Paul, 7th & 8th Grade Math Teacher
In the Middle School years, students will transition from concrete to more abstract thinking, but they travel this journey at different rates. Some students have a natural preference for one over the other and, hence, might start earlier and progress more quickly through this developmental stage. In mathematics, this process is helped along by bridging concrete and abstract thinking through representational thinking.
In pre-algebra and algebra, students mostly work with visual representations, rather than physical manipulatives. For example, when performing operations on integers, students no longer need to model this with plastic chips of different colors that represent positive and negative numbers. They can visualize a mental bag in which they deposit amounts of positive and negative values and develop and apply rules for the various operations that way.
Later, when learning to use variables and equations to solve situational problems, students might initially set up a table to work through the numerical process that describes the problem using a guess-and-check approach. Once that process is clear, students will replace their guesses with an unknown and work through the same steps to arrive at an equation that they can use to obtain a solution algebraically. Over time, students will no longer need to set up a table to represent the process underlying a problem because they will have developed the ability to formulate an abstract, algebraic model directly.
Similarly, students in algebra use visual representations containing squares and rectangles to model factoring quadratic expressions with “differences of squares” or “perfect square” trinomials. Once the general rules have been learned and understood, students are able to classify future problems and apply the rules to solve them using abstract thinking. There is no longer a need for a visual representation.
At the same time, certain visual representations will continue to be useful. When studying linear and quadratic functions, students will continue to use graphs and tables to illustrate their abstract thinking. While they might find the important features of a graph, such as intercepts, extrema, or intersections, algebraically, they will frequently summarize all the information in a graph. On the other hand, when trying to make sense of certain aspects of a problem, they might use a graphing calculator or a graphing program, such as Desmos, to create a visual representation, get an idea for or arrive at a solution, and then use algebraic methods to find or verify it.
Mathematics as a subject has the potential to advance students’ reasoning abilities. The rate at which a student develops from concrete through representational to abstract thinking does not have to be fixed. Providing students with the right experiences can accelerate their progress on the so-called “ladder of abstraction.” Seeing other students’ examples and participating in discussions allow students to make connections between what they already know and can do and what they are learning. As such, transitioning from one level of abstraction to the next might be a slow process, but it is essential for further learning and academic work.