob: So I could write it like this . . . He wrote 18 x 6 /3 . . . right? I: Right. Bob: And that kind of division line [pointing to the slash] doesn't need brackets, right? I: Right. Bob: So? I: You're right: you don't need those brackets. I tell students to put them in every time because otherwise they forget when they're really necessary. Bob: Like when i t ' s a plus or a minus. Bob's technique was not exact ly "proof by c a l c u l a t o r , " as i s shown by h i s ignor ing of the c a l c u l a t o r i n h i s second "proof," but Using C a l c u l a t o r s to Teach Algebra 94 i t could c e r t a i n l y be c a l l e d "proof i n c a l c u l a t o r language." From then on, Bob seemed to p r i d e himself on l e a v i n g out any brackets he c o u l d . On another occas ion , he was asked to t e l l what buttons he would punch on h i s (a lgebraic) c a l c u l a t o r to c a l c u l a t e the fo l l owing : 1 .35 2 + 2 1 . 5 - 3.7 + 1.9' 1 .795 He had wr i t t en 1.5 - ( 3 .7 + 1.9 y x 3 ( ( 1.35 x 2 + 2 ) -=- ( ( 1 4- 2 ) x 2 ( 1 + 3.2 ) y x 3 -r 1 .795 ) ) x 2 ) = and was comparing h i s answer with mine: 1.5 - ( 3 .7 + 1.9 y x 3 x ( ( 1.35 x 2 + 2 ) ( ( ( 1 * 2 ) x 2 x ( 1 + 3.2 ) y x 3 ) -r 1.795 ) ) x 2 ) = t t As he saw, he had l e f t out the x s igns and the two brackets marked with arrows. Bob: Well, I can leave out times signs on my calculator. And 1 don't need those brackets [pointing to the ones marked above with arrows] 'cause that's just a multiply [pointing to the multiply marked with an arrow]. And I don't really need those brackets, either [pointing to the ones marked with arrows below] 4, 4, 1.5 - ( 3 .7 + 1.9 y x 3 x ( ( 1.35 x 2 + 2 ) -r ( ( ( 1 * 2 ) x 2 x ( 1 + 3.2 ) y x 3 ) v 1.795 ) ) x 2 ) = t t I: Why not? Using C a l c u l a t o r s to Teach Algebra 95 Bob: 'Cause that's just a divide [pointing to the + marked with an arrow above] ... just wait a minute ... He ran h i s f inger slowly backward along the l i n e s from the marked -r u n t i l he reached the + marked above with an arrow. Bob: ... yes, I do, 'cause of that plus. I: What about this plus [pointing to the one marked below]? 1 . 5 - ( 3 . 7 + 1 . 9 y x 3 x ( ( 1 . 3 5 x 2 + 2 ) ( ( ( 1 2 ) x 2 x ( 1 + 3 . 2 ) y x 3 ) -r 1 . 7 9 5 ) ) x 2 ) = t Bob: That's already in brackets, see? [pointing to the brackets around (1 + 3.2)]. I: OK. So did you do the question right? Bob: [Triumphantly] Yep! I was s u r p r i s e d that he would choose to determine the necess i ty or otherwise of brackets by looking at the long l i n e of "buttons he would punch" instead of the o r i g i n a l numerical express ion, which to me i s f ar eas i er to read . I can e x p l a i n i t only by supposing that the p i c t u r e i n h i s mind was of what a c a l c u l a t o r would do, or would have done, or needed to be t o l d , at each stage along the l i n e . A l l of the above anecdotes demonstrate that students can become f luent i n c a l c u l a t o r language and that c a l c u l a t o r language can be used to express mathematical ideas economically - a fundamental c h a r a c t e r i s t i c of any mathematical language. Non-Algebraic C a l c u l a t o r Language In f a c t , some students even argued the r i g h t of c a l c u l a t o r s Using C a l c u l a t o r s to Teach Algebra 96 to determine or modify the language they used, as i f they had to communicate with c a l c u l a t o r s only and not people, as the fo l lowing two anecdotes demonstrate. Kyle (Ma 9A - A ) , who had a non-a lgebraic c a l c u l a t o r , had been asked to solve the fo l lowing equation for x. 2 = 1 3 According to what I had taught him, he should have w r i t t e n the fo l l owing : x * 3 + 5 -r 6 2 1 3 ( 1 3 1 / 2 x 6 - 5 ) 3 = x C h e c k : 4 9 . 8 9 9 9 2 2 9 6 1 4 9 . 8 9 9 9 2 2 9 6 5 + = 1 3 Instead, he wrote x -r 3 + 5 -r 6 rx 2 1 3 c x 5 + 3 = 1 3 1 3 y x 1 / 2 x 6 - 5 x 3 x = 4 9 . 8 9 9 9 2 2 9 6 C h e c k and then argued with me afterwards thus: Kyle: How come I didn't get full marks for this question? I: Because you left out brackets in the second column, you didn't write out the second column normally and you didn't write Using C a l c u l a t o r s to Teach Algebra 97 out the check properly. Kyle: I don't need brackets on my calculator. I: I would let you get away with leaving brackets out of the second column if you'd put them in when you wrote it out normally, but you didn't even write it out normally. Kyle: [Belligerently] Why should I write it out normally when I've got a non-algebraic calculator? Writing it out normally is for algebraic calculators. I: But this [pointing to the second column] isn't the standard way to write math. No one outside this class would know what it meant. Kyle: So? No one outside this class has to read it. You know what it means, don't you? I: Yes. Kyle: So? I: But Kyle, I'm trying to teach you math. Part of math is how you write it down. You may have another teacher next year who doesn't understand this way. Kyle: Aren't you going to teach Math 10A next year? I: I don't know. Kyle: If you don't I just won't take Math. I'll leave it until you teach it. I: You can't do that. Kyle: And this check [pointing to his long arrow]. Are you telling me nobody would know what that meant? I: No, I must admit that's pretty clear. Using C a l c u l a t o r s to Teach Algebra 98 Kyle: So why should I write out the whole of the first equation again with the number for x? I: [resignedly] OK, I'll give you the marks for the check. Kyle: [challengingly] I do know how to write out the second column, you know. I: I know you do. I was disappointed when you lost so many marks for not doing it. Kyle: So will you let me do it now? I: OK ... and put the brackets in the second column as well. In more moderate fash ion , Matt (Ma 11A - A) presented me with a s i m i l a r argument when he was s o l v i n g compound i n t e r e s t problems. He had been given the values of A, P, n, t and u and was s o l v i n g t h i s equation nt/u A = P ^ + 1 100 for r . He wrote r A -f n -r P 4 100 y x 1/(nt/u) + 1 - 1 y x (nt/u) x 100 x p x n A r and then reached for h i s c a l c u l a t o r . I: Aren't you going to put brackets in and write out the second column normally? Matt: No - I've got a non-algebraic calculator, so it's easier to do it this way [running his finger down the second Using C a l c u l a t o r s to Teach Algebra 99 column]. I: Could you write it out if you had to? Matt: Oh yeah - I did it in the formulas unit. But there's no point in this unit - you just want the answer. I: Of course, you can't really just go down the column, can you? I mean, you have to work out 1/(nt/u) first and put it in the memory. Matt: Oh yeah, but that's easy. I do that first, and then just go down the column and press memory recall when I get to the y x . Both of these students r e a l i z e d that the w r i t t e n no ta t ion of the "reversa l" column was s u f f i c i e n t and even perhaps more appropriate than conventional nota t ion for a non-a lgebra ic c a l c u l a t o r . I t can be argued that for students with non-a lgebra ic c a l c u l a t o r s to "write out the r e v e r s a l column normally" i s indeed superf luous; a student who r e a l i z e s t h i s can be s a i d to have been mathematically c r e a t i v e and to be j u s t i f i e d i n defy ing the conventions, as Kyle d i d . Order of Operations Kyle and Matt were not the only ones to quest ion convent ional w r i t t e n n o t a t i o n . Yvonne (Ma 8A - C+) a l so showed that she would pre fer what might be c a l l e d "sequential" nota t ion when she was asked to wri te numbers under the operations i n t h i s express ion to show the order i n which they must be done. 12 + 3 x 4 7 6 - 12 Using C a l c u l a t o r s to Teach Algebra 100 Yvonne: [Petulantly] Why do mathematicians write it like this when they want you to do this [pointing to the *] before this [pointing to the +]? I: How would you like them to write it? Yvonne: Write the first one first. I: I see what you mean. I don't know. I guess I'm so used to thinking of the usual order of operations that I can't imagine it any other way. Yvonne: Well, it seems dumb to me. I: [Thinking of "columns"] Later on, in Unit Two, I'm going to let you write it your way, because it is easier sometimes. Yvonne: Oh, good. I: Unfortunately, you have to change it back to the "proper" way in your final answer, because nobody else does it the way I do. Yvonne: [Resignedly] Whatever. There can be no doubt that conventional wr i t t en n o t a t i o n , taken i n conjunct ion with the conventional order of operat ions , does not communicate to many students the s t r u c t u r e that mathematical people see at a glance, as Rick (Ma 8A - B) showed when he was asked to (a) number the operat ions i n the fo l lowing express ion, (b) t e l l what buttons he would punch on h i s c a l c u l a t o r to get the answer, and (c) g ive the answer to two decimal p lace s . Rick: [handing in his quiz] Can I ask you about a question -Using C a l c u l a t o r s to Teach Algebra 101 I'm handing it in: I'm not going to change it. I: Yes. Rick: First I did it this way ... He copied the quest ion on to another piece of paper and numbered the operations thus ( 7 1 - 6 ) d h > 2 3 1 ...But then I remembered you said multiply before divide, so I did it this way. He erased the 2 and the 3 and wrote them i n aga in , r e v e r s i n g t h e i r p o s i t i o n s . I: That's right. Rick: Well, I thought I'd try it on my calculator - look ... He ran and got, not h i s own c a l c u l a t o r , which was non-a l g e b r a i c , but Borzoo's , which was a l g e b r a i c , and he ld i t so that I could see the d i s p l a y . ... watch [he punched 71.6 x, then stopped]. You don't need brackets around the 3, right? I: Right. Rick: 'Cause it's just a single number. I: Right. Rick: Now, watch [he punched 3 +, then held the calculator in front of me] - see? [The calculator showed 214.8.] That means it's done the times, right? I: Right. Rick: So what you said was right. Using C a l c u l a t o r s to Teach Algebra 102 He laughed deprecat ing ly , as i f he was a f r a i d of seeming rude i f he questioned what I had s a i d . Rick: Of course, I knew that anyway. But when I look at this question [he pointed to the original problem] it looks as though you should do the divide first. I: Why? Rick: 'Cause it looks a bit like a fraction. I: I see what you mean. Actually, in this case you would get the same answer if you did do the the divide first. If I had written the question this way . . . I wrote 7 - b ( 7 1 - 6 > . . . you would have got exactly the same answer. Try it, if you like. Rick: [After trying it] Yeah. I: The reason I tell you to do multiplies and divides left to right is that sometimes - as in a question like this . . . I wrote 3 -r 6 x 9 ifc would give you a different answer if you did the multiply before the divide. In this question you have to do the divide before the multiply. So i t ' s easier just to remember to do multiplies and divides left to right. Rick: OK. I: Why did you use Borzoo's calculator instead of yours? Using C a l c u l a t o r s to Teach Algebra 103 Rick: 'Cause mine's non-algebraic. I: So? Rick: So it doesn't tell you the order of operations. A f t e r t h i s , Rick would probably have agreed that sequent ia l no ta t ion was super ior to conventional n o t a t i o n . Thi s i s another example of "proof by c a l c u l a t o r . " I found i t i n t e r e s t i n g that Rick had accepted the c a l c u l a t o r as something he could consul t to f i n d out what was r i g h t when he was not sure . He was communicating with the c a l c u l a t o r i n a way I had not seen before: I had taught them that an a lgebra ic c a l c u l a t o r "knew" what order the operat ions should be done i n , even when you punched them on to the c a l c u l a t o r i n the order i n which they were normally w r i t t e n ; I had never suggested that an a l g e b r a i c c a l c u l a t o r could tell us what order they should be done i n . But Rick had apparently assumed that i f the c a l c u l a t o r "knew" what to do, i t could be made to " t e l l us" what to do. In t e l l i n g students that an a lgebra i c c a l c u l a t o r "knew" what order the operations should be done i n , I had always s tressed that the order was merely convent ional , an emphasis which perhaps bo l s t ered K y l e ' s and Matt ' s expression of preference for the sequent ia l no ta t ion of a column. Many students , I have found, do not understand, even when they know how to apply i t , that BEDMAS (a wel l known acronym for brackets , exponents, d i v i s i o n , m u l t i p l i c a t i o n , a d d i t i o n , subtract ion) has the a r b i t r a r y nature of a convention. "You mean I don't have to understand BEDMAS?" a student asked once. " A l l Using C a l c u l a t o r s to Teach Algebra 104 these years I 've been t r y i n g to understand i t and i t ' s jus t an agreement among a l o t of mathematicians? I'm not as dumb as I thought." He added, humourously, "I f e e l I can say that here, i n t h i s c l a s s . . . I mean, i t ' s l i k e Mathematics Anonymous, r i g h t ? " Far from accept ing the order of operations as "merely" conventional and, l o g i c a l l y , replaceable with any other convention, Bob (Ma 8A - A) was convinced that he knew why the convent ional order had been chosen. Bob: I know why they do multiplies and divides before adds and subtracts. I: Why? Bob: Because multiply and divide are higher. I: What do you mean? Bob: They're higher. I: Do you mean they give you higher answers? Bob: Yes. You learn add and subtract first, and then you learn multiply and divide. Then you learn powers. I: I don't see what that's got to do with it. Bob: You always do higher operations first. I: Do multiply and divide always give you higher answers? Bob: [After a pause] They don't have to, I guess. But they're still higher operations. Besides the obvious ideas that a d d i t i o n and subtrac t ion are "easier" than m u l t i p l i c a t i o n and d i v i s i o n , and that m u l t i p l i c a t i o n can give higher answers than a d d i t i o n , was there i n Bob's mind, perhaps, a sense of a h ierarchy among the operat ions , Using C a l c u l a t o r s to Teach Algebra 105 perhaps the h ierarchy due to the fac t that m u l t i p l i c a t i o n can be descr ibed i n terms of a d d i t i o n and powers i n terms of m u l t i p l i c a t i o n ? Whatever was i n Bob's mind about the necess i ty of BEDMAS, Matt , Kyle and even Yvonne showed signs of c r e a t i v i t y i n the f i e l d of mathematical no ta t ion , and I would say that i n each case c a l c u l a t o r s had been the c a t a l y s t . (If nota t ion sounds l i k e a minor matter, consider the breakthrough i n elementary p a r t i c l e phys ics made poss ib l e by Feynman diagrams.) Use of Equals Sign As I showed i n Appendices 3 and 4, my students use the symbol = as i t i s used i n ar i thmet i c and c a l c u l a t o r language and continue to use i n t h i s way when they l e a r n how to solve equations by the r e v e r s a l method. The fo l lowing two anecdotes i l l u s t r a t e how students use the = s ign to designate "the answer." When I f i r s t s t a r t e d showing students how to so lve the fo l lowing equation for x 2 x + 3 = 8 I wrote on the board s t a r t x x 2 + 3 e n d 8 Kyle: Are you always going to tell us the answer? I: No, we have to work out the answer. Kyle: But if we don't know the answer, how do we know what to Using C a l c u l a t o r s to Teach Algebra 106 end up with? I: Oh, I see what you mean. You mean this [pointing to the 8 in the original equation], right? Kyle: Yeah, the answer. I: Yes, I'll always tell you what to end up with. On another occas ion, Kyle was s o l v i n g the fo l lowing equat ion: 3 . 1 6 ( 4 . 9 5 + x ) 2 = 3 5 . 2 J : What's the answer, Kyle? Kyle: 35.2. I: No, the answer for x. Kyle: Oh . . . -1.61244992. I though you meant the answer. I: Isn't -1.61244992 the answer? Kyle: No, that's x. 35.2's the answer to the equation. You give us the answer. We have to find out what you started with. I: I'm asking you a question: what's x? Isn't -1.6124492 the answer to that question? Kyle: Yeah, I guess you could say that. But it would be confusing. For many students, "confusing" i s an exact d e s c r i p t i o n of teachers ' attempts to change the f a m i l i a r a r i t h m e t i c meanings of equals and "the answer." Another anecdote i l l u s t r a t e s an even stranger use of the = s i g n . I expected my students, given a word problem l i k e the fo l l owing : A b a n k p a y s s i m p l e i n t e r e s t o n t e r m d e p o s i t s a t a r a t e o f 1 0 . 5 0 % . I f y o u e a r n $ 1 0 0 0 . 0 0 i n o n e m o n t h , ( a ) h o w m u c h h a v e y o u i n v e s t e d a n d ( b ) h o w m u c h d o y o u e n d u p Using C a l c u l a t o r s to Teach Algebra 107 with? together with the fo l lowing equations and d e f i n i t i o n s P r t / u I = 100 A = P + I P p r i n c i p a l r i n t e r e s t r a t e t time u how many time u n i t s per year I i n t e r e s t A t o t a l amount to set up a tab le thus p r t u I A and to enter the data from the problem under the appropriate headings before attempting to c a l c u l a t e the values of the unknown v a r i a b l e s , which they would then record i n the c o r r e c t p lace i n the t a b l e . In order to make sure that the students knew what they had c a l c u l a t e d , I t r i e d to i n s i s t that they wri te out the f i n a l answer i n the form of a sentence on the grounds that otherwise they were simply asking me to choose the r i g h t answers from t h e i r t a b l e . However, the students t r i e d hard to th ink of ways around t h i s requirement. Matt ' s way (appl ied to the problem above) was to i n s e r t "(a) =" i n the tab le under P and "(b) =" under A. However, students do not merely see the symbol = as des ignat ing the answer: they a l so th ink i t should be w r i t t e n before and not a f t e r the answer. I found that students were accustomed to f i n d i n g the s i n g l e Using C a l c u l a t o r s to Teach Algebra 108 number i n an equation on the r i g h t , as (of course!) the answer should be, so i n w r i t i n g equations I d e l i b e r a t e l y s t a r t e d a l t e r n a t i n g between t h i s way 3 x + 6 . 5 = 1 0 and t h i s way 12 = 1 4 x - 5 . 4 Kathy (Ma 8A - C): [Pointing to the second kind] Why do some of them have the answer first? I: Because I want you to know that mathematicians often write them that way. Kathy: But we're not mathematicians. I: No, but you're studying math. Kathy: Do we have to do that kind backwards? I: Well, you have to read it backwards. You always start with x, and you always end with the answer, so you read it like this: x times 14 minus 5.4 equals 12. Kathy: Can you read an equals backwards? I: Yes. Kathy: Well, it seems dumb to me. Given her view of the = s ign as what you wri te before you wri te the answer, Kathy's quest ions and comments were reasonable . I always, when I s t a r t e d teaching how to solve equations, used the words "start" and "end" i n column 1, but wi th in a few days I would encourage students to leave them out. However, a number of students wrote (and continued to write) = at the end of each column. Others, l i k e Rosa (Ma 11A - A ) , wrote a long h o r i z o n t a l Using C a l c u l a t o r s to Teach Algebra 109 l i n e above the l a s t l i n e ("the answer") i n each column. Something, students seem to f e e l , i s necessary to d i s t i n g u i s h "the answer." In an extreme example, S a l l y (Ma 11A - C+) i d e n t i f i e d "the answer" with an = s ign even before she s t a r t e d . I always gave an equation and the v a r i a b l e to solve for i n the fo l lowing form p = a p i + b pf By h e r s e l f , S a l l y s t a r t i n g always put a double l i n e under the v a r i a b l e at the r i g h t before w r i t i n g anything e l s e . X ; Why do you do that? Sally: It tells me what I have to start with. I: OK. Sally: Besides, it looks like an equals sign, and that's what I have to end up with ... you know, equals. However, as the fo l lowing anecdote i l l u s t r a t e s , I observed some evidence that students could broaden t h e i r concept of the = s ign and "the answer" that fol lows i t . For example, Pat (Ma 10A - A) was s o l v i n g the fo l lowing ("double-star") equation for x. T m y k " D - i - i x Pat: What's the answer in this equation? I: It depends on what you start with. Pat: I'm starting with 1 [pointing to the 1 immediately to the right of the = sign]. I: So how would you read that equation? Pat: What do you mean? Using C a l c u l a t o r s to Teach Algebra 110 J : [Indicating the whole equation] Read out what it says, starting with 1. Pat: [Hesitantly] One minus one over x equals . . . Is this whole group the answer [sketching a circle around the left side of the equation]? I: Yes. Pat: [Slowly] OK. I: [After a pause during which he appeared to be thinking] What if I had asked you to solve for m? Where would you start then? Pat: m. I: What would you end up with? Pat: This [sketching a circle around the right side of the equation]. I: So it depends on where you start. Apparent ly , students do indeed view the = s ign as "what you wri te before the answer." The ir view can be modif ied somewhat, as Pat and Kathy showed, but i t i s e n t i r e l y appropriate to a r i t h m e t i c and c a l c u l a t o r language and i t s extension in to a lgebra v i a the r e v e r s a l method of s o l v i n g equations. Di f ferences Between Input and Execution In Chapter 2 I showed that one s i g n i f i c a n t d i f f e r e n c e between ar i thmet i c and algebra i s the fact that the operat ions i n a lgebra are " v i r t u a l " - they cannot be c a r r i e d out - and yet they s t i l l determine the s t ruc ture of an express ion . To be success fu l i n Using C a l c u l a t o r s to Teach Algebra 111 a lgebra , there fore , students have to be w i l l i n g to accept and work with s o - c a l l e d open expressions such as a + b, accept ing a delay between the w r i t i n g of the operat ion and i t s execut ion. The execution can be c a r r i e d out only when numbers are s u b s t i t u t e d for the l e t t e r s and the delay may be extended i n d e f i n i t e l y . C a l c u l a t o r s can play a r o l e i n persuading students to accept t h i s de lay , as the fo l lowing anecdotes show. Jack (Ma 10A - C-) had learned the convent ional order of operat ions and I was now t r y i n g to exp la in to him that the convent ional order was not the order i n which he should type the fo l lowing problem on to h i s a lgebra i c c a l c u l a t o r . 2 ( 1 . 0 1 + 1 . 0 0 1 ) 3 He had been asked to (a) p lace numbers under the operat ions to show the order i n which they must be done, (b) wr i te down what buttons he would push on h i s c a l c u l a t o r to get the answer, and (c) do the problem on the c a l c u l a t o r and wri te down the answer. He had wr i t t en 2 ( 1 . 0 1 + 1 . 0 0 1 ) 3 (a) 3 7 2 (b) ( 1.01 + 1.001 ) y x 3 x 2 = ( c ) 1 6 . 2 7 and I had marked (b) wrong because he had an a l g e b r a i c c a l c u l a t o r . Jack: So what's wrong with this? I got the right answer, didn't I? I: Yes, but you're not using your calculator the way it's meant to be used. Jack: [challengingly] So? Using C a l c u l a t o r s to Teach Algebra 112 J ; I'm trying to teach you how to use your calculator as efficiently as possible, not just get the right answers. Jack: [resignedly] OK . . . show me how I'm supposed to do this one. I: On your calculator, you type in the problem exactly as i t ' s written. I wrote 2 x ( 1 .01 + 1.001 ) y x 3 = Jack: I'm supposed to do this first [pointing to the x]? I: Yes. Jack: But that's number three. I: I should have said you type it on to your calculator first. Your calculator won't do it first, but you type it in first. Jack: But doesn't the calculator do it when I type it in? I: No; your calculator is programmed not to do a times [pointing to the x] when it sees that the next operation is in brackets [pointing to the +]. Jack: You mean, it waits to see what's coming up next? I: Yes. Jack: So brackets mean "wait." I: Yes; that's exactly what this kind of bracket [pointing to the ( bracket] means. But as soon as you punch this [pointing to the ) bracket] the calculator knows that it can do the +, so it does it. Jack: [facetiously] The long wait's over. Using C a l c u l a t o r s to Teach Algebra 113 J : Yes. Jack: So now it can do the times. I: Well . . . can it? The next operation [pointing to the y*] is a power. Remember, your calculator is programmed to do operations in the right order even when you type them in in a different order. Jack: Oh . . . I see . . . [thoughtfully] a power . . . No, I guess it s t i l l has to wait . . . I: . . . until you press =; when you press = the calculator does all the operations you've typed in. That's why you never press = on an algebraic equation until you get to the end. Jack: Now I see . . . doing an operation doesn't mean i t ' s done. I: You tell the calculator what to do, but it doesn't always do it right away. Jack: OK. So why do you tell people with algebraic calculators to put in the order of operations? I: Because I know you're going to need it in Unit II. Jack: So why do you teach it now? I: Because the non-algebraic people do need it now, and i t ' s easier to teach it to everybody at the same time. Jack: That makes sense. (Notice how Jack t r a n s l a t e d between E n g l i s h and c a l c u l a t o r language on h i s own: "Brackets mean wait" and "The long wa i t ' s over.") Thi s delay between the input of an operat ion by a person on Using C a l c u l a t o r s to Teach Algebra 114 to an a l g e b r a i c c a l c u l a t o r and the execution of the operat ion by the c a l c u l a t o r may be the f i r s t h i n t students get that what Pimm (1987) c a l l e d "the powerful prompt" (p. 167) of an operat ion s ign cannot always be obeyed immediately. A s i m i l a r delay - although very short - e x i s t s even on a non-a l g e b r a i c c a l c u l a t o r , enabl ing students to c o r r e c t mis-punched operat ions provided they recognize t h e i r e r r o r before the operat ion i s c a r r i e d out by the c a l c u l a t o r . I had been exp la in ing to a Mathematics 8A c l a s s that i f students with non-a lgebraic c a l c u l a t o r s punch the wrong operat ion , they can correc t i t by punching i n the correc t operat ion immediately afterward. However, I had warned, students with a l g e b r a i c c a l c u l a t o r s should not attempt the same t h i n g . ( A c t u a l l y , you can, on an a lgebra i c c a l c u l a t o r , sa f e ly replace a power operat ion with x or and x or -r with + or - . However, i t would not be safe , for example, for students to attempt to rep lace + or - with x or - because press ing + or -might cause the c a l c u l a t e to execute prev ious ly punched +'s or - ' s which would have been fur ther delayed i f the c o r r e c t operat ion had been punched.) However, Bob (Ma 8A - A) wanted to see for h imse l f . Bob: You said you can't make corrections on an algebraic calculator, but you can. I just tried it. Look ... He punched 2 + x 3 = and showed me the 6 on h i s d i s p l a y . I: Yes, but you have to be careful. Bob: Why? Using C a l c u l a t o r s to Teach Algebra 115 J: [Writing down 11 + 6 x 5 - 4] Well, suppose you punched a - there [pointing to the *] by mistake. You couldn't just change it to a x . Bob: Why not? I: Well, think about what the calculator would do when you pushed the - by mistake. Bob: [After a pause] It would do this plus [pointing to the + in the question]. Oh, I see, you couldn't get the 11 and the 6 apart again. I: Right. Bob: But if I did a x there [pointing to the -] I could change it to a -. I: Why? Bob: Because the x would make the calculator do the x [pointing to the x between the 6 and the 5], but that's OK, because it's supposed to be done next. I: [Summarizing] So you have to be careful. I found h i s statement that "you cou ldn ' t get the 11 and the 6 apart again" p a r t i c u l a r l y i n t e r e s t i n g ; i t was almost as though he were seeing that the immediate c a r r y i n g out of an operat ion could at times have disadvantages. The concept of a delay between input and execution of operat ions ev ident ly stayed i n Bob's mind, because he rever ted to the point when he and two other students bought programmable c a l c u l a t o r s at my recommendation. While they were not yet programming them, they had discovered that when they had to Using C a l c u l a t o r s to Teach Algebra 116 evaluate a numerical express ion, the c a l c u l a t o r "pr inted out" the e n t i r e express ion on the d i s p l a y without any eva luat ion u n t i l they pressed the button marked EXE (there was no button marked =). Bob (Ma 8A - A): This is great: the calculator doesn't do anything until you tell it to, so you can go back and change something if you've done it wrong. Borzoo (Ma 8A - A): Even if the calculator's done it, you can s t i l l go back and do it again. Bob: Yes, I know, just punch EXE again. Borzoo: No, punch this button [pointing to a left-pointing arrow]. Bob: What does that do? Borzoo: It shows you the whole problem again. I saw that the ambiguous use of the word "do" was going to cause problems. I : Doing the problem and executing the problem are not the same thing. Vince (Ma 8A - A): What d'you mean? I: First you punch the problem on to the calculator - all the numbers and all the operations. Then, when you're sure you've punched it in right, you tell the calculator to execute the instructions. Vince: I thought execute meant, you know, electrocute or something. I: Actually, "execute" means "carry out" or "make happen." First the judge gives the sentence: he says what should happen to Using C a l c u l a t o r s to Teach Algebra 117 the criminal. Then, later on, the sentence is executed, or carried out. So first you say what should happen to the numbers and then you tell the calculate to execute your instructions. Vince: Ohhhhh . . . J see. Bob: So really, this is like a third type of calculator. I: What do you mean? Bob: Non-algebraic, algebraic, and this kind. I: This kind is algebraic, you know. Bob: I know, but non-algebraic calculators execute - is that what you say? - every operation as soon as you punch the next operation. Algebraic calculators execute some operations right away . . . like they don't execute a plus if they see that a times is coming next, so they wait, but if, say, a minus comes next they execute the plus. But this kind of calculator doesn't execute anything until you tell it to. Again , the d i s t i n c t i o n between "doing a problem" i n the sense of w r i t i n g i t out or typing i t on to a c a l c u l a t o r l i k e Bob's and "executing the problem" i n the sense of c a r r y i n g out the operat ions i s a foreshadowing of the d i f f erence between what Pimm (1995) c a l l s the " v i r t u a l " operations (p. 109) i n a l g e b r a i c expressions and the ac tua l or executable operat ions i n a r i t h m e t i c express ions . An a p p r e c i a t i o n of t h i s delay on a c a l c u l a t o r can even he lp students to accept mathematicians' habi t of t a l k i n g about a v a r i a b l e with an undetermined or even indeterminate value ("before they know what i t is") as Kyle (Ma 8A - A) impl ied by h i s Using C a l c u l a t o r s to Teach Algebra 118 i n s i s t e n c e on the c o n d i t i o n a l mood for the verbs i n the fo l lowing exchange. Given that m = 2, a = 6, and c = 6 i n the fo l lowing equation m(b + a) = c I had e s tab l i shed that i t was b that we were t r y i n g to f i n d and was now demonstrating how to f i n d i t . I had w r i t t e n s t a r t b + a x m end c saying as I wrote, "If we s t a r t with b, then add a, then m u l t i p l y by m, we end up with c ." Kyle: How can you start with b when you don't know what it is? I: Why shouldn't we? Kyle: 'Cause you don't know what b is, so how can you start with it? Logically, you can't even talk about it until you know what you're talking about. [Cheekily] Always think before you speak, Miss Murphy. I: But, Kyle, I'll be able to figure out what b is in a minute. Can't you let me talk about it for a second when you know I'm going to work it out? Kyle: If you did know what b was, you could start with it. Then if you did add a and did multiply by m, we would end up with c. I: OK. Kyle: [Challengingly] Well, say that, then. Using C a l c u l a t o r s to Teach Algebra 119 However, perhaps the u l t imate i n wi l l ingness to delay execution of operat ions was exh ib i ted by Bob (Ma 8A - A ) . Given 3 1 . 8 2 + 3 . 9 " 2 f _ _ 2 _ _ 2 _ _ »1 5 3 1 6 + 9 5 on a q u i z , he was required to (a) number the operat ions to show t h e i r convent ional order , (b) t e l l what buttons he would punch on h i s c a l c u l a t o r to get the answer, and (c) give the answer to two decimal p lace s . Bob: [Having completed parts (a) and (b) of the question] Do I have to punch this out? I: Yes, if you want the other mark. The question's out of eight. You get one for the order of operations, six for telling me the buttons, and one for the answer. Bob: 'Cause I don't feel like punching this out. I: Then you'll lose one mark. Bob: Why do I have to punch this out? I: To get the answer. Bob: But if I've told you how I'm going to punch it out, isn't that the same? I: No. Bob: But if I've written down how to punch it out, anyone in the world could do it. I: So? Bob: So it's just the same as if I've done it. Using C a l c u l a t o r s to Teach Algebra 1 2 0 I: If the bank sent you a statement, and they didn't tell you how much you actually had at the end of the month, would you be satisfied? Bob: Yes - if they told me how to calculate it, like I've done. I: Would you calculate it if they told you how to? Bob: Yes. I: Why? Bob: To find out how much I had. I: Wouldn't you want to check that the bank had got the same answer? Bob: I guess so. Anyway, I'm not going to do it - I'm late for a rugby meeting. S i g n i f i c a n t l y , he c a l l e d back as he ran o f f , "Just push E X E . " I t was a l so Bob who, i n the middle of the exam at the end of the c a l c u l a t o r u n i t , announced loudly and i n a tone of great f i n a l i t y , "I'm - not - going - to punch - any - more - buttons . I jus t won't. I won't punch another button" - thereby s a c r i f i c i n g a s i g n i f i c a n t number of marks on the exam. I t appears that c a l c u l a t o r s , by de lay ing the execution of an operat ion , can introduce to students the idea that the operat ion has an existence apart from i t s ar i thmet i c execut ion, an idea that i s e s s e n t i a l to a lgebra . Conclus ion Otherwise unmathematical students are capable of becoming Using C a l c u l a t o r s to Teach Algebra 121 f luent i n ar i thmet i c and c a l c u l a t o r language. They can be observed to use i t i n two-way communication with c a l c u l a t o r s and with other people, to express and explore t h e i r i n s i g h t s i n t o a r i t h m e t i c and a lgebra , and even to create mathematics for themselves. Using C a l c u l a t o r s to Teach Algebra 122 Chapter 7 Br idg ing the Gap Between Ar i thmet i c and Algebra The anecdotes inc luded i n t h i s chapter f a l l i n t o the second h a l f of my twelve categor ies : importance of operat ions as opposed to numbers, t r a n s i t i o n from numbers to l e t t e r s , nature of v a r i a b l e s , s t ruc ture of express ions, groups, and a u t h o r i t y . In genera l , they demonstrate that unmathematical students can be helped to bridge the gap from ar i thmet i c to a lgebra . Operations Versus Numbers The fac t that one of c h i l d i s h des ignat ions for a r i t h m e t i c i s "numbers" i s s i g n i f i c a n t , for i t i s the numbers, not the operat ions , that f i r s t engage the a t t e n t i o n of students when they begin a r i t h m e t i c , as, for example, while they are memorizing the number fac t s of a d d i t i o n and m u l t i p l i c a t i o n . Many of my students , when they f i r s t begin numbering operations to show t h e i r convent ional order , wri te the order ing numbers under the numbers ins tead of under the operations u n t i l they are c o r r e c t e d . Learning to use a c a l c u l a t o r as I teach i t encourages students to stop focuss ing on the numbers and s t a r t focuss ing on the operat ions ins tead . When he f i r s t heard a summary of what I teach i n my c a l c u l a t o r u n i t and the language i n which I teach i t , Pimm s a i d t e l l i n g l y , "The numbers have already dropped out ." Indeed, a number of students do "drop them out" i n a very r e a l sense, as the fo l lowing anecdotes demonstrate. Using C a l c u l a t o r s to Teach Algebra 123 Kathy (Ma 8A - C) had been asked on a qu iz to so lve the fo l lowing equation for x: Kathy: How come there are two fives in this equation? I: Why shouldn't there be? Kathy: Then how do you know . . . Oh, I guess the operations tell you. Apparent ly , she answered her own quest ion by her r e a l i z a t i o n that the two f ives were i n fac t d i s t i n g u i s h a b l e from each other . I n t e r e s t i n g l y , she d i d not suggest that i t was because of t h e i r p o s i t i o n s i n the equation (e.g., l e f t and r i g h t of the equals s ign , one higher and the other lower on the page, or part of the "question" as opposed to part of the "answer"), but ra ther because of "the operations" (among which, I suspect, she was i n c l u d i n g the = .) For Steve (Ma 11A - B) , the numbers had "dropped out" so completely that he abso lute ly refused to wri te them down. Given the express ion 1 3 . 6 + 4 . 5 ( 1 . 4 2 - 0 . 4 1 9 9 ) 1 4 2 "11 .2 _ 13.24" 1 .45 2 . 5 on a q u i z , he was supposed to t e l l me what buttons he would punch on h i s (a lgebraic) c a l c u l a t o r to get the answer. I expected him to wr i te ( 1 3 . 6 + 4 . 5 x ( 1 .42 - 0 . 4 1 9 9 ) y x 14 T ( ( 1 1 . 2 * 1 .45 - 1 3 . 2 4 -r 2 . 5 ) x 2 ) = Using C a l c u l a t o r s to Teach Algebra 124 but ins tead he wrote ( + x ( - ) y x 7 ( ( v - H X 2 ) = I: These aren't the buttons you would punch on your calculator. Steve: Yes, they are. I: What about the numbers? Steve: Well, of course I would punch the numbers - but you know that. You just want to know if I know what order to punch the operations in. I: What if the same operation comes twice in the same question? Steve: [Sarcastically] I'll colour them different. He d i d . In ra ther milder s t y l e , G a i l (Ma 8A - A ) , showed that she was ignor ing the numbers thus: Gail: What would I do if . . . say you had a plus here, and a minus here, and then maybe a T there . . . She wrote, as she spoke + . . . would I need brackets here [pointing to the space between the - and the +]? For Conrad (Ma 11A - C - ) , who had spent the previous day programming c a l c u l a t o r s , i t was ev ident ly the c a l c u l a t o r ' s way of c a r r y i n g out programmed operations that impressed him: when Marc, who had missed the previous day's c l a s s , s a i d , "What's so great Using C a l c u l a t o r s to Teach Algebra 125 about programmable c a l c u l a t o r s ? " Conrad r e p l i e d , "They remember operat ions ." I found i t i n t e r e s t i n g that he d i d not a l so inc lude the l e t t e r s , which he could not type on to h i s own (non-algebraic) c a l c u l a t o r but which he had typed on to the programmable c a l c u l a t o r together with the operations the day before . Perhaps he was used to h i s own c a l c u l a t o r remembering numbers and d i d not th ink remembering l e t t e r s to be s u f f i c i e n t l y more impressive to make i t worth mentioning. The a b i l i t y of Bob (Ma 10A - A) to see that not only numbers, but a l so l e t t e r s , could be "dropped out" i n Pimm's sense l ed to my own undoing one day. The bas i c equations students were r e q u i r e d to solve for x i n Math 10A contained three of the four operat ions + , - , x, and -f, e.g. a = b(x + c) - d or 45x - 54.6 T^—7 = 10 12.6 In order to give students s u f f i c i e n t p r a c t i c e i n s o l v i n g equations conta in ing l e t t e r s only , I had made up three worksheets each i n c l u d i n g 48 bas i c quest ions . Since there are only 64 combinations of the three operat ions , the three worksheets contained r e p e t i t i o n s , which I had d i sgu i sed by changing the l e t t e r s randomly and, more often than not, by rearranging the quest ions . (This t a c t i c meant that I d i d not have to make up a new key for each worksheet.) On one occas ion, however, I had not had time to rearrange the quest ions , as i t would have meant Using C a l c u l a t o r s to Teach Algebra 1 2 6 reformatt ing the worksheet. Bob: Worksheet 15C is just the same as 15A! I: No, it isn't. Bob: Yes it is - look [aligning the two worksheets side by side and pointing from one to the other]. I: Those aren't the same - they're different letters. Bob: [In the tone of one rallying a child who is trying to pull a fast one] Miss Murphy, they're the same! I: How can they be the same when they've got different letters? Bob: Yeah, I know, but they've got the same operations. I: So are you telling me that . . . I wrote x + a - b = c . . . is the same as I wrote x - a + b = c . . . just because they've got the same operations? Bob: [After a long pause, during which he looked from one to the other of the two equations I had written] No . . . because they've changed places. But [returning to the two worksheets] here you didn't change them. You've got the same operations in the same order on both worksheets. I: [Trying to sound incredulous] And you're telling me that if I have the same operations in the same order i t ' s the same question, even if i t ' s different letters? Using C a l c u l a t o r s to Teach Algebra 127 Bob: Of course it is! The letters don't matter. I: [Breaking down and smiling] You're right, but don't tell the others. By us ing a c a l c u l a t o r , students hand over to the c a l c u l a t o r r e s p o n s i b i l i t y for anything that depends on the i d e n t i t y of the numbers. I t remains t h e i r r e s p o n s i b i l i t y to g ive the c a l c u l a t o r the c o r r e c t numbers, but that i s easy: every h igh school student can read numbers and copy them on to a c a l c u l a t o r . However, i t a l so remains t h e i r r e s p o n s i b i l i t y to give the c a l c u l a t o r the c o r r e c t operat ions i n the correc t order , and that i s new to them; hence t h e i r s h i f t of focus from numbers to operat ions . T r a n s i t i o n from Numbers to L e t t e r s The t r a n s i t i o n from ar i thmet i c to algebra invo lves most no t i ceab ly a s h i f t from numbers to l e t t e r s , a s h i f t the students are ready to make, even when no numbers are given for t h e i r va lues , once the numbers have t r u l y "dropped out ," as shown i n the fo l lowing anecdotes. For Mathematics 8A students' f i r s t venture in to a lgebra , I had given them a worksheet on which each problem cons i s t ed of an express ion made up of l e t t e r s only , e.g. d(a + be) and asked them to (a) p lace numbers under the operat ions to show the order i n which they must be done and (b) t e l l what buttons they would punch on t h e i r c a l c u l a t o r to get the answer. These were exact ly the same i n s t r u c t i o n s I had given them on the Using C a l c u l a t o r s to Teach Algebra 128 previous worksheet, except that now there were l e t t e r s ins tead of numbers and I was not asking them to c a l c u l a t e the "answers." Choon (Ma 8A - A): How do we do this? I: The same as the last worksheet, except that now you can't calculate the answer. Choon: But how do we do it with letters? I: Each letter stands for a number. Just do it the same as you would if they were numbers. Choon: But we can't put letters on a calculator. Borzoo (Ma 8A - A): Just pretend your calculator has letters on it instead of numbers. Choon: Oh - OK. This i s not to say that some students, l i k e Mark, d i d not have reservat ions about t h i s new k ind of mathematics. Mark (Ma 8A - C-): [Quietly and privately] I don't want to be rude, but isn't this rather dumb? I: Why? Mark: I mean, this isn't math. I: my not? Mark: Math is numbers. You can't get the answer with letters. I: You can if I tell you what numbers the letters stand for. Mark: Yeah, but you haven't told us. I: But I could, so this is s t i l l math. The more you do math, the more you'll use letters. Mark: You're kidding. Using C a l c u l a t o r s to Teach Algebra 1 2 9 J ; No. Look, here's my Physics 12 book [getting it out of my bag and opening it at random]. See, this page has no numbers on it at all, just letters. Mark: But how can the kids get the answer? I: This [I pointed to F = ma] tells the kids that if they want to calculate F, they have to find out what number m is, find out what number a is, and then multiply them together and the answer is what F i s . Mark: [In the tone of "If you say so"] OK. Students a l s o , I found, had d i f f e r e n t ideas from me about how to represent v a r i a b l e s which had r e a l meanings. For example, when I introduced problems conta in ing regu lar p r i c e and sa le p r i c e , I explained that although we would l i k e to use "P" for p r i c e to remind us of what i t meant, we had to have d i f f e r e n t v a r i a b l e s for regu lar p r i c e and sa le p r i c e . I: So what we do is use different subscripts. We'll use P r for the regular price and P s for the sale price. Kathy (Ma 8A - C): Why don't you use R p and S p ? I: We could . . . I guess . . . but I think that makes it look as though they're totally different variables; don't you? Kathy: No. I think it looks more like what it means. I: That's not what mathematicians usually do. For example, in the physics I was just teaching this morning, we had two masses, so we called them [I wrote] mi and m2, not [I wrote] 1 m and 2m. Besides -I'm sorry - I've already made up the worksheets. Using C a l c u l a t o r s to Teach Algebra 130 Kathy: (Resignedly) Whatever. On the other hand, students soon adopted mathematicians' informal conventions for v a r i a b l e s . I had w r i t t e n three worksheets on each new top ic i n s o l v i n g equations, and although I had used almost a l l the l e t t e r s of the E n g l i s h alphabet , I had developed the habi t of asking students to solve for x on Worksheet A, y on Worksheet B, and z on Worksheet C. However, i n order to make a p a r t i c u l a r p o i n t , I wrote a worksheet where the f i r s t few quest ions looked l i k e t h i s : 1. a + b + c + d = e a 2 . a + b + c + d = e b 3 . a + b + c + d = e c 4 . a + b + c + d = e d with the students asked to solve for the v a r i a b l e at the r i g h t . Yvonne (Ma 8k - C+): [Pointing to the column of variables at the right] What's this? I: Those are the letters you have to solve for. Yvonne: [With exaggerated surprise] Solve for a and b? I: Yes; why not? Yvonne: I guess so ... But we always solve for x or y or z. I: Well, it's good to change. Yvonne: I guess so ... but it's hard to remember what you're solving for when you keep changing it. These i n i t i a l d i f f i c u l t i e s with l e t t e r s over, however, students soon began asking whenever they were presented with a new worksheet, "Is i t l e t t e r s or numbers?" In no time at a l l , they were groaning when i t was numbers and cheering when i t was Using C a l c u l a t o r s to Teach Algebra 131 l e t t e r s . Portia (Ma 8A - C+): I like letters better than numbers. I: Why? Portia: 'Cause you can see what you're doing better. Plus you don't have to work out the answer. The students' de s i re to cut down on the amount of work, p a r t i c u l a r l y w r i t i n g , they had to do l ed to some s u r p r i s i n g connections between v a r i a b l e s and c a l c u l a t o r s and some i n s i g h t s in to the nature of v a r i a b l e s that I found very va luable for my own t h i n k i n g , as revealed i n the next three anecdotes. I had always i n s i s t e d that when the students had so lved for the value of x i n a numerical problem, they check that the value they had obtained a c t u a l l y s a t i s f i e d the o r i g i n a l quest ion and that they show me that they had done so. One student, asked to solve 2 = ( 3 + z ) 7 for z , wrote z 2 2 - - 3 = z = - 2 . 7 1 4 2 8 5 7 1 4 + 3 -r 7 7 x 7 - 3 2 z C h e c k : 2 = ( 3 + ) 7 The same student, who had a Texas Instrument c a l c u l a t o r , when asked to solve ( 5 + z ) ( 3 ) ( 7 ) = 5 for z , wrote Using C a l c u l a t o r s to Teach Algebra 132 5 5 + 5 . 7 f x 3 v 3 — 5 = z = - 4 . 7 6 1 9 0 4 7 6 2 (STO) x 7 - 5 5 z C h e c k : (5 + R C L ) ( 3 ) ( 7 ) = 5 ("STO" i s the button that s tores a number i n the memory and "RCL" the button that r e c a l l s i t . ) Notice how "RCL" has replaced "z" i n the f i n a l equat ion. Larry (Ma 9A - A ) , asked to solve the fo l lowing equation for x, 7(x + 2) = 1 5 . 5 had w r i t t e n the fo l lowing: x 1 5 . 5 1 5 . 5 + A . 7 = x = 0 . 2 1 4 2 8 5 7 1 4 x 7 - 2 1 5 . 5 x He knew that he was now expected to write ( C h e c k ) 7 ( 0 . 2 1 4 2 8 5 7 1 4 + 2) = 1 5 . 5 and to check on h i s c a l c u l a t o r that t h i s was i n fac t t r u e . Larry: I hate having to write these big numbers [pointing to the 0.21485714] over and over again. I: Well, if that's what x works out to, I don't see what else you can do. Larry: You can't? And you're a math teacher? I: Larry! Larry: No, but can't you think of something so we don't have to write it all down? It takes ages, and besides, sometimes I don't copy it right or punch it in right and then I get the wrong answer when I've really got the right answer. Using C a l c u l a t o r s to Teach Algebra 133 I: What about putting it in your memory? Larry: Can I? I: Yes - just push x->M when you 've got x and then RM when you want x back again. [His calculator was a Sharp.] Larry: Yeah, but I s t i l l have to write it down. I: Well, you'll have to write it down once or I won't know whether you've got the right answer for x, but maybe you could tell me somehow that you've put it in the memory and pulled it out again and then you won't have to write it out in the check. Larry: OK. By himsel f , Larry gradual ly s t a r t e d w r i t i n g x 1 5 . 5 1 5 . 5 2 - y ^ = x = 0 . 2 1 4 2 8 5 7 1 4 x-»M x 7 - 2 1 5 . 5 x ( C h e c k ) 7 ( R M + 2 ) = 1 5 . 5 Larry: Doing this [he pointed to the x-*M and the RM] is just like letters again. I: What do you mean? Larry: It just like changing from x to M. I: I see what you mean. Larry: So what's the point? I: Well, it does tell me that you've put the number into your memory and recalled it again. Larry: [In a "have it your own way" tone of voice] OK - I guess you like M better than x 'cause of your name. I began to r e a l i z e for the f i r s t time what a very good image of a v a r i a b l e a c a l c u l a t o r ' s memory i s : something that can "hold" Using C a l c u l a t o r s to Teach Algebra 134 a v a r i e t y of numbers. This image i s even be t t er i f the c a l c u l a t o r i s programmable, f or then the memories are designated by the l e t t e r s A - Z , as I r e a l i z e d a f t e r the fo l lowing exchange. Borzoo (Ma 8A - A ) , had jus t suggested to another student, "Pretend your c a l c u l a t o r has l e t t e r s on i t ins tead of numbers." Then he turned to me. Borzoo: Are there any calculators that do letters? I: Yes. Borzoo: Can we buy some? I: We could, but they're expensive and they're not going to help you that much. Borzoo: But then we could just punch in the letters. I: Yes, but you'd still have to tell the calculator what numbers went with the letters before it could calculate the answer. Borzoo: So? That would be fun! I: So a programmable calculator won't save you any time -unless you're doing the same question again and again with different numbers. Your worksheet isn't like that - each question is different. Borzoo: Why don't you make a worksheet for us like that? I thought of the i n t e r e s t problems I was in tending to g ive them i n Unit 3. J : Later on, I will. Borzoo: So can we buy calculators with letters now? I went shopping that night to inves t iga te programmable Using C a l c u l a t o r s to Teach Algebra 135 c a l c u l a t o r s and found that a d i scont inued l i n e - the Casio fx-4500P - was being s o l d cheaply. Unl ike any other programmable c a l c u l a t o r I have seen ( i n c l u d i n g , unfortunate ly , the one that has replaced i t ) , i t d i d not demand that the program conta in steps to i d e n t i f y each v a r i a b l e and ask for i t s va lue . The program cons i s ted simply of a formula typed i n thus I = PRT/U/100 When the program was run , the c a l c u l a t o r would d i s p l a y P? as we l l as the value c u r r e n t l y i n memory P. The students could press ENTER to accept t h i s value or change i t to some other value f i r s t . Three students - Borzoo, Bob, and Vince - bought these c a l c u l a t o r s from me and began to make even deeper connections between c a l c u l a t o r s and v a r i a b l e s , which are descr ibed i n the next subsect ion . Nature of V a r i a b l e s A f t e r t h e i r i n i t i a l s u r p r i s e , the students accepted l e t t e r s as part of mathematics. They even began to pre fer them because they found them eas i er to wri te and to copy c o r r e c t l y and because they "didn' t have to work out the answer" when the quest ion contained only l e t t e r s . F i r s t "the numbers had dropped out"; now I began to f i n d the i d e n t i t y of the l e t t e r s "dropping out" as w e l l , as students began to conceptual ize v a r i a b l e s as having an exis tence def ined by t h e i r r e l a t i o n to the re s t of an express ion , independent of the i d e n t i t y of the l e t t e r . For example, i n Mathematics 8A, which I was teaching for the Using C a l c u l a t o r s to Teach Algebra 136 f i r s t t ime, I was w r i t i n g three worksheets on each new t o p i c i n s o l v i n g equations, and there was n e c e s s a r i l y some r e p e t i t i o n . To hide the r e p e t i t i o n (for I knew that students would object to "doing the same questions again"), I wrote a program for my computer which would change the l e t t e r s to other l e t t e r s according to a pa t t ern which looked random. Thi s e n t a i l e d f i r s t changing the l e t t e r s to symbols such as ©, © , V , • , * , * , e t c . and then changing the symbols back to d i f f e r e n t l e t t e r s . On one occas ion , however, my program f a i l e d , and I copied a qu iz for the students without r e a l i z i n g that i t had hearts ( v ) on i t . When the students pointed i t out to me, I t o l d them to change a l l hearts to x ' s . Mark (Ma 8A - C-): [Cheekily] Why should we? I: I meant to put x's. Mark: Why can't we leave them as hearts? I: You can, I guess. Mark: I mean, it could be anything. I: That's true. Mark: [Getting to work] I like hearts! At the same time, as mentioned i n the l a s t subsect ion , the three students who had bought programmable c a l c u l a t o r s began to see connections between l e t t e r s and c a l c u l a t o r s which I had never thought of . One day Vince (Ma 8A - A) and Borzoo (Ma 8A - A) were t a l k i n g to each other about t h e i r programmable c a l c u l a t o r s , which, when running a program, would d i s p l a y the value c u r r e n t l y i n each memory ( l e t t e r e d A-Z) and ask for a pos s ib l e new value at the same Using C a l c u l a t o r s to Teach Algebra 137 t ime. Vince: What's A right now for you? Borzoo: 10.5. Vince: Mine's 16. I: That's not really what the variable A is - it's just what the calculator has got stored in memory A. Vince: Same thing. It says A = 16 [showing me]. In my mind, A was the name of a memory which could ho ld a value which could be changed at w i l l ; i n the students' minds, A was c u r r e n t l y 16; I could see why they thought my i n t e r j e c t i o n a mere q u i b b l e . Bob apparently kept t h i s image of a v a r i a b l e with him r i g h t up to Grade 11. When my Mathematics 11A students were programming i n t o the schoo l ' s TI-82 ' s the long formulae for saving and f inanc ing (see Appendix 1), the l a s t two l i n e s of the i n s t r u c t i o n s I had given them read as fo l lows: Buttons to punch Calculator shows PRGM • 3 2nd A L P H A " A " E N T E R :Disp "A " PRGM • 3 A L P H A A E N T E R :Disp A The f i r s t l i n e prompts the c a l c u l a t o r to d i s p l a y the l e t t e r A, while the second l i n e prompts i t to d i s p l a y the number c u r r e n t l y h e l d i n memory A. In programming h i s c a l c u l a t o r , Jack (Ma 11A - C-) had omitted the quotat ion marks. Bob, who was used to h i s own programmable Using C a l c u l a t o r s to Teach Algebra 138 c a l c u l a t o r , was exp la in ing to him why h i s program wasn't working and why the quotat ion marks were e s s e n t i a l . Bob: When you put the quotation marks in, it means the letter. When you leave them out, it means the number that letter i s . I: Actually, Bob, it means the number that is in the memory that has that letter. Bob: Yeah, but the memory's just the same as the letter. Jack: But you just said, not if it doesn't have quotation marks. Bob: [turning back to Jack] Don't listen to Miss Murphy . . . when you put the quotation marks in, it means the letter . . . like this [writing the letter A] . . . this letter, right here, right? When you leave them out, it means the number for that letter, OK? Jack: OK. With quotation marks, i t ' s a letter; without quotation marks, i t ' s a number. The anecdotes i n the l a s t two subsections show that students can not only use c a l c u l a t o r language to he lp each other over the i n i t i a l d i f f i c u l t i e s of s u b s t i t u t i n g l e t t e r s for numbers, desp i te the fac t that most c a l c u l a t o r s can be given numbers but not l e t t e r s , but a l so to b u i l d for themselves an image of a v a r i a b l e as high as (e) on the l i s t of Kiicheman and C o l l i s (1975). S tructure of Expressions By r e q u i r i n g my students always to "put i n the order of operat ions ," I encourage them to th ink of a l g e b r a i c expressions as Using C a l c u l a t o r s to Teach Algebra 139 i n s t r u c t i o n s which they should fo l low. As a r e s u l t , many of them remain mathematically at the l e v e l of c o n s t r u c t i o n workers who p lod through t h e i r jobs step by step, with no thought of the f i n i s h e d b u i l d i n g , asking themselves at each stage, "What should I do next?" or , as my students would put i t , "May I do this before this?" However, a s i g n i f i c a n t number of students begin to behave more l i k e supervisors who, even though they are fo l lowing i n s t r u c t i o n s , keep the f i n i s h e d b u i l d i n g i n mind. One of the l a t t e r was Matt (Ma 11A - A ) , who had been asked to put i n numbers to show the order of operat ions i n the fo l lowing two express ions: 5 + 6 5 + 6 3 + 2 3 + 2 Matt: These two questions are the same. I: No, the lengths of the divisions lines are different. Matt: So how do you do them? I: Well, first of all, put in the brackets that come with the long division lines, starting with the longest. Matt: [Doing it] Oh, OK; I never do that because I can just see where the numbers go. I: In this kind of problem it helps. A f t e r Matt had put i n brackets c o r r e c t l y thus: Using Calculators to Teach Algebra 140 (5 + 6) (5 + 6) 7 7 9 9 (3 + 2) (3 + 2) I put my pencil point on the + in the bottom line and started asking, "May I do this before this?" Matt: [Interrupting] I don't need to do that. I can see it now. I just needed the brackets. To continue my construction analogy - Matt behaved like workers who, a former electrician assures me, never read the instructions at the beginning of a job, but only when they reach a point where they cannot "see" what is to be done next. Another student who clearly began to see the larger picture was Sheila (Ma 11A - B), who was solving the following equation for a: Sheila: How come [pointing to the original equation] this has . . . you know . . . a nice . . . kind of . . . you know, shape [cupping her hands around the left side of the equation] and when I make a the subject it doesn't? Apparently, Sheila had noticed the symmetry on the l e f t side of the original equation and the lack of symmetry in the f i n a l I: OK. a b + c d e f + g h = 3 She ended up correctly with j ( e f + g h ) - c d b = a Using C a l c u l a t o r s to Teach Algebra 141 equation - not n e c e s s a r i l y the symmetry induced by the commutativity of a d d i t i o n and m u l t i p l i c a t i o n , but probably the p h y s i c a l symmetry of the wr i t t en express ion. However, Iago (Ma 11A - B) appeared to apprec iate a deeper symmetry, as he showed when he had to solve the fo l lowing equation for x: mn + np = z x - a Iago: I can't solve for x right away, right? I: Right. Iago: So I have to solve for something else first. I: Right. Iago: But what? I: The nearest thing to x that isn't under the divide line. Iago: Does it matter which? I: Yes, you have to pick the nearest. Which is the nearest? Iago: That one [pointing to p] ... I guess. I: Wouldn't you say that one [pointing to m] is really nearer to x? Iago: Not really. I: Isn't it? I measured the p h y s i c a l d is tances from m to x and from p to x between my f inger and thumb. Iago: [Rallyingly] Come on, Miss Murphy. That's not what you meant, is it? I: [Smiling] No. I just wanted to see what you thought. Using Calculators to Teach Algebra 142 lago: But really, which is the nearest? I: Why did you say p was nearest? Iago: 'Cause when you read the question, like [pointing as he read] m n plus n p divided by x minus a, p comes right before x. I: So why did you ask me? Iago: I wondered if it really made any difference. I: Actually, in this case, you could start with m, n, or p and it wouldn't make any difference. Do you know how you can tell? Iago: How? I: Suppose I wrote the question like this. I wrote mn + pn — = z x - a J : Would that mean the same thing, or is that different? Iago: It's different. I: How? Iago: You turned these two around [pointing to the p and the n]. I: If I gave you numbers for p and n, would it make any difference to the answer? Iago: Oh, I see . . . no. I: Why not? Iago: 'Cause 2 times 3 is the same as 3 times 2. I: Right. So p and n are interchangeable in this question. Iago: So it doesn't matter whether I start with p or n? Using C a l c u l a t o r s to Teach Algebra 143 J ; Right. Could you start with m? Iago: [After a pause] Yes. I guess so. I: How do you know? Iago: 'Cause I could write it . . . He wrote pn + mn = z x - a . . . and that would work out the same when you put numbers in. I: So if you can change around two letters without making any difference to the answer, then it doesn't matter which of those two letters you start with. Iago: OK. Dave (Ma 11A - B) made observations s i m i l a r to Iago's , but i n a s l i g h t l y d i f f e r e n t context . He had jus t solved the equation a + b m - n c d j k = P for a and b, obta in ing P + m - n j - k (c + d) - b = a and P + m - n j - k (c + d) - a = b Dave: Look, these are both the same except the a and the b changed places [pointing alternately to the b and the a in his second answer]. Using C a l c u l a t o r s to Teach Algebra 144 J : Yes; it's neat, isn't it? Dave: Does that always happen? I: No. Dave: How do I know when it's going to happen? I: Why do you want to know? Dave: 'Cause then I can just write down the answer. I: OK. Look at the original equation. Would it make any difference if I changed the a and the b around? Dave: What do you mean? ... Oh, no, I guess not. I: If the a and the b are interchangeable in the original equation, then they're interchangeable in the answers, too. Dave: Oh ... OK! Kyle (Ma 8A - A) not iced not only t h i s k ind of symmetry, but a l so the asymmetry introduced by d i v i s i o n . The students had been us ing the fo l lowing equation to solve simple i n t e r e s t problems for I , P, r , or t when they were given the values of the other v a r i a b l e s . Prt/u 1 00 (The v a r i a b l e "u" takes the value 1, 12, 52, or 365, depending on whether the time i s given i n years , months, weeks, or days.) I not i ced that Kyle was s i t t i n g apparently doing nothing . I: Kyle, have you finished your work? Kyle: I don't have to do any more. They're all the same. I: No, they're not. What do you mean? Kyle: Well, look. Using C a l c u l a t o r s to Teach Algebra 145 He showed me h i s paper, on which he had w r i t t e n , i n the course of s o l v i n g for P, r , and t : n o o u n o o u t — — = r n o o u Kyle: See? They're all the same, so I'm not going to do any more. But how come you never ask us to find u? I: Well, it wouldn't he very realistic, would it? I mean, if I tell you t, of course you know what u is. And if I ask you to find t, of course I have to tell you what units I want the time in. So I never really have to ask you for u. Kyle: I made up a problem like that ... I: Did you? Kyle: ... but I couldn't solve for u. I: Why not? Kyle: 'Cause when you start with u you have to divide backwards. I: Good for you! I wasn't going to teach you how to do that kind until Grade 10. Kyle: [obviously pleased with his own acumen] Hey, wow, man! So how do you do it? I: You use one of the other equations you've already worked Using C a l c u l a t o r s to Teach Algebra 146 out. [Looking over his work] See, in these equations [I pointed to the ones he had derived] u doesn't come after a divide; it comes after a multiply, and you can multiply backwards. So you could solve these for u. But I'm not going to give you any questions like that. Kyle: OK . . . [to his neighbour] Hey, did you hear? I figured out a Grade 10 problem! Bob (Ma 10A - A) developed an a b i l i t y to see the s t r u c t u r e of an express ion that was unusual i n my students . I t began when he complained that he was bored and I chal lenged him to so lve the equations on that day's worksheets as I myself d i d , without w r i t i n g down e i t h e r of the two columns. Accord ing ly , asked to solve the fo l lowing equation for b p P = 1 + a 1 + a [ l + a ( b + 1 ) m - c ] d he wrote i n the order of operations thus 2 6 • 1 + a|~1 + a ( b + 1 ) m - c l d 8 J- 4 3 1 5 J 9 P = 1 + a 12 11 and then wrote P - 1 1 I / P 1/n 1 1/rr + d - 1 + c - 1 - 1 saying to himself as he wrote, "End up with P, so s t a r t with P, Using C a l c u l a t o r s to Teach Algebra 147 minus 1, d i v i d e d by a, brackets , y to the x 1 over p, p lus d , minus 1, d i v i d e d by a . . . " Here he s t a r t e d to wr i te the symbol " / " , but scratched i t out, saying "No, there ' s a minus there . . . and a p l u s , so I need a long one . . . brackets , y to the x one over n, p lus c, minus 1, long d i v i d e d by a, brackets . . . you know, you can p r a c t i c a l l y guarantee that i f there ' s a y to the x coming up, y o u ' l l need brackets . . . y to the x 1 over m, minus 1, equals b . " I: Practically guarantee? Bob: Seems like it. I: Why? Bob: 'Cause the calculator won't do anything before y to the x unless you put brackets. I: "Anything" or "practically anything"? Bob: Anything, I guess. I: So are you always going to need brackets? Bob: Yes . . . No, not if i t ' s a single letter before the y to the x. I: Good! That's worth remembering. (Notice again how Bob expressed h i s i n s i g h t in to the need for brackets i n c a l c u l a t o r language.) The equations I chal lenged Bob to solve i n t h i s way had only a s i n g l e l e t t e r as the object of each operat ion . Unfortunate ly for h i s marks, he then attempted to solve every equation, even those i n c l u d i n g groups and mul t i - s t ep s o l u t i o n s , i n the same way, and was not succes s fu l . As a r e s u l t , h i s f i n a l mark i n both Grade 10 and Grade 11 was B instead of A. I attempted on a number of Using C a l c u l a t o r s to Teach Algebra 148 occasions to persuade him to wri te down the columns at l ea s t for quizzes and exams, p o i n t i n g out ( t r u t h f u l l y ) that even I , a teacher, could not e a s i l y solve a mul t i - s t ep problem i n my head and would at l eas t wri te down the intermediate equations, but he always answered that he "should be able to do i t " and continued to t r y . Audrey (Ma 8A - A) saw perhaps more deeply i n t o the s t r u c t u r e of equations, as opposed to express ions, than any other Grade 8 student. For fun, I made up the fo l lowing equation for her c l a s s one day when some of them s a i d they had f i n i s h e d and were bored 1/h - P = M u and t o l d them to solve i t for x. They soon found (Mur + p ) h y = x and apprec iated the joke. Without any more help from me, some of them then s t a r t e d to t r y doing the same th ing with t h e i r own names, and severa l succeeded, apparently by t r i a l and e r r o r . The next day Audrey a r r i v e d i n c l a s s with a s i m i l a r problem already made up and asked i f she could put i t on the overhead p r o j e c t o r for everyone to so lve . I agreed, and she then t r i e d to e x p l a i n to the c l a s s how she had constructed the equat ion. Audrey: You have to make up the answer first. I: Then what do you do? Audrey: Then you reverse it. I: And what does that give you? Using C a l c u l a t o r s to Teach Algebra 149 Audrey: That gives you the question. I: When you've made up the answer, what do you solve it for? Audrey: [After a pause] The first letter in your name. Kakal (Ma 8A - A): I've got two k's in my name . . . Oh, I guess I should make the first one a capital. I: Why? Kakal: 'Cause if I don't you won't know which k I mean. I: And when you've got the question, you give it to somebody else. What do you tell them to solve for? Audrey: x. Apparently - to continue my cons truc t ion metaphor - i f you b u i l d and take down enough houses step by step, seeing only one step at a time, you begin to see more than one step at a time and f i n a l l y the whole s t r u c t u r e . The same th ing happens - as I can a t t e s t - i n fo l lowing an unfami l iar k n i t t i n g or bead-weaving p a t t e r n . Eventua l ly the un i t of the pat tern i s reified, to use S f a r d ' s word, and i t i s ready to become a u n i t i n a l a r g e r p a t t e r n . Groups: S t a t i c A lgebra ic Expressions The c l o s e s t students come i n my courses to reifying expressions i s i n t h e i r treatment of what I c a l l groups (see Appendix 4), i n which, at the very l e a s t , students have to consider a + b as "the answer to a + b ." Even before I speak of groups, however, I no t i ce a few students reifying numerical expressions for the purposes of Using C a l c u l a t o r s to Teach Algebra 150 c a l c u l a t o r s . For example, Kyle (Ma 8A - A) was c a l c u l a t i n g the answer to the fo l lowing: 3 5 3 " 6 . 9 - 4 On h i s (non-algebraic) c a l c u l a t o r , he had a lready punched out 6 . 9 - 4 M+ 3 -r MR MC x = = = = M+ 3 - MR MC M+ 3 -r when he stopped. I: Go on. Kyle: [After a pause] Oh . . . of course, that's in the memory. I: What is? Kyle: That [pointing with his pencil approximately towards the lower part of the question]. I: What exactly? Kyle: [In an exasperated tone] That! He very d e l i b e r a t e l y drew a heavy black l i n e around the express ion thus L 6- 9 - 4J Kyle: [sarcastically] OK? Got it now? I f I had been d e s c r i b i n g the same "thing" to him, I would have s a i d , "The answer to tha t ," as I drew the l i n e . By l eav ing out the i t a l i c i z e d words, Kyle had c l e a r l y made a "thing" of the e n t i r e express ion . Perhaps he could ignore i t s i n t e r n a l s t r u c t u r e because he had a lready performed those operat ions , turn ing the Using C a l c u l a t o r s to Teach Algebra 151 boxed express ion in to a s i n g l e number - c l e a r l y a "thing" - and s t o r i n g i t i n h i s c a l c u l a t o r ' s memory. I d e n t i f y i n g "what we have to keep together as a group" proved to be d i f f i c u l t for many students, who at f i r s t d i d not r e a l i z e that what c o n s t i t u t e d a group i n any given express ion depended on what v a r i a b l e they were s o l v i n g f o r . Hoping to he lp them see t h i s p o i n t , I made up a worksheet on which they had to so lve the same equation for a number of d i f f e r e n t v a r i a b l e s . For example, they had to so lve the fo l lowing equation f i r s t for a, then for b, then for c and then for d: a ( b + c ) + d = e Kelly (Ma 10A - C+): [in an exasperated tone] How do you know when it's a group and when you can break it up? I: Which question are you on? Kelly: This one [pointing to the one which asked her to solve the equation for a]. I said this is a group [sketching a circle around b + c] but Christine said it's this [sketching a circle around a(b + c)]. Which is it? I: It depends on what letter you're solving for. Kelly: [With surprise] It does? I: Yes. What's Christine solving for? Christine (Ma 10A - C+): d. I: Well, Christine's right for her question and you're right for your question. Kelly: How come? I: Put in the order of operations. Using C a l c u l a t o r s to Teach Algebra 152 K e l l y wrote a ( b + c ) + d = e 2 7 3 I: What are you starting with? Kelly: a I: OK; start with a. She wrote a J ; Now what? Kelly: Times, right? I: Right. She wrote a x J : Times what? Kelly: I don't know. I: That times was operation 2, right? Kelly: Right. I: Then any operation which has a lower number than 2 is already supposed to be done by the time you get to it, and that means it has to be kept in a group. Kelly: So b plus c is a group. I: Yes. Kelly: Not b plus c plus d. I: No, because the plus d is operation number three and three is supposed to be done after two, so you write plus d on the next line. Using C a l c u l a t o r s to Teach Algebra 153 K e l l y f i n i s h e d the problem c o r r e c t l y . Kelly: Don't go away. Let me do the one Christine was doing. She wrote d + Kelly: This [pointing to the + she had just written and looking back at the original equation] . . . is number 3. So [sketching a circle around a(b + c) in the original equation] all this has to be kept as a group. I: Right. Kelly: Wow! She f i n i s h e d that problem c o r r e c t l y as w e l l . Kelly: So if I start with a letter that's inside a group, the group gets broken up. I: Yes. Kelly: But if I start outside a group, it doesn't get broken up. I: Yes. What you call a group depends on which letter you're starting with. Kelly: Now I got it! In t h i s exchange, K e l l y learned to i d e n t i f y as a group any express ion i n s i d e which the operations had lower numbers than the operat ions outs ide . She showed no s igns of reifying groups, but rather i d e n t i f i e d them "opera t iona l ly ." In contras t , Larry (Ma 10A - C+), who always wrote very s lowly and neat ly , showed signs of reifying a l g e b r a i c expressions Using C a l c u l a t o r s to Teach Algebra 154 i n order to cut down on the amount of w r i t i n g he had to do. He was s o l v i n g the fo l lowing equation for v i : a i V i + a 2 v 2 = M a 2 v 2 - a 3 v 3 Larry: Can't I just put a square instead of writing a group out all the time? I: Show me what you mean. He had w r i t t e n V i x a i + a 2 v 2 - • M and now he pointed to the square. Larry: 'Cause the square is just like brackets anyway. I: Yes, that's OK, provided you don't forget what's inside the square. And of course you have to write it out in your final answer so I can tell. Larry: Can I put a box instead of this group, too [pointing to a2V2? I: You can, but won't you get mixed up about which box is which? Larry: I'll use a triangle for the other one. He eventual ly developed a system of drawing c i r c l e s , rectangles and t r i a n g l e s around groups i n the o r i g i n a l equation that he could see were going to remain unchanged. In the two columns he would draw only the shapes; i n the f i n a l equation he would wr i te out what the shapes contained or represented. Using C a l c u l a t o r s to Teach Algebra 155 Unfortunate ly , i n mu l t i - s t ep problems, he often attempted to avoid i d e n t i f y i n g the contents of the shape i n the intermediate equations, and thus became confused i n the l a t e r stages of the s o l u t i o n . L a r r y ' s reification of the expressions i n s i d e h i s shapes f i t s the d e s c r i p t i o n given by Sfard (1991) exac t ly : "Seeing a mathematical e n t i t y as an object means . . . being able to recognize the idea 'at a glance' and to manipulate i t as a whole, without going i n t o d e t a i l s . . . . " In contras t , K e l l y viewed a group as a "potent ia l ra ther than a c t u a l e n t i t y , which comes in to existence upon request i n a sequence of act ions" (Sfard, 1991, p . 4) . The fo l lowing two samples of students' work, which g ive c l e a r evidence of students' reification of groups, are t y p i c a l of a very large number of o thers . On the quiz from which they were d e r i v e d , for example, seven other students out of a c l a s s of 33 a l so l e f t out brackets i n intermediate s teps . Asked to solve the fo l lowing equation for x b b - c = a x - p + n A l i s o n (Ma 10A B) should have wr i t t en for the f i r s t step b -r (x - p + n) * b a - c + c - + c (x - p + n) = b x b x (x - p + n) b a but she l e f t out one set of brackets thus: Using C a l c u l a t o r s to Teach Algebra 156 -r ( x - p + n ) - c x b a a -r b + c x ( x - p + n ) b — + c ( x - p + n ) = b b However, at the beginning of the next step she wrote x - P + n a b + ° j u s t as i f she had not omitted the brackets . She ended up with the c o r r e c t expression for x, so I gave her f u l l marks. However, as I handed back her q u i z , I l a i d i t down on the desk i n f ront of her , pointed immediately to the r e s u l t of her f i r s t s tep, and asked her to "put i n the order of operat ions ." She wrote 4— + c ( x - p + n ) = b b 5 3 2 1 as I had expected. When I pointed out the discrepancy between t h i s order and what she had wr i t t en i n the next s tep, she at f i r s t appeared confused and then s a i d , d i smiss ing the problem, "Well , I guess I knew what I was doing yesterday!" S i m i l a r l y , Rosa (Ma 11A - A ) , asked to solve z n y t/s = b for y, wrote the fo l lowing (notice e s p e c i a l l y how she a l t erna te s between l eav ing out brackets and p u t t i n g them i n , even around the same express ion, but not i ce a l so her use of long h o r i z o n t a l l i n e s Using C a l c u l a t o r s to Teach Algebra 157 to separate "what she does" from "what she gets"): b b(t/S) + n/y = (e/x) z e/xz - n/y V t/S x t/S + n/y e / x n * y + b(t/S) e/x z e/ x" - b(t/S) n [e/x z- b(t/S)]y = n n n x e/xz - b(t/S) = y -r e/xz - b(t/S) n - b(t/S) I t seems that students mentally reify groups as they are w r i t i n g them, even i f they do not wri te i n the brackets that are necessary to i d e n t i f y the expression as a group i n the next s tep . Provided they read what they have wr i t t en immediately afterward, the group i s s t i l l reified i n t h e i r minds and they read what they meant, not the conventional i n t e r p r e t a t i o n of what they wrote. S i m i l a r l y , one of my f r i e n d s , who i s a newspaper e d i t o r , t o l d me that i t i s common, when he complains that a sentence i s not c l e a r , f or the w r i t e r of the sentence to read i t aloud with what he c a l l s "ora l punctuation" which makes the meaning c l e a r , l i k e the student who i s s a i d to have wr i t t en "John where Jane had had had had had had had had had had had the approval of the examiner" and to have read i t thus: "John, where Jane had had ' h a d , ' had had 'had had ' ; 'had had' had had the approval of the examiner." L i k e A l i s o n and Rosa, the w r i t e r knows what he or she meant when the Using C a l c u l a t o r s to Teach Algebra 158 sentence was wr i t t en and continues to read i t that way for some t ime. In p r a c t i c e , students w i l l e x p l i c i t l y use "ora l punctuation" to reify groups, as N i k k i (Ma 11A - C+) d i d when I asked her , from the front of the classroom, to t e l l me, from her desk, how to solve the fo l lowing equation for p . Nikki: Start with p ... I wrote P Nikki: Multiply by b plus c ... I d i d not know whether she wanted to me to wr i te P x b + c or P x ( b + c ) I: Tell me exactly what to write. Nikki: I told you. I: I'm not sure whether you mean times b plus c, on two lines, or times b plus c in brackets on one line. Nikki: I told you. I said times [short pause] b plus c [saying "b-plus-c" quickly as though it were one word]. One of the most s i g n i f i c a n t reifications by student occurred i n a new context when Matt (Ma 11A - A) had to solve the fo l lowing equation for x: Using C a l c u l a t o r s to Teach Algebra 159 a + b c - d + = z x - m y - b Matt: Can I start with this whole group [sketching a circle around "a + b"]? I: Yes. Matt: OK. I had not suggested t h i s technique to the c l a s s , having i d e n t i f i e d groups o p e r a t i o n a l l y , as having lower operat ion numbers than the operat ion jus t performed. Matt f i n i s h e d the problem on h i s own. Matt: But I got a different answer. [Looking at the answer sheet] No I didn't ... yes I did. I: Let me see. My answer was b + a - + m = x c - d 2 " T^> while Matt ' s answer was a + b ^~Ta 2 " 7 ^ Matt: [Before I could start to explain] Oh, I guess it doesn ' t make any difference. I: Why not? Matt: Because a plus b is the same as b plus a. I: Right. Matt: [Curiously] But why did you get b plus a? I: I was following my own rules very carefully. I suppose Using C a l c u l a t o r s to Teach Algebra 160 most students would call b the closest thing to x that you can start with, and so I started with b. But you started with a plus b, all together, so your answer looks a bit different. Matt wrote b + a Matt: So your way, b and a got separated. I: Right. Matt: But how did they get back together again? I: In the next step. See, my answer to the first step would be . . . I wrote I: Then I would go . . . I wrote x b - m + a - [] - a b See? Matt: [Slowly] I see. I l e f t him examining the quest ion . Matt: [After a while] I think my way's better. I: Why? Matt: 'Cause it doesn't take as many steps. Using C a l c u l a t o r s to Teach Algebra 161 I: Show me. Matt: Here ... He pointed to the end of the f i r s t column of h i s second s tep, which looked l i k e t h i s : ... 'cause I didn't have to break up this group [sketching a circle around a + b]. You had minus a and then ended up with b on another line. I: Yes. The bigger the group you start with, the fewer operations you have to reverse. Matt: 'Cause you just end up putting them back together again in the next step anyway. I: Right. Just make sure that you don't start with a group in the last step. Matt: Why not? Oh ... 'cause then you wouldn't end up with the letter you're supposed to. I t would be d i f f i c u l t to argue that such students have not taken the "leap" whereby "a process s o l i d i f i e s in to an objec t , i n t o a s t a t i c s tructure" and becomes "detached from the process that produced i t , " as Kieran s a i d (1992, p. 392). Nevertheless , mathematics educators must not forget the need for a "lengthy t r a n s i t i o n per iod i n moving from opera t iona l (procedural) to s t r u c t u r a l conceptions" (Kieran, 1992, p . 395). Nor can we forget people l i k e K e l l y , for whom the leap "is x - m X a + b Using C a l c u l a t o r s to Teach Algebra 162 inherent ly so d i f f i c u l t " that "the s t r u c t u r a l conception w i l l remain p r a c t i c a l l y out of reach whatever the teaching method" (Kieran, 1992, p . 412). Author i ty Behind "The Right Answer" Sometimes students know, simply and beyond argument, even beyond the quest ion of marks, that what they have w r i t t e n or s a i d i s t rue ; they know that they are i n touch with a r e a l i t y or a t r u t h which i s independent of the teacher: i t does not depend on the teacher for v a l i d a t i o n . Occas iona l ly such students r e v e a l t h e i r inner c e r t a i n t y by arguing with the teacher, as the fo l lowing anecdote i l l u s t r a t e s . I had been teaching my f i r s t Grade 8 c lasses that they could do add i t ions and m u l t i p l i c a t i o n s backwards, but not subtrac t ions and d i v i s i o n s . The ir worksheet looked l i k e t h i s : Without using your calculator, t e l l whether you w i l l get the r i g h t answer (R) or the wrong answer (W) to the question i f you press the following buttons on your c a l c u l a t o r . Question Buttons Answer R or W 1. 142.45 + 35.2 35.2 + 142.45 = 142.45 + 35.2 = Kyle (Ma 8A - A) got zero on the q u i z . J ; [After handing it back to him] I thought you understood this! Kyle: I do! This [stabbing his pencil on to the paper for emphasis] ... is ... one ... hundred ... percent ... correct. I: It's not; it's one hundred percent wrong. Using C a l c u l a t o r s to Teach Algebra 163 Kyle: [Stabbing at the paper] If this is the question, and I press these buttons, I'll get the right answer. I: So why did you put W? Kyle: [Slowly, as if trying to teach me] W ... R ... I... T ...E. I: [Wondering whether he was trying to hoax me] So how do you spell wrong? Kyle: [In the same manner] R ... 0 ... N ... G ... But [as the bell for the end of class rang] I don't care. I know I got it all right. I: I see what the problem is. But didn't you notice when you checked the answers on the back of the worksheet that you were getting them wrong? Kyle: I didn't look. I knew I was getting them right. I: Well, I'll give you 100% for the math - but you'd get zero for spelling. Kyle: I don't care. [Striking a pose] I know, and God knows. (The fo l lowing year, I changed "R" and "W" on the worksheet to check marks and x ' s . ) Annoying as such episodes may be, teachers should value them as s igns that the student has t r u l y learned whatever i s i n quest ion and no longer needs the teacher. Science teachers have the same experience when students t e l l them what they a c t u a l l y observed i n an experiment instead of asking , "Is that what's supposed to happen?" Using C a l c u l a t o r s to Teach Algebra 164 Conclus ion C a l c u l a t o r s can help students to s h i f t t h e i r focus from numbers to operat ions and to make the t r a n s i t i o n from numbers to l e t t e r s which i s demanded as they move from ar i thmet i c to a lgebra . Moreover, c a l c u l a t o r s can help students develop an image of a v a r i a b l e as an e n t i t y whose numerical value can vary . Moreover, the language of c a l c u l a t o r s as I employ i t i n s o l v i n g equations by the r e v e r s a l method can enable students to understand and descr ibe v e r b a l l y the s t ruc ture of a l g e b r a i c expressions and can even help some students to make the leap c a l l e d reification. Using C a l c u l a t o r s to Teach Algebra 165 Chapter 8 Conclusion and Consequences I share with other teachers what Pimm (1995) descr ibes as a "fear of apparent s o p h i s t i c a t i o n of performance unrooted i n understanding, and the perennia l de s i re J . . to be able to read comprehension from success fu l p r a c t i c e " (p. 94). Accord ing ly , i n "regular" mathematics c lasses I have t r i e d to teach for understanding, not jus t p r a c t i c e . On the whole I have f a i l e d , not only to teach the majori ty of my students to understand, but a l so even to complete the c u r r i c u l u m . I remember we l l the p lea u t tered by a Mathematics 11 student to the e f f ec t that she was going to be a nurse, that she needed c r e d i t for Mathematics 11 to enter nurs ing schoo l , and that p lease , would I stop t r y i n g to make her understand mathematics and jus t t e l l her what she had to do to pass. In p r a c t i c e , most h igh-school mathematics teachers make, to var ious degrees, the compromise t h i s student was p lead ing f o r . The r e s u l t i s descr ibed by Kieran (1992): In r e p o r t i n g the r e s u l t s of the fourth mathematics assessment of seventh- and e leventh- grade U . S . students by the Nat ional Assessment of Educat ional Progress (NAEP), Brown et a l . (1988) concluded that "Secondary school students genera l ly seem to have some knowledge of bas ic algebra and geometric concepts and s k i l l s . However, the r e s u l t s of t h i s assessment i n d i c a t e , as the r e s u l t s of past assessments have, that students often are not able to apply t h i s knowledge i n problem-solv ing s i t u a t i o n s , nor do they appear to understand many of the s tructures under ly ing these mathematical concepts and s k i l l s (pp. 346-347)." However, to cover t h e i r lack of understanding, i t Using C a l c u l a t o r s to Teach Algebra 166 appears that students r e s o r t to memorizing r u l e s and procedures and they eventual ly come to be l i eve that t h i s a c t i v i t y represents the essence of a lgebra . Brown et a l . reported that a large majori ty of the students i n the NAEP study f e l t that mathematics i s ru l e -based , and about h a l f considered that l e a r n i n g mathematics i s mostly memorizing. These f indings are not r e s t r i c t e d to the NAEP eva lua t ion . They have been reported i n countless s tudies conducted i n other c o u n t r i e s , (p. 390) Who needs to be able to do h igh-school a lgebra as i t i s taught i n the "regular" courses? Not nurses, typese t t ers , or bankers. Not average c i t i z e n s , who do not even balance t h e i r cheque books, count t h e i r change, or f i l l out t h e i r own income tax r e t u r n s . C e r t a i n l y not students for whom the essence of a lgebra i s "memorizing r u l e s and procedures ," who f e e l that "mathematics i s rule-based" and consider l earn ing mathematics to be "mostly memorizing" (Kieran, 1995, p. 390). C e r t a i n l y not a newspaper layout person I know, who c a l c u l a t e s the percentage of i t s o r i g i n a l s i z e to which a photograph should be reshot i n order to f i l l the space a l l o t t e d on something that looks l i k e a c i r c u l a r s l i d e r u l e (I have never a c t u a l l y examined i t ) . On one occas ion, her i n s t r u c t i o n s to the photographer who was going to do the reshoot ing read "50%, then 92%" instead of the usual s i n g l e percentage. When I i n q u i r e d why, he explained that the o r i g i n a l photograph had been so large and the space a l l o t t e d so small that she had s a i d the two numbers would not f i t on her s l i d e r u l e . She had not known what to do, so he had suggested that she halve the s i z e of the o r i g i n a l f i r s t . "Doesn't she r e a l i z e that t h i s i s 46%?" I asked him. "Probably Using C a l c u l a t o r s to Teach Algebra 167 not ," he r e p l i e d . "She probably thinks I ' l l shoot i t f i r s t at 50% and then reshoot the r e s u l t at 92%." C e r t a i n l y bankers do not need high school a lgebra . As treasurer of the North Vancouver High School Education Foundation, I have to c a l c u l a t e the simple i n t e r e s t I can expect to be p a i d i n t o our account from some t h i r t y - t w o investments i n about twelve f i n a n c i a l i n s t i t u t i o n s annual ly . Eleven of them always pay exact ly the amount I expect; the twel f th i s always s l i g h t l y too high or too low, apparently at random, by up to two percent of what I expect, and no one i n the bank or i t s head o f f i c e s knows why. Handy (1994) probes the meaning of i n t e l l i g e n c e and l i s t s nine forms - f a c t u a l , a n a l y t i c a l , l i n g u i s t i c , s p a t i a l , mus ica l , p r a c t i c a l , p h y s i c a l , i n t u i t i v e , and in terpersona l (pp. 204-205) -none of which, he says, i s "necessari ly connected with any other" (p. 203). When I read out t h i s l i s t and gave examples of each k ind of i n t e l l i g e n c e to students i n two Mathematics 8A c lasses and a Mathematics 9A c l a s s i n October 1994, the students were i n t e r e s t e d , but they had only two d i s t i n c t responses - both of them, I th ink , t r a g i c : "Can I have a copy of that l i s t to show my parents?" and "What i f you don't have any of those?" Handy says, of a n a l y t i c a l i n t e l l i g e n c e , "When t h i s i n t e l l i g e n c e i s combined with f a c t u a l i n t e l l i g e n c e , examinations come easy. When we descr ibe someone as an i n t e l l e c t u a l , i t i s often t h i s combination we have i n mind" (p. 204) - that i s , a combination of jus t two of the n ine . Using C a l c u l a t o r s to. Teach Algebra 168 I myself do not shine at a t h l e t i c s or a r t . Yet I never f e l t put down i n school because of these i n a b i l i t i e s . I t must be very d i f f e r e n t for students who do not shine i n mathematics, i f one can judge from the excuses made by adul ts for lack of success i n mathematics: "I had a bad teacher i n Grade F i v e , " "I missed a s ec t i on of Grade Nine and never understood anything af terward," "I never saw how I could use math." How often do people admit openly, without shame or excuse, that they "cannot do math?" One of the people who does i s my school p r i n c i p a l , who t o l d me that he had simply memorized h i s way through mathematics courses . With h i s permiss ion, I t e l l my students at the beginning of each course that some otherwise i n t e l l i g e n t people are not good at mathematics and c i t e him as an example. I t ev ident ly makes an impression on them; he t e l l s me that a s i g n i f i c a n t number l a t e r mention to him what I have t o l d them. I exp la in to my physics students that they cannot get A i n my courses unless they can apply the p r i n c i p l e s they have learned to so lve problems they have not seen before, because phys ics i s not p r i m a r i l y a "memory" subject , but an "understanding" or "thinking" subjec t . Although i t i s a p r e s t i g i o u s subject , students know that they are not a l l expected to be good at phys i c s . C r e d i t i n Physics 11 or 12 i s not requ ired for h igh-school graduat ion or for fur ther t r a i n i n g except i n r e l a t e d f i e l d s . I f students cannot succeed i n phys ics , they need not take the courses . But students are requ ired to pass Mathematics 8-11 courses i n order to graduate from high school i n B r i t i s h Columbia. The Using C a l c u l a t o r s to Teach Algebra 169 i m p l i c a t i o n i s that any "normal" person can succeed i n the "regular" math courses (despite the fac t that a large percentage of them simply memorize r u l e s ) . I f we want our mathematics courses to be p r i m a r i l y "thinking" courses and at the same time want them to be passable by students who c l e a r l y (on a l l other grounds) deserve to graduate from h igh schoo l , then we must al low students to work at the l e v e l at which they can th ink mathematically and accla im them for doing so. For example, when one of my Grade 9 students worked out for h e r s e l f that i f she knew the p r i c e of one item, she could "add two more on" to f i n d the p r i c e of three , she was t h i n k i n g mathematical ly . I t would be p o i n t l e s s to argue that a Grade Nine student should be able to do be t ter than that . To t e l l her to "mult iply by three" would be to subs t i tu te my r u l e for her t h i n k i n g . Conclus ion 1. I t i s pos s ib l e to s h i f t the a t t e n t i o n of students from numbers to operations by the use of e l e c t r o n i c c a l c u l a t o r s . 2. I t i s p o s s i b l e to s h i f t the a t t e n t i o n of students from operat ions to the o v e r a l l s t ruc ture of a l g e b r a i c expressions and equations by the use of e l e c t r o n i c c a l c u l a t o r s . 3. The responses of students who have been taught a lgebra i n c a l c u l a t o r language show that they understand a lgebra . The mathematics courses I have designed use c a l c u l a t o r language, which i s a s p e c i a l i z e d form of the language of Using C a l c u l a t o r s to Teach Algebra 170 a r i t h m e t i c , and continue to use t h i s language i n s o l v i n g equat ions . Students who are unsuccessful i n "regular" mathematics courses can perform t h i s mathematics s u c c e s s f u l l y . Moreover, they understand i t and can v e r b a l i z e t h e i r own mathematical i n s i g h t s . They can even create i t . Despite the fac t that the emphasis of my courses i s on success fu l p r a c t i c e , students from a l l grades and a l l l e v e l s of achievements show by what they wri te and say spontaneously that they have an understanding of what they are doing . Moreover, they of ten know when they have "done i t r i g h t " without wai t ing for my approval -evidence that they do not think they are simply obeying a r b i t r a r y , meaningless r u l e s . The experience of students who take my mathematics courses i s summed up i n the fo l lowing l e t t e r , wr i t t en to me by a g i r l a f u l l semester a f t e r she had f i n i s h e d Mathematics 8A (the s p e l l i n g and punctuation are hers ) : Dear Miss Murray: How are you? I'm doing f i n e . My second semester has been a l r i g h t . Even though I havn't been a l l that good I would l i k e to thank you, for what you d i d for me. I'm t a l k i n g about making math look so easy. I had a r e a l good time l a s t semester i n math c l a s s . It was on of the c lasses , I looked forward to going too, l a s t semester. I a l so d i d hate math before , a l l those numbers seemed to be bor ing , for me. When I f i r s t came in to your c l a s s , I thought i t would be a long and bor ing c l a s s and semester. A f t e r a couple of c l a s se s , I s t a r t e d understanding i t . Then i t s t a r t e d becoming s imple . I never thought that I would ever get a B i n math. I always got C or lower. I would r e a l l y l i k e to thank you for a l l your he lp . Using C a l c u l a t o r s to Teach Algebra 171 Consequences Mathematics as I teach i t , which may be c a l l e d opera t iona l a lgebra , can be o f fered to students as an end i n i t s e l f ; that i s , as mathematical a c t i v i t y that i s worth engaging i n for i t s own sake, l i k e s k i i n g , because i t i s enjoyable and because, l i k e any other a b i l i t y , e s p e c i a l l y t r a i n e d a b i l i t y , i t enhances l i f e . I t appears p o s s i b l e that , with further research, my success could be genera l i zed to other students who have had l i t t l e or no success i n previous "regular" mathematics courses . There i s much less evidence that my courses could serve as a br idge from ar i thmet i c to "regular" a lgebra . F i r s t , I have never attempted to use them i n t h i s way. Second, I know of only two students who have switched from my courses to "regular" courses . Both of them were very success fu l i n Mathematics 8A, one f i n i s h i n g with 94% and the other with 83%. The f i r s t one has s ince passed Mathematics 9, 10, and 11 with marks s l i g h t l y above 50%, while the second has bare ly passed Mathematics 9 and 10, having f a i l e d each course once before he passed i t , and i s c u r r e n t l y f a i l i n g Mathematics 11. Nevertheless , taken i n conjunct ion with S f a r d ' s (1991) a n a l y s i s of reification, descr ibed i n Chapter 3, the s igns of reification among some of my students suggest that fur ther research along these l i n e s might be p r o f i t a b l e . Accord ing ly , I c la im that the r e s u l t s of my case study have consequences f o r : 1. research in to how mathematics i s learned, taught, Using C a l c u l a t o r s to Teach Algebra 172 conceptual ized , and performed - a top ic which i n t e r e s t s some educators for i t s own sake, apart from i t s a p p l i c a t i o n to teaching . 2. the design of high school mathematics courses for the unmathematical, who can l e a r n algebra without reification. 3. the design of high school mathematics course for the mathematical, who might be able to use a lgebra without reification to br idge the gap between ar i thmet i c and "regu lar ," or reified, a lgebra . Using C a l c u l a t o r s to Teach Algebra 173 Bibl iography Augustine of Hippo (1993). Tracta te 29. In J . R e t t i g (Trans . ) The Fathers of the Church (Vo l . 88, pp. 14-21). Washington: The C a t h o l i c U n i v e r s i t y of America Press . ( O r i g i n a l work publ i shed c . 416) Balachef f , N. (1988). Aspects of Proof i n P u p i l s ' P r a c t i c e of School Mathematics. In D. Pimm ( E d . ) : Mathematics, Teachers, and Children (pp. 216-235). London: Hodder and Stoughton. Behr, M . , Erlwanger S . , & Nicho l s , E . (1976). How C h i l d r e n View E q u a l i t y Sentences. PMDC Technical Report 3. Ta l lahassee : F l o r i d a State U n i v e r s i t y . Compact Edition of the Oxford English Dictionary (1980). Oxford: Oxford U n i v e r s i t y Press . Computer Mathematics: An Introduction. (1985). New York, NY: New York C i t y Board of Educat ion. Davis , P . , & Hersh, R . , (1981). The Mathematical Experience. Boston: Birkhauser . Fauve l , J . & Gray, J . (Eds) . (1987). The History of Mathematics: a Reader. London: Macmil lan. Fowler, H.W. (1984). A Dictionary of Modern English Usage. H e r t f o r d s h i r e : Omega Books L t d . G i a n c o l i , D. (1980). Physics: Principles With Applications. Toronto: P r e n t i c e - H a l l of Canada L t d . Handy, C. (1994). The Age of Paradox. Boston: Harvard Business School Press . Herscovics , N . , & K i e r a n , C. (1980). Construct ing Meaning for the Concept of Equat ion. Mathematics Teacher 73, 572-80. K i e r a n , C. (1989). The E a r l y Learning of Algebra: A S t r u c t u r a l Perspec t ive . In Wagner, S . , & K i e r a n , C. ( E d s . ) , Research in the Learning and Teaching of Algebra (pp. 33-56). V i r g i n i a : Nat iona l Counc i l of Teachers of Mathematics. K i e r a n , C. (1992). The Learning and Teaching of School A lgebra . In Grouws, D. ( E d . ) , Handbook of Research on Mathematics Teaching and Learning pp. 390-419. Toronto: Maxwell Macmillan Canada. Lakatos, I . (1976). Proofs and Refutations: the Logic of Mathematical Discovery. Cambridge: Cambridge U n i v e r s i t y Press . Using C a l c u l a t o r s to Teach Algebra 174 Lewis, C. (1952). Mere Christianity. Glasgow: Wi l l i am C o l l i n s Sons & Co. L t d . Lewis, C. (1969). V a r i a t i o n i n Shakespeare and Others . In Hooper, W. ( E d . ) , Selected Literary Essays pp. 74-87. Cambridge: Cambridge U n i v e r s i t y Press . Pimm, D. (1987). Speaking Mathematically: Communication in Mathematics Classrooms. London: Routledge and K. P a u l . Pimm, D. (1995). Symbols and Meanings in School Mathematics. London: Rout1edge. Rotman, J . (1991). Arithmetic: Prerequisite to Algebra? Paper presented at the annual convention of the American Mathematical A s s o c i a t i o n of Two-Year Col leges , Sea t t l e , WA Schoenfeld, A. & A r c a v i , A. (1988). On the Meaning of V a r i a b l e . Mathematics. Teacher 80, 420-427. S f a r d , A. (1991). On the Dual Nature of Mathematical Conceptions: R e f l e c t i o n s on Objects and Processes as D i f f e r e n t Sides of the Same C o i n . Educational Studies in Mathematics, 22, 1-36. Shukla, K. ( E d . ) . (1976). Aryabhatiya of Aryabhata. New D e l h i : Indian Nat ional Science Academy. Sutherland, R. (1991). Some Unanswered Research Questions on the Teaching and Learning of Algebra . For the Learning of Mathematics 11(3), 40-46. von G l a s e r s f e l d , E . (1988). Cognition, Construction of Knowledge, and Teaching. Washington DC: Nat ional Science Foundation. Using C a l c u l a t o r s to Teach Algebra 175 Appendix 1 Mathematics 11A F i n a l Examination bas i c 108 * 60 ** 92 A 221.0 B 142.8 C+ 91.8 C 75.6 C - 54.0 D 27.0 E 0 Simple Interest I = P r t / u 1 0 0 A = P + I Compound Interest r/n A = 1 0 0 (nt/u) + 1 I P Lb oo 'j I = A - P 1 MOO Saving 00m m/n + 1 + —-—x Financing P = "F (C - D ) r / 1 0 0 ( n - m ) 2 0 0 - m 1 r/n Too + 1 (nt/u) - 1 I = pmt/u - C + D . [ [ ( n . . ) | ^ . . ] l ^ + c . D ] [ [ W | + l ] - ' - ' . 1 ] + c . D A = I + C (nT/u) P p r i n c i p a l d deposi t C cost D downpayment r i n t e r e s t ra te t t o t a l time T time a lready spent paying u how many time u n i t s per year n how many times i n t e r e s t i s compounded per year m how many deposi ts or payments per year p payment I i n t e r e s t (f inance charge) A t o t a l amount a amount s t i l l owing e equivalent simple i n t e r e s t ra te Using C a l c u l a t o r s to Teach Algebra 176 Ins truc t ions 1 . Round o f f i n t e r e s t rates to two decimal p laces and times to zero decimal p laces . 2. Show your working as neat ly as you can i n the space prov ided . 3. In any quest ion where you use a program i n your c a l c u l a t o r , you must show how to change the subject , i f that would be necessary, i f you want f u l l marks. 4. Don't forget $ s igns and % s igns . Part 1 Do the problem on your c a l c u l a t o r and wri te down the answer to the number of decimal places i n d i c a t e d i n the brackets . 1 . 5 2.3 1.81 + 1.5 2 . 7J (13.75 - 12.702) 13.6 - 1.35 (3) (2) *3. 1.1 + 2.1 [1.5 + 0.7[0.85 + 0.1(1 + 1/14)"]] (0) c*4- (103 + 1 ,4) 2 1 .3 _ 1_J5 1 .7 1.2 1.5 + 1.6 1 .407 (0) 0.698 Part 2 Evaluate the v a r i a b l e whose value i s not given and round o f f your answer to two decimal p laces . 5 b A = — + c(a - b) A = 15, a = 3.4, b = 1.2, q = 15 q 6. g - h c = ^ — c = 132.56, g = 1.4, h = 4.5 b *7. a + 2b a - 2b r = •=: + ^ ^ r = 2.953, a = 2, b = 3.4, d = 9 b - c b - d Using C a l c u l a t o r s to Teach Algebra 177 Part 3 9. I f you invest $200.00 for 8 mo and end up with $216.45, (a) how much i n t e r e s t do you make? and (b) at what i n t e r e s t ra te i s your money deposited? 10. I f you leave $750.00 i n the bank at 15.00% from May 1, 1990 to Jan 25, 1991, (a) how much do you make? and (b) how much do you end up with? 11. Jenny puts $1500.00 i n a savings account paying 7.25% compounded monthly. (a) How much does she have i n the account a f t e r 6 mo? (b) How much i n t e r e s t has she made? 12. F ind the present value of a T - b i l l that w i l l be worth $3000.00 i n 256 da i f i t pays 8.75% compounded d a i l y . 13. F ind the equivalent simple i n t e r e s t ra te of 15.25% compounded weekly. 14. A c e r t a i n savings account pays 9.50% compounded monthly. How much must you pay in to the account every h a l f month i f you want to end up with $25 000.00 i n 5 a? 15. You pay $50.00 every month in to your savings account, which pays you 8.75% compounded q u a r t e r l y . How much do you have i n t o t a l by the end of 10a? 16. You borrow $5 000.00 and pay i t back i n monthly ins ta l lments of $250.00 each. I f you are being charged 15.00% i n t e r e s t compounded monthly, c a l c u l a t e how much you s t i l l owe when you have been paying for 15 mo. 17. You make a downpayment of $25 000.00 on a house which costs $150 000.00. You pay the re s t i n monthly ins ta l lments over 24 a at an i n t e r e s t ra te of 12.50% compounded semiannually . F ind (a) the s i z e of each payment, (b) the f inance charge, and (c) the t o t a l b i l l . *18. You deposi t $350.00 i n the bank on Feb 1, 1992. You get 12.00% i n t e r e s t and you want to make $50.00. On what date should you take the money out? *19. You can buy a l o t for $80 000.00 or $10 000.00 now and another $100 000.00 i n 5 a. If the banks are paying 11.50% compounded q u a r t e r l y , which i s the bet ter deal? How much do you save now by taking the bet ter deal? *20. Show which i n t e r e s t rate on your savings account i s be t t er over 1 a - 10.25% compounded semiannually or 10.50% compounded d a i l y . Using C a l c u l a t o r s to Teach Algebra 178 *21 . You buy a house worth $185 000.00 for $35 000.00 down. You amortize your mortgage over 25 a, making monthly payments and being charged 14.00% i n t e r e s t compounded semiannually. How much do you s t i l l owe a f t er 10a? **22. A f t e r 7 a i n an account which compounds i n t e r e s t q u a r t e r l y , your p r i n c i p a l has t r i p l e d . F ind the rate at which i t was inves ted . **23. How many years and days does $3000.00 have to be invested at 10.25% compounded d a i l y to make $1200.00 i n t e r e s t ? **24. What i n t e r e s t rate compounded semiannually i s equiva lent to 12.00% compounded monthly? **25. You pay $750.00 twice a month to pay back a loan of $120 000.00 on which the i n t e r e s t ra te i s 15.00% compounded monthly. F ind how many years and months i t takes you to (a) reduce your debt to $60 000.00 and (b) (b) pay back the loan . **26. I f you deposi t $150.00 every h a l f month i n an account that pays 8.75% compounded monthly, how many years and months does i t take to accumulate $25 000.00? Using C a l c u l a t o r s to Teach Algebra 179 Appendix 2 Students' Records i n Previous Mathematics Courses Tables 2-B, 2-C, and 2-D conta in the high school mathematics records of three c lasses of Mathematics 11A students , one taught by me, the others by another teacher i n my school us ing my m a t e r i a l s . Students' records are d i v i d e d from each other by h o r i z o n t a l l i n e s and each designated by a d i f f e r e n t l e t t e r . These records show the lack of success experienced by these students i n previous courses . F a i l i n g grades are p r i n t e d i n b o l d type and the symbols • , v, * , and * are used to designate courses other than the "regular" mathematics courses . L e t t e r grades are to be i n t e r p r e t e d according to the fo l lowing scheme, which i s standard i n B r i t i s h Columbia schools : 86% £ A £ 100% 73% < B < 86% 67% < C+ < 73% 60% < C < 67% 50% < P < 60% 0% £ F < 50% Table 2-A expla ins the meanings of the other symbols. Table 2-A Symbols Used i n Students' Records Symbol Meaning I course work incomplete N no mark assigned W , withdrawn from course SG standing granted F-»P, SG-»P second mark obtained at summer school • in troductory course i n t e r p o l a t e d between "regular" courses V modified course * taken at an a l t e r n a t i v e school * for "spec ia l" students Using Calculators to Teach Algebra 180 Table 2-B Records of Students Taught By Terry Baker Between September 1991 and January 1992 Grade 8 Grade 9 Grade 10 Grade 11 Ma 11A A Ma 8 B Ma 9 B Ma 10 C c+ B Ma 8 F Ma 9 P Ma 10 P •IMa 11 F c+ C Ma 8 C VMa 9C F Ma 10 C B D Ma 8 53% VMa 9AB N VMa 10A P B E VMa 8C C+ VMa 9A P VMa 10A P P F VMa 8C P Ma 8 F Ma 8 P Ma 9 F Ma 9 F VMa 10A C B G Ma 8 P Ma 9 P VMa 10B F Ma 10 C B H VMa 8C C VMa 9C C VMa 10A P Ma 11A F c+ I Ma 8 F VMa 10A P C+ J Ma 8 F Ma 8 F VMa 9C C VMa 10A C P K Ma 11 F Ma 11 F Ma 11 F B L Ma 8 P Ma 9 P Ma 10 45* •IMa11 F C+ M •IMa8 P VMa 9A C C N Ma 9 F Ma 9 F-»P VMa 10A P F 0 Ma 8 SG Ma 9 SG VMa 10A P Ma 10 F B P VMa 8C C VMa 9A C VMa 10A P C+ Q Ma 8 P Ma 9 P Ma 10 F Ma 10 F C+ R •IMa8 C VMa 9A C VMa 10A P C S Ma8 61% Ma 9 40% Ma 10 P Ma 11 F A T VMa 8R A Ma 8 P VMa 9AB P VMa 10A P B U Ma 8 F Ma 8 P VMa 9AB P VMa 10A P B V C+ W VMa 8C C VMa 9C P VMa 10A SG Ma 11A F F X Ma 8 F Ma 8 P Ma 9 F VMa 10A F F Y Ma 8 F Ma 8 P Ma 9 P VMa 10A C+ B Z Ma 8 F Ma 8 P Ma 9 F Ma 9 P Ma 10 F VMa 10A C C+ a VMa 8C P Ma 8 SG VMa 9A F VMa 9A F VMa 10A C C+ Using C a l c u l a t o r s to Teach Algebra 181 Table 2-C Records of Students Taught By Terry Baker Between September 1992 and January 1993 Grade 8 Grade 9 Grade 10 Grade 11 Ma 11A A Ma 8 B Ma 9 C+ Ma 10 F Ma 10 P C+ B •Ma 7/8 C Ma 8 F Ma 8 P VMa 9A SG VMa 10A C+ C C C+ D VMa 8C P VMa 9A F VMa 9A F VMa 10A P C+ E Ma 8 F VMa 10A SG F F Ma 8 C Ma 9 F Ma 9 I Ma 9 W VMa 10A B c+ G VMa 10A SG P H Ma 8 P Ma 9 SG VMa 10A C c+ I Ma 8 50% VMa 9A 66% VMa 10A 46% •IMa11 F B J Ma 8 C Ma 9 P VMa 10A B B K VMa 10A C C+ L Ma 8 SG->P VMa 9A P VMa 10A P Ma 11A F C+ M Ma 8 B Ma 9 P VMa 10B F C+ N Ma 8 C Ma 9 F VMa 10A B B 0 Ma 8 A VMa 9A A B P •Ma 7/8 F Ma 8 B Ma 9 P VMa 10A F VMa 10A P C+ Q •Ma 7/8 F Ma 8 F VMa 9AB F VMa 9AB C VMa. 10A P C R Ma 8 F Ma 8 F Ma 9 F VMa 10A C+ C S Ma 8 F Ma 8 F VMa 9A F VMa 9A P B T Ma 9 P B U VMa 8A 53% Ma 8 46% Ma 8 63% Ma 9 F > Ma 9 P Ma 10 F C+ V Ma 10 F Ma 10 F Ma 10 F C+ w Ma 8 F Ma 8 C Ma 9 F Ma 9 P Ma 10 F Ma 10 F VMa 10A C+ C+ X Ma 8 C Ma 9 F VMa 10A F •Ma 10K B+ C+ Y •Ma 7/8 B Ma 8 C VMa 9B C VMa 10B F Ma 10 F VMa 10A P B Z Ma 8 P Ma 9 F->P Ma 10 F P a Ma 8 P VMa 9B F VMa 10A C C+ , Using C a l c u l a t o r s to Teach Algebra 182 Table 2 -D Records of Students Taught by Maureen Murphy Between January 1993 and June 1993 Grade 8 Grade 9 Grade 10 Grade 11 Ma 11A A •Ma 7/8 Ma 8 C+ r Ma 9 F C+ B Ma 8 p VMa 9B F-+P VMa 10A C C C C+ D Ma 8 68% •Ma 9K A VMa 10A C+ C+ E *SkBdMa2 C+ •SkBdMa B •SkBdMa C B F Ma 8 P Ma 9 P Ma 10 F B G B H Ma 8 VMa 8T F N VMa 9A F VMa 10A F B I VMa 8ME 71% VMa 9 A 50% VMa 10A 14% F J Ma 8 F VMa 10A C B K •Ma7/8 Ma 8 C F-»P Ma 9 F • VMa 10A P C+ L •Ma 7/8 Ma 8 C+ C Ma 9 P VMa 10A C+ B M *SkBdMa2 C+ *SkBdMa B *SkBdMa B A N •Ma 7/8 Ma 8 Ma 8 C F F VMa 9 A C+ VMa 10A P C+ 0 Ma 8 P VMa 9 A C VMa 10A C C+ P Ma 8 P VMa 9B P VMa 10B P Ma 11 F A Q Ma 9 Ma 9 Ma 9 F F P Ma 10 Ma 10 F F C+ R Ma 8 Ma 8 F P Ma 9 F-*P C+ S Ma 9 C C+ T Ma 8 30% Ma 9 C+ VMa 10A N Ma 11A F C+ U •IMa 8 P VMa 9A C •Ma K P B V VMa 9A C+ VMa 10A C B W Ma 8 C+ VMa 9A C+ VMa 10A B C+ X Ma 8 B Ma 9 P Ma 10 P A Y Ma 8 F VMa 9 A P VMa 10A C+ B Z VMa 8R Ma 8 F F VMa 9AB F C+ a •Ma 7/8 Ma 8 C+ P Ma 9 P Ma 10 F B 183 Appendix 3 Use of the E l e c t r o n i c C a l c u l a t o r In order to make the reader f a m i l i a r with my c a l c u l a t o r language, I here present the lessons as I would present them to the students i n wr i t t en form. In p r a c t i c e , I present them o r a l l y , and consequently they involve a great deal of quest ioning and r e p e t i t i o n . Whenever i t has seemed appropriate or necessary to e x p l a i n to the mathematical reader the reasoning behind my i n s t r u c t i o n s , I have inc luded i t i n i t a l i c s . Order of Operations What i s the answer to the fo l lowing problem? 2 + 5 x 3 I f you do the a d d i t i o n f i r s t , you get 21. I f you do the m u l t i p l i c a t i o n f i r s t , you get 17. However, mathematicians do not l i k e to have more than one r i g h t answer, so they have agreed among themselves always to do a m u l t i p l i c a t i o n before an a d d i t i o n . I f mathematicians want you to do the a d d i t i o n before the m u l t i p l i c a t i o n , they put brackets around the + s ign and the two numbers that are being added, thus: (2 + 5) x 3 Mathematicians place brackets around an operat ion to make us do one operat ion before another when we would not otherwise do so. Brackets are not operat ions; we cannot "do" brackets . Rather, Using C a l c u l a t o r s to Teach Algebra 184 brackets t e l l us that we must do the operat ions i n s i d e them before the operat ions outs ide them. Nor do brackets mean " m u l t i p l y . " Mathematicians have had to make a number of other r u l e s l i k e the one about a d d i t i o n and m u l t i p l i c a t i o n i n order to ensure that every problem has only one r i g h t answer. Mathematicians c a l l anything you can do to a number, l i k e add, subtrac t , m u l t i p l y , d i v i d e , or r a i s e to a power, an operation. You have to obey the r u l e s about the order of two operat ions when the two operations involve the same number, as the + and the x do i n the example above: they both invo lve the 5. In other words, you have to obey the ru le s when two operat ions are adjacent, or next to each other . However, you do not have to obey the r u l e s when two operations have another operat ion between them. For example, i n 2 x 3 + 10 2 i t does not matter whether you do the m u l t i p l i c a t i o n f i r s t or the power f i r s t ; as long as you do them both before the + you w i l l get the same answer, 106. J have found that students interpret BEDMAS, an acronym for brackets, exponents, division, multiplication, addition, subtraction, which they have usually been taught by the time they reach Grade 11, to mean that in any given expression there is a unique permitted order, determined by going through the problem from left to right and top to bottom "doing the brackets," then going through the problem again the same way doing the multiplications, etc. Using C a l c u l a t o r s to Teach Algebra 185 For example, when confronted with the expression ( 1 . 0 0 9 + 0 . 0 0 1 ) 4 4 ( 1 . 1 - 0 . 9 5 ) 2 1 . 1 2 + 1 . 3 9 5 1 . 4 7 5 9 students tend to the operations in the order indicated by the italicized numbers thus: 1 3 ( 1 . 0 0 9 + 0 . 0 0 1 ) 4 4 2 4 • 5 ( 1 . 1 - 0 . 9 5 ) 2 5 1 .1 2 7 1 . 3 9 5 8 1 . 4 7 5 9 If a non-algebraic calculator, which executes operations in the order in which the operation buttons are pressed, is employed to evaluate this expression by performing the operations in this order, 1.009 will have to be added to 0.001 and the result written down or stored in the calculator's memory; then 0.95 must be subtracted from 1.1 and the result written down (most non-algebraic calculators still have only one memory); then the first result must be raised to the 44th power and the result written down or stored, etc. Instead, I wanted students, in order to utilize their calculators most efficiently (that is, with a minimum of writing) to perform the operations in the following order: 7 8 ( 1 . 0 0 9 + 0 . 0 0 1 ) 4 4 3 4 1 ( 1 . 1 - 0 . 9 5 ) 2 F 1 .1 2 „ -5 + 1 . 3 9 5 6 1 . 4 7 5 9 so that nothing would have to be written down except the final Using C a l c u l a t o r s to Teach Algebra 186 answer. When you are dec id ing which of two adjacent operat ions to do f i r s t , always do the l a s t one f i r s t i f you are allowed to , because that works out best for a c a l c u l a t o r , but you must obey the fo l lowing r u l e s : 1. I f one operat ion i s i n s i d e brackets and the other i s outs ide , you must do the one inside brackets first. 2. I f one operat ion i s a power and the other i s not , you must do the power first. 3. I f one operat ion i s a m u l t i p l y or d i v i d e and the other i s an add or subtrac t , you must do the multiply or divide first. 4. I f both operations are add or subtrac t , you must do them left to right. 5. I f both operations are mul t ip ly or d i v i d e , you must do them left to right. It will be seen that rules 4 and 5 are more stringent than is necessary to ensure "one right answer. " For example, 12+6-2 = 16 whether the addition or the subtraction is performed first. However, 12-6+2=8 if the subtraction is performed first and 12 -6 + 2 =4 if the addition is performed first. The situation is similar with multiplication and division. Rather than teach students when the order makes a difference and when it does not, I have found it easier to write simple rules that always apply and make special provisions for more knowledgeable or insightful students when necessary. Before I can do an example, I must mention the fo l lowing Using C a l c u l a t o r s to Teach Algebra 187 p o i n t s : 1. Mathematicians often leave out m u l t i p l i c a t i o n s igns . M u l t i p l i c a t i o n s igns are the only ones they leave out, and therefore you can assume that there i s an i n v i s i b l e m u l t i p l i c a t i o n s ign between any two numbers which have no operat ion s ign between them. 2. Leaving out a m u l t i p l i c a t i o n s ign between two numbers could make the two numbers run together and be misunderstood. Put t ing i n a r a i s e d dot l i k e t h i s • i s dangerous because i t can be mistaken for a decimal p o i n t . To avoid confus ion, mathematicians of ten put i n brackets . For example, i f they do not want to wr i te 2 x 3 they cannot wri te simply 2 3 but must wr i te 2 ( 3 ) or ( 2 ) ( 3 ) o r , l e ss commonly ( 2 ) 3 ins t ead . However, these brackets are not "rea l" brackets ; they have no operat ion i n s i d e them, but only a s i n g l e number. They are "fake" brackets , which you can leave out i f you put the m u l t i p l i c a t i o n s ign back i n . 3. The symbol / means the same as the symbol -f. You read i t from l e f t to r i g h t . Using C a l c u l a t o r s to Teach Algebra 188 4. The h o r i z o n t a l l i n e i n , for example, \ means the same as the symbol + . You read i t from top to bottom. 5. A long d i v i s i o n l i n e l i k e t h i s one 2 + 4 3 - 6 does the job of brackets as wel l as meaning d i v i d e . The l i n e has i n v i s i b l e brackets attached to i t s ends above and below i t l i k e t h i s ( 2 + 4 ) ( 3 - 6 ) so the express ion above means the same as ( 2 + 4 ) -=- ( 3 - 6 ) It will be seen that the upper brackets would not be necessary if the addition in this example were replaced with a power, a multiplication or a division. On the other hand, the lower brackets could be omitted only if the subtraction were replaced with a power. Again, I have found it easier to teach students one simple rule. 6. Eva luat ing an expression means doing the operations i n i t . We cannot do the numbers. If I gave you t h i s problem on a tes t 2 8 3 you would not know what to do. I t i s only when I w r i t e , for example 2 8 3 + 4 5 that I am t e l l i n g you to do something. Therefore , i f we are numbering the operat ions to show t h e i r order , we place the numbers Using C a l c u l a t o r s to Teach Algebra 189 i n d i c a t i n g the order of operations under the operations and not under the numbers. However, there i s no symbol for a power l i k e the symbols for the other operations ( + , - , x, H-, / , — ) , so we simply place the number i n d i c a t i n g the order of a power near the exponent. The example that fol lows shows how these r u l e s are a p p l i e d . I t may appear tedious , but you w i l l probably f i n d that you w i l l soon be able to do i t without asking yourse l f a l l the ques t ions . Example 7 ~2 + 32" 1 - 7 To place numbers under the operations to show the order i n which they must be done, f i r s t put i n the brackets impl ied by the long d i v i s i o n l i n e . "(2 + 32)1 ' (1 - 7) Look at the l a s t p a i r of operations on ly . "(2 + 32)1 * T Ask yourself: May I do the power before the -? Answer: No, because the power i s outs ide brackets and the - i s i n . Look at the next p a i r of operat ions . "-> ( 2 + 3 2 ) l 7 (1 - 7) T Using C a l c u l a t o r s to Teach Algebra 190 Ask yourself: May I do the - before the -r? Answer: Yes, because the - i s i n s i d e brackets and the -f i s not . Do: Put 1 under the - . ( 2 + 3 2 ) (1 - 7) 7 Look at the l a s t p a i r of operat ions . ( 2 + 3 2 ) (1 - 7) 7 Ask yourself: May I do the power before the •=•? Answer: No, because the power i s outs ide brackets and the -r i s i n . Look at the next p a i r of operat ions . ( 2 + 3 2 ) •L 7 2 (1 - 7) 7 Ask yourself: May I do the -r before the power? Answer: No, because the -r i s not i n brackets and the power i s . Look at the next p a i r of operat ions . ( 2 + 3 2 ) (1 - 7) 7 Ask yourself: May I do the power before the +? Answer: Yes, because powers must be done before +. Do: Put 2 over the power, 3 over the +, 4 at the s ide of the - r , and 5 under the power. 3 2 ( 2 + 3 2 ) (1 - 7) 7 Using C a l c u l a t o r s to Teach Algebra 191 Two D i f f e r e n t Types of C a l c u l a t o r Press the fo l lowing buttons, i n t h i s order , on your c a l c u l a t o r : 2, +, 3, x, 5. what answer do you get? If you got 25, your c a l c u l a t o r i s non-algebraic. I f you got 17, your c a l c u l a t o r i s algebraic. You have to l e a r n to use your c a l c u l a t o r . Non-algebraic c a l c u l a t o r s work d i f f e r e n t l y from a lgebra i c c a l c u l a t o r s and you have to use them d i f f e r e n t l y to get the r i g h t answer. I f you have an a l g e b r a i c c a l c u l a t o r , sk ip the i n s t r u c t i o n s for non-a lgebra ic c a l c u l a t o r s . Non-Algebraic C a l c u l a t o r s Before I can teach you how to use a non-a lgebra ic c a l c u l a t o r , I have to teach you how to do a power. Most non-algebraic calculators do not have a power button (usually designated y x , x v , or a x ) . The only non-algebraic calculators with power buttons are business calculators, which tend to be more expensive. Example To c a l c u l a t e 2 5 , type 2 x = = = =. That i s , t e l l the c a l c u l a t o r that you are s t a r t i n g with 2; then t e l l i t that you want to m u l t i p l y . However, s ince you are going to m u l t i p l y by 2 four more times, you do not have to press 2 again . Just press = four t imes. Using C a l c u l a t o r s to Teach Algebra 192 Example To c a l c u l a t e 1 .009 1 9 2 , f i r s t d i v i d e 192 by 2 u n t i l you reach an odd number. Just press 192 T 2 = = = . . . and wri te down the numbers you get thus: 1 9 2 9 6 4 8 2 4 1 2 6 3 Now press 1.009 x = =. That w i l l c a l c u l a t e 1.0093 (the 3 i s the 3 on the bottom l i n e ) . For each of the other l i n e s , press x =. Al together , you w i l l press 1.009 x = = x = x = x = x = x = x = . Example To c a l c u l a t e 1 .009 3 6 4 , f i r s t d i v i d e 364 by 2 u n t i l you reach an odd number. I f t h i s number i s too high to press = that many times, subtract 1, and then continue d i v i d i n g by 2, s u b t r a c t i n g 1, e t c . , u n t i l you reach a convenient ly low number. Every time you subtract 1, wri te the r e s u l t on the same l i n e . Every time you d i v i d e by 2, wri te the r e s u l t on the next l i n e down. 3 6 4 1 8 2 91 9 0 4 5 4 4 2 2 11 1 0 5 Now press 1.009. Then press M+, s ince you are going to go sideways to get back to 364 as wel l as up. (This puts the number 1.009 i n t o the c a l c u l a t o r ' s memory.) Then press x = = = = to get Using C a l c u l a t o r s to Teach Algebra 193 I . 009 s . Then press x = to go up to 10, then x MR to go over to I I , then x = to go up to 22, then x = to go up to 44, e t c . Al together , you w i l l press 1.009 M + x = = = = x = x M R x = x = x M R x = x M R x = x = . (I am indebted for this last technique to Val Greig, who served as an aide in my classes for a number of years.) Here are the r u l e s for us ing a non-a lgebraic c a l c u l a t o r : 1. Number the operations as you have been taught and then perform the operat ions i n that order . 2. You may do addi t ions and m u l t i p l i c a t i o n s forward (reading l e f t to r i g h t ) or backward (r ight to l e f t ) because that does not change the answer. For example, i f you do 2 x 3 on your c a l c u l a t o r , you get the same answer as i f you do 3 x 2 I f you do 2 + 3 on your c a l c u l a t o r , you get the same answer as i f you do 3 + 2 Therefore , i f you f i n d yourse l f wanting to do an a d d i t i o n or a m u l t i p l i c a t i o n backward, do so. 3. However, you must do subtract ions forward (reading l e f t to r i g h t ) and d i v i s i o n s forward (reading + or / l e f t to r i g h t and — top to bottom) or you w i l l get the wrong answer. For example, i f you do 8 -r 2 Using C a l c u l a t o r s to Teach Algebra 194 on your c a l c u l a t o r , you get 4, but i f you do 2 -r 8 you get 0.25. S i m i l a r l y , i f you do 8 - 2 you get 6, but i f you do 2 - 8 you get -6, which looks s i m i l a r to 6 but means something very d i f f e r e n t . (Which would you rather have as your bank balance?) Therefore , i f you f i n d yourse l f wanting to do a s u b t r a c t i o n or a d i v i s i o n backwards, stop, s tore what you have done so far i n the c a l c u l a t o r ' s memory, go to the number i n front of the s u b t r a c t i o n or d i v i s i o n s i gn , and do the subtrac t ion or d i v i s i o n forward, r e c a l l i n g what you put i n the memory when you need i t . 4. A l s o , i f you f i n d that you have to s k i p over one operat ion to do another operat ion f i r s t , stop, s tore what you have done so f a r i n the c a l c u l a t o r ' s memory, do the next operat ion i n order , and r e c a l l what you put i n the memory when you need i t . 5. To s tore what you have done i n your c a l c u l a t o r ' s memory, press the M+ button. You do not have to press = f i r s t ; the M+ button does the equals for you. However, the memory must be empty. I f there i s a number i n the c a l c u l a t o r ' s memory a lready , press ing M+ w i l l add the number you are t r y i n g to s tore to the number that i s a lready there . 6. To r e c a l l a number from your c a l c u l a t o r ' s memory, press MR. 7. To c l e a r your c a l c u l a t o r ' s memory, press MC. On many Using C a l c u l a t o r s to Teach Algebra 195 c a l c u l a t o r s , MC i s on the same button as MR. With t h i s k ind of c a l c u l a t o r , you should get in to the habi t of always pres s ing MC immediately a f t e r you have pressed MR; simply press the same button twice. On such a calculator, clearing the memory entails recalling the stored number first, and recalling the stored number entails losing the currently displayed number. Only if the memory is cleared immediately after a stored number is recalled can the memory be used twice in the same problem. Example C a l c u l a t e 2 + 3' 1 - 7 The order of operations i s as fo l lows: 3 2 •> (2 + 3 2 ) (1 - 7) 7 Therefore , on a non-a lgebra ic c a l c u l a t o r , you would press the fo l lowing buttons to get the answer: 1 - 7 M + 3 x = + 2 v M R M C x = = = = = = The reason why I ask students to do later operations first if possible can now be made clear; if you numbered the operations thus 2 1 ( 2 + 3 2 ) (1 - 7) 3 you would start by pressing the following buttons: Using C a l c u l a t o r s to Teach Algebra 196 3 x = + 2 M + 1 - 7 . . . at which point you would be faced with doing the division backwards. A l g e b r a i c C a l c u l a t o r s Not ice that an a lgebra i c c a l c u l a t o r has two power buttons , one marked y x , x v , or a x and the other marked x 2 . Not ice that i t a l so has two bracket buttons, marked ( and ). Use your power buttons as fo l lows . Example To c a l c u l a t e 2 2 , press 2 x 2 . Use t h i s button only i f the exponent i s 2. Example To c a l c u l a t e 1 . 0 0 9 1 9 2 , type 1.009 y x 192 =. Use t h i s button for every exponent except 2. Here are the r u l e s for us ing an a lgebra i c c a l c u l a t o r . 1. In general , type the expression on to your c a l c u l a t o r j u s t as i t i s w r i t t e n , from top to bottom and from r i g h t to l e f t . 2. However, (a) do not type i n any fake brackets (that i s , brackets around a s i n g l e number). (b) do type i n any unwritten m u l t i p l i c a t i o n s igns . (This may not be necessary i n most circumstances on some c a l c u l a t o r s , but i t i s sa fer to type i t i n . ) Using C a l c u l a t o r s to Teach Algebra 197 (c) do type i n the brackets that come with long d i v i s i o n l i n e s . Example C a l c u l a t e ~2 + 321 1 - 7 F i r s t put i n the brackets that come with the long d i v i s i o n l i n e . " ( 2 + 3 2 ) ] 7 ( 1 - 7 ) Then, on an a lgebra i c c a l c u l a t o r , you would press the fo l lowing buttons to get the answer: ( ( 2 + 3 x 2 ) - ( 1 - 7 ) ) y x 7 = It will be seen that students do not need to know the conventional order of operations to use an algebraic calculator. However, I insist that students with algebraic calculators insert numbers indicating the order of operations as outlined above for two reasons. One is that they need to be able to order the operations correctly when I teach them to solve equations. The second is that the better students need to know the order of operations in order to evaluate long expressions in the midst of which an algebraic calculator can indicate an error, as shown in the following example. Using C a l c u l a t o r s to Teach Algebra 198 Example 3 . 6 + 5 . 2 12 - 6  3 .1 " 3 7 l ] ] The order of operations i s as fo l lows: 3 . 6 + 5 .2 x 12 - 6 x 6 5 4 3 3.1 - 7 2 371 1. Type the problem on to your c a l c u l a t o r as u s u a l , but watch the d i s p l a y for an e r r o r message and note the exact po int i n the c a l c u l a t i o n at which the e r r o r message appears. Assume that the e r r o r message appears immediately a f t er you type i n the second subtrac t ion s i g n . Note t h i s p o i n t . 3 . 6 + 5 . 2 x ( 12 - 6 x ( 3 .1 - 6.1 * 371 ) ) = t You have typed the operations numbered 6, 5, 4, and 3 on to your c a l c u l a t o r , but your c a l c u l a t o r has not done them ye t . I t knows i t cannot do operat ion 6 u n t i l i t has done operat ion 5, and i t cannot do operat ion 5 u n t i l i t has done operat ion 4, e t c . Instead, i t has to remember these operat ions . When i t g ives you the e r r o r message, i t i s t e l l i n g you that i t cannot remember any more operat ions . 2. C lear your c a l c u l a t o r . Now back up from the e r r o r p o i n t , say to the previous bracket , and, beginning there , under l ine a natural unit from the middle of the problem, as shown. 3 . 6 + 5 . 2 x 6 5 [ 12 - 6 x 4 3 3.1 - 7 2 371 Using C a l c u l a t o r s to Teach Algebra 199 A natural unit inc ludes only consecutive operat ions and may not omit one of a p a i r of brackets . 3. Type i n the under l ined part of the problem and s tore the answer i n your c a l c u l a t o r 1 s memory. ( 3.1 - 6.1 T 371 ) M 4. Type i n the whole problem again, r e c a l l i n g the s tored part from the memory when you get to i t . 3.6 + 5.2 x ( 12 - 6 x MR ) = Using C a l c u l a t o r s to Teach Algebra 200 Appendix 4 Reversal Method of So lv ing Equations In order to make the reader f a m i l i a r with the language of the r e v e r s a l method of s o l v i n g equations, I here present the lessons as I would present them to the students i n w r i t t e n form. In p r a c t i c e , as with c a l c u l a t o r lessons, I present them o r a l l y , and consequently they involve a great deal of ques t ion ing and r e p e t i t i o n . Again, whenever i t has seemed appropriate or necessary to e x p l a i n to the mathematical reader the reasoning behind the i n s t r u c t i o n s , I have inc luded i t i n i t a l i c s . Formulae I f you put some money i n the bank, the bank w i l l pay you i n t e r e s t . The amount of i n t e r e s t the bank pays you depends on how much money you put i n the bank ( c a l l e d the " p r i n c i p a l " ) , what the i n t e r e s t ra te i s , and how must time you leave the money i n the bank. To c a l c u l a t e how much i n t e r e s t they owe you, the bank s t a r t s with the p r i n c i p a l , m u l t i p l i e s by the i n t e r e s t r a t e , and then m u l t i p l i e s by the time. I f we wri te the p r i n c i p a l as P, the i n t e r e s t ra te as r , the time as t and the i n t e r e s t as I , we can wri te down the formula the bank uses thus: P x r x t = I Using C a l c u l a t o r s to Teach Algebra 201 Usual ly we leave out the x's because they look l i k e x's, and so the formula looks l i k e t h i s : Prt = I Any formula i s a set of mathematical i n s t r u c t i o n s for c a l c u l a t i n g the value of one v a r i a b l e from the values of other v a r i a b l e s . Since t h i s i s a formula for c a l c u l a t i n g I , I i s c a l l e d the subject of the formula. We can use t h i s formula as fo l lows . Suppose I put $200 i n the bank at 10% i n t e r e s t for 5 years . That means P = 200, r = 0.10 (because i t i s a percentage) and t = 5. The formula t e l l s us what to do with these numbers: on our c a l c u l a t o r s , we press 200 x .10 x 5 = and we get 100. That i s the value for I . I = 100. The amount of i n t e r e s t the bank w i l l pay us i s $100. The advantage of us ing l e t t e r s i n a formula i s that the formula s t i l l works even when we change the numbers. We c a l l the l e t t e r s P, r , t , and I variables because t h e i r values - the numbers we subs t i tu te for them - can vary. Reversing a Formula However, we can a l so use the formula Pr t = I i n another way. Suppose I t e l l you that I l e f t my money i n the bank at an i n t e r e s t ra te of 10% for 5 years and got $100 i n t e r e s t and ask you what my p r i n c i p a l must have been. In t h i s case, I am t e l l i n g you that r = .10, t = 5 and I = 100 and asking you what P must be. You know that P must be 200, because that i s the value of P that gave us 100 for I i n the f i r s t p lace . But how could we work Using Calculators to Teach Algebra 202 out the value of P i f we did not know i t already? What i f I told you that r = .10, t = 5, and I = 200; what would P have to be then? Here i s a method by which you can change the subject of a formula. To see how i t works, calculate the following without using your calculators: Start with 2 ... add 3 ... multiply by 4 . . . subtract 5 ... divide by 5 ... add 6 ... you end up with 9. We can write this as follows: s t a r t 2 + 3 x 4 - 5 -r 5 + 6 end 9 J write the operations in a vertical column instead of a horizontal row because students, having been taught the conventional order of operations, are unwilling to perform the operations in the given order if I write them all on the same horizontal line. I write "end" instead of "=" to distinguish 9 from the other numbers. In shape, "=" looks very like an operation sign. Now undo what you have just done by starting at the end, doing the opposite operations in the opposite order, and ending up where you started. Start with 9 •. .. subtract 6 (because subtract is the opposite of add) ... multiply by 5 (because multiply i s the opposite of divide) ... add 5 ... divide by 4 ... subtract 3 ... and you end up with 2. We can write this as follows: Using C a l c u l a t o r s to Teach Algebra 203 s t a r t 2 9 + 3 - 6 x 4 x 5 - 5 + 5 - r 5 "T 4 + 6 - 3 end 9 2 We can use the same reasoning to change the subject of the formula I = Pr t from I to P. The order of operat ions i n the o r i g i n a l formula i s as fo l lows: P r t = I 72 The o r i g i n a l formula says that i f we s t a r t with P, then m u l t i p l y by r , and then m u l t i p l y by t , we w i l l end up with I . We can wr i te t h i s as fo l lows: s t a r t P x r x t end I Then we can wr i te the reverse as fo l lows: s t a r t P I x r -r t x t -r r end I P Notice that the second, or reverse , column t e l l s us what to do with I , t , and r to get P. However, t h i s i s not the usual way of w r i t i n g formulae; the usual way i s as fo l lows: I This i s a formula for c a l c u l a t i n g P. We have changed the subject of the o r i g i n a l formula from I to P. I f we do on our c a l c u l a t o r s what the formula t e l l s us to do with the values above: Using C a l c u l a t o r s to Teach Algebra 204 100 T 5 T .1 =, we get 200, as we expected. Brackets Make b the subject of t h i s formula. + c F i r s t put i n the order of operat ions . A reverse brackets write normally-start b A A v a - c - c (A - c ) a = b + c x a x a e n d A b b Decide where to put brackets according to the fo l lowing r u l e : Look down the operations i n the reverse column. Wherever a + or a - i s immediately above a x or a •*-, put a l i n e between the + or -and the * or + and make i t in to a box around the top of the column. You must put brackets around whatever i s i n the box, so that the + or the - and a l l the operations which come before i t are i n s i d e brackets and the x or the -r i s l e f t ou t s ide . These brackets are necessary to make people do the + or - before the x or - 7 . D i v i s i o n Lines As Brackets Make b the subject of t h i s formula. p = a i b - c F i r s t put i n the order of operat ions . Using Calculators to Teach Algebra 205 s t a r t b x a i - c e n d p reverse P + c v a , b p = a i b - c 7 2 brackets P + c write normally * a i b If the operation immediately after the + or - is a T, then a long division line w i l l do the job of brackets. Be sure you make the division line long enough to go under everything that i s supposed to be in brackets. You may leave out the words "start" and "end." Reversal of Powers Make a the subject of this formula. R = Q [ ( a - b ) 2 + 3 ] V m + n F i r s t put in the order of operations. Q[~(a -L 7 1 1/m b ) 2 + 3 4 + n 2 3 J 6 a 8 a - b y x 2 + 3 y x 1 /m x Q + n • f c y x a reverse R y x 1/a x C - n * Q y x m Using Calculators to Teach Algebra 206 brackets write normally y~ 1/2 + b a R y x 1/a x c - n * Q y x m - 3 R 1 / ac - n i V2 - 3 + b = a y x 1/2 + b a Write y x for power operations i n the columns because there i s no other symbol for a power and that i s what the c a l c u l a t o r c a l l s i t . However, do not write y x when you write i t out normally: simply write the exponent i n a r a i s e d p o s i t i o n , x 1 I y. y and y1'-"- are the reverses of each other. To see t h i s , punch the following buttons: 2 y x 5. The answer you get i s 32. Now punch y x ( 1 + 5 ) =. You get back to 2. When there are powers i n the formula, you also need brackets under the following circumstances: Look down the operations i n the reverse column. Wherever any operation i s immediately above a y x , put a l i n e between the other operation and the y x and make i t into a box around the top of the column. You must put brackets around whatever i s i n the box, so that the other operation and a l l the operations which come before i t are insi d e brackets and the y x i s l e f t outside. These brackets are necessary to make people do the other operation before the y x . Using Calculators to Teach Algebra 207 Groups Make a the subject of this formula. A = a(b + c) + d F i r s t put in the order of the operations. A = a(b + c) + d 2 1 3 reverse brackets write normally A - d a A x (b + c) - d A - d (b + c) = a + d -r (b +' C ) -r (b + C ) A - d A a a : = a b + c If you want to make a the subject, you must start with a . But i f you start with a , you have to write the x next. What does the original formula t e l l you to multiply by? b? No, by the time you do the multiply (operation 2), b is supposed to have been already added to c (operation 1). c? No; there i s the same objection. By the time you do the multiply, b and c no longer exist; they have been added together. A l l there i s now is the answer to b + c. Does that mean you cannot start with a and then multiply? The answer i s that you can, provided somebody has already worked out the answer to b + c for you and stored i t in your memory, ready for you to multiply a by. You do not know the answer to b + c, because you do not know what b and c are. So write b + c to t e l l people you want them to add b and c (that i s , get the answer to b + c) and put brackets around i t to make sure people do the addition and get the answer to (b + c) before they multiply by i t . Using C a l c u l a t o r s to Teach Algebra 208 When you reverse , do not change the + i n (b + c ) , but only the x i n front of i t . In the f i r s t column you multiplied by the answer to (b + c ) ; i n the reverse column you must divide by the same number, namely the answer to (b + c ) . When you wri te out the reverse column normal ly , you w i l l f i n d that the long d i v i s i o n l i n e can do the job of the brackets around the (b + c ) , so you can leave out the brackets . We c a l l (b + c) i n t h i s example a group. Groups can have any number of operat ions i n s i d e them, as the next example shows. In fu ture , when you are w r i t i n g out the f i r s t column, whenever you come to an operat ion whose number i s lower than the operat ion you have jus t wr i t t en down, keep the group conta in ing the lower-number operat ion on the same l i n e , i n brackets . When you reverse , do not change any operat ions i n s i d e the group. When you wri te out the new formula normal ly , take out any brackets that you can take out without changing what the formula means. Groups can be of any s i z e . For example, make m the subject of t h i s formula. A = a + c j + m p + r + e + g F i r s t put i n the order of the operat ions . A = 5 a + c e + g 4 6 + 2 j + m 7 p + r 3 Using Calculators to Teach Algebra 209 m + j * (P + r) a + c + e + g reverse A a + c e + g x (p + r) - j m brackets a + G e + g x (p +r) - j m A -write normally a + cl e + g A - a + c e + g (p + r) - j = m (p + r) - j = m Solving f o r an Exponent Make n the subject of t h i s formula. P = 100 [(1 + r t ) n r- 1] Put i n the order of the operations. P = 100(~(1 + r t ) n - 1] L 2 7 3 4 J 5 F i r s t i d e n t i f y the group that the exponent "applies to" - the group that i s being m u l t i p l i e d by i t s e l f n times - and make that group together with the exponent the subject f i r s t . This group contains a l l the operations to the l e f t of the exponent which are supposed to be done before the power. reverse brackets write normally (1 + r t ) n P none - 1 T 100 -2— +1 = (1 + r t ) n 100 x 100 +1 P (1 + r t ) n Now take the logarithm of the group on the l e f t side of the equals sign and div i d e i t by the logarithm of the group on the r i g h t side without the exponent. (We usually write " l o g " instead of "logarithm.") That gives you the exponent you want. log 100 Using C a l c u l a t o r s to Teach Algebra 210 + 1 = n log (1 + r t ) I t i s very d i f f i c u l t to understand t h i s example. However, the fo l lowing may he lp . To see what the log button on your c a l c u l a t o r does, punch the fo l lowing buttons: 10 y x 3 =. The answer i s 1000. Now punch your log button. The answer i s 3. You can see that the log button t e l l s you what exponent you used to get 1000. Whenever you want to make an exponent the subjec t , you w i l l have to use your log button. M u l t i - S t e p Solut ions Make a the subject of t h i s formula. A = b e - a d Put i n the order of the operat ions . A = b e - a d 2 3 1 I f you s t a r t with a, you w i l l m u l t i p l y i t by d . . . but then you would have to do the subtrac t ion backwards, and that i s not what the o r i g i n a l formula says. Subtract ions have to be done forward, from l e f t to r i g h t . But that means you cannot s t a r t with a r i g h t away. Instead, you have to do an extra s tep . Step 1 : s t a r t with the nearest l e t t e r to a that i s not a f t e r the s u b t r a c t i o n . Using C a l c u l a t o r s to Teach Algebra 211 c x b - (ad) A reverse A + (ad) •f b c brackets A + (ad) write normally A + (ad) -i- b c b A + ad = c = c Step 2: Now use this formula to make a the subjec t . Not ice that the - i n the o r i g i n a l formula which made i t impossible to s t a r t with a has now been reversed to a +, and +'s can be done backwards. F i r s t put i n the order of the operat ions for the new formula. 2 1 A + ad 3 = c a x d + A •5- b c reverse c x b - A -r d a brackets write normally cb - A ; = a The same th ing w i l l happen with l e t t e r s a f t e r d i v i s i o n s , because d i v i s i o n s cannot be done backwards e i t h e r . In fu ture , i f the l e t t e r you want to make the subject comes a f t e r a - s i g n , a / s ign or a -r s i gn , or i f i t comes under a long d i v i s i o n l i n e , you cannot make i t the subject i n one s tep. Instead, choose the nearest v a r i a b l e which does not have t h i s problem and make that l e t t e r the subject ins tead . Then use the new formula to make the o r i g i n a l l e t t e r the subject . I f the l e t t e r you want to make the subject comes a f t e r one or more subtract ions and one or more d i v i s i o n s , i t may take three or more steps, as you w i l l see i f you make b the subject of t h i s formula. V = Using C a l c u l a t o r s to Teach Algebra 212 d + c a - 1 /b Put i n the order of operat ions . a - 1 / b 2 1 b i s a f t e r a / s ign , a f t er a - s i gn , and under a long d i v i s i o n l i n e . In other words, there are three "problems" with making i t the subject . I f one "problem" means two steps , then three "problems" mean four s teps . The nearest l e t t e r to b that i s not a f t e r a subtrac t ion or a d i v i s i o n i s c, so Step 1 i s to make c the subject f i r s t . c + d * ( a - 1 /b ) V reverse V x (a - 1 lb) - d c brackets none write normally V ( a - Mb) - d = c Put i n the order of operations for t h i s new formula. V ( a - 1 /b ) - d = c 3 2 1 4 The nearest l e t t e r to b that i s not a f t e r a subtrac t ion or a d i v i s i o n i n t h i s new formula i s a, so Step 2 i s to make a the subjec t . brackets s t a r t a - (1 / b ) x V - d e n d c reverse c + d •f V + d / b ) a c + d write normally c + d ^ V + d / b ) a V c + d V + (1 / b ) = a + 1 /b = a Put i n the order of operations for t h i s new formula. Using C a l c u l a t o r s to Teach Algebra 213 2 c + d 3 + 1 / b = a V 4 7 The nearest l e t t e r or , i n t h i s case, number, to b that i s not a f t e r a subtrac t ion or a d i v i s i o n i n t h i s new formula i s 1, so Step 3 i s to make 1 the subject . reverse brackets write normally "c + d l 1 - r b c + d a a c + d c + d V V a - b = 1 x b 1 x b 1 a -c + d b = 1 Put i n the order of operations for t h i s new formula. 7 c + d a - 2 3 V b = 1 4 F i n a l l y , the l e t t e r you were t r y i n g to make the subject , b, no longer comes a f t e r a subtrac t ion or a d i v i s i o n . I t may look as though i t comes a f t er the subtrac t ion , but the l e t t e r s i n brackets w i l l be a group i f we s t a r t with b, so you w i l l not have to "do" the subtrac t ion at a l l ; i f i t i s i n a group, you assume that i t has been done. reverse 1 a -c + d a -c + d V brackets write normally none 1 , = b a -V 1 It will be noticed that this method of solving equations would not work if the variable being solved for occurred more than Using C a l c u l a t o r s to Teach Algebra 214 once in the formula. That difficulty, however, does not arise in the formulae I use in the rest of the course. Using C a l c u l a t o r s to Teach Algebra 215 Appendix 5 Two Methods of Poet ic D e s c r i p t i o n In March 1781, Thrale and Boswell presented Samuel Johnson with a problem: had Shakespeare or M i l t o n drawn the more admirable p i c t u r e of a man? The passages produced on e i t h e r s ide were Hamlet's d e s c r i p t i o n of h i s father (Hamlet, I I I , i v , 55) and M i l t o n ' s d e s c r i p t i o n of Adam (Paradise Lost, IV, 300). The two passages are reproduced here. Hamlet's D e s c r i p t i o n of His Father See, what a grace was seated on t h i s brow; Hyperion's c u r l s , the front of Jove himsel f , An eye l i k e Mars, to threaten and command, A s t a t i o n l i k e the hera ld Mercury New-lighted on a heaven-kiss ing h i l l , A combination and a form indeed, Where every god d i d seem to set h i s s e a l , To g ive the world assurance of a man. M i l t o n ' s D e s c r i p t i o n of Adam His f a i r large front and eye sublime d e c l a r ' d Absolute r u l e ; and hyacinthine locks Round from h i s parted fore lock many hung C l u s t e r i n g , but not beneath h i s shoulders broad. Using C a l c u l a t o r s to Teach Algebra 2 1 6 Appendix 6 Out l ine of Mathematics 8A-11A In the fo l lowing o u t l i n e of the courses I have designed, * designates the work attempted by more able students (those capable of ach iev ing B) and ** designates the work attempted by the most able students (those capable of achiev ing A ) . The terminology of t h i s o u t l i n e i s explained i n Appendices 3 and 4. In order to save space, "operation" has been abbreviated "op." For example, what i s wr i t t en under "Solving Equations" for "Ma 10A" means that the "star" work i n Mathematics 10A involves s o l v i n g equations comprising four operations and groups conta in ing one operat ion , with up to three steps requ ired for the s o l u t i o n , but there are no powers which have to be reversed and no logarithms; e.g., s o l v i n g the fo l lowing equation for x: a + b + c = n m - x Using C a l c u l a t o r s to Teach Algebra 217 Table 6-A Contents of Mathematics 8A-11A C a l c u l a t o r s So lv ing Equations A p p l i c a t i o n s Ma 8A eva luat ing numerical expressions with 2 operat ions 2 ops . ; no power r e v e r s a l , groups, m u l t i - s t e p , logs markdown tax *evaluat ing numerical expressions with 3 operat ions *3 ops . ; no power reversa1, groups, m u l t i - s t e p , logs **evaluat ing numerical expressions with < 6 operat ions i n c l u d i n g complex f r a c t i o n s **4 ops . , power r e v e r s a l ; no groups m u l t i - s t e p , logs Ma 9A eva luat ing numerical expressions with z 3 operat ions 3 ops . , groups with 1 o p . ; no power r e v e r s a l , m u l t i - s t e p , logs markdown tax simple i n t e r e s t *evaluat ing numerical expressions with s 4 operations *4 ops . , groups with 1 op . ; no power r e v e r s a l , m u l t i - s t e p , logs **evaluat ing numerical expressions with s 7 operat ions i n c l u d i n g complex f r a c t i o n s **7 ops . , power r e v e r s a l , groups with 1 or more ops . ; no m u l t i - s t e p , logs Ma 10A eva luat ing numerical expressions with < 3 operat ions 3 ops . , groups with 1 o p . , two steps; no power r e v e r s a l , logs markdown tax simple i n t e r e s t compound i n t e r e s t *evaluat ing numerical expressions with s 4 operations *4 ops . , groups with 1 o p . , s 3 steps; no power r e v e r s a l , logs **evaluat ing numerical expressions with s 7 operations i n c l u d i n g complex f r a c t i o n s **7 ops . , power r e v e r s a l , groups with > 1 o p . , i 4 steps; no logs Ma 11A eva luat ing numerical expressions with s 4 operat ions s 5 ops . , m u l t i - o p . groups, < 2 steps; no power r e v e r s a l , logs simple i n t e r e s t compound i n t e r e s t saving with regu lar depos i ts f inanc ing with regu lar payments *evaluat ing numerical expressions with s 10 operat ions *s 10 ops . , m u l t i - o p . groups, s 3 steps; no power r e v e r s a l , logs **evaluat ing numerical expressions with < 15 operat ions , complex f r a c t i o n s **s 15 ops . , power r e v e r s a l , m u l t i - o p . groups, s 5 steps, logs 

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