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I'm not a math teacher but am actively involved in teaching my children mathematics (elementary age). I learned the traditional "long" approach to multiplication, but the school systems now emphasize (or perhaps require) the use of the partial products method. And, I was also recently introduced to the term "grid" method.

I would like to know if these other newer methods of teaching multiplication have good evidence to show improved learning outcomes? (I consider them "newer" only in the sense that they were not part of my elementary education which was a looooong time ago.)

Edit: I would be very interested to know whether there is a "systematic review" or, preferably, a "meta-analysis" on this topic. From an empirical standpoint, is there a consensus to support one method or another on any particular learning outcome?

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When you say "grid multiplication", do you mean lattice multiplication or grid method multiplication? I'd be curious to see how the former stacks up against the usual technique; the latter looks really uninteresting, sort of random, and more primitive compared to it. –  Ryan Reich Jun 19 at 16:09
    
@RyanReich To be frank, I don't know the difference between "lattice" and "grid method." I am wondering if the new methods are based on philosophy and theory, as opposed to actual evidence. If evidence supports one method over another, then I am motivated to learn and teach that method. Otherwise I am inclined to teach what I know and what seems to have worked for me. –  Brian P Jun 19 at 16:13
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Evidence supports that kids are able to learn and execute the "long" approach to multiplication without understanding multiplication. So, disagreement on approach may be rooted in what counts as an outcome. In other words, making kids better calculators vs. helping them to become better mathematical thinkers. What this means for your question is that it might help to clarify what outcomes you're interested in. –  JPBurke Jun 19 at 16:25
    
@JPBurke I have heard that these new methods promote better understanding. But, I haven't found the research that supports these outcomes. Math education is outside of my area, so I may not be using the right search words. Seeing the evidence as opposed to hearing the theory and philosophy is my interest. Thanks for the response (+1). –  Brian P Jun 19 at 16:29
    
@BrianP - you need to look at the links in Ryan's comment. I suspect you mean lattice, given the choice. –  JoeTaxpayer Jun 19 at 19:10

4 Answers 4

up vote 10 down vote accepted

You will probably not find precisely what you're looking for if you want to compare one algorithm to another for effectiveness. The reason is, the problem that was identified in the 70's (Erlwanger, 1973) was not about the unsuitability of the algorithm itself. It could better be thought of as a problem of focusing on algorithm.

Erlwanger (1973) demonstrated that mastery of the content did not imply understanding. So, researchers began to consider whether teaching for understanding was possible.

Is there evidence that a shift to teaching for understanding "worked?" Yes, in the sense that students were able to achieve computational fluency not at the expense of conceptual learning and problem solving. An example of a study that demonstrated this is this longitudinal study of helping teachers teach for understanding (Fennema et al., 1996).

But rather than write a lit review on teaching for understanding, I'll point you to Schoenfeld's chapter (2008) in Powerful Learning. He does an excellent job explaining mathematics learning as a sense-making endeavor, and includes quite a bit of a bibliography to the research that counts as evidence for the approach. It's a great resource, I think, for parents who want to understand some things that have changed about mathematics education as a result of real research.

Another great resource is Jo Boaler's How To Learn Math online EdX course. She mentions her book (What's math got to do with it?) as well. But I believe the EdX course is free. She's a researcher and educator who has done influential work.

It's not what you were looking for, but here is an article that is a snippet of the Powerful Learning book, by some of the authors. It doesn't have references, but it does discuss why learning for understanding is a thing.

Reorganizing into points:

  1. Newer approaches to mathematics education that value conceptual understanding are often going to involve more sense-making on the part of the students (ideally). So algorithms take a back seat, meaning that there isn't a competition between algorithms.
  2. However, there is a reluctance to teach an algorithm that backgrounds sense-making because a student may not be motivated to work toward conceptual understanding if they know they can just use the standard algorithm. Analogy: algorithms are technology, like calculators. If you knew the calculator (or algorithm) was going to give you the answer, and it was ever-present, you might feel the urge skip the conceptual work and just use the calculator.
  3. There is evidence that a lack of conceptual understanding could hide in procedural fluency. This was discovered in the 70's.
  4. Use of alternative algorithms may help students not because the alternative algorithm is better. Other reasons are paramount.For instance, an approach may allow teachers to see student mathematical reasoning and intervene (formative assessment). It may allow flexibility in approach, changing an activity from procedural practice to actual problem solving based on number sense and other mathematical knowledge.

I hope this is helpful.


Cited:

Erlwanger, S. H. (1973). Benny’s conception of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behavior, 1(2), 7–26.

Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A Longitudinal Study of Learning to Use Children’s Thinking in Mathematics Instruction. Journal for Research in Mathematics Education, 27(4), 403–434.

Schoenfeld, A. H. (2008). Mathematics for understanding. In L. Darling-Hammond, B. Barron, P. D. Pearson, A. H. Schoenfeld, E. K. Stage, T. D. Zimmerman, … J. L. Tilson, Powerful Learning: What We Know About Teaching for Understanding (1st ed., pp. 113–150). San Francisco, CA: Jossey-Bass.

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I tutored a homeschooled boy who loves math. He would do multiplication in his head, and didn't know how to write it down. I was not tutoring in any conventional way, and never tried to get him to write it down. But when we were working through Introduction to Number Theory, by Matthew Crawford (an AOPS book), he had a bit of trouble doing multi-digit base 8 multiplication in his head.

After watching him think for almost a minute one day, I asked him to describe what steps he was taking in his head. It turned out he was going left to right (which works better for most people than right to left, when doing things in one's head), and using the top number's digits times the bottom number - everything was backwards from the conventions. It would have worked, but there was too much. I wrote it down for him a few times, and then he knew how to write what was in his head.

His method and the conventional method are equally good algorithms. My opinion, if you're not trying to all do it the same in a classroom, is that the method the students likes best is the best method for them.

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I also do left to right in my head. –  Ryan Reich Jun 20 at 13:28

JP's answer brings up some disturbing issues. "a student may not be motivated to work toward conceptual understanding if they know they can just use the standard algorithm." 100% true, and tough to navigate around. Disturbing, in my opinion, because many countries are pushing the use of standardized testing which in my opinion, ignores that conceptual understanding JP highlights.

In my opinion, the times table should be memorized up to 12 X 12. Then the 'old' method of lining up the numbers under each other and multiplying by hand. The lattice method, shown below, is an interesting way to multiply by hand, but avoid some of the confusion of carrying.

enter image description here

We are multiplying 327 by 586 here, and it should be obvious that this is just doing single digit math, with the lattice helping to avoid any carry, as each unit place has two cells. My daughter was taught this method in 7th grade (age 12/13) and dismissed it as unnecessary, the old way worked fine. I was tutoring a high school (age 14/15) freshman who was having multiplication issues, and pulled out this method. His eyes lit up. Regretting my inability to use a Vulcan Mind Meld, I came to accept that this alternate method could, and should be used as a secondary way to explain what we'd call simple multiplying of numbers beyond 12 X 12. My time with the students is always constrained, I suspect the lattice helped clarify the 'carry.' At least that's what I hoped. To JP's point, the level of analysis would be tough to produce. My anecdotal evidence suggests that it's a worthwhile tool to offer students who are struggling with multiplication, but not useful as a replacement for the standard method we all learned.

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I like how you contextualized the lattice method. Personally, I think for some students it may make a great puzzle as well. If used appropriately "why does this work?" could be quite an exercise. I'm not sure where, but I think the question is compelling. –  JPBurke Jun 20 at 13:50

Whatever is the "best" way of doing it (or teaching it) is mostly irrelevant for your case. Just go with the way your children are taught at school. Teaching them another method will just confuse them, get them into trouble when they use the "better" method in class/exams, and create useless arguments with their classmates over the matter.

[If you ask me, I'll get them a cheap calculator and teach them what to do with the operations, instead of drilling them in algorithms they'll hardly ever use in earnest in later life. But that's just me, not your neighborly school board.]

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His kids may not go to school. –  Sue VanHattum Jun 20 at 4:22
    
@SueVanHattum right. Not legally possible here, so I didn't even think about the possibility –  vonbrand Jun 20 at 7:27

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