In response to: Knowing and Teaching Elementary Mathematics by Liping Ma
The two chapters of the book analyze the various methods used in elementary mathematics. The author looks very deep in the different methods used in the U.S. and China that are very different when compared to each other. The subtraction and multiplication problems that the author uses are identical but are explained very differently using various methods from both countries. The chapters overview teachers’ understanding of certain math concepts and their knowledge of regrouping and decomposing numbers.
I liked the information in these chapters; it was very, very interesting for me. I thought I would end up just skimming through the 50 pages; I ended up reading precisely every word to word. I felt that it did challenge my beliefs. I used to think that teaching kids was all about telling them how to do it and teach them certain tricks so they would remember. For example, when multiplying; I thought a teacher must just explain that zeros are a necessity and end the conversation at that. However, the conversations brought up about explaining the zeros and knowing why the zeros are there made me realize that it is mandatory to explain to students exactly when they are doing and why. It makes it ore interesting for them to work on the math problems when they know exactly what they are doing.
Many methods of subtracting and multiplication are used in the two countries. One of the methods discussed is regrouping. With regrouping, you can regroup numbers in many different ways. 143 can be regrouped into three: 100, 40, and 3. Another method is composing numbers; it is a nice method has great wording: composing, decomposing. It makes sense that you can compose or decompose a number. When a number in the ones place isn’t big enough, you can decompose a number in the tens place to get 10 ones. This method, compared to the traditional method is much more focused on mathematics instead of a “neighbor” that one borrows from. For multiplication, there is the distributive law method, which explains why the zeros should be in the multiplication problem. There is also the Place Value method where teacher explain that when a student is solving for the tens place, it makes sense to put the answers also in the tens place.
The traditional methods are also discussed in the chapters. It looks like the traditional method for subtraction isn’t very helpful since it uses the word “borrow.” It isn’t very mathematical to teach children to “borrow” from numbers when they really aren’t going to give it back. It is said that using the composing and regrouping method is much more focused on mathematics. For multiplication, the distributive method was discussed and seemed very helpful to explain to students why the zeros must be included in the problem. However, the students responded that they like the traditional method for multiplication, especially for higher numbers.
The information given in the chapters was similar to what we learn in class. We learned that the idea of “borrowing” from another number isn’t very mathematical at all. We also talked about the regrouping of numbers in order to subtract or even multiply and about the distributive property that I have never heard of. It was great extra information as well.
The use of manipulative section was very interesting to me, because manipulatives and number charts don’t really teach children about regrouping and decomposing numbers when counting and subtracting with them. However, some teachers came up with ideas like using dimes and pennies or putting a rubber band around 10 sticks as a 1 ten. Their ideas seemed very reasonable.
I enjoyed reading the different math methods; a lot of it actually surprised me. It was very surprising to read about the composition method since it was my very first time reading about it. I enjoyed reading about the multiplication problem (students not putting the zero for a place holder) and how teachers respond to it and tried to think of ways to better explain it to students. The information was great.
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