College Math Teaching

March 7, 2010

Why Some Students Can’t Learn Elementary Calculus: a conjecture

This semester, I am teaching two 30 student sections of a course called “brief calculus”: it is your classical “calculus light” course that is taken by business majors and (sadly) by some science majors.

Throughout my career, I’ve noticed that many students struggle because concepts such as “the derivative”, “slope” and “rate of change” really don’t make sense to them. They can memorize and repeat verbatim, but struggle when they have to combine concepts.

Here is an example: I have a student who fully understands “how to take the derivative of x^3 + 1 and who can tell you that “the derivative df/dx gives you the slope of the tangent line” but completely fell apart when asked to “find where the slope of the tangent line to the graph of f(x) = x ^3 + 1 is equal to 12. She didn’t know where to start, and evidently, most in the class didn’t know either.

So, in an effort to help the students understand that they had to work on understanding the concepts as well as the calculations if they were to learn the stuff, I did an in class exercise with both sections (9 am and 2 pm):

1. I told them to pay attention and to put down their pens or pencils and to have a blank sheet of paper available.

2. I wrote two sentences on the board, one above the other:

I aligned the letters as shown and wrote in large, capital letters.

3. I asked them “do you see this?” I counted to 5 (internally) and then erased the board completely.

4. Then I asked them to reproduce what they saw on their paper and then to turn it in.

I then asked “which of the two sentences was easier to reproduce”? Most said “sentence 2”; the reason was “it made sense”. I noted that both sentences had the same number of 3 letter words, 4 letter words and, in fact, the same number of letters.

I explained: if the course material doesn’t make sense to you, you won’t be able to do well on an exam; you’ll get confused and make errors that reveal a lack of understanding.

But on a whim, I decided to do some data analysis by looking at what they wrote on the paper. I wondered if there was a difference in performance on this exercise between students who were doing well in the class versus those who were doing poorly. The students had taken one “hour” exam so far; hence I decided to write their (uncurved) scores from the first exam and I decided to classify their attempts at reproducing the sentences into two different categories:
I: they got almost all of “the dog ate the bone”; they either got it fully right or got “the dog ate” or “dog ate the bone” without inserting extra unrelated words or words from the first sentence.
II: they got almost none of the second sentence (two or fewer words) or added words from the first sentence into the second, or just made stuff up.

I then ran a statistical t-test on the mean of the scores on the first exam from group I versus the mean of the scores on the first exam from group II with the null hypothesis: “the exam one scores from group I were equal to the exam one scores from group II”

This is what I found:

t-Test: Two-Sample Assuming Equal Variances

Variable 1 Variable 2
Mean 61.65384615 48.71428571
Variance 438.0753846 206.3736264
Observations 26 14
Pooled Variance 358.8089936
Hypothesized Mean Difference 0
df 38
t Stat 2.060670527
P(T<=t) one-tail 0.023113899
t Critical one-tail 1.685954461
P(T<=t) two-tail 0.046227797
t Critical two-tail 2.024394147

That is, there was a statistically significant difference on the performance on exam one between those who were able to reproduce the sentence “the dog ate the bone” and those who weren’t able to; those who could reproduce the sentence scored, on the average, 13 points higher!

I thought: “ok, this wasn’t a proper experiment as the venues were different (chalk board for the 9 am class versus white board for the 2 pm class), different time of day; perhaps I gave one group more time than the other, etc.”

So I decided to test the differences within each class (correct reproducers versus incorrect reproducers in the 9 am class, then again in the 2 pm class).

Here were the results:
9 am class:

t-Test: Two-Sample Assuming Equal Variances

Variable 1 Variable 2
Mean 54.16666667 50.375
Variance 509.969697 259.6964286
Observations 12 8
Pooled Variance 412.6412037
Hypothesized Mean Difference 0
df 18
t Stat 0.408944596
P(T<=t) one-tail 0.343702458
t Critical one-tail 1.734063592
P(T<=t) two-tail 0.687404916
t Critical two-tail 2.100922037

Aha! No statistically significant difference in the 9 am class!

But then I ran the 2 pm class:

t-Test: Two-Sample Assuming Equal Variances

Variable 1 Variable 2
Mean 68.07142857 46.5
Variance 314.8406593 162.7
Observations 14 6
Pooled Variance 272.5793651
Hypothesized Mean Difference 0
df 18
t Stat 2.677670073
P(T<=t) one-tail 0.007681222
t Critical one-tail 1.734063592
P(T<=t) two-tail 0.015362443
t Critical two-tail 2.100922037

Holy smokes! Here p = .015, and the spread was 21.5 points!

Note: on exam one, the 9 am section had a mean of 51.5 and a median of 53; the 2 pm class had a mean of 65.0 and a median of 60.

Ok, “n” is too small for this to be a proper study, and the conditions were not tightly controlled. But this gives me reason to wonder if there is something to this: maybe the poor performing students really couldn’t make sense of “the dog ate the bone” quickly!

I’d love to see a proper experiment that would test this.


  1. […] Educational Issues Mathematics education: I wonder if many students who do poorly in college calculus courses do poorly because they simply lack the ability to do bette… This posts describes an exercise which shows that students who did poorly on their first exam of […]

    Pingback by 7 March, education edition (and other topics) « blueollie — March 7, 2010 @ 9:01 pm

  2. No matter how many times I practice, I’m still an F student! that drive me nuts. I don’t know what to do anymore.

    Comment by Nell — March 28, 2012 @ 4:41 pm

  3. I have gotten an A in every course I’ve taken between my 2 years at my Junior college, to the 1 year I’ve spent at a private university except for one class; pre calculus (C). I’m taking calculus now, and even though I’ve never even gotten a B I’m almost certain not only will I not pass, but I may not score a single point in this class.
    More to the article, I thought the impromptu study was pretty cool.

    Comment by Timmy — August 27, 2012 @ 12:31 am

    • Obviously, a short little experiment like the one I ran in class is insufficient to establish anything. I do have one guess though: I’ve known a few students who need to UNDERSTAND something in order to be able to do it (their brains are always asking “why”). Such students need to study very differently than those who can perform procedures that they’ve memorized but can’t understand.

      Comment by collegemathteaching — August 27, 2012 @ 1:05 am

  4. So what is the solution? How did you figure out how to get these students to understand the material and do well in the class? I am one of these students. I was a 3.75 student and the highest grade I got in pre-calculus was 2.2. I worked so hard and that was the grade because I just couldn’t comprehend the subject at all.

    Comment by Shannon — January 15, 2013 @ 5:18 am

    • Good question. I don’t think that there is one “magic bullet” answer. One technique (that worked for some) is to do at least a few homework problems with the book and notes closed. THAT will show you where you are lost (where your understanding ends). Another is to do the problems in jumbled order; lots of times 75 percent of the battle is knowing which technique to use.

      Of course, there are no guarantees; I wish that there were.

      Comment by collegemathteaching — January 15, 2013 @ 6:47 pm

  5. I did poorly in High School Math and expected to do poorly in college. However, my college had a course called “Remedial Math” which was meant to get Veterans back into the learning system. I started with Remedial Math, Algebra and Trigonometry, all with the same, FANTASTIC teacher. He started with 1 + 1 = 2, base 10, base 2, base 13, etc. and made us REALLY UNDERSTAND math. I went on through Calc-1, 2 &3, plus Vector Analysis, all straight A’s. Then I had a TERRIBLE TEACHER for Differential Equations and got a C. I think the secret is to get the BASICS from a good teacher.

    Comment by Bob Ojala — July 3, 2013 @ 8:49 pm

    • That is something that worked for you. However, you had the motivation to learn the basics and the capacity to remember the basics.
      As far as the DE course? Who knows; it is easy to blame bad instruction. It could also be that the material really was a step up. I can’t tell because I wasn’t in your class.

      Comment by blueollie — July 3, 2013 @ 10:07 pm

  6. Wtf

    Comment by Alex — February 14, 2015 @ 10:29 pm

  7. I like this! But something seems to have gone wrong with your description of categories I and II: as stated I would expect most of the students to fall in both categories at the same time: being able to reproduce the first sentence but not the second. From the remainder of the text I get the impression the groups were defined by how well they could reproduce the dog-sentence and the other sentence did not play a role in the experiment anymore, is that correct?

    Comment by Vincent — June 4, 2015 @ 8:57 am

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