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The relationship between various pupil characteristics and performance on mathematics laboratories Dilley, Grace 1976

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THE RELATIONSHIP BETWEEN VARIOUS PUPIL CHARACTERISTICS AND PERFORMANCE ON MATHEMATICS LABORATORIES by Grace D i l l e y B. Ed. (Secondary) University of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILLMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Mathematics Education We accept t h i s thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA June, 1976 0 G r a c e D i l l e y , 1 9 7 6 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th i s thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my writ ten permission. Department of t"<jlu.cation The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date I^IU i i Abstract The purpose of t h i s study was to examine the r e l a t i o n s h i p between certa i n p u p i l c h a r a c t e r i s t i c s and performance on mathe-matics laboratories. The four classes of grade s i x students involved i n the study were c l a s s i f i e d by sex, field-dependence-independence, re f l e c t i v e - i m p u l s i v e tempo, past performance i n mathematics, present performance i n mathematics, and i n t e l l i -gence. Eight mathematics laboratories designed and used i n the study were categorized t o p i c a l l y into number theory or geometry laboratories. Each laboratory a c t i v i t y was designed to allow pupils to manipulate materials while exploring an idea and c o l l e c t i n g data. In part two of a laboratory a c t i v i t y , which included a te s t section, pupils were required to analyze data, make a pred i c t i o n , and v e r i f y the prediction using manipulative materials before extending a pattern or rul e . Laboratories were randomly assigned to classes. Results showed that a l l the selected c h a r a c t e r i s t i c s except sex had a s i g n i f i c a n t r e l a t i o n s h i p with performance on mathematics laboratories. Sex showed a s i g n i f i c a n t r e l a t i o n s h i p only to the geometry laboratories. An analysis of covariance was performed using past achievement as the covariate. The r e s u l t s indicated that there was a s i g n i f i c a n t difference i n performance only on the geometry laboratories between boys and g i r l s and between field-dependent and field-independent students. The differences were found to be i n favour of the g i r l s and the field-independent students. i i i The re s u l t s of t h i s study suggested that further research i s necessary to determine the most e f f e c t i v e means of using mathematics laboratories. i v TABLE OF CONTENTS Page LIST OF TABLES i v Chapter 1. THE PROBLEM 5 Background of the Study 5 The Problem 7 D e f i n i t i o n of Terms 7 J u s t i f i c a t i o n for the Study 7 S t a t i s t i c a l Hypotheses 8 2. REVIEW OF THE LITERATURE 11 Introduction 1 1 General Opinions on Mathematics Laboratories 1 1 Research Relevant to Mathematics Laboratories 15 Research Relevant to Pupil C h a r a c t e r i s t i c s 18 Research of S p e c i f i c Relevance to the Problem 23 Summary of Research Findings 25 3. DESIGN AND PROCEDURE 26 Design 26 Controls 26 Subjects , 26 Laboratories 27 Measurement of Individual C h a r a c t e r i s t i c s 29 Procedure 31 S t a t i s t i c a l Procedure 32 4.. RESULTS OF THE STUDY 34 Results 34 Discussion of the Results 38 5. SUMMARY AND CONCLUSIONS 42 The Problem 42 The Findings 42 Implications 43 Limitations 44 Further Research 46 BIBLIOGRAPHY 49 APPENDICES 56 V LIST OF TABLES Table Page 1. R e l i a b i l i t y C o e f f i c i e n t s for Number Theory Laboratories 28 2. R e l i a b i l i t y C o e f f i c i e n t s for Geometry Laboratories 29 3. R e l i a b i l i t i e s of Test Instruments 34 4. Correlations Between Pu p i l C h a r a c t e r i s t i c s and Mathematics Laboratories 35 5. Means and Adjusted Means 36 6. Analysis of Covariance: Sex vs Field-Dependence-Independence on Number Theory Laboratories 37 7. Analysis of Covariance: Sex vs Field-Dependence-Independence on Geometry Laboratories 37 8. Analysis of Covariance: Sex vs Field-Dependence-Independence on A l l Mathematics Laboratories 38 CHAPTER I THE PROBLEM Background of the Study In the present North American educational system, many d i f f e r e n t i n s t r u c t i o n a l methods are advocated and used. This variety of methods can be p a r t i a l l y a t t r i b u t e d to the desire of educators to respond to what they perceive as changes i n the demands of society. Many educators believe that the pres-ent demands of society require that the products of the educational system should be able not only to respond auto-matically to a known s i t u a t i o n , but also to apply t h e i r know-ledge to new situations (Reys & Post, 1973, P. 6). In mathematics such a perceived demand has caused a s h i f t i n emphasis away from a heavy concentration upon computational e f f i c i e n c y and toward an 'understanding 1 of the underlying structures of mathematics (Scott, 1966, p. 193). Educators have also advocated a var i e t y of teaching methods because of t h e i r desire to respond to i n d i v i d u a l differences. Differences i n mental a b i l i t y , previous achievement, s o c i a l and emotional maturity, personality, and environmental back-ground have influenced the placement of pupils into s p e c i a l classes or groups and have caused teachers to try a variety of teaching methods (Jones & Pingry, 1960, pp. 121-148). Brown concluded that: At present teachers do not know which students should be exposed to what 6 methods. The successful teacher t r i e s many methods u n t i l he finds one that seems to work with a p a r t i c u l a r student or group of students (1968, p. 411). He also suggested that there i s "no one answer to a l l problems with a l l students" (Brown, 1968, p. 411). Currently there are several teaching methods which attempt to promote 'discovery' on the part of the student. Since d i s -coveries are often the outcome of experiment, i t has become common to use a s e t t i n g which simulates that of a laboratory. Bruner (1962, p. 85) i s convinced that "the greater the p a r t i c i p a t i o n i n the learning process on the part of the stu-dent, the greater w i l l be the transfer of t r a i n i n g and the more l i k e l y w i l l be the development of i n t u i t i v e thinking". In addition, Bruner supported the use of a mathematics laboratory when he hypothesized four major benefits from using a discovery approach to teach mathematics. Bruner (1961) claimed that there w i l l be an "increase i n i n t e l l e c u t a l potency, ... the s h i f t for e x t r i n s i c to i n t r i n s i c rewards, ... learning the h e u r i s t i c s of discovering, ... and an aid to memory processing" (p. 23). Barson (1971, p. 565) also supported the use of mathematics laboratories when he observed that a "mathematics laboratory i s a c t i v i t y centered; the c h i l d i s placed i n a problem-solving s i t u a t i o n and through s e l f - e x p l o r a t i o n and discovery provides a s o l u t i o n based on h i s experience, needs, and i n t e r e s t s " . 7 The Problem It i s the purpose of t h i s study to explore the r e l a t i o n -ship between some pu p i l c h a r a c t e r i s t i c s and the performance of pupils i n mathematics laboratories. Selected c h a r a c t e r i s t i c s of pupils were sex, previous achievement i n mathematics, present achievement i n mathematics, i n t e l l i g e n c e , r e f l e c t i v e - i m p u l s i v e tempo, and field-dependence-independence. D e f i n i t i o n of Terms Mathematics Laboratory s h a l l mean a context i n which a pupil i s presented with a mathematics problem which requires manipulation of physical objects to generate data, and i n which, a f t e r the c o l l e c t i o n and organization of data, the pup i l i s expected to detect a mathematical r e l a t i o n s h i p , extend the r e l a t i o n s h i p by i n t e r p o l a t i o n and extrapolation, and try to state a generalization. Reflective-Impulsive Tempo i s a facet of conceptual tempo which involves the slow or f a s t decision-time of students. Field-Dependence-Independence i s a facet of cognitive s t y l e known as perceptual s t y l e . In a field-dependent mode, perception i s strongly influenced by the organization of the f i e l d . In a field-independent mode, parts of the f i e l d are perceived as disc r e t e from the organization of the f i e l d . J u s t i f i c a t i o n for the Study Control, as a goal of science, i s defined by Van Dalen (1966, p. 43) as "The process of manipulating c e r t a i n of the 8 e s s e n t i a l conditions that determine an event so as to make the event happen or prevent i t from occurring". Control, as a goal i n education, i s the process of manipulating the method of i n s t r u c t i o n for p a r t i c u l a r students. This analogy suggests that investigations should be undertaken concerning the type of student that w i l l benefit from a p a r t i c u l a r i n s t r u c t i o n a l method. An advocate of t h i s kind of research i s Cronbach (1966, pp. 91-92) who suggested that "we have to explore a f i v e - f o l d i n t e r a c t i o n -- subject matter, with type of in s t r u c -t i o n , with timing of i n s t r u c t i o n , with type of p u p i l , with outcome". The outcome of research that explores these aspects should provide s u f f i c i e n t information to enable teachers to say "that a fourth grader with one p r o f i l e of attainment needs discovery experience, whereas another w i l l move ahead on a l l fronts i f teaching i s d i d a c t i c " (Cronbach, 1966, pp. 90-91). Additional support for t h i s type of research i s given by Brousseau (1973) when he stated that: ' There i s a need for knowledge of not only the methods involved, but also of c l a s s i f y i n g and coding the educational and learning c h a r a c t e r i s t i c s of pupils i n order that one might f i t the method of i n s t r u c t i o n to the in d i v i d u a l student (p. 103). S t a t i s t i c a l Hypotheses Nine hypotheses were tested i n t h i s study: H^: There w i l l be no s i g n i f i c a n t r e l a t i o n s h i p between the sex variable and performance i n mathematics laboratories. (a) There w i l l be no s i g n i f i c a n t r e l a t i o n s h i p between Verbal I.Q. and performance on mathematics labora-t o r i e s . (b) There w i l l be no s i g n i f i c a n t r e l a t i o n s h i p between Non-verbal I.Q. and performance on mathematics laboratories. H^: (a) There w i l l be no s i g n i f i c a n t r e l a t i o n s h i p found between past achievement i n mathematics and per-formance on mathematics laboratories, (b) There w i l l be no s i g n i f i c a n t r e l a t i o n s h i p found between present achievement i n mathematics and performance on mathematics laboratories. : There w i l l be no s i g n i f i c a n t r e l a t i o n s h i p between f i e l d -dependency-independency and performance on mathematics laboratories. H^: There'will be no s i g n i f i c a n t r e l a t i o n s h i p between r e f l e c -tive-impulsive tempo and performance on mathematics laboratories. H,: (a) The mean score for g i r l s on number theory laboratories b w i l l not be s i g n i f i c a n t l y higher than for boys when I.Q. i s co n t r o l l e d . * (b) The mean score.for boys on geometry laboratories w i l l not be s i g n i f i c a n t l y higher than for g i r l s when I.Q. i s c o n t r o l l e d . * * See page 36. 10 (c) The mean score for boys and g i r l s w i l l not be the same on a l l laboratories when I.Q. i s con t r o l l e d . * H^: The mean score for impulsive g i r l s w i l l not be d i f f e r e n t from the mean score f o r r e f l e c t i v e g i r l s on mathematics laboratories. H c: The mean score for impulsive field-dependent students o w i l l not be d i f f e r e n t from the mean score for r e f l e c t i v e field-dependent students on mathematics laboratories. H n: The difference between the means of field-dependent boys and field-dependent g i r l s w i l l not be s i g n i f i c a n t l y d i f f e r e n t from the difference between the means of field-dependent boys and field-dependent g i r l s when I.Q. i s c o n t r o l l e d . * * See page 36. 11 CHAPTER 2 , • • REVIEW OF THE LITERATURE Introduction In the l i t e r a t u r e of mathematics education, the term 'mathematics laboratory' i s not well-defined. The two most popular connotations, however, seem to be as follows: (i) a p a r t i c u l a r methodology for learning, ( i i ) a p a r t i c u l a r place i n which learning occurs. Reys & Post (1971, p. 9-10) stated that the term laboratory encompasses the physical f a c i l i t i e s and educational intent, but should be extended beyond the use of materials and f a c i l -i t i e s to incorporate the aspects of inquiry through exploration and discovery and to create a problem solving atmosphere. General Opinions on Mathematics Laboratories One c a l l for the i n i t i a t i o n of mathematics laboratories was given by Moore i n 1902 when he addressed the American Mathematical Association. Moore suggested that: t h i s program of reform c a l l s for the development of a thorough-going laboratory system of i n s t r u c -t i o n i n mathematics and physics, a p r i n c i p a l purpose being as far as possible to develop on the part of every student that true s p i r i t of research, and an appreciation, p r a c t i c a l as well as theoretic, of the fundamental methods of science .... (p. 250) 12 I t seems,'however,to have taken over f i f t y years for the use of the laboratory method of teaching mathematics to take hold i n North America. An a r t i c l e by Davidson and F a i r (1970) indicated not only a possible structure for a laboratory, but also i t suggested some of the possible outcomes from using a mathematics labor-atory. Their suggestions to p u p i l s , used, while t r y i n g to e s t a b l i s h the tone for t h e i r laboratory, indicate that they hold these b e l i e f s : (i) the use of manipulative materials can act as a vehicle for learning mathematics; ( i i ) the mathematics laboratory method involves invest-i g a t i o n , exploration, hypothesizing, and looking for patterns; ( i i i ) hypotheses can be checked by using manipulative materials; (iv) laboratory experiences can be related to s p e c i f i c mathematics concepts, to problem-solving techniques, or to modes of mathematical thinking, (p. 107) Other outcomes suggested by users of mathematics labora-t o r i e s usually include items such as improvement of pupils' attitudes towards mathematics, increased i n t e r e s t i n mathematics, increased self-confidence, and improvement shown i n areas of concepts and concept applications (Beuthel & Meyer, 1972; Schaeffer & Mauthe, 1970; Vance & Kiernan, 1971). One further 13 suggested learning outcome from the use of mathematics lab-oratories i s the development of problem-solving s k i l l s (Kidd, Myers & C i l l e y , 1970). Kidd, Myers and C i l l e y (1970) expanded t h e i r opinion concerning the development of problem-solving s k i l l s by stating that: The surroundings of physical objects aid him i n s i z i n g up the problem and i n s e l e c t -ing sound and f e a s i b l e methods of attack. He also receives t r a i n i n g i n s e l e c t i n g data relevant to the question that has been posed and to his methods of attack. Furthermore, there are often b u i l t - i n v e r i f i c a t i o n s that give him feedback on the soundness of his method, the accuracy of his data, and the correctness of his computations, (p. 28) Most mathematics laboratories seem to begin by having students manipulate physical materials and then abstracting concepts from t h e i r manipulation (Fitzgerald, 1972, pp. 2-3). Copeland (1974) described a mathematics laboratory as "a classroom designed to allow children to i n d i v i d u a l l y perform the necessary p h y s i c a l manipulations or concrete operations that are necessary for r e a l learning of mathematical concepts" (pp. 327-328). He concluded that he sees the mathematics laboratory as contributing to the "nature of learning as envisioned by Piaget" (p. 360). 14 Kiernan and Vance (1971) suggested that the a c t i v i t i e s used i n a mathematics laboratory could be used not only to develop concepts, but also to permit discovery of a r e l a t i o n -ship. Laboratory a c t i v i t i e s are designed to lead to the development of a concept or the d i s -covery of a r e l a t i o n s h i p . The concrete materials serve not only to create i n t e r e s t and motivate learning but to provide a r e a l -world s e t t i n g for the problem to be solved or the concept to be investigated. The students use physical objects or manipula-t i v e devices to perform experiments and • c o l l e c t data r e l a t i n g to a problem, and they record t h i s information i n a table or on a graph i f possible. On the basis of these observations, hypotheses are formulated and tested. F i n a l l y , generalizations are stated. The newly discovered rule might then be used to answer ad d i t i o n a l questions or to do prac- / t i c e exercises for the purpose of c o n s o l i -dating learning, (p. 586) Davidson and Walter (1972) defined the mathematics lab-oratory as ... an approach to learning materials rather then a p a r t i c u l a r place i n a building. Such an approach encompasses exploring, 15 investigating, hypothesizing, experimenting, and generalizing. I t means that students are a c t i v e l y involved i n "doing" mathematics at a concrete l e v e l . I t provides abundant opportunities" for them to manipulate objects, to think about what they have done, to discuss and write about t h e i r findings, and to b u i l d necessary s k i l l s . Problems are re l a t e d to the children's own experiences and often emerge from the natural surroundings. The teacher acts as a c a t a l y s t and resource person, be-coming an active investigator along with the students, (p.222) Research Relevant to Mathematics Laboratories Within the f i e l d of mathematics education, the recent trend has been toward involving students a c t i v e l y i n learning. Shipp and Deer (-I960) , upon in v e s t i g a t i n g the percent of class time spent on developmental a c t i v i t i e s and on p r a c t i c e work, found that there was a trend toward higher achievement at a l l l e v e l s of a b i l i t y when the time spent on developmental a c t i v i -t i e s was increased (pp. 117-121). Also i t seemed that more than f i f t y percent of class time should be spent on develop-mental a c t i v i t i e s . In a l a t e r study, Shuster and Pigge (1965) investigated the e f f e c t s of devoting twenty-five percent, f i f t y percent, and seventy-five percent of class time on 16 developmental-meaningful a c t i v i t i e s . They concluded that f i f t y to seventy-five percent of class time has to be spent on developmental a c t i v i t i e s to.show an increase i n achieve-ment (pp. 24-31). The r e s u l t s of a study done by Ku h f i t t i n g (1974) involving manipulative aids seemed to imply that: the benefits derived from using the type of aids that students can manipulate them-selves tend to be l o s t when the expository i n s t r u c t i o n a l method i s employed, while independent work with such aids may lead to discovered i n s i g h t s , (p. 108) He also suggested that i f t h i s f i nding i s substantiated by further studies, the increased use of the mathematics lab-oratory i s to be highly recommended (p. 108). Wilkinson (1970) reported that he found that students who were taught geometry by a laboratory method did as well as students tuaght by the conventional teacher-textbook approach on a geometry achievement test. In a study comparing the laboratory method and i n d i v i d u a l i z e d i n s t r u c t i o n when teaching metric geometry, Whipple (1972) found that the math-ematics laboratory classes did better both on a conventional t e s t and i n the computing of areas and volumes which required the use of actual objects. Silbaugh (1972) found s i g n i f i c a n t d ifferences between c r i t e r i o n scores for groups using labora-t o r i e s , groups not using laboratories when laboratory f a c i l i t i e s were i n the school, and groups not using laboratories when laboratory f a c i l i t i e s were not a v a i l a b l e i n the school. Cohen (1970) compared laboratory and conventional methods to teach f r a c t i o n s to underachievers. He found that the students using the conventional method did better. He noted also that the laboratory method required more time than the conventional method. Ropes (1970) was able to use a laboratory method with pupils once a week for fourteen weeks. He reported that there was no s i g n i f i c a n t change i n attitude on the part of the students i n the laboratory s e t t i n g . The findings also showed that the lab students demonstrated no greater competency than the non-participants i n the a b i l i t y to solve problems or i n the a b i l i t y to c l a s s i f y and formulate class concepts. The lab students, however, did score as well as the non-participants on a standardized t e s t even though the lab students had spent 20% less time under conventional i n s t r u c t i o n . Nowak(1972) also did a study which compared the e f f e c t s of mathematics laboratories on achievement and a t t i t u d e s . She found that attitude changes were non-significant, that f i f t h and s i x t h grade students did better i n a laboratory than i n a non-laboratory program, that an i n d i v i d u a l i z e d laboratory program was more e f f e c t i v e than an i n d i v i d u a l i z e d non-labora-tory program, and that fourth grade students did better i n the conventional mathematics classes than those i n the laboratory s e t t i n g . 18 Research Relevant to Pupil C h a r a c t e r i s t i c s Previous studies relevant to t h i s study include those which re l a t e general mental a b i l i t y and mathematics performance. As a variable, mental a b i l i t y i s frequently employed i n studies that consider what type of student p r o f i t e d by the t r a i n i n g . Results of the studies u t i l i z i n g the general mental a b i l i t y variable usually show either that the l o w - a b i l i t y students performed as poorly as before and the h i g h - a b i l i t y students did very well, or that the l o w - a b i l i t y students did exceptionally well and the h i g h - a b i l i t y students performed about as well as usual. Because of such r e s u l t s , Anderson (1967) suggested that: i n t e r a c t i o n s , more expressly c o r r e l a t i o n s , between aptitude measures and performance af t e r t r a i n i n g , contain information that can be employed to improve t r a i n i n g by matching the kind of t r a i n i n g to the kind of person i n one of several possible ways, or by modifying the t r a i n i n g so that those with low aptitude scores achieve better, (p. 87) A study by J a r v i s (1964) , which demonstrates these t y p i c a l findings, compared a b i l i t y differences and performance i n elementary school mathematics. Ja r v i s found that: 1. The bright boys were found to be superior to t h e i r peer group g i r l s i n both reasoning and fundamentals. 19 2. A l l c l a s s i f i c a t i o n s of male students .excelled the female students i n t h e i r a b i l i t y to perform arithmetic reasoning functions. 3. A l l c l a s s i f i c a t i o n s of g i r l s were superior to boys i n t h e i r a b i l i t y to execute the arithmetic fundamental operations with the exception of the bright group, (p. 659) In the above study, general mental a b i l i t y Was claimed to be an i n d i v i d u a l difference which a f f e c t s p u p i l performance. The second variable to be considered i n t h i s study i s one involving conceptual tempo. A conceptual tempo known as the r e f l e c t i v e - i m p u l s i v e dimension ref e r s to the slow or f a s t decision-time when response uncertainty i s present. Tests have been designed and considerable research has been done by Jerome Kagan and his associates at the Fels I n s t i t u t e using the r e f l e c t i v e - i m p u l s i v e tempo va r i a b l e . From the studies undertaken, Kagan (1965) has suggested that the tendency: for r e f l e c t i o n increases with age, i s stable over periods as long as 20 months, manifests pervasive generality across varied task s i t u a -tions, and i s linked to.some fundamental aspects of the c h i l d ' s personality organization, (p. 134) Also he suggested that the tendency "to show f a s t or slow de-c i s i o n times was not highly related to verbal a b i l i t y " (p. 140) 20 and "to r e f l e c t over a l t e r n a t i v e hypotheses generalizes not only across tasks where a l l response alt e r n a t i v e s are given but also shows generality on tasks where the c h i l d must gen-erate his own alt e r n a t i v e s " (p. 141). The observed behavioral c h a r a c t e r i s t i c s of impulsive students are also of i n t e r e s t . Impulsive students were l i k e l y to d i s -play momentary lapses of attention during involvement i n a school task; they were more l i k e l y to look out a window, gaze at a peer, or orie n t to a sound during a period when they were working at an .'.academic task. These behavioral obser-vations are i n complete agreement with the problem-solving behavior displayed by children i n the laboratory s e t t i n g . (Kagan, 1964, p. 29) A further study done at the Fels I n s t i t u t e examined the r e l a t i o n -ship of the relective-impulsive conceptual tempo and persistence. The findings of t h i s study revealed that: r e f l e c t i v e boys spent more time with the hard tasks than impulsives and t h i s d i f -ference was greater for low-verbal than for high-verbal boys Among g i r l s , verbal s k i l l was a more c r i t i c a l v ariable i n determining free play performance, and the brighter g i r l s spent more time with 21 hard tasks. As with boys, the impulsive g i r l s with low verbal a b i l i t y spent less time on hard tasks, but the impulsive bright g i r l s spent the longest time on the d i f f i c u l t tasks. (Kagan, 1964, p. 158) Kagan r e a l i z e d that many teachers were not aware of the d i f -ference i n conceptual tempo between ch i l d r e n . Most teachers are not attuned to t h i s : .1 behavioral aspect of the c h i l d . Teachers are apt to categorize the c h i l d as bright or d u l l , obedient or disobedient, timid or outgoing; but r a r e l y do they notice whether the c h i l d i s impulsive i n his conceptual approach to problems. When they do acknowledge t h i s dimension there i s a tendency to c l a s s i f y the r e f l e c t i v e c h i l d as 'slow' and less bright than the impulsive, quick c h i l d with the same i n t e l l i g e n c e score and s o c i a l - c l a s s back-ground, (pp. 158-159) Kagan also suggested that r e f l e c t i v e - i m p u l s i v e tempo could have an e f f e c t on p u p i l performance i n subjects where the program u t i l i z e s the discovery method of i n s t r u c t i o n (Kagan, Pierson & Welch, 1966, p. 594). Whenever reports of research are examined, one frequently finds reference made to the difference i n performance between 22 the sexes. As already seen, both Kagan and Jar v i s i n t h e i r studies made d i r e c t reference to the d i f f e r e n t l e v e l s of attainment achieved by boys and g i r l s . Fennama (1974), a f t e r exploring the depth the l i t e r a t u r e since 19 60 concerning mathematics learning and the sexes, remarked that: although generalizing r e s u l t s from so many divergent studies that analyzed d i f f e r e n t aspects of mathematics learning hides subtle and important v a r i a t i o n s , i t appears safe to conclude that i n o v e r a l l performance on tests measuring mathematics learning that there are no s i g n i f i c a n t differences that consistently appear between the learning of boys and g i r l s i n the fourth to ninth grade. There appears to be a trend, however, that i f a difference does e x i s t , g i r l s tend to perform better i n tests of mathematics computation ... and boys tend to perform better i n tests of mathematical reasoning... (p. 135) Further studies which demonstrate the analysis of sex differences i n learning are those done by Witkin concerning the field-dependence-independence mode of perception. Witkin (1962) found that " t h i s dependence of females on the v i s u a l f i e l d was apparent at a l l ages from the adult, l e v e l to the eight year old l e v e l and indicates that from an early age females are less a n a l y t i c a l than males" (p. 14). Waetjen 23 (1962) suggested that i f g i r l s are less a n a l y t i c a l than boys, then schools need to be aware of t h i s v a r i a b l e . Since g i r l s are more field-dependent and more sensi t i v e to people than boys, i t would seem that they should fare better i n the usual classroom where the teacher and the curriculum content are major part of the f i e l d . ... being more a n a l y t i c a l , boys may create d i f f i c u l t i e s for themselves by making too many decisions on t h e i r own rather than responding to suggestions and dir e c t i o n s from the teacher (Waetjen, 1962, p. 14). Waetjen's conclusions suggested that boys w i l l perform better i n a set t i n g where they can make many of t h e i r own decisions. If t h i s i s so, then boys may be more successful i n lessons using the laboratory method where they are able to work more inde-pendently. Research of S p e c i f i c Relevance of the Problem Two studies were found which considered the r e f l e c t i v e -impulsive tempo and performance i n mathematics. A study by Cathcart and Liedtke (1969) which considered the r e f l e c t i v e -impulsive conceptual tempo and achievement i n mathematics found that r e f l e c t i v e students obtained higher mathematics achievement scores i n the dimension of basic facts and problem 24 solving, but showed no s i g n i f i c a n t difference i n the applica-t i o n of concepts (p. 594). Upon examination of the r e f l e c t i v e -impulsive tempo and performance i n elementary school mathematics, Rebhun (1973) found that r e f l e c t i v e children generally did obtain higher scores than did impulsive children on each of the Metropolitan Mathematics Tests involving computation, concepts, and problem solving. After examining predictors of seventh grade achievement i n mathematics, Fuys (1974) recommended that a measure of p r i o r achievement be used for s e l e c t i n g students for a u n i f i e d modern mathematics program. He also found that a measure of p r i o r achievement and the Chapter One te s t for the program predicted success i n the program as e f f e c t i v e l y as a l l s i x of his pre-d i c t o r s . H o l l i s (1972), who used a diagnose-prescribe model of a mathematics laboratory, found that both slow and g i f t e d students at the f i f t h and s i x t h grade leve l s showed a s l i g h t l y increased academic achievement i n mathematics when compared to the cor-responding control groups. Also, the laboratories f a c i l i t a t e d an increased p o s i t i v e attitude toward mathematics with the s i g n i f i c a n t increase occurring i n the school located i n a deprived area. There was no s i g n i f i c a n t difference i n the achievement scores between laboratory and control groups. 25 Summary of Research Findings After reviewing the available l i t e r a t u r e concerning mathematics laboratories, i t appears that although mathe-matics laboratories are perhaps no better than conventional methods of teaching mathematics, the achievement ratings of pupils involved i n research studies tend to suggest that the pupils taught using mathematics laboratories perform as well as those students taught using conventional methods. 26 CHAPTER 3 DESIGN AND PROCEDURE Design The present study employed mathematics laboratories and four classes of students. A l l subjects performed a l l of the laboratories. Information was 'gathered for each subject con-cerning i n d i v i d u a l c h a r a c t e r i s t i c s as follows: sex, Verbal IQ, Non-verbal IQ, performance i n mathematics during the pre-vious and the current school year, f i e l d dependence-indepen-dence, and the ref l e c t i v e - i m p u l s i v e tempo. Controls Precautions were taken to control teaching s t y l e and Hawthorne-type e f f e c t s . No attempt was made to sel e c t teachers with a s i m i l a r teaching s t y l e . However, upon questioning the teachers, i t was found that none had had previous experience i n the lab-oratory approach when teaching mathematics. Although the novelty of doing mathematics laboratories was a factor i n the experiment, a l l students experienced the same effect, and therefore i t can be assumed that a l l students were affected equally. Subjects The population chosen was Grade Six students from some 27 B r i t i s h Columbia schools i n the lower mainland. Four classes were selected i n two schools which would provide the study with pupils of varying a b i l i t y and with d i f f e r e n t teachers. Laboratories were assigned on a random basis to each c l a s s . A necessary r e s t r i c t i o n on the assignment of the laboratories was that no two classes could do the same laboratory on the same day due to the necessity of sharing equipment. Laboratories Ideas for the laboratories were selected from number theory and geometry. An attempt was made to choose topics which would, i n a l l p r o b a b i l i t y , be unfamiliar to the p u p i l s . This was done i n order to avoid the need to pretest for p r i o r knowledge. Each laboratory had two parts. The f i r s t part was designed to allow pupils the opportunity of using mani-pulative materials to explore the idea and to c o l l e c t data. Teachers were permitted to a s s i s t pupils i n handling the materials i n a manner appropriate to the problem. In the second part, which was used as the tes t section, pupils were required to analyze t h e i r data, make a pr e d i c t i o n , and v e r i f y the p r e d i c t i o n by using manipulative materials before extending the pattern or rule discovered. (See Appendix I) I n i t i a l l y the laboratories were prepared for a p i l o t study using two s i x t h grade classes. A f t e r completion of the laboratories, part two of each laboratory was scored. A r e l i a b i l i t y c o e f f i c i e n t was then obtained for each laboratory 28 using the Kuder-Richardson Formula (KR20). Since each lab-oratory could be rela t e d to either number theory or geometry, the laboratories with t h e i r r e l i a b i l i t y c o e f f i c i e n t s are l i s t e d i n the two categories. Table 1 R e l i a b i l i t y C o e f f i c i e n t s for Number Theory Laboratories Laboratory R e l i a b i l i t y C o e f f i c i e n t Divisors of a Number .52 Figurate Number .54 Squares i n a Square .74 (sums of square numbers) Primes and Composites .82 Common Divisors . 86 Least Common Multiple .93 29 Table 2 R e l i a b i l i t y C o e f f i c i e n t s for Geometry Laboratories Laboratory R e l i a b i l i t y C o e f f i c i e n t Networks .49 Traceable Networks .54 Euler's Formula .67 C l a s s i f i c a t i o n of Triangles .66 Triangles i n Polygons .69 Making Polygons using Triangles and Squares .73 Pic's Theorem .77 Diagonals of Polygons .85 In each category the four laboratories with the highest r e l i -a b i l i t y c o e f f i c i e n t s were selected for use i n the experiment. (See Appendix I) For reference when marking laboratories i n the main study, a record was kept of the responses of patterns and generalizations that were accepted i n the p i l o t . Measurements of Individual C h a r a c t e r i s t i c s Sex and Mathematics Performance Some measures were gathered from each subject's Permanent Record Card. Schools i n B r i t i s h Columbia are required by the Department of Education to keep th i s record card. The card 30 contains not only information about a student and his family, but also a record of the r e s u l t s of a l l standardized tests given to the student and the r e s u l t s of his achievement i n school subjects at each grade l e v e l . The core subjects, which include Language, S p e l l i n g , Reading, S o c i a l Studies, Arithmetic,. and Science, are graded on a seven point l e t t e r grade system. This Permanent Record Card was used to check the sex and age of each subject and to f i n d the subject's achievement l e v e l i n Mathematics for Grade 5 and Grade 6. The Grade 6 achievement l e v e l , l a t e r r e f e r r e d to as present performance, was an extremely current r a t i n g since the record cards were i n the process of being completed for the school year at the time data were gathered. IQ Scores Verbal and Non-verbal IQ scores were obtained from the Canadian Lorge-Thorndike I n t e l l i g e n c e Test which was admini-stered and scored by the experimenter. Field-Dependence-Independence Scores for t h i s c h a r a c t e r i s t i c were obtained from the Group Embedded Figures Test administered and scored by the experimenter. This- t e s t was designed by P h i l l i p K. Oltman, Evelyn Raskin, and Herman A. Witken (1971) and published by the Consulting Psychologists Press, Inc. 31 Reflective-Impulsive Tempo Subjects completed a questionnaire which was intended to indicate the r e f l e c t i v e - i m p u l s i v e tempo of each p u p i l . This questionnaire was a s i m p l i f i e d version of Part C of a questionnaire used i n K i l p a t r i c k (1967) "Analyzing the Solution of Word Problems i n Mathematics: An Exploratory  Study". Procedure Before the experiment began the experimenter met with the teachers involved to explain t h e i r r o l e . Emphasis was placed on the need to help students on the f i r s t part of the laboratories only. Arrangements were made for d e l i v e r i n g and c o l l e c t i n g the laboratories each day and for administering the tests and questionnaire following the laboratory sessions. Each day during the experiment the laboratories and required materials were delivered to the teachers. Time was provided for the teacher to try some of the questions i n the lab and to ask any questions concerning the lab or the pro-cedure. The previous day's laboratory and equipment was picked up at t h i s time. Following the completion of a l l eight laboratories the teachers administered the r e f l e c t i v e - i m p u l s i v e tempo question-naire. Three add i t i o n a l days were then required for the admin-i s t r a t i o n of the Lorge-Thorndike IQ Test and the Group Embedded Figures Test, by the investigator.:. . 32 Part two. of each laboratory was marked. The answers r e l a t i n g to patterns and generalizations were marked with reference to the acceptable responses made during the p i l o t study. Subjects' scores were kept i n the two categories of number theory and geometry. To make the summing of scores possible, the scores on each lab were translated to normalized standard scores. Due to absence, some of the subjects did not complete a l l the labs. The t o t a l score for each category was calculated only for those subjects who had completed a l l the laboratories i n the category. Also a grand t o t a l score was calculated for those students who had completed a l l eight laboratories. The r e f l e c t i v e - i m p u l s i v e questionnaire, the Lorge-Thorndike IQ Test, and the Group Embedded Figures t e s t were hand marked and scored by the investigator. Results were recorded for each p u p i l . Again, due to absence, not a l l subjects wrote a l l t e s t s . S t a t i s t i c a l Procedure The analysis to follow i s considered i n terms of seven p u p i l c h a r a c t e r i s t i c s and the scores on the mathematics lab-o r a t o r i e s . The pu p i l c h a r a c t e r i s t i c s were Verbal IQ, Non-verbal IQ, r e f l e c t i v e - i m p u l s i v e tempo, field-dependence- inde-pendence, mathematics performance i n Grade 5, mathematics performance i n Grade 6, and sex. The mathematics laboratories were used as a whole unit- and were subdivided into the two 33 categories of number theory oriented labs and geometry oriented labs. Since not a l l tests were validated on groups s i m i l a r to that of the study, the r e l i a b i l i t i e s of the laboratories, the Group Embedded Figures Test, and the re f l e c t i v e - i m p u l s i v e tempo questionnaire were calculated i n terms of a Cronbach Alpha. The Cronbach Alpha was u t i l i z e d i n order to decide whether i t would be reasonable to sum the i n d i v i d u a l laboratory scores to get Number Theory and Geometry subtest scores. A l l data were analyzed at the University of B r i t i s h . Columbia Computing Centre using a r e l i a b i l i t y of test program, a c o r r e l a t i o n program, and an analysis of covariance program. 34 CHAPTER 4 RESULTS OF THE STUDY Results R e l i a b i l i t y figures i n terms of a Cronbach Alpha are given i n Table 3 for a l l the laboratories, the Number Theory laboratories, the Geometry laboratories, the Group Embedded Figures t e s t (GEFT), and the re f l e c t i v e - i m p u l s i v e question-naire (RI Quest). Table 3 R e l i a b i l i t i e s of Test Instruments Test Cronbach Alpha A l l Laboratories .77 Number Theory Labs .67 Geometry Labs .53 GEFT .85-RI Questionnaire .02 Because of the low r e l i a b i l i t y of the RI Quest (.02), no analysis was performed involving the ref l e c t i v e - i m p u l s i v e tempo c h a r a c t e r i s t i c . Although some of the laboratory r e l i a b i l i t i e s appear to be low, i t was decided to continue the planned analysis, based on Davis 1 i n t e r p r e t a t i o n of r e l i a b i l i t i e s . Davis suggested that r e l i a b i l i t i e s as low as .5 are serviceable when measuring c h a r a c t e r i s t i c s of c l a s s -s i z e groups, and even lower r e l i a b i l i t i e s may y i e l d useful information when the group i s larger (Davis, 1964, p. 24). The c o r r e l a t i o n between performance on the Number Theory labs and the Geometry labs was .67. Therefore i t was decided to consider these at t r i b u t e s separately as-well as.in t o t a l . The appropriate c o r r e l a t i o n between each p u p i l char-a c t e r i s t i c and the mathematics laboratories i s shown i n Table 4. Table 4 Correlations Between Pupil C h a r a c t e r i s t i c s and Mathematics Laboratories C h a r a c t e r i s t i c s Laboratories No. Theory Labs Geometry Labs A l l Labs Sex -.07 .22* .10 Verbal IQ .41** .52**. .49** Non-verbal IQ .47** .48** .51** Math Performance Grade 5 .51** .48** .58** Math Performance Grade 6 .49** .34** .49** GEFT .43** .47** .51** * S i g n i f i c a n t at the .05 l e v e l ** S i g n i f i c a n t at the .01 l e v e l 36 The table shows that a l l p u p i l c h a r a c t e r i s t i c s except sex were s i g n i f i c a n t at the .01 l e v e l . Since the Lorge Thorndike IQ test was administered a f t e r the completion of the laboratories, t h i s variable could not be used as a covariate. Past achievement, therefore, was selected as the covariate. The re s u l t s of the analysis of covariance follow i n Tables 5 to 8. Table 5 Means and Adjusted Means Cha r a c t e r i s t i c Laboratories A l l No. Theory ^Geometry Labs Boys Mean 1 9 8 . 3 1 9 0 . 5 3 8 9 . 9 •Adjusted Mean 1 9 1 . 7 1 8 0 . 5 3 7 2 . 1 G i r l s Mean 2 0 0 . 9 2 0 3 . 7 4 0 8 . 8 •Adjusted Mean 2 0 4 . 9 2 0 0 . 6 4 0 7 . 7 Field-Dependent Mean 1 9 4 . 2 1 9 0 . 5 3 8 9 . 1 •Adjusted Mean 1 9 6 . 8 2 0 0 . 6 4 0 6 . 2 FieId-Independent Mean 2 0 4 . 9 2 0 3 . 7 4 0 9 . 6 •Adjusted Mean 2 0 5 . 0 2 0 6 . 8 4 1 1 . 5 •Means were adjusted by using previous mathematics achievement as a covariate. . ' 37 Table 6 Analysis of Covariance Sex vs Field-Dependence-Independence on Number Theory Laboratories Source d.f. F P Sex 1 0.252 0. 623 GEFT 1 3.523 0.062 Sex X Geft 1 0. 242 0.630 Error 58 Total 61 Table 7 Analysis of Covariance Sex vs Field-Dependence-Independence on Geometry Laboratories Source d.f. F P Sex 1 5.199 0 .025* GEFT 1 5.010 0 .028* Sex X Geft 1 1.443 0 .233 Error 57 Total 60 * p<.05 38 Table 8 Analysis of Covariance Sex vs Field-Dependence-Independence on A l l Mathematics Laboratories Source d.f. F. P Sex 1 3.308 0.072 GEFT 1 3.368 0.070 Sex X Geft 1 2.101 0.150 Error 46 Total 49 1 Discussion of the Results The data presented i n Table 4 indicated that a l l the pupil c h a r a c t e r i s t i c s except the sex a t t r i b u t e was s t a t i s t i -c a l l y s i g n i f i c a n t at the .01 l e v e l . This l e v e l of si g n i f i c a n c e was attained whether the mathematics laboratories were taken as a whole or subdivided into the categories of number theory laboratories and geometry laboratories. The sex c h a r a c t e r i s t i c s showed si g n i f i c a n c e at the .05 l e v e l with respect to the geometry laboratories only. Since the correlations at the .01 l e v e l of signicance were . 3 4 — r ^ . 5 8 , the variance a t t r i -buted to these c h a r a c t e r i s t i c s ranged from . 1 2 — r ^.34. The . s t a t i s t i c a l hypotheses r e s u l t s follow. 39 Hypothesis H^ which predicted no s i g n i f i c a n t sex r e l a t i o n -ship i n performance on mathematics laboratories had mixed r e s u l t s . With respect to number theory laboratories or a l l laboratories, the hypothesis was not rejected. But with respect to geometry laboratories, the hypothesis was rejected at the .05 l e v e l of s i g n i f i c a n c e . This difference i n per-formance on geometry laboratories was i n favour of the g i r l s . Hypothesis H 2(a) which predicted no s i g n i f i c a n t r e l a t i o n -ship between Verbal IQ and performance on number theory lab-o r a t o r i e s , geometry laboaratories, or a l l laboratories was rejected at the .01 l e v e l of s i g n i f i c a n c e . Predictably t h i s was i n favour of higher Verbal IQ pu p i l s . Hypothesis (b) which predicted no s i g n i f i c a n t r e l a t i o n -ship between Non-verbal IQ and performance on number theory laboratories, geometry laboratories, or a l l laboratories was rejected at the .01 l e v e l of s i g n i f i c a n c e . This was i n favour of pupils with higher Non-verbal IQ.\ Hypothesis H^(a) which predicted no s i g n i f i c a n t r e l a t i o n -ship between general mathematics performance during the past year and performance on number theory labor a t o r i e s , geometry laboratories, and a l l laboratories was rejected at the .01 l e v e l of s i g n i f i c a n c e . This difference was i n favour of higher general mathematics performance. Hypothesis H^b) which predicted no s i g n i f i c a n t ^relation-ship between general mathematics performance during the current year and performance on number theory laboratories was rejected 40 at the .01 l e v e l of s i g n i f i c a n c e . This difference was i n favour of high general mathematics performance. Hypothesis H^ which predicted no s i g n i f i c a n t r e l a t i o n -ship between field-dependence-independence and performance on number theory laboratories, geometry laboratories, and a l l laboratories was rejected at the .01 l e v e l of s i g n i f i -cance. This difference was i n favour of the field-independent p u p i l s . Hypothesis H,. which predicted no s i g n i f i c a n t r e l a t i o n -ship between a measure of ref l e c t i v e - i m p u l s i v e tempo and performance on laboratories was not tested because the r e l i a b i l i t y of the ref l e c t i v e - i m p u l s i v e questionnaire was too low. Hypothesis Hg(a) which predicted no s i g n i f i c a n t difference between the mean score for g i r l s and boys on the number theory laboratories when past achievement was cont r o l l e d was not rejected. Hypothesis H,(b) which predicted no s i g n i f i c a n t difference between the mean score for g i r l s and boys on the geometry laboratories when past achievement was cont r o l l e d was rejected at the .05 l e v e l of s i g n i f i c a n c e . This difference was i n favour of the g i r l s . Hypothesis Hg(c) which predicted no s i g n i f i c a n t difference between the mean score for g i r l s and boys on a l l laboratories •when past achievement was cont r o l l e d was not rejected. Hypotheses H^ and Hg could not be tested because both hypotheses required the use of the re f l e c t i v e - i m p u l s i v e tempo questionnaire which was un r e l i a b l e . 41 Hypothesis H^ which predicted no s i g n i f i c a n t difference between the means of field-dependent boys and field-dependent g i r l s and the means of field-independent boys and f i e l d -independent g i r l s when past achievement i s co n t r o l l e d on a l l mathematics laboratories was not rejected. 42 CHAPTER 5 SUMMARY and CONCLUSIONS The Problem The purpose of t h i s study was to f i n d some c h a r a c t e r i s t i c s of pupils that related well to t h e i r performance on mathematics laboratories. The c h a r a c t e r i s t i c s selected were sex, i n t e l l i -gence, fiieJ-'d'-dependence-independence , previous . mathematics achieve-ment, and current mathematics achievement. The Findings The r e s u l t s of the data analysis indicated that, i n general, there was a r e l a t i o n s h i p between the c h a r a c t e r i s t i c s of i n t e l l i -gence, field-dependence-independence, previous mathematics achievement, and current mathematics achievement. The range of the c o r r e l a t i o n s was . 34< r . 58. A l l these co r r e l a t i o n s were s t a t i s t i c a l l y s i g n i f i c a n t at the .01 l e v e l . The better performance on the mathematics laboratories was related to higher i n t e l l i g e n c e , greater field-independence, and higher mathematics achievement. When the laboratories were subdivided t o p i c a l l y into number theory laboratories and geometry laboratories, the sex c h a r a c t e r i s t i c showed si g n i f i c a n c e only on the geometry laboratories. The correlation, of . 2 2 , which was s t a t i s t i c a l l y • s i g n i f i c a n t at the .05 l e v e l , was i n favour of the g i r l s . 4 3 Implications Although most of the corr e l a t i o n s between the selected p u p i l c h a r a c t e r i s t i c s and the performance on mathematics lab-oratories were s t a t i s t i c a l l y s i g n i f i c a n t , the corr e l a t i o n s were too low for many implications to be drawn as a conse-quence of t h i s study. However, the low corr e l a t i o n s between both Verbal and Non-verbal IQ and the performance on the mathematics labor-atories indicate that only approximately 25% of the variance i s accounted for by the i n t e l l i g e n c e c h a r a c t e r i s t i c s . There-fore, grouping on the basis of i n t e l l i g e n c e for mathematics laboratories may be inappropriate. I t was also noted that the c o r r e l a t i o n between pupi l achievement and performance on mathematics laboratories re-lated to geometry dropped from .48 with past achievement to .34 with present achievement. Some of t h i s difference may be at t r i b u t e d to the increase i n computational work i n the curriculum between years f i v e and s i x . As a r e s u l t ' o f the increased load with respect to computation i n year s i x , geometry i s forced to play a more minor r o l e , and the achieve-ment ra t i n g for t h i s year was probably mainly a r e f l e c t i o n of the pupils' computational s k i l l . If geometry i s intended to be a i n t e g r a l part of the elementary mathematics program, the teachers need to be encouraged to place more emphasis on geometry at the year s i x l e v e l . Limitations The l i m i t a t i o n s of t h i s study f a l l i nto two categories: those that the investigator found unavoidable and those that appeared while the study was i n progress. The unavoidable l i m i t a t i o n s involve the population, the materials, the timing and teacher c h a r a c t e r i s t i c s . Limitations which were unexpected but appeared while the study was i n progress involved v a r i a -tions i n the teachers' a b i l i t y and a t t i t u d e , teacher-pupil i n t e r a c t i o n , and previous learning experiences of p u p i l s . The population sample was. r e s t r i c t e d . Although four classed of students were used, these students came from only two schools i n s i m i l a r economic areas. Clearly a more representa-t i v e sample of the population could have produced d i f f e r e n t r e s u l t s . Because the mathematics laboratories were experimenter-designed and because the r e l i a b i l i t i e s of the tests varied considerably, the accuracy of the scores on the laboratories may be questionable. Laboratory assignments could have been scheduled more e f f e c t i v e l y i n order to achieve maximum psychological e f f e c t . Each day the pupils were given a lab involving a d i f f e r e n t topic. The pupils received no feedback on the previous lab-oratory related either to the acceptable responses or to t h e i r own performance. This created anxiety for some pupils and may have affected the performance of pupils on the laboratories. 45 Two areas of weakness with respect to the p a r t i c i p a t i n g teachers were found. F i r s t , the teachers generally were un-s k i l l e d i n using manipulative materials with students. A l -though informal inservice t r a i n i n g did take place, formal ins e r v i c e t r a i n i n g involving the use of manipulative materials could conceivably have influenced the r e s u l t s of the study. Second, i t was found that hal f of the pupils involved i n the study had teachers who f e l t that using manipulative materials was a waste of time. The performance scores on the labora-t o r i e s for these pupils were probably affected by the attitude of the teachers. The structure of the laboratories allowed for no d i s -cussion either between teacher and pu p i l or between pu p i l and pup i l during the time that the pupils were attempting to f i n d a pattern and t r y i n g to formulate a generalization. This loss of i n t e r a c t i o n was probably too severe for pupils con-sidering t h e i r age and t h e i r previous experience at genera l i -zing i n mathematics. Again, the scores were probably affected. Many pupils had d i f f i c u l t y i n formulating generalizations. Even the i n i t i a l steps were beyond the c a p a b i l i t y of some pupils. If the data were natu r a l l y organized by the format of the laboratory, many pupils were able to detect the pattern and extend i t . I f , however, the c o l l e c t e d data required organi-zation, some pupils were unable to organize i t and the data therefore, -were u n l i k e l y to recognize the pattern. For pupils who were able to organize the data, detect a pattern, and extend i t , many seemed unable to decode the pattern which would allow 46 for the expression of a generalization. Even the simple decoding s k i l l s of looking at differences or renaming numbers were not obvious to many pupils. Pupils who were able to decode the pattern then had d i f f i c u l t y expressing what they had discovered either i n statement form or i n a mathematical sentence. Further Research Directions for ..further research. are suggested .by some of the l i m i t a t i o n s of t h i s study. Certainly i t would be desirable to have a broader sampling of the population. Perhaps the sample could span ages as well as economic background. In addition, an inservice program for teachers involving the use of mani-pulative materials would be b e n e f i c i a l . This would not only improve the manner i n which the materials were used, but also provide the experimenter with an opportunity to improve teacher attitude concerning manipulative materials. A f t e r seeing the weakness i n the s k i l l s of organizing data, processing data, and communicating a generalization, i t appears that some pre-teaching of these s k i l l s should be done. Scandura (1971) examines the s k i l l s for detecting r e g u l a r i t i e s and suggests techniques for teaching pupils to describe ideas (pp. 6-40). I f these s k i l l s were pre-taught, then a c r i t e r i o n l e v e l t e s t could be incorporated into the. i n v e s t i g a t i o n . * Considerable scope e x i s t s for variables which could be correlated with performance on mathematics laboratories. Some 47 possible c h a r a c t e r i s t i c s that could be used are transience of students, learning environment, mixed l a t e r a l i t y , convergent-divergent thinking, family size and b i r t h order, and type of family housing. Certainly the structure of the laboratories could be changed, and they could be c l a s s i f i e d d i f f e r e n t l y . Labora-t o r i e s could be related to a topic i n the curriculum rather than the topics being extensions of the curriculum. The laboratories could a l l be r e l a t e d to a single topic, or, i f time permitted, be related to a number of topics. C l a s s i f y -ing the laboratories by the type of generalization involved or by the processing s k i l l s involved i n detecting the general-i z a t i o n are two other possible methods of categorizing. The laboratories could be used i n a d i f f e r e n t manner and preferably over a longer period of time. Groups using the laboratories could be compared to groups unexposed to t h i s method of teaching. Groups taught using only a laboratory method could be compared to groups where the laboratories are used only when introducing a topic or compared to groups where laboratories were interspersed with lessons using various other teaching methods. Another possible study using group comparison i s one which involves groups of d i f f e r i n g a b i l i t y and groups of s i m i l a r a b i l i t y . One further p o s s i b i l i t y i s a study comparing groups where the mathematics laboratory has been used with an ent i r e c l a s s , with small groups, and as independent study work. 48 C l e a r l y more inv e s t i g a t i o n i s needed to enable educators to determine the value of using a laboratory method when teaching mathematics. The manner i n which laboratories can be used most e f f e c t i v e l y should be examined more i n t e n s i v e l y . Thus, i t would appear that considerable scope e x i s t s for useful research involving the mathematics laboratory. 49 BIBLIOGRAPHY Anastasi, A. Individual Differences. New York: John Wiley and Sons, Inc., 1965. Anderson, R.C. Individual differences and problem solving. In R.M. Gagne (Ed.), Learning and Individual Differences. Columbus: Charles E. M e r r i l l Books, Inc., 1967. American Educational Research Association. Review of Education  Research, 1969, 39(3). American Educational Research Association. 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The Arithmetic Teacher, 1973, 2_0 (6) , 425-436. 54 Unkel, E.R. A Study of the Laboratory Approach and Guided Discovery i n the Teaching Learning of Mathematics by Children  and Prospective Teachers. Boca Raton, F l o r i d a : F l o r i d a A t l a n t i c University, 1971. Vance, J.H. & Kiernan, T.E. Mathematics laboratories - more than fun? School Science and Mathematics, 1972, 72 (7) , 617-623. Vance J.H. & Kiernan, T.E. Laboratory settings i n mathematics: what does research say to the teacher? The Arithmetic Teacher, 1971, 18 (8) , 585-589. Van Dalen, D.B. Understanding Educational Research. New York: McGraw-Hill Book Company, 19 66. Waetjen, W.B. Is learning sexless? NEA Journal, 1962, 51, 12-14. Weaver, J.F. Seductive shibboleths. The Arithmetic Teacher, 1971, 18(4), 263-264. Westcott, A.M. & Smith, J.A. Creative Teaching of Mathematics  i n the Elementary School. Boston: A l l y n and Bacon, Inc., 1967. Whipple, R.M. A s t a t i s t i c a l comparison of the effectiveness of teaching metric geometry by the laboratory and individua-l i z e d i n s t r u c t i o n approaches. (Doctoral d i s s e r t a i o n , Northwestern University, 19 72). D i ssertation Abstracts  International, 33, 2699A-2700A. (University Microfilms No. 72-32, 611). Wilderman, A.M. & Kr u l i k , S. On beyond the mathematics labor-atory. The Arithmetic Teacher, 1973, 20(7), 543-544. Wilkinston, J. The Laboratory Method as a Teaching Strategy i n Grades 5-8. Paper presented at the NCTM Meeting, Winnipeg, Manitoba, October 1970. Wilkinson, J.D. Teacher-directed evaluation of mathematics laboratories. The Arithmetic Teacher, 1974, 2^(1), 12-24. Witkin, H.A. A cog n i t i v e - s t y l e approach to c r o s s - c u l t u r a l research. International Journal of Psychology, 1967, 2^(4), 233-250. Witkin, H.A. Some implications of research on cognitive s t y l e for problems i n education. Professional School Psychology, 1969, 3, 198-227. 55 Witkin, H.A., Goodenough, D.R. & Karp, S.A. S t a b i l i t y of cognitive s t y l e from childhood to young adulthood. Journal  of Personality and S o c i a l Psychology, 1967, 7_(3) , 291-300. Witkin, H.A., Oltman, P.K., Raskin, E. & Karp, S.A. A Manual  for the Embedded Figures Test. Palo A l t o : Consulting Psychologists Press, Inc., 1971. Wright, J.C. & Kagan J . (Eds.). Basic cognitive processes i n c h i l d r e n . Monographs of the Society for Research i n Child Development, 1963, 28_, 2. 56 APPENDIX I 57 A NUMBER PATTERN Materials: Graph Paper, Ruler, P e n c i l , Cubes Part I: 1. A square l x l drawn on the graph paper shows exactly one square shape. 2. Draw a 2 x 2 square on the graph paper. How many d i f f e r e n t square shapes can you see i n the drawn square? Record t h i s information on the table below. 3. Repeat question 2 for squares that are 3 x 3 , 4 x 4 , and 5 x 5 . Record the information for each si z e of square on the table below. Size of Square No. of Squares l x l 2 x 2 3 x 3 4 x 4 5 x 5 i Part I I : 1. Without using the graph paper, add to the table the informa-ti o n for a 6 x 6 square. Now check your r e s u l t using graph paper. 2. Now add to the table the information for a 7 x 7 square and an 8 x 8 square. 3. Examine the column i n the table that gives the number of squares. • (a) What would the 20th terms of t h i s sequence be? (b) Write a mathematical sentence for the pattern of thi s sequence. 58 Use cubes to help you f i n d the sequence of numbers for the number of cubes i n a cube. Write the f i r s t f i v e numbers i n the sequence on the l i n e below. 59 PRIMES AND COMPOSITES Materials: 30 Counters Part I: 1. A p a r t i c u l a r number of counters can be arranged into d i f f e r -ent rectangular arrays. Sixteen counters can be arranged as shown below. xxxxxxxxxxxxxxxx xxxxxxxx xxxx X X X xxxxxxxx xxxx X X X 1 by 16 xxxx X X X 2 by 8 xxxx X X X X X X 4 by 4 X X X X X X X X X X 8 by 2 X X x X X X X 16 by 1 2. Record on a table l i k e the one below information about the rectangular arrays for each whole number from one to t h i r t y . Number Arrays Number of Arrays 1 1 by 1 ' 1 2 3 4 1 by 4, 2 by 2, 4 by 1 • 3 5 30 60 Look at your table and record below any numbers that have (a) only one array (b) only two arrays ._. (c) more than two arrays (d) an " Q b y Q " array A number having only two arrays i s c a l l e d a PRIME number. A number having more than two arrays i s c a l l e d a COMPOSITE number. A number having an " • b y D " array i s c a l l e d a SQUARE number. Part I I : Find the f i r s t prime number afte r 30. Find the f i r s t composite number af t e r 30. Find the f i r s t square number a f t e r 30. Think of the square number greater than 1. Are they prime or composite? Is 42 a prime or composite number? Why ; L i s t a l l the arrays for 50. 61 C. D.'s Materials: 1/2" Grids, Scissors, Ruler, P e n c i l , Coloured Pencils Part I: 1. From one sheet of gridded paper cut out some l x l , 2 x 2 , 3 x 3 , 4 x 4 , and 5 x 5 squares. These squares w i l l be c a l l e d t i l e s . 2. On another sheet of gridded paper draw a rectangle 4 x 8 to represent a f l o o r plan for a kitchen nook. Without cutting any of the t i l e s , can you completely cover the f l o o r plan with your l x l t i l e s ? 3. Record on the chart below the t i l e s i z e that could be used for t h i s kitchen nook. 4. Complete the chart below for the other room sizes given. Use your gridded paper and cut t i l e s to help you. Perhaps you can f i n d the reason why only some t i l e sizes work. Floor Size LENGTH of side of each size of square t i l e used 4 x 8 6 x 8 7 x 12 10 x 15 10 x 4 18 x 12 11 x 7 Part I I : 1. The number pairs given below could be thought of as room sizes . Without using your t i l e s and gridded paper, give a l l the d i f f e r e n t sizes of t i l e s that could be used for t i l i n g these f l o o r s . (a) 9, 15 (b) 5, 7 Now check your r e s u l t s using the gridded paper and t i l e s . 62 The sizes of t i l e s that can be used are the d i v i s o r s that are the same for both numbers. Divisors that are the same for two or more numbers are c a l l e d ' COMMON DIVISORS. Find the common d i v i s o r s for the pairs of numbers below. (a) 9, 18 (b) 30, 24 (c) 25, 18 (d) 48, 20 Find the common d i v i s o r s for the two groups of numbers below. (a) 12, 36, 16 (b) 50, 30, 15 .  63 L.C.M. Materials: Gridded Paper (1/4"), P e n c i l , Ruler, Coloured Pencils, Scissors. Part I : 1. Cut out several rectangles from the gridded paper that are 4 x 2 , 1 x 3 , 2 x 1 , 3 x 4 , 5 x 1 0 , 2 x 3 , 9 x 6 , 5 x 3 , and 4 x 6 . 2. Take the cut out rectangles that are 4 x 2 . Arrange the rectangles to form a square. Find the smallest sized square that you can. Record the size of the smallest square that you can make on the chart below. Repeat the above for each set of rectangles that you have made. Record the size of the smallest square that you can f i n d for each set. Size of Rectangle Size of Smallest Square 4 x 2 2 x 1 3 x 4 5 x 10 2 x 3 9 x 6 5 x 3 4 x 6 I Part I I : 1. Predict the smallest size of square that can be made using rectangles 2 x 5 . Check your answer using gridded paper. The smallest square that can be made from a set of rectangles the same size i s the square whose edge length i s the smallest number that both dimensions of the rectangle w i l l d ivide. Example: For a rectangle 4 x 6 , the smallest sized square i s 12. The smallest square i s 12 because 12 i s the smallest number that both 4 and 6 w i l l divide. 12 i s c a l l e d the LOWEST COMMON MULTIPLE for 4 and 6. Find the lowest common multiple (smallest sized square) for the pairs of numbers below. (a) 1, 3 (b) 4, 8 (c) 5, 6 (d) 8, 12 (e) 6, 3 (f) 7, 2 There i s a lowest common multiple for a group of three or more numbers also. Find the lowest common n u l t i p l e for the groups of numbers below. (a) 3, 2, 4 (b) 5, 20, 8 (c) 12, 8, 3 TRIANGLES IN POLYGONS Materials: Set of Plane Figures, Ruler, Pen c i l Part I: ' -Draw l i n e s to cut the polygon into only triangular pieces. Find the lea s t number of triangular pieces that a shape can be cut into. Record t h i s information on the chart below. Number of Sides 3 4 5 6 7 8 9 10 11 12 13 14 15 Number of Triangles Part I I : What i s the l e a s t number of A for an 11-sided polygon? Test your answer by drawing an 11-sided polygon on the back and show the l i n e s . Complete your table. Predict the lea s t number of tri a n g l e s for a 48-sided polygon. Can you f i n d a pattern for the l e a s t number of tri a n g l e s for a polygon? I f you can, write a number sentence about i t . Would your sentence hold for the two polygons below? Draw i n the necessary l i n e segments on the polygons below to show the lea s t number of triangular pieces. 67 MAKING POLYGONS Materials: Patterns for two r i g h t isoceles t r i a n g l e s and two squares, p e n c i l , s t r a i g h t edge, s c i s s o r s , and large pieces of newsprint. 1. Cut out the shapes on the back page to use as patterns. 2. You are going to be asked to make polygons by tracing a-round two or more of your patterns. A rule that must a l -ways be followed i s that when two shapes are f i t t e d toget-her, the edges that meet must be the same length. This would be a l l . r i g h t . This would not be r i g h t . 3. Using just two t r i a n g l e s , we can f i t them together to make another t r i a n g l e Try to make a t r i a n g l e by f i t t i n g together just (a) 3 tria n g l e s (b) 4 triang l e s (c) 5 t r i a n g l e s . Show your r e s u l t s on the large sheet of newsprint. 4. Try to make a square using just (a) 2 tri a n g l e s (b) 3 tria n g l e s (c) 4 tri a n g l e s (d) 5 t r i a n g l e s . 5. Try to make a rectangle using just (a) 2 tri a n g l e s (b) 3 tri a n g l e s (c) 4 tr i a n g l e s (d) 5 t r i a n g l e s . 6. Try to make a square using just (a) 2 squares (b) 3 squares (c) 4 squares .(d) 5 squares (e) 6 squares. 7. Try to make a rectangle using just (a) 2 squares (b) 3 squares (c) 4 squares (d) 5 squares (e) 6 squares. 8. Make as many d i f f e r e n t shapes as you can using j u s t (a) 1 t r i a n g l e and 1 square (b) 2 tr i a n g l e s and 1 square (c) 2 tri a n g l e s and 3 squares. Part I I : 1. Show two d i f f e r e n t ways of f i t t i n g squares and triangles.together to make the shape shown at the r i g h t ... (Use your s t r a i g h t edge and pe n c i l to draw l i n e s on each to show how you would do i t . ) Part I: 68 By making one st r a i g h t cut on each of the figures on the l e f t , you could f i t the r e s u l t i n g pieces together to make the figures on the r i g h t . Use your straight edge and pe n c i l to draw a l i n e on the l e f t figure to show where you would cut i t i f you had to. Where would you cut thi s to make t h i s Examine the figure below c l o s e l y . Using only the l i n e segments given, count and record the maximum number of each named polygon that you can see i n the diagram. (a) t r i a n g l e (b) square (c) rectangle Use your st r a i g h t edge and pe n c i l to draw l i n e s on each diagram to show how you would cut i t i f you had i t , to make what i s asked f o r . You must use a l l of the shape given i n the diagram. (a) How would you cut th i s rectangle to make 3 triangles? (b) How would you cut th i s q u a d r i l a t e r a l to make a rectangle and 2 triangles? (c) How would you cut t h i s rectangle to make a square and a rectangle? 69 PIC'S THEOREM Part I: 1. Construct a polygon with 4 pegs on i t s boundary and no  pegs inside. :, The area of t h i s polygon i s . Construct other polygons with 4 pegs on t h e i r boundaries and no pegs inside. What i s the area of each polygon? 2. 4. 2. Construct three d i f f e r e n t polygons each with 5 pegs on t h e i r boundaries and no pegs in s i d e . What i s the area of each polygon? F i l l i n the table below, where A means number of pegs on the boundary and Q means area (in square units) A 1 2 • If two students each make a polygon that has 8 pegs on i t s boundary and no pegs ins i d e , would the area of the two poly-gons be the same? Part I I : Predict the area for a polygon with 10 pegs on i t s boundary and no pegs in s i d e . Check your prediction using the geoboard. If you construct a polygon with f i f t e e n pegs on i t s boundary and no pegs ins i d e , what w i l l be the area of t h i s polygon? 3. Add the information about polygons with 10 to 15 pegs on t h e i r boundaries to your table above. 4. Examine the table carefully. Which number sentence below gives the r e l a t i o n s h i p between the A (number of pegs on the boundary) and Q (area)? C i r c l e the number sentence. (a) & = 2 + • (b) 1 A - 2 = • (c) 1 A " 1 = • 2 2 5. Now construct f i v e polygons on your geoboard that have exactly 1 peg inside i t s boundary.. Count the pegs on the boundary of each and. fi n d the area.of each polygon. 70 Record the information about the polygons i n question 5 on the table below. Remember that A means number of pegs on the boundary and Q means the area of the polygon. A D Write a number sentence using /\ and Q to show the r e l a -tionship between the number of pegs on the boundary and the area when the polygons have 1 peg ins i d e . 71 DIAGONALS OF POLYGONS Materials: Cardboard, Pins, String, P e n c i l Part I: 1. Make a q u a d r i l a t e r a l on the piece of cardboard by marking the ver t i c e s with pins. Use the pieces of s t r i n g to mark the diagonals. Find the (a) number of sides for t h i s polygon (b) the number of diagonals from one vertex (c) the t o t a l number of diagonals. Record t h i s information about the q u a d r i l a t e r a l on the chart below. 2. Repeat part #1 for the t r i a n g l e , pentagon (5 sid e s ) , hexagon (6 sides), and heptagon (7 s i d e s ) . Polygon Number of Sides No. of Diagonals from 1 vertex Total number of Diagonals Triangle Quadrilateral Pentagon Hexagon Heptagon Part I I : 1. Predict the t o t a l number of diagonals for an octagon (8 sides) Now check using your cardboard, pins and s t r i n g . Add t h i s information to your chart. 2. Add to your chart the polygons nonagon (9 sides) and decagon (10 s i d e s ) . Complete the information about these polygons. 3. How many diagonals would a 20-gon have? * 4. Write a mathematical sentence for finding the number of diagonals for a 20-gon. 72 APPENDIX I I . Table 1 - Sex vs Number Theory Labs 249.0 227o0 205 o0 183.0 161*0 139*0 H 249.0 227.0 205.0 183.0 .1 161.0 J 139.0 boys g i r l s Table 2 - Sex vs Geometry Labs 258.0 230*0 202.0 174.0 146.0 118.0 258.0 230.0 202.0 174.0 J 146.0 118.0 boys g i r l s T a b l e 3 - S e x v s A l l L a b s 5 0 0 . 0 4 5 0 „ 0 4 0 0 . 0 3 5 0 . 0 3 0 0 . 0 2 5 0 . 0 5 0 0 . 0 4 5 0 . 0 A 4 0 0 . 0 b o y s g i r l s T a b l e 4 - V e r b a l I Q v s N u m b e r T h e o r y T I 2 4 9 . 0 2 2 7 . 0 2 0 5 . 0 1 8 3 . 0 1 6 1 . 0 1 3 9 . 0 J 2 4 9 . 0 2 2 7 . 0 2 0 5 . 0 J 1 8 3 . 0 1 6 1 . 0 1 3 9 . 0 5 9 ' 7 3 8 7 1 0 1 1 1 5 1 2 9 Table 5 - Verbal IQ vs Geometry 258.0 230.0 202.0 174.0 146.0 118.0 4 258.0 J 230.0 ] 202.0 174.0 146.0 i 118.0 59 73 87 101 115 129 Table 6 - Verbal IQ vs A l l Labs 500.0 450.0 400.0 350.0 300.0 250.0 500.0 A 450.0 400.0 350.0 300.0 1 250.0 59 73 87 101 115 129 Table 7 - Nonverbal IQ vs Number Theory 249.0 227.0 205.0 183.0 161.0 139.0 249.0 227.0 205.0 183.0 161.0 4 139.0 64 76 88 100 112 124 Table 8 - Nonverbal IQ vs Geometry 258.0 230.0 202.0 174.0 146.0 118.0 258.0 230.0 202.0 174.0 J 146.0 J 118.0 64 76 88 100 112 124 T a b l e 9 - N o n v e r b a l I Q v s A l l L a b s 500.0 450.0 L 400.0 350.0 300.0 250.0 1 V I 1 - • - • - • - • - -_ _ J • • • 1 1 — 500.0 450.0 1 400.0 350.0 300.0 250.0 64 76 88 100 112 124 T a b l e 10 - M a t h P e r f o r m a n c e ( G r . 5 ) v s N u m b e r T h e o r y - i r 249.0 227.0 205.0 183.0 161.0 139.0 249.0 227.0 205.0 1 183.0 1 161.0 t139.0 1(E) 5.8 7(A) Table 11 - Math Performance (Gr.5) vs Geometry 258.0 230.0 202.0 174.0 146.0 118.0 258.0 230.0 202.0 174.0 4 146.0 118.0 1(E) 2.2 3.4 4.6 5.8 7(A) Table 12 - Math Performance (Gr.5) vs A l l Labs 500.0 450.0 400.0 350.0 300.0 250.0 4 500.0 4 450.0 400.0 4 350.0 300.0 1 250.0 1(E) 2.2 3.4 4.6 5.8 7(A) Table 13 - Math Performance (Gr.6) vs Number Theory 249.0 227.0 205.0 183.0 161.0 139.0 J 249.0 >27.0 1 205.0 1 183.0 161.0 139.0 2 ( B ) 3 6 7(A) Table 14 - Math Performance (Gr.6) vs Geometry 258.0 230.0 202.0 174.0 146.0 118.0 258.0 230.0 202.0 174.0 146.0 118.0 2(D) 3 5 6 7(A) Table 15 - Math Performance (Gr.6) vs A l l Labs 500.0 450.0 400.0 350.0 300.0 250.0 500.0 450.0 400.0 350.0 300.0 250.0 2(D) 3 6 7(A) Table 16 - GEFT vs Number Theory 249.0 227.0 h 205.0 183.0 161.0 139.0 249.0 227.0 205.0 183.0 161.0 139.0 0 3.6 7.2 10.8 14.4 18 Table 17 - GEFT vs Geometry 258.0 230.0 202.0 174.0 146.0 118.0 258.0 230.0 202.0 174.0 146.0 118.0 0 3.6 7.2 10.8 14.4 18 Table 18 - GEFT vs A l l Labs 500.0 450.0 400.0 350.0 '300.0 250.0 4 500.0 450.0 400.0 -I 350.0 300.0 I 250.0 0 3.6 7.2 10.8 14.4 18 APPENDIX I I I 83 QUESTIONNAIRE ON PROBLEM SOLVING Name ; T e a c h e r F o r each o f t h e f o l l o w i n g s t a t e m e n t s p u t an x i n the sp a c e on t h e l i n e which shows the degree t o which you a g r e e o r d i s a g r e e w i t h t he s t a t e m e n t . 1. I u s u a l l y guess t he answer t o a problem b e f o r e w o r k i n g i t o u t i n d e t a i l . a g r e e s t r o n g l y 2. T a l k i n w o r k i n a g r e e m o d e r a t e l y g about i d e a s g on a r i t h m e t i a g r e e a l i t t l e i n mathemat -c problems. d i s a g r e e a l i t t l e cs i s much rr d i s a g r e e m o d e r a t e l y o r e e n j o y a b l e d i s a g r e e s t r o n g l y t o me than a g r e e s t r o n g l y 3. I woul a g r e e m o d e r a t e l y d r a t h e r ask 1 ag r e e a l i t t l e "or h e l p on i d i s a g r e e a l i t t l e * d i f f i c u l t p d i s a g r e e m o d e r a t e l y r o b l e m than wo d i s a g r e e s t r o n g l y rk on i t a l o n e a g r e e s t r o n g l y 4. I woul ma then a g r e e m o d e r a t e l y d r a t h e r resp( l a t i c s than t o ag r e e a l i t t l e Dnd s l o w l y t gamble on m d i s a g r e e a l i t t l e o a q u e s t i o n y answer. d i s a g r e e m o d e r a t e l y my t e a c h e r has d i s a g r e e s t r o n g l y ; asked me i n ag r e e s t r o n g l y a g r e e m o d e r a t e l y a g r e e a l i t t l e d i s a g r e e a l i t t l e d i s a g r e e m o d e r a t e l y d i s a g r e e s t r o n g l y 5. I have always t h o r o u g h l y e n j o y e d w o r k i n g on m a t h e m a t i c a l games and p u z z l e s . a g r e e a g r e e a g r e e d i s a g r e e d i s a g r e e d i s a g r e e s t r o n g l y m o d e r a t e l y a l i t t l e a l i t t l e m o d e r a t e l y s t r o n g l y 84 6. I can t e l l w i t h o u t c h e c k i n g whether an answer I have g o t t e n i s c o r r e c t . a g r e e a g r e e a g r e e d i s a g r e e d i s a g r e e d i s a g r e e s t r o n g l y m o d e r a t e l y a l i t t l e a l i t t l e m o d e r a t e l y s t r o n g l y 7. I have found t h a t a p e r i o d o f s i l e n c e b e f o r e a n s w e r i n g a q u e s t i o n does not h e l p me. a g r e e s t r o n g l y 8. I t i s than 1 a g r e e m o d e r a t e l y more i m p o r t a n l :o be a b l e t o v a g r e e a l i t t l e : t o be a b l e /ork a v a r i e di s a g r e e a l i t t l e t o t a l k a b o i ty o f mathemc d i s a g r e e m o d e r a t e l y i t i d e a s i n ma1 i t i c a l problems d i s a g r e e s t r o n g l y :hematics • • a g r e e s t r o n g l y 9. I woul a g r e e m o d e r a t e l y d r a t h e r g i v e a g r e e a l i t t l e a wrong ans\ d i s a g r e e a l i t t l e ver t h a n no c d i s a g r e e m o d e r a t e l y tnswer a t a l l . d i s a g r e e s t r o n g l y a g r e e s t r o n g l y 10. The mc a g r e e m o d e r a t e l y )st i m p o r t a n t r a g r e e a l i t t l e *eason f o r 1( d i s a g r e e a l i t t l e e a r n i n g mathe d i s a g r e e m o d e r a t e l y j m a t i c s i s i t s d i s a g r e e s t r o n g l y u s e f u l n e s s . a g r e e s t r o n g l y 11. The be l o t s c a g r e e m o d e r a t e l y ;s t way t o Tear )f them. agre e a l i t t l e *n how t o so d i s a g r e e a l i t t l e Ive mathemati d i s a g r e e m o d e r a t e l y cs problems i s d i s a g r e e s t r o n g l y , t o s o l v e a g r e e s t r o n g l y 12. My f i r m a t i c s a g r e e m o d e r a t e l y *s t c h o i c e o f c a g r e e a l i t t l e i s u b j e c t t o d i s a g r e e a l i t t l e s t u d y i n hie d i s a g r e e m o d e r a t e l y jh s c h o o l woulc d i s a g r e e s t r o n g l y 1 be mathe-a g r e e s t r o n g l y a g r e e m o d e r a t e l y a g r e e a l i t t l e d i s a g r e e a l i t t l e d i s a g r e e m o d e r a t e l y d i s a g r e e s t r o n g l y 85 13. F r e q u e n t l y the answer t o a p r o b l e m o c c u r s t o me a f t e r I have gone on to do something e l s e . a g r e e s t r o n g l y 14. I have a g r e e m o d e r a t e l y a h a r d time c a g r e e a l i t t l e : oncentratin< d i s a g r e e a l i t t l e 3 on my mathe d i s a g r e e m o d e r a t e l y m a t i c s homewor d i s a g r e e s t r o n g l y k. a g r e e s t r o n g l y 15. I l i k e a g r e e m o d e r a t e l y to d i s c o v e r J agr e e a l i t t l e . e v e r a l d i f f d i s a g r e e a l i t t l e a r e n t ways t c d i s a g r e e m o d e r a t e l y > do t h e same p d i s a g r e e s t r o n g l y i r o b l e m . a g r e e s t r o n g l y 16. I p r e i a g r e e m o d e r a t e l y : e r d o i n g t o t\ a g r e e a l i t t l e l i n k i n g . d i s a g r e e a l i t t l e d i s a g r e e m o d e r a t e l y d i s a g r e e s t r o n g l y a g r e e s t r o n g l y 17. I o n l ^ a g r e e m o d e r a t e l y i r a i s e my han< a g r e e a l i t t l e i when I kno d i s a g r e e a l i t t l e w I have the d i s a g r e e m o d e r a t e l y c o r r e c t answer d i s a g r e e s t r o n g l y a g r e e s t r o n g l y a g r e e m o d e r a t e l y a g r e e a l i t t l e d i s a g r e e a l i t t l e d i s a g r e e m o d e r a t e l y d i s a g r e e s t r o n g l y 18. I would r a t h e r ask f o r h e l p on a d i f f i c u l t m athematics problem than have someone t e l l me t h e answer. agre e s t r o n g l y 19. I don a g r e e m o d e r a t e l y t l i k e t o disc a g r e e a l i t t l e :uss the pro d i s a g r e e a l i t t l e blems on a te d i s a g r e e m o d e r a t e l y j s t a f t e r I've d i s a g r e e s t r o n g l y f i n i s h e d i t . a g r e e s t r o n g l y a g r e e m o d e r a t e l y a g r e e a l i t t l e d i s a g r e e a l i t t l e d i s a g r e e m o d e r a t e l y d i s a g r e e s t r o n g l y 86 20. I t ' s b e t t e r t o be s l o w e r and r i g h t a l l o f the time than t o be f a s t e r and r i g h t p a r t o f t h e t i m e . a g r e e s t r o n g l y 21. I alwa answer a g r e e m o d e r a t e l y y s guess on a a g r e e a l i t t l e m u l t i p l e - c h c d i s a g r e e a l i t t l e ) i c e t e s t whe d i s a g r e e m o d e r a t e l y n I don't know d i s a g r e e s t r o n g l y t he r i g h t a g r e e s t r o n g l y a g r e e m o d e r a t e l y a g r e e a l i t t l e d i s a g r e e a l i t t l e d i s a g r e e m o d e r a t e l y d i s a g r e e s t r o n g l y 

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