Can You Prove It?

The process of learning is the polar opposite of a passive experience. Actively constructing knowledge, Constructivism, is rooted in Piagetian theory. Children construct meaning, building upon what they already know. Students are not empty vessels to be filled with our magic. They are their own shining stars that perform at center stage. As Kloosterman and Gainey suggest, students are “thinking individuals who try to make sense of new information.” (ch1, p.5). The cumulative effect of building upon prior knowledge has implications in all content areas, especially mathematics. According to the NCTM, Notional Council of Teachers in Mathematics,

    “The mathematics curriculum should include the investigation of mathematical connections…describe results using mathematical models…and use a mathematical idea to further their understanding of other mathematical ideas.”

Stated simply, this boils down to two simple words, explain and justify. Exploring problems together as a whole class and in small groups, a student’s job is to investigate and make connections, (to real life, to other mathematical models through a spiraling curriculum, and to a multiplicity of solutions). When we shape an environment that encourages sharing solutions and explaining what students believe and why they think the way they do, we help them recognize relationships and add meaning to their conceptual knowledge. 

Approaching mathematical problem solving in this way mimics the method of scientific inquiry. This process of discovery, followed by elaboration, discussion, and articulating a cohesive conclusion, provides structure for students to prove their thinking with evidence. It is hard for me to imagine students blindly following rules without reasons, yet, in some classrooms, I know it happens every day. I expect more from my students than the ability to get something correct and memorize a formula. They must be able to justify, explain, and prove, how they know what they know. I expect my students to engage in active learning, to process then apply their content knowledge, to question, share and reflect as they learn. When students reach this level of understanding in mathematics, they validate their own beliefs as they construct relationships to past and future conceptual meaning. Ah… but, can they prove it?