Monday, June 17, 2013

Wanna race?

It depends. I do not agree to challenges without more information. To do so would be, in my opinion, irresponsible. Sort of like using blanket statement to make some point.

For me, it seems to come down to phronesis: what's available, and what's worth doing?

Take, for example, the car to the right. It was built to get you from point A to point B over land or water. It seems to be an effective means of travel, and it could probably beat most regular modes of transportation in a race that involved a combination of land and water. But there are times when it would not be the best choice in a race. What if the race was all on land (or all water)? The best method of transportation depends on the conditions.

The ability to choose from a variety of methods came to mind today as I read The Faulty Logic of the 'Math Wars' (here). In the fourth paragraph, Crary and Wilson write:
The most efficient algorithm for addition, for instance, involves stacking numbers to be added with their place values aligned, successively adding single digits beginning with the ones place column, and “carrying” any extra place values leftward.
If I understand their assertion correctly, then I would like to challenge them to a race. First one to compute the following sum wins. They can use their efficient algorithm and I'll use some other method.




999,999 + 41,562


  1. I literally laughed out loud when I read your question. I guess that means that I get your point! Hah!

  2. Brilliant repartee! I read that same article this morning and cringed all the way through it, though for other reasons.

    Here's another race you could challenge them to and win. They can even use a calculator if they want:

    411^2 - 410^2.

  3. Yep. I'm curious about the background of the authors. The title sounds like they're going to rise above the two sides, but that wasn't the case at all.

  4. No! No! No! You're missing their brilliant point! Your strategy to solve your problem would be progressive! You'd be "promoting" 999,999 to 1,000,000 (unjustly, I'm sure) while "deprecating" 41,562 to 41,561.

    Although, now that I think about it, maybe the authors would see this problem as an example of the Matthew effect in action, and 999,999 SHOULD be promoted while 41,562 should be deprecated. In which case they could still argue that they're using the "standard algorithm" after promotion/deprecation.

  5. I do love you!

    Short and sweet. And very kind.

    I'm reading that garbage and thinking all sorts of swear words and getting my blood pressure up. And you so succinctly and gently make the point.

    (From my brain and my overworked heart)


  6. I'm curious as to what the authors of that NYT piece think is the "most efficient" means of doing different sorts of arithmetic, particularly multiplication. Because if they're thinking that the multiplication algorithm we all learned in school is the most efficient, they'd be wrong. The Karatsuba algorithm isn't even the fastest multiplication algorithm out there and it still beats long multiplication handily in terms of computational complexity.( For that matter, the "Russian peasant's algorithm" for multiplication dates back to the ancient Egyptians and is faster than schoolbook long multiplication.

    The assumption in the piece is that the "standard algorithms" are standard because they're the "best" (most elegant, most efficient, etc.). But this is not necessarily so. Sometimes the standards are standard because "we've always done it this way".

  7. I once a young teacher-to-be in class who grew up in Europe, and during our class discussion comparing our "standard algorithm" for subtraction with the equal additions method, she was quick to point out that the algorithm she grew up with was equal additions.

  8. I wanted to know more about the authors, so I looked. Wilson is the mathematician. Here's a readable piece on his position. Crary is the philosopher. She's a feminist. I would have expected her to be sensible. (Just shows that agreeing in one arena doesn't make us agree in another.) Not sure why she paired up with him on this. Maybe she's another parent who wasn't happy with her kid's math class.

  9. Wilson: "The standard algorithms for adding, subtracting, multiplying, and dividing whole numbers are the only rich, powerful, beautiful theorems you can teach elementary school kids, and to deny kids these theorems is to leave kids unprepared."

    How does one come to this conclusion?

  10. The Crary & Wilson quote, about the most efficient algorithm for addition, has no evidential basis so far as I can tell- I think they just made it up.

    On the other hand, most algorithms have a best, worst, and average number of steps (not to mention resources). The Euclidean algorithm for the GCD of two integers, for example, does worst when the inputs are Fibonacci numbers. But that worst case is not typical. Equally, some sorting algorithms do worst when the input data is already sorted. So there has to be a decision as to how to assess the efficiency of an algorithm.

    For that reason I do not like David's example - it's plucked out of the air to be cute, but it does not represent what is typical.

    I would like to see references - if they exist - to the algorithmic efficiency of various means of addition. But then, I would like to see elementary school kids learning how to code these algorithms, not practice poorly what a machine can do well.

    (And, BTW, what elementary teacher can prove the correctness of the "standard" algorithms- or any algorithm for that matter? )

    1. Gary, what I saw as David's point (while it 'was' a cute example) was that we should be teaching students to evaluate each problem presented to them, so they start by saying 'what is the best approach in THIS situation'? If we only teach one mechanism, then they don't do that. Obviously in this case it was a straightforward alternative, but getting students to consider what the best algorithm for any problem is, as a step in the solving process, would be a very helpful skill.

  11. Of course a bigger problem is that they set up a 'strawman' (strawperson?) when they claim that kids are not being pointed to a good algorithm. (I hope that's a false argument.)

    What I think should happen: kids explore both meanings and procedures, when they seem clear on the meanings and have seen how the procedure work to do the job, they begin to use one or two algorithms that work best for humans. Each kid eventually chooses one, but can follow classmates' work done in another.

    Efficiency is not the best standard for that, in my opinion. Efficiency and transparency together might be. For example, we use division so rarely, it is helpful for it to be completely transparent. Many of my college students did not remember the division procedure. I wanted to be sure they understood it as we worked on understanding polynomial division, both long form and synthetic. So I used a modified division algorithm that stacks complete numbers on top. It's much easier to see why this one works (than the standard algorithm I learned), and it worked fine for them.

    (The comment thread on that article is pretty lame, but maybe someone should point out this lovely discussion. Or we should get the New York Times to hire one of us to write a rebuttal. (Hard to do well, because it's hard to know what happens in classrooms all around the country.) Or, better than a rebuttal would be a description of a good classroom that uses the methods the authors are criticizing.)

  12. I think it's way past the time when we should move from ox and plow method of arithmetic, and let algorithms and machines do the work. I really don't care if a kid knows 8x9=72, or can divide 97 by 19. Machines do these things way better and have done for a long time. We don't plow fields with an ox and a plow any more, we don't write by chipping marks in stone, and I think it's time we stopped doing silly calculations that a machine does better. It is not mathematics, and it is one of the compelling reasons why kids hate what they think is mathematics.

    It's assertions such as those of Crary and Wilson that convince me there is very little mathematics being taught in school.

    BTW, I agree with Sue VanHattum that efficiency is not always paramount and Crary and Wilson use it as a red herring (a red herring and a straw person !!). If we want to do very large calculations and many of them then we would be sort of silly not to care about efficiency. But if we're carrying out just a few calculations who cares about efficiency?

  13. >I really don't care if a kid knows 8x9=72, or can divide 97 by 19.

    Gary, I think we disagree. I don't want to torture kids with arithmetic. But I do think we need facility with basic multiplication in order to understand factoring, prime numbers, and other structural features of the natural numbers. This is a vital piece of mathematics in my opinion. I don't think we need to be able to use the standard algorithm for the division problem you posed, but we should have facility in thinking about ways to solve it without technology.

    1. It's an interesting question Sue. Suppose a group of kids were not fluent with multiplication and decided to investigate prime numbers - how many there are? how often they occur? and then investigate different factorizations of numbers. Would they need to become familiar with multiplication tables? I claim not, for the simple reason that almost none of us knows our 13, 14, 15, 16, 17 times tables, let alone our 97 or 523 times tables, yet we certainly would seem to need to to investigate anything but a few simple primes and a few simple facotrizations.

      As to "we should have facility in thinking about ways to solve it without technology." - I agree entirely. Thinking is the key skill. When thinking leads to devising algorithms it's called computer science. I object to the Crary & Wilson overemphasis on the mindless practice of carrying out algorithms that machines do better.

  14. "Over 70 years ago in Manchester, New Hampshire, children learnt no formal arithmetic until grade 6 (about age 11). The program's creator, Superintendent Louis Benezet, describes it like this:

    In the fall of 1929 I made up my mind to try the experiment of abandoning all formal instruction in arithmetic below the seventh grade and concentrating on teaching the children to read, to reason, and to recite - my new Three R's. And by reciting I did not mean giving back, verbatim, the words of the teacher or of the textbook. I meant speaking the English language. I picked out five rooms - three third grades, one combining the third and fourth grades, and one fifth grade."

    1. I think Benezet expected them to learn standard algorithms in sixth grade, though I have no evidence for that belief. I was actually thinking of him when I said we shouldn't torture kids with this. ;^)

  15. Here's what I think......the over use of manual calculations through application of algorithms is a waste of time; it has virtually no value in terms of furthering one's knowledge of mathematics (use technology to do the "number crunching").
    Having said that, understanding WHY various algorithms work DOES have value and they could be explored to that end. I've seen cases where students "engage" in drill work for weeks on related rates and various rules found in introductory calculus; huge waste of time as far as I'm concerned. This time would be far better spent by developing theory behind WHY those rules work and then designing solutions to various problem scenarios that would call upon those rules. Bring Wolfram Alpham in from there.
    Thank you.

  16. Thanks to everyone for your comments. I appreciate that this has been a respectful and deep discussion complete with links to references. I have stayed out of the conversation thus far because, quite honestly, I am more interested in your thinking than mine. But with the flow of comments ebbing, I thought I would add my two cents. This is personal/anecdotal, so be forewarned.

    When I taught middle school, I was assigned several remedial math classes. Many of the students in these classes struggled with the traditional algorithms. Although they had been taught these algorithms for years, something just didn't click. I decided that it would be more efficient to focus on estimation and number sense so that these students might use the available technology in meaningful ways.

    As I teach preservice elementary teachers, these remedial math students remain foremost in my mind. When these future teachers complain that a particular computational approach is not the way they learned it and therefore not as easy/efficient as the standard algorithm, I wonder what will become of their students who struggle with what Crary and Wilson call the "privileged method." (This phrase, more than anything else in the article, prompted me to write this post.)

    You see, this is real to me. I know my choice of numbers was not representative of what is typical, but the students I taught were also not representative of what is typical. I came across a quote from Grover Cleveland recently: "We are confronted by a condition, not a theory." In theory, the standard algorithms might be the most efficient, but I want students and teachers to make decisions based on the conditions.

    I look forward to your thoughts on this.

  17. I hope that the focus on number sense might facilitate having the algorithms "click." When I read about theoretical students exploring prime numbers and facility with times tables not providing the entire solution , it just makes me cringe a bit. My adult students are not exploring prime numbers. They're trying to pass remedial math classes. They have extremely varying degrees of basic number sense and fact fluency, and it's sometimes fascinating to witness the effect that has on their response to our instruction and course requirements.
    Many are ingrained in procedural learning; the woman this morning who was completely frustrated by factoring found it all a heck of a lot easier when she focused on what factors *were* -- those things that you multiply to get that trinomial. She'd had parentheses and plus signs all tangled up in each other... oh, but boy-oh, knowing those times tables well rather helps with factoring... but knowing what factors *are* is also awfully important.

  18. SG, I think you're referring to my comment, but I totally agree with you - we work with students where they are.

    Dave, I'm curious what changes you saw in those students' understanding. You reminded me of a post I saw long ago (or maybe a real-life conversation?). This guy taught some struggling middle-school students how to work in base 5 so the times tables would be easier. They actually got it, and loved it.

    1. That was a long time ago, Sue, but my memory of it is that we were ALL happier to be building new learning rather than focusing on their areas of struggle. There's actually some interesting research on this (especially at the college level) that I will try to find and share with you. In fact, in some cases focusing on the things students struggle to learn/apply actually exacerbates the problem.

  19. This was such fun to read. I wrote a post about this piece too ( and used an almost identical example - using the standard algorithm to multiply 999 x 123. I totally agree that it's an issue of choice - there are of course times when the standard algorithm will be the most efficient. But I love to see kids assess the conditions, as you say, before they select their method. That moment of choice is such a rich one, metacognitively.