A Peer Reviewed Journal

Volume 3, Number 2

Nancy Pon , Fall 2001

© 2001 Nancy Pon and EGallery

EGallery grants reproduction rights for noncommercial educational purposes with the provision that full acknowledgment of the source is noted on each copy.

http://www.ucalgary.ca/~egallery

Constructivism in the Secondary Mathematics Classroom

An Independent Inquiry - Semester I

by Nancy Pon

for Michele Jacobsen

Introduction

A seed of my identity as a teacher was planted in the second week of the case seminars in the Master of Teaching program, when we examined "The BIG Four Learning Theory-Isms" - behaviourism, cognitivism, humanism and constructivism. Constructivism sounded like "real" learning to me. As I continue to experience, research and observe constructivism in action, my vision as a teacher has begun to grow. I eventually hope to be a teacher in full bloom, facilitating students' mathematical understanding in a secondary classroom grounded in constructivism.

In this paper I discuss the nature of teaching and learning based on the pedagogical applications of constructivism in secondary mathematics education. Based upon an examination of the advantages and disadvantages of this approach to education, I present a strong case for embracing constructivism for teaching mathematics. The following questions are addressed in this paper:

What Is Constructivism?

Constructivism is defined as "that philosophical position which holds that any so-called reality is, in the most immediate and concrete sense, the mental construction of those who believe they have discovered and investigated it" (Saunders, 1992, p. 136). From this perspective, learning is understood to be a self-regulated process of resolving inner conflicts that become apparent through concrete experience, discussion, and reflection (Brooks & Brooks, 1993).

There are three major tenets of a learning theory based on constructivism (Saunders, 1992).

        1. Meaning is constructed in the mind of the learner as a result of the learner's interaction with the world and cannot be communicated by a teacher to a student. "Wisdom can’t be told" (p. 136).
        2. "The construction of meaning is a psychologically active process which requires the expenditure of mental effort" (p. 137).
        3. "Cognitive structures are sometimes highly resistant to change, even in the face of observational evidence and/or formal classroom instruction to the contrary" (p. 138).

Ideas about constructivism are not new. Aspects of constructivism can be found as far back as 470-320 B.C. as Socrates, Plato and Aristotle described the formation of knowledge (Crowther, 1997). Vico, however, first formalized constructivism when, in 1710, he explained that, "the truth is the same as the made" (von Glasersfeld, 1987a, p. 27). In the eighteenth century, Locke and Kant (Crowther, 1997) further detailed constructivism. They believed that no person’s knowledge could go beyond their experience. Ceccato and Dewey again expanded upon these ideas in the twentieth century (von Glasersfeld, 1987a).

However, the major development of constructivism as a philosophy is generally credited to Jean Piaget (Crowther, 1997; Opper, 1979) working in the twentieth century. Piaget’s constructivist learning model or cognitive constructivism (Saunders, 1992) begins with a flow of information from the student’s senses into the structure of their mind. When the student’s expectations or predictions don’t coincide with their experience (i.e. observations), the student is in disequilibration. The student then has three choices: (1) don’t believe the observations, (2) don’t care one way or the other, or (3) alter understanding so the observations fit the predictions. This latter choice is called cognitive restructuring or meaningful learning. Piaget’s research indicated that if cognitive restructuring is to occur, the student must be provided with repeated, exploratory, inquiry-oriented activity that will demonstrate that his previous understanding is no longer useful (Opper, 1979). Therefore, the goal of learning is to acquire meaning and undergo cognitive restructuring so that the mind is more consistent with the world.

Piaget argued that students learn by constructing their own understanding while doing hands-on tasks that are developmentally appropriate, and by moving from concrete to more abstract ideas. He postulated that students actively construct an understanding of a static body of knowledge at developmentally appropriate times, and use language to express their thoughts.

Social constructivism is a philosophy developed later in the 1960s, primarily due to the efforts of Ernest, Goldin, Lerman, Bauersfeld and Vygotsky (Ernest, 1994). Proponents of social constructivism believe that social interactions play a major role in constructing understanding, language forms thought and that mathematics is not a static body of knowledge, but a socially constructed and evolving way of thinking. Constructivism, based on the definition cited from Saunders (1992) and Brooks and Brooks (1993), amalgamates the cognitive restructuring concept of Piaget’s constructivist model with these tenets of social constructivism. Students need to actively construct meaning in relationship to what is already known, to each other and to their own experiences. Constructivism, however, does not consider the zones of proximal development entailed in social constructivism, or the developmental stages of the brain involved in cognitive constructivism.

The conceptualization of constructivism continues to evolve today to include radical constructivism, and constructionism. Radical constructivism, proposed by von Glasersfeld (1987b), extends the idea of constructing knowledge to truly radical limits, stating that knowledge is totally subjective, since it is constructed in the minds of individuals based on their unique personal experiences. Constructionism, as defined by Harel & Papert (1991), acknowledges that learning is an active and socially dependent construction unrestricted by age or developmental stage, but emphasizes the need to engage students in the design and/or actual construction of personally significant projects.

Is Education Based On Constructivism Today?

Constructivism has had profound implications for teaching. During the last two decades, pedagogical applications of constructivism have been endorsed extensively throughout the United States (particularly in Minnesota, Oregon and Connecticut), the United Kingdom, Germany (Saunder, 1992) and Taiwan (Pirie & Kieren, 1992). The United Kingdom was the first country to nationally mandate constructivism in schools with the publication of the Cockcroft Report in 1982 and the National Curricula in 1989 and 1991 (Boaler, 1998). However, since 1994, this country has experienced a "back-to-basics" campaign away from constructivism (Boaler, 1998)!

In 1989, the National Council of Teachers of Mathematics issued its Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), which endorsed constructivism as the standard for the United States. This document was viewed as a "world-class standard" (Stein et al, 1998, p. 17). Alberta Learning (1996) has now adopted the NCTM standards for mathematics. The Alberta Assessment Consortium (1997) states that Canada is "in the midst of an important shift in our thinking about learning and teaching. The traditional view of the learner as a passive recipient of information has given way to an emerging view of the learner as an active participant" (p. 4).

Cobb et al (1992) believe that "math educators almost universally accept that learning is a constructivist process" (p. 4). Boaler (1998) however, believes that pedagogical applications of constructivism in math classes are "extremely rare in schools" (p. 42). I have also found this to be true in the few schools I have observed— particularly in secondary math classes. I was only able to find one high school math teacher who actively practiced constructivism, and s/he knew of no others (Personal interview, 1999). Does this reflect a need for better communication between local teachers or is this a telling state of affairs?

How Does Constructivism Differ From Traditional Teaching?

Traditional teaching, as it is used here, is defined as objectivist, teacher-as-information-giver, content-based, scheme-led or formal logical teaching. The ‘classical didactical triad model’ represents this traditional way of teaching, reflecting the idea that students, teachers and subjects are mutually independent entities (Cobb et al, 1992). This model represents learning as "a process of acquiring accurate mental representations of fixed mathematical structures, relationships and the like, that exist independently of individual and collective activity" (Cobb et al, 1992, p. 29).

Many will recognize or remember the typical pattern exhibited in a traditional math lesson.

        1. The teacher reviews by correcting the homework.
        2. The teacher introduces a new concept or procedure by doing examples on the board while the students take notes.
        3. The teacher directs the class through some practice problems.
        4. The students work individually on practice problems.

Teaching and learning based on constructivism looks quite different! The major differences are outlined in Figure 1 by comparing behaviours in classrooms based on constructivism and objectivism.

Figure 1. A Comparison of Pedagogy Based on Objectivism and Constructivism
(Source: Yager, 1991)

More Objectivism

Behaviour

More Constructivism

Teacher

Identifies the issue/topic

Student

No

Issue is seen as relevant

Yes

Teacher

Asks the questions

Student

Teacher

Identifies written and human resources

Student

Teacher

Locates written resources

Student

Teacher

Contacts needed human resources

Student

Teacher

Plans investigation and activities

Student

No

Varied evaluation techniques used

Yes

No

Students practice self-evaluation

Yes

No

Concepts and skills applied to new situations

Yes

No

Students take action

Yes

No

Math concepts and principles emerge because they are needed

Yes

No

Extensions of learning outside the school is evidence

Yes

 

This comparison illustrates how the student takes a much more active role in learning in a class based on constructivism. This motivates students to learn, to identify and resolve their personal misunderstandings, and to apply what they are learning to situations relevant to their own lives. This enables the student to better understand and develop strengths, weaknesses and interests, which in turn provides for life long learning and career opportunities. The teacher facilitates this process to maximize the potential of each student.

What Does Constructivism Look Like In The Classroom?

The Teacher's Role

Many researchers believe that teacher quality is by far the single most important determinant of student performance. This is even more significant when applying constructivism (Darling-Hammond & Falk, 1997). The teacher’s knowledge, beliefs, and actions all influence the success of the learner.

Recent research has documented the positive relationship between student success and teachers’ knowledge, both of math and student cognition (Swaffort et al, 1997). Further, teacher knowledge of the learner has been shown to be a more powerful predictor of student success in math than the teacher’s knowledge of problem solving or number fact strategies (Mercer et al, 1996). Brooks & Brooks (1993) believe that the most valuable quality of a teacher applying pedagogy based on constructivism is the "instantaneous and intuitive vision of the pupil’s mind as it gropes and fumbles to grasp a new idea" (p. 20).

Teachers must also develop a deep and thorough understanding of the mathematics curriculum, which enables them to pace and direct experiences so that the curriculum is covered. This knowledge enables teachers to know which questions to expand upon and which directions to move toward. Teachers must also be highly flexible risk-takers (Confrey, 1990). Together, these skills allow teachers to delight in unexpected questions and deviations, which then allows students to build on their previous learning and relevant experiences.

In terms of teacher beliefs, Pirie and Kieren (1992) explain that a teacher must have the following basic underlying beliefs to successfully embrace constructivism.

        1. All students will not achieve the same goal.
        2. There are many paths to get to the same understanding.
        3. Everyone holds a different understanding.
        4. For each topic, students can be at any one of eight different levels of mathematical understanding (i.e. primitive knowing, image making, image having, property noticing, formalizing, observing, structuring, and inventing.)
        5. Learners construct their own knowledge.

Embracing this understanding of the learner and the math curriculum, the teacher is then prepared to take action. These actions could be as varied as the number of students in the class! Generally though, teachers should correct or warrant the knowledge a learner constructs, thus promoting the development of powerful and effective constructions (Confrey, 1990; Ernest, 1994). They must appropriately direct the student to provide experiences that can question or expand upon their previous learning. Noddings (1990) explains that teachers must also continuously reassure students that they are doing things right, that their thinking has power, and their errors are correctable. Teachers should allow students to choose activities, ask students to explain answers and prompt all students to be involved (Mikusa & Lewellen, 1999). A teacher can use invocative acts, in which students are encouraged to think at less sophisticated levels of understanding, or provocative acts; students are faced with dilemmas or challenges that push them forward in their understanding (Pirie & Kieren, 1992).

Hanley (1994) summarizes the role of the teacher in a class based on constructivism. The teacher should: become one of many resources that the student may learn from, not the primary source of information; engage students in experiences that challenge previous conceptions of their existing knowledge; allow student responses to drive lessons and seek elaboration of students’ initial responses; allow students some thinking time after posing questions; encourage the spirit of questioning by asking thoughtful, open-ended questions; encourage thoughtful discussion among students; use cognitive terminology such as ‘classify,’ ‘analyze’, and ‘create’ when framing tasks; encourage and accept student autonomy and initiative; be willing to let go of classroom control; use raw data and primary sources, along with manipulative, interactive physical materials; promote student leadership, collaboration, location of information and taking actions as a result of the learning process; encourage students to suggest causes for events and situations and encourage them to predict consequences; extend learning beyond the classroom; not separate knowing from the process of finding out; insist on clear expression from students because when students can communicate their understanding, then they have truly learned.

Researchers (Mikusa & Lewellen, 1999) and experienced teachers (Personal interview, 1999) caution that teachers should thoughtfully organize and reflect upon their knowledge, beliefs, and action plans, before committing to an approach based on constructivism and that, once committed, they maintain this approach for at least the year. Since this pedagogy involves a new way of thinking about teaching and learning, it takes time, effort and commitment for the teacher, the students and the parents to adapt.

Although the role of the teacher in this new approach certainly is critical and more challenging, it also sounds very stimulating and exciting! As one teacher said after beginning to create an environment based on constructivism, "The transformation I’ve witnessed in my students [has been] worth all the efforts and risks" (Buerk, 1994, p. 20).

The Lesson and Curriculum

There would appear to be an abundant source of ideas and activities designed to increase the effectiveness of the pedagogical application of constructivism in a math classroom. I began my own search by consulting Dr. Constructivist herself! Dr. C was a fictional, female authority on mathematics education, created by Drs. Mikusa and Lewellen (1999), who provided an online forum for discussing constructivism. When people sent questions or comments to the website, Dr. C responded and then published the correspondence in a refereed journal to reach a wider audience. Unfortunately, since this publication, Dr. C has left the internet.

I wrote Dr. C and was delighted to receive the following advice on what a lesson based on constructivism looked like. "When you practice listening to students — as opposed to listening for THE response to your questions - you will begin to understand that constructivism isn’t about a method of teaching, but rather a way of thinking about teaching and learning" (Mikusa, 1999). This response made me realize that there was no cookbook approach to lesson planning to teach based on constructivism. Instead, constructivism involved a new way of thinking about the relationship between teachers and students and between knowledge and learners.

However, some researchers have suggested very broad guidelines for implementing constructivism in the math classroom. Buerk (1994) first suggests that students participate in group-building activities to develop skills important to successfully embrace constructivism. These include cohesiveness, effective interaction and good communication skills. Erickson’s (1989) book of group math puzzles is excellent for this purpose.

Hanley (1994) outlines four major stages to consider when planning lessons based on constructivism: (1) engage the students’ interests in a topic, (2) provide a means whereby students encounter data that doesn’t fit with their understanding, (3) divide the students into small groups to decide how to resolve the dilemmas, and finally, (4) reunite the whole class and share ideas to come to some conclusion.

To obtain a real sense of this process in action, I spent a morning in a Calgary public school pure math 10 class which was grounded in constructivism. The teacher explained (Personal interview, 1999) that s/he couldn’t predict exactly where the students would lead the class that day. Since s/he taught the curriculum as the topics arose, the order and exact content varied each year. It was amazing to watch this teacher react to the students’ questions and ideas, while still directing the learning!

The class was just beginning to study statistics. The teacher began by having students read through some newspapers in small groups and then brainstormed in a large group about the types of articles they had found that contained statistics. Students were certainly enthusiastically engaged in these tasks and many genuine questions arose! They were then most eager to produce their own statistical newspaper articles, in a format of their choosing. The teacher provided handouts, which posed questions to solve in the process of producing these newspapers. Even though the students were as yet unfamiliar with the statistical terms in the handouts, they were very motivated to handle the task at hand. They divided themselves into smaller groups and immediately began to apply their interests and strengths to learn about statistics by producing a newspaper. They planned to use many resources, including each member of the group, the teacher, textbooks, computer resources, etc. They were eager to present their final products to their classmates and families. This discovery based project made learning statistics relevant to the students’ own lives, mathematically meaningful and memorable.

Many other examples of math lessons based on constructivism can be found in the literature. Buerk (1994) had students actively construct an understanding of the slope-intercept form of the equation of a line and right-triangle trigonometry and reports many resources for cooperative mathematical learning. O’Callaghan (1998) developed an innovative computer-intensive algebra curriculum grounded in constructivism. Burz & Marshall (1996) outlined a performance-based curriculum for mathematics using constructivism to teach the seven strands of high school math. Constructivism has also been applied using "benchmark" (diSessa & Minstrell, 1998) and process-based (Boaler, 1998) lessons. Boaler (1998) refers to at least thirteen comprehensive math curriculum programs that have been designed to address constructivism (Boaler, 1998).

Several math textbooks which embody constructivism, such as Minds on Math (Addison-Wesley, 1996), are used in Calgary high schools. Compared to traditional texts of the past, these new books cover different material (for example, data analysis and logic), are organized by problems instead of topics and have ideas presented in a different sequence. Math concepts are embedded in applications and provocative problems are presented. These texts help students to develop models, examine their implications, revisit the situation and then confirm the models. Math models are also related to technology and real life situations. Visual representations of the math, the graphing calculator and computer websites are also frequently referred to throughout these texts.

The Classroom

Traditionally, desks are placed individually in rows facing the teacher. To facilitate the new approach based on constructivism, seating must be flexible, as students need to work in small groups of varying size, as well as in large groups when and where information is amalgamated.

In planning small groups of students, research indicates that four is the optimal number of members, that students learn better in groups of different ability levels, that high achievers excel when also given an opportunity to work in small groups with each other, that roles in the group should switch, that students should be assigned both individual and group objectives, and that the teacher should help when requested by the students (Buerk, 1994; Leikin & Zaslavsky, 1999).

The ‘exchange-of-knowledge’ and ‘jigsaw’ methods detail how groupings can change during a lesson to facilitate constructivism (Leikin & Zaslavsky, 1999). For example, students in each group can work on different task cards and then mix with students from other groups to share and expand on their results.

To further promote constructivism in my classroom, I want to post the "Ten Myths of Mathematics" (see Table 2) prominently on the bulletin board for discussion, along with student work, motivational messages, pictures of nature that have been mathematically modelled, math web addresses, common mathematical formulae and symbols, and timely problems that students and researchers raise. Designing the learning environment in this way may provide opportunities for students to start with what they know and construct new mathematical meanings, as well as enhancing mathematical communication and connections. I would invite students to bring in math periodicals or interesting articles or stories with mathematical applications, or biographies of mathematicians to share with other students in a class library. Live plants and seashells would also be in the class to remind students of the real world, which math is created to model.

Table 2. Ten Myths of Mathematics (Source: Buerk, 1994, p. 13)

Myth 1: Mathematics is true or false, right or wrong.

Myth 2: Mathematics involves mostly calculations and equations.

Myth 3: Every question has an answer.

Myth 4: Every question has only one correct answer.

Myth 5: There is only one way (or at least a preferred way) to find that answer.

Myth 6: This solution probably involves a formula, a step-by-step procedure, or a trick.

Myth 7: Someone has to tell you what to do.

Myth 8: Real mathematics is formal: definitions, theorems, proofs, formulas and so on.

Myth 9: Mathematics is a static body of knowledge compiled long ago.

Myth 10: Mathematics is only marginally related to the real world.

Cobb et al (1992) discuss factors to consider when using math manipulatives (such as computers, calculators, 3-D models, blocks, play dough, scales, math tiles, etc.) to symbolize mathematical thinking. These should simply be available for use if the students decide they would be helpful. Teachers should be cautious not to use apparently transparent manipulatives in the belief that they are required to facilitate constructivism. The meaning of models created with such manipulatives (i.e. Dienes blocks) may only be self-evident to the expert. For example, some students still count blocks, grouped in tens, by twos (Cobb et al, 1992). Cobb et al (1992) provide a meaningful alternative to blocks using their Candy Factory Model. The workers (i.e. students) in the candy factory must figure out how to box the candy into bags of ten, boxes of hundred, and crates of a thousand and then deliver the required number to different stores.

Research indicates calculators and computers enhance the success of constructivism. Heid (1997) argues that, "the single most important catalyst for today’s math education reform movement is the continuing exponential growth in personal access to powerful computing technology" (p.5). When computers or calculators are used, the epistemological authority becomes the student, not the teacher or the text. Hundreds of studies (Heid, 1997) have shown that the appropriate use of technology promotes exploration, communication, connections, number sense, and problem solving skills, which allow students to construct their own meanings from what they already know.

From this general outline of the characteristics of teachers, lessons and classrooms, constructivism certainly sounds feasible, exciting and purposeful. However, are there any good reasons why a teacher should choose to go against the grain and (re)train to use this new approach? Is there actually any documented research indicating it provides any advantages to the learner when compared to traditional teaching?

What Are Advantages Of Constructivism?

When I asked Dr. C to explain the advantages of using constructivism to teach, she indicated that researchers are just beginning to determine and evaluate how students learn more abstract math at the high school level. Piaget’s theories of learning, in fact, lumped all students over twelve years old in one category — the "formal operations period" (Opper, 1979). It would appear then that more research is certainly needed to further validate the success of constructivism, particularly at the upper grade levels (Bruer, 1997).

However, many researchers, teachers and students are acknowledging that conventional approaches to "teaching as transfer" have major disadvantages. Even if students (taught in the traditional way) score well on standardized tests, they are often unable to use memorized facts and formulae in real-life applications outside school (Boaler, 1998; Yager, 1991), are unaware where school knowledge is applicable to the real world (Saunders, 1992) or simply forget their rote-learning over time (Yager, 1991).

Math teachers suggest that students are unable to use methods and rules learned in traditional schools because they do not fully understand them. "Educators relate this lack of understanding to the way that mathematics is taught" (Boaler, 1998). Schifter (1996) writes:

In the last four years I have begun to question just what it means to be an effective teacher. I have discovered that my students, who appeared to be successful, were often lacking insight and true understanding of the very concepts that I thought they had…learned. This realization caused me to question what it means to teach for understanding (p. 1).

There have been several studies done in the last decade (Boaler, 1998) that indicate pedagogical applications of constructivism lead to students who are more motivated, more excited about math, able to apply math to real-life situations, are more gender accommodated, and more able to problem solve, than traditionally taught students. They are also equally competent in math skills. Boaler (1998) studied students in Grade 9 to 12 math classes at two different schools. One school was grounded in constructivism and the other in traditional pedagogy. Students in the former classes didn’t know more math than students in the latter school, but were more willing and able to perceive and interpret different situations and develop meaning from them, had a deeper understanding of the procedures so they knew when to use them in real situations, and were more confident to use math and change procedures to fit new situations. Boaler (1998) concluded that the "traditional textbook approach that emphasizes computation, rules, and procedures, at the expense of depth of understanding, is disadvantageous to students, primarily because it encourages learning that is inflexible, school-bound and of limited use" (p. 60).

The term "situated learning," refers to the way learning is linked to the situation or context in which it takes place (Boaler, 1998). If we want students to be able to use math in their world, we have to have them learn math in their world — i.e. use situated learning.

Caprio (1994) compared community college math classes based on traditional pedagogy with those based on constructivism. Students in the two types of classes had similar academic ability, prerequisites and career aspirations. When given the same exam, however, the students in the latter class had significantly better grades, were more confident of their learning, enjoyed class more, had more energy to work and took more responsibility for their learning.

Carey et al (1989) rated Grade 7 students on how they thought science was investigated, before and after a unit on the scientific method was taught based on constructivism. Before the class, students viewed science as a way of understanding facts about the world. After the class, most of the students more appropriately believed that scientific inquiry is guided by questions and ideas, and understood the difference between an idea and an experiment.

By switching from a traditional class environment to a computer-intensive curriculum for college algebra based on constructivism, O’Callaghan (1998) demonstrated that students were better at problem solving, modelling, interpreting and translating and developed a richer understanding of the concept of variable, while still attaining an equally proficient level of skill development.

The teacher I observed applying constructivism in his/her classroom (Personal interview, 1999) reported that the math 10 students did just as well on the same tests as other students who were taught by more traditional math teachers in the same school. In addition, s/he felt that s/he and the students could cover three times the material! This math teacher was certainly connecting with the students as s/he was voted teacher of the year by the student body.

Another benefit of constructivism is the alternative assessment style that should accompany it. Using this type of assessment, students are not labelled, but construct their own evaluation of themselves. Self-assessment allows students to know themselves better and take more responsibility for their learning and career direction. Concepts and skills can also be evaluated separately, giving students and parents a better idea if a tutor might be helpful (Personal interview, 1999).

Research shows that businesses today want employees who can problem solve, work in teams, and deal with and communicate quantitative information (NCTM, 1989). Since mindless, repetitive tasks are now being done by machines, we are better preparing our students for the real world by preparing students to be able to construct their own understandings, use their previous knowledge, and leverage community resources.

Despite the many concrete advantages of constructivism, I think the most significant justification for this new way of thinking about teaching and learning, is the reaction of teachers, researchers, employers and students who have experienced it. After embracing constructivism, they discover "the beauty, the joy and the power of mathematics…Math is alive, flexible and inherently interesting…fundamentally obvious…common sensically accessible" (Buerk, 1994, p. 13).

When I asked the math teacher I observed using constructivism (Personal interview, 1999) why s/he chose this approach, the first reason was that, when s/he switched methods, his/her students finally recognized the exciting power of math. Von Glasersfeld (1987a) says we must "introduce children to the art, the mystery, and the marvellous satisfaction of numerical operations" (p. 4). Math "brings beauty, harmony and order into our lives" (Fusheh, 1997, p. 25).

The Master of Teaching Program at the University of Calgary is unique in Canada in that it is based very much on constructivism, and helping those who are preparing to teach to understand this new way of thinking about teaching and learning. It is exciting, powerful and progressive! In one of our Master of Teaching lectures, a guest principal commented on how articulate, analytical and enthusiastic the first year teachers were that s/he had hired from this program. When learning and teaching engenders reactions like this, the pedagogical formula must be right!

 

Arguments Against Constructivism

Not everyone, however, sees the beauty and value in constructivism. This educational reform in the sciences, especially math, has started a "Science War." Morrison (1997) expresses the opinion of many parents, scientists and even educators that, "a harmful vision of science has been steadily taking over education in schools and universities" (p. 1). Morrison believes that scientific knowledge is not merely a subjective belief system but objective truth, that school standards are too low, that reflection and verbal expression are not part of math, and/or that students don’t have the time or ability to construct math from scratch.

Listed below are the principal arguments against constructivism. My commentary will demonstrate how each of these arguments may be flawed or could be dealt with to make constructivism a feasible approach for teaching high school math. Skeptics believe that pedagogical applications of constructivism: (1) incorrectly transform mathematics from an objective and static body of knowledge to a subjective and socially constructed wasteland; (2) allow students to believe "falsehoods"; (3) do not allow enough time to complete the curriculum; (4) do not garner support of parents and students; (5) are not compatible with the assessment structure of today’s society; (6) demand too much of the teacher; (7) are not possible for students of all ability levels.

First, in response to the claim that pedagogy based on constructivism incorrectly transforms mathematics from a respectable, objective and static body of knowledge to a subjective and socially constructed wasteland, there are many compelling arguments indicating that math knowledge is indeed a social construction that can be subjective. Ernest (1991) believes new math knowledge is subjective, since it is created by an individual. This knowledge then becomes objective knowledge when it is published, reviewed, discussed, reformulated and accepted. This objective knowledge is then internalized and reconstructed by other individuals when it again becomes subjective knowledge. "Mathematical entities have no more permanent and enduring self-subsistence than any other universal concepts such as truth, beauty, justice, good, evil or even such obvious constructs as money or value" (Ernest, 1991, p. 57). "The basis of math knowledge is linguistic knowledge: conventions and rules, and language [are] a social construction; interpersonal social processes are required to turn an individual’s subjective mathematical knowledge, after publication, into accepted objective mathematical knowledge; and objectivity itself [is] understood to be social (Ernest, 1991, p. 42).

In a class guided by the principles of constructivism, all math knowledge is open for revision, is mutable and is a social construct. The goal of learning mathematics is not to have each student memorize predetermined facts from a fixed discipline in order to get right answers on tests, but to continue to create mathematics based on a deep understanding of the history of the discipline and the students’ own interactions with the world. This does not weaken mathematics, but keeps it alive by allowing it to grow and evolve with humanity. Indeed, it is just such a process that has allowed us to finally accept once forbidden mathematical notions such as zero, infinity or non-Euclidean geometry and has fuelled the phenomenal expansion in the number of fields of study within mathematics.

The second disadvantage associated with constructivism is the misconception that students can construct and believe falsehoods (Morrison, 1999). Cobb et al (1992) emphasize that students are NOT free to construct their own private truths about math. The goal is to have students construct TRUE math understanding in the sense that the individual, class and societal knowledge becomes a consensus of the best representation of the real world. By expressing their own understanding initially, however, misunderstandings are clearly identified, skills to solve dilemmas are developed, and self-confidence and peer support are nurtured.

Third, Mikusa & Lewellen (1999) and many others (Personal interview, 1999) believe that embracing constructivism actually does NOT reduce the amount of curriculum that can be covered. But even so, is a bigger curriculum necessarily better? "One of the major findings in the TIMSS report was that our curriculum…is a mile wide and an inch deep" (Mikusa, 1999). The American curriculum actually covers three times more material than that covered by countries, such as Japan, that have adopted constructivism, yet United States and Canada are outperformed in mathematics by these same countries! Might it be better to cover less anyway, and better learn what remains?

Teachers who teach through constructivism do not argue with the fact that students needn’t reconstruct all mathematical knowledge. Students need only begin constructions where they no longer can explain a situation or where the teacher identifies that a misconstruction has been made. But some construction is necessary. Dr. C’s maxim is, "Reinventing the wheel is stupid if you are interested only in wheels" (Mikusa & Lewellen, 1999, p. 158).

Fourth, re-educating students and parents to a new way of thinking about schooling is indeed a problem that the educational community still needs to address. Teachers report that it takes many weeks or months to retrain students to accept constructivism (Personal interview, 1999). Once they are retrained, however, students do like the new approach better. When the traditionally trained students graduate from Grade 12, and the math reform movement expands, lack of public acceptance may no longer be an issue. In the meantime, it is important that the benefits of constructivism be discussed with the public to address their misconceptions and doubts. The teacher I observed provides parents with the first page of the Alberta Learning math curriculum to illustrate that communicating, reflecting and constructing mathematical understanding is a required part of the curriculum. S/he also believes that having a math celebration evening, to which parents are invited to come and discuss and view student projects, allows them to see how effective constructivism is.

The fifth perceived problem with constructivism is working around the standardized government tests. This problem will be alleviated if Alberta continues to move in the direction of alternative assessment, which is consistent with the reform philosophy. Unfortunately, parents and students are still so concerned about their Grade 12 math marks that every teacher I have spoken to still believes s/he would only teach Math 30 using traditional pedagogy. The public again needs to be reminded that the TIMSS (1995) math scores of countries using constructivism are higher than those that use traditional teaching and assessment strategies. However, until the assessment structure of our education system changes, alternative, performance-based assessment could be done along with traditional testing, to strengthen student learning.

The sixth disadvantage associated with constructivism is that teachers must be well trained or retrained to develop the beliefs and skills to facilitate this approach. One teacher (EDC, 1999) expressed the challenges of constructivism by saying, "You really have to move around. You have to be there to ask the right question at the right time, to push children’s thinking along in certain ways."

Some mechanisms that allow teachers to explore constructivism are: (i) shared planning time, (ii) regularly scheduled math discussion groups, (iii) more published research, (iv) more resources and (v) more opportunities to observe colleagues. Teachers have to be taught how to engage their students in exploration, to use new technology and manipulatives and to assess students’ understanding. Many web sites, similar to that maintained by the Educational Development Center, Inc (1999), have been developed to help teachers understand the pedagogical implications of constructivism. A University of Calgary based professional development group, The Galileo Educational Network (1999), also shares mathematics education information on the net, which facilitates constructivism. It is my opinion that it will take many years before I feel familiar enough with the curriculum and student learning styles to fully utilize the power of constructivism. I do have faith though, that I can surmount this challenge, drawing from a genuine concern for the learner!

Finally, many teachers, parents and researchers believe that constructivism would not be successful with all types of learners. Mercer et al (1996) summarize the attributes, skills and knowledge required of a student in a class based on constructivism and how students with learning disabilities often lack these characteristics. This difficulty has led to a growing acceptance of an expanded interpretation of constructivism. A continuum of explicit and implicit instructional approaches to math has been suggested as illustrated below in Table 3. The degree of teacher assistance, student independence, and types of skills varies with student ability.

Table 3. Continuum of Explicit and Implicit Constructivism
(
Source: Mercer et al, 1996, p. 148)

Exogenous Constructivism

Dialectical Constructivism

Endogenous Constructivism

Explicit instruction

Intermediate instruction

Implicit instruction

Most teacher assistance

Collaborative

Least teacher assistance

Emphasis on complex skills and prerequisite skills including basic facts

Use of scaffolding to help learners acquire basics and begin to explore mathematical relationships and uses

Emphasis on mathematical concepts, relationships and applications

Teacher regulation of learning

Shared regulation of learning

Student regulation of learning

Behavioral

Interactive

Holistic

Directed discovery

Guided discovery

Self-discovery

 

Endogenous constructivism assumes that knowledge is developed entirely within the learner and that the teacher should facilitate learning without explicit instruction. Exogenous constructivism addresses learners who require much teacher support and guidance to facilitate self-discovery. Such practices could include directed modelling, teaching to mastery, scaffolding, thinking aloud and using mnemonics as described by Mercer et al (1996). In between these extremes, dialectical constructivism is a collaborative enterprise in which instruction includes both explicit and implicit methods.

Student and curriculum content factors would determine which instructional options along the continuum, were appropriate. If the student had an adaptive or flexible motivational system and significant prior knowledge of the concept, the teacher should begin with an implicit approach. Explicit instruction would be used if content to be taught is complex, critical to subsequent learning, poorly defined, factual instead of conceptual and requires a task-specific strategy instead of a general problem-solving strategy. It could also be used if time is limited, a priority on mastery exists (i.e. diploma exams), or a hazardous task is involved. Mercer et al (1996) outline the format of a math lesson for diverse learners as well as perceptions of the learner, content, interactions, motivation and assessment to be expected by the teacher and student when using this continuum of constructivism.

However, I would agree with researchers who believe that sitting somewhere on the fence, between explicit and implicit constructivism, is not beneficial to the student or to the discipline of mathematics. Cobb et al (1992) believe that students may develop a conflict between what sense of the world is acceptable and what is not, or experience conceptual anomalies and learning paradoxes. Also, the idea that math meanings are socially and culturally situated is ignored. I would also anticipate that parents would certainly have difficulty accepting this variable approach. Mercer et al (1996) do caution us though, that "narrow views keep educators from the work of educating: Ideologies do not own effective practices" (p. 147).

Conclusion

Just twenty years ago, it was generally accepted that educators put knowledge into students’ heads and educational researchers found better ways of making this transfer happen (von Glasersfeld, 1987a). Now, more and more people are thinking about teaching and learning in terms of constructivism. It is my hope that this inquiry has demonstrated the positive and significant implications constructivism has for education, especially in mathematics.

Constructivism certainly does have the potential to revolutionize teaching for all grades and subjects. However, more research, more public awareness, stronger direction toward authentic assessment, and more teacher education are needed, before constructivism can become more widely used in high school math classes.

I am extremely optimistic and excited to begin my journey into constructivism so that I too can share the joy and power of math with my students. What an exciting time to be embracing the teaching profession!

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