# “Good” and “Bad” student intuitions when teaching and learning mathematics

I'm a college math/science tutor and I'm really interested in STEM education. I'm currently starting work on a project I hope to present in a couple of months at a tutoring conference and I was wondering if you could point me in the direction of useful research. The project is about mathematical intuition.

I'm aware of at least a couple of ways to define the thing:

1. As the innate, untutored sense some have for certain mathematical concepts. I'm aware (in passing) of research that claims children have an innate sense of number, for example. As my students are college-age, I'm not sure this is the definition I want to focus on.
2. As the sense, usually gained from experience, of the basic nature of the solution to a problem, without having formally attempted to solve it.

It is this later definition that piqued my curiosity. In my experience, while many of my students can do calculations just fine, they struggle to decide whether or not their calculation makes sense. In Chemistry for example, students have a hard time figuring out which units are bigger than which, frequently making errors like: $$10\,miles < 50\,meters$$

Or: $$number\,of\,atoms\,in\,5\,moles\,of\,F < 10^{23}$$ As far as mathematics goes, I frequently have to remind students that variables follow the same rules "normal" numbers follow, for example that: $$\frac{x}{2x}+\frac{x}{x^{2}}$$ Would be solved in a similar way as $$\frac{1}{3}+\frac{3}{5}$$ is solved.

Personally, I find this one of the most frustrating parts of my job, as it suggests we (both the professors and myself) have issues communicating both conceptual understanding and a successful problem-solving heuristic, and instead are training students in symbol manipulation.

This second definition of mathematical intuition brings both Polya and Fermi to mind, as they both were at least partly focused on making plausible approximations. Further, I'm also interested in the "bad" intuitions students bring to a problem. Here I am thinking of things like the gambler's fallacy which is so prevalent in beginning statistics classes.

My search for relevant articles has turned up little that relates to higher education, and more that relates to either intuition as it applies to professional mathematicians or primary/secondary education. My questions are as follows: Do you know of any research on the common mathematical misconceptions and intuitions of college students/college courses, where they come from and how to combat them? Do you know of any research on strategies to develop helpful intuitions?

I would also be very curious to know if you have encountered "bad" intuitions, and about what you personally do to cultivate "good" intuitions.

• This is rather speculative, but I've heard it suggested the problem is that there aren't enough examples of "when not to do X" to go along with the examples of "when to do X", which pushes people to develop overly permissive intuitions about doing X. – user797 Mar 4 '17 at 2:45

By no means a direct or definitive answer, just three references: the 1st on estimating in college-level engineering, the 2nd more speculative at the research level in mathematics and theory of computation, the 3rd just for fun (at middle-school level).

(1) Rebecca Bourn, Sarah Baxter. "Developing Mathematical Intuition by Building Estimation Skills." 2013 ASEE Annual Conference & Exposition. Download link

By thinking critically about the validity and purpose of particular numbers and applying them outside the bounds of their standard usage, students show mastery of comprehension and move on to the pedagogical realms of application and creation.

(2) Richard Lipton (blog). Mathematical Intuition—What Is It?. Oct. 2010.

Open Problems. Is intuition simply built up by learning more and more about an area? Or is intuition something that is separate from just being an expert in an area? Can you be quite strong in an area and still have weak intuition, or is that impossible?

(3) Robert Kaplinsky, 2013: How Much Money IS That?!.

Assume all US\$100 bills. About$220,000,000, i.e., nearly a quarter-billion US dollars.

• Thank you! I'm currently reading Fischbein's work on intuition and it's my impression that intuition (as per the second definition) and estimation are intimately linked. – Jorge Medina Mar 10 '17 at 2:27

Tversky and Kahneman's work showing that people rely on heuristics might be in the right direction. Their research showed that peoples intuition for statistics is often bad due to our heuristics that can ignore critical information about a problem.

I think mathematical intuition especially with college level concepts on up is the second definition you used, that "good" mathematical intuition is mostly a matter of practice and being familiar with a problem and "bad" mathematical intuition is not enough experience or heavily biasing one piece of information i.e: $2^{2^{2^{2^{2}}}}$ looks like a small number because it's made of 2s but if you know that stacking exponents make numbers incredibly large it's not surprising to find out that it's in the order of $10^{19728}$

Furthermore symbol manipulation is great for problems that can't be understood concretely. Few people can intuit a 4 dimensional surface and following a rote procedure can allow a mathematician to manipulate that space even if they don't "understand" it.

• Thank you! I've been reading some review work by Ben-Zeev that suggests that suggests bad intuitions might in part be the result of faulty schemata students develop when classifying problems. Students, for example, can read a problem and misclassify it in their mind as being of one type when it in fact isn't. I'm still interested in what you yourself do to foster good intuitions – Jorge Medina Mar 1 '17 at 6:15
• I think the best way to foster good intuitions is to encourage practice and if possible self-motivated practice. I know that for me personally I'm much more likely to remember an answer and the process for getting that answer if I had to seek it out myself than if it was just handed to me. That in turn builds a better, more complete schemata. – user3362964 Mar 1 '17 at 7:23