I have a student who is struggling in math. What she is able to do is so interesting. It is even more interesting when I consider her struggles. It seems like the number sense that she should have gained in second grade is NOT intact. For example, she cannot move around the hundreds chart with ease. If asked what 24 plus 30 is she would have two strategies: standard algorithm and counting on by thirty singles! The problem with her standard algorithm use is that she has memorized the procedures with no understanding. Therefore, 26 + 6 is sometimes 86! She cannot count by ten from any number. She has little command of facts that sum to ten so she has a difficult time knowing that n = 6 when 34 + n = 40. She cannot respond accurately when asked, "what would you add to 8 to get a sum of 18?"
Here is where it gets interesting. She knows her multiplication facts COLD. She can play "Big Array/Small Array" covering a larger array with two smaller arrays. She can generate algebraic notation showing her thinking (ex: 18 x 7 = (10x7) + (8x7) 18 x 7 = 70 + 56 and finally, 18 x 7 = 126)
When given a problem to solve like the one shown in the assessment below, she struggled. Even though she works proficiently with the array cards, she is having a difficult time moving from the concrete to a more symbolic representation. Today, she wanted to decompose the 7. I asked her how she would break seven apart. She said she would break it into a 3 + 4. I asked her what her next move would be. She froze. At this point I wish I had her grab the array cards and work to find two array cards that would cover an 18 x 7 array. Instead I showed her the smaller array she was left with. The one we discussed was an 18 x 3. I asked her if she knew her three facts up to 18 or if she knew her 18 facts. There was a long pause. I was even hoping that she might suggest adding 18 three times but that suggestion didn't come. My next move was to ask her if there was another option, other than decomposing the 7. She was able to suggest decomposing the 18. When I asked her how she might break it up, she struggled. I asked what 7 facts she knew by heart. She suggested 7 x 5. I asked if she knew any larger facts. She knew 7 x 10. I asked her to partition the array so that one part was a 7 by 10. She was able to do this. When I asked the length of the remaining side, she couldn't tell me. After many attempts, she counted on from 8 and settled on ten as her answer. Once she was able to decompose the large array and label the array's dimensions, she was on her way and completed the rest of the assessment independently.
I wish I always knew the best path to help students work through problems. I ask a lot of questions. I give think time. Still, I wish I had grabbed the array cards because she may have been able to make the connection between the physical arrays that she has worked with successfully and today's more symbolic representation. I wish I had a math coach or another adult who loves math as much as me, by my side in the classroom, so that we could strategize. I'd love to have more confidence in the math moves I make.
I still have to think about how to fill some of those number sense gaps. I'll begin with an Investigations game called "Capture 5" and go from there. I hope that this game will help her to add ten to any number with ease and to learn facts that sum to ten and make the connection between these facts and figuring out missing addend problems or subtraction problems where knowing those critical facts is helpful. This is what I'm thinking. I just wish I was more certain.