The questions and suggestions below pertain to the way multiplication is taught to kids. Examples pertain to multiplication problems involving two two-digit numbers.
Question One: Do you teach kids to multiply right to left or left to right? The standard multiplication technique when one has pencil and paper is right to left.
When you are sitting in a class room or meeting or taking a test this problem can be solved much quicker in your head by noting that 10 x 37 + 7 x 37 = 629.
The steps are actually the same for the two methods but the second process clarifies that multiplying 37 by 17 is the same as multiplying 37 by 10 and 37 by 7 and adding the two products. The second method is essentially an example of the distributive property.
An experiment: One teacher should teach only the first method right to left. The second teacher should present both methods.
Teachers should give a test of the following form
15 x 12
43 x 24
65 x 18
95 x 21
25 x 17
The teachers should review the material and give a second test. This time the teacher who initially only taught the first method might also teach the second method. Our assessment of the two methods should also consider improvements between the first test and the second one.
It might be useful to time these tests because i suspect that the use of the distributive property when multiplying could increase speed as well as accuracy. Speed is important on tests, in informal situations where paper, pencil and calculators are unavailable and when analyzing complex multi-step problem.
Question Two: How do you teach kids to multiply a fraction and a number?
Move the decimal point two places to the left and get 9.0.
10% of 60 is 6.0.
1% of 60 is 0.6.
5% of 60 is 5 x 0.6 which is 3.0.
15% of 60 is 6.0+3.0 =9.0.
Thankfully, we get the same answer both ways but the thought process differs. The first method usually requires pencil and paper. The second may be better in a test situation or in a meeting where neither pencil or paper nor a calculator are readily available.
A second experiment: Again, it would be useful to compare student performance for classes that are only taught method one to classes that are taught both methods. After the first test teach both classes the second method and give a second test. Measure both absolute performance and improvement for the two groups. Consider applying a time constraint because both accuracy and speed are important.