First, I have a quiz for you. If you speak and write Hindi, you can skip this quiz. For the rest of you, look at the symbols below. Can you tell which are letters and which are numerals?
Symbols 1 and 4 are numbers, while symbols 2 and 3 are letters. How did you do? Did you see any distinguishing characteristics that helped you make your determination? I sure don’t. They all look the same to me. What’s this got to do with math? I once saw a very short video clip, lasting no longer than about 10 seconds. It was an adorable little boy facing the camera. All he said was this: I used to think 5 was a letter. Now I know it’s a number. Think about that for a minute. Until a child knows that 5 means 5 fingers, or 5 dots, or 5 toy cars, or 5 years old, the symbol doesn’t look much different than a letter. It has some curved lines and some straight lines, like the letters a, b, or p.
You may have heard of a little country called Singapore and its successful math instruction. Twenty years ago, Singapore occupied the space squarely at the bottom of the world rankings in math. Now, they are among the top. What changed? They adopted a teaching philosophy that is built on the concrete, representational, abstract (CRA) sequence of instruction. They call it CPA, with the P standing for pictorial. Regardless of the letters used, this sequence of instruction is based on the research of Jerome Bruner. It says, quite simply, that students must experience and interact with a concept to develop a true understanding.
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Think of the picture at the top of this post. Can a person truly understand an apple at a deep conceptual level without holding and tasting one? I don’t think so. Assume a child has never seen, held, or tasted an apple or a blueberry. Only seen the words. Now ask the child to compare the two fruits. The child might say that a blueberry is bigger because it has more letters. Sound a little far-fetched? Ask a child which number is larger, 3.5 or 1.389, and I guarantee you a high percentage will say 1.389. Why? Because it has more digits. If a child has not had concrete, hands-on, place value experiences he will not understand that the first number is larger because it has more ones. Next, ask a student, 3rd through 5th grade, which fraction is larger, 1/8 or 1/4, and many will tell you that 1/8 is larger. Why? Because 8 is larger than 4, of course. If, however, a child has had many experiences folding fraction strips, using fraction tiles, working with fractions on a number line, and cutting up brownies or cookies into equal parts, there is no earthly way they would say that 1/8 is larger than 1/4.
My point here is not only the importance of the CRA sequence of instruction, but also that the majority of our students are getting far too little concrete and pictorial experiences with key concepts, and that is a major cause of their lack of understanding.
How important is CRA? Here in Texas, it’s actually written into the standards–the Texas Essential Knowledge and Skills (TEKS). Common Core, not so much, and that’s actually one thing I think Texas got right. Let’s look at multiplication, for example:
2.6.A Model, create, and describe contextual multiplication situations in which equivalent sets of concrete objects are joined (concrete)
3.4.D Determine the total number of objects when equally-sized groups of objects are combined or arranged in arrays up to 10 by 10 (concrete)
3.4.E Represent multiplication facts by using a variety of approaches such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting (representational)
3.4.F Recall facts to multiply up to 10 by 10 with automaticity and recall the corresponding division facts (abstract)
A few comments about these standards. Should 2nd-grade students be drilling multiplication facts? No. According to the wording of this standard, all of the multiplication experiences in second grade should be concrete, because you are developing the basic understanding that multiplication is joining equal groups. This shift from additive thinking (3 + 2) to multiplicative thinking (3 x 2) is huge. A lack of concrete experiences with multiplication in 2nd grade becomes painfully obvious in 3rd grade. Now, what about 3rd-grade? The bulk of your instruction should be on the concrete and pictorial learning described in 3.4.D and 3.4.E to support the understanding required for 3.4.F. Skip or limit 3.4.D and 3.4.E, and it should be no surprise that students can’t recall with automaticity their multiplication facts, because they have no picture in their head of 3 x 2.
Three final notes.
- The CRA sequence of instruction is for all grade levels, not just the primary grades. Two great examples are decimal place value and computation with fractions. These are both skills introduced in the upper elementary grades and are skills that simply can’t be well understood without hands-on and pictorial experiences. Check your standards–CRA is in there!
- 4.2.E Represent decimals, including tenths and hundredths, using concrete and visual models and money (concrete and representational)
- 4.2.F Compare and order decimals using concrete and visual models to the hundredths (concrete and representational)
- 4.3.B Decompose a fraction in more than one way into a sum of fractions with the same denominator using concrete and pictorial models and recording results with symbolic representations (concrete, representational, and abstract)
- 5.3.I Represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models (concrete and representational)
- 5.3.J Represent division of a unit fraction by a whole number and the division of a whole number by a unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using objects and pictorial models, including area models (concrete and representational)
- You can’t assume that students have had the concrete experiences coming to you that they require to understand your current concepts. As a 4th grade teacher, for example, you might have to go back and provide the concrete experiences students missed in 2nd or 3rd grade. You might find that the majority of your class needs additional concrete experiences to support current concepts, or you might have only a handful of students requiring concrete or pictorial reinforcement through small group instruction.
- CRA must be part of the remediation process. A great example is addition or subtraction with regrouping. If a child can’t subtract with regrouping, additional abstract practice–even if you are explaining the place value implications–will not be as powerful as breaking out the base-10 blocks and connecting the concrete to the abstract.
As the school year winds down, I challenge you to reflect on your current teaching practices and ask yourself how well you understand (1) your standards, and (2) how to incorporate CRA into your mathematics instruction. These are two of the most important ways you can improve student understanding and achievement. Commit to spending a portion of your summer months digging into your standards and learning more about how to provide concrete and representational experiences for your students, and I know you will be happy with the results!
You might want to check out this follow-up post with a video showing how to connect concrete learning with the abstract algorithm for subtraction with regrouping.