I really am in the Christmas spirit! David finished putting up the tree yesterday. I'm not doing any heavy duty baking because the lad is very close to his goal weight for finalizing his acceptance to his college of choice, The Citadel. I've been heavily involved in our church Christmas programs like Operation Christmas Child, the Christmas party for the local nursing home we adopted, special music, and our musical "Where Are You, Christmas?" (which you can't find anywhere because our choir director put the music together and wrote her own script). Tonight, we're going to a Christmas bon fire. Last night, I had a blast creating a video from the still shots from our program and I dare you to watch it and not get into the Christmas spirit! Yesterday, I attended a neighbor's informal Christmas party and what was on her mind? Finding a math tutor for her grandson over the holidays!
While some children memorize their times tables very quickly, others struggle! My son continued to use his "cheat sheet" (a table of multiplication facts) until eighth grade. Simply drilling him with flashcards like I did as a child wasn't effective. I figured, with enough use, he would internalize them, and he did.
I should have employed more strategies in helping him nail them down earlier. In his article on the bogus dichotomy in mathematics education, Hung-Hsi Wu writes, "It is the fluency in executing a basic skill that is essential for further progress in the course of one’s mathematics education. The automaticity in putting a skill to use frees up mental energy to focus on the more rigorous demands of a complicated problem." Making a times table chart available enabled him to make progress, but at a slower pace. While many students take algebra in eighth grade, we waited until ninth for David. All wasn't lost. He made it all the way through precalculus (and earned a high B in that class) and did well enough on the SATs to get accepted by his Plan A and Plan B colleges.
Helping Students Learn Multiplication Combinations has a great list of strategies:
- Using repeated addition (3 x 5 = 3+3+3+3+3)
- Skip-counting by multiples (2, 4, 6, 8, 10, . . .)
- Patterns found in multiples (11, 22, 33, 44, 55, . . .)
- Doubling (double your 3s to get your 6s)
- Using partial products (6 x 9 = 5x9 + 1x9)
- Using five-times and ten-times (9 x 12 = 10x12 - 12)
- Patterns found in 9s (For two digit numbers, add the digits and you must get 9. Also, look for the symmetry in 09, 18, 27, 36, 45, 54, 63, 72, 81, 90.)
- Properties of mathematics: commutative (8 x 2 = 2 x 8), distributive/partial products (8 x 6 = 8x5 + 8x1), identity (any number times one is itself)
- Using known facts to find unknown facts (I know my 2s and 5s can use them to get 7s)
Last year, I profiled Greg Tang's The Best of Times last year. Pamela and I read it to help her view calculation in more flexible ways. You can do 69 x 12 in your head if you realize that 69 x 12 = 70x12 - 1x12 = 840 - 12 = 828. This book covers many of the strategies listed above as it goes through basic multiplication facts. You can do some mental math by kicking up his ideas a notch.
- Double is repeated adding: you add a number to itself: 56 x 2 = 56 + 56 = 112.
- Triple is also repeated adding but it might be easier to double it and add the original number to the double. 38 x 3 = 38x2 + 38 = 76 + 38 = 114.
- Quadruple is repeated adding, of course, but you may want to double the double: 49 x 4 = 49x2 x 2 = 98 x 2 = 196.
- Struggling with skip counting for 5s? Take half of the 10s: 92 x 5 = 92x10 / 2 = 920/2 = 460.
- Can't do 6s? Triple the double (or double the triple): 66x2 x 3 = 132 x 3 = 396.
- 7s not heaven? Add the 5s and 2s: 84 x 7 = 84 x 5 + 84 x 2 = 420 + 168 = 588.
- Bet you can guess 8s . . . Double the double of the double: 78x2 x 2 x 2 = 156x2 x 2 = 312 x 2 = 624.
- What about 9s? No, it's not triple the triple, but you can do that if you like. Do 10s and subtract the number: 99 x 9 = 99x10 - 99x1 = 990 - 99 = 891.
Of course, you don't want to sock a child with all of these at once. If I could have a do-over for David and his times tables, I think I would have done a multiplication intervention. First, I would randomly zip through flashcards of all multiplication facts through 12 to figure out which ones he has down cold. Then, I would create a chart of all of his knowns and analyze his known to pick the strategies that had the most bang for the buck. For example, suppose he consistently knew facts like 4 x 8 = 32 but not 8 x 4 or 6 x 12 = 72 but not 12 x 6. That would tell me he need to learn that multiplication is commutative (a x b = b x a). Of course, I would not put it that way to him. Rather, I would set up paired problems for him to figure out by making arrays of pennies until he saw the pattern and convinced himself this property of multiplication was true. As Wu put it, "Children always respont to reason when it is carefully explained to them," which sounds a lot like Charlotte Mason ("Arithmetic becomes an elementary mathematical training only in so far as the reason why of every process is clear to the child. 2+2=4, is a self-evident fact, admitting of little demonstration; but 4x7=28 may be proved" (Page 255)).
Once David mastered one strategy, I would study his known chart and find another that would help him fill in several holes in one swoop and go on to demonstrate each strategy, one at a time. As Daniel Willingham (Page 18) puts it, "New concepts must build upon something that students already know." After we covered enough to build up his confidence, I would quiz him with flashcards again and record any new knowns into the chart. What is beautiful about these strategies is they illustrate what Wu calls "heeding the call of the indispensable mathematical principle to always break down a complicated problem into simple components" which sounds a lot like Mason, "Nothing can be more delightful than the careful analysis of numbers and the beautiful graduation of the work, 'only one difficulty at a time being presented to the mind'" (Page 262).
Teaching children in such a thoughtful manner requires insight and creativity, something one might not associate with math instruction. The problem is many of us were poorly taught as were many mathematics teachers today! Wu writes that rote learning takes place "when the teacher does not possess a deep enough understanding of the underlying mathematics to explain it well. The problem of rote learning then lies with inadequate professional development and not with the algorithm." And guess who wrote something similar a century ago, "Mathematics depend upon the teacher rather than upon the text-book and few subjects are worse taught; chiefly because teachers have seldom time to give the inspiring ideas" (Page 233).
All is not lost. The internet is full of inspiring ideas and meaningful ways to demonstrate math. While much of online mathematics is fancified rote learning, you can find gems that give the kind of strategies to help our children reason more clearly and with greater confidence.