The idea that problems can be “related but differ in what is known and what is unknown” is a new idea to me (Carpenter, 34). When reading three different problem types I first saw them as the same “problem”. In all the problems, no matter what the unknown was, I was able to differentiate the numbers to create the same problem.
(On page 34 of CGI)
Ex. Megan has 5 bags of cookies. There are 3 cookies in each bag. How many cookies does Megan have all together.
5 x 3 = ?
Ex. Megan has 15 cookies. She puts 3 books in each bag. How many bags can she fill?
3 x ? = 15
Ex. Megan has 15 cookies. She put the cookies into 5 bags with the same number of cookies in each bag. How many cookies are in each bag?
5 x ? = 15
These problems are all similar to me, but to a child first learning mathematics they have trouble differentiating the different types of problems. Therefore, it is through this book that I learned that teachers need to pay particular attention to the type of word problem they are giving to their students.
Another aspect of CGI that I found interesting is that “there are god reasons for introducing multiplication and division word problems early in the mathematics curriculum.” I had always thought of multiplication and division as being something that was taught at the upper elementary grades, but in Children’s Mathematics: Cognitively Guided Instruction they state “With experience, many kindergarten children can solve simple multiplication and division problems by using counters to model the groups described in the problems. By first and second grade, many children use a variety of strategies to solve multiplication and division word problems” (44). This came as a shock to me, but through thinking about younger children being able to do multiplication and division I have realized that the different strategies available to children (and that children known) have an impact on their ability to solve the problems.
Even with the knowledge that young children can do multiplication and division problems, I question how much this is done in elementary schools. In my field I have seen the kindergarten children doing mental math with addition, but wonder if structurally this is what the children should be doing. The other issue is that in kindergarten the children cannot read quite yet. This causes issues with the children being able to do word problems without the help of an adult. I am wondering what you see during your time in the field and if you feel that the children are as advanced as what is shown in the CGI book?
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I also really like this idea of introducing multiplication and division earlier than we would expect in our teaching. In my field placement, my second grade students are working on addition and subtraction of two digit numbers. They use a combination of manipulatives and algorithms. I am really curious to see how students do when I introduce a multiplication story problem. I have confidence that if they use manipulatives many of them will be able to figure it out. I plan to include some multiplication problems in my student interviews.
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