This week I reintroduced multiplication to my students and the patterns for the multiples of 2, 5, and 9. In the beginning a lot of the students had a difficult time really being able to show me that they understood the pattern by using it to explain their answer.
For example I would prompt them with this question: "Using the pattern for multiples of 5 how do you know 5 x 4 is NOT 24." That was a multiplication fact they all knew and they all knew the proper product was 20 not 24. However, because these students have never had to use reasoning for math it was hard to answer the question by using the pattern to explain. One of my students answers was the following,
"I know it is not 24 because it is 20. I know this because the fives count up by 5s and that has a 4."
Although this student is correct in the fact that the multiples of 5 count up by 5, you could say that for any multiple of any number however, the distinct pattern for the multiples of 5 that we discussed and they wrote down was that the number in the ones place of the product always ends in a zero or a five. This student was able to tell me this pattern, write this pattern down, explain the pattern to me however, when it came time to applying the pattern they could not do it. I think this is because they have not fully internalized the pattern and do not fully understand how they can explain their mathematical thinking using the facts and patterns they know.
This is a wonderful "sample of student work", since it comes from a conversation, but is quite illuminating. Also important to note is how this information was revealed because of the good question that you asked (a very simple, but very incisive mathematical reasoning question). Certainly I encourage you to keep asking questions like this as often as you can. It is these types of questions that will help you facilitate a whole-class discussion where students defend their reasonings to one another.
ReplyDeleteThat leads to another point, which is that there is certainly more than one way to answer this question. Clearly, then, the most valuable next step is for students to compare / defend their reasonings to one another, which should scaffold a great deal of growth in understanding among all the students.