It originally occurred to me to start this blog when 2 things connected in my mind: one is that I met the everyone-knows-she’s-a-rockstar Kate Nowak at a conference over the summer, along with the also-a-rockstar-and-looking-forward-to-a-time-when-everyone-knows-this Jesse Johnson, and they met each other, which inspired Jesse to start her blog, and got me thinking that might be cool. The other is that I was independently thinking hard about forcing myself to create an annotated bibliography on math education research, to force myself to process what I read. Which seemed like a worthwhile but totally exhausting and daunting task. When it finally occurred to me to do the bibliography *as* a blog, it was too obvious; I had to.

That said, I kind of haven’t really done it yet! I imagined myself forcing myself to read something every week so I could post on it but so far, I’ve only written about stuff I’d already read before I started blogging! (To be fair, I did read most of the original book by Oskar Pfungst for the first time for the Clever Hans post, and every post has caused me to reread at least key parts of whatever I was writing about.) So in a way, I’m beginning from scratch with this one. I hope you stay interested 😉

“Strategy Use and Estimation Ability of College Students”

Deborah Levine, *Journal for Research in Mathematics Education*, 1982, Vol. 13, No. 5, pp. 350-359

“Computational Estimation Strategies of Professional Mathematicians”

Ann Dowker, *Journal for Research in Mathematics Education*, 1992, Vol. 23, No. 1, pp. 45-55

Both of these articles are available from JSTOR but I can’t seem to find them for free on the Internet.

Bottom line: When asked to estimate the answers of semi-difficult multiplication and division problems, mathematicians were very successful, used a wide variety of strategies tailored to the different problems, and often used different strategies when posed the same problem again months later. They made very little use of standard algorithms. When college non-math majors were given the same task, they were much less successful, and were much more likely to use standard algorithms. Also, the ones who were least successful tended to be the ones who adhered most to the use of standard algorithms.

Lesson for educators: unclear. Food for thought, though, for sure. (More at the very end.)

Details:

In 1982, Deborah Levine took 89 non-math majors at a New York City college, gave them 10 multiplication and 10 division problems, and asked them to estimate the answers, and to think aloud as they did so. Here are the problems:

76×89; 93×18; 145×37; 824×26; 187.5×0.06; 482×51.2; 64.6×0.16; 424×0.76; 12.6×11.4; 0.47×0.26;

9,208÷32; 4,645÷18; 7,858÷51; 25,410÷65; 648.9÷22.4; 546÷33.5; 1,292.8÷71.2; 66÷0.86; 943÷0.48; 0.76÷0.89

In addition to the 10 multiplication and 10 division problems, which were created expressly for the study with some care, piloting and refinement, she also gave the students another separate test called the *School and College Ability Test (SCAT) quantitative subtest* which appears to be a standardized test used – possibly produced? – by the Center for Talented Youth. Levine refers to the results of this test as the students’ “quantitative ability.” (Her purpose was to control for this variable so she could isolate relationships between estimation strategies and estimation success that were “independent of quantitative ability.” She didn’t find any.) She didn’t provide any details about this second test so I have no idea what it actually measures. Consequently I have put the phrase “quantitative ability” in scare quotes throughout. In any case, results on this test were strongly correlated with success on the estimation task. I wish she had skipped this whole bit and just analyzed the estimation data.

She then rated the students’ answers to the estimation task by accuracy and categorized the strategies they used. Then she asked:

1) Are some strategies more commonly used than others?

2) Is there a relationship between the students’ “quantitative ability” and the type of strategies they used?

3) Is there a relationship between the students’ “quantitative ability” and the number of strategies they used?

She found that yes, yes, and yes. She had 8 strategy categories which she called “fractions,” “exponents,” “rounding both numbers,” “rounding 1 number,” “powers of 10,” “known numbers,” “incomplete partial products/quotients,” and “proceeding algorithmically.” She found that two categories – “proceeding algorthically” and “rounding both numbers” – accounted for 61% of all responses. “Proceeding algorithmically” alone accounted for 34%. On any given task, the students who used “proceeding algorithmically” strategies had the poorest “quantitative ability” scores, especially when compared with students who used “fractions” strategies. Also, students with lower “quantitative ability” scores used fewer strategies overall.

Levine also asked:

4) Is there variation in the success of different estimation strategies that is not accounted for by variation in “quantitative ability?”

And found that no, not especially.

In 1992 Ann Dowker gave the exact same estimation task to 44 pure mathematicians (ranging from 3 graduate students on the verge of their PhDs to 7 members of prestigious professional organizations like the Royal Society). She also rated their responses by accuracy and categorized them by strategy. (Her categorization, also into 8 categories, was based on Levine’s but slightly different, reflecting the presence of strategies used frequently by the mathematicians but not by the college students.) She gave 18 of them the exact same task again six to nine months later. She found that:

1) The mathematicians did way, way better than Levine’s college kids. (No shocker.) More interestingly:

2) They displayed a striking variability in their strategies. There was no problem for which the mathematicians, taken as a whole, used fewer than 7 distinguishable strategies, spanning 3 of the eight strategy categories; and in nine of the twenty problems, the mathematicians used at least 16 distinguishable strategies, spanning at least 6 of the eight categories. On every single problem, at least one mathematician used a strategy that none of the other mathematicians used (and on all but one problem, at least *four* did).

3) The mathematicians who were given the task a second time did not especially do the problems the same way they had the first time.

4) In stark contrast with Levine’s data, the mathematicians almost never used an algorithmic approach. (Specifically, they used such an approach 4%, as opposed to Levine’s 34%, of the time.) They used “fractions” approaches 40% of the time, and “rounding both numbers,” and “known numbers,” 15% of the time each.

5) The mathematicians seemed to be guided by aesthetic considerations while solving the problems.

#5 is illustrated by this awesome anecdote from Dowker’s paper (p. 53). One of the mathematicians was estimating 1292.8÷71.2:

`He said, "Divide by 4; 323.2÷17.8. That's 32x10.1÷(72÷4). [Pause] I don't like not being able to do something with the 323.2!" He then solved the problem successfully by rounding both numbers to 1300÷70 and estimating 18, but still seemed disappointed at not having managed to use the number 323.2`

Thoughts:

I imagine you could be thinking about many things right now. Here’s what I’m thinking about:

1) Multiplication and division are a lot of fun if you’ve got access to a rich set of number relationships to approach them with. My favorites among Levine’s estimating problems are 12.6×11.4, 64.6×0.16, and 4645÷18. (Guess why?) Looking for ways into the problems, chosen around the particular details of the numbers involved, is a fun, creative activity. A goal of elementary work in multiplication and division should be the cultivation of this sense of creativity.

2) Knowing how to execute the standard algorithms is really a paltry shell next to what’s possible. It cannot be regarded as the primary goal of elementary-level work on multiplication and division. In fact, a student’s reliance on the algorithms as their only (or even their primary) method is a sign that something’s wrong. It strikes me as more or less precisely like reliance on a calculator. Note: I am *not* saying the algorithms shouldn’t be taught. I love them. For doing large-number computations they may have been rendered somewhat passe by the calculator, but they are still deeply relevant as *sources of insight about numbers and operations*. For example, the long division algorithm is used to prove that every rational number has a repeating or terminating decimal expansion.

1. I got stuck at the beginning — by concentrating on “estimate” I cut off “calculate” and made myself very slow. 93 x 18 ummm… I decided after a few that calculating, in places, was ok, and 93x2x3x3 takes but a moment.

2. Knowing single digit addition facts is absolutely crucial to all of this. Single digit multiplication facts is a clear number 2. And then knowing the standard algorithms a distant #3, but well ahead of anything else.

Ironic, no, that those who choose to use the standard algorithms least are those who know them and understand them best?

Jonathan

As usual, jd2718, I find myself responding to you at some length – hope you read it all:

1. The two studies seemed to treat the question of whether the subjects

hadto estimate, rather than calculate exactly, differently. Levine reminded subjects who appeared to be calculating exact answers to estimate instead. Dowker seemed to allow subjects to calculate exactly if they felt like it. Like you I would definitely have been mad if they didn’t let me calculate some of them exactly. (In particular, 12.6×11.4, as below.)2. Clearly knowledge of single digit addition and multiplication facts was a key component of success for everyone who performed successfully in both Levine’s and Dowker’s studies. In fact, the mathematicians often made use of a more extensive set of similar facts, such as the powers of 2, and decimal approximations for common fractions.

However I think you’re wrong that “knowing the standard algorithms is a distant #3, but well ahead of anything else,” at least as far as these mental estimation problems go. In fact I think that is a clear finding of the studies. The algorithms were used very infrequently by the people most effective at the task. How can they be “well ahead of anything else” as far as usefulness for the task goes?

Maybe you’re saying that mastery of the algorithms was an important step for the mathematicians, and you and I, to develop whatever other skills were needed to effectively do the estimation problems. The data at hand don’t speak to that thesis one way or the other. But personally I really don’t think so and I think a level-headed reflection on the question would’t lead one to think so. Most of the tricks the mathematicians, and I, and probably you, used in solving the estimation problems were very tailored to the specifics of the numbers in the problems, whereas the standard algorithms proceed in exactly the same way regardless of the numbers. Relatedly, Dowker reports that some mathematicians in her study made comments suggesting that they saw what they were doing as unrelated to standard methods. “Occasionally some of the mathematicians in the study explicitly commented, ‘I’m sure that’s not how we were taught to do things at school!'” (p. 53)

Let me step back from the particular question and comment that I think that the conversation is being hindered here by the rigid dichotomies of the math wars. In the language of the math wars, the importance of algorithms and of single-digit multiplication and addition facts are identified with the same “side” (the traditionalist side). The idea that kids should be allowed to use calculators a lot is on “the other side.” So if you hear someone say something critical of the importance of algorithms, in the logic of the math wars it’s safe to assume that they are also low-key about learning the single-digit facts, and that they probably are into students using calculators a lot.

This logic suppresses important pedagogical differences between knowledge of the algorithms and knowledge of the single-digit facts, and it suppresses important pedagogical similarities between calculators and algorithms. (For example, the parallel I suggested above between the problems for kids with over-reliance on either.)

The relevance to the present conversation is that when I argued against giving too central a place in the elementary curriculum to knowedge of the standard multiplication and division algorithms, you responded by defending the importance of the single-digit facts (along with the algorithms). The importance of the single-digit facts wasn’t at issue. Although technically off topic, the everpresent, distant but persistent background buzz of the math wars made it entirely reasonable for you to bring it up. However, I’d like to be able to have the algorithm conversation on its own merits. This strikes me as an example of how the math wars have made it harder for us all to communicate clearly with each other.

To reengage the actual conversation about algorithms, having gotten this out of the way:

As I said in the post, I love the (standard multiplication and division) algorithms and I think they have a right to an honored place in the elementary curriculum. However, I do not like the role that they seem to have played in the education of middle- and high-schoolers I’ve taught. I’m reading between the lines because I wasn’t in the room when these kids were taught the algorithms, but it seems to me, as a generalization, that they learned the multiplication algorithm fairly early on in the development of their multiplication technique, and learned it as a go-to method. Indeed, they tend to think of it as what it means to multiply so I presume that’s how they were introduced to it. They see the division algorithm the same way, but they learned it later and badly, so they think of themselves as sucking at division.

I believe that this is a bad role for the multiplication (and division) algorithm to occupy. This belief didn’t originate with reading the studies I describe above, but I take them to support it. They are one illustration of the problem. Three more anecdotal illustrations come from the year I taught middle school: student A was asked what was 70 times 60, took out a piece of paper and enacted the entire multiplication algorithm – “0x0 is 0; 0x7 is 0; 6×0 is 0; 6×7 is 42; 00 plus 4200 is 4200” – before answering. Student B, who showed excellent number sense when reasoning about addition, subtraction and percents, and sometimes exhibited creative problem solving when given totally novel tasks, was posed several 2-digit by 2-digit multiplication problems like 98×23 and told to solve them mentally. He couldn’t think of anything to do but to draw them invisibly with his finger on the wall and enact the multiplication algorithm in invisible finger-ink. My final anecdote is that I had a significant number of students who, when given a sequence of multiplication problems like 3×20, 6×20, 30×20, 36×20, 36×21, could reliably get all the answers via the algorithm, but drew a blank when asked how the problems were related or why I might have asked them in this order.

I think that much better justice to the algorithms would be served by treating them as

grand technological breakthroughs in the art of multiplying and dividing, much like calculators, but with the advantage that their inner workings can be exhaustively studied by elementary students. Here’s a way I’d love to see it go:a) You’d learn the multiplication algorithm a bit later than currently, and only after a significant amount of work developing various techniques that are less general and better adapted for mental (as opposed to paper-and-pencil) use. First and foremost, commanding all the 1-digit facts, obviously, and multiplication by powers of 10. Other examples are using associativity i.e. multiplying highly factorable numbers a factor at a time; using distributivity e.g. as 98×23 = 100×23 – 2×23; using proportionality and simple fraction ideas, e.g. multiplying by 5 as half of multiplication by 10, or multiplication by 25 as a quarter of multiplication by 100; etc. Engagement with this broad set of techniques would familiarize students with all the relevant properties of multiplication, along with a large collection of specific number relationships that come in handy – not only the times tables but the low powers of two, whole/half/quarter relationships between common numbers, etc.

b) When the algorithm comes in (say, at the end of 4th grade, or the beginning of 5th), the kids would already be good multipliers of 2 digit by 2 digit numbers with these other specific-to-the-numbers techniques. The algorithm would be discussed as a powerful historical breakthrough that made multiplying large numbers much easier and allowed a single technique to encompass all multiplication problems. I’d like to see teachers actually talking about how the invention of the modern numeral system in India and its further development in the Middle East made this breakthrough possible – maybe even mention historical figures like Al-Khwarizmi, the Persian mathematician from whose name the word “algorithm” is derived. Nobody would be left with the misimpression that the algorithm exhausts what it means to multiply. Anybody presented with a multiplication problem for which the algorithm is an inefficient method (such as 99xanything) would recognize it as such. Everybody would appreciate as a breakthrough the algorithm’s power to operate in the same way on every number.

c) In addition to mastery in executing the algorithm, a central goal in its study would be understanding its inner workings – what properties of multiplication, and of place value, is it making use of? Why do you execute it from right to left rather than the reverse? Why do you line up the numbers as you do? How is it affected by decimals and why? Students’ ability to have these conversations would be supported by their work in (a). Students would not only have the conversations but be accountable for coherent answers to these questions, just as they are accountable for procedural mastery of the algorithm.

All the same goes for division.

Virtues of this context for study of the algorithms, as I see it:

*Consolidate students’ understanding of the operations and place value.

*Support the future use of the algorithms in developing more advanced ideas such as polynomial multiplication and proof that rational numbers have repeating decimal expansions.

*Set the stage down the line for logarithms as the

nextbig historical breakthrough in the art of multiplying.I’m not aware of a curriculum that does this. I’ll be excited if somebody is. As I’ve mentioned before, Catherine Twomey-Fosnot has a new curriculum out and I expect it does an awesome job with (a) but I don’t know about (b) and (c).

and 12.6 x 11.4 I would die estimating, but can calculate effortlessly as (12 + .6)(12 – .6)

.16 is awfully close to 1/6

I don’t know why you like division by 18. Divide by 2, then 3, then 3?

Actually I was enjoying 64.6x.16 as being close to 64x.16 = 1/100th of 64×16=2^6×2^4; but the 1/6 is awesome too.

I liked 4645/18 because I saw multiples of 9: 4545/18 + 100/18 = 505/2 + about 5.5 = about 258.

And yes, the immediate exact answer of 12.6×11.4 as (12+.6)(12-.6) was what I had in mind.

It didn’t occur to me before but a further cuteness for 4645/18 is to take the 9’s thing all the way:

4645/18 = 4545/18 + 100/18 = 4545/18 + 99.99…/18

= 505/2 + 11.11…/2 = 252.5 + 5.55… = 258.055… for an exact answer.

Another really cute one that didn’t occur to me before is that 66/0.86 is totally 66 divided by (6/7), i.e. 66 times 7/6, i.e. 77.

Reading this conversation a year-and-a-half later. (Missing both your voices lately.) I’ve got to get a better way of following comments…