Student Work 4

This week, the students were learning how to multiply multi-digit numbers by numbers that have one or more zeros at the end (i.e., 45 x 500). I was intrigued by the fact that my mentor teacher simply explained how to do it by saying "First you count the zeros, put them in the answer and then multiply as if they aren't even there." Some of the students caught on right away, but there were others who were clearly uncomfortable with the idea of just adding the zeros to the answer and moving on and they didn't understand why they could just do that. Because of this, I decided to look in the book at the problems they were assigned and look at how the book approached this type of problem. As you can see in the blue box, the only instruction the book offers is the first step, "Multiply by the ones. Place a zero in the ones place (0 ones x 52)." Then step two, "Multiply by the tens (3 tens x 52)." I find it very surprising that the book does not even attempt to explain how these problems relate to place value and why it is necessary to add the zeros at the end. Beyond the math that the students have already been doing (multiplying multi-digit numbers by one-digit numbers), this new type of problem does not require much more learning at all. I feel that the way my mentor teacher and the book have presented these problems, the students will still think of zero as "nothing" rather than a place holder for larger numbers and this will lead them to struggle when they learn to multiply more than one multi-digit number that do not include zeros at the end.

In order to say how I would anticipate the students doing these problems, I am going to analyze the way that I would think students would learn to do this. I would anticipate that students would learn how to multiply multi-digit numbers together before learning zeros, that way they could learn to simply add the zeros to the end as a "short-cut" after they have discovered how to do it using the multiplication algorithm. I would expect students to be confused about putting an entire row of zeros, then multiplying out the next number, but in the end they would understand better because when they move on to multiply the tens place, they need a place holder zero for the addition of the ones and tens. If the students were to learn how to do it in this order, I feel as though it would make more sense to the students to add the placeholder zero and also why it is okay to simply add the zeros to the end of the answer and keep going with the multiplication. When the students finish their work in class, they bring it to me to check it over, and as can be predicted, the students who simply accepted the method finished quickly and had no errors. The biggest errors I found in their work was in placement of commas and in the multiplication of the numbers larger than zero. I think the fact that the students asked for explanations of why it works to add the zeros at the end shows that they are understanding that math has a purpose and doing math serves a purpose. I also think that the fact that they seemed uncomfortable with adding the zeros to the end seeming "too easy" indicates that the students are predisposed to thinking that mathematics is difficult at first, until you understand.

Since the students did learn this before they learned to multiply two two-digit numbers together, I feel as though that is the next logical step. I think that even the students who have been doing well with placing their zeros will then understand why it works, but the students who didn't understand will benefit the most because they will understand as well. Once they have learned about using zero as a place holder to multiply by the tens column, I think they will understand a lot more about zero and its multiple functions in mathematics.