What causes student errors on a problem involving differentiating a factorial?

Question

Consider the following maths problem, assuming only a high-school level knowledge of calculus:

If $f(x) = x!$, find $df/dx$

Almost all of my respondents seemed to switch on their "mental differentiation pipeline" and tried to expand $x!$ as a product and got stuck. The right answer is that $x!$ is undefined for any non-integer, so $f$ is discontinuous, and $f'$ doesn't exist!

Is there a name in psychology for the phenomenon that causes one to think along the wrong lines in the problem above?

Is it the same as the one leading to incorrect answers to the bat and ball problem? Is this a test of your thinking style? For, perhaps an intuitive thinking style will lead one to expand $x!$ as a product and attempt to differentiate term-wise. Is there a "cognitive remedy" for wrong thinking?

Answer 1

I imagine this question is tricky for students for a several reasons.

1. Question wording: The question may suggest to the student that it is possible to differentiate $x!$. Or they may assume from the wording that some meaning is meant where it is possible to differentiate. For example, alternatively, you could ask a set of questions, one for each functions, about whether a given function was differentiable (e.g., $x^2$, $2x+3x^3$, $log(x)$, $x!$). This would clearly be easier, because it puts the potential that the function might not be differentiable into the student's mind. I'm not saying what method of questioning is better from a pedagogy perspective; I'm just noting that question wording influences the difficulty of the problem, and that in this case, the question might be labelled a "trick question".

2. Analogy to solve novel problems: If students think that there is an actual derivative as an answer, but no obvious solution exists, it is natural to apply strategies that seem relevant to the problem. For example, they might know how to calculate $x!$ for any given positive discrete value of $x$. Therefore they might assume that working out the derivative involves interpolating (e.g., something like a gamma function) or they might rely on what they know about the product rule or some such. In general it is common for people to try to solve problems that they don't understand by drawing on strategies that seems relevant.

3. The question relies on the ability to apply a key fact that discrete functions are not differentiable. Thus, students need to know that fact, and they need to know that the factorial is a discrete function, and they need to link these two ideas. Thus, from a knowledge perspective, this question relies on a piece of knowledge that students might not possess.

So, in summary, I think the question exemplifies the kinds of challenges that students encounter when learning and demonstrating mathematical skills. I think a lot can be learnt about students cognitive representations of a problem by the kinds of errors they make. However, I do not think failing to solve this problem correctly says anything fundamental about your thinking style.

For further discussion of cognitive science in relation to mathematics education, perhaps check out Robert Siegler's work.

References

• Siegler, R. S. (2003). Implications of cognitive science research for mathematics education. In Kilpatrick, J., Martin, W. B., & Schifter, D. E. (Eds.), A research companion to principles and standards for school. mathematics (pp. 219-233). Reston, VA: National Council of Teachers of Mathmatics. PDF

Answer 2

If you pose the question only to students who have a perfect knowledge of high school calculus, without any knowledge gaps, then they should all give the correct answer. The "problem" is, that you can graduate from high school without having understood all the details about differentiation. Most students don't finish high school with the best mark in maths, so obviously they must have some knowledge gaps. And even many who finish with the best possible mark have only understood the format of the questions, not everything that's behind them. And then there are those for whom this detail was not part of the curriculum. Because even with an identical syllabus, there will enough leeway for a teacher to focus on different aspects and only superficially touch upon others or leave them out altogether.

In short: Assuming that everyone with a high school degree has the same depth of mathematical understanding is somewhat naive.

I graduated from high school and never have understood even the basics of differentiation, despite the fact that it filled one year of my curriculum :-)