# Knowing mathematics does not translate to knowing to teach mathematics. Why?

Many brilliant mathematicians seem to make average or even poor classroom teachers. Is this an accurate assessment? Has there been any research to explain the phenomena?

What is the difference between knowing mathematics and knowing to teach mathematics? Can you provide any examples (either abstract or concrete) of this phenomenon occurring?

• I don't mean this to demean this question; indeed, I'm curious to see responses. But I don't think this phenomenon is at all unique to mathematics! – Brendan W. Sullivan Mar 15 '14 at 3:46
• @brendansullivan07: Indeed! Everyone reading this site knows how to read. How many of us know how to teach someone to read? – Mark Meckes Mar 15 '14 at 8:53
• You've gotten some excellent answers so far, but original body of this question failed several of the criteria for great subjective questions. I've tried my hand at rewriting it. It could still use an edit to dig into the question a bit more. – Jon Ericson Mar 15 '14 at 14:13
• @John, Mara asked for examples, which people offered in the answers. It looks like you took that out which will make the answer thread confusing. I'd prefer to see the request for examples left in. (Or else please explain why it was removed.) – Sue VanHattum Mar 15 '14 at 15:10
• I don't accept the premise of the question. Most of the good mathematicians I have encountered were also excellent teachers. – Dan Fox Oct 7 '19 at 19:43

There is yet another issue here, it is related to "expert blind spot", but not the same.

To be a good teacher, one needs to understand the pupils and their issues with the topic. This becomes hard if the teacher thinks in a qualitatively different way. He may know exactly what causes the problem and why, but cannot explain it well, and his messages don't reach the students, because his explanations align with his thought process, not the students'. Such a teacher might refer to intuitions that are alien to students and thus not convincing. On the other hand it's hard for him to assume student's thought-process, because he might see it as deeply flawed and missing some essential characteristic of considered structure.

One example where this happens frequently is Category Theory. Learning it involves some "brain rewiring" and after it happens that person speaks a different language even if it appears to be normal. The most useful introductory texts I saw were

• written by people during the process of learning the subject,
• stuffed with examples that would let the reader build their own intuitions, independent of authors explanations.

Of course, after "it clicked" my opinion on them was different. I didn't have to be an expert to see them as unnecessarily verbose, skewed or having too many trivial, pointless examples. It was even funny how little it took to change the perspective and how hard it was to begin that change.

I hope this helps $\ddot\smile$

• Thanks for teaching us the \ddot \smile LaTeX trick! $\ddot \smile$ – Joseph O'Rourke Apr 29 '14 at 1:12
• And this is why "experts" are not qualified to choose textbooks for introductory courses. – DavidButlerUofA May 8 '15 at 21:12
• Man, what a great answer! I'm especially appreciative of the observation that the best intro. texts are written by people learning the topic! I have felt that my teaching seems to become "worse and worse" the older and more experienced I get. Sometimes my intuitions about well-traveled undergraduate topics feel so disconnected from my students' understanding (e.g. the use of basic logic…especially involving quantifiers) that trying to explain things to students begins to feel quite pointless. The advice about examples here is the only remedy that seems to work in light of this phenomenon... – Jon Bannon Feb 24 '16 at 2:42

There's two parts to this question. So, I'll give them separately:

Why? There is a lot of education and psychology research on expertise that points to the idea of an "expert blind spot". An easy way of thinking about it is that someone who has been studying mathematics for a very long time has likely forgotten the struggles she went through when she initially learned the material. This can make it very difficult to understand the issues that students have and why they have them. It often presents itself in the feeling "This is trivial! I have no idea why they didn't get this correct." Here's a paper that explains this phenomenon in more detail.

It's probably worth noting that a shallow understanding of a topic can account for students' abilities to get perfect scores but not be able to explain their reasoning to someone else. This could be relevant to your question if you're interested in why the best students in your class might not be very good at teaching or communicating with the other ones.

Examples. I have no idea what level or area of math you're looking for here. So, I'll try to give some varied examples and/or ways in which this can happen.

Teaching Induction. Often, when mathematical induction is being introduced for the first time, the induction step is stated as something like "Prove $P(n) \implies P(n+1)$". Students often respond to this with something like "but that's just assuming the conclusion!", because internally they aren't fixing $n$ to be a particular natural number--to them, it's more intuitive to just assume it for every natural number. Understanding this distinction (and this common misunderstanding) might lead to an instructor highlighting what $n$ is, rather than just saying it and moving on.

Assuming Prior Knowledge. Someone who uses group theory regularly might try to motivate or teach any number of theorems (Fermat's Little Theorem, Wilson's Theorem, any variety of counting problems) using either generalizations or proofs that rely on thinking abstractly about mathematical operations (e.g. Fermat's Little Theorem is a trivial consequence of Euler's Theorem). Someone else might assume that the bijection between $\mathcal{P}(\mathbb{N})$ and functions from $\mathbb{N}$ to $\{0,1\}$ that uses binary is trivial to students who might be new to bijections or never have seen binary before.

Ignoring "Trivial" Steps. Another very common issue is that instructors sometimes accidentally leave out steps or details (because they view them as "obvious"). Someone who has been differentiating functions for a very long time might not explicitly call attention to uses of the chain rule, product rule, etc. It's often an oversight rather than the intention. One could imagine an extension of this where the students are supposed to take a partial derivative, as well.

• @ adamblan Adam, thanks for the paper, I have not seen that one before now. Good examples, too! I was not searching for any particular level. Rather, I am interested in all levels :) Somewhat philosophically, what makes some mathematicians great teachers while others successful mathematicians struggle explaining their thinking? Questions extends into ongoing challenges of schools of education to "produce" (need better word here) effective mathematics teachers. – Mara Mar 15 '14 at 3:52
• I think 'develop' more effective math teachers is a better word. Also, I think that the concept behind these examples can be applied to all levels. Ignoring trivial steps happens when teaching Algebra, Geometry, DiffEQ, etc. It doesn't depend on a high level of math. Just that the instructor forgets that the students cannot always follow the logical jumps. – David G Mar 15 '14 at 5:35
• Absolutely! I tried my best to give kind of a "schema" and then give an example of that schema. This way there's some abstract and some concrete! – adamblan Mar 15 '14 at 15:30
• I'm not familiar with any research on the subject, but my personal experience suggests that another aspect of the "expert blind spot" is that not only has the expert forgotten her initial struggles with the material, she may not have had the same struggles that many students do. For whatever reason, some things really just are easier for some people than for others. – Mike Shulman Mar 15 '14 at 20:51
• Mathematics education researchers have a name for what Adam is describing here; pedagogical content knowledge. There is the knowledge of the mathematics, and there is the knowledge of the alternate approaches to understanding the mathematics, and there is the experience to know which approach is useful now, with this student. – David Wees Apr 7 '14 at 10:48

A good math educator is good not only at math but also at psychology - reading the students' body language to figure out what their response is, good at facilitating groups, good at encouraging students to work hard, good at understanding where someone is stuck (and different students do get stuck in different places), and how to explain not one way but two or three.

A wonderful book that explains what more a math educator needs to know to explain math ideas (at elementary level) is Liping Ma's Knowing and Teaching Elementary Mathematics.

• Thanks Sue, Love that book! I have been also exploring the concept of pedagogical content knowledge (coined by Shulman) related to mathematics which is also focused on scaffolding and understanding what kind of knowledge helps us be more effective mathematics educators. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4- 31. – Mara Apr 6 '14 at 18:01
• This old question got bumped up to the site's front page, and I was struck by the first item listed under psychology, "reading the students' body language." I feel that this is one of the main losses caused by on-line teaching during the coronavirus crisis. Because of privacy concerns, I don't require students to have their video cameras on, and teaching people whose reactions I can't see just doesn't fell right. – Andreas Blass Sep 6 '20 at 12:58

Yours is an important question in mathematics education, and especially in mathematics teacher preparation. Subject area content knowledge (as mentioned in the question) is of some importance, as well as generalized pedagogical knowledge (reflecting the general knowledge about what it means to be a good teacher.

However, Shulman (1986) identified other forms of knowledge important to teaching. For example, Pedagogical Content Knowledge:

A second kind of content knowledge is pedagogical knowledge, which goes beyond knowledge of subject matter per se to the dimension of subject matter knowledge for teaching. I still speak of content knowledge here, but of the particular form of content knowledge that embodies the aspects of content most germane to its teachability. (p. 9)

Knowledge of representation is an important form of PCK. For example, a mathematician may not need to know the sorts of visual models that help a 4th grader grasp aspects of ratio and proportion as a part of their learning fractions (as opposed to teaching them to manipulate fractions).

Within the category of pedagogical content knowledge I include, for the most regularly taught topics in one's subject area, the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations—in a word, the ways of representing and formulating the subject that make it comprehensible to others. (p. 9)

This is not just knowledge of students. There is legitimate mathematical content knowledge that is more important for teaching than it is for doing mathematics. For instance, few mathematicians have a reason to distinguish between quotative and partitive division. Yet they are distinct mathematical conceptions of division that also happen to make it easier to understand how young students see division.

This page goes into some detail on the difference between these conceptions, including a video example of work with a student.

Shulman also discussed "Curricular Knowledge" which specifically refers to instructional materials (and would be of little interest to experts in the subject area).

Other researchers focusing on teacher knowledge have broken the categories down further. Hill, Ball, and Schilling (2008) proposed an overall domain map known now as "MKT" (Mathematical Knowledge for Teaching). There is some evidence linking forms of knowledge described in this model to classroom practice (Shechtman, Roschelle, Haertel, & Knudsen, 2010). It breaks down into many, very specific, categories that researchers continue to investigate (Common Content Knowledge, Specialized Content Knowledge, Knowledge at the Mathematical Horizon, Knowledge of Content and Students, Knowledge of Content and Teaching, and Knowledge of Curriculum).

Even more recently, Schoenfeld has asked questions about how teachers make decisions (2010) and how we should observe the quality of a particular mathematics classroom based on what we see going on there (2013). The latter has resulted in a theoretical model of classroom quality involving multiple dimensions that are related to what he sees as going into educational decision-making (teacher knowledge, teacher orientations, routines, etc.). Not surprisingly, the classroom itself is quite a bit more complex than what goes into teacher decision making.

In any case mathematical content knowledge is distinct from what makes a good math teacher. Others in their responses here have covered the idea that expertise sometimes makes it harder to actually bridge a communication gap in understanding (referred to by @adamblan as a blind spot). The differences go well beyond that, and even beyond knowledge of students and teaching. Some of the references here should help you delve into how researchers investigate (and have theorized about) the differences alluded to in your question.

Cited:

Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers’ topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.

Schoenfeld, A. H. (2010). How We Think: A Theory of Goal-Oriented Decision Making and Its Educational Applications. Studies in Mathematical Thinking and Learning Series. ERIC.

Schoenfeld, A. H. (2013). Classroom observations in theory and practice. ZDM, 45(4), 607–621.

Shechtman, N., Roschelle, J., Haertel, G., & Knudsen, J. (2010). Investigating links from teacher knowledge, to classroom practice, to student learning in the instructional system of the middle-school mathematics classroom. Cognition and Instruction, 28(3), 317–359.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4.

One of the points is that teaching is an expertise on its own. Part of it is innate, much can be learned.

Important questions, that during your traditional undergraduate/graduate curriculum (at least if not directed at pedagogy) aren't ever mentioned, and which many of us never really considered, include:

• How do I design an exam? What kinds of questions should I include? How do I make the result easy/unambiguous to grade? How do I make sure a dumb mistake early on doesn't invalidate later work (and perhaps requires extra work grading, following a completely different path)?
• What should be the homework's contents?
• What proposed exercises (for self study) should be given?
• Which ones are the best examples for classroom use?
• A good underlying assumption you have here is that expertise in teaching includes knowing your learner; knowing learners/students/pupils contextualizes these questions and it also points at the complexity of the teaching and learning processes. – Mara Mar 21 '14 at 18:33

As for examples: I taught gifted students for many years. Some were ahead of others and I would sometimes ask them to work together and teach each other. Some students were natural teachers but others struggled. The ones who struggled would sometimes give up and say they can't explain anything and at other times just give step by step instructions of what to do.

I have heard that if you can't explain math - you don't really understand it; if you can explain it you have a deeper understanding. This idea has been .repeated at workshops I've attended, by my administration, and throughout my elementary curriculum. From experience- I have seen gifted math students who can't explain what they are doing (and therefore can't teach) However through repeated questioning their understanding and ability to explain will come through.

There are many aspects to this, and there are already some good answers here. Just on one point: I never heard of the "expert blind spot" before, but based on my own experience I have often told people that the hardest thing about teaching is remembering what it was like before you understood something.

• This! After a near 30 year career, I retired, and starting in '13, I work as an in-house high school tutor. There are times the kids ask me why my style is different than their actual math teachers. I tell them I never went to 'teaching school'. "You have me, remembering what learning this stuff was like, 40 years ago." I remember hating the radical symbol, preferring exponent notation, and the confusion multiple notations used in calculus caused me. I'd just heard it articulated this way before. – JTP - Apologise to Monica Oct 5 '19 at 19:56

From my understanding if you aren't consciously learning to teach then you just "learn" things without really concentrating upon the various aspects associated with it. For example there are two distinct universes, mathematical and this real one. Mathematical Universe has things which are a representation of real world things. A teacher must know that for a student to understand and place a new knowledge properly in the Mathematical Universe, student must know the connections of this new thing with things ALREADY known, so that a student is able to properly place this new knowledge. For an experienced teacher it is very easy to move between Abstract Mathematical world and Real Physical World but not at all for a student. So the connection between the things in Mathematical Universe and Real Universe must be made very clearly such that student is able to follow his/her teacher in both the universes.

Also if a student is not able to bring himself into the Mathematical Universe then the teacher must realise that and take Mathematical Knowledge to the Student's world and explain Math using things in student's world in a language student understands(language here means manner of speaking). Different students understand things differently based on so many things like personality, social background etc. Identifying all of this has got nothing to do with "knowledge of subject" rather a human art form.

A person good in Mathematics may be lacking in soft skills required to communicate the knowledge to the student. That's why good knowledge of subject doesn't necessarily translate into good teacher of the subject...