Student Work #4

The purpose of this activity was to see the student’s level of understanding of the concept of fewer and more than. This activity is under the common core standard “N.ME.00.02: Use one to one correspondence to compare and order sets of objects to 30 using phrases such as “same number”, “more than”, or “less than”; use counting and matching.” It is designed to elicit student thinking because the questions asked are done orally and then the student uses his own method of thinking in order to solve the problem. There are also follow-up questions asked based on the student’s answer to get him thinking further.

            For the activity, there were two piles of unifix cubes, one with 17 and the other with 9. I asked the student, “Without counting, which one of these piles do you think has fewer cubes?”. Then, I will ask the student, “how many fewer does this pile have than the other?” to observe how he solves it.

            The student could approach this problem in a couple ways. The student could count in his head to find out how many fewer one pile had. The student could also line up the blocks (one group on top, the other below it) and see how many more the larger group had. I anticipate that *James will solve this problem the second way, lining up the blocks to see which one has more and then counting how many are left over. My reasoning for this is that the numbers in the group are larger (done intentionally to challenge the student) which would make it harder for the student to do in his head. Possible errors *James might demonstrate are adding instead of subtracting, since subtraction is a newer concept they are learning.

Steps taken to approach the problem:

Me: “Without counting, which one of these piles do you think has fewer cubes?”

James: “(points to smaller one) This one.”

Me: “How do you know that?”

James: “Oh because it looks like this one has more (pointing to bigger pile) because it’s a bigger circle, it spreads out more on the desk. (cups hands around cubes in larger pile)

Me: “You’re right. It’s taking up more space on the desk. Now, can you tell me how many fewer this pile has than the other one?”

James: “Well this pile has 9 and this pile has 1, 2, 3, 4,…17. This one has 17. So, (pause, looks at fingers for a couple seconds)…8. You need 8 more in that pile to get to 17.”

Me: “That’s right, so how many fewer does this pile have than that one?”

James: “8 because I just counted.”

Me: “Can you tell me how you counted them? What did you count in your head?”

James: “I started at 9 and then just said 10, 11, 12, 13, …17. So 1 for 10, 2 for 11, 3 for 12, and then I just kept counting.”

Me: “Wow! Good job! You’re right. Do you know any other ways you could have solved it?”

James: “I could just make matches with the cubes and count the ones with no matches.”

Me: “You could. Will you show me how you would do that?”

James: “Yeah. (pairs the cubes up and counts the left over blocks) There’s 8 with no matches so that pile has 8 more.”

Me: “Yes, it does have 8 more. So this pile (pointing to smaller one) has 8 fewer.

            It was somewhat surprising that James solved this problem in his head. Most of the students in this class would not be able to do a subtraction problem with that high of numbers in their head. It was interesting to see that he kept referring to the larger pile as having more than the smaller one, rather than what the question was asking (focusing on fewer). I can hypothesize that James’ mathematical understanding of this concept is at grade level, because he answered the question correctly, using the counting up from largest strategy. I could also hypothesize that James is able to compare sets of objects using phrases like “more than” and “less than” using counting and matching.

            To advance James’ thinking, I could ask him more specific questions about having fewer than. I could ask him to take out 10 blocks. Then, ask him to show me 4 fewer blocks. I could do multiple examples of this, switching back and forth from showing me fewer than and showing me more than. I could also ask more abstract problems to see his general understanding of the words “fewer” and “more than”. For example, I could ask him questions such as, “which is fewer, the number of students in this school, or the number of teachers in this school”? These might advance his thinking because there is no way for him to count specific numbers, but will show me if he is strong in understanding the concepts.