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Constructivism and Mathematics Teaching

Dr. Steve Atkins

The purpose of this article is to delineate a personal philosophy of mathematics teaching

that is consistent with research and theoretic literature on radical and social constructivism. This

eclectic approach to mathematics teaching places equal importance on students cognition and

their socio-cultural contexts. Cobb et al. (1992) suggested that coordinating findings that derive

from each perspective has led to advancements in the field of mathematics teaching and is

necessary to understanding learning. Key components of an eclectic teaching philosophy based

on combining radical and social constructivism have included: (a) Knowledge is actively

constructed, not passively received; (b) teaching should focus on student thinking; (c) teaching

should facilitate conceptual change; (d) teaching should be interactive; (e) teaching should

support individual and collaborative learning arrangements; (f) teaching should involve

authentic tasks; and (g) teaching should encourage multiple representations and connections.

The article is organized into two parts: (a) an overview of radical and social

constructivism; and (b) a detailed description and rationale for the key components of the

teaching philosophy listed above based on theoretic and research findings in mathematics

education.

Constructivism

Over the past two decades researchers have been developing a new view of what it means

to teach mathematics. Constructivist philosophies have contributed to shaping these efforts by

providing researchers and mathematics educators useful ways to understand learning.

Constructivism encompasses a range of beliefs about learning. One extreme is radical

constructivism, based on the views of Piaget (1978), which views knowledge as individually

constructed; and the other extreme is social constructivism, consistent with the views of

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Vygotsky (1962), which asserts that learning occurs solely through socio-cultural interactions.

Simon (1995) claims that the difference depends on the focus of the observer. The radical

constructivist emphasizes a cognitive orientation, focusing on the psychology of how an

individual integrates knowledge; whereas social constructivists focus on social interactions and

the individuals cultural context (Cobb, 1989).

Many researchers see either extreme as too limiting (Cobb, 1995, Simon, 1995). Cobb

(1994) states that each perspective, the socio-cultural and the cognitive, equally contributes

toward understanding learning. He states that cognitive and social influences are complimentary

in that social interactions implicitly assume an actively constructing child....cultural practice is

the background against which self-organization comes to the fore. (p.17). Despite the

differences in the locus of cognitive development, radical constructivists such as Piaget (1970)

do not deny the importance of others in the development of cognition, nor did Vygotsky (1987)

deny the primacy of the active construction of knowledge by the cognizing individual.

Key Components of Constructivist Teaching Practices

Knowledge is Constructed, Not Received

Kilpatrick (1987) states that constructivist epistimologies consists of three major

premises: (1) knowledge is actively constructed, not passively received or transmitted, (2)

coming to know is an adaptive process of organizing ones experiences, and (3) learning does not

involve discovering an independent pre-existing world outside the mind of the knower. These

statements together form an epistemology that is in sharp contrast with the traditional view that

sees learning as an encoding of an objective reality out there. Acceptance of these premises by

the mathematics education community has consequently resulted in a shift from behaviorist and

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positivist models of learning toward cognitive and interpretive models (Mathematical Sciences

education Board [MSEB], 1989).

Much of constructivism is based on the work of Piaget (1978) who viewed learners as

builders of knowledge, not passive receptors or tabula rasas. From this perspective, learners

take in knowledge and relate it to previously assimilated knowledge in order to fit it within

their existing mental models (Simon, 1995). Hence, thinking is considered an interpretive act,

which contrasts significantly with the more traditional view of learning that sees concepts as

transferable and ready-made from teachers to students. Thus, constructivist theory suggests

that, rather than simply accepting new information without question, students interpret

everything they see and hear and then remember new information in relation to what they

already know. Students enter the classroom with their own cognitive structures constructed from

their own past experiences. This idiosyncratically constructed knowledge may be viable and

useful, invalid, incomplete, or it may be hinder students subsequent learning. Simon (1995)

states that the knowledge we have constructed helps us to make sense of, interpret, and predict

events in our experiential world.

The assumption that humans have no access to an objective reality independent of

their way of knowing it seems counterintuitive to many mathematics educators (Wilson et al.,

1995). Some educators believe that an acceptance of non-dualistic thought, where learners arrive

at different but legitimate understandings, lead inevitably toward solipsism or an anything goes

mentality (Simon, 1995). In response, Wilson (1995), a social constructivist, suggests that

learners negotiate the meaning of their experiences by interacting with others to arrive at shared

understandings. Von Glaserfeld (1987), states that student understanding of a concept derives its

validity not from its accuracy in describing the real world, but whether or not it works. A concept

works or is viable to the extent that it does what it needs to do: to make sense of our perceptions,

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to make predictions accurately, or to solve a problem. Wilson et al. (1995) conjecture that the

denial of dualism may result in classroom practices that are more sensitive toward personal

meanings and multiple perspectives of the world.

Social constructivists contend that the manner in which students learn is inseparable from

the content (Schoenfeld, 1985). This is quite different from the traditional didactic education that

assumes a separation between knowing and doing that treats knowledge as a self sufficient

substance. A view that knowing mathematics is doing mathematics suggests that

mathematics is not a fixed body of procedures, but rather a growing and changing science of

quantity and pattern, whose practice involves experimentation, reasoning, and argumentation

(MSEB, 1989). Wilson et al. (1995) suggest that instruction should recognize the dynamic nature

of knowledge and encourage students to engage in a continuing search for improved

understanding.

Past and current research supports the claim that knowledge is constructed and not

passively received. Research conducted by Brownell (1935) supports this view. At the beginning

of the school year Brownell gave a group of third graders (n=57) a test in 100 simple addition

combinations. They were taught these combinations in grades one and two by methods which

agree with drill theory. Thirty-two children were chosen for interviews, ten students were

selected that had the highest score, thirteen who had average scores, and nineteen who had low

scores. Interviews attempted to discover how each child calculated sums and thought of the

numbers and processes employed. The interview revealed that 22.7% of the combinations

involved counting; that 14% were solved indirectly; that 23.8 % were incorrectly guessed; and

that only 39.5% were known as memorized associations. The fact that students demonstrated

strategies that were quite different from the learning intended by the teacher supports the notion

that students do construct their own knowledge.

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The study by Brownell demonstrates the inadequacy of teaching methods that are based

on transmission models where teachers seek to transfer knowledge to passive students.

Constructivists such as Blais have described how these methods effect the learner. Blais (1988)

states that traditional teaching in mathematics that involves giving students a maximum of

explanation implies the creation of a listener-follower role for students. Such a role, he

conjectures, contributes to dependence, fosters the growth of learned helplessness, and

encourages students to memorize information instead of construct robust knowledge structures.

Blais contends that traditional mathematics courses focused too much on transmitting content at

the expense of thought processes, and he felt that the focus should be on the development of

thinking skills and conceptual understanding.

That all knowledge is constructed differently and can interfere with subsequent learning

has been an important focus in constructivist teaching. Luchens (1942) shows that students tend

to solve problems using a single approach if the approach was used extensively prior to solving

new problems, even though a different way would be more productive. For example, Luchens

found that 64% of the college students in an experimental group were unable to solve the

following problem: Measure 25 cups using jugs with 28, 76 and 3-cup capacities. The

experimental group had worked a previous set of problems by applying a rule - the problems

could all be solved by adding two of the first jug listed and subtracting one of each of the other

two jugs. However, only 5% of a control group who were given no rules failed to solve this

problem. This illustrates how (a) relying on algorithms for solving problems can inhibit students

abilities to solve similar problems when the learned algorithm does not apply, and (b) giving

students algorithms for solving problems does not necessarily produce viable knowledge.

Norman and Pritchard (1994) have described how students prior

conceptualization of functions as processes rather than objects causes difficulty for these

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students in calculus. Being able to see a function as an object is essential in calculus for gaining a

basic understanding of several concepts such as function composition, the chain rule, and

integration (Ferrini-Mundy & Graham, 1991; Dubinsky, 1991). Norman and Pritchard state that

students tendency to process is carried over into calculus. They offer evidence based on

interviews with college students, that this limited intuition is often a cognitive obstacle toward

students gaining a comprehensive understanding of the derivative and integral concept. During

interviews students were asked to find F(a) when F(x) is defined by F(x) = ax f(t)dt. Most

students responded that F(x) doesnt exist unless a formula for f(t) is known. Norman and

Pritchard assert that students tendency to think of functions as processes impeded their ability to

solve this and similar problems during interviews.

Teaching Should Focus on How Students Think

Taken together Brownell (1935), Luchins (1942), and Norman and Pritchard (1992)

demonstrate that students do construct their own knowledge, and that students prior

constructions influence subsequent learning. Knowing what mistakes students are likely to make

and common errors in understanding will help teachers assist students in meaningful

constructivist learning activities. For example, Carpenter et al. (1989) recruited 40 first-grade

teachers to participate in a month long workshop on Cognitively Guided Instruction. Cognitively

Guided Instruction (CGI) involves having teachers learn about research on childrens thinking

processes. At the workshop Carpenter shared findings from a previous study, showing how five-

and six-year-olds can solve many types of complex addition and subtraction word problems.

Teachers then discussed and debated the meanings of these findings and also interviewed

children to see for themselves whether children actually solved problems as reported by the

researchers.

One year following the workshop, workshop participants demonstrated stronger beliefs

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that children construct, rather than receive mathematical knowledge; and, to facilitate this

construction, instruction should not center around presentations but around childrens own

knowledge and developing understandings. At the end of the school year assessments were made

including a pretest and post-test of student achievement with various measures of problem-

solving ability, measures of student confidence, and student interviews. Students in CGI teachers

classes did better than students in control teachers classes on written and interview measures of

both complex problem solving and number fact knowledge.

Conceptual Change and Teaching

It has been shown that research supports the view that knowledge is actively constructed,

that these constructions effect new learning in desirable and sometimes undesirable ways, and

that teachers knowledge of student constructions can impact student learning. Implications are

that mathematics teachers must be knowledgeable not only in their content areas, but also in the

research literature on how students learn and on instructional strategies that foster learning. A

more detailed description of how conceptual growth occurs within individuals, from a cognitive

and social perspective, will serve as a framework for discussing the research on what teachers

might do to affect conceptual change.

Simon (1995) states that students modify their understanding by directly confronting

misconceptions. Piaget (1978) asserts that when misconceptions occur cognitive development

takes place through the mechanism of equilibration. When presented with a new idea, something

to learn, a learners cognitive equilibrium is disturbed. Either the student will ignore the new

idea, and return to a state of equilibrium or modify ones cognitive structures so that the new idea

is incorporated. Reflection on successful adaptive operations (reflective abstraction) leads to new

or modified concepts. Piaget (1970) states that an unstimulating environment would result in

infrequent states of disequilibrium providing the learner with little or no reason to learn.

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Nussbaum and Novick (1982) state that constructivist researchers have focused on the

identification of misconceptions in the cognitive structures of learners. They contend that

standard methods such as lecture, demonstrations, and class tests are poor methods to identify

and correct many kinds of misconceptions. Given the research findings that reveal how

misconceptions hinder student ability to process information properly (Norman & Pritchard,

1992), models are needed that support ways in which teachers can identify and assist students in

affecting change in their conceptual frameworks, thereby eliminating misconceptions in favor of

more viable schemas.

According to Fischbein, (1990) beliefs are often robust, but can be addressed by inducing

cognitive conflict during instruction. Research conducted by Skane (1993) investigated the

efficacy of a conceptual change model in amending college students misconceptions about the

distributive law - overgeneralizations of linearity that produce such errors as log (x+3) = log (x)

+ log(3). To amend this misconception in a community college precalculus course (n=68), the

author implemented a method called a constructivist teaching sequence consisting of five stages

of instruction: orientation, elicitation, restructuring, application, and review (reflection). During

the elicitation phase students were given questionnaires containing problems that often involve

student overgeneralization errors. Restructuring activities involved students discussing their

answers to the questionnaires, negotiating reasons and answers until consensus was achieved,

and noting in writing any changes in their understanding. This was followed by whole class

discussion, with students presenting as many different reasons as possible and observing

similarities and differences in items. Finally, the students reflected on the whole exercise and

responded in writing to statements such as I used to think... with the least step being the most

important one. Results indicate that the conceptual change model was effective immediately and

over time as a teaching strategy to amend misconceptions. Pretest and posttest results

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demonstrated positive amendment of distributive law misconceptions. Student comments were

positive about the experience.

In summary, research reveals that students have well-entrenched misconceptions. In order

to change these misconceptions constructivists theorize that students must undergo conceptual

change by restructuring their mental schemas. Skanes models contribute toward our

understanding of how conceptual change can be facilitated within the context of whole class

instruction. Other approaches such as cooperative learning (Yackel et al, 1991), discussed later in

this paper, have also been used to facilitate amendment of misconceptions. Models of teaching

have been designed to facilitate conceptual change through classroom discourse.

Planning and Classroom Discourse

The teachers role in the classroom from a constructivist teaching perspective is seen as

an interactive one. It views communication in the classroom as a two-way street, where teachers

ask probing questions to assess or to facilitate students understanding and where students are

encouraged to question the teacher. Hence, the teacher is seen as a facilitator of learning rather

than a transmitter of knowledge. Communication is vital to constructivist teaching in order to

recognize misconceptions when they arise and to provide opportunities to challenge students

misconceptions through discourse.

Piaget (1980) and Vygotsky (1962) provide reasonable explanations for concept

development occurring as a consequence of student-teacher interactions. Piaget emphasized that

communication provides opportunities to take different points of view into account which, he

claims, offer the potential to produce cognitive conflicts. Resolving the conflicts leads directly to

cognitive development (Piaget & Inhelder, 1967). Vygotsky (1987), in contrast, asserts that

learning develops between a learner and a more competent person who moves the learner from

where he is to where he can be with help (zone of proximal development). By actively engaging

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in discourse with teachers and more competent peers Vygotsky believed that learners internalize

the methods of more expert problem solvers.

Simon (1995) has developed a model for teaching consistent with a cognitive approach

that elaborates on the teachers responsibilities in the mathematics classroom. His model was

developed from analyzing data from a whole-class constructivist teaching experiment he

conducted to explore the pedagogical implications of constructivism. Simons model of teaching

involves three components (1) teachers learning goal, (2) teachers plan for learning activities,

(3) and teachers hypothesis of students learning processes. The basis of the design is to

hypothesize what the learner might learn and finding ways of fostering this learning. Class

lessons consist of small-group problem solving and teacher led whole-class discussions. The goal

is based on relating the teachers mathematical understanding to the teachers hypotheses about

students learning.

The hypothesis of student learning refers to the teachers prediction as to how learning

might proceed. The generation of hypothetical learning is made prior to classroom instruction

and is the process by which the teacher develops a plan for classroom activity. Simon states that

although learning proceeds along idiosyncratic paths, mathematical activity often occurs in

predictable ways, and that many students in the same class can benefit from the same

mathematical task. The hypotheses of students learning processes is based on information from

a variety of sources: experience with students in a related area, pretesting, past experience with

similar groups, and research data on difficulties students have with particular concepts. Simon

uses a rich set of problem situations that challenges students understanding and exposes their

misconceptions and conceptual difficulties.

During a teaching episode Simon poses questions to attempt to create disequilibrium with

his students. He states that as he interacts, listens to and observes students, he constantly assesses

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students thinking, which may in turn lead to a new or modified hypothetical trajectory of how

learning will proceed. He continually engages in adjusting his hypothesis to better reflect his

enhanced knowledge, and poses questions to help eliminate misconceptions that were exposed

during the teaching process.

In a similar study Tzur (1995) conducted a teaching experiment to account for possible

relationships between classroom discourse and cognitive and social-emotional aspects of their

fraction learning. Fifty-one teaching episodes were conducted with fourth and fifth grade

children. The episodes were videotaped for later analysis.

From his analysis of the data Tzur developed a constructivist model of teacher-learner

interaction. The model stresses that during planning periods, the teacher infers the childrens

fraction schemes and operations, and plans initial, reflective, and anticipatory tasks that are likely

to activate the childrens schemes and engaging the whole class in mathematical inquiry. During

interactive periods, the teacher poses the tasks and interacts with the children to facilitate child-

initiated and child sustained iteration. According to the childrens work and responses the teacher

modifies the tasks to help them in neutralizing perturbations by modifying their fraction

schemes. Simons (1995) and Tzurs (1995) models of constructivist teaching and the

contributions of other researchers in mathematics education provide a useful framework for

considering the teachers role in constructivist teaching. Both models involve an initial

assessment of students thinking prior to instruction. Lampert (1990) states that initial planning

involves using a variety of methods that includes pre-testing, engaging in whole-class discussion,

or using a task with the potential to reveal a wide variety of students thinking. After this initial

assessment teachers plan tasks and situations to engage students in active learning. During the

planning periods, the teacher infers students knowledge and operations from initial assessments;

and then plans tasks that are likely to activate and challenge childrens current mental structures.

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During interacting periods the teacher poses tasks, problems, or situations and interacts

with learners to facilitate conceptual change (Simon, 1995). Teachers assist students in a

supportive role as they explore concepts together instead of directing their learning autocratically

(Brown, 1991). Hence, the teacher plays the role of facilitator, coach, or colleague (Cobb, et al.

1992) who listens continually to students ideas as they are allowed and encouraged to explore,

make conjectures, generalize and explain and justify their answers. Teachers show flexibility in

class allowing for the teaching moment - where students thinking results in venues for

subsequent dialogue or activity (NCTM, 1991).

In a study which examined the role of discourse in the mathematics classroom

Rosenblum (1992) employed a constructivist framework for classroom interaction consistent

with Simons and Tzurs models. The role of communication was examined in helping children

make connections between what they know and what they are trying to understand. A sample of

20 classes were randomly selected and randomly assigned to treatment and control groups. The

participating students were second and third graders. The treatment groups received a discourse

intervention allowing students to solve problems many ways and to determine answers through

negotiation. The control group received traditional instruction.

The intervention took place for 30 minutes, three times a week for five months.

Dependent variables included reasoning ability, self-esteem, computation and creativity.

Multivariable analysis of covariance was performed on the four dependent variables using pretest

scores as covariates. Computation and creativity contributed to a significant main effects for

second and third graders respectively. No significant grade by group interaction was found.

Benefits in the mathematical discourse according to reports from teachers included (a) better

understanding and evaluation of student mathematical ability, and (b) improved student

willingness to try difficult problems.

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Theoretic and research literature on constructivism stress the importance of teachers

interacting with the students in ways similar to the description provided by Simon (Steffe &

Gale, 1995; Cobb et al., 1992; Tzur, 1995). While Simons and Tzurs models provide important

insight into how constructivist theory can be concretized in the classroom, few models of

mathematics teaching consistent with constructivism are located in the literature. Most offer little

more than anecdotal evidence regarding their effectiveness. Much research remains to be done

to understand the implications of models related to classroom discourse within the context of

constructivism.

Teaching Should Support Collaborative and Individual Constructions

A growing body of research on collaborate or cooperative learning supports both

Vygotskys and Piagets views of the advantages of social interactions for concept development.

Yackel et al. (1991) conducted an experiment in which all instruction in a second grade

mathematics classroom was replaced by small group problem-solving strategies for an entire

school year. The goal was to foster collaborative learning and conflict resolutions skills. Children

were grouped in pairs and spent half of each math period working together to solve math

problems. The cooperative learning approach was used for all aspects of instruction including

computation activities, and other traditional topics such as time and measurement, as well as

instructional activities that were problematic situations. Work was not graded so that external

reward systems were not a factor in the experiment. Pairs were formed that were homogeneous

with respect to mathematical development and other pairs were heterogeneous in this regard.

Whole group discussions followed paired activity. In groups where children were at different

conceptual levels the teachers sometimes modified the obligation so that students could reach

consensus about solution methods. It was acceptable for the most advanced child in a

heterogeneous group to construct a solution method that the other could not understand provided

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the weaker child had also constructed a solution.

Yackel et al.(1991) concluded that the interactions provided students opportunities for

learning that resulted directly from the interactions. Further, attempts to resolve conflicts led to

opportunities for students to reconceptualize problems and construct other solution methods.

Cobbs et al. study provides anecdotal evidence to support constructivist teaching models and

provides a framework for implementing constructivist teaching.

Qin (1993) examined the effectiveness of achieving higher-order learning tasks in

cooperative learning compared with competitive learning through a meta-analysis. He looked at

63 studies comparing the effectiveness of cooperative learning with competitive learning in

different types of problem-solving. The major hypothesis of the study was that learners are more

successful in problem-solving under cooperative learning than under competitive learning.

Findings were: (a) overall learners who engaged in cooperative learning were more successful in

problem-solving than students involved in competitive learning arrangements; and (b) there is a

tendency for older learners to do better than younger learner in three types of problems:

linguistic and well-defined problems, but not in ill-defined problem.

Studies conducted by Yackel.et al. (1991) and Quin (1993) support constructivist teaching

by providing evidence that social learning arrangements can provide contexts for constructing

more viable schemas. They also provide evidence that cooperative learning may be differentially

effective across learners.

An important aspect of constructivist teaching requires that students become more

reflective thinkers. von Glaserfeld (1987) discusses mathematical knowledge as a product of

conscious reflection. Vygotsky (1962) recognized the importance of writing on reflective

thought, he states, that in written speech, we are obliged to create the situation, to represent it to

ourselves (p. 99). He believed that writing encouraged learners to analyze, compare facts, and

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synthesize information. Moreover, he believed that language and thought were dialectic in

nature, and were both transformed in the act of representation. Kennedy (1980) and Connelly

(1989) state that writing requires students to think about a topic, focus on and construct

important concepts, and make those concepts to some degree their own.

A study conducted by Rose (1989) explored, the role of writing to support the learning of

mathematics. In her study, students were encouraged to think on paper rather than communicate

information or persuade an audience. The setting for the study was a calculus course for business

majors at a small private college. As the researcher taught this course, she used autobiographical

narratives, in-class focused writing, and spontaneous writing. In-class writing focused on specific

mathematical content currently being covered. Typical writing assignments included: What is a

composite function?; or What is the procedure for solving a related rate problem? Qualitative

research methods were used on both what the students wrote and what they said about the

writing experience to ascertain perceived benefits. An analysis of data revealed that (a) writing

helped students retain concepts, (b) promoted independent learning, and (c) writing had transfer

benefits to other courses. Students believed that throughout the dialogue created with the journal,

a more supportive atmosphere was created in the classroom. Rose states that the activities

fostered dialogue and helped produce a more personal, cooperative, and active learning

environment.

Powell (1989) used journal entries and free writing in a study of college freshmen

enrolled in a developmental computations class. Powell asked students to write for five minutes

during each class about what they have done, discovered, invented, or felt. Although students

were free to write on any topic or issue, Powell offered questions to help stimulate thought and

reflection. He found that journal writing enabled students to become more involved in actively

constructing and making sense of mathematics; and that students benefited because they were

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able to monitor their own learning. Powell concluded that writing caused students to reflect

critically about their mathematics learning; and, that a high correlation existed between improved

problem solving performance and the extent to which reflective and descriptive writing activities

were used during the course.

Students full use of language are vital aspects of constructivist teaching. Researchers

provide evidence that collaborative learning and writing are ways in which teachers can utilize

language to facilitate knowledge construction. The nature and quality of tasks need further

elaboration.

Authentic Tasks

Tasks in the classroom from a constructivist teaching point of view should be authentic

in the sense that students are allowed to solve meaningful problems in much the same manner as

experts or mathematicians. That is students should be allowed to discuss problems with others, to

conjecture, explore, invent, and to use all of the tools and resources that are common practice

outside of school. (Brown, et al, 1991). The philosophy of constructivist teaching endorses the

use of authentic tasks, viewing them through the lenses of both Piaget (1978) and Vygotsky

(1962), as opportunities for students to modify their schemas and as opportunities to become

enculturated into authentic practice through activity and social interaction.

Authentic tasks are related to the idea of situated cognition. Collins (1991) states that

situated cognition is based on the social constructivist claim that knowledge is coded by and

connected to the activity and environment in which it was developed. The implications are that if

students learning is embedded in a mathematics classroom which resembles the context of

expert practitioners, then their cognition will be closer to that of an expert. Hence, constructivist

teaching supports the use of authentic activities that reflect how problem solving is performed

and how expertise is developed in out-of-school contexts. Schoenfeld (1987) states that

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authentic practice implies the creation of a mathematics culture in the classroom that

encourages students to think of mathematics as an integral part of their everyday lives.

Brown et al. (1989) state that, to be truly authentic, tasks need to immerse students in all

aspects of how problems are solved in out-of-school contexts. He states that real-world problem

solving involves verbal and nonverbal interaction, technology including calculators and

computers, writing, graphs, charts, tables, physical objects, learning strategies, computers, and

calculators. The implications are that in-school mathematics that restrict students from using all

of the tools and practices common to the community of expert practitioners causes students to

become encultured into an unrealistic situation that will foster expertise only in in-school

contexts (Schoenfeld, 1989). Vygotsky (1962) states that engaging in progressively more

complex activities and tasks and internalizing social interaction and psychological tools bring

about habits of the expert. The role of the teacher as participant and guide helps the student learn

to solve problems and to eventually gain expertise (Vygotsky, 1987).

Support for the situated cognition movement is found in Saxes (1991) analysis of the

computational algorithms developed by child candy sellers in Brazil. Saxe reported that the

candy sellers mathematical procedures were closely linked to their participation in the monetary

system. He reported that the children could work with numerals only if they were those written

on familiar bill or coins. Choi (1995) conducted a study that corroborated these findings. The

purpose was to determine the effects of contextualization and complexity of a situation on

achievement, transfer, and attitude toward mathematical problem solving. A total of 101 fifth

grade students from an elementary school participated in the study. In general, students who

studied complex, contextualized problems performed the best and students who studied complex,

decontextualized problems performed the worst. Students who studied simple problems in

decontextualized situations performed best on one-step questions (both context-rich and context-

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poor), while students who studied complex problems in contextualized situations performed best

on multi-step questions (both context-rich and context-poor). Results indicate that situated

instruction can promote learning by providing meaningful and complex problems rather than

decontextualized problems.

An environment where students are allowed to explore real-world problems using

cooperative learning and technology brings classroom practice closer to authentic math practice

than traditional approaches that involve students working alone on textbook word problems.

Johnson (1992) examined the impact of constructivist learning activities on students problem-

solving skills and attitudes in a university-level applications mathematics course. Four sections

were taught using a constructivist approach; and six sections were taught using traditional lecture

methods. There were a total of 331 students. Constructivist activities included working in groups,

having students generate problems, having students present problems and solutions to the class,

and discussing concepts and the interpretation of problems and solutions. Control classes used

the traditional lecture and discussion methods with nonroutine problem solving as time

permitted. Non-routine problems were added to final exams to collect problem-solving skills

data and math SAT scores were used as the control variable in an ANOVA.

Results indicated that students in constructivist classes had significantly greater problem-

solving achievement than students in the control classes. A t-test revealed that course grades

were significantly higher for the treatment group, and the mean differences between pretest and

posttest scores for an attitude measure, Aiken Revised Math Attitude Scale, demonstrated

significantly improved attitudes for the treatment group.

Brown et al. (1989) state that school mathematics disregards most of the inventive

heuristics that students bring into the classroom. Hence, he claims that it implicitly devalues the

whole process of inventive problem solving that is valued in out-of-school mathematics.

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Anbanithi (1993) examined the effect of using a constructivist approach in the classroom that

allowed students to discover, invent, and negotiate their own problem solving strategies. One

hundred and six subjects were assigned to four experimental conditions, (1) constructivist

teaching, (2) explicit strategy instruction, (3) explicit strategy teaching plus a constructivist

environment, and (4) control. After an intervention of fourteen days, various measures were

administered to assess students levels of conceptual understanding, beliefs about reasons for

success in mathematics, use of deep processing strategies, ability to think and communicate

mathematically, and perceived competence in mathematical problem solving.

Four main findings emerged. First, students exposed to a constructivist environment

performed superior to those in the explicit instruction group for transfer problems that measured

deep conceptual understanding. Second, students exposed to a constructivist environment held

stronger beliefs and more positive goals in math about the importance of understanding and

collaborating. Third, constructivist students reported the use of deep processing strategies than

students in the explicit instruction group. Fourth, constructivist students were able to

communicate more effectively during mathematics problem solving than students in the direct

explanation group.

The studies conducted by Johnson (1992) and Anbanithi (1993) support the situated

cognition approach for problem solving. They suggest that to help students develop

mathematical understanding, they should engage in tasks of inquiry, reasoning, and problem

solving that reflect real-world mathematical practice. They also reveal that constructivist

approaches which emphasize communication, reasoning and problem solving have a positive

effect on development of beliefs, problem solving ability, and promote conceptual understanding.

Teaching Should Encourage Multiple Representations and Connections

Both theoretic and research literature on constructivism suggest that teaching should

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emphasize multiple representations and connections in the classroom. Bruner (1960) and Dienes

(1977) suggest that concept learning is enhanced when a concept is presented in a variety of

physical contexts. Bruner contends that students be allowed to construct different representations

of concepts, and he claimed that representations the learner has constructed for himself will be

more accessible for retrieval and more likely to be applied correctly in appropriate situations.

Skemp (1977) states that students who are able to connect mathematical symbols to appropriate

referents (representations) possess relational schemas that assist them in solving problems even

when they forget the rules for conducting operations.

Research on cognition has shown the importance of helping students develop a well-

connected body of representations. Rosenshine (1995) states that the size of a persons mental

structure, the number of connections between pieces of knowledge, the structure of the

connections, and the organization and richness of the relationships are all important for

processing information and problem solving. She contends that it is easier to assimilate new

information and easier to use prior knowledge for problem solving when knowledge structures

are large and well connected. Hiebart and Carpenter (1992) state that a major difference between

an expert and novice is that an experts knowledge structures has a larger number of knowledge

items, more connections between items, stronger links and connections, and a better organized

knowledge structure. Hence, it is important to help students see the interrelations of topics in

order to connect entities.

The introduction of graphing calculators and computers has been a catalyst for research

on representations and connections. Much of the research available indicates that students who

use the graphing calculator and software to explore, propose, and build connections among the

numeric, graphic, and algebraic representations of functions have a better understanding of

functions and related concepts. Melin-Conjeros (1992) states that relationships that are unclear to

21

students in mathematical equations or tables of data become more distinct in graphical form.

Melin-Conjeros investigated the effects of doing calculus homework assignments in a

mathematics laboratory equipped with a Computer Algebra System (CAS) on students'

achievement and attitude towards mathematics. Interviews revealed that students who had used

the CAS for their homework had a better understanding of selected concepts - increasing and

decreasing functions, asymptotes, concavity, limits and continuity - and that students taught with

computer-generated representations were better able than students in a control group to recognize

derivatives and integrals in different forms and to relate the forms to algebraic and graphical

representations of functions. Goldenburg (1988) showed that the most successful students in

advanced mathematics are those with the flexibility to work in more than one representation

(graphic, numeric or symbolic) and that students who are limited to one representation are less

likely to solve a wide range of problems. Research conducted by Browning (1990) corroborates

the empirical studies which suggest that students in technology based courses gain a better

understanding of connections between graphical, numerical, and algebraic representations than

students in a traditional classroom.

In summary, constructivist teaching offers a different view of teaching from the

traditional positivist/behaviorist paradigm of teaching. The center of classroom teaching shifts

from teacher-centered to a focus on student knowledge and experiences. The constructivist view

requires a teacher to adjust his or her role as knowledge dispenser to a role of co-investigator,

competent peer, or guide and interlocutor. They probe students understanding and help them

resolve conflict between new concepts and previous mathematical knowledge. The most

important attribute of the constructivist teacher is the ability to pose questions, tasks, and have a

broad understanding of how students learn mathematics. Further, he or she must continually

analyze plans and instructional approaches to fit students ways of knowing.

22

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