Algebraic Reasoning: Concrete to Abstract

Today, while practicing teaching best practices of negative integers, I learned (1) the importance of teaching numerous strategies for solving math problems, and (2) the effectiveness of moving from concrete to the abstract when teaching algebraic reasoning.  What is subtracting, after all, but adding the opposite.  To quote a colleague, “Negative numbers doesn’t mean it doesn’t exist. It forces the value in an opposite direction.” An abstract thought indeed. In the real world, we may relate this to the debt side of an accounting spreadsheet, or a host of engineering and science career paths. Yet, how do we teach? We begin with the concrete method of manipulating black and white beans. We point to and count aloud along a number line, or walk facing forward and then walk backward along a masking tape line on the floor. We can list the pattern in a series of related numeric equations, so kids can “see” the answer in a new way. We can manipulate poker chips; base ten blocks, rods, cubes and sticks. We can flip over algebraic tiles from positive to negative as we move from concrete demonstration and practice, to the lattice method, and finally to the abstract “Foil” method (first, outer, inner, last). Using multiple strategies can help students of different learning dispositions to solve, factor and determine the product of two binomials.

Getting kids to open up and explain their thinking is one way to welcome a host of solutions into the classroom. “What do you think about __________’s idea?” or “Next?” This encourages reflection and conversation which is student-centered. Reinhart, S. (2000) shares, “I concluded that if my students were to ever really learn…they would have to do the explaining, and I the listening.” The implications are that kids not only get it, but also thrive to become teachers themselves. The result is more guided teaching in the class, less direct instruction and more demonstration of student knowledge.

The uncomfortable struggle all students face, is showing confidence in the face of the unknown. Will the teacher call on me? Will I open my mouth and stick my foot squarely in it? Can we set the environment for a safe caring classroom making it okay to say, “I’m stuck.” I plan to offer excessive wait time in my classroom to gather more thoughtful response. Seeming like an eternity, I will move on and offer, “(student) I know you have valuable ideas to contribute, will you promise me that you will raise your hand on another question when you have something to share?” It is important to validate the effort. Struggle makes our brains grow!