Mathematics Education

Models and Processes

Lyn D. English, Graeme S. Halford

• Publisher: Routledge
• Year: 2012
• ISBN:
  9781136481628
• Edition: 1
• Topic: Education
• Subtopic: Education General

To define better techniques of mathematics education, this book combines a knowledge of cognitive science with mathematics curriculum theory and research. The concept of the human reasoning process has been changed fundamentally by cognitive science in the last two decades. The role of memory retrieval, domain-specific and domain-general skills, analogy, and mental models is better understood now than previously. The authors believe that cognitive science provides the most accurate account thus far of the actual processes that people use in mathematics and offers the best potential for genuine increases in efficiency. As such, they suggest that a cognitive science approach enables constructivist ideas to be analyzed and further developed in the search for greater understanding of children's mathematical learning. Not simply an application of cognitive science, however, this book provides a new perspective on mathematics education by examining the nature of mathematical concepts and processes, how and why they are taught, why certain approaches appear more effective than others, and how children might be assisted to become more mathematically powerful. The authors use recent theories of analogy and knowledge representation -- combined with research on teaching practice -- to find ways of helping children form links and correspondences between different concepts, so as to overcome problems associated with fragmented knowledge. In so doing, they have capitalized on new insights into the values and limitations of using concrete teaching aids which can be analyzed in terms of analogy theory. In addition to addressing the role of understanding, the authors have analyzed skill acquisition models in terms of their implications for the development of mathematical competence. They place strong emphasis on the development of students' mathematical reasoning and problem solving skills to promote flexible use of knowledge. The book further demonstrates how children have a number of general problem solving skills at their disposal which they can apply independently to the solution of novel problems, resulting in the enhancement of their mathematical knowledge.

1

Cognitive Psychology and Mathematics Education

The discipline of psychology and, more recently, cognitive science has had a seminal influence on how mathematics is taught and learned (Kilpatrick, 1992). From the early part of the century, the teaching and learning of mathematics has received considerable attention from psychologists, as can be seen in E. L. Thorndike’s (1922) book, Psychology of Arithmetic. Thorndike’s work made a significant impact in many a classroom where the chanting of tables (reflecting Thorndike’s “law of exercise”) was a regular occurrence. Psychology has continued to show an interest in mathematics education, as can be seen from the large number of papers presenting a cognitive analysis of children’s mathematical learning (e.g., J. S. Brown & VanLehn, 1982; Collis, 1974; R. B. Davis, 1984; Greeno, 1992; Halford, 1993; Siegler & Jenkins, 1989; Sweller, 1989). Mathematics has been a popular domain of psychological research largely because of its importance in the school curriculum and because its hierachical structure facilitates the construction of tasks of varying levels of difficulty (Kilpatrick, 1992). Mathematical knowledge also lends itself to representation in the computational models of cognitive science.

Changes in psychological thought over the century have been reflected in the major reform movements in mathematics education. These movements have been characterized by changes in the content of the curriculum as well as in the methods used to impart this knowledge. Thorndike’s era, for example, was noted for its emphasis on arithmetic, with drill and practice activities reflecting Thorndike’s primary laws of learning. Likewise, the current emphasis on problem solving and mathematical thinking processes, with children’s active involvement in their own learning, reflects ideas from the constructivist paradigm and from cognitive science, including information processing. In the first part of this chapter, we trace the main developments in psychological thought and their impact on mathematics education. These include the period of drill and practice, the period of meaningful learning, and the “new math” phase. We then review the psychological and societal forces that have shaped the current scene in mathematics education. We conclude with our views on a psychological theory of mathematics education and present an outline of the approach we adopt in this book.

Historical Developments in the Psychology of Mathematics Education

The Period of Drill and Practice

Arithmetic is … a set of rather specialized habits of behavior toward certain sorts of quantities and relations (Thorndike, 1922, p. 73) … learning arithmetic is like learning to typewrite … [it] is in some measure a game whose moves are motivated by the general set of the mind toward victory -winning right answers (pp. 283–284). Drill and practice was the primary focus of mathematics education during the first 30 years of this century. Edward Thorndike was the main proponent of this approach, with his theory variously termed, “connectionism,” “associationism,” and “S-R bond theory.” Thorndike (1922) maintained that, by means of conditioning, specific responses are linked with specific stimuli. He believed that “almost everything in arithmetic should be taught as a habit that has connections with habits already acquired and will work in an organization with other habits to come” (p. 194). Instruction that focused on the formation of necessary bonds and habits was considered particularly important for elementary school children who were not thought to have the ability to deduce the rules of arithmetic from examples and previously learned rules.

Thorndike established three primary laws of learning: the law of exercise or repetition, the law of effect, and the law of readiness. The first law states that the greater the number of times a stimulus-induced response is elicited, the longer the response (i.e., learning) will be retained. This means that each S-R bond that is to be established requires many practice exercises. According to the law of effect, responses that are associated with satisfaction are strengthened and those associated with pain are weakened. Thorndike’s third law, that of readiness, associates satisfaction or annoyance with action or inaction depending on a bond’s readiness to act.

These laws led to a fragmentation of arithmetic into many small components of facts and skills to be taught and tested separately. Bonds were presented in a carefully programmed way so that the important bonds were practiced often, and the less important, less frequently. “Propaedeutic” bonds were used to promote the learning of new concepts and would be practiced temporarily but would subsequently fall away through lack of use. For example, to calculate four 2s, the child would be taught initially the propaedeutic bond for counting by 2s (i.e., 2,4,6, 8). This would be replaced later by the bond, “four 2s are 8.” Because bonds were considered to have an effect on each other, Thorndike (1922) maintained that “every bond formed should be formed with due consideration of every other bond that has been or will be formed” (p. 140). It was not considered appropriate to teach closely related facts for fear of establishing incorrect bonds.

The teacher’s role was to identify the bonds that comprised a particular body of mathematical content and then organize them so that learning the simpler bonds would assist the student in learning the more difficult bonds that occurred later in the teaching sequence. The teacher would then arrange for practice on each set of bonds, with this practice being just sufficient to avoid errors occurring when the next, more difficult set of bonds was introduced. Despite his strong empahsis on drill and practice, Thorndike nevertheless highlighted the importance of making arithmetic problems enjoyable and interesting for children and relevant to their everyday experiences. In this way, children would be more likely to make stronger connections and be better able to make the appropriate connections at the appropriate times. Applying a correct series of connections at the proper time indicated an ability to perceive the structure of a problem, to choose the appropriate series of connections, and to produce all of the connections in series (Nik Pa, 1986).

Thorndike’s arithmetic texts reflected the drill sequences he recommended, however as Resnick and Ford (1984) noted, his rules for generating them were largely intuitive. Thorndike did not systematically address issues such as the amount of practice that is considered adequate for establishing a particular bond or the best way to organize practice on different kinds of bonds. Nevertheless, his theory had a big impact on mathematics education and is no doubt still alive in many classrooms today. Although Thorndike emphasized the content of specific subject matter, he had little to say about the nature or structure of thinking and learning (Resnick & Ford, 1984). This was left to members of the structural school of thought, which came into being during the progressive education era in the 1930s.

The Period of Meaningful Learning

The progressive education movement of the 1930s and 1940s emphasized “learning for living” (Kroll, 1989). There was a shift away from meaningless drill methods where speed and accuracy were the criteria for measuring learning, to a focus on developing mathematical concepts in a meaningful way. There were two schools of thought here. Some educators (e.g., Wheeler, 1935) stressed the social utility aspect of arithmetic learning where it was believed that children learn all the required mathematics through incidental experience rather than systematic instruction. Others however, did not consider such an approach to be the most effective and advocated teaching the structure of mathematics. William Brownell (1935,1945) was one of the major proponents of this view.

William Brownell. Brownell (1945) argued that “meaning is to be sought in the structure, the organization, the inner relationship of the subject itself” (p. 481). The aim was to impart to students the strucure of arithmetic, that is, the “ideas, principles, and processes” of mathematics. The test of learning was not “mere mechanical facility in ‘figuring’ but an intelligent grasp upon number relations and the ability to deal with arithmetical situations with proper comprehension of their mathematical as well as their practical significance” (Brownell, 1935, p. 19). For example, a child who promptly gave the answer “12” to the fact “7 + 5” would not be regarded as having demonstrated a knowledge of the combination unless she understood why 7 plus 5 equals 12 and could convince others of its correctness. This would require the child to have an understanding of the mathematical principles and patterns underlying computations, this being one of the four categories of arithmetic meanings identified by Brownell (1947).

It is worthwhile reviewing these categories because they are still relevant to today’s curriculum. Brownell’s first category comprises basic concepts such as the meaning of whole numbers, fractions, decimal fractions, and ratio and proportion. The second group of meanings includes an understanding of fundamental operations. Here, children must know when to add, subtract, multiply, and divide. They must also know what happens to the numbers used when a given operation is applied. Brownell’s third category comprises the more significant principles, relationships, and generalizations of arithmetic such as, “the product of two abstract factors remains the same regardless of which factor is used as multiplier” (Brownell, 1947, p. 257). The final category pertains to an understanding of the decimal number system and its application in rationalizing computational procedures and algorithms.

The use of concrete materials and practical applications played an important role in Brownell’s school. Drill and practice had a place but was to be utilized only after children had understood the ideas and processes that were to be reinforced. Without a meaningful understanding, drill activities would encourage students to view mathematics as a collection of unrelated ideas and independent facts. Investigating children’s cognitive abilities was considered an essential aspect of curriculum and instruction, with adjustments to the curriculum being made to suit the child’s mental capacities.

Henry Van Engen. Another prominent advocate of meaningful arithmetic was Van Engen who belonged to the operational school. According to this school, arithmetic is concerned with the operations that can be performed on groups (1949). Van Engen expressed concern over the previous era, stressing that drill activities are only a means to an end, and that drill alone will do little more than teach children how to manipulate numbers. He advocated a change in curriculum content that was to be accompanied by an entirely different concept of instruction. An emphasis on the semantic meanings of arithmetic, in contrast to Brownell’s strong syntactical focus, was considered essential. Van Engen argued that Brownell had not addressed meanings that involve associating a symbol with an operation, rather, he had concentrated on the syntactical meanings that involve the formation of relationships between operations. Van Engen’s analysis of meaning was more advanced than Brownell’s in that the child was seen to actively give meaning to symbols. The pupil acquired meaning by “reading into” a symbol, realizing that the symbol is a substitute for an object.

In contrast to Brownell, Van Engen did not consider the roots of arithmetical meaning to reside in the structure of the subject matter, rather he viewed physical or sensory-motor activity as its source (Nik Pa, 1986). Although Van Engen (1953) maintained that manipulative activities are of utmost importance in the development of children’s number concepts, he nevertheless warned of the dangers of inappropriate use of visual aids. His advice is timely: “Too many visual aids in use today do not highlight the essential features of the concept they are supposed to teach. In many cases the essential features are too embedded in the total situation. In still others, it is merely a visual aid, there is no relevance to the development of the concept” (p. 93).

Van Engen’s (1958/1993) emphasis on developing students’ ability to recognize problem structure is also pertinent today. He believed that the schools of his time had failed to develop students’ ability to detect patterns in similar and seemingly diverse situations and had been directing children to the wrong element in problem solving, namely the answer. Rather, the child should grasp the structure of a problem before she looks for the answer. Differences in ability to recognize structure were seen to distinguish good problem solvers from poor.

Gestalt Theory. The Gestaltists also expressed concern for the development of the complex behaviors of problem solving and reasoning. The Gestalt or “field” theory grew out of a reaction against both the structuralist and connectionist doctrines (Dellarosa, 1988; Fehr, 1953). The German word Gestalt means an organized whole in contrast to a collection of parts. Learning, according to the Gestalt psychologists, is a process of identifying relationships and of developing insights. It is only when the relationship of a part of a situation to its whole is perceived that insight occurs and a solution to a problem can be produced. In contrast to Thorndike’s approach, the Gestaltists would attempt from the outset to bring all the elements of a problem together.

The Gestaltists however, did not manage to overthrow their contemporaries’ hold on psychological investigation. This was largely because they relied on the subconscious and failed to develop a comprehensive theory of behavior or cognition (Dellarosa, 1988; Schoenfeld, 1985b). Their body of work consisted mainly of descriptions of phenomena that did not serve as the basis for theory building but as evidence of the inadequacy of the connectionist and structuralist models. Nevertheless, the Gestaltists did make some significant contributions in the area of perception, problem solving, and thinking. For example, they pointed out how the qualitative aspect of an optical illusion could not be reduced to its components, supporting their view that “the whole is greater than the sum of its parts.”

In the area of thinking and problem solving, the Gestaltists highlighted the active, constructive nature of the processes involved. The Gestalt “model” of problem solving involves (a) saturation, that is, working on a problem until one has reached the end of conscious resources; (b) incubation, where the problem is put aside while the subconscious mind works on it; (c) inspiration, in which the solution appears (the “Aha” experience); and (d) verification (Wallas, 1926). Because problem solving was considered to take place in the subconscious, the Gestalt model was not widely accepted. The Gestaltists could offer few suggestions as to how the teacher might foster students’ problem-solving skills. Despite this, the Gestaltists did lay the foundation for some of the later work on problem solving and thinking. For example, Duncker (1945) used “think aloud” protocols of subjects solving problems to conclude that problem solving is a top-down, goal-oriented process rather than a bottom-up, stimulus-driven process of trial and error (Dellarosa, 1988).

Another prominent Gestaltist, Max Wertheimer, addressed the issue of productive thinking and provided useful suggestions for fostering such thinking in the classroom. In his classic text, Productive Thinking, Wertheimer (1959) distinguished between productive and reproductive thinking. A productive thinker grasps the structural relations in a given problem or situation and then combines these parts into a dynamic whole. A person who thinks reproductively fails to see relations among subparts and simply repeats learned responses to individual subparts. According to Wertheimer, productive thinking can be encouraged in the mathematics classroom by avoiding, where possible, the giving of ready-made steps.

A good example of fostering productive thinking is Wertheimer’s approach to teaching area, in particular, the area of the parallelogram. Wertheimer related the classroom example of how children who had been taught the standard rule had difficulty when presented with an “upside down” parallelogram. Because they did not understand the derivation of the rule, they could not apply it to this new situation. This understanding can be readily developed by making use of the functional equivalence between the parallelogram and the rectangle, as indicated in Fig. 1.1 (Resnick & Ford, 1984).

Fig. 1.1. Derivation of the rule for the area of a parallelogram.

Many of Wertheimer’s ideas are directly relevant to mathematics education today. For example, students can be guided in deriving for themselves, the formulae for finding the area of the basic plane shapes rather than blindly following given rules. Once students know how to find the area of a rectangle, they can use this knowledge in generating the formulae for the square, triangle, circle, and parallelogram.

The Period of the “New Math”

The 1960s witnessed major changes to the mathematics curriculum. These changes were the result of severe criticism of the American education system in the postwar years. The restructuring of the mathematics curriculum followed the 1959 Woods Hole conference in which recommendations for reform were outlined. It was considered essential that students possess a knowledge of the fundamental structures of mathematics. This knowledge would enable students to reconstruct mathematical facts should they forget them.

The teaching of mathematical structure in the new math era tended to overemphasize the logical aspect (Ernest, 1985). This was evident in the explicit teaching of set theory and the laws of arithmetic, as well as in the teaching of traditional Euclidean geometry in the secondary mathematics curriculum. Numerous texts reflected this emphasis, with activities based on set theory, systems of numeration, and number theory. The explicit teaching of the notation and algebra of sets and the general laws of arithmetic were, by their very nature, abstract. The new math ideas were presented in a spiral format where previously taught concepts were revisited at higher grade levels and extended and elaborated upon. There were mixed reactions to the notion of introducing young children to mathematical topics that were formerly reserved for secondary school and college students. One strong supporter of this approach however, was Jerome Bruner (1960), whose text, The Process of Education, was widely acknowledged.

Jerome Bruner. Bruner was one of several postwar psychologists who had an interest in the cognitive proceses involved in learning and thinking. He was particularly concerned with the ways in which children represent the concepts and ideas they are being taught. Building in part on the ideas of Piaget, he proposed that children move through three levels of representation as they learn: the enactive, the iconic, and the symbolic. These stages are considered to be developmental, with each mode building on the previous. In the enactive stage, the child directly manipulates objects. In the next stage, the child moves to the realm of mental imagery where she visualizes an operation or concrete manipulation. In the final phase, the child manipulates symbols rather than objects or images of these objects. Many advocates of the new math organized children’s learning experiences according to these modes; that is, children were introduced to a new concept or procedure through the manipulation of concrete materials. The concept was then represented pictorially, and finally, in the symbolic form. This approach has been adopted by other theorists (e.g., Watson, Campbell, & Collis, 1993) and is st...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Dedication
• Contents
• Preface
• 1. Cognitive Psychology and Mathematics Education
• 2. Cognition and Cognitive Development
• 3. Cognitive Models and Processes in Mathematics Education
• 4. Numerical Models and Processes
• 5. Elementary Computational Models and Processes: Addition and Subtraction
• 6. Elementary Computational Models and Processes: Multiplication and Division
• 7. Advanced Computational Models and Processes
• 8. Problem Solving, Problem Posing, and Mathematical Thinking
• 9. Reflections and Recommendations
• References
• Author Index
• Subject Index