Another thing that comes to mind about my student is that once he noticed the unifix cubes in front of him, he never looked back! Even though he was more than likely exposed to the standard algorithm to set up multiplication/division problems, he didn't use them unless it was a last resort and the manipulatives were confusing him. If you look at the pictures below, you will see that M is solving a problem with the cubes. In short, the problem says there are 45 students that will be on 3 teams, how many people on each? In terms of a standard algorithm, this is a division problem; 45/3=15. However, M didn't see it that way. He counted out all 45 cubes, then grouped them in to threes and then counted how many groups of three he had. In all actuality, this is a very complex strategy. It appears that he realized that he is trying to see how many times 3 goes in to 45 because he separated the blocks into groups of 3 rather than making three large groups and dispersing the blocks equally. See below...
I think that there are a couple of different ways other students may have solved this problem. The first is by separating the cubes in to 3 different "teams". Count out and put cubes one cube in each group until they are all gone. Once they are all gone, re count (to double check) how many cubes are in each group: 15.
Another way a student could solve this problem is by first counting out all 45 cubes then subtracting 3 from the group until the all the cubes are gone. Once all the cubes are gone, count all the groups. This strategy is very similiar to M's strategy but it is turning the problem into subtraction (and eventually addition) instead of putting them in to groups of three from the beginning.
I think that it is nice that your kid was familiar with the cube manipulatives provided. They are a very helpful tool. My kid was very reluctant to use the cubes for me. The student would set up the problem on pencil and paper and then get stuck and not know what to do. It was not until after some convincing on my part that the student tried working with the cubes. She was able to solve the problem when visually representing the math. I really like the possibilities that these cubes allow students. They will be one of the first purchases for my classroom!
ReplyDeleteEric
I definitely agree that cubes can be slightly confusing to students who have not been exposed to them. I have also been very surprised in the comfort that it can provide many students. Rather than requiring a student to recall a specific algorithm and to know when to use which one, cubes allow the student to find the answer without depending on recall of facts. This is especially important for students with disabilities who often struggle with memorization.
ReplyDeleteWe were never allowed to think of a strategy that made sense to us because we were taught using, "the only way," to do something method! I like how we are being asked to come up with different ways to solve word problems and other related forms of problem solving. I realize now that one way or one strategy (i.e. algorithm) will work for all of our students. Just last night I was having to change my word problem lesson to help refocus a student on the reading part of word problems, and get her to try using other strategies to act out the word problem and solve for the correct answer. Not every student needed this alternative perspective of the 'action' of word problems, but I thought it would work for her and the limitations she was experiencing. In the end we are having to do more work, trying to explore alternative and inventive strategies, but ultimately we are doing this to help future generations and how they approach math! (Math beyond mad minutes!)
ReplyDeleteI remember using manipulatives as a kid, but they weren't "pushed" as much as they are now, for sure, and we still did "mad minute" and "around the world". I just wish my middle schoolers would try to use manipulatives to help them understand more basic facts, but they think they're too cool for that. It's too bad manipulatives get the rep as being for "preschoolers", because I think it would really help them.
ReplyDeleteIts funny that you said that your child used so many different methods. With one of the kids I work with, I wish he would get to learn more than just direct modeling. I think he just works too fast. I also liked how your child didn't hesitate to use blocks. With the child I videotaped at Houston Elementary, he refused to use blocks when I tried made the suggestion. Your child likes to explore new things and thats good. I think this will help you when teaching him/her new stuff.
ReplyDeleteSomething that jumped out at me for some reason is the realization that given a problem, students have been able to determine the appropriate operation to solve the problems. Your student figured out that he needed to divide the large number by the little number in using the cubes and making 3 groups. No algorithm needed.
ReplyDelete~Claudia