Yesterday I was teaching a grade one girl and she had some trouble to figure out the answers with questions such as ? + 5 = 11 but when given the question 6 + 5 to her, she could do it with ease. So clearly the reverse thinking presents a bit difficulty to her. She does not have any learning disability and is eager to learn question. In 2-hour lesson she sit in front of me, she never asked for any breaks but continually asked me for more problems to work on. At the same time I was thinking about a boy who was about grade 3 and could get the following answer in his brain and his mom could not figure out how sometimes he got answers.
The question is something like A + B = 10, B + C = 16, C + A = 20, what is A, B, C?
Asked how he got the answer, he told me that because A + B = 10 which is smaller than B + C so he knew C is bigger than A by 6 so he substituted C = A + 6 into C + A =20 to get A which is 7. I do not know if he really figured the answer all by himself or he already had some training before he came to see me but still it is impressive he could do it all in his head and found out the relationship. Some grade 11 students in high school have trouble to work out this kind of problem. His mom told me that he is gifted.
I remember a boy who is 4 years old and could already do additions, subtractions, and multiplications all in his head but had difficulty to "see" what shall be the next move on the chessboard (even made wrong moves). Why? Do our brain process numbers and images (chessboard) differently? A grade 3 girl who came to my class and just could not do multiplication but finally could do it very fluently at grade 6 and I asked her what happened so she told me she decided to memorize at grade 6 and the same story happened to a boy, he told me that he finally "decided" to memorize it at grade 6 etc.
My purpose of recording many of my teaching examples above is to say that there are still a lot that we do not know on how children learn and how to effectively motivate them and get their interest in learning and how we teach them effectively. The most surprising I discovered is we have not taught children math in an effective way and perhaps in the wrong way. I will explain my thinking and observations below.
1. Other than gifted children, the math skills need to be taught to children and also learned by children. The examples of Pascal who could add from 1 to 100 without being taught and the boy cited above who could do simultaneous equations in his head are examples that some children could do math in their head with a method. How they possess these "methods" in their head without being taught is interesting. What happens to those children who were not born with these "methods' in their brain? We can train them but how?
2. Most math worksheets today are not designed in a way to require children to do a lot of thinking and basically are done with repetitions to get proficiency and fluency. Many times, we see word problems require children to do backwards calculations but do we have basics worksheets to encourage backward calculations or reversing thinking?
3. I did an experiment on the first girl I mentioned that I gave her some training on ? + 5 = 11 similar type of questions then eventually she was happy since she could do all kinds of similar problems with different numbers but the interesting experiment is the minute I changed the question to the following circle + triangle = 11, triangle = 5 and what is the value of circle? She got confused but we know all I did was simply replace numbers by symbols and she got into trouble again. The next level will be to replace symbols by variables to go into algebra. From this experiment, I learned that it is very important to train children with symbols at very earlier age so that they are not afraid about symbols. The problem is most worksheets one can buy is not designed this way and does not foster the thinking to go from "figures" or pictures to symbols.
4. The biggest problem is that we are failing to teach how to do reverse thinking and how to solve word problems. It seems strange to children that once they master their basics computation skills they face another hurdle that is they still have to learn another set of skills to do word problems. This is the problem that we as educators have not come up with a transition way to show children on how to go from "computation" to "word problem".
5. Many math teachers are making mistakes by using the methods of explaining the concepts to do calculations as well. For example, the method of using "tiles" to explain how to do integers is fine but it is entire different matter to use the "tile method" to do actual computations. It is too slow and cumbersome and children will lose confidence if to use the tile method every time to do integers calculations. The same problems with times table using skip counts or subtractions using count back, all these are good for explaining concepts but to get the fluency children required to move ahead, then children can not rely on count back every time to do subtractions. I ask one child who is using finger method to do multiplication by 2 hands (he is grade 5), "Are you going to put your pencil down and so you can use your 2 hands to do calculations at grocery counter with you wife and children standing in front of you?" If not then what good is the finger method when it actually allows children to use outside instrument (similar to the idea of using a calculator), not their mental power to help them to develop their brain?
As I observe more of how my students learn and how they react to the problems I present to them, the more strong feeling that I know that the there are problems with our math worksheets, for example, these conventional worksheets do not train children on how to do word problems. The worst is that some math capability some of these gifted are born with but can be trained are not provided in the worksheets for a normal child to learn those skills. I believe that if educators realize these problems then we will have more happy children who at least will not hate math.
At Ho Math and Chess, I am leading a team to continuously develop revolutionary worksheets, which I hope to achieve 3 purposes. The first one is a very realistic one that is it must be helpful to children's school math so the end result is our worksheets must boost their school marks. To achieve this purpose I need to observe where are the bottlenecks of worksheets and their achievement and how we can improve our worksheets so I do "experiment" on my students using our invented worksheets. The second one is these worksheets must be interesting and this is most difficult to achieve with pencil and paper type of worksheets but is easier with computer interactive mode but we still need children to work on pencil and paper to see how they arrive answers so we can help children. With our invention of Symbolic Chess Language, we have successfully integrated math and chess so we are on the right track. The third is I try to develop worksheets, which also improve their brainpower. This has been achieved with our puzzle-like intriguing math computation problems.
More information on Ho Math and Chess teaching method can be found at www.mathandchess.com