"What methods do we use to help children become math scholars?"
This article is a response to a question parents ask us frequently: "what methods do you use to help children become math scholars?"
First, no matter what topic we teach in math we always use the 'from simple to complex' and/or the "scaffolding" teaching methods (The former is an ancient philosophy and the latter is by psychologist N. Chomsky).
To clearly illustrate our point we are going to use a problem from the international math olympiad, Kangaroo, levels 1 & 2: Terence gives Donald 8 of his toys. Donald gives Terence 6 of his toys. Now each of them has 13 toys. How many toys did Terence have before the exchange?
Before we even begin to discuss the process of solution it is important to teach children how to separate the information in the problem into
"needed" and "not needed". In other words, here, in this particular problem, children should be asked to pay attention to which information they would not use to find the solution. In fact, they will not use the information that, “after the exchange the amounts were equal.”
Furthermore, this point leads to another very important concept: teaching children how to think mathematically by scaffolding, or in layers!
In the first layer all we do is help children understand that though both boys end up having the same amount of toys after the exchange this was just extra information.
In the upper levels, and in the older grades, math students will need to learn the following concepts when dealing with problem solving:
a) insufficient information
b) identity solution
c) multiple solutions
d) no solution
At this higher level of learning the students do not only need to examine the information in the problem, but also reason through it in order to reach the conclusion that the information in a particular problem sometimes either would not be sufficient to obtain one correct solution, or there would be more than one solution. During my 20 years of teaching math to middle and high school students I have observed that a great number of them had struggled with these concepts. Therefore, it is very important to teach this skill going from "simple to complex" and as early as possible.
We are now going to discuss the process of solution to this problem, and we want to stress, once again, that we use the Singapore math System to help children develop mental images to decipher verbal sentences in a math problem. We begin this process by visualizing a unit of 13 and labeling it 'Terence'. Then, we alter this unit using 'backward strategy' or work in the opposite direction to what is described in the problem: Terence now has 13 which should be added to 8 after which 6 should be subtracted from the sum to represent the amount that Terence had before the exchange, 13+8-6=15.
Once again, it is important to stress to children that we focused our solution on Terence only, because that was all that was asked and that we did not need to use the information: "both boys had an equal amount after the exchange.” In addition, it is very important to check the answer in the reverse order of the above operation and compare to the answer, 15, and check if we get back to 13.
In conclusion, we must first, use simple examples and use a 'scaffolding' method when we teach complex concepts. In addition, we teach children to visualize the information in mental images to reason with it and to use various math strategies, as for instance in this discussion we mentioned a 'working backward' strategy. Lastly, we teach children to always check their solutions. Thus, we conclude that if a child acquires all of the above strategies he or she can eventually become a math scholar.