Frequently, teachers of mathematics appeal to this platonic ideal of “intuition” to explain where concepts come from, or even as steps in proof-sketches. But, what really is “intuition?” What properties of this word allow it to be so incredibly powerful? Is intuition something you are born with, or does one gain intuition with experience? And, if intuition comes from experience, why do teachers rely so much on students to have experience in a field they are meant to be learning?
The following is a conversation inspired from interactions I have had with professors, teachers, and advisors of mathematics. Many of the linguistic structures used by the “Professor” part are near exact replicas of those I have previously experienced.
[Somewhere, during class, a professor gives an example of a structure they just defined for the class.]
Professor: So, let me give an example. Take M to be any set and f to be a map from M to N. Constructing this map requires some intuition, but once constructed, the proof is simple…
[After the professor finishes their explanation, they pause for questions.]
Student: Professor, I am not sure I understand how you decided to build the map the way you did. Would you mind going into that a little bit?
P: Well, I will do my best, but that construction really just requires some intuition into the subject.
S: I’m sorry, but I am still not sure I understand. What do you mean by “intuition?” Is that a mathematical concept?
P: No, intuition is more of something you have that helps you solve problems. Having some intuition about a problem is very helpful, and really is imperative to being a mathematician.
S: Okay, I see. So, is intuition like a talent or skill that mathematicians have? Like something they are born with?
P: Not really. Great mathematicians may be born with it, but the rest of us (who are not great) have to learn from experience.
S: Oh, I see. So, I really just need more experience to understand how you knew that was the right map? How can I gain that experience?
P: If you are struggling with this concept, you may want to revisit your materials from previous courses.
S: Okay. I received good grades in my past classes, and I feel that I understood the material well. Maybe I did not find all of the information I needed in those classes to develop this intuition.
P: I would not think so. This is a natural (normal, pedagogical, etc.) progression of courses. If you do not understand this concept, see me during office hours. Otherwise, I would suggest going over previous course materials. That is the best advice I can give right now.
[The professor returns to the lecture at this point.]
Alright, to understand what happened here, we will break this down in question-and-answer couplets. Our setting is at any university, in nearly any mathematics class. Our professor has just given us the statement of some definition or theorem, and now they will give an example. They say,
So, let me give an example. Take M to be any set and f to be a map from M to N. Constructing this map requires some intuition, but once constructed, the proof is simple…
In this specific interaction, the professor immediately uses the word “intuition” without giving a good idea what it means, or really what this intuition would entail. Now, some may read this and say that “intuition” is a colloquial term, and that we all really know what means. To those readers, I would let them know that, per their own definition, that they just suggested we all have knowledge of intuition intuitively, and I would ask them to read on. Now back to the professor…
Generally, the professor would probably now give a brief description of the map in question, and would proceed with the proof, giving relatively little attention to what makes the map intuitive. In this case, “intuition” allows the professor to bypass a complex or confusing aspect of the proof in question, by appealing to the audience’s previewed collective knowledge. Such bypassing is a very poor rhetorical device and develops a weak argument, especially in the world of mathematics where proofs are meant to be logically rigorous. Whatever that means.
Moving forward, the professor states that after applying “intuition,” the rest of the proof becomes “simple.” This is an example of a microinvalidation, a form of microaggression. Microinvalidations dismiss individual and group experiences, bypassing the speaker’s need to explain concepts more fully to the individual or group, or even bypasses the speaker’s need to view the individual or group as equal. At best, this microinvalidation dismisses any subsequent confusion on the part of students. At worst, this microinvalidation is intended as an insult and is meant to segregate “learned” people from “ignorant” people (the latter is an example of a microinsult).
Next, we see the student’s question, and the professor’s response,
Professor, I am not sure I understand how you decided to build the map the way you did. Would you mind going into that a little bit?
Well, I will do my best, but that construction really just requires some intuition into the subject.
Our student asks for further clarification of the map our professor described as intuitive. Rather than responding to the student’s question by finding an alternative explanation of the map, the professor decides to appeal to intuition once again. Once again, the professor places the onus (the responsibility associated with responding to the student’s question) upon the use of intuition. As such, the professor’s response here is problematic for the exact reasons the professor’s initial statement was problematic.
Now, I will say that my experience here goes two ways. Either a professor simply appeals to intuition without further explanation, as our constructed professor did here, or the professor will still appeal to intuition, but will include further explanation. While the latter case does include the explanation the student asked for, the professor still invalidates the student’s need for further clarification. This, at best, serves no educational purpose and, at worst, discourages the student from asking further questions.
Let us turn our attention to the next couplet,
I’m sorry, but I am still not sure I understand. What do you mean by “intuition?” Is that a mathematical concept?
No, intuition is more of something you have that helps you solve problems. Having some intuition about a problem is very helpful, and really is imperative to being a mathematician.
Our student asks for a clarification of “intuition,” which admittedly would almost certainly never happen for fear of reprisal. Our professor responds that intuition is not a mathematical concept, but is some intrinsic aspect of our selves, and that we need this intrinsic aspect of ourselves to be a mathematician.
To begin, our professor’s description of intuition is not a definition. It is an ambiguous characterization of an abstract concept our student is already confused about that really only serves to cause further confusion. Then, the statement about intuition being imperative to being a mathematician falls down into the “no true scotsman” fallacy. “No true scotsman” refers to appealing to a most pure or ideal sort of person or thing to bypass the proper defining of that person or thing. Here, “no true mathematician lacks intuition.” Further, anyone who does not have intuition could not be considered a mathematician. Again, this is invalidating of the student’s experience, and their question, and does not answer the student’s question.
Okay, I see. So, is intuition like a talent or skill that mathematicians have? Like something they are born with?
Not really. Great mathematicians may be born with it, but the rest of us (who are not great) have to learn from experience.
Now, the student’s question shifts its form rather radically. Rather than asking for clarification due to a desire to understand the subject, our student asks out of concern for their future as a mathematician. Are they good enough, or were they not born with the skill they need to be a mathematician? Rather than totally mollify the student’s concern, our professor distinguishes great mathematicians from common, inferior, etc. mathematicians. To appear humble and understanding, the professor lumps themself in the group of common, inferior, etc. mathematicians, but again, this is just an invalidation of the student’s questions.
Interestingly enough, our professor does say that we are meant to learn intuition from our experiences. This actually gives us a good understanding of what it means when we appeal to intuition; we are appealing to lived experience, and the very subjective feeling we get in response to solutions to proofs we are working on that resemble proofs we have seen in the past. So, intuition is built up by giving students an ample data set of correct, complete proofs which the student can emulate in the future and draw from in complex and non-linear ways to produce their own correct proofs.
The next two couplets are taken together, as the first couplet simply builds up to the second,
Oh, I see. So, I really just need more experience to understand how you knew that was the right map? How can I gain that experience?
If you are struggling with this concept, you may want to revisit your materials from previous courses.
Okay. I received good grades in my past classes, and I feel that I understood the material well. Maybe I did not find all of the information I needed in those classes to develop this intuition.
I would not think so. This is a natural (normal, pedagogical, etc.) progression of courses. If you do not understand this concept, see me during office hours. Otherwise, I would suggest going over previous course materials. That is the best advice I can give right now.
In couplet 1, the student asks how they can gain the experience they need to build up intuition. This question was included in the dialog because school and university, and of course their associated work, are generally considered to be the experience required to master the subjects in which one acquires their degree. So, how is a student meant to find these experiences if they are being deprived of the opportunities to develop those precise experiences?
In response to this question, the instructor shifts any blame for the student’s misunderstanding upon the student themself and rejects any blame to be placed on our professor, or on the mathematical pedagogy. This rhetorical device is incredibly common in response to these questions, and is incredibly fallacious. This specific fallacy is the “tu quoque” or the “you also” (I prefer “no you”) fallacy is defined as a speaker reflecting criticism away from themselves and back on the person asking the question. This is not a strong argument, and does not even approximate an appropriate response.
The second couplet only builds on the first with our professor invalidating any of the student’s studies in pre-requisite courses as misunderstood, appealing to the divine authority (fallacious) of the math department, or normalcy to do so. The professor then shuts down any further conversation, stating that this is their best advice. But, why, if this is their best advice, would they suggest the student see them again during office hours? Well, mostly because this is not seen as a question to be had during lecture.
Questions like the above are dismissed by the mathematical community as uninteresting, meaningless, philosophical, or even as attacks directed at the professor and/or standard pedagogies. Such dismissal is unacceptable, as it is microaggressive, deters students from entering and reteurning to the field of mathematics, gives the false assertion of mathematics only being for the skilled, etc. In fact, the notion here of mathematics being only for the skilled is not only ableist, but classist.
Students who have trouble visualizing data, understanding “simple,” “trivial,” “definitional,” etc. proofs, or understanding lecture or text material are seen as less-than, their concerns dismissed. This is an example of the bandwagon effect, where those in the know become teachers, never ask to understand others lived experience, believe their experience is the same as everyone’s experience, and then impose that experience upon everyone else. Further, their experience and presumption gets confirmed by other professors who followed the same path. Such behavior is ableist, because it dismisses any neurodivergent experience of the material as non-existent. Such behavior is also classist, the self-developed intellect is established as being that much higher than any intellect developed with external assistance.
It is high time that “intuition” as a proof and teaching tool, and its ableist, invalidating, and classist roots, be totally eradicated within the mathematical community. It is not difficult to add an extra definition to a lecture plan, if said definition is so simple or intuitive. Furthermore, it is not difficult to state fully the definitional or trivial case of a proof, in stating that proof, by definition. There is no argument supporting the continued usage of ableist, classist, microaggressive language that consistenly supports the right of a student to learn in a safe and appropriate environment. As mathematicians, it is time we adopt this logical next step, the only logical next step, and fully remove this language from our vocabulary.