The math problem: All rote, no reasoning

Most new community college students are placed in remedial math, despite having completed all their high school math requirements. They may have memorized a handful of procedures, but they don’t understand what they’re doing, writes Nate Kornell in Psychology Today.

For example, new research, building on studies in 2011 and 2010, show few community college students can place -o.7 and 13/8 on a number line that runs from -2 to 2. Asked which is greater, a/5 or a/8, 53 percent answered correctly, barely beating a coin toss.

Some fell back on procedural knowledge, probably because that’s the only knowledge they had about fractions. For example, seeing two fractions near each other apparently triggered an urge in some students to use the cross-multiplication procedure they had memorized.

Students also did addition and subtraction mindlessly.

In an interview one student was asked if he could think of a way to check whether 462+253 = 715. He smartly subtracted 253 from 715 and came out with 462. So far so good. But when he was asked whether he could have subtracted 462 from 715 instead, he said he did not think so.

Students could not take advantage of relationships between problems to find easy solutions, such as going from 10 x 3 to 10 x 13 to 20 x 13. Most worked out every problem, frequently making errors that defied common sense. Here’s how one student solved a series of related multiplication problems:

10 ×  3 = 30
10 × 13 = 130
20 × 13 = 86
30 × 13 = 120
31 × 13 = 123
29 × 13 = 116
22 × 13 = 92

Asked what it means to be good at mathematics, remedial community college students said math is “all memorization” or “just all these steps.”  One said, “In math, sometimes you have to just accept that that’s the way it is and there’s no reason behind it.”

If you think there’s no reason behind math, then there’s no reason the answer to 20 x 13 can’t be smaller than the answer to 10 x 13.