Source: Math is Fun |

In the last couple weeks she has developed her own strategy for finding answers to equations like these. She does it by adding it all up on her fingers. We did the roll-the-dice grouping game a bunch of times over the course of a month (January, I think?) and she seemed to catch on to what it was all about. But now, it seems, counting on her fingers works just fine except that when she gets to numbers over six or seven it can take a long, long time to find the answer. She's not noticeably perturbed by the effort, and from my point of view it's awesome that she understands that she's counting nine, eight times to get the answer. That's the main point of what multiplication is at this stage, right? .

Although, there she was, still trying to figure out the answer. So, I offered, "How about I go get the multiplication table?" She agreed but when I brought it back she said, "But I don't know how to use it!"

Visually tracking columns and rows is difficult, even for an adult sometimes, so I hit upon two strategies, which I showed her:

The first is to put your pencil in the first column, in this case the 0, and leave the tip just above the nine row, then move the pencil column by column until it's in the right one for the equation, in this case the eight. The tip points directly at your answer.

The other strategy is to just find the nine in the shaded area on the left and, using your finger, point to each box in that row and say "zero, one, two, three..." until you get to eight and there's your answer. Easy, right? But the girl had a different reaction:

"THAT'S CHEATING! If I do it this way, how am I going to really understand it????!!!"

I must admit I was a tiny bit impressed with this statement but I was also more than a bit flummoxed about how to respond. I mean, what's there to really understand? You learn what multiplication means (which I think she's got), you learn your facts and then you use them when you need them. I finally said:

"Well, addition is the most important thing to know how to do in your head. When you add in your head you are learning how numbers combine and recombine to make other numbers. It's an important skill to have and that's why they don't have a facts chart for it. It's not quite the same for multiplication -- what people usually do with the times table is memorize it. There are all sorts of fun number patterns to find in this chart..."

At this point she was still absolutely convinced that using the chart to find any answer constituted some kind of unlawful activity. Trying a different approach, I said, "Okay, I have an idea. If you really want to understand how and why multiplication works, you can skip count. You already know how to skip count zero, ones, twos, fives and tens. So all you need to do is learn how to do that with threes, fours, sixes, sevens, eights and nines. It's like adding..."

That was the best I could do in the heat of the moment. My daughter is prone to ginormous reactions and sometimes it's hard for me to think clearly under duress. But, the question still remains -- is it 'cheating' to use the multiplication chart? I should add that she seems to have no interest in memorizing the facts either.

After writing this all out I'm starting to think that that maybe all this (her finger counting strategy, eschewing the chart, etc.) is just her way of mastering the content. All the same, I am wondering what kind of understanding she wants. Maybe she's looking for familiarity? Maybe at some point she'll finally figure out on her own that memorizing the times tables facts is actually pretty helpful? Am I missing anything here?

Sigh. If you have any thoughts, including helpfully pointing out any errors you find in my explanations and reasoning, I'd love to hear them...

No thoughts, no problems with anything you said. I just want to say that I love your daughter's persistence.

ReplyDeleteI think it's great that you've inspired such a huge interest in multiplication. Looking at that chart is a bit overwhelming for me. 100 numbers to memorize? When I look at the charts I'm usually looking for number patterns. Have you already colored in patterns on the chart? If she's not into the chart maybe she could write the numbers 3 to 30 by 3's and then go back to the chart and see if her numbers and the chart match?

ReplyDeleteJulie

highhillhomeschool.blogspot.com

A case of cans of worms opening here, you know. :)

ReplyDeleteMultiplication is not repeated addition, though for some subsets of the real numbers, the right answer to a multiplication problem can be obtained through repeated addition (if you have the patience to go through, say, 49 addends of 163).

Obviously, from an efficiency perspective, it's not sensible to compute that way, given reasonable alternatives (which include, of course, technology, which includes, of course, multiplication tables). The more operations, the more the likelihood of error and consumption of time (however, I always like to point out that if I screw a problem up doing it quickly, that may be less efficient and more costly than doing it right the first time with a slower algorithm. There are lots of things to be traded off).

However, my comment isn't going to cut a lot of ice with your daughter at this point, as long as she is only looking to get "the answer" to a (comparatively) simple and decontextualized computation like the ones you described.

The problem comes in when we ask: What, if anything, comprises the differences between additive and multiplicative reasoning?

If there really is no fundamental difference worth noting, then the MIRA (Multiplication IS Repeated Addition) lovers are right and people like Keith Devlin (Stanford professor of mathematics, NPR's "The Math Guy," and author of many excellent books on mathematics and mathematics education for the educated public (his recent one on video games and the future of mathematics education is currently in the group of books I'm reading) and elementary mathematics education researchers Terezinha Nunes and Peter Bryant are spitting into the wind. My money happens to be on Devlin, Nunes, Bryant, and others, rather than on those in the MIRA camp.

One place your daughter might care to look is at area. Why do the units change when we compute the area of a rectangle (or any other figure in the plane or surface in 3-space), but not when we add lengths in 2- or 3-space? Is that significant? Or is it telling us something about the respective natures of addition and multiplication?

ReplyDeleteWhen we have a group of 5 apples and add 3 more apples to it, we get 8 apples. But if we add 3 oranges to our 5 apples, there's an issue: do we have 8 apple-oranges? Few children, even at six, would likely agree. They MIGHT agree that we have 8 "pieces of fruit," but the unit just changed from "apples" and "oranges" to "pieces of fruit" to accommodate an otherwise meaningless result.

On the other hand, take 5 bags with 4 "apples per bag" (note that this latter mixed unit is different from merely "4 apples"). When we add 3 more bags with 4 "apples per bag," no problem - we get 8 bags with 4 apples per bag. But if we multiply 5 bags times 4 apples/bag, the result, happily, is 20 apples. Where did the bags go? Well, we actually had 20 bags * apples/bags or with some rewriting, 20 apples * (bags/bags) = 20 apples.

Putting 'unit analysis' into the picture (or equation) suggests (to some of us, anyway) that, as Keith Devlin opined several years ago, "Multiplication ain't no repeated addition."

Again, that is NOT a claim that there is no connection between the two sorts of operations or thinking, or that you can't get the same numerical answer through repeated addition (whatever, exactly, that is: it's not a well-defined operation, or so it seems when you push it hard enough).

Devlin prefers the notion of "scaling" to repeated addition as a way to think about multiplication. At least one mathematics educator, Gary Ernest Davis , with whom I am in vehement disagreement, claims that the scaling concept would reinforce the incorrect assumption kids emerge from early grade arithmetic with: that multiplying always makes things larger. I questioned his 'evidence' (he offers none) for this assertion. Given that the vast majority of kids learn multiplication IS repeated addition AND come out of the early grades with the misconception that a x b must be bigger than a and b (even if you point out to them that

(a x 1) = a and (a x 0) = 0, and that if a > 1, a x -1 = -a which is < 0 (and less than a, of course), and take that whole set of exceptions out of the conversation, most kids (and adults) will still bite on the notion that if a and b are greater than 0 and neither is equal to 1, that a * b > a and also a * b > b. Not until kids have to come to grips with proper fraction multiplication are they forced to face the fact that multiplication can make things smaller and division can make things larger (and in our curriculum, fractions are generally taught before integers (a middle school topic, I believe) and the arithmetic of both is highly problematic for many kids regardless).

Please note that neither Devlin nor I suggest that we should be asking young kids to grapple with the terminology (e.g., scalars and scaling), but I'm reasonably confident that kids that young have some real world experience with things being scaled and are not necessarily unable to grasp what is meant by scaling up and scaling down. Making that more mathematically precise is tricky, however, in no small part due to natural language problems. What is meant by twice as large? Twice as small? Half as big? 50% larger? The list goes on, and it's routine to encounter vast confusion about these notions among adults, taken advantage of all the time by advertisers and salespeople (though they, too, may merely be completely ignorant).

As I said to start: a case load (perhaps a tractor-trailer load) of cans of worms here. Do a quick search on multiplication repeated addition and have fun.

Hi Julie, That all sounds great but she's not having any of it. I tried earlier today to see if she wanted to fill in an empty chart and look some number patterns that way, but no go. I think she's got her approach (using fingers) and she's sticking to it until she's ready to move on. I have been specifically bringing up other things related to numeracy (understanding place value, challenging her skills with mental arithmetic, etc.) and she's been fine with that. She discovered multiplication all by herself, so I guess this is just her thing and I'll leave her to it until she tells me she's ready.

ReplyDeleteThanks for your comment -- it's helpful to have this kind of conversation. I understand the situation better now after writing it out in this comment and also from some other comments from friends on FB. :)

(And, Sue, as I've always said, she's nothing if not persistent. ;-)

Hey Mike -- I dipped into that can of worms a few months ago when the kid first brought up multiplication and I recognize a lot of what you've said as being in that can from my reading and my conversations on the Living Math forum.

ReplyDeleteThe things I really appreciate about your comments are the points about scaling and multiplying fractions. It's great to have this bigger picture to keep in view for down the road when she's older. For now I can imagine the kid would probably love playing around with concepts like that if I could figure out a good way to do it. I will definitely dip back into your comments here when I figure it out. As you know, I am learning all this along with her -- I had a feeling there was more to all of this, and you've given me a really good picture of what it is!

The important thing here is that my daughter does seem to understand the units concept. She's always just played around with writing random equations -- little tests she gives herself. She's been doing it since kindergarten with addition and subtraction and I see it as just her particular way of 'getting to know' numbers. The fact that she can repeatedly add 9 on her fingers all the way up to 72 indicates that she does indeed understand the grouping concept and the understanding of units (even though they appear to be out of context.) Also, a month or so ago we were at the grocery store and she saw that there was a case of wine with a sign that said $10/bottle. On her own volition she wondered how much the whole case cost and proceeded to skip count by 10's. I think that's probably a good example of her understanding -- skip counting is just a tool to help her answer questions she has. So while the MIRA folks and Devlin, et al battle it out, here in the other trenches I'll keep an eye on whether my kid is understanding what to do with the numbers she has around her. ;-) Really -- I'm so pleased you took the time to lay it all out for me here. I can't tell you how helpful it is!!!

Always glad if what I've offered is helpful. I should add on the "cheating" issue that I went through that with my son in early elementary grades until I could convince him that it's one thing to use outside aids to do your work/thinking for you (not that I'm opposed to that route in general: I use technology - paper and pencil included with electronic gizmos and the Internet - when it suits me to do so, and I feel not the slightest guilt), and quite another to use calculators and answer pages to check your results (with the caveat that they, too, can and do err, so there's no excuse for NOT estimating and doing mental arithmetic whenever possible).

ReplyDeleteFinally, if the research done by Nunes & Barnes and some others is significant (of course, some folks dismiss it and there may be grounds for their objections that aren't strictly ideological, but I've not had time to check out some things I bumped into recently along those lines), then we do a real disservice to kids if we insist on a rigid MIRA approach. As a lot of folks get absolutely nuts when Devlin is mentioned, I tend to think they doth protest too much and refuse to consider that they may have erred as educators (of their own children or other people's).

I was fascinated with a learned debate in comments. What I told daughter is that a multiplication table is simply "a shortcut". You get the results faster that way. She is still adding though, just like your daughter, for the facts that she didn't memorize, but she gets more and more of them memorized as she comes along. Some books like 2x2=Boo were pretty good helping her visualize multiplication.

ReplyDeleteI wouldn't be in too much of a hurry to get to memorizing the multiplication table. There are a lot of things you can learn along the way, like:

ReplyDeleteTo find 4x a number, you can double it and then double it again. (associative law of multiplication)

To find 6x a number, you can do 5x the number and then add the number on (after you have memorized the 5x table, which is one of the easier ones to learn)

To find 9x a number, you can do 10x the number and then subtract that number (9 6's is one fewer 6 than 10 6's). (distributive law)

These aren't things you _have_ to know to learn multiplication, but if you do learn them, you'll find a lot of places later on where those tricks turn out to be useful. Which doesn't mean that multiplication fact learning is the only place where you learn those tricks--it's just one place where you can.

Thanks Lsquared! It's interesting, but in the intervening month since I wrote this post both my daughter and I have come to terms with multiplication. She's realized that it's a lot about memorization (which is easier than counting on her fingers, lol!) and I've realized that there are more important thing to pay attention to about number multiples. Similar to what you've described, we've been looking at doubles and halves of numbers as it comes up (which is a lot, actually). The concept of 'doubles and halves' seems to be a simple yet effective way to deepen her/our understanding of relationships between numbers.

ReplyDeleteIn the end I guess the reason I was pushing the 'times tables' was because that's all I really knew about multiplication, and that's the way I was taught. I think I've made peace with my past and moved on now, lol!

Thanks again for your thoughts.