Processing through my dissertation data has reinforced my belief that students need to play or “tinker” with mathematics. Play is important because students are allowed to control part of the learning narrative. According to Seymour Papert, “I am convinced that the best learning takes place when the learner takes charge.” Play can take a variety of forms through manipulatives, projects, or a well-constructed task. Or it can be a simple shift in changing a practice problem into a more open question or having students interpret a mathematical situation.
Recently, I created a Desmos activity involving polynomial addition. Practicing adding and subtracting polynomials is a skill that is needed in algebra as it is part of manipulating the language of algebra, which can be useful in problem-solving. However, practicing a skill without exploration or connections can restrict student understanding to procedural fluency or a programmed series of steps. Programming without context is limiting at best. Below is an example of a practice problem with no context, useful in developing skills but can curtail student thinking and student input.
In the same activity, before the practice problem above, I made a slide where students move tiles into different piles to create a given sum (seen below). The shift of having students create the problem by playing with digital manipulatives offers opportunities for different strategies. There is cogitative strength in constructing mathematical connections. Students can make connections among manipulatives, algebraic notation, and quantities through play.
Interpreting A Situation
Functions are essential tools in mathematics as they allow for the construction of models to explore relationships. Defining functions and then having students practicing determining if something is a function or not or even classifying the type of functional relationship (linear, quadratic, exponential) is a valuable skill. Yet, functions without context limit student mathematical understanding. George Pólya encouraged teachers to have students interpret mathematical situations and restate them in their own words as a strategy for solving problems. Play is one way students interpret situations and can tell a mathematical story.
Below is a slide I created as part of exploring the sketch feature in Desmos. The premise is simple, “What happens when you sketch?” Students can play with different types of sketches and determine what happens. Feedback through technology helps shape student understanding through the multiple ladder racers on screen. The speed at which the racers climb, if they are going up or down, where they start on the ladder, which racer is in first place, and if multiple racers appear on a single ladder all depend on the sketches students create. This type of visual feedback is one example of how using technology to play can create meaningful learning conditions for students. When students interpret a situation, they give meaning or purpose to mathematical concepts.
While practice can support student learning in mastering basic skills and applying knowledge, play can create the conditions for student driven discovery and for student centered learning.