The mathematics of students is a powerful tool for a teacher. It allows a teacher to hypothesize what is happening in the mind of a child and plan a next step that will allow that child to construct more sophisticated understanding. Today I want to look at the mathematics of a student we call Grace (pseudonym). I love telling Grace’s story because she moved from solving problems like 8+4 to 98+4 in our 6 week study!
When Grace was interviewed in December, we noticed that she could solve problems even when counters were not in front of her. She used her finger patterns to help her solve these problems. In my experience working with teachers before joining the AIMS Center, I would have described Grace’s behavior simply as she needed to use her fingers. I had no idea that how a child uses their fingers can be an indicator of very different understandings and mental material. Children use their fingers in a variety of ways to solve problems, and it is helpful when you can use the unique way they use their fingers to interpret their mathematical understandings. .
Please watch the following video and write down a few things that you notice.
Grace did not explain her process right away by using her fingers, but after a prompting from one of the research associates she explained her solution by using a simultaneous finger pattern (lifts all fingers at the same time) and then counting additional fingers. Let’s look at how she solved the first problem, which was seven under one cloth and four under a second cloth. She starts by lifting a finger pattern for seven simultaneously. Then she counts while lifting a finger every time she says a number, “1, 2, 3, 4.” She explains that the solution has to be eleven because she didn’t have enough fingers and had to imagine one more.
Grace is using her recognized finger pattern for seven to help her create something to count. She knows what ‘seven’ looks like on her fingers without having to count them one at a time. In the second problem she solves the problem in the same manner and even states that she solved it like the last problem. Did you notice that detail the first time you watched the video? Watch it again. Can you solve 8+5 using Grace’s method? We refer to mental images, movements, or words as “figurative material”. It is something the student generates (not concrete) to make it possible to solve the problem. Given that fact that she did this without her fingers being visible (they were under the table), I think Grace is definitely moving away from working with concrete material to working with figurative material. Students have to work with figurative material before they can construct abstract material. And constructing abstract material is the goal.
In my next few blog posts I will write about the limitation of this method that Grace used and I will show what tasks I presented her to foster opportunities to construct more sophisticated methods. In the meantime if you’d like to hear more about the mathematics of students, head over and listen to an interview I did with Dr. Chris Brownell and our ZPC Podcast.