Children’s Math Strategies

When I was in elementary school, I learned that there was only one way to do math problems, and that was the way the teacher taught us. In my current math class, I am learning the opposite. There are many ways to approach math problems, and often the way they’re done is more important than their answers. I would never have believed this a few years ago, but today I am learning to see math in a very different way.

I had the opportunity to do two different one-on-one assessments with two different third graders, in two different classes, and at two very different levels. I was told that I could not help the children and only evaluate their strategies for finding basic mathematical facts. We were given half an hour to do four questions, but they were all challenging ones.

My first child, Mr. J, was very friendly and worked very hard trying to please me, but he had a very difficult time understanding the basic directions and needed a lot of coaching to keep him on task. He exhibited a low attention span, but I loved his attitude. Though he was unsuccessful at solving any of the problems, he did his best when he was asked to divide 10 large cookies between 6 children. He would give each child one cookie and the other four to his “brother, sister, dad and mom.” It made a lot of sense to me, and I hated to tell him that he couldn’t give the other cookies to his family and could only divide the 10 cookies among the six children. From there, though, it went downhill. He then tried to divide the cookies all in half but couldn’t figure out what to do with them when he ended up with 20 half-cookies.

My recommendation for Mr. J was to work on early math facts using direct modeling. Because the story problems were too abstract for him, he wasn’t able to find a formula for them, and I recommended direct modeling using counters or manipulatives to help him to see the concepts.

My other child, Mr. C, was very different. His single-minded focus was to finish the work, and his focus was intense. On his first problem, he read it and immediately wrote a mathematical equation out and solved the problem. In fact when I asked him why he did the problem that way, his response was “Because that’s what you’re supposed to do.” That sounded like something I would have said and done when I was his age. Ironically, he was able to solve all the problems except the cookie problem. Like Mr. J, he too wanted to start by giving each child one cookie, but after that he differed. He worked very hard to divide the rest of the cookies between the children and finally resorted to drawing it out. Interestingly enough, he was able to draw the division accurately but did not know how to convert it into numerical measurement. I believe he is moving into the counting strategy. He further demonstrated this on another problem when he successfully divided 246 pieces of candy putting 10 in each box. He solved this by continuing to subtract 10 numerically from the total separately each time until he no longer had 10 to subtract.

Earlier Mr. C had told he did not like math. I could see why as I watched him struggle and work so hard to complete his work. I wondered how it could be easier for both Mr. J and Mr. C and think about my earlier blog on subitizing and how seeing patterns can make math easier. A different child in Mr. C’s class knew that there were 10 tens in 100. That meant there were 20 tens in 200 and 24 in 240. Knowing that would have saved Mr. C a lot of time and perhaps made math a whole lot more fun for him. I’m looking forward to teaching children how to see more of these patterns and helping them to see that math can be fun.

Subitizing

What are some of the different ways students can add up groups of dots? Finding that out is the intent of the exercise called Quick Images. Two of us worked with a small group of third graders assessing their abilities to subitize, find the total number without giving them enough time to count each one individually. These dots were on flash cards and each flash card showed a different grouping. There were groups of twos, threes, fours, and fives. We flashed them an image for only a few seconds and then asked them to visualize the dots in their heads. We then gave them a chance to check their answers by flashing the dots in front of them once more for another three seconds. After this we asked for their totals.

The students did have one advantage. This exercise had been demonstrated to them, and they were now alert for patterns, and they found them. I was amazed, however, at how many different patterns they could find. They circled the ways they had grouped them, so the rest of us could see how they were able to come to their answers. Because they were asked to explain their methods to the rest of us, such as dividing the dots into groups of threes or sixes, it became evident to the students while explaining themselves if they were correct or not, and we all learned from each other.

One interesting development I noticed was that the students who were struggling with other mathematical concepts were excelling in this exercise. These students saw this exercise as more of a game than an assessment, and they were having fun. This was evident in their joyful faces and the excited way that they participated. Gone were the anxious faces. They were in their element, and it was like the playing field had been made even for them at that moment.