>2) Through all of this I am reminded of one of those "eye opening" experiences >I had with my own son a couple of years ago. He had just finished 4th grade >and had a teacher who DEMANDED that all of her students learn ALL of the "basic >facts", including multiplication through the 12s. He had been successful. >That summer we were traveling in Mexico when he saw a road sign indicating that >we were 120 kilometers from our destination. He wanted to know how far that >was in miles. (Wow, I thought, a chance for an authentic application of some >math during the summer!!) So, I told him that if you divide by eight and >multiply by five you can get pretty close. In my mind I am seeing the division >problem and saying "12 divided by 8 is 1 remainder 4, 40 divided by eight is 5, >multiply by 15 by 5, 75 miles". He is still thinking (no paper or calc >around). I asked him to talk to me about his thinking and he says: Well, I >know there are 3 40s in 120 and that 40 divided by 8 is 5, so 5 times 3 is 15 >and then 15 times 5 is 75. That must be the answer. That is mentally much >easier than my method. But, it seems to me that only a child with a facility >with the "basic facts" would look for ones he knows (3x4, 40/5) to accomplish >this problem. I see him do this all the time (now going into 7th grade), but I >rarely see my "factless" students even try. I am sure there are other factors >involved with my students, but without knowing the facts, how do they have a >chance? > >Just some thoughts. > >Rosemary Beck
It is comforting to know that there are students out there who can do such mental calculations. And, certainly, it would have been almost impossible had your son not known his multiplication/division facts. But, nearly every student is forced to learn those facts (often through memorization), while only a handful seem competent at the mental calculations your son displayed. So, while knowing those facts is important, perhaps there is a missing link.
Of course, in my opinion, that missing link is a knowledge of the properties of numbers and their operations. Your son indicated a type of proportional thinking that is far more complex than simply knowing the multiplication tables. He was able to see that rather than dividing 120 by 8, he could adjust 120 by dividing it by 3, divide the result by 8, and re-adjust the result by multiplying by 3. Clearly there is much more going on here than simply knowing the multiplication tables.
Now, *I* happen to remember that 120/8 is 15, so my method would have been the shortest of all. Does this imply that perhaps we should learn our multiplication tables up to 15 so that we can solve such problems? Hardly. Rather, by knowing a few *properties* along with a few *facts*, we can turn what appears to be a difficult computation into a much easier computation (as you pointed out, a proper balance is critical). What if your son hadn't known that 40/8 = 5? I suspect, based on your anecdote, that your son still would have found a way to solve the problem. He just would have used the *properties* to lead him toward facts that he *did* recall. So, in my opinion, his successful mental computation relies much more heavily on an understanding of properties than on a knowledge of those traditional facts.
BTW, on an only slightly related note, here is a fun way to convert kilometers to miles:
1) Write the number of kilometers as a sum of distinct, non-consecutive Fibonacci numbers (there will be only one way to do it).
120 = 89 + 21 + 8 + 2
2) Change each Fibonacci number to the preceding Fibonacci number:
89 -> 55 21 -> 13 8 -> 5 2 -> 1
3) Add the new Fibonacci numbers together:
55 + 13 + 5 + 1 = 74
So, 120 km is about 74 mi.
Actually, this method is usually more accurate than dividing by 8 and multiplying by 5 (even though it is clearly not as easy to do mentally).