When I first started this semester, I was able to observe the students in my placement work on several basic skills (such addition, subtraction, and multiplication). During this time, the students worked on addition problems, and I noticed them using a method I feel I was not familiar with. They would expand the numbers in the equation, add their place values separately, and then together to find the answer. For example, 71+54=? The students would separate the tens (70, 50), then the ones (1,4), and then add them (70+50), (1+4). Then they would take those totals and add them together (120+5=125).
Another way a student can solve this problem is by adding 1 to 54, and changing 71 to 70. So then the problem would become 70+55= 125. Going off the expansion example, another strategy would be to subtract 1 from 70, 4 from 54 and add 70+50, 1+4. Then add both sums together. A different way would be to subtract 4 from 54, and add those 4 to 71 (causing 71 to become 75, and 54 to become 50). Then 75 and 50 could be added together to find the answer. Finally I know that 30+30=60, so if i took 60 from 71, I would have 11 left. Then I could add those 11 to 54, causing 54 to become 65. Now, I can easily add 60+65 because I know in the tens value 60+60=120 (because I know 6+6= 12 so I just add on a zero), and in the ones 5+0=5. So when I combine 120+5, I get 125.
When working on a different problem, (21*2) I noticed that one of the students changed the problem so that the smaller number was on top and the bigger number on bottom. I noticed the student would do this for all their double digit by single digit multiplication. One strategy that could have been used would have been to add, 21+21, instead of multiplying. Another strategy would be to expand 21 (20, 1) and multiply each digit by 2 (20*2, 1*2) and then add the sums. To make the numbers smaller, I could also split 21 (10, 11) and then multiply 10 by 2, 11 by 2 and then add those sums together to get the same answer. Those are a few ways I can think of how to solve a problem like above using invented strategies.
(pictures to be posted soon)
I'm not sure what grade(s) you are working with, but I saw this expansion in my 2nd grade GED setting last fall. They seemed to find it much easier to add the tens together. I feel this is because it's a "nice" number; no regrouping required! Breaking it down by tens+tens and ones + ones seems to help the students because the ones are more than likely going to be "quick fact" recall and the tens are "nice" numbers as mentioned above.
ReplyDeleteMy students also used the strategy of taking some from one number and moving it to another. This strategy also demonstrates the student's desire for "nice" numbers. However, I have found that at times the students become confused on what they did or didn't take away and how much from what number. I think I would loose track too!
I love how you took this opportunity to talk about your students rather than the problem solving interview. By now it is safe to assume that we know are students pretty well and can fully discuss their strategies. Where as with the interview we had to make inferences and assumptions if we didn't ask a lot of questions at the time. Hindsight is always 20/20!
Ashley,
ReplyDeleteOnce again, I was surprised to see the strategy of adding the tens together and then the ones in action. I am finding that I really like it, and wished I had used something like that when I was first beginning to learn math.
I also like your other strategies for solving math problems. It's funny, but I had never really thought about changing the numbers or breaking them down to compute (other than simply rounding them) before I read this chapter and started reading through the blogs.
Jennifer
Ashley,
ReplyDeleteI like to look at problems and see how many ways that they can be solved. Especially when problems like those above may seem like an easy process to us now.
I also like to see if I can figure out problems left to right instead of right to left. I still remember the math video we all saw in class about how different cultures solve math differently. The movie we saw and what we are experiencing in the classroom really shows how there is many possibilities to solving a math problem.
I really love the fact that you've seen so many different student-invented strategies! This is exciting that they have been able to have a teacher that fostered using this strategies, not only relying on traditional algorithms. This definitely gives the students more understanding of the work they are doing because they are backing up the problem with knowledge they already own, and it gives them a chance to take ownership of it. Being able to apply a strategy that makes sense to them allows for the chance to be more personally invested in their learning.
ReplyDeleteOne example of a different student-invented strategy that the students could have used for the problem 71+54 could be a combination of the examples you gave. You could change 71 to 70 and 54 to 50, which turns it into 70+50. Then, we could split the numbers into parts (70 -> 30, 30, 10; 50 -> 20, 20, 10), and work from there to get to 120 (I found that it was easier to do 30+30=60 -> 60+20+20=100 -> 100+10+10=120). After you get the answer 120, we account for what we took away at the beginning when we rounded (1+4=5), so we get 120+5=125!
In the end, it's all about what's best for the student and what they feel comfortable using. We want them to understand their strategy and take ownership of it! :)