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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
20
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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