The math problem: All rote, no reasoning

Community college students placed in remedial math — a large majority — may have memorized a few procedures, but they don’t have a clue what they’re doing, according to researchers.

In one study, few could place -o.7 and 13/8 on a number line from -2 to 2. Asked which is greater, a/5 or a/8, 53 percent answered correctly, barely beating a coin toss.

“Seeing two fractions near each other apparently triggered an urge in some students to use the cross-multiplication procedure they had memorized,” writes Nate Kornell on Psychology Today. If all you’ve got is a hammer, everything looks like a nail.

About Joanne


  1. Which is greater a/5 or a/8?

    If a == 0, then they’re equal
    if a > 0, then a/5 is greater
    if a < 0, then a/8 is greater

    • This seems to continue the firm tradition of “Math Education Specialists” not actually understanding the math, but seeming to know what all the problems are about how to teach it.

      • Maybe I’m underestimating them here, but exactly how many remedial math students at a community college are going to be thinking about cases where a <= 0? There might be a handful, but as far as I can tell from the paper, none gave the answer "it depends".

        ". The students were also asked to explain how they got their answer. About a third (36%) could not come up with an explanation (half of these had answered correctly, the other half incorrectly). The ones who provided some sort of explanation tended to summon some rule or procedure from memory that they thought might do the trick. Many of the students said that a/8 is larger because 8 is larger than 5. Not surprisingly, another group claimed just the opposite, having remembered that the lower the denominator the larger the number. Some students tried to perform a procedure: some found common denominators, though often they made mistakes and got the wrong answer anyway. Others cross-multiplied (something they apparently believed you can do whenever you have two fractions).
        Only 15% of the students tried to reason it through. These students said things such as: If you have the same quantity and divide it into 5 parts, then the parts would be larger than if you divide it into 8 parts. Assuming you have the same number of these different sized parts, then a/5 must be larger. Although it is discouraging that only 15% took this approach, it is interesting to note that every one of these students got the answer correct. "

    • Some information got lost in the various levels of referencing.

      (1) The blog post (that we are commenting on) is quoting a blog post (also posted by Joanne).

      (2) The blog post is quoting a paper titled “Teaching the Conceptual Structure of Mathematics”. That paper can be found here:

      (3) The paper “Teaching the Conceptual Structure of Mathematics” is summing up a finding from a paper titled “What Community College Developmental Mathematics Students Understand about Mathematics”. This paper can be found here:

      (4) The problem as posted in the original paper read: “If a is a positive whole number, which is greater: a/5 or a/8?”

      So the original question was fine. The summary of the question in “Teaching the Conceptual Structure of Mathematics” dropped the qualifications that made it a good question.

  2. More info: In the paper at @, the question is asked “If a is a positive whole number, which is greater: a/5 or a/8?”

    But in, on the other hand, it says merely “Which is greater? a/5 or a/8/” – a very different question.

  3. Well, considering that my home state of Nevada only managed to graduate 62% of high school students in 2010-2011, and only 50% of african american students (those actually passing the exit exam in this state), and a remediation rate of 40% it does not surprise me that the students cannot even handle remedial math.

    Perhaps because we do such a lousy job of teaching math in grades K-5 that our students do so poorly in math in the later grades (where it’s often TOO late to fix the problem).


  4. “Given that U.S. students are taught mathematics as
    a large number of apparently-unrelated procedures that
    must be memorized, it is not surprising that they forget
    most of them by the time they enter the community

    When was the last time students had any sort of traditional math curriculum? My son had MathLand over ten years ago, and it had been in use for quite a while. It was replaced by good ol’ “trust the spiral” Everyday Math. Do we need to wait just a few more years for this pedagogy to really begin to work? This rote learning meme doesn’t hold.

    Back when I taught college algebra (in a college not much above a community college), students could not pass with rote skills. One slightest variation in a problem would cause them to flunk. It’s quite another thing to claim that knowledge is not flexible enough, but solving that problem takes more than a fuzzy beginning conceptual understanding level. It requires a continuing effort appled to all of the homework problems, not just the first few in the set. It’s almost as if these people have never cracked open a good math textbook. You can’t do the problems with just a rote understanding.

    The examples here reflect basic competence, not rote learning versus understanding. What do the best students do on these questions? They had the same math. Why are they successful? Gee, they could ask their parents.

    I thought we were way past this sort of simplistic analysis. This is a classic case of “research shows” what they expect to see.