Thursday, September 29, 2011

Quick Image Math - Fluency in Basic Addition

Interested in helping your students (at home or school) become fluent in basic addition? I just read a fabulous article on the topic in the Sept. 2011 issue of the National Council of Teachers of Mathematics (NCTM) magazine, Teaching Children Mathematics.

In the article, author Gina Kling asks "what exactly does fluency mean, and how might fluency differ from having instant recall of each and every basic fact?" Great question. And one with which many teachers and parents struggle.

NCTM defines computational fluency as efficient, accurate and flexible ways of computing. Kling writes, "Traditionally, learning basic facts has focused on rote memorization of isolated facts, typically through the use of flash cards, repeated drilling, and timed testing. However, ...drill alone does not develop mastery of single-digit combinations."

The article goes on to describe strategies that fluent students use and suggests several ways that teachers might help students to become more fluent. She talks about ten frames and games. But the concept that jumped out at me was subitizing.

Subitizing is "instantly seeing the quantity." Kling suggests using quick images to show students a representation of a number "with the expectation that they will retain a mental picture of what they saw and then use that image in some way." The key? " flash the image quickly enough so that students [can] not rely on counting to determine their answer."

I decided to try it this morning. I put pegs into a mat, gave a 5yo a quick image, and asked him to then replicate the number of pegs in his own mat. The mathematical thinking is apparent in the video below as he decomposes the number four, seeing it as "these two and these two make four." As you can see in the video, he doesn't count the numbers. He knows that it is four because he recognizes 4 as a double of 2 or 2+2=4. Note that we've done no formal work with addition; I've never heard him say,"two plus two makes four."

In the representation at right I asked, "How do you know it's 5?"

"Because 2 of these  and 2 of these [he pointed at two on one side, then two on the opposite side] and 1 of this [pointed to middle] makes 5."

It's quite exciting to see his mathematical reasoning.

We're on our way to learning basic facts. I'd like to try other methods using subitizing. The author suggests one could flash a pattern of dots (think of 5 or 10 frames) or a certain construction of unifix cubes. Watching my boys this morning, it'd actually make a great partner game in a math station at school or in a workbox at home. Each student could make a quick image "puzzle", give their partner a quick peek, challenge the partner to replicate it and then check the work by explaining the mathematical thinking involved. For instance, each child could have a five frame, put pennies on it (anywhere from 0-5) and have the partner try to replicate it after a quick peek. What an engaging way to develop fluency! As I was jumping around the web I noticed that DreamBox Learning has a whole category of games/activities for quick images. (I registered--free--and could immediately access.) Also, while this 5 Frame exercise isn't presented as a quick image, it's also a great resource for fluency practice.

P.S. Want to check out something similar for bigger people? Check out "What's Your Number Sense?" on Mathwire and go take the Panamath Test and check your own Number Sense! It's fascinating!

P.P.S. Just found a couple links:
What is Subitizing and Why Teach It?
Where Will You Land? A Game Focusing on Subitizing
Subitizing: What Is It? Why Teach It?


  1. I made a really quick center for subitizing -- because I, too, am fascinated with using it to help develop fluency and better number sense. I took a poster board, and evenly spaced 24 velcro pieces. I laminated 20 black circles, and attached velcro to the back. Students can put as few or as many as they want and then show a partner to see if they can subitize it. The idea is that they will develop skills not only in linear patterns of subitizing, but in circular, and abstract.

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  3. Fluency is exactly what we've been aiming for! We've done some dot pattern recognition work but I also like the idea of the child using pegs to replicate and explain his work. I noticed that developing number sense may be developmental as our daughter was able to make huge leaps when she was nearing 5 year old. Her ability to work abstractly and find patterns increased during this time. Thanks for posting this!

  4. This is why I loved using Math Their Way in the early grades. We use Dreambox and I've been very impressed with the visual models it uses.


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