Invented Math Strategies

How do children develop their own strategies?
I have come to believe that children develop their own strategies. Sometimes, it happens as a result of confusion. A student could be taught one strategy from one teacher and another strategy from another teacher. This could cause one child to incorporate two ways into one, resulting in confusion and the wrong answer. This is with a student at my intern school.We were working on regrouping with subtraction. I showed them a way of subtracting that includes showing the number of 10's in the tens place. For example, with the the number 43, there are 4 ten's. These tens would be wrote on the bottom left side of the math problem. I asked him to cross out a ten when he is adding it to the one's place. When he got the chance to use this method, with no help, he would take the ten away but would add it to the number that is being subtracted. He would put that in the place of the regrouped tens place (I hope that makes sense). I taught him how to demonstrate the adding of 10 to his tens place and another teacher taught him to add that ten to the ones place ( I realized this when asking him to subtract and regroup before I tried to teach him any strategies). I acknowledge the fact that I taught him this strategy poorly. But I wrote about this to show that students create strategies from ones that they've been taught. He also incorporated using blocks for his problems to help him maintain the placement of the numbers.

An example: 21 - 7 = 14. In this picture, he got it wrong at first. It showed another mistake that he was making, but I will discuss this later in the blog. He first added 10 to the 7 down at the bottom of the paper. You can't really tell because he erased his first response. He eventually added the 10 to the 1 in the tens place and gotten the problem correct.

When subtracting, one of the students would use tally marks. This was a strategy used by another student. With 2 of the 4 problems given using tally marks, he had gotten wrong (picture below). Noticing this, I introduced him to a new strategy. I called it the 'Count Up' strategy. When subtracting smaller numbers, take the subtracted number and count up until you get the bigger number (I can't remember the proper name for the subracting the number and the 'original number'). In other words, if c - b = d, I told him he could take 'b' and count up (not including b) until he got to 'c'. The number it took to get to 'c' was the answer or 'd'. This was also a problem that the other student was making in the first picture. I showed him the 'Count Up' strategy as well. (Below) His tally marks caused him to get the problem wrong. I tried to help him perfect this way, but it was much easier to explain the 'Count Up' method to both of them.

Math Talk Moves

What was the specific mathematical content being taught? This week I taught them a lesson on how to use a specific method for word problems. After I modeled filling out this special I had them to read a word problem and underline any information that they thought would be important. After that, we began to fill out the Math Problem Solving Guide. With this guide, we answered three basic questions and followed one command; what do I need to know, what do I know, what do I need to do, and check your answer.

Any specific talk moves being used? Were they effective? I noticed that I used a lot of wait time. There was one student who caught on really fast. When I realized this, I tried to spread out my questions among the other students. By doing this, I had to use more wait time because they didn't catch on as fast. As the book stated, some of them needed more time to get their thoughts together. I believe they were effective. The other student contributed with correct responses. Their responses gave me an opportunity to use two of the other Talk Moves. After a student would suggest something, I would ask them if other had more to add to the other comments. I also restated their suggestions to make sure that I understood what they were saying before I wrote them down on the board. I didn't allow them to say if they agreed or not because I wasn't sure if they could respectfully disagree. I wasn't sure if any of them would get offended if someone openly disagreed with them.


If I could redo this lesson, what Talk Moves would I use? I would probably allow them to agree or disagree with one another. Before I did that, I would have made sure that they know its OK for others to disagree with them. I am curious to see if it would make them want to put out more comprehensive suggestions if they knew someone would be listening. This would help when we are discussing extra information that is word problems. I can ask the question 'is this something we need to know'. Based on their answers, other students can comment on whether or not they believe it should be written on our Problem Solving Guide. After that, I would probably ask them to restate what another person had just said. I believe putting it in their own words would help them in understanding their peers' comments as well as other students. This could be helpful when I am asking 'why' should a statement be on our guide. When they present a rationale, I could ask another student to restate what the other student just said to clarify.