What is CRA? It stands for Concrete-Representational-Abstract and it is the progression by which the CCSS math standards were written.
Concrete is often called the “doing phase”. This is the phase in which students are manipulating objects (e.g. unifix cubes, base ten blocks, two color counters) to make sense of and solve problems.
Representational (sometimes referred to as “pictorial”) is the “seeing phase”. Students working at this level of learning are drawing representations of their understanding (e.g. part-whole models like number bonds and tape diagrams, area models, arrays).
Abstract is where most of us did all of our math learning as students. This is the work done with numerals and symbols (e.g. number recognition, expressions, equations).
Decades of research have supported the notion that conceptual mathematics understanding is achieved through this three phase process. In times past, teachers told us that the numeral 5 is a “five” and we spent time memorizing it. Now, rather than being told it is a “five”, we are encouraging children to develop a deeper sense of “fiveness”. Build five blocks, see five dots, count five cars, etc. The ultimate goal is still to see a five and name it as such, but to shift is for students to do so with understanding. As we progress in the grade levels, we see evidence in the math standards of the CRA approach leading up to the introduction and use of all standard algorithms. The chart below delineates the phases of CRA throughout the grade levels and how they progress to the use of the standard algorithm.
Addition/ Subtraction | Multiplication | Division | |
TK/K | Objects or drawings | ||
1st | Concrete models or drawings and strategies | ||
2nd | Concrete models or drawings and strategies | ||
3rd | Strategies and algorithms | Objects, drawings, strategies | Objects, drawings, strategies |
4th | Standard algorithm | Drawings, strategies, illustrations, explanations | Drawings, strategies, illustrations, explanations |
5th | Standard algorithm | Strategies, illustrations, explanations | |
6th | Standard algorithm |
No longer do students mindlessly “carry a one” or use a “place holder”. Students now recognize through experience and sense making that these actions are based on number sense and place value understanding.
One of my favorite approaches to implementing the CRA model into my teaching is through the use of a “Build it, Draw it, Write it” activity. Just recently I was teaching a second grade lesson focusing on standard 2.NBT.7, which states that students should be able to add and subtract within 1000 using concrete models or drawings using various strategies and relate the strategy to a written method. Moreover, this lesson focused on the second half of this standard…the dreaded “borrowing”! The language of the lesson emphasizes the that it is “sometimes necessary to compose or decompose tens or hundreds”. During the lesson we focused on the language of compose and decompose rather than the term borrow. Let’s face it…you don’t give the ten back! It’s truly only borrowing if you return it.
During the lesson, students were given a place value mat and place value disks, both of which were simply copied on to cardstock.
We recorded our thinking using this Build it, Draw it, Write it (BDW) document.
Students worked to make sense of subtraction through the use of the concrete place value disks, in which many problems provided opportunities for students to see that a ten could be decomposed into ten ones in order to solve. Students were then asked to draw a representation of building they had done. Finally, students were asked to write an equation to match the work they had done both concretely and representationally.
CONCRETE PHASE ~ doing phase
REPRESENTATIONAL PHASE ~ seeing phase
In this final photo, you can see the student beginning to make sense of decomposing the ten into ten ones and how you would represent that in a drawing. As you can see, the student does not quite understand how to do that *YET*, but the concept is beginning to emerge. How exciting!
At the end of this lesson, not only did I see evidence that the students were building a strong foundation in place value subtraction, but I also saw high levels of engagement from students. As they left for recess, one student said to me, “Thanks for playing with us today!” That’s when it hit me…the students didn’t see what we did as learning. They saw it as playing. And that was when I knew I had done my job as a teacher!