Like chapters 1-3, I found chapters 4-6 to be extremely interesting because of the presentation of diverse types of problems that can be used within a classroom setting. Though I have heard of the terms Multiplication, Measurement Division, and Partitive Division, it has always been difficult for me to adequately distinguish the difference between them. I know realize that when the number of groups are unknown it is a Measurement Division problem, while when the number of items in each group are unknown (or parts in each group) it is a Partitive Division problem. This knowledge is extremely helpful because it allows one to establish which type of problem may be more advanced and which would be more suitable to a child performing at or below grade level content expectations. For example, it is much easier for a child to be given a Partitive Division problem because the child can then establish how many groups there are are and equally distribute the given quantity of items one by one into each group until there are none left. This strategy was used by Ellen on page 37 of our CGI textbook. I also became aware of the fact that some students use separate objects or counters to represent each group. For example if the problem reads, Mr. Franke baked 20 cookies. He gave all the cookies to 4 friends, being careful to give the same number of cookies to each friend. How many cookies did each friend get? A child may first count out 20 objects to represent the cookies but then also count out 4 additional separate objects to represent the 4 friends while diving the 20 cookies. I found this to be extremely interesting but also wondered if this strategy may confuse children who choose to use it. Although the book suggesting it does not, I am still a bit skeptical.
Another concept I found extremely interesting, is that children can compute two and three digit numbers even if they have a limited notion of grouping by tens. I always thought quite the opposite in that children must have a keen understanding that the number 21 is really 2 sets of ten joined with the single digit 1. However, the book states that this assumption is invalid and "as long as children can count, they can solve problems involving two digit numbers even when they have limited notions of grouping by ten." This knowledge is extremely helpful because had I not known this I think that I would have prolonged teaching students how to compute two and three digit numbers in the future if I did not feel that they fully grasped the concept of grouping and regrouping tens. Now that I know it is acceptable to advance regardless of their knowledge of this concept, I will be able to do so in the future.
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I too found these chapters really interesting. As they broke down the multiplication and division problems, I actually found them easier to understand. I never realized there was so much to each problem, but once you dissect it and make it simpler, I think it's easier to understand and solve. I had the exact same feelings on using a representative object for each group. I didn't necessarily see the need for it if a child was making groups, but I figured it must help with some students understanding. I tried to think as a young student and could see them accidentally counting the extra one. When we watched that video in class though, I started to see how it could help. I thought it was interesting that Alex used this method in his two problems. I liked how he built his group onto the representative cubes for the second problem. I thought that was clever because he wouldn't be as tempted to possibly count it. I think for younger children, it might also be the idea of the number of groups. For instance, in the first problem Alex does, it asks for three bags of cookies. This could have been a way for Alex to remember what he was looking for.
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