Constructivism and Mathematics Teaching by Steve Atkins

 

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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 e qual importance on students‟ cognition cognition and their socio-cultural contexts. Cobb et al. (1992) suggested that coordinating c oordinating findings that derive from each perspective has led to advancements a dvancements in the field of m mathematics athematics teaching and is necessary to understanding learning. Key components compone nts 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 should focus on student student thinking; thinking; (c) teaching should facilitate conceptual change; (d) teaching should be interactive; (e) teaching should support individual and collaborativ collaborativee learning arrangements; (f) teaching should involve authentic tasks; and (g) teaching should encourage multiple representations 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 ke y 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 be en developing a new view of wha whatt it means to teach mathematics. Constructivist philosophies have contributed to shaping these the se 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 a sserts 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 individual‟s cultural context (Cobb, 1989).  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 self-organization comes to the fore.” (p.17). (p.17). Despite the differences in the locus of cognitive development, radical constructivists such as Piaget (1970) do not deny the importance importan ce of others in the development o off cognition, nor did V Vygotsky ygotsky (1987) deny the primacy of the active construction of knowledge by the cognizing individual.

Key Components of Constructivist Co nstructivist T Teaching eaching 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, transmitted, (2) coming to know is an adaptive process of organizing one‟s 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 a n encoding of an objective reality „out there.‟ the re.‟ Acceptance of these premises by by the mathematics education community has consequently resulted in a shift from behaviorist and

 

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 positivist models of learning toward cognitive cognitive and interpretive models (Mathematical Sciences education Board [MSEB], 1989). Much of constructivism is based on the work of Piaget Pia get (1978) who viewed learners as  builders of knowledge, not passive pa ssive receptors or “tabula rasas.” From this perspective, pe rspective, 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“ready-made” from teachers to students. Thus, constructivist theory suggests that, rather than simply accepting new information without question, qu estion, students interpret everything they see and hear hea r and then remember new ne w 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 hinde r 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 o off their way of knowing it seems counterintuitive counte rintuitive to many mathematics educators (W (Wilson ilson 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, respon se, Wilson Wilson (1995), a social constructivist, suggests that learners negotiate the meaning of their experiences e xperiences by interacting with others to arrive at shared understandings. Von Von Glaserfeld (1987), states that student understanding o off 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 e xtent 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). 19 85). 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. view. At the beginning of the school year Brownell gave a group of third graders (n=57) a test in 10 100 0 simple addition combinations. They were taught these combinations in grades one and two by methods which agree with drill theory. 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 c hild 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 a s 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 inadequac y of teaching methods that are based on “transmission” models where teachers seek to transfer knowledge to passi pa ssive ve students. Constructivists such as Blais have described how these methods eeffect ffect 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 muc h 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 interf ere 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.  prob lem. 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 esse essential ntial in calculus for gaining a  basic understanding of several concepts such as function composition, the chain rule, and integration (Ferrini-Mundy & Graham, 1991; Dubinsky, Dubinsky, 1991). Norman and Pritchard state that students‟ tendency tendency to process is carried over o ver 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 d derivative erivative and integral concept. During x

interviews students were asked to find  F(a)  F(a) when  when F(x)  F(x) is  is defined by F(x) by  F(x) =  = a  f(t)dt . Most students responded that F(x) that  F(x)  doesn‟t exist unless a formula for  f(t)  f(t) is  is known. Norman and Pritchard assert that students‟ tendency to think of functions func tions as processes impeded their th eir 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 student‟s prior constructions influence subsequent learning. Knowing what mistakes students are likely to make and common errors in understanding unde rstanding 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 lon g workshop on Cognitively Guided Instruction. Cognitively Guided Instruction (CGI) involves having teachers learn about research on children‟s thinking  processes. At At the workshop Carpenter shared findings from a previous study, study, showing how fivefive and six-year-olds can solve many types of complex com plex addition and subtraction word problems. Teachers then discussed discussed and debated de bated the meanings of these findings and also interviewed children to see for themselves whether children actually solved problems as reported rep orted 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, a nd, to facilitate this construction, instruction should not center around presentations but aaround round children‟s own knowledge and developing understandings. unde rstandings. At the end of the school year assessments were made including a pretest and post-test po st-test of student achievement with various measures of problemsolving ability, 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 co nstructions 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 w within ithin individuals, from a cog cognitive nitive 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 learner‟s learner ‟s cognitive cognitive equilibrium is disturbed. Either the student will ignore the new idea, and return to a state of equilibrium or modify one‟s cognitive structures so that the new idea is incorporated. Reflection on successful adaptive operations ope rations (reflective abstraction) leads to new or modified concepts. Piaget (1970) states that an a n 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 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 m ethods 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‟ mis misconceptions conceptions 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, a pplication, 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, Finally, the studen students ts 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 amen d 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 constructivis co nstructivists ts theorize that students must undergo conceptual change by restructuring their the ir mental schemas. Skane‟s models contribute toward our understanding of how conceptual concep tual 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 teacher‟s role in the classroom from a constructivist teaching perspective is seen as an interactive one. It views communication communica tion 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 que stion the teacher. Hence, the teacher is seen as a facilitator of learning rather rathe r than a “transmitter” of knowledge. Communication is vital to constructivist te teaching aching 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 con concept cept 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 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 proxim al development). By actively engaging

 

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in discourse with teachers and more competent co mpetent peers Vygotsky 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 teache rs responsibilities in the mathematics class classroom. room. His model was developed from analyzing data from a whole-class constructivist teaching experiment he conducted to explore the pedagogical implications of constructivism. Simon‟s model of teaching involves three components (1) teacher‟s teacher ‟s learning learning goal, (2) teacher‟s teacher ‟s plan plan for learning activities, (3) and teacher‟s teach er‟s hypothesis of students‟ learning learning processes.  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 whole -class discussions. The goal is based on relating the teacher‟s teacher‟s mathematical understanding to the teacher‟s hypotheses about students‟ learning.  learning.  The “hypothesis of student learning” refers to the teacher‟s teache r‟s prediction as to how learning might proceed. The generation of “hypothetical learning” is made prior to to classroom instruction and is the process by which the teacher develops a plan for classroom c lassroom activity. activity. Simon states that although learning proceeds along idiosyncratic 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 hav havee with particular concepts. Simon uses a rich set of problem situations that challenges challe nges students‟ und understanding erstanding and ex exposes poses their misconceptions and conceptual c onceptual difficulties. 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 o bserves 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, kno wledge, and poses questions to help eliminate misconceptions that were exposed during the teaching process. In a similar study Tzur (1995) conducted a teaching te aching experiment to account for possible relationships between classroom classroom discourse and cognitive and ssocial-emotional ocial-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 d uring planning periods, the teacher infers the children‟s fraction schemes and operations, and plans p lans initial, reflective, and anticipatory tasks that are likely to activate the children‟s schemes and engaging the whole class in mathematical inquiry. During interactive periods, the teacher poses the tasks and interacts with the children to facilitate childchil dinitiated and child sustained iteration. According According to the children‟s work and responses the teacher modifies the tasks to help them in neutralizing ne utralizing perturbations by modifying their fraction schemes.

Simon‟s (1995) and Tzur‟s (1995) models of constructivist constructivist teaching and the

contributions of other researchers in mathematics education provide a useful framework for considering the teacher‟s role in constructivist teaching. Both models involve an initial assessment of students‟ thinking prior to instruction.  instruction.  Lampert (1990) states that initial planning involves using a variety of methods that includes pre-testing, pre -testing, engaging in whole-class discussion, or using a task ta sk with the potential p otential to reveal a wide variety of students‟ thinking. After this initial assessment teachers plan tasks and situations to engage students in active ac tive learning. During the  planning periods, the teacher infers students‟ knowledge and operations from initial assess assessments; ments; and then plans tasks that are likely to activate activa te and challenge children children‟s ‟s current mental structures.  structures. 

 

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During interacting periods the teacher poses tasks, problems, or situations and interacts with learners to facilitate conceptual c onceptual change (Simon, 1995). Teachers‟ ass assist ist students in a supportive role as they explore concepts together instead of directing their learning autocratically (Brown, 1991). Hence, the teacher te acher 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 gene ralize and explain and a nd 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 e mployed a constructivist framework for classroom interaction consistent with Simon‟s and Tzur‟s 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, ability, self-esteem, computation co mputation and creativity. 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 importance of teachers interacting with the students in ways similar to the description provided by Simon (Steffe (Ste ffe & Gale, 1995; Cobb Cobb et al., 1992; Tzur, 1995). While Simon‟s and Tzur‟s models provide important insight into how constructivist con structivist theory can be „concretized‟ „con cretized‟ 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 cclassroom lassroom discourse within within the context of constructivism. Teaching Should Support Collaborative and Individual Constructions A growing growing body of research on o n collaborate or cooperative learning supports both Vygotsky‟s ygotsky‟s and Piaget‟s views of the advantages ad vantages of social interactions for concept development. Yackel et al. (1991) conducted c onducted an experiment in which all instruction in a second grade mathematics classroom was replaced by small group problem-solving 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 a nd 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 Work was not graded so that external e xternal reward systems were not a factor in the experiment. e xperiment. 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. 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 acce ptable 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. Cobb‟s et al. study provides anecdotal anecd otal evidence evidence to support constructivist teaching models and  provides a framework for implementing constructivist 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 problems:: linguistic and well-defined problems, but not in ill-defined problem. p roblem. Studies conducted by Yackel.et Yackel.et al. a l. (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 Vygotsky (1962) recognized the importance of writing on reflective thought, he states, that “in written s peech, 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 a nd

 

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synthesize information. Moreover, 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 b y Rose (1989) explored, the role of writing to support the learning of mathematics. In her study, 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 c ourse 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 rela ted 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 inde pendent 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 produ ce 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 c omputations 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 co rrelation 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. Researche Researchers rs  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 ma manner nner 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 th that at are common practice outside of school. (Brown, et al, 1991). The philosophy of constructivist constructivist teaching endorses the use of authentic authe ntic tasks, viewing them through the lenses of both Piaget (1978) and Vygotsky 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 develope developed. d. The implications are that if students‟ learning learning is embedded in a mathematics classroom which resembles the context of expert practitioners, then their cognition will be closer c loser to that of an expert. Hence, constructivist c onstructivist teaching supports the use of authentic authe ntic activities that reflect how problem solving is performed and how expertise is developed in “out-of“out -of-school” school” contexts. Schoenfeld (1987) states that

 

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authentic practice authentic  practice implies the creation of a “mathematics culture” 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 a ll aspects of how problems are solved solved in out-of-school contexts. He sstates tates that real-world problem solving involves verbal and nonverbal interaction, technology tec hnology 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 expertise only in in-school contexts (Schoenfeld, 1989). 1 989). Vygotsky 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. e xpert. The role of the teacher as participant and guide helps the student learn to solve problems and a nd to eventually gain expertise (Vygotsky (Vygotsky,, 1987). Support for the situated cognition movement is found in Saxe‟s (1991) analysis of the computational algorithms developed by child candy c andy sellers in Brazil. Saxe reported that the candy sellers‟ mathematical procedures were closely linked to their th eir participation in the monetary mone tary system. He reported that the children could work with numerals only if they were those written on familiar bill or coins. Choi (1995) conducted con ducted 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 mathem atical problem solving. A total of 101 fifth grade students from an elementary school participated in the study. study. In general, students who studied complex, contextualized problems performed the best be st and students who studied complex, decontextualized problems performed the worst. worst. Students who studied simple probl problems ems 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“real-world” problems using cooperative learning and technology technolo gy brings classroom practice closer to authentic math practice than traditional approaches that involve students working alone on “textbook” word problems. Johnson (1992) examined the im pact of constructivist learning learning activities on students‟ problemproblem 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 a nd 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 a nd math SAT SAT scores were used as the control variable in an ANOVA. ANOVA. Results indicated that students in constructivist classes had significantly greater problemsolving 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 treatme nt 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-of-school” 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 understanding,  beliefs about reasons for success in mathematics, use of deep processing strategies, ability to think and communicate c ommunicate mathematically, and perceived competence in mathematical problem solving. Four main findings emerged. First, students exposed to a constructivist con structivist environment  performed superior to those in the explicit instruction group 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 a nd 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 inquir y, reasoning, and problem solving that reflect real-world 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, ability, and promote conceptual understanding. Teaching Should Encourage Multiple Representations and Connections Both theoretic and research literature on constructivism suggest that teaching teac hing 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 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 app lied correctly in appropriate situations. Skemp (1977) states that students who are able ab le 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 wellconnected body of representations. Rosenshine (1995) states that the size of a person‟s person‟s 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 expert‟s 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

 

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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, 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 showed that the most success successful ful 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 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, summary, constructivist teaching offers o ffers 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 knowled ge and experiences. The constructivist view requires a teacher to adjust his or her he r 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, que stions, tasks, and have a  broad understanding of how students learn mathematics. Further, Further, he or she must continually analyze plans and instructional approaches to fit students ways of knowing.

 

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