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.)


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


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.