Start with concrete manipulatives, progress to drawing those representations and finally, represent the mathematical thinking abstractly through symbolic notation. In either case, the big idea is the same. ![]() Some know this idea as concreteness fading, while others have called this progression concrete, representational, abstract (CRA). The goal here is that when students use the symbolic notation, they can visualize what the concrete representation of that mathematical statement represents. When students have built an understanding both concretely and visually, we can then begin moving to the final stage called abstraction where we use symbolic notation. By introducing these visual drawings of the concrete representations students are creating, it will be easier for students to shift away from concrete manipulatives and towards visual (drawn) representations when they are ready. While students are working with concrete manipulatives, it is helpful for the teacher to model visual representations of the student work for all to see. When a student is able to “look up” as if they are peering into their mind to visualize their math thinking, we know students are thinking conceptually rather than simply following a memorized procedure. The goal is not to burden students with a big bag of manipulatives that they must carry around with them anytime they are required to do any mathematical thinking, but rather to ensure that they can build their spatial reasoning skills physically – through the manipulation of concrete objects – so they can begin to visualize mathematics in their mind. When we intentionally start with concrete manipulatives to learn new math concepts, our goal is to help students better construct an understanding of the mathematics in their mind. Concreteness Fading: Concrete, Representational, Abstract (CRA) Approach When we teach in this way, we minimize the level of abstraction so students can focus their working memory on the new idea being introduced in a meaningful way. A great place to start new learning is through the use of a meaningful context and utilizing concrete manipulatives that students can touch and feel. However, if we consider that new learning requires the linking of new information with information they already know and understand, we should be intentionally planning our lessons with this in mind. ![]() Like many other teachers, I was just teaching in a very similar way to that how I was taught. ![]() By the end of the lesson, I could help many students build an understanding, but there was always a group I felt who I would leave behind. Then, I would spend the remainder of the lesson attempting to help my students make sense of these very new and often abstract ideas. To make things more challenging for my students, I would simultaneously introduce the symbolic notation used to represent those ideas. During the first half of my teaching career, I would spend what seemed to be the first half of a math lesson teaching a new math concept by sharing definitions, formulas, steps and procedures.
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