CGI Ch. 4-6 Reflection-- Erin

Despite my initial negative feelings of reading about division and multiplication, I actually found these three chapters to be both interesting and helpful. I have found that this book explains math in a way that you have never thought about before. I personally find it confusing as I am reading it, but when I sit back and think about what I read, it actually ends up simplifying the subject for me. Like the previous chapters, the chapter on multiplication and division taught me more about the structure and types of problems than I ever thought possible. One of the more interesting points I found in the chapter was something I don't believe was made to be a big idea. On page 439 the book is discussing skip-counting as a counting strategy. I found it so interesting that children generally skip-count using three. I can see five or two but I was surprised with the number three. I tend to believe that sets of three tend to be more difficult than these children obviously see them. As this chapter went on, we see on page 42 that children sometimes have to figure out what to count by. They explain that trial and error is used in a situation like this. As they explain on the same page, "... it is significantly more difficult to use a counting strategy for a Partitive Division problem than for a Multiplication or Measurement Division problem". I found myself wondering when children tend to grow out of it and have this realization. Does it take showing other techniques and strategies or do they naturally realize that there are better ways to figure it out? I also found myself wondering whether there is a correct order to teach area and array problems. If one should be taught before the other one, how do you build into and relate to the other one?
As the book moved into multidigit number concepts, I found myself being even more amazed by all the different problems and strategies. I like the idea of base 10 because I know that it can simplify problems to the point where children can truly understand. I remember feeling like I could do math so fast when I learned to count by tens. In reality, I'm sure it was no faster than I should have been doing it, but the confidence allowed me to enjoy math more in elementary school. What did surprise me a lot in this chapter, was all the different students abilities. For some students, they understood right away with no manipulatives, others could get it with some direct modeling and others were not able to use the idea of base ten at all. I was also surprised to see that kindergartners were working with base ten. In the kindergarten class I am currently placed in, some students can not even count to a hundred. I wonder though if I were to ask them to count by groups of ten to one hundred, if they would be able to do it. Throughout this chapter they look at so many different modeling strategies for all of addition, subtraction, division and multiplication. I do not understand though if there is a correct order to teach all of this in. For instance, there is a large section on algorithms. The invented algorithms are obviously something that come to the children naturally since it is their way of looking at a number. They go through so many positives that come from using these algorithms but where and when do they start using them? Is there something that must be taught before they can begin to implement them? How early do these kind of rationalizations come to children? Although these chapters brought me many questions, it also once again calmed me. I feel like I am slowly grasping a better idea of what math is and how children see it.