In a recent blog post, Kyle Pearce (@mathletepearce) raised the question “good at math or good at memorization?” Unlike most middle school math teachers, Kyle asks his students to wrestle with complex, unstructured, real-world problems. In doing this, he has found that many of his students who previously considered themselves “good at math” often struggle when asked to work with these inquiry-based challenges. He wants his students to understand the why of math beyond just the execution of a series of steps, so students who previously succeeded based on their ability to memorize formulas now find themselves grappling with greater challenges.
From the Behaviorism of Procedures to the Constructivism of Understanding
For me, the timing of Kyle’s article was perfect as I had just read Behaviorism, Cognitivism, Constructivism: Comparing Critical Features from an Instructional Design Perspective from Ertmer and Newby (1993). The authors present pedagogies as a continuum from Behaviorism to Constructivism rather than as siloed learning theories. They state that these pedagogical theories should be used to guide instructional design and allow educators to make predictions about the types of student learning that will result based on the applied approach. In this light, Kyle can reasonably predict that because many of his students had previously learned math through behaviorist, rather than constructivist approaches, they will excel at applying procedures but struggle with constructing meaning. Similarly, he can assume that students who truly understand the reasoning behind mathematical concepts will then be able to demonstrate the mechanics and procedures required to solve various algorithms.
Most people recall the story of Pavlov’s Dogs when thinking about the concept of behaviorism. Pavlov rang a bell and the dogs would salivate as a conditioned response. While educators don’t like to consider students in the same light, at the heart of behaviorism lies the concept that learning is an observable change of performance in response to an environmental stimulus. In the classroom, a teacher asks a math problem (stimulus), and a student performs the correct procedure in order to reach the desired solution (response). As highlighted by the video below, too often math instruction stops there.
However, individuals “cannot afford the ‘luxury’ of restricting themselves to one theory” (Ertmer & Newby, 1993, p. 52). Behaviorism may be effective for more low-level cognitive skills while cognitive theories focus on making knowledge meaningful. In the case of the YouTube video, a cognitive approach to math would have students engage in reasoning and problem solving so as to better consider when to apply the mathematical principles rather than formulaically crank through an “Order of Operations” based on the PEMDAS acronym.
Kyle wants his students to construct their own reality about math. In his class, students cannot march through procedures without understanding why that procedure should be applied in the first place. Learning occurs as students create meaning from experience and then apply their knowledge. By using a Constructivist approach, Kyle asks his students to build their own sense of reality based on their understanding of the problems presented.
Learning to Learn in a Constructivist Classroom
This week, I worked with a group of math teachers using two specific tools: Formative and GeoGebra. Unfamiliar with the latter tool, I reached out to Chris Harrow (@chris_harrow), Mathematics Chair at the Hawken School and my high school math teacher. First, Chris gave me a series of procedures to complete. This behaviorist approach helped me to understand the basic functionality of the tool. Then, he shifted to a cognitive approach and asked me to recall prior knowledge and connect the procedures to previously constructed mental schema about geometry (which were admittedly trapped way in the back of my Long Term Memory!) As Chris continued to challenge me, I manipulated the tool and dynamically watched the geometry unfold. Through this constructivist process, I started to make connections not only to previous learning in school but also to how I might want to structure my workshop.
Chris treats his math class more like a science laboratory. Leveraging Jo Boaler’s work at Stanford University, he aims to develop his students Mathematical Mindsets. Rather than present solutions and the steps to get there, he asks students to explore math concepts using tools such as GeoGebra, construct their own reality based on their observations, and then defend their hypotheses. The students seek out new solutions rather than apply procedures to existing ones.
Given these ideas, I decided to structure my workshop in a similar manner: use behaviorist principles to teach the procedural knowledge of the tools and then constructivist ideals to engage in deeper inquiry. Towards the end of the day, one teacher felt frustrated by the process and made a comment about her prior experience. “They don’t see the patterns,” she said.
This statement made me think about Kyle’s students. Those who struggled did not always see patterns because they had previously memorized a solution without learning how to first deconstruct the problem. In his article, Kyle comments, “I try to help the student and the parents understand that knowing procedures alone is like being a ‘mathemagician’ with a limited number of math tricks that are only useful in very specific situations. Helping students develop a deep understanding by thinking critically on a daily basis is the only way students can truly understand math and apply their knowledge to unique situations.” I hope that this particular teacher does not give up on the potential to push her students to truly construct their own understanding. Her students may not see the patterns right now because they have never been asked to look.
Most adults, and many of today’s students, associate memorizing algorithms and working through clearly defined procedures with math classes. Constructivist approaches such as those used by Kyle and Chris break that vision and disrupt the notion of how math should be taught. However, the ability to solve unstructured problems and apply mathematical thinking to new contexts are rapidly becoming critical skills for students in today’s students. I hope that more classrooms - whether math or any other subject - begin to resemble those of these two educators.