Student Work 5

 Last week, I posted about how my mentor teacher chose to introduce multiplying by multi-digit numbers by first selecting only numbers that have one or more zero(es) at the end of a single digit. For example, she would have the students do set of problems that all look similar to: "19 X 500 / 54 X 20 / 33 X 6,000 / 65 X 50,000 etc." I mentioned that I found it interesting that she did this before teaching numbers that have multiple integers, because this way of teaching does not show place value. This week when I went to my placement, the students had moved on to multiplying two two-digit numbers together. The problem that my mentor teacher found that her students were having was with lining up the numbers in their proper places. For this reason, she had the students finish the assignment they had started the day before, but instead of using just lined notebook paper, she showed them how to use graph paper to better keep track of how to line the numbers up. I chose to upload two pictures to show the differences that I saw with my students attempting to use the graph paper to think about where to place their numbers while multiplying.

A couple different ways that the students could work on these problems are skip counting, using memorized multiplication facts, or using their fingers to count the multiples. However, because the problems are in the algorithm form, there is really only one way to complete the algorithm, and the methods I mentioned are only for solving the individual single-digit multiplication facts involved in completing the algorithm. One of the other problems that the students had with multiplying two two-digit numbers other than lining up the digits in the correct places, was that they didn't understand why they had to always put a zero in the ones place before moving on to multiplying the tens place. As you can see in the bottom photo, this student did a great job of lining up her numbers, but in the second to last problem, she got the answer wrong because she did not use a tens placeholder. As I was worried about in last week's post, the students did not understand that the placeholder was needed because the problem is representative of multiplying by the ones first then by the tens, meaning that there will never be an answer involving ones when multiplying tens. I tried explaining this to the student, but she did not seem to understand what I meant by that. For this reason, I showed her the distributive property of multiplication and split the problem. She was able to solve both of the simple problems when it was separated, but was still struggling when she returned to doing the whole problem altogether. I think that the fact that the students don't understand how zero works as a placeholder and why it is necessary reveals their current mathematical knowledge about the number zero. I learned in a previous math class that one of the most difficult things for children is to understand that once you get to a certain level of operating with zero, it is no longer just "nothing," but instead has a purpose and has a meaning within the mathematics that they need to understand. I think it is clear that these students do not yet have an understanding of zero deeper than "zero means nothing," at least not explicitly. I think it would be worthwhile for my mentor teacher to spend some time talking with the students about their current conceptions of zero and then discuss how to take those understandings deeper, so that it is more meaningful to the students that there needs to be a zero place holder in the second step of the two-digit multiplication algorithm. It seems as though if she explained it further and made it more clear as to why the zero is needed and why it makes sense to put it there, the students would be more comfortable with the algorithm and feel more confident that they understand what they are doing and why.