Constructivism in the Mathematics Classroom: Towards a New Approach

Paper written for Dr. Samson Nashon, ETEC 530, March 2006

The mathematics education systems in many jurisdictions in North America, including the province of British Columbia (BC), follow the standards set by the National Council of Teachers of Mathematics (NCTM, 2000). The following information from the BC Education Mathematics K-7 Rationale (BC Education, 1996) reflects the tone and intent of the NCTM standards:

Constructivist goals, including “problem-solving, reasoning, critical thinking, and the active and reflective use of knowledge” (Driscoll, 2005, p 393) are well-suited to the new philosophy of teaching and learning mathematics, and many educators are trying to use constructivist teaching strategies in their classrooms. Despite the upsurge in the use of constructivism, it is not the answer to all of our “math woes”. This paper examines selected studies that compare a constructivist approach with more “traditional” or other approaches. In these studies, researchers examined math anxiety, achievement in the acquisition of basic math skills, and use with low-achieving students. Finally, an alternate version of constructivism, one that is true to its socio-historical roots, is proposed.

Math Anxiety

Alsup (2004) reported a study in which three classes of preservice elementary teachers received mathematics instruction in either a constructivist or traditional mode. The students involved were two classes of Math Concepts I and one class of Math Concepts II. One of the Math Concepts I courses was taught in a traditional manner (the control group), while the other Math Concepts I and the Math Concepts II courses were taught in a constructivist manner (the experimental groups). The researcher compared pre- and post-course levels of “math anxiety, mathematics teaching efficacy beliefs, and perceptions of autonomy or empowerment” (p. 4).

While all students were taught emphasizing a problem-solving approach, the experimental classes had considerably more input and control over the way the learning took place. In the constructivist classes, the instructor spent a short time at the beginning of the class lecturing on the day’s content, and then the students were assigned problems to work on in groups. After working through the problems, the students presented their solutions and strategies to the rest of the class. Once during the course, each group of four students presented the lecture and homework problems for the rest of the students. Throughout these classes, the instructor acted as facilitator rather than teacher. In contrast, the control group was taught using the traditional lecture and assignment format using only instructor-directed activities and problem-solving strategies.

Post-course results showed that all students’ math anxiety decreased and the level of self-efficacy increased. The clearest result, however, was that the autonomy scores of the experimental Math Concepts I group increased dramatically, although the students in the control group also improved in this area. Students in the Math Concepts II course actually regressed in terms of math course anxiety, with worse scores at the end of the study. The researcher himself noted some reasons for the seemingly contradictory results: Math Concepts I is more of a review course, with very few new topics, whereas Math Concepts II introduced topics that some students had never studied before; and some students brought with them from high school a negative attitude toward geometry, one of the topics covered in Math Concepts II.

One of the criticisms of this study is that there was no control group for Math Concepts II, where the instructor would have used traditional teaching techniques. The best explanation for the differences is that the students were more comfortable with constructivist teaching techniques when the material was somewhat familiar to them, but with new material, the open-endedness of the problem-solving process and lack of closure for some problems caused anxiety.

Basic Mathematics Skills Achievement

Does constructivist teaching improve results when learning basic mathematics skills? Not in the short term, according to a study by Chung (2004). Two groups of grade three students were taught basic multiplication facts using either a Constructivist or Traditionalist method. With the Constructivist method, students were taught first with concrete materials, then pictures, and then numbers, following the ideas of Bruner (in Chung, 2004). With the Traditionalist method, the lessons were based on the teachers’ text for the school district math curriculum. The main teaching strategies were explanation followed by practice worksheets.

Students were evaluated using a pre-test and post-test on three different measures of multiplication knowledge, including an researcher-designed survey which consisted of open-ended questions asking the children to demonstrate multiplication facts with concrete materials and numbers. Results of the post-test showed that there was no statistical difference between the Constructivist classes and the Traditionalist classes, and that both groups had improved their multiplication skills.

An interesting point to note in this study is that even though the classroom teacher who taught with the Constructivist strategy had a Master’s degree focussing on Constructivist methods and manipulatives, this teacher expressed concern that “… the novelty and change in routine caused the children to become more active than usual or desirable” (p. 276), while the classroom teacher using the Traditionalist approach was pleased with the progress of the students. This seems to indicate that even though the Constructivist teacher was educated in constructivist methods, she was not regularly using them in teaching.

Basic Mathematics Skills with Low Achieving Students

In a study of the acquisition of multiplication skills by 265 low achieving students aged 8 to 11 years, Kroesbergen, Van Luit & Maas (2004) compared constructivist teaching techniques with explicit instruction and regular classroom instruction. In the constructivist group, students discussed strategies, using materials and manipulatives if necessary. The teacher asked questions in order to promote discussion, but did not demonstrated particular strategies. In the explicit instruction group, the teachers gave direct instruction on appropriate strategies for different kinds of problems, using concrete materials such as blocks, or number lines if appropriate. Students then practiced with several problems using the strategies that the teacher had taught. In the control group, students received instruction on multiplication in their regular classrooms. Students in both experimental groups received two sessions of 30 minutes of instruction per week, done at the same time as their regular math classes, while students in the control group also received approximately one hour of instruction per week in their regular classes.

Results of the post-tests and follow-up testing three months later showed that both the constructivist and explicit instruction groups improved significantly over the control group in tests of automaticity. However, students in the explicit instruction group fared better than those in the constructivist group on tests of problem-solving and number of strategies used. The researchers believe that low-achieving students may benefit from having a small number of correct strategies to choose from, rather than a large number of strategies, some of which may be incorrect, as generated by students in a constructivist classroom.

A concern of this study, as noted by the authors themselves, revolves around the number of students in the treatment groups vs. the control groups. Students in the constructivist and explicit instruction treatments were taught in small groups of four to six students, while control group students were taught in the regular classroom, resulting in more instructor attention for the students in the treatment groups. A concern not noted by the researchers, but equally valid, is the effect on the treatment group students of being removed from the classroom for “special” lessons.  Both of these factors may have skewed the treatment groups scores in a positive direction.

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A new approach

If the current constructivist teaching techniques leave some mathematics students anxious and floundering, do not produce any better outcomes in achievement than traditional methods, and do not necessarily work for low-achieving students, one might question the value of using these techniques in the classroom. Although the studies reported here did not prove the superiority of constructivist methods over any other, many of the researchers still expressed their confidence in using constructivist methods over traditional transmissive techniques.

A criticism of constructivism as it is used presently is that it has strayed too far from its roots. Socio-historical constructivism was proposed by Lev Vygotsky, the Russian psychologist, in the early 1930s, and his ideas, along with other theories, provide the basis for the constructivism that is used today. According to Vygotsky’s theories, socio-historical constructivism anchors instruction within the social and cultural milieu within which the students learn, and language is one of the tools by which children internalise the body of knowledge that has been created by their culture (Driscoll, 2005).

In contrast, the constructivist techniques currently in use reflect a more radical constructivist view. This philosophy is epitomized in the words of von Glasersfeld: “It starts from the assumption that knowledge, no matter how it be defined, is in the heads of persons, and that the thinking subject has no alternative but to construct what he or she knows on the basis of his or her own experience” (1996, p.1). To this end, teachers use concrete materials and manipulatives in the math classroom, but may fail to connect students’ constructions and algorithms with those developed through many years of mathematical work in the students’ own culture. Kroesbergen, Van Luit & Maas note, “… if students do not discover a strategy on their own, it is not discussed within the group” (2004, p. 240), and this would also apply to the traditionally used, or standard, algorithms as well. Schmittau (2004) notes, “constructivists highly value children's actions with concrete materials, and since it is often difficult to establish connections between children's actions and standard algorithms, they assume that if algorithms are taught they must of necessity be learned by rote” (p. 25). This often leads teachers to deliberately not teach the standard algorithms, fearing, perhaps, that to do so will lead them back to transmissive teaching techniques. In contrast to this, Schmittau (2004) claims that students must be able to connect their actions with the standard algorithms: “In abandoning the teaching of algorithms or failing to connect them to the meaningful mathematical actions that gave rise to them, we fail to pursue the development of a concept through to its complete historical fruition, and consequently to its full mathematical power” (p. 25).

An alternative for the constructivism presently in use is a mathematics curriculum developed by V.V. Davydov (Schmittau, 2004), which is based on Vygotsky’s work. In this program, developed for children in grades 1 – 3, students start by learning about quantity instead of number, the concept of which is not even introduced for the first three months. Using materials such as liquid in jugs and measuring devices such as cups, students develop the ideas of equality and definitions of addition and subtraction. Once the concept of “number” is introduced, students use “^” notation to develop several other ideas: “fact families”; the decomposition of numbers in base 10 or other bases; measurement using multiple units; and missing addend or missing first term problems (Schmittau, 2004). (See figure 1).

fact families and missing first term problem

Figure 1.

Once students master the basic ideas, the distributive property becomes important in helping students develop algorithms for multiplication and division. The result of Davydov’s program is that students in grade 3 are able to solve problems that are troublesome for high school students using the regular curriculum.

This curriculum is in use in about 10% of the schools in Russia (CRDG, 2006) and in selected schools in the United States. In one instance, an innovative program called “Measure Up”, which is an updated version of Davydov’s curriculum, has been in place since 2001 in Hilo, Hawai'i, sponsored by the University of Hawai'i Education Laboratory School, and research is being carried out on selected students in the program who were chosen to represent the diversity of the area. Researchers in the program expect that students will have taken a rigorous algebra course by the end of grade 6 (CRDG, 2006), something that would be impossible using the standard North American curriculum.

Conclusion

The current constructivist approach to teaching and learning, however flawed, is better than the transmissive teaching techniques that were promoted in teacher education until the 1980s. However, we must recognize that these techniques may not necessarily be adequately addressing issues such as math anxiety, better achievement or assistance with low achieving learners. An alternative curriculum, such as the one proposed by Davydov, may hold some hope for helping our learners make connections between their actions and the rich cultural heritage of mathematics.

References

Alsup, J. (2004). A comparison of constructivist and traditional instruction in mathematics. Educational Research Quarterly, 28 (4), 3-17. Retrieved February 22, 2006, from Academic Search Premier database.

BC Education. (1996). Rationale. Mathematics K-7 Integrated Resource Package (IRP). Retrieved March 1, 2006, from http://www.bced.gov.bc.ca/irp/mathk7/mk7ratio.htm

Chung, I. (2004). A comparative assessment of constructivist and traditional approaches to establishing mathematical connections in learning multiplication. Education, 125 (2), 271-278. Retrieved February 22, 2006, from Academic Search Premier database.

Driscoll, M.P. (2005). Psychology of Learning for Instruction. USA: Pearson Education, Inc.

Kroesbergen, E.H., Van Luit, J.E.H., & Maas, C.J. (2004). Effectiveness of explicit and constructivist mathematics instruction for low-achieving students in the Netherlands. The Elementary School Journal, 104 (3), 233-251. Retrieved February 22, 2006, from Academic Search Premier database.

National Council of Teachers of Mathematics. (2000). Standards for School Mathematics. Retrieved March 1, 2006, from http://www.nctm.org/standards/standards.htm

Schmittau, J. (2004). Vygotskian theory and mathematics education: Resolving the conceptual-procedural dichotomy. European Journal of Psychology of Education, 19 (1), 19-43. Retrieved February 22, 2006, from Academic Search Premier database.

University of Hawaii Curriculum Research and Development Group (CRDG). (2006). Russian Math in Hawaii: Measure Up. Retrieved March 5, 2006, from http://www.hawaii.edu/crdg/promos/measureup/index.html

Von Glasersfeld, E. (1996). Radical constructivism: A way of knowing and learning. London: Falmer Press. Retrieved March 5, 2006, from Questia at http://www.questia.com/PM.qst?a=o&docId=103919722

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