Should we teach the standard algorithms for arithmetic?

Just posted this comment on this article lamenting the loss of the standard algorithms in Mathematics classrooms.

Should we teach the standard algorithms for arithmetic? Absolutely, but they shouldn’t be the only algorithms kids learn.

Why exactly is the ability to add, subtract, divide and multiply large numbers so critical? It seems clear to me that these are useful skills for numbers we will encounter in our day to day lives, and that it is useful to know that algorithms exist to work with larger numbers, but your other connections seem tenuous to me at best.

You’ve argued that without practice using algorithms, students will not be able to remember them to use them later, and this I agree with. It is a basic tenet of education that spaced repetition helps students remember how to use knowledge.

The question is, what type of knowledge is critical for students to remember? Does knowing how to multiple 39835 by 2338383 or any other arbitrarily large number assist the typical person in their life? Does it even contribute to a greater understanding of advanced mathematics? Has the number of people completing advanced mathematics degrees dropped? Statistics Canada data from 2007 suggests that it has dropped very slightly (see http://www.statcan.gc.ca/pub/81-004-x/2009005/article/11050-eng.htm) but not by an alarming amount.

Regarding your achievements as a PHD in mathematics, don’t forget, the plural of anecdote is not data. You can’t generalize from your one experience to what is useful for all of society.

Understanding how to use the algorithm seems sensible to me, but I think it is even more important that people understand algorithms (emphasis on the plural) which is probably lacking in the current curriculum as it is constructed.

One problem is that all across our society, at many different age groups, we have a lack of people using any advanced mathematical thinking to solve problems. If you look at how people solve problems similar to what they learned in school, but in a different context (see Jean Lave’s work), you find that it is rare for people to use the standard algorithms they learned in life, despite the fact that the standard algorithms are much more efficient than the various algorithms people construct for themselves. This suggests that even though the standard algorithms are more efficient, they may still not be the best algorithms to teach.

It seems to me that if over the course of a lifetime, some knowledge is going to be forgotten, the skill of learning is more important than what specific knowledge is learned.

Update: I’ve had another conversation with the author of the blog post above, and it seems I’ve over-reacted a bit. We have more in common than we disagree about.

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6 Comments

  • David:
    I believe we spend too much time beating on kids to learn how to do computation and not enough with the big concepts. I bet you can find kids that can do long division who can’t tell when to divide or even explain what division is. Every year seems to start with multiplication facts. I ask you, if you owned a business and caught an employee doing computations by hand, what would you do? I would tell them if I caught them doing that again, they could look for another job. I would compliment them for knowing that they had to add, subtract, multiply, or divide. Bounce this off your colleagues and let me know what they think.
    Keep up the good work.
    Doug

  • Deacon k wrote:

    I can see both sides to this. I’m beginning to lean more towards seeing the bigger concepts now though. With the advancements in tech, and availability of calculators I’m glad they have those tools available to use. BUT, I still think people should be able to do the basic operations with minimal problems. That’s what has me pulling out my hair now. Is students know what they need to do but can’t compute the basic facts without a calculator. I shouldn’t have to wait for a student to pullout a calc to multiply 6 times 7. That is what frustrates my students, is that math is more work for them because they dont know their facts.

  • I have taught everything from high school math down to grade 3. I think that it is very important to teach both sides of this mathematical coin. The most frustrating times as a math teacher was when I was teaching 6th and 7th grade and could not teach the bigger concepts because the kids were stuck on the basic facts. They didn’t know them, became frustrated, and gave up before they got to the problem solving thinking. Now I teach grade 3 and feel frustrated by my math consultant and my new math program that both seem to disregard basic math facts and algorythms as something kids don’t need to master because they can use a calculator. If we were to focus more on these in the primary end, I think it would open the door for middle years and high school math teachers to delve deeper into the problem solving and bigger math concepts.

  • Your closing line is clutch:

    It seems to me that if over the course of a lifetime, some knowledge is going to be forgotten, the skill of learning is more important than what specific knowledge is learned.

    As is Doug’s line in his reply:

    I ask you, if you owned a business and caught an employee doing computations by hand, what would you do?

    It is far more important to understand math than to know how to do it unassisted.

  • I wish this conversation you’re having was required reading for all pre-service math teachers. I’ve been struggling with this issue personally and depending on the day, you’ll find me on either side of the argument.
    Ideally, I’d like to start with the big picture, have students develop their own algorithms while I assist them with that process and ensure that they figure out when to use one of their algorithm tools. I’d also like to develop a strong estimation sense within my students to avoid the ‘calculator must be wrong’ scenarios. My worry is that under the time constraints that I’ll be facing as a new teacher, I may end up focusing on the practical applications at the expense of the standard algorithms – thereby losing some of the big picture.

  • Elizabeth Lyon wrote:

    Roy,
    the time that you save by NOT having to “reteach” the same algorithms that the students “learned” last year (and the year before, and the year before that…) will give you time to guide math conversations in your class so that they develop a strong number and estimation sense. Then, once they have developed and understand their own processes, you can show them the standard algorithm. They may find that it’s more efficient or, in many cases, find that their own process makes more sense!

    (ie: our standard algorithm for multi-digit addition completely disables their sense of place value.)

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