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)**

- 4.2.E Represent decimals, including tenths and hundredths, using
- 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.

This is my heartfelt philosophy that I live each and every day! I say Amen!

Fabulous post. So informative with excellent examples and suggestions. Thank you!

I wonder, what about those students who can do the math in their head? In 2nd grade, I had a student who could accurately add and subtract 3 digits in his head. He actually got stuck when he had to show his work for regrouping. This may well further support your point for the need of that deeper understanding.

Also, what about the brighter than average students? Rather, what is the best way to discern when kids have this true understanding as opposed to just being strong in mAth?

Wow, I have so much to say! First, I definitely agree with your comments. Students learn the abstract by understanding the concrete. Using manipulatives is essential, and students need to experience a big variety of them at all grade levels. Using color tiles, pattern blocks, tangrams, pentominoes, geoboards, dice, playing cards, dominoes, color cubes, in addition to the normal manipulatives offered by textbooks is very important–and, I might add, FUN! I have thought of this as the constructivist philosophy, whereby students must construct their own knowledge to understand and master it. Second, while I agree with the CC person who listed some CC objectives that relate to CRA, I couldn’t help ;but notice how wordy and complicated many of them are compared to the CRA. This last year I tried to correlate an algebra test to the CC standards, and it was awful to try to pinpoint a particular skill to a particular CC standard. 3rd, using real life experiences is also very helpful. When comparing freactions, for example, use things like a pizza or shooting free throws. If you like pizza, would you rather have one slice of a pizza cut into 8 slices or one slice of an equal sized pizza divided into 4 slices? Who did a better job of shooting free throws? John made 1 out of 4, while Todd made 1 out of 8? Using those kinds of analogies together with manipulatives helps students develop a better understanding of comparing fractions. I also want to put in a plug for cooperative learning, in which students can experiment with manipulatives and discuss them in a group setting.

So true! Remember Math Their Way? It total works this way as well. As a MTW instructor, we used to give the teachers math problems with made up symbols, even fun dot to dot problems. ALL the teachers were frustrated & could feel what the kids feel AND as adults we have prior knowledge to make sense of the made up symbols. A great activity to show parents these stages also.

Are there any good resources for teachers regarding CRA?

Van de Walle’s Teaching Student-Centered Mathematics is an excellent resource, Tonya! It is not specifically on CRA, but the instructional techniques used throughout the book show the progression through the stages.

Thank you Donna.

What a great post. I love the beginning. I’ve been working with parents, albeit a small group, showing them how important it is to develop a conceptual understanding before they can learn and master procedural skills/algorithm. Am definitely going to bookmark your post to share.

I agree that the CRA approach to mathematics in crucial to a deep understanding. In California, we are still using Common Core standards, where I believe the CRA approach is represented coherently and in a strong progression. I worry that your comment about it not being there was a bit misleading to the teachers who are also using CCSS.

The deep level of understanding developed only through CRA is definitely a part of CCSS, Jamie. And it’s clearly included in many unpacking documents. My comment referred to the actual wording of the standards. The TEKS use specific wording, such as objects, pictures, concrete, etc., in many of the standards, while I don’t see that specific wording in the CCSS. It is more implied in the CCSS, versus specifically stated in the TEKS.

Just thought I’d chime in here… Donna, here are some examples of CCSS standards that use specific wording that can be tied to CRA. We have been very clear about the intent for CRA instruction throughout the CCSS in our professional development with K-5 teachers here in California! :

K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). (“objects or drawings…”)

1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. (“objects, drawings, and equations”)

1.NBT.4 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. (“concrete models or drawings…relate the strategy to a written method…”; similar wording for subtraction in 1.NBT.6)

2.NBT.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction (“concrete models or drawings…”)

There are also many mentions of using visual fractions in the NF domain for grades 3-5.

Thanks for all the great work that you do! 🙂

Yes, those are great examples, Chrissy! So glad that your PD is heavily emphasizing CRA. 🙂

I believe that the Common Core does not give enough time to the “C” or the “R” steps. My 7th grade math students are expected to do a lot of abstract thinking without the proper basis. Does anyone else feel this way?

My 3/4th grade students are definitely having challenges learning fractions. We have no manipulatives. What can I give them that they won’t loose?

Fractions is an unfortunate label. Should simply by called parts! Unfortunately, we get stuck by our own use of abstract terms, ie fractions. Fractions should b introduced as parts of a whole, ie, 3/4 is really 3 out of 4 pieces of anything. So, 3/4 + 2/4= 5 one-fourth pieces, etc

That’s exactly how it is introduced in 1st and 2nd grade. In 3rd grade, students learn formal fraction notation, but they need manipulatives to connect meaning to the abstract symbols/terms.

Get some play doh and have kids divide circles cut with the play doh can. This is very effective.

Chaya, Donna’s paper version of the plastic fraction tiles is great. Another idea is to have your students make their own fraction sets with regular paper. Give each one of them 8 different colors of paper. Have them divide (first in pencil and check it!) one color into 2 equal pieces, another into 3 equal pieces, another into 4 equal pieces, and so on for 5, 6, 8, 10, and 12 (If you follow the CCSS, 3rd graders expectation are limited to fractions with denominators of 2, 3, 4, 6, and 8). Then they can cut the fraction pieces apart and each student can store their set in a plastic bag.

Chaya, that is a major bummer that you have no manipulatives. Have you looked into Donors Choose to try to get funding to buy some? In the meantime, check out this blog post where you can download paper versions of plastic fraction tiles. As far as students not losing them, I might recommend keeping their paper tile sets in a zippered baggies that you can hand out easily to students