Counting and adding seem like a such simple tasks to adults. We learned how to count and add so long ago that we don’t even remember the thinking that took place to make sense of these mathematical tasks. Here at the AIMS Center we are reading the research by Dr. Les Steffe that shows us just how complex and detailed the mental process is for students learning numbers, counting, and adding.
For the past two weeks you have heard from my colleagues Elin and Beverly regarding their trip to CMC South Conference and the presentation they made regarding “counting-on” as a non-teachable strategy. I am going to connect to both of their conversations and bring in what I have written about before in my previous blog regarding awareness.
Beverly shared examples of a student, Chloe, who struggled to solve problems when she did not have enough fingers to complete an addition problem. We argue/infer that she had empty procedures for counting-on.
The students that I will focus on have their own meaningful ways of solving problems and are at different stages based on Dr. Steffe’s research. They all are aware of visible or imagined items when they set out to solve a problem posed by their teacher. Take a minute and watch the students solve the problem given to them: add 27 and 8.
In the video clip, both Ally and Grace had different ways of establishing or making the first number, 27. They are aware of counting up to 27 in a way that is meaningful to them.
- Watch Ally again. How does she count-up to 27?
- Now watch Grace again. Does she count-up to 27 in the same way as Ally?
They both had a way of making 27 before they added 8 to the 27.
- Ally ran through the number words from 1 to 27 before she added the 8, keeping track of her 8 added counts on her fingers to know when to stop counting.
- Grace needed something more than just the words to make the first number 27. She counted her finger movements one at a time as she counted from 1 to 27. The movement of her fingers, in addition to saying the number words aloud, helped her establish 27 before she added the 8 with her fingers.
These subtle differences tell us a lot about where students are in their development of number concepts. Ally is aware of the words to make 27, while Grace needs movement and words to establish that same number. Chloe (from Bev’s post) did not have enough fingers to establish the number 27. Both Ally and Grace have progressed further along this continuum than Chloe, as they don’t need to see 27 objects to count them. The process a child goes through to establish 27 with less and less material prepares them to make a sense of counting-on.
Whew! Can you remember doing all that when first learning to add two numbers? Most of us don’t. We are so far removed from this experience that we seem to have forgotten this complex process. By taking time to observe students we can learn and understand much more about how to support them in their mathematical construction of number.