A few weeks ago, a cohort member and I had the opportunity to meet with four 3rd grade students from Woodmoor Elementary; Shay, Allen, Kindle and Gladys (not their real names). Our objective, during this meeting, was to attempt to understand their mathematical thinking. We did this by asking students to quickly count the number of objects they were shown for a period of three seconds. We were trying to find out how they organized the items in the picture to count them. We noticed that each student was able to come up with the “correct” number of objects but they each grouped the items differently.
This reinforced the idea that there can be many ways to look at a problem. Some of the students grouped the objects horizontally, while others grouped them vertically. One student, interestingly enough, actually “borrowed” an item from a bottom row, to add to a middle row, to match the number of items in a top row. All of the students, however, started grouping the items they saw from the left side of the page and then moved to the right side (much like reading, left to right.) The students, however, didn’t always start at the top of the page, sometimes they started in the middle.
This experience was very eye-opening. It wasn’t as easy to “extract” information from the students that I thought it would be. It, at times took a lot of effort to get the student to tell me his or her thinking. They took for granted that I “knew” what group of three they meant when he or she said, “I grouped the three objects together”. My cohort member and I spent a lot of time repeating and pointing and circling the groupings they made.
Another thing we noticed was that all the students added their groups of items together. None of them noticed “x” amount of groups and then multipled that number by the number of items in each one. From what I could tell, the students were “adding” their groups together, rather than multiplying them. By doing this exercise with the students, we were able to make this assessment.
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