# Place value and the Lattice Algorithm (or A Simple Task, Years in the Making)

I’m going to assume you know the lattice algorithm for multidigit multiplication. If you do not, and if you would like a primer, here is one.

This post isn’t really about the lattice algorithm, but it’s the context for what I’m really trying to say, which is this: It is worth the time to craft classroom tasks carefully.

I have used the lattice algorithm for years with my future elementary teachers. We learn the steps in class, they go off and practice it. And then they write about it, using the ideas of the course to analyze the algorithm.

After a number of semesters of this, I became tired of reading in their work some variant of the following claim,

The lattice algorithm is very good for teaching place value because you have to pay attention to the places as you work with it.

I could not disagree with this claim more strongly. As I work the lattice, I am going digit-by-digit. I am absolutely NOT thinking about the values of those digits. And I suspect most children are not either. This makes it an efficient algorithm.

Last semester I decided to put that claim to the test. If these future teachers thought the lattice algorithm exposes important ideas of place value, then what task could I give them to demonstrate that it does not?

Well, they have been analyzing the algorithm; they have written papers about it. So if it teaches place value, they should be able to ace any place value task involving the lattice, right?

So here’s the task: Perform the lattice algorithm to multiply 7,343 by 1,568. When you are done, use a marker to highlight each and every tens digit in the lattice.

No follow-up or clarification questions allowed. If the premise is that the lattice helps us to learn place value, then we should know enough about place value to make a commitment to the meaning of a tens digit.

Can you guess which of the answers below is the more popular in my classroom?

When both are presented, a really useful discussion of the algorithm and its position with respect to place value ensues. And that discussion helps to explain the really clever “slide trick” for placing the decimal point (as seen about 2:30 into this video).

But back to my point. I can tell my students that the lattice doesn’t bring place value understanding along for free. Or I can show them. Showing requires carefully crafted tasks. But I find it’s worth the time.

When I have the choice between telling and showing, I nearly always choose to show.

Which is why I’m always running behind on content coverage.

I made my peace with that years ago.

### 10 responses to “Place value and the Lattice Algorithm (or A Simple Task, Years in the Making)”

1. Laura

As an 8th grade teacher, I hate the lattice algorithm. The students who use this NEVER, and I mean NEVER, understand what place value means. I assume because the elementary school teachers who taught them the method didn’t implicitly teach the concept. It makes it much harder when dealing with percentages and such.

2. I have a lot to say on the subject of lattice multiplication, but probably don’t have time to get to it until at least Sunday. I will say that it’s appalling that future teachers think that the purpose of algorithms is to get you to think: just the opposite is the case – they hide the thinking that went into creating the algorithm and minimize thinking necessary on the part of the user. And therein, of course, lies the danger: mindless teachers teaching mindless algorithms mindlessly to kids as if THAT was what MATHEMATICS was about. Is it any wonder this country produces very few non-mathematicians who have the slightest clue what mathematics actually is?

I think a task worth giving is to ask teachers to explain the similarities and differences among several algorithms for the same task (in this case, multiplication). In curricular materials where the lattice method is one of those offered, it’s rare in my (somewhat limited) experience to find a teacher who understands either how lattice multiplication actually works or how it relates to the other algorithms offered in the text. The result is often that the teacher teaches these algorithms as unrelated to one another, much as I’ve seen teachers teach fractions, decimals, and percents as if they were at best marginally related. Incredible, yet true.

I do like what you chose to do to challenge the beliefs of your students and make them confront their baseless assumptions/claims about place-value. Have you had any students state the an advantage of lattice is that it does a better job of organizing the numerals and thus cutting down on computational mistakes due to misalignment? While I’ve never had an elementary student put it that way, I know that many prefer this method and claim it’s “easier,” but explaining WHY it’s easier from their perspective is not so easy for them to articulate. In any event, if someone can’t multiply two single-digit numbers, the lattice approach isn’t any more magical than the algorithm most people my age were taught. And I expect that you know some of the history behind lattice (or “gelosia”) multiplication, how far back it goes, why it fell into disuse, etc. It pisses me off royally to read some of the incredible lies and insinuations about this approach that have popped up thanks to the Math Wars. Nothing like having a sense of history no longer than your own life.

3. @Laura: you have data or just impressions for your claim? Because place value isn’t something that gets taught just a multiplication algorithms are introduced. It is introduced with addition and subtraction. So your implication that somehow lack of place value understanding is grounded in use of the lattice method simply doesn’t make sense. Do you think the folks who used it in the pre-printing press era mostly didn’t understand place value? Abacus users? Knotted-string calculators?

The fault is not in our algorithms, but in how many of our elementary school teachers are unfit to teach math, and what a poor job we seem to do at both recruiting stronger candidates to teach at the K-5 level (not that I think we don’t need to do better at the 6-12 level, too), and ensuring that they either get a real understanding of elementary mathematics or refrain from teaching the subject to poor innocent children (I’m starting to sound like some of the Math Wars folks I dislike, but I can’t help it). One solution would be to allow for more specialization at the elementary level, at least starting in 2nd grade, though I have no problem with it starting earlier where possible. That makes me a heretic among the elementary ed experts, for the most part, but I can live with that.

4. Laura

It is just my impression. The truth of the matter is the symptom is probably the cause: students who use lattice use it because they don’t understand place value, not they don’t understand place value because they use lattice.
The major issue I have with lattice is the time consumption by students who get easily frustrated. They spend time (these are 13 year olds) drawing boxes and diagonals to multiply things like 23 x 7.
I’m not writing about all of my students, but enough of my 125 students do this that I cannot find the time to meet with them individually, or even in small groups, to help them all transition to long multiplication. I am only able to ask for volunteers who would like to learn to do this faster. Those students often don’t know what I’m talking about when I say ones, tens, etc… Again, no research, just anecdotal.

5. Thanks, Christopher. (I wish I were teaching future teachers!)

6. @Laura: appreciate your honesty. You might want to check the history of this method: it was predominant in Europe until the rise of the printing press. It fell into disfavor for the simple reason that it was difficult to print given the relatively primitive methods of typesetting then available. Other than that, it might well still be how most people in the West do multiplication.

But you have an opportunity to do for your students what I would bet was NOT done for them by their elementary school teachers. You can show HOW this method works and how it relates to other methods, including “long multiplication,” what I call expanded notation (probably the wrong term, but I’ll live with that, too) where for example a two digit x two digit problem uses the distributive property to make 37 x 46 into (30 + 7)(40 +6), etc., and even the area model on 10 squares/in. graph paper, replete with coloring in hundreds, tens, and ones squares and rectangles with different colors (a beautiful DEMO that isn’t intended to be used more than a couple of times to give a visual underpinning and reinforcement for how multi-digit multiplication works). If you take the time to do this, you’ll give your kids a much firmer grounding in a host of things, including place-value, and you’ll give them a legitimate basis to make INFORMED choices about which algorithm or algorithms they want to use.

I agree that for compactness, students would do well to learn and use “long multiplication,” but it does have its own pitfalls. It is easy to mis-align columns, for one thing, something I’m sure we’ve all done when trying to do our figuring quickly or in cramped space. But regardless, when students really understand multiplication, they’ll do what makes sense to them: either they’ll get a lot more efficient with their algorithm of choice or they will pick another. But they need to make that choice from an informed perspective, and they need to make it on their own.

7. This is really nice. I strongly agree with the benefits of showing (rather than telling). It takes ingenuity and careful attention to detail to create anything that “shows” well. You’ve done it very well here.

8. Keita

I have been a fan of Lattice Mulitiplication since I was shown by a student of mine that came from Mexico. That was 12 years ago. I am convinced that the reason for the ‘standard’ multiplication is not just because of typesetting. I have done many races and tests using 3rd grade to 11 grade and even special ed. Lattice Multiplication is ALWAYS more accurate and about 90% of the time faster. I will have to make videos of this and post them. I really thought this conversation was finished. Lattice Multiplication is faster and more accurate. I have been teaching now 13 years and proponent of this “new” old algorithm. In fact, my daughter who is 21 still uses this since I showed her when she was having trouble multiplication in the 6th grade. Evidence and lots of evidence is coming out supporting this and things are going to change. The old way of confusing people is dying. Understanding is coming. This is only one example.