As we work to help students develop deeper conceptual understanding in math, CRA can be helpful. CRA is a system of instruction from the concrete to the abstract where students are first introduced to a concept through concrete experiences. In the area of math that means using concrete objects or manipulatives. Instruction then moves to representational experiences where drawings are used. After the concrete and representational levels are practiced and understood, then students are asked to work with abstract symbols, notation, and numbers in math. The abstract level is what we know as standard algorithms or standard notation. Through the instructional experience of CRA, students can make stronger connections and develop deeper understandings.

When we use concrete experiences with our math students, we help them to develop foundational understandings of a skill or a concept. Think about when students are first introduced to measurement. We do not put a ruler in their hands right away. Students often have experiences measuring with counters, cubes, or links. Using these concrete manipulatives helps to develop an understanding of length. Concrete experiences should be embedded in instruction when new concepts/skills are introduced.

The representational level, sometimes referred to as semi-concrete, is when students draw what would be represented by the concrete objects/manipulatives. For example if students are using counters to make groups when learning about multiplication at the concrete level, students could then draw Xs, dots, or even use mini-stampers (What fun!) to create a visual representation of counters. The representational level helps to bridge understanding from the concrete to the abstract for students.

Following representational experiences, students can work with a math concept/skill abstractly. Symbols, numbers, and notation are loaded with meaning in math. If students jump to the abstract level before having the concrete and representational experiences there can be a breakdown in understanding. Consider the different standards below and how the activities/strategies focus on the different stages of the CRA framework. Click on the image to grab a copy.

When considering using technology with students, choose programs that are at the appropriate readiness levels for students. View the chart below to see how you can structure technology use at the different stages of understanding for a concept/skill. Click on the image below to download a copy. Visit some of the sites to see the progression of understanding from the concrete to the abstract.

How do you weave these different levels of instruction into your daily routines?

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