Constructivist Mathematics
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Descriptions, definitions, synonyms, organizer terms, types of
Synonyms or Related Topics: constructionist teaching, discovery method
Constructivist mathematics is a method of teaching that is suggested as a way of reforming the ways students are taught mathematics. This need for reform in mathematics teaching has been called for due to students in the United States struggling to maintain levels of thinking comparable to other countries. It has been found by Dosey that, “Most students, even at age 17, do not possess the breadth and depth of mathematics proficiency needed for advanced study in secondary school mathematics” (cited in NCTM, 1990,p. 1).
Constructivist mathematics is a subject specific from of constructionist teaching. The idea behind constructivist mathematics is to formulate a plan for learning that meets the needs of students as found by how students learn and internalize mathematical concepts. The Principles and Standards for School Mathematics published by NCTM suggests that “leamers be provided with the autonomy to select activities that blend with their interests and prior experiences to build mathematical connections through active leaming using concrete materials” (cited in Chung, 2004, p. 272). This is the main difference between constructivist and traditional teaching. Constructivisim relies on activies and building mathematical concepts through hands-on activities while traditional teaching relies on the ability of the teacher to convey material to the students while the students absorb the teacher’s knowledge.
According to Brooks and Brooks (1993) as quoted in Hau and Yuen (2006), there are four guiding principles to constructivist teaching:
(1) posing problems of emerging relevance to students (2) structuring concepts from whole to part (3) valuing students’ points of view and addressing students’ suppositions (4) assessing student learning in context.
Application in classrooms
The application in the mathematics classroom can range from simply adjusting way certain concepts are taught to implementing entire programs entirely based around constructivist learning.
An example of a constructivist teaching method is described below as taken from the National Council of Teachers of Mathematics (1990):
The Madison Project introduced positive and negative numbers by using an activity called Pebbles in the Bag, where a bag,
initially containing an unknown number of pebbles, has pebbles added to it or removed from it. The question is
never "How many pebbles in all are in the bag?"-that remains unknown-but rather "How many more pebbles are now in the
bag?" or "How many fewer pebbles are there in the bag?" Thus, 6-5 would correspond to "putting 6 pebbles into the
bag, and then removing 5." We would not know how many pebbles were in the bag at that point, because we did not know how
many were in the bag at the beginning, but we would know that there was one more pebble in the bag than there had
been when we started" (p.95-96).
The idea here is that students are not simply memorizing rules in addition or subtraction, but rather, they are able to visualize the concept of addition and subtraction since they can picture pulling pebbles from a bag and replacing them (NCTM, 1990).
In addition to providing visualizations to help internalize mathematical concepts, there are entire curriculums that are dedicated to constructivist mathematics views. One such program created by the University of Chicago is entitled Everyday Mathematics. This is a program designed for elementary school through eighth grade that has a "focus on real-life problem solving, balance between whole-class and self-directed learning, emphasis on communication, facilitation of school-family cooperation, and appropriate use of technology" ("Everyday Mathematics," n.d.). More on this program and some sample lessons can be obtained from the official website.
Evidence of effectiveness
The entire idea of constructivism is championed by cognitive theorists including Jean Piaget, Jerome Bruner, Zoltan Dienes, and Lev Vygotsky. Particularly, both Jean Piaget's intellectual development stages and Jerome Bruner's learning modes describe the progression of children through constructivist levels from concrete to abstract (Chung, 2004,p. 272).
In a study done on 28 elementary education students who were participating in an experimental mathematics class, the idea was “to improve students’ mathematical problem-solving ability and deepen their understanding of mathematics” (Morrone et al. 2004, p. 25). The instruction in the class was focused on using a constructivist method where students worked in groups on complex problems revolving around algebraic thinking and number sense, and all groups were required to share their findings with the entire class at the end of the day. The results of the experimental class were stated as follows:
We have seen students start as passive learners and become highly active participants in their own learning. We have
seen reluctant students become involved and willingly begin to offer their thoughts and answers to mathematics problems.
We have seen students with little understanding of even the most basic mathematical principles, learn how to explore a
concept, formulate a theorem and then give the correct mathematical explanation for that theorem (Morrone, et al. 2004,
p. 19).
Critics and their rationale
Not all studies have had such a positive view on the effects of constructivist mathematics. In a study of third grade students, discussed in Chung (2004), the effect of constructivst versus traditional teaching was evaluated based on students' skills and ability to grasp concepts in multiplication. The students taught using the constructivist method had an experienced classroom teacher who was trained and held a graduate degree involving manipulatives in mathematics education, while the students taught in the traditional approach had a teacher who held a Bachelor’s degree. Both styles of lessons were written by a researcher with the constructivist method focusing on three levels of representation: action, visual pictures, and use of abstract words and numbers. The traditionalist lessons mainly used introduction and explanation methods paired with practice worksheets. At the end of this study it was concluded that each method of teaching improved both skills and concepts in multiplication. As stated in Chung (2004), “a Diagnostic Inventory of Essential Mathematics, and the Researcher-Made Test of open-ended questions (Multiplication Survey)-revealed no differences (p > 0.05) between the mean scores on all measures in the Constructivist and Traditionalist approach groups. The repeated measures of Analysis of Variance (ANOVA) showed there were no statistical significant differences between two approaches (p. 274).
Other studies, such as one by Smalley and Moch (1999), have found that classroom teachers feel the use of manipulatives to construct a students’ own learning requires more time than the minimal benefits support. The same study also revealed that teachers had more difficulties with classroom management when teaching in constructivist methods than with more traditional mathematics (Chung, 2004).
Alternative explanations due to Diversity considerations
When considering the method of contructivist mathematics, there are some questions that one should consider when evaluating the results of studies. These include:
- Most of the studies are in elementary schools, what about the results in secondary schools?
- What about low SES areas?
- How do you acclimate students who are not well socialized into working into groups?
- What about differences due to teacher personalities and classroom management styles?
- There are many different types of students. What role do their mathematical learning styles play in the debate of methods?
Signed “life experiences”, testimonies and stories
References and other links of interest
Chung, I. (2004). A comparative assessment of constructivist and traditionalist approaches to establishing mathematical connections in learning muliplication. Education 125(2). Retrieved April 23, 2009, from EBSCO Host database.
Everyday Mathematics. (n.d.). Retrieved May 1, 2009, from http://everydaymath.uchicago.edu/.
Hau, K., & Yuen, K. Constructivist teaching and teacher-centered teaching: A comparison of students’ learning in a univeristy course. Innovations in Education and Teaching International, 43(3). Retrieved April 23, 2009, from EBSCO Host database.
Morrone, A. S., Harkness, S. S., D'Ambrosio, B., & Caulfield, R. (2004). Patterns of instructional discourse that promote the perception of mastery goals in a social constructivist mathematics course. Education Studies in Mathematics, 56. Retrieved April 23, 2009, from EBSCO Host database.
National Council of Teachers of Mathematics. (1990). Constructivist views on the teaching and learning of mathematics. Journal for Research in Mathematics Education, 4.
