I got an email from a reader last week asking about the impact of technology on the CRA sequence of instruction, so I thought I’d blog about it. First, let me remind you that CRA stands for concrete, representational, and abstract. From Van de Walle (pg 99):
“…this model reflects a sequence that moves from an instructional focus on concrete representations (manipulative materials) and models to semi-concrete representations (drawings or pictures) and images to abstraction (using only numerals or mentally solving problems).”
Lots of problems occur when we skip steps in this instructional model or move too quickly through them. For example, before students start practicing with traditional flashcards (very abstract), they need to have lots of concrete and representational experiences to develop understanding.
Van de Walle makes a great point, however, that the stages need to overlap. That is, we need to connect the abstract to the concrete and representational along the way.
“…it is essential that there is parallel modeling of number symbols throughout this continuum to explicitly relate the concrete models and visual representations to the corresponding numerals.”
What this means is that while you are using base-10 blocks to model addition or subtraction with regrouping, you need to also show the connection to the algorithm or students will never move past the concrete or representational stage.
Now back to the original question about technology. Are virtual manipulatives, such as the ones you find at the National Library of Virtual Manipulatives or NCTM’s Illuminations website, concrete learning? My thinking is that virtual manipulatives actually fall between concrete and representational. They are clearly not as concrete as holding manipulatives in your hands, yet because of the interactive nature, they are more concrete than representational. My advice? Use virtual manipulatives, but not to the exclusion of true hands-on learning. Also, consider the age and developmental stage of the child.
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| The Factorize interactive activity at NCTM’s Illuminations website |
I think what’s important is that when choosing technology, whether it’s computer programs, websites, or apps, you have to consider the CRA stages, just like you would when choosing any instructional activity.
I hope this all makes sense. I would love to hear your comments!
Hi Donna- I totally agree. I teach first grade and my team mate and I have spent the entire year building number sense. We just introduced fact tests because we know the children will be expected to do them next year. Guess what…they flew through them. It was very affirming. We are the only grade level at our school that has embraced common core this year and we love it.
Awesome! I love hearing success stories like yours. I’ll bet you won’t be the only grade level embracing it next year with results like that! 🙂
I like the CRA approach you mention and I love that virtual manipulatives seem to be as effective as tangible, in hand ones. Can you talk more about the “R” stage and how you go about having kids do that phase? Do you just have the kids do their own drawings each time or do you have them use stamps, etc.
Great question! It could be either a creating their own picture/model or using and interpreting pictures. So, for example, they could draw a model of multiplication, like circles and stars. Or they could use cards in a math workstation that show models of multiplication and, say, match them to a fact, which bridges representational to abstract.
The CRA approach is vital to student understanding. I feel like to often kids who don’t get it are pushed to practice more at the abstract level but what they really need is to go back to the concrete and/or representational level. I do love the national library of virtual manipulatives. I have used it a great deal especially with older kids because they often feel like they are beyond manipualtives. The number line bars- fractions has completely revolutionized how I teach fraction division.
Tara
The Math Maniac
You are so right, Tara! And we need to think about CRA when we remediate as well. If you’ve got a 5th grade who can’t subtract with regrouping, giving them a bunch of abstract problems isn’t going to help. You’re going to need to bust out the base-10 blocks and go back to the concrete level.
Absolutely agree. Anytime we are working with addition or subtraction in math groups using manipulatives or dominoes we are modeling the number sentences and giving the language to go with it. I do this all year on calendar math because it makes it more familiar when we get to the unit in the spring. There are few things we do that involve only one of these at a time. Great way to put it into perspective!
Right, Keri! I think sometimes teachers forget to connect the stages, and that creates a disconnect–kiddos who can only do it with concrete materials, for instance, because they don’t understand how what they’ve been doing ties to the abstract.
Can you suggest any books that do a good job explaining the CRA sequence? I would like to do some pd with my teachers.
You know, that’s an interesting question, Tamara. Of all the books in my rather extensive library (kind of an obsession), there aren’t really any that specifically address CRA. My go-to book for professional development, though, is Van de Walle. The link in the post above is to the professional development version of his popular series of books, and it’s excellent. If you look at how he talks about developing concepts in his book, it totally follows CRA.
This site explains it really well: http://fcit.usf.edu/mathvids/strategies/cra.html
Thanks for sharing! 🙂
I think these virtual manipulatives will be a very useful tool if we do have to have distance learning this year. While I agree with your thought that they are between concrete and representational, they are a great tool for kids at home. They may be necessary if we have to limit contact with tools in the classroom – maybe for those who are moving more toward abstract than the others.
I believe that we need to gather more information about this topic. With online learning becoming more prevalent, young students who need more concrete practice might not be getting everything they need to develop strong mathematical understanding. I agree that the virtual manipulatives have a place. I have been curious about this for sometime and have not seen much research. I met someone when I was at one of Jo Boaler’s YouCubed classes. He worked with research at the University of Chicago. I am wondering if they have done any research on this topic.
Do you know?
Thank you for raising this question! I don’t know of specific research, but what I’ve read indicates that virtual manipulatives should NOT be the first experience children have with a concept. Their first experiences should still be concrete.