Constructivism is a learning theory that basically states students learn by building on previous knowledge.  This knowledge has been gained through experience.   As Land and Hannafin state, “Constructivist environments scaffold thinking and actions in order to deepen understanding.”   One way that math teachers try to introduce constructivism into the class room is by using open-ended rich math tasks.  A rich math task is a math problem that has multiple ways to come to a solution.  For example a problem could be solved using a physical reconstruction of the problem, a graph, algebraically, or even using calculus.   Each method leads to the same result and to a discussion about the concepts behind each method.  Applying constructivism into a math class in the form of rich math tasks has some pros and cons.

                One pro of using rich math tasks in a math class is in retention on knowledge.  According to Sarah Lubienski (2007), “Students who approach mathematics by merely memorizing rules are less likely to retain what they have learned than are students who have deep understandings of mathematical concepts and relationships (and can then reconstruct any formulas they have forgotten).”  Proponents of rich math tasks have found that these tasks allow students to create a solution to a problem using what they already know, one of the main tenants of constructivism.  Since student’s are actually engaging in a problem it takes math out of the realm of simply memorizing facts and regurgitating them on the page and focuses on deeper understanding of the concept which is usually remembered longer

In education there tends to be a divide among students of different socio-economic status.  Using rich math tasks has been shown to help reduce the divide between students based on financial factors ( Lubienski 2006).  Much of the success of rich math tasks is ascribed to the multiple entry points of the problem.  Students at any level can work on the problem; if they aren’t good with equations they can use visuals to help solve the problem.    This type of open ended problem levels the playing field for all students regardless of skill level and socio-economic status.

There are two cons that go with open ended problems.  The first con is that many students are not used to a class that uses open ended problems.  As Land and Hannafin state, “Teachers have traditionally possessed the required knowledge, determined what is correct and what is incorrect and set and enforced grading standards.”  With open ended problems there is no right way or wrong way to do the problems. The second issue is that sometimes what students have as previous knowledge is false.  Students can’t understand the new concept because the new concept disagrees with their previous knowledge.   The only solution to this problem is to deal with each student individually, and try to figure out what misconceptions they are bringing to the problem.

The constructivist learning theory is easily described using rich math tasks.  These problems demonstrate the ability of using prior knowledge to construct new usage and add meaning to a problem.  They are very useful in allowing all students to engage in a math problem while at the same time giving the students a concrete example to refer back to and remember.  While there are issues with rich math tasks, they are a useful way of teaching mathematics.

Refernces

Hannafin, M. J., & Land, S.M. Student-Centered learning environments.  In D. Jonassen, & S. Land (Eds.), Theoretical foundations of learning environments (pp.1-23). New Jersey: Lawerence Erlbaum Associates

Lubienski, S. T. (2006). Examining instruction, achievement, and equity with NAEP mathematics data. Education Policy Analysis Archives, 14(14). Available: http://epaa.asu.edu.libproxy.boisestate.edu/epaa/v14n14

Lubienski, S. (2007). What We Can Do About Achievement Disparities. Educational Leadership, 65(3), 54. Retrieved from EBSCOhost.