What the PISA Results Really Say About Pure and Applied Math

The Hechinger Report asks, “Is it better to teach pure math instead of applied math?”:

In the report, “Equations and Inequalities: Making Mathematics Accessible to All,” published on June 20, 2016, researchers looked at math instruction in 64 countries and regions around the world, and found that the difference between the math scores of 15-year-old students who were the most exposed to pure math tasks and those who were least exposed was the equivalent of almost two years of education.

The people you’d imagine would crow about these findings are, indeed, crowing about them. If I were the sort of person inclined to ignore differences between correlation and causation, I might take from this study that “applied math is bad for children.” A less partisan reading would notice that OECD didn’t attempt to control the pure math group for exposure to applied math. We’d expect students who have had exposure to both to have a better shot at transferring their skills to new problems on PISA. Students who have only learned skills in one concrete context often don’t recognize when new concrete contexts ask for those exact same skills.

If you wanted to conclude that “applied math is worse for children than pure math” you’d need a study where participants were assigned to groups where they only received those kinds of instruction. That isn’t the study we have.

The OECD’s own interpretations are much more modest and will surprise very few onlookers:

  • “This suggests that simply including some references to the real-world in mathematics instruction does not automatically transform a routine task into a good problem” (p. 14).
  • “Grounding mathematics using concrete contexts can thus potentially limit its applicability to similar situations in which just the surface details are changed, particularly for low-performers” (p. 58).

BTW. I was asked about the report on Twitter, probably because I’m seen as someone who is super enthusiastic about applied math. I am that, but I’m also super enthusiastic about pure math, and I responded that I don’t tend to find categories like “pure” and “applied” math all that helpful. I try to wonder instead, what kind of cognitive and social work are students doing in those contexts?

BTW. Awhile back I wrote that, “At a time when everybody seems to have an opinion or a comment [about mathematics education], it’s really hard for me to locate NCTM’s opinion or comment.” So credit where it’s due: it was nice to see NCTM Past President Diane Briars pop up in the article for an extended response.

Featured Comment:

Chris Shore:

What is often overlooked in these kind of studies is the students who are enrolled in the various courses. The correlation between pure math courses and higher level math exists because higher achieving students are placed in the pure math classes, while lower performing students are placed in applied math.

Same thing is true for studies that claim that students who take calculus are the most likely to succeed in college. No duh! That is because those who are most likely to succeed in college take calculus.

The course work does not cause the discrepancy, the discrepancy determines the course work.

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I’m Dan and this is my blog. I’m a former high school math teacher and current head of teaching at Desmos. More here.

13 Comments

  1. I am not familiar with the PISA items … but … math scores on WHAT? If it is math scores on pure math items then this is an utterly ridiculous result to even report.

  2. Seems to me this gets at an issue some of us with a background in quantitative work have with math education; it does not address quantitative problem solving as a subject in its own right.

    For a useful discussion we need a common nomenclature: I understand what you call “pure mathematics” as “purely” or “detached” mathematics where quantities are always dimensionless numbers and the problem is to find solutions to equations whose variables are these numbers. As I understand “applied” here, it refers to what we see in school tests and text books: detached mathematics wrapped in a wordy shell. Let’s agree we would rather talk about quantitative problems, plausible as real and interesting, like yours.

    It is not the case that such problems are best approached through detached mathematics. A study that shows that is, is not plausible: either the problems in the test are the artificial ones, or the “applied” math students have not been taught problem solving.

    It is actually easy to see detached math is not an effective way to address real problem. If it were mathematics majors and mathematicians would be the ones hired to do electrical engineering and theoretical physics.

    From what I can tell it, there is no good general quantitative problem solving material available at the school level. Could someone tell me I am wrong?

  3. Dear Mr. Fuller,

    Teaching with “dimensional analysis” as much as possible and then stressing that “x, and y” are abstractions to save time and to see how different problems are attacked similarly works well for my students. Think Chemistry conversions!

  4. Linda Richard

    June 29, 2016 - 5:34 am -

    I think “applied versus pure” is less interesting than “concrete versus abstract.” I think one of the important outcomes of math education is teaching kids how to think abstractly, how to strip away the concrete referents but still hold the concepts in their minds. I had students who were excellent concrete thinkers but struggled mightily as soon as abstraction was introduced. The applied-to-pure progression is one way to help teach abstract thought to concrete thinkers.

  5. Pisa involves way too many factors for it to be a reliable analysis tool. They re comparing the most competitive kids in Shanghai that go thru rigorous private schools and after school academies preparing for the national test that will make or break their life with kids in say Canada who simply live in a completely different universe.

    If I forced my applied students into the theoretical route and exposed them to more advanced math that already makes little sense to them they would fail and shutdown like I’ve seen happen many times . Not perform better.

    Eg recently I had one grade 10 student who was a wonder in grade 10 applied class and when her mom pushed her to go into the purely theoretical grade 11 instead of applied grade 11 the girl dropped to the lowest possible score because nothing made sense anymore.

    I think the reality is that Pisa is not a very good indicator of what math should be taught at all and if traditionally China and Korea taught theoretical math and their kids performed better than more applied American kids it’s simply an indicator of different culture not a cause effect factor.

  6. I am so excited about this last post. Not for reasons most others might be excited for this post though. I am planning on having my AP Stats read this blog to allow us to discuss experiments and show them a “real life” example instead of some contrived story problem. My inner geek is squealing today. Thanks for posting.

  7. Karen L:

    I am not familiar with the PISA items … but … math scores on WHAT? If it is math scores on pure math items then this is an utterly ridiculous result to even report.

    Hi Karen, as the article mentions, “First, the PISA exam itself is largely a test of applied math, not equation-solving.” So this isn’t a case of pure math students just doing well because they were given a pure math exam.

    Dick Fuller:

    From what I can tell it, there is no good general quantitative problem solving material available at the school level. Could someone tell me I am wrong?

    Statway and Quantway are two programs that more than fit the bill for “good general quantitative problem solving material,” I think. Give them a look and see what you think.

    Kelly Berg:

    I am so excited about this last post. Not for reasons most others might be excited for this post though. I am planning on having my AP Stats read this blog to allow us to discuss experiments and show them a “real life” example instead of some contrived story problem.

    Awesome. Have at it.

  8. I read the article in USNews and interpreted the findings to show that teaching for conceptual understanding should be the focus of mathematics instruction – regardless of whether it’s “pure” or “applied”. Teaching either without building a foundation of understanding will result in poor transfer skills, and students will not be able to apply their knowledge. I think the title “Is it Better to Teach Pure Math instead of Applied Math?” is a faulty one and leads to poor conclusions. It’s not a matter of which one is better, but rather how to each each one effectively.

  9. Yeah, agreed, I had to look hard for the pure v applied point in the study itself. It isn’t prominent. Two demerits for US News.

  10. What is often overlooked in these kind of studies is the students who are enrolled in the various courses. The correlation between pure math courses and higher level math exists because higher achieving students are placed in the pure math classes, while lower performing students are placed in applied math.

    Same thing is true for studies that claim that students who take calculus are the most likely to succeed in college. No Duh! That is because those who are most likely to succeed in college take calculus.

    The course work does not cause the discrepancy, the discrepancy determines the course work.

  11. First of all, I really love your work. The other day I saw a thing in the world that made me think of the 101 questions and I took a photo of it specially. Haven’t submitted it yet, but I feel enriched for this way of looking at the world.

    Thank you for taking on PISA. David Spiegelhalter says some interesting things about the overall ratings. The UK recently dropped in rankings (though I suspect this is not one of their main worries just now) and he pointed out that the movement is within the margin of error, and it is simply the fact that they use rankings that makes some countries look as if they are doing much worse. Now that would be an interesting thing to simulate!

    I love aspects of pure maths. Discrete maths such as networks, permutations and combinations and suchlike are such fun. But I emphasise statistics as I think it has such a wide applicability to citizenship and social justice. Basically though, any mathematical science or branch of maths, taught well, is going to enrich the lives of our people.

    Keep up the inspiring and thought-provoking work.