Chapter 2




Theories of mathematical learning and understanding


According to Romberg (Grouws, 1992), there is no general agreement on the definition of learning, how learning takes place and what constitutes reasonable evidence that learning has taken place. Some say it is observable changes in behavior, others that it means acquiring new knowledge, and other say that it is the creating of a disequilibrium.


Psychologists have made different philosophic assumptions about the nature of the learning process. Those who hold that learning is determined by the forming connections between the environment stimuli and useful responses are called associationist. A representative of this view, E.B. Thorndike (1922), recommended that in mathematics, for example, students perform much drill and practice on correct procedures and facts to strengthen correct mental bonds. Associationists also argued that curricula should be structured to keep related concepts well separated, so that students did not form incorrect ties.

By 1943, the behaviorists were maintaining that a real science of education could be built only on direct observation. Absent from the research and discourse of behaviorists were "thinking", "meaning" or other such unobservable and possibly nonexistent phenomena. Though behaviorists, led by B.F. Skinner, denied the theory of "mental bonds" that associationist had put forth, their prescriptions for mathematics teaching were similar: drill and practice, with reinforcement by reward for desirable behavior in the form of correct answers and punishment for undesired behavior. The behaviorists brought to the educational scene programmed learning curricula and new standardized testing techniques. In the study of teaching "process-product" researchers searched for types of teaching behavior that led to greater student achievement.


During that same time there existed other views of knowledge and learning. In 1916 Dewey said that "It is that reconstruction or reorganization of experience which adds to the meaning of experience, and which increases ability to direct the course of subsequent experience" (p.89). In another occasion Dewey (1938) wrote that "I use the word understanding rather than knowledge because ...knowledge to so many people means 'information.' Information is knowledge about things (it is static), and there is no guarantee that understanding-the spring of intelligent action-will follow from it"(p.48). Brownell (1935) maintained that although incidental learning could help counteract the practice of teaching mathematics as an isolated subject, it did not provide an organization in which "the meaningful concepts and intelligent skills requisite to real arithmetical ability" could be developed. Brownell wrote about a theory of instruction in which making "sense" of what was learned was the central issue in arithmetic instruction. Piaget and his coworkers who interviewed hundreds of children, proposed that in learning, children pass through developmental stages and that the use of active methods which gives scope to spontaneous research by the child help him rediscover or reconstruct what is to be learned "not simply imparted to him" (Piaget, 1973, p.23).


Piaget's research and theory, is called developmental constructivism (Romberg, 1969), and maintains that children acquire number concepts and operations by construction from the inside and not by internalization. Piaget (1968) pointed out that every normal student is capable of good mathematical reasoning if attention (and care) is directed to activities of his interest, and if by this method the emotional inhibitions that too often give him a feeling of inferiority in lessons in mathematics are removed.


In contrast to Piaget's explanation of construction, Vygotsky (1986) presented an alternate theory where imbalance and not equilibrium is considered normal.


Jean Piaget's Theory of Learning


According to Jean Piaget (1979), human intellectual development progresses chronologically through four sequential stages. The order in which the stages occur have been found to be largely invariant, however the ages at which people enter each higher order stage vary according to each person's hereditary and environmental characteristics.

Piaget defined intelligence as the ability to adapt to the environment. Adaptation takes place through assimilation and through accommodation, with the two processes interacting throughout life in different ways, according to the stage of mental development.


In assimilation, the individual absorbs new information, fitting features of the environment into internal cognitive structures. In accommodation, the individual modifies those internal cognitive structures to conform to the new information and meet the demands of the environment. A balance is maintained through equilibration, as the individual organizes the demands of the environment in terms of previously existing cognitive structures. A child moves from one stage of cognitive development to another through the process of equilibration, through understanding the underlying concept so that the understanding can be applied to new situations. Equilibration is a balance between assimilation and accommodation.


The stages of cognitive development that Piaget distinguished are four: (Piaget, 1968)

Sensorimotor (0-2 years of age) - children begin to use imitation, memory and thought. They begin to recognize that objects do not cease to exist when they are hidden from view. They move from reflex actions to goal-directed activity.

Preoperational (2-7 years) - Children gradually develop language and the ability to think in symbolic form. They are able to think operations through logically in one direction and they have difficulty seeing another persons point of view.

Concrete operational (7-11 years) - Children are able to solve concrete (hands-on) problems in logical fashion. They understand the laws of conservation and are able to classify and seriate. They also understand reversibility.

Formal operational (11-15 years of age) - Children are able to solve abstract problems in logical fashion. Their thinking becomes more scientific, they develop concerns about social issues and about identity.


Piaget suggested that when children do not understand or have difficulty with a certain concept, it is due to a too-rapid passage from the qualitative structure of the problem (by simple logical reasoning -e.g. a ball existing physically) to the quantitative or mathematical formulation (in the sense of differences, similarity, weight, number, etc.). Conditions that can help the child in his search for understanding according to Piaget is the use of active methods that permit the child to explore spontaneously and require that "new truths" be learned, rediscovered or at least reconstructed by the student not simply told to him (Piaget,1968). He pointed out that the role of the teacher is that of facilitator and organizer who creates situations and activities that present a problem to the student. The teacher must also provide counterexamples that lead children to reflect on and reconsider hasty solutions. Piaget argued that a student who achieves a certain knowledge through free investigation and spontaneous effort will later be able to retain it. He will have acquired a methodology that serves him for the rest of his life and will stimulate his curiosity without risk of exhausting it.


A third type of knowledge that Piaget suggests is social or conventional knowledge. He said that it is always through the external educational action of family surroundings that the young child learns language, which Piaget (1973) called is an "expression of collective values." Piaget pointed out that without external social transmission (which is also educational) the continuity of collective language remains practically impossible.

There are three types of feelings or emotional tendencies, according to Piaget, that affect the ethical life of the child, that are first found in his mental constitution. In the first place is the need for love, which plays a basic role in development in various forms from the cradle to adolescence. There is a feeling of fear of those who are bigger and stronger than himself, which plays an important role in his conduct. The third is mixed, composed of affection and fear at the same time. It is the feeling of respect that is very important in the formation or exercise of moral conscience.


Noddings (1990a) points out certain characteristics that constructivist teachers must have an ethical commitment to inquiry in order to aid students in their investigations, and the receptivity and responsiveness of an ethic of care which involves sharing and listening to students, taking interest in their purposes as well as in those of the teachers' truth.


A constructivist view of knowledge implies that knowledge is continuously created and reconstructed so that there can be no template for constructivist teaching (Peterson & Knapp, 1993). Since this point of view holds that learning involves student's constructing their own knowledge, this leads to a redefinition of the teachers' role to one of facilitator. This also leads to teaching that emphasizes the importance of listening to and valuing students' perception, even when their understanding differs from conventional knowledge (Cochran, Barson & Davis, 1970).


Mathematical understanding and number sense


Cognitive scientists and mathematics educators who favor the cognitive science approach have moved well beyond Piaget in describing the way the mind operates. There has been a shift from an organic language of Piaget to a language "highly colored" of computers, (Noddings 1990b) with words such as networks, connections, paths, frames, etc.


As a cognitive position, constructivism maintains that all knowledge is constructed, as Piaget's theories hold. Not only are intellectual processes themselves constructive but are themselves products of continued construction. It can be said that the construction and subsequent elaboration of new understandings is stimulated when established structures of interpretation do not permit or accept a new situation or idea. This clash (not understanding) produces a disequilibrium that lead to mental activity and the modification of previously held ideas to account for the new experience (Simon & Schifter, 1991).


Hiebert and Carpenter (1992) propose a framework for considering understanding from the constructivist

perspective which would shed light on analyzing a "range of issues related to understanding mathematics." They make a

distinction between the external and internal representation of mathematical ideas, pointing out that, to think and

communicate mathematical ideas, people need to represent them in some way. Communication requires that the  representations be external, taking the form of spoken language, written symbols, drawings or concrete objects.

Mathematical ideas becomes tangible when people can express them. By learning to express their ideas to one another,

students can begin to appreciate the nuance of meaning that natural language often masks, but that the precise language

of mathematics attempts to distinguish (Lo, Wheatley, & Smith, 1994; Silver, Kilpatrick & Schlesinger, 1990; Lesh,

Post & Behr, 1987).


The Curriculum and Evaluation Standards for School Mathematics of the National Council of Teachers of Mathematics (1989) point out in the Standard on Communication that understanding mathematics can be defined as the ability to represent a mathematical idea in multiple ways and to make connections among different representations. In order to think about mathematical ideas these need to be represented internally but these mental representations are not observable. This has led cognitive science to consider mental representations as a field of study (Ashcraft, 1982; Greeno, 1991; Hiebert & Carpenter, 1992) .


Connections between external representations of mathematical ideas can be constructed by the learner (Hiebert & Carpenter, 1992) between different fortes of the same idea or between related mathematical ideas. These connections are often based on relationships of similarity or of differences. Connection within the same representation are formed by detecting patterns and regularities.


The relationship between internal representations of ideas constructs a network of knowledge. Understanding then is the way information is represented, so that a mathematical idea, procedure or fact is understood if it is part of an internal network. Networks of mental representations are developed gradually as new information is connected to the network or new ties are constructed between previously disconnected information. Understanding grows as the networks become larger and more organized and can be limited if connections are weak or do not exist becoming useless (Hiebert & Carpenter, 1992).


Although the image of adding to existing networks is appealing in its simplicity, it may turn out that the image is too simple. Studies have suggested that students in the act of building understanding reveal a much more chaotic process (Hiebert, Wearne & Taber, 1991). There have been a number of studies in which the process of learning and understanding are of central interest: Cobb, Wood, Yackel, Nichols, Wheatley, Trigatti & Perlwitz (1991); Carpenter, Fennema, Peterson, Chiang & Loef (1989); Doyle (1988); Baroody, (1985); Hiebert & Wearne (1988).


Children are natural learners and the environment both social and physical offers them many opportunities to acquire notions of quantity. Even in very poor or diverse cultures, races or classes, children have the opportunity to acquire quantitative notions (Gelman 1980; Ginsburg, Posner & Russel1,1981; Ginsburg & Russell, 1981).


Each healthy human brain, no matter the age, sex, race or culture, comes equipped with a set of unique features: the ability to detect patterns and to make approximations, a capacity for various types of memory, the ability to self-correct and learn from experience and external data and self reflection, and an great capacity to create (Came & Caine 1994). Because of this predisposition of the brain, children and adults constantly search for ways to make sense and make connections. This can be translated into a search for common patterns and relationships as Hiebert and Carpenter (1992) propose.


Caine and Caine (1994) argue that brain research confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching. They add that the brain is designed as a "pattern detector" and that the function of educators should be to provide students with the kind of experiences that enable them to perceive "the patterns that connect."


Children from a very young age are sensitive to quantity. They perceive differences in number; they see correlation among different numbers of events; their actions contain quantity and they use words referring to basic mathematical events (Gelman, 1980; Ginsburg, 1989). Various studies (Ginsburg & Baron, 1993; Starkey & Cooper, 1980; Van de Walle & Watkins, 1993) have pointed out the importance of taking into account children's informal mathematical connections as building block for formal mathematics. Ginsburg (1989) suggests that students need to learn that it is acceptable, "even desirable", for them to connect conventional arithmetic with their own informal knowledge, intuition and invented procedures.

In a study by Carpenter, Ansell, Franke, Fennema and Weisbeck (1993), the results suggest that children can solve a wide range of problems, including problems involving multiplication and division, much earlier than is generally presumed. With only a few exceptions, children's strategies could be characterized as representing or modeling the action or relationships described in the problem. These researchers conclude that young children's problem-solving abilities have been seriously underestimated. They suggest that if from an early age children are motivated to approach problem solving as an effort to make sense out of problem situations, they may come to believe that learning and doing mathematics involves solving problems in a way that always makes sense.


The use of manipulatives in mathematics


For years mathematics educators have advocated using a variety of forms to represent mathematical ideas for students. Physical three-dimensional objects are often suggested as especially useful. Despite the intuitive appeal of using materials, investigations of the effectiveness of the use of concrete materials have yielded mixed results (Bednarz & Janvier, 1988; Bughardt, 1992; Evans, 1991; Hestad, 1991; Hiebert, Wearne, & Taber, 1991; Simon, 1991; Thompson,J., 1992).


P. Thompson (1994) suggests that the apparent contradictions in studies using manipulatives are probably due to aspects of instruction and students' engagement to which the studies did not attend. Evidently, just using concrete materials is not enough to guarantee success according to Baroody (1989). The total instructional environment must be looked into to understand the effective use of concrete materials. In a project by Wesson (1992) for grades 1 and 2, which emphasized exploratory activities with manipulatives, the results suggested that while a much wider range of content than in standard books or tests was covered, there was no loss of arithmetic skills.


Children understand when using concrete materials if the materials are presented in a way that helps them connect with existing networks or construct relationships that prompt a reorganization of networks. It is important to consider then, the internal networks that students already carry with them and the classroom activities that promote construction of relationships between internal representations (Hiebert et al, 1991). Manipulatives then can play a role in students' construction of meaningful ideas. Clements and McMillan (1996) and others suggest they should be used before formal instruction, such as teaching algorithms. Clements and McMillan propose that concrete knowledge can be of two type: "sensory-concrete" which is demonstrated when students use sensory materials to make sense of an idea; and "integrated concrete" which is built through learning. Integrated concrete thinking derives its strength from the combination of many separate ideas in an interconnected structure of knowledge. When children have this type of interconnected knowledge, the physical objects, the actions they perform on the objects, and the abstractions they make are all interrelated in a strong mental structure.

Ross and Kurtz (1993) offers the following suggestions when planning a lesson involving the use of manipulatives. He suggests that the mathematics teacher should be certain that:


1. manipulatives have been chosen to support the lesson's objectives;

2. significant plans have been made to orient students to the manipulatives and corresponding classroom procedures;

3. the lesson involves the active participation of each student;

4. the lesson plan includes procedures for evaluation that reflect an emphasis on the development of reasoning skills.


Invented strategies and number sense


In the last few years there have been studies about the idea of students' constructing their own mathematical knowledge rather than receiving it in finished form from the teacher or a textbook (Carpenter, Ansell, Franke, Fennema, Weisbeck, 1993; Markovits & Sowder, 1994). A crucial aspect of students' constructive processes is their inventiveness (Piaget, 1973). Children continually invent ways of dealing with the world. Many of the errors they make can be

interpreted as a result of inventions (Ginsburg & Baron, 1993; Peterson, 1991). Similarly, in school mathematics, students rely many times on invented strategies to solve a variety of problems (Carpenter, Hiebert, & Moser, 1981; Carraher & Schliemann, 1985; Ginsburg, 1989). Kamii and Lewis (1993) and Madell (1985) have reported successful work in programs where children are not taught algorithms, but are encouraged to invent their own procedures for the basic operations. Treffers (1991) suggests a similar program in the Netherlands and Baker & Baker (1991) in Australia.


Various studies have been made in the area of invented strategies. Cook and Dossey's (1982) findings show that children learn number facts easily and quickly and recall them better when using a strategy approach than when using a learned algorithm, drill or practice approach. Browne (1906); Howe and Ceci (1979); Kouba (1989); Rathmill (1978);

Sowder and Wheeler (1989) have done studies on strategies used for calculation. Carpenter and Moser (1984) found that children in the United States ordinarily invent a series of abbreviated and abstract strategies to solve addition and subtraction problems during their first four years in school. Romberg and Collis (1987) found that even though some children are limited by their capacity to handle information, most are able to solve a variety of problems by inventing strategies that have not been taught. English (1991) observed that in a study of young children's combinatoric strategies, a series of six increasingly sophisticated solution strategies were identified. A significant number of children independently adopted more efficient procedures as they progressed on the task.


A study by Markovits and Sowder (1994) examined the effect of an intervention in the instruction of seventh grade students for the purpose of developing number sense. Instruction was designed to provide diverse opportunities for exploring numbers, number relationships, and number operations and to discover rules and invented algorithms. Measures taken several months later revealed that after instruction students seem more likely to use strategies that reflected number sense and that this was a long-term change. Rathmill (1994) suggests that planning for instruction that promotes the development of children's thinking and reasoning about mathematics not only helps them make sense of the content they are studying, but also helps them learn ways of thinking that later will enable them to make sense of new content. Lampert (1986) proposed that "a sense-making" atmosphere is necessary and that arithmetic should make sense in terms of children's own experience. Reynolds'(1993) study suggests that children's imaging activity is at the heart of their sense making and problem solving. Silver, Shapiro and Deutsch (1993) found that students' performance was adversely affected by their dissociation of sense making from the solution of school mathematics problems and their difficulty in providing written accounts of their thinking and reasoning.


In the Everybody Counts document from the National Research Council (1989) the major objective of elementary school mathematics is the development of "number sense". The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) also includes number sense as a major theme throughout its recommendations. Greeno (1991) interprets number sense as "a set of capabilities for constructing and reasoning with mental models." This perspective he argues, provides reasons that support considering various aspects of number sense as features of students' general condition of knowing in the area of numbers and quantities, rather than skills that should be given specific instruction. The term number sense refers to several important but elusive capabilities according to Greeno, capabilities including flexible mental computation, numerical estimation and quantitative judgment. Flexible mental computation according to Greeno involves recognition of equivalence among objects that are decomposed and recombined in different ways.


Reys et al. (1991) describe number sense in the following manner:

         Number sense refers to an intuitive feeling for numbers

         and their various uses and interpretations; an

         appreciation for various levels of accuracy when

         figuring; the ability to detect arithmetical errors,

         and a common-sense approach to using numbers... Above

         all, number sense is characterized by a desire to make

         sense of numerical situations (pp.3-4)


Sowder and Schappelle (1994) suggest that there are common elements found in classrooms that help children acquire good number sense:

1.     Sense-making is emphasized in all aspects of mathematical learning and instruction.

2.     The classroom climate is conducive to sensemaking. open discussions about mathematics occurs both in small groups and with the whole class.

3.     Mathematics is viewed as the shared learning of an intellectual practice. This is more than simply the acquisition of skills and information. Children learn how to make and defend mathematical conjectures, how to reason mathematically and what it means to solve a problem.


Mental Computation


Mental computation according to Trafton (1986) refers to nonstandard algorithms for computing exact answers. It is also referred to as the process of calculating an exact arithmetic result without the aid of an external computational or recording aid. (Hope, 1986; Reys, 1986). It is recognized as both important and useful in everyday living as well as valuable in promoting and monitoring higher-level mathematical thinking (Reys et al., 1995). It has been recognized in the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989) that increased attention should be given to mental computation. A National Statement on Mathematics for Australian Schools (Australian Education Council and the Curriculum Corporation, 1991) was released in 1991 recommending substantial change in emphasis among mental, written and calculator methods of computation and between approximate and exact solutions. A major objective is to redirect the computational curriculum in schools to reflect a balance in the emphasis on methods of solution. Before the Statement, the curriculum was divided as: 75% written computation, 25% Calculator, Estimation, Mental Computation. With the new statement it would be 25% for each method of computation.


According to Boulware (1950) mental arithmetic has its origin during the second quarter of the nineteenth century. The idea of building a broader foundation of meaning and understanding in arithmetic gave rise to Mental Arithmetic as it was known in the middle of the nineteenth century with Warren Colburn (1841) considered as pioneer in the field of mental arithmetic. Before his time, arithmetic had reached a point of extreme abstraction according to Boulware. The second half of the century witnessed the decline in interest and understanding of the purpose of mental arithmetic. With the coming of more writing paper, cheap pencils, with the rise of industry and its accompanying needs for persons skilled in computation, the practical or computational phase of arithmetic took on importance around the turn of the century. The emphasis in arithmetic at that time was the teaching of isolated facts, followed by drill upon these facts. High among the purposes stated for the study of arithmetic many authors of the time placed speed, memory and accuracy by mechanical rules. There was an emphasis in arithmetic on drill for perfection and automatic response at the expense of meaning and understanding. In 1950, a dissertation by Boulware is representative of the quest for the development of "meaning" in mental computation stirred by Brownwell (1935), who urged that meaning and seeing sense in what is being learned should be the central focus of arithmetic instruction. Boulware's conception of mental computation is as follows: Mental arithmetic deals with number as a unified, consistent system, and not as an aggregate of unrelated facts. [It] consists of methods of dealing with numerical situations whereby a clear concept of the number system may be conceived and utilized in quantitative thinking. It proceeds to the analysis of number combinations by processes of meaningful experiences with concrete numbers, reflective thinking in number situations, seeing relationships, and discovery of new facts as an outgrowth of known facts (pp.7-8). In 1960, in an article by Sister Josefina there seems to begin interest in mental computation and in the 1978 NCTM yearbook on computational skills there appears an article by Trafton (1978) where the need for including proficiency with estimation and mental arithmetic as goals for the study of computation is presented. A good number of studies and articles about mental computation appeared in the period of the 1980s (e.g. Reys, R.E., 1984, 1985; Reys, B. J., 1985a, 1985b; Madell, 1985; Hope, 1985, 1986, 1987; Reys & Reys, 1986; Langford, 1986; Markovits and Sowder, 1988; Baroody, 1984, 1985, 1986, 1987, 1989 and others) leading up to the statement of the inclusion of mental computation as an area where increased attention is needed in school mathematics by the NCTM (1989). With the increase of studies in cognitive skills and number sense (e.g.Simon, 1979; Resnick, 1986; Silver, 1987; Schoenfeld, 1987; Greeno, 1980; Sowder, 1988) and more recent studies mentioned in this chapter, mental computation is suggested to be related to number sense, needed for computational estimation skills and considered a higher order thinking skill.


In a study by Reys, Reys and Hope (1993) they argued that the low mental computation performance reported in this study most likely reflected students' lack of opportunity to use mental techniques they constructed based on their own mathematical knowledge. The study of Reys, Reys, Nohda and Emori (1995) assessed attitude and computational preferences and mental computation performance of Japanese students in grades 2, 4, 6, and 8. A wide range of performance on mental computation was found with respect to all types of numbers and operations at each grade level. The mode of presentation (visual or oral) was found to significantly affect performance levels, with visual items generally producing higher performance. The strategies used to do mental computation were limited, with most subjects using frequently a mental version of a learned algorithm.


In a study by G.W. Thompson (1991) about the effect of systematic instruction in mental computation upon fourth grade students' arithmetic, problem-solving and computation ability, a significant difference favored the group taught mental computation, with girls improving more than boys.


According to Markovits and Sowder (1994) it would seem reasonable that if children were encouraged to explore numbers and relations through discussions of their own and their peers' invented strategies for mental computation, their intuitive understanding of numbers and number relations would be used and strengthened. Okamoto (1993) found that children's understanding of the whole number system seemed to be a good predictor of their performance on word problems.

Cross-cultural research has identified a variety of mental computation strategies generated by students, (e.g. Hope & Sherill, 1987; Markovits & Sowder, 1988) and the difference in mental computation in an out of school and in-school context (e.g. Ginsburg, Posner, & Russell, 1981; Pettito & Ginsburg, 1982). Sribner (1984) points out that individuals develop invented procedures suited to the particular requirements of their particular occupations.


In a study on individuals who are highly skilled in mental arithmetic (Stevens 1993), forty-two different mental strategies were observed. Efficient, inefficient and unique strategies were identified for each of five groups (grade 8). Dowker (1992) describes in a study the strategies of 44 academic mathematicians on a set of computational estimation problems involving multiplication and division of a simple nature. Computational estimation was defined as making reasonable guesses as to approximate answers to arithmetic problems, without or before actually doing the calculation. Observing people's estimation strategies, Dowker suggests, may provide information not only about estimation itself, but also about people's more general understanding of mathematical concepts and relationships. From this perspective Dowker concludes that estimation is related to number sense. Sowder (1992) who agrees with this position points out that computational estimation requires a certain facility with mental computation.


In a study by Beishuizen (1993), he investigated the extent to which an instructional approach in which students use of the hundreds board supported their acquisition of mental computation strategies. In the course of his analysis, he found it necessary to distinguish between two types of strategies for adding and subtracting quantities expressed as two digit numerals as follows:


1. 1010 strategy - 49 + 33 -> 40 + 30 -> 9 + 3 = 12


70 + 12 = 82


2. N10 strategy - 49 + 33 -> 49 + 30 -> 79 + 3 = 82


Beishuizen's analysis indicates that N10 strategies are more powerful, but that many weaker students used only 1010 strategies. The study's findings also suggest that instruction involving the hundreds board can have a positive influence on a student's acquisition of N10 strategies. Fuson and Briars (1990) and others have also identified these strategies.


Hope (1987) points out that because most written computational algorithms seem to require a different type of reasoning than mental algorithms, an early emphasis on written algorithms may discourage the development of the ability to calculate mentally. Lee (1991) recommends that perhaps it is time to investigate changing our traditional algorithms for addition and subtraction to left-to-right procedures.


According to Reys et al. (1995) there have been many studies that suggest the benefit of developing mental computation strategies. Mental computation has also been highlighted in the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989).


Mental computation can be viewed from the behaviorist perspective as a basic skill that can be taught and practiced. But it can also be viewed from the constructivist view in which the process of inventing the strategy is as important as using it. In this way it can be considered a higher-order thinking skill (Reys et al., 1995).


Addition, subtraction and teaching strategies


The Curriculum and Evaluation Standards for School Mathematics (NCTM 1989) recognizes that addition and subtraction computations remain an important part of the school mathematics curriculum and recommends that an emphasis be shifted to understanding of concepts. Siegler (1988) indicated how important it is for children to have at least one accurate method of computation. In a study by Engelhardt and Usnick (1991) while no significant difference between second grade groups using or not using manipulatives was found, significant differences in the subtraction algorithm favored those taught addition with manipulatives. Usnick and Brown (1992) found no significant differences in achievement between the traditional sequence for teaching double-digit addition, involving nonregrouping and then regrouping, and the alternative, in which regrouping was introduced before non-regrouping examples in second graders.


Ohlsson, Ernst, and Rees (1992) used a computerized model to measure the relative difficulty of two different methods of subtraction, with either a conceptual or a procedural representation. The results of the use of the model suggested that regrouping is more difficult to learn than an alternative augmented method, particularly in a conceptual representation, a result that contradicts current practice in American schools. Dominick's (1991) study with third grade students suggested that students' confusion with the borrowing algorithm centered around a misunderstanding of what was being traded. Evans (1991) found that groups taught with pictorial representations or by rote learned to borrow in significantly less time than did a group using concrete materials in grades 2 and 3.


Sutton and Urbatch (1991) recommended the use of base-ten blocks, beans and bean sticks or beans and bean cups to serve as manipulatives to use for trading games and with the "transition board". (A modified version of the base ten board). They also emphasized that attempting to teach addition and subtraction without initially preparing the student with trading games could be counterproductive and result in lack of understanding due to lack of preparation. In a study which analyzed individual children's learning of multidigit addition in small groups in the second grade, results suggested that rarely did a child spontaneously link the block trades with written regrouping (Burghardt, 1992). Fuson and Briars (1990) and P.W. Thompson (1992) found that the base ten blocks could be a helpful support for children's thinking, but many children do not seem spontaneously to use their knowledge of blocks to monitor their written multidigit addition and subtraction. The Fuson and Briars study suggested that frequent solving of multidigit addition or subtraction problem accompanied by children's thinking about the blocks and evaluating their written marks procedure, might be a powerful means to reduce the occasional trading errors made by children. The study also suggested that counting methods that use fingers, are not necessarily crutches that later interfere with more complex tasks.


Fuson and Fuson (1992) found that in all of the groups studied, children were accurate and fast at counting up for subtraction as at counting on for addition. This contrasts with the usual finding that subtraction is much more difficult than addition over the whole range of development of addition and subtraction solution strategies. Sequence counting on and counting up according to Fuson and Fuson are abbreviated counting strategies in which the number words represent the addends and the sum. In both strategies the counting begins by saying the number word of the first addend. For example: 7 + 5, a child would say 7 pause 8, 9, 10, 11, 12 (up to five numbers, the last number of the sequence is the answer) and 12 - 5 would be, 5, pause 6, 7, 8, 9, 10, 11, 12, seven numbers were counted, which is the answer. Thornton's (1990) study provides evidence that children who were given an opportunity to learn a counting up meaning for subtraction as well as counting down (counting back from minuend), preferred the counting up meaning.


In a series of studies by Bright, Harvey and Wheeler (1985) they defined an instructional game as a game for which a set of instructional objectives has been determined. These instructional objectives may be cognitive or affective and are determined by the persons planning the instruction, before the game is played by the students who receive the instruction in it. The results of the studies suggest that:


  1. games can be effective for more than drill and practice and for more than low level learning of skills and concepts,
  2. games can be used along with other instructional methods to teach higher level content such as problem solving,
  3. games should probably be used relatively soon before or after instruction planned by the teacher for the same material,
  4. the use of more challenge, fantasy or curiosity might enhance the effectiveness of instructional games.


Hestad (1991) found that the use of a card game was effective for third grade students in introducing new mathematical concepts and maintaining skills.


In a study by Cobb (1995) the use of the hundreds board by second graders' in a classroom where instruction was broadly compatible with recent reform recommendations (NCTM, 1989, 1991) was investigated. The role played by the use of the hundreds board over a 10-week period in supporting the conceptual development of four second graders was studied.


Particular attention was given to the transition from counting on to counting by tens and ones. The hundreds board

is a ten-by-ten grid from either 0 to 99 or 1 to 100. The results indicated that the children's' use of the hundreds board did not support the construction of increasingly sophisticated concepts of ten. However, children's use of the hundred board did appear to support their ability to reflect on their mathematical activity once they had made this conceptual advance. The utility of the hundreds table in teaching computation has been also recognized by Beishuzen (1993);Hope, Leutzinger, Reys and Reys (1988); Thornton, Jones and Neal (1995) and Van de Walle and Watkins (1993) .


Teachers' Pedagogical Beliefs about Mathematics Teaching Learning and Assessment


We can learn more about how invisible components in the teaching and learning situation can contribute to or detract from the quality of the mathematical learning that takes place by focusing on the culture according to Nickson (1992). It is important, he points out, in exploring the mathematics classroom from the perspective of the culture, it generates, to remember that we are concerned with the people in the setting and what they bring to it. Nickson adds that we must increase our sensitivity to the importance of their hidden knowledge, beliefs, and values for mathematics education.


One of the major shifts in thinking in relation to teaching and learning of mathematics in recent years has been with respect to the adoption of differing views about the nature of mathematics as a discipline. The view of mathematics that has informed and historically transfixed most mathematics curriculum has been, according to Lakatos, (1976) one of considering that mathematics as consisting of "immutable truths and unquestionable certainty". Such a view does not take into account how mathematics changes and grows and is waiting to be discovered (Nickson, 1992). Brown and Cooney (1982) note that the intensity of the teachers' beliefs is very important in the classroom culture. The traditional detachment of mathematics content from shared activity and experience, so that it remains at an abstract and formal level, constructs barriers around the subject, according to Nickson, that sets it apart from others areas of social behavior. The message conveyed is that is has to be accepted unquestioningly and from which no deviation is permitted. The classroom culture will mirror this unquestioning acceptance. The visibility and acceptance of what is done or not done in mathematics are factors in stopping teachers from engaging in activities that they may instinctively feel are appropriate but might challenge the supposedly inviolable essence of mathematics as they themselves were taught.


In investigating the relationship between what teachers believe about how children learn mathematics and how those teachers teach mathematics, A. Thompson (1992) points out that studies have examined the congruence between teachers' beliefs and their practice and findings have not been consistent. Researchers such as Grant (1984) and Shirk (1973) have reported a high degree of agreement between teachers' professed views of mathematics teaching and their instructional practice, where as others have reported sharp contrasts (e.g. Carter, 1992; Cooney, 1988; Shaw, 1989; Thompson, 1984).

It has been argued (Nickson, 1988; Ball, 1993) that bringing teachers into the arena of research activity can be an important step in increasing their understanding of research processes and results and their relation to classroom practice. Each mathematics classroom will vary according to the actors within it. The unique culture of each classroom is the product of what teachers bring to it in terms of knowledge, beliefs, and values, and how these affect the social interactions within that context. The daily experiences of students in mathematics classes of teachers with positive attitudes were found to be substantially different from those of students in classrooms of teachers with negative attitudes in a study by Karp (1991). Overall, teachers with negative attitudes toward mathematics employed methods that fostered dependency and provided instruction which was based on rules and memorization, relied on an algorithmic presentation, concentrated on correct answers and neglected cognitive thought processes and mathematical reasoning, whereas teachers with positive attitudes were found to encourage student initiative and independence. Swetman (1991) found no significant relationship between teachers' mathematics anxiety and students' attitude toward mathematics in grades 3 to 6. Attitude toward mathematics however, became more negative as grade increased in teachers and students.


Teacher influence on student achievement


At the time of a study by Good and Grouws, (1977), comparatively few studies had included observational measures that detail how the teacher functions as an independent variable in order to influence student achievement. Teacher effectiveness (as operationally defined in their study) appeared to be associated strongly with the following clusters: student initiated behavior; whole class instruction, general clarity of instruction, and availability of information as needed, a non-evaluative and relaxed learning environment which is task focused; higher achievement expectations; classrooms that are relatively free of major behavioral disorders. Brophy's (1986) study found that most investigative efforts had focused on curricular content and students' learning without careful consideration of teachers' instructional practices. Loef (1991) found that more successful teachers (in grade 1) represented differences among addition and subtraction problems on the basis of the action in the problem and the location of the unknown, and they organized their knowledge on the basis of the level of the children's understanding of the problem in context. Hiebert and Carpenter (1992) note that it seems evident that procedures and concepts should not be taught as isolated bits of information, but it is less clear what connections are most important or what kind of instruction is most effective for promoting these connections.


Teachers' influence on class content


Even though some researchers have concluded that textbooks determine the content addresses in classrooms (Barr, 1988; Barr & Dreeben; 1983) others provide evidence to challenge that assertion (Freeman .& Porter, 1989; Stodolsky,1989) as Sosniak and Stodolsky (1993) have pointed out. In mathematics, Barr (1988) found that seven out of nine fourth-grade teachers used their textbooks by moving lesson by lesson through the book. In contrast, Freeman and Porter (1989) and Stodolsky (1989) found most mathematics teachers to be selective in their use of textbook lessons, problem sets, and topics, although topics not included in the texts were only occasionally added to the instructional program.

Research suggests that teachers are "gatekeepers" (Thornton, 1991) who make their own decisions about which parts of a textbook to use and which ways to use them (Barr & Sadow, 1989) and such decisions may not necessarily lead according Brophy (1982) to close adherence to the textbook material.


Sosniak and Stodolsky (1993) found in a study of four fourth-grade teachers that the influence of textbooks on teachers' thinking and on instruction was somewhat less than the literature indicates. Their results suggest that patterns of textbook use and thinking about these materials were not necessarily consistent across subjects even for a single teacher, and that the conditions of elementary teachers' work encouraged selective and variable use of textbook materials.

In a study by Stigler, Fuson, Ham, and Myong (1986), an analysis is made of addition and subtraction word problems in American and Soviet elementary mathematics textbooks. The data suggests that American children entering first grade can solve the simple kinds of addition and subtraction word problems on which American texts spend so much time.

Another study on text books is one by Ashcraft and Christy (1995) in which they study the frequency of arithmetic facts in elementary texts. The study tabulated the frequency with which simple addition and multiplication facts occur in elementary school arithmetic texts for grades 1-6. The results indicated a "small-facts bias" in both addition and multiplication. "Large" facts, with operands larger than 5, occurred up to half as frequently as those with operands in the 2-5 range. As was also found in an earlier tabulation for grades K-3, facts with operands of 0 and 1 occurred relatively infrequently, except for patterns like 1+2 and 1x3 which had a high frequency. The small facts bias in the presentation of basic arithmetic, at least to the degree observed, probably works against a basic pedagogical goal, mastery of simple facts. It may also provide a partial explanation of the widely reported problem size or problem difficulty effect, that children's and adults' responses to large basic facts are both slower and more error prone than their solutions to smaller facts.


In a study by Porter (1989) elementary school mathematics is used as a context for considering what could be learned from careful descriptions of classroom content. Teachers log and interviews show that large numbers of mathematics topics are taught for exposure with no expectation of student mastery: much of what is taught in one grade is taught in the next, skills typically receive 10 times the emphasis compared to either conceptual understanding or application, and depending on school and teacher assignments, mathematics instruction a student receives may be doubled or halved. Porter argues that "ultimately teachers must decide what is best for their students and within the limits of their own knowledge, time and energy." (p.15)


Teaching practices and their effects


Koehler and Grouws (In Grouws, 1992) have suggested that teachers' behavior is influenced by their knowledge of: the mathematics content being taught, how students might learn or understand that particular content and of the methods of teaching of that particular content. Also influencing teachers' behaviors are teachers' attitudes and beliefs about teaching and mathematics.


Bush (1991) in a study about factors related to changes in elementary student's anxiety found that mathematics anxiety tended to decrease as teachers in grades 4-6 spent more time in small group instruction, had more years of experience, and took more post-bachelor's mathematics courses. According to a study by Tangretti (1994), findings indicated that the elementary teachers that participated in the study were not adequately prepared to meet NCTM expectations. Their teaching focus was found to be an algorithmic approach with emphasis on numeration and computation. Lack of confidence in content areas beyond arithmetic were reported as contributing to the lack of preparedness of elementary teachers to implement innovative curriculum. Wood, Cobb and Yackel (1991) report that after participating in a study, changes occurred in a teacher's (second grade) beliefs about the nature of mathematics (from rules and procedures to meaningful activity), about learning (from passivity to interacting) and about teaching (from transmitting information to guiding students' development of knowledge). A similar result was reported in a study by Zilliox (1991). In-service elementary school teachers felt they were teaching more and better mathematics lessons, were more comfortable with student use of hands-on materials and with managing small groups, and had a different sense of student capabilities and different expectations for student behavior after participating in the study.


A study by Carpenter, Fennema, Peterson, Chiang, and Loef (1989) investigated teachers' use of knowledge from research on children's mathematical thinking and how their students' achievement is influenced as a result. Although instructional practices were not prescribed, the teachers that participated in the treatment activities taught problem solving significantly more and number facts significantly less than did control teachers. Treatment teachers encouraged students to use a variety of problem-solving strategies, and they listened to processes their students used significantly more than did the control group teachers and knew more about individual students problem-solving processes.


Teachers and assessment issues


A view of learners as passive absorbers of facts, skills, and algorithms provided by teachers is a basis for much of the most current use of measures (Stenmark, 1991). Standard achievement tests according to Kamii and Lewis (1991) measure students' abilities to recall and apply facts and routines presented during instruction. Some items require only the memorization of detail; other items, although designed to assess higher-level learning outcomes, often require little more than the ability to recall a formula and to make the appropriate substitutions to get the answer (Lambdin, 1993). Test items of this type are consistent with the view of learning as a passive, receptive process, a process which is additive and incremental. The

practice of scoring answers to items of this type (right or wrong) is consistent with the view that "bits" of knowledge or skills are either present or absent in the learner at the time of testing. Under this approach, diagnosis is simply a matter of identifying the missing pieces of knowledge in the student,thereby creating a need for remedial teaching. In the constructivist view of teaching, the student is a participant in building his own understanding. The learner does not absorb new ideas and data but rather constructs his own version and relates it to existing information (Wilson, 1992). In order to help the student construct firm connections in the sense of the constructivist theory, the teacher can contribute by facilitating time for auto-evaluating, reflection on processes and ideas, auto-monitoring procedures like journals, portfolios, rechecking work (Sanford, 1993; Stenmark, 1989, 1991; NCTM, 1995). These are all metacognitive processes that can be strengthened through these practices. Metacognition refers to one's knowledge of one's own cognitive processes and products, and of the cognition of others. It also refers to self-monitoring, regulation and evaluation of the cognitive activity (Silver & Marshall, 1990). According to Beyer (1988), metacognition involves thinking about how one thinks as well as thinking to make meaning. For assessment and monitoring of student learning, an implication of the constructivist view is that teachers must measure understanding and models that individual students construct for themselves during the learning process (Webb & Romberg, 1992) . Accordingly, achievement could be better defined and measured not in terms of number facts and procedures that the student can reproduce, but in terms of best estimates of his or her level of understanding of key concepts and interrelated underlying principles (Wilson, 1992) .


A survey to investigate teacher awareness of alternative assessment of students in mathematics of (n=126) public school teachers in primary (K-2), elementary (3-5), middle school (6-8) and high school (9-12) showed that significant differences in awareness of alternative assessment practices exists among the four levels (Drury, 1994). In another study, (Watts, 1993) which is a description of the implementation of the Curriculum Standards for School Mathematics (NCTM, 1989) in grades K-3, it was found that teachers used in their tests, knowledge level items significantly more frequently than higher level items, and items with manipulative materials significantly more than items without manipulatives. Alternative assessment formats were considered significantly more difficult to use.


In a study on the influence of district standardized testing on mathematics instruction for grades 3 and 8 (Kolitch, 1993) it was reported that in two school districts, the curriculum was aligned to test content; in a third district with an innovative mathematics program, the district test had little influence on mathematics instruction, but the program was in jeopardy because of decreasing computation scores. Kamii and Lewis (May 1991) also report a similar finding of achievement testing in primary mathematics as perpetuating lower-order thinking. According to an achievement test, traditional instruction produced results as good as or better than, a constructivist program in second grade. Such tests were created within a framework of mathematics which Kamii and Lewis argue does not measure understanding.


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