I was particularly drawn to Chapter 10 of the text where the author discussed about the developmental process on how students gain mastery of the basic facts in Mathematics. The three phrases in the process of learning facts are as follows (p. 172):

*Phase 1 - Counting strategies: Using object or verbal counting to arrive at an answer. *(That was exactly what I did when we were asked to figure out the 99th letter in Ban Har's name!)

*Phase 2 - Reasoning strategies: Using known information to logically determine an unknown combination. *(I know that 4 x 12 = 48, so 4 x 13 is 4 more, which is 52. )

*Phase 3 - Mastery: producing answers efficiently (fast and accurately) *(Being able to provide the answer almost immediately. What is 9 x 9? It's 81!)

I am now wondering if the reason I did well in my Secondary school mathematics was most likely because I was pretty good with memorizing and figuring out which formula to use to solve each particular problem. I don't think I really understood the problem itself. One great example would be when Ban Har posted us with the problem *How many halves are there in 3/4? *I stared at the problem for the longest time and thought it was 6. It turned out that I was giving an answer to another question - *How many eighths are there in 3/4?.* It was only after explaination that I realized that the initial problem was simply 3/4 divided by 1/2. If the problem was posted to us as
¾ ÷ ½*, *I knew what to do right away - ¾ x 2 = 1½ . But is that really the Mastery of facts? I was able to solve it not becuase I understood the mathematical content. I was able to solve it because I was 'trained' to do so by my mathematics teacher - Don't ask why. And that was probably why I never did well in fractions. I skipped the entire reasoning part of the mathematical content.

Mathematics is not only about giving the right answers. It is also about the process one has to go through when solving the problem. There is no ONE correct way to solve the problem. In all, for children to be able to gain mastery of the basic facts in mathematics, they need to be given countless of opportunities in using effective reasoning strategies in the classroom.

Take for example 51 - 7. Using the standard subtraction algorithm doesn't require much thinking as long as you know they way to do it. But think about the many reasoning strategies that one can use to solve the problem without the use of the subtraction algorithm! I can count down to 50 and take 7 way (43) and add 1 more to 44. Another child may decide to use the Take from the 10 strategy, 50 is 10 + 41. 10 - 7 is 3. Adding 3 back to 41 would also make 44. I am glad to know that they have decided to move away from teaching children to use the standard algorithm to solve problems. Because like what Ban Har says, that's what something brainless like a calculator can do. Not that algorithm should not be taught at all but it should come in only after children master the basic facts in mathematics. We as teachers, need to recognize the importance of promoting the use of effective reasoning strategies to help children gain mastery in mathematics.