The more time I spend subbing in classrooms, the more I am discovering that I had some pretty amazing elementary math teachers in the 1960s. They taught us the whys and the hows. They also taught us to check the reasonableness of each answer. In fact, I remember learning to look at a problem to determine about what the answer would be before solving it.
I think the focus of math instruction in some classrooms is more on accurate calculation and less about understanding numbers, the procedures, and the processes. Last week, I tried using logic to help a 4th grader understand why his answer was not correct. The problem: 258 - 199. His answer: 159. He knew that 199 was only 1 less than 200 and that 258 - 200 = 58, yet he could not understand how that affected his answer. I've also tried to help students think logically about multiplication facts. If they know that 5 x 8 = 40, they should realize that 3 x 8 can't be 36.
While working on equivalent fractions with middle schoolers, they told me that to find equivalent fractions you do the same thing to the top as you do to the bottom. I asked them why I could do that without changing the value of the fraction. Not one of them knew, so I taught them what I had been taught about equivalent fractions. I could see the lightbulbs go on. They understood that when you multiply both the numerator and denominator of a fraction by the same number, you are really multiplying by 1, so the resulting fraction is equivalent to the original fraction. In my own class, I would take the time to write 4/9 x 4/4 = 16/36, and let them talk about it to see if they would discover for themselves that 4/4 = 1.
After watching Conrad Wolfram's TED talk, I believe he is 100% correct - "Stop teaching calculating, start teaching maths."
Computers can do the calculations. Students need to be able to ask the right question, turn it into a math problem, and check the reasonableness of the computer's answer. If we must use multiple choice assessments, perhaps we could check for understanding by posing questions like this:
I have a garden that is almost 38 feet by just less than 32 feet. About how much planting area do I have?
a) 120 ft b) 1,200 sq ft c) 120 sq ft d) 1,200 ft