Something I found interesting in chapter 3 was how the Mayans and Egyptians dealt with a problem similar to the sub problem from earlier in the book. I thought it was interesting that their thinking allowed them to arrive at a similar solution as the students who had the sub problem. They both used the concept of unit fractions to divide the sub. I think this concept is interesting because it almost seems instinctive to divide a sub this way before you have knowledge about other ways to use fractions. I am still wondering, however, why 2/3 seemed to be the magic number. Why didn't the Myans/Egyptians divide bread into halves so there would be 5 halves to distribute with a remaining half that they could have divided into tenths? This seems like it would have been easier, especially when using unit fractions.
In Chapter 2 I was interested in the part where James was trying to divide a strip of paper into thirds. Made 3 pieces and threw the extra piece away. This idea ties into my math lesson that I completed yesterday. My lesson was on part-part-whole addition and subtraction. In the beginning of the year, my students could not do a part unknown subtraction problem. After doing the lesson, I can see that a lot more of the students understand the relationship of parts to a whole, but I am interested in observing how the think when they start to use fractions. I am hoping to observe some fraction practice before the end of the year so I can reflect on this later.
I also thought Tanya's case of comparing fractions was interesting. When she got to the point of comparing 2/3 and 5/8 she really struggled because she did not have a consistent whole. Her bar graph seems to be useful and it makes sense to her, however, the problem lies in the fact that she is comparing fractions of two different wholes. I think this is a common misconception that students have. She saw the 3 in the denominator of 2/3 and realized that 3 is smaller than 8 (5/8) so she made the 5/8 bar larger than the 2/3 bar. This seems to make sense, but to help scaffold, I think it would be useful to give Tanya two concrete objects such as strips of paper that are the same size (so the whole is equivalent) and have her divide them into 2/3 and 5/8 so she can see that the whole has to be the same. You could even compare her drawing to a real life situation. You could ask her how big her whole is and eventually scaffold to the idea that her representation is similar to comparing 5/8 of a large sub to 2/3 of a small sub.
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Anne,
ReplyDeleteThe interesting thing about reading the books we have in TE 402 this semester is that we were able to study different math problems and the students mathematical thinking while working through the problem. I feel that this has given us a lot of insight about the different types of answers children may find when solving different problems. I am also very interested in how children cut off part of the whole to make equal parts. Due to the fact that I have an understanding that the whole matters this seems odd to me. But now knowing that this concept does not come easy to children I have a greater understanding of why a child I am working with may choose to change the whole.
Overall, I have many questions about the different things you posted about as well. I feel that working with children and reading case studies is the best way to gain more knowledge about children’s thinking. It is always important to remember that what we are teaching children took people hundreds of years to figure out. It does not necessarily come natural to people and we have to be patient and understand why a student may think of a problem in a particular way.
I hope this was helpful!
Kathryn