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After a conversation with a "well respected mathematician who was heavily involved with K-12 mathematics education," W. Stephen Wilson re-analyzed the results of the arithmetic test he gave his Calculus III students at Johns Hopkins in 2007. The unnamed mathematician had told Wilson that fewer than 1% of college students would be unable to work a multiplication problem by hand, so Wilson took a look:
He was a little off on his estimate.For passers-by, here's a quick run-down of Wilson's original observations:
In the fall of 2007 I gave a 10 question arithmetic test to my 229 Calculus III (multi-variable calculus) students on the first day of class. Among other things, this means they already had credit for a full year of Calculus. The vast majority of these students were freshmen and ... and the average math SAT score was about 740.
[snip]
Seven of [the problems involving multiplication] were each missed by 8% or more of my students and 69 students, or 30%, missed more than 1 problem.
These are high achieving, highly motivated students (remember the 740 average SAT math score). These are disturbing numbers for them, but I suspect the numbers are much much worse among college freshmen with an average math SAT of 582, and, from Table 145 of Digest of Education Statistics 2009 we see that the average SAT score for the intended college major of engineering is 582 in 2008-2009.
Anyway, there is no real purpose to this paper except as a resource for me. It does suggest, very strongly, to me, that we have lost the pro-arithmetic war. This is a revelation to me and it calls into question what I will do next year with my big service course. I now feel compelled to assume that [my students] are chronically accident prone or they really are arithmetically handicapped. It isn’t clear that there is a difference. The question remains, how can I teach serious college level mathematics to students who are ill-prepared?
As another experiment, Wilson gave a short test of basic math skills at the start of his Calculus III class in 2007. The results predicted how students later fared on the final exam. Those who could use pencil and paper to do basic multiplication and long division at the beginning of the semester scored better on the final Calc III material. His most startling finding was that 33 out of 236 advanced students didn’t even know how to begin a long division problem.
Back to Basics for the "Division Clueless"
DECEMBER 6, 2010 | BY LISA WATTS
9 comments:
I tried doing something similar once at the beginning of the semester - I suspected that there would be a positive correlation between the ability to do basic arithmetic and a student's final grade in the course (general education math, for non-majors). I even stated that the quiz wouldn't count.
Oh, the whining about no calculators, though. You would have thought I had asked them to dig a ditch with a spoon.
"Anyway, there is no real purpose to this paper except as a resource for me. It does suggest, very strongly, to me, that we have lost the pro-arithmetic war."
I see a big difference between the fraction problems and the arithmetic problems without calculators.
There seems to be an argument lately about some sort of correlation (or not) between basic arithmetic and more abstract math, like algebra or calculus. This is not defined well if you mix up these two problems on the test.
"9. Divide 51.072 by 0.56. (answer 91.2) 33% of my students missed this, i.e. 76."
"10. Add 2/3 and 1/2, and divide the result by 5/12. (answer 14/5) 5% of my students missed this, i.e. 11."
The ability to do fractions is critical to upper level math courses, but not the arithmetic ones by hand. I find it annoying that my son can't do long division by hand as fast as I can, but I'm not worried. I would, however, be extremely worried if he couldn't do the fraction problems.
"These are high achieving, highly motivated students (remember the 740 average SAT math score)."
What was their average SAT II scores?
"These are very good students, and if they know fractions but are just really careless, that is just as much of a problem as not knowing fractions."
They can't be careless if they average 740 on the SAT. I would analyze the tests to see exactly why they got the problems wrong. How much importance did they place on the test "on the first day of class"?
There is not enough information here to come to any real conclusion. There are some educators, however, who will see this as a sign that mastery of skills is not tied so directly to doing well in college math.
My view is that you can get away with missing problem 9, but not problem 10. He should redo the test and allow calculators for the arithmetic problems, but not for the fraction problems. He should warn them on the first day of class and then test them on the second, telling them that it will count as a grade.
Tests like this are so vague that they allow people to confirm whatever they want to see.
If the test didn't count, perhaps some students didn't think doing tedious arithmetic problems to be worth their time and effort. Perhaps the higher achieving students are also the ones who were also more compliant or motivated to do well no matter the task.
There are many examples of mathematicians who, for whatever reason, had difficulty with rote computation as children. (As an aside, this is one reason I think using the Quantitative subtest of the CogAT as the sole determiner of *mathematical* ability (and therefore be identified as "gifted" in mathematics) is a crock. It's much more a test of processing speed, working memory, and motivation to manipulate numbers.)
"They can't be careless if they average 740 on the SAT."
The SAT is multiple-guess, though. If this is free response, I can see how the kids could do much worse than on a test where you have a built-in partial check of not finding your answer.
-Mark Roulo
"The SAT is multiple-guess,..."
The SAT has a free response section in math.
SteveH says:
"The ability to do fractions is critical to upper level math courses, but not the arithmetic ones by hand."
A quote from the blog post says:
"Those who could use pencil and paper to do basic multiplication and long division at the beginning of the semester scored better on the final Calc III material."
It would seem that arithmetic is hugely important in higher level math.
"It would seem that arithmetic is hugely important in higher level math."
It could be that these were just the better students. Were any of these good Calc III students bad at fractions? What is it about the actual standard algorithms that causes this success?
I'm all for ensuring mastery of the standard arithmetic algorithms. It's an indicator of whether students can handle detailed algorithms, and it's an indicator that they are not put off by hard work that doesn't seem to have any real payoff, especially when you have calculators. Long division has other benefits in terms of estimation and synthetic division in algebra, but I would never use Wilson's test to determine any sort of causal link between arithmetic and higher level math.
I always thought that the Math Wars were too focused on issues over the standard arithmetic algorithms. You can see it in the CCSS standards. Look for the word "fluency" and it really only appears when they talk about the standardized algorithms. They don't talk about fluency when it comes to fractions or percents or many other things. They don't talk about ensuring mastery on a grade-by-grade basis.
In terms of pedagogy, the existence (or lack thereof) of a focus on standardized algorithms is usually a give-away for the philosophy or expectations of a math curriculum. It could be that being good in arithmetic means that the students went throught a good math program.
The problem is that with CCSS and their requirements for fluency in the standardized algorithms, they still won't get the job done. In the future, you might find that students who good at the standarized algorithms are bad at higher-level math. That's why all of my eggs are not in the arithmetic basket.
You can add standardized algorithms to the K-6 curriculum, but it won't change what's in the hearts of educators. That is full inclusion, child-centered learning, and low expectations for mastery. We will keep Everyday Math and its trust the spiral, but it will have some units on the standardized algorithms.
How many of the students who got a problem wrong made a careless error rather than an error indicating poor understanding of the underlying concepts? I was much more concerned when my DD inverted the wrong fraction in a fractional division problem than when she forgets to add a carried number. The latter type of careless mistake I simply have her correct, while the fractional division mistake required me to go over the topic with her.
This paper came about from a conversation I had with a well respected
mathematician who was heavily involved with K-12 mathematics education.
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