I teach for a company along with a large group of teachers, almost all of which are people who have graduated with the standard Bachelor level education in Education and Science/Mathematics. I am currently teaching senior high school and undergraduate mathematics whilst studying towards becoming a professional mathematician.

However, a very large number of my coworkers, especially the junior high school teachers, refer to themselves as 'mathematicians' to both other staff members and to students. These students go into senior high school and enter my classes, where I hope to get them interested in studying mathematics beyond the high school level. However, when talking to them about mathematics beyond high school, I often hear something along the lines of

"Being a mathematician seems a bit boring and easy. My Grade 9 teacher is a mathematician and it really doesn't seem like an interesting or difficult occupation. There's only so much you can learn about fractions and trigonometry.

This bothers me quite a bit, as I feel that it trivialises what mathematicians really do. It also turns students away from the field even before they get a chance to do it properly with me. Sometimes I have to resort to saying

"Teacher $X$ is not a mathematician, despite what he/she claims"

which does not feel like a very nice thing to say.

As you know, a professional mathematician actually works with mathematics for a living and at the very least have doctorates in mathematics. Many of these so-called 'mathematicians' are unable to teach senior high school, let alone award themselves such a label. You could not call a Biology teacher a 'Biologist'. You would not call an Economics teacher an "Economist". Similarly, you would not call a mathematics teacher a "Mathematician".

I know precisely which teachers are making these claims, but I am not sure how to approach them in a nice way about this. I am getting a bit tired of having to justify and explain my career choices to these misinformed students. Should I just bite the bullet and let it go?

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    $\begingroup$ There is a nice piece by Cuoco, Goldenberg, and Mark on mathematical habits of mind in which they use the word mathematician (p. 384) and foot-note it as follows: "Of course, by mathematicians, we mean more than just the members of AMS; we mean the people who do mathematics. Some mathematicians are children; some would never call themselves mathematicians." I am, personally, inclined to agree with this trio out of EDC; so I quite disagree with your claim that one "would not call a mathematics teacher a 'Mathematician'." I would call them exactly that. $\endgroup$ Commented Apr 22, 2016 at 9:34
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    $\begingroup$ Instead of trying to convince your colleagues that they are not mathematicians, perhaps you could make a distinction for your students between mathematicians who are high school educators and mathematicians who work as professionals with higher mathematics. This would prevent a fight with your colleagues and instead give your students a chance to learn about a different kind of mathematician and mathematics. $\endgroup$
    – Amy B
    Commented Apr 22, 2016 at 9:54
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    $\begingroup$ I have friends with doctorates in mathematics who are not mathematicians by any standard. I have friends with no doctorates who are published and accredited mathematicians. I think you need to reconsider what characteristics you value a 'mathematician' to have. $\endgroup$
    – Sloan
    Commented Apr 22, 2016 at 13:36
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    $\begingroup$ I quite agree with the OP that his colleagues are not mathematicians. For example, the Wikipedia definitions of expected education and responsibilities (for teachers who are mathematicians) align with my intuitions. However, confronting the junior-high colleagues about this will likely make them more entrenched, and the situation even worse. That's a societal/institutional problem beyond your personal capacity to fix. $\endgroup$ Commented Apr 22, 2016 at 15:46
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    $\begingroup$ If the students conclude "Teacher X is a mathematician. What teacher X does is boring. Therefore what all mathematicians do is boring" they seem to fail already on simple syllogisms ... $\endgroup$ Commented Apr 23, 2016 at 14:38

8 Answers 8


I do not know if there is an accepted definition of what a mathematician is. There are teachers of mathematics and professors of mathematics, for example, and most people agree that people of the latter category are mathematicians while I am not sure if there is any consensus about the former category.

So my suggestion would be to not got into the trouble of arguing who is a mathematician and who is not but, instead answering to

"Being a mathematician seems a bit boring and easy. My Grade 9 teacher is a mathematician and it really doesn't seem like an interesting or difficult occupation. There's only so much you can learn about fractions and trigonometry."

something like

"Teaching mathematics is just one thing a mathematician can do. There are many many other things mathematicians do. If you finish a university degree in mathematics, you may enter research mathematics which is an incredibly large field. You may also work as a mathematician in companies doing very different things like data analysis, technical development, code development but also management, strategy development and other things."

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    $\begingroup$ beat me to it, i was going to give essentially the same answer. You will never be able to change the behavior of your colleagues so the best bet is to show your students that while some "mathematicians" are teachers, there are many other things that they can do. Just a quick web search found this list math.uh.edu/~tomforde/Web/Jobs.html and i am sure there are many other resources for careers in mathematics $\endgroup$
    – celeriko
    Commented Apr 22, 2016 at 13:15
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    $\begingroup$ Also, after all, teaching can occur at all levels, from grade-school through graduate-level stuff, whether it's teaching of mathematics or something else. And teaching can be done badly and boringly, or not, at all levels. $\endgroup$ Commented Apr 22, 2016 at 13:35
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    $\begingroup$ "Maybe your mother is a singer in church. Maybe your father is self-employed which makes him the CEO of a company. I don't doubt they are good at what they do, but they are no Taylor Swift or Steve Jobs. Mathematicians can be the same way. I don't doubt that your 9th grade teacher knows mathematics. What I wonder is if he has ever experienced the /Taylor Swift/ magnitude of mathematics. Mathematicians are frequently the ones that "discover" new particles long before anyone else has ever considered them. Mathematics paves the way for new technology. Are you ready to be part of that?" $\endgroup$ Commented Apr 22, 2016 at 19:04
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    $\begingroup$ @Keeta: Taylor Swift is not a great singer at all, although she might a marketing success. A singer in a church could be way better than her. Just saying. $\endgroup$ Commented Apr 22, 2016 at 20:43
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    $\begingroup$ You could also tell them about the mathematicians who crashed Wall Sreet in 2008 or are in the NSA, or about* Sergey Brin* who co-founded Google with Larry Page. $\endgroup$ Commented Apr 24, 2016 at 2:48

My favourite definition of a mathematician is the one given by Ian Stewart in his book Letters to a young mathematician (on page 32).

What is Mathematics? In despair, some have proposed the definition "Mathematics is what mathematicians do." And what are mathematicians? "People who do mathematics." This argument is almost Platonic in its circularity. But let me ask a similar question. What is a businessman? Someone who does business? Not quite. It is someone who sees opportunities for doing business when others might miss them.

A mathematician is someone who sees opportunities for doing mathematics.

I like this definition because when I stop to consider what makes a mathematician in this light, then I am forced to ask what it means to see an opportunity for doing mathematics. I am sure it will be a revelation to many students that this is a valid way of looking at mathematics and mathematicians.

I am also confident that the businessman example will resonate with them, because it immediately clicked for me. I have absolutely no business sense in me, and I can clearly recognise it in those people around me who possess it. The idea that one can recognise a mathematician with the same (or similar) sense is an exciting idea when you are first introduced to it.

You could also try making an analogy with sportspersons if your students are athletically inclined. For instance, even if a person goes through the motions properly during training, if they do not possess a keen insight on the playing field, then they will come up short against one who has that insight.

The final thing that I like about this definition is that by accepting it you ensure that you're not taking the easy way out. The student can no longer say that Mr. X says he's a mathematician, so he is. The student must apply their own judgement based on their own understanding of what seeing an opportunity for doing mathematics means.

And that is the best one can hope for, in my opinion. The students with sufficient mathematical intuition will be able to pick the true ones out, and they are the ones who are ideally placed to take up mathematics later on, anyway. So, rather than hoping to convince everyone that real mathematics is interesting and not like what your colleagues do, it would be better to place an effective tool in your students' hands that they can apply by themselves to decide what mathematics is and who mathematicians are.

  • $\begingroup$ A mathematician is someone who has to tell you what he thinks a whole number is. $\endgroup$ Commented Oct 19, 2023 at 4:05

You probably can't change the behavior of your colleagues, for a number of reasons (one vs. many; people get defensive/entrenched; you're likely only part-time; etc.).

But if this is an ongoing irritation for you, then you should be sensitive to that, and think about how you can ju-jitsu this into a learning opportunity for your students. Perhaps make this a point of discussion/assignment, and remediate it head-on early in the semester with your new students. "I find that many people have incorrect impressions of what it means to be a mathematician. What do you think? How can we investigate that?"

Here's a resource: "What Do Mathematicians Do?", by the AMS. The page is short; consider assigning that as a reading to your students. They point out that "mathematician" was identified as the #1 job in America a number of times (on Careercast.com, where it is defined as "Conducts research to develop and understand mathematical principles"; currently #6, with the top #1-2 jobs now being "data scientist" and "statistician"). The AMS page also says:

Many people are familiar with mathematicians in academia, but mathematicians also work in many other fields, including:

  • Astronomy and space exploration
  • Climate study
  • Medicine
  • National security
  • Robotics
  • Animated films
  • and in a wide range of businesses...

The AMS page emphasizes members who have advanced graduate degrees like the doctorate (likely not your junior-high school colleagues):

How Many Mathematicians Are There in the U.S.?

There are over 35,800 individual members of the four leading professional mathematical sciences societies in the U.S.---the AMS, the Association for Women in Mathematics, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. Most would call themselves mathematicians; many received their doctoral degrees outside the U.S. There are at least 10,000 more members of the societies who are graduate students or in other categories, and there are also mathematicians who are not members of any of these societies.

Although they have advanced degrees in mathematics, many of those employed in academia might call themselves professors instead of mathematicians, and similarly, those in industry and government may not have "mathematician" in their job title...

Note that the title of the AMS page is taken by an essay from A.J. Berrick (one of several links on the AMS page), part of whose thesis is that we can't simply answer what mathematicians are (like many other disciplines, according to target of study), but rather what they do (according to specific strategies applied). He writes in his conclusion, emphasizing the goal of new discoveries:

The discovery of a unifying pattern can be like lightning flashing from one discipline to another. The difference is that it can illuminate both subjects forever. So there is a simple message for the nonmathematical researcher reading this article. When all seems cloudy, contact a mathematician (as broadminded as possible). Then stand by for flashes of lightning!

Surely, that doesn't sound boring; and it also doesn't really sound like what your colleagues do. More generally: As a college math teacher, I think you'll find that a large part of your career will be spent remediating/fixing hideous misinformation that K-12 teachers have foisted upon their students. Best to use this for practice in that regard now. Good luck!


A quarterback and a center have completely different jobs, but they are both football players. "A mathematician" covers a lot of territory. I think they are mathematicians and they deserve some respect for the level of competence that they have achieved - however small compared to, say, Euler or Hofstadter or Angel Muleshkov.

You are upset about the wrong thing. No one is going to become a mathematician because he read a good definition of what a mathematician is. And there is no polite way to tell someone who thinks of himself as a mathematician that you don't think he is. If you want to get your students interested in mathematics, explain to them what mathematics is about.

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    $\begingroup$ I assume you mean Hofstadter? One could equally well question whether Hofstadter is a mathematician (other options include cognitive scientist, computer scientist, physicist, and very fine writer about mathematics...). $\endgroup$ Commented Apr 24, 2016 at 18:56
  • $\begingroup$ @AlexKruckman - I have always, seriously, said that a mathematician is a person who does mathematics and mathematics is what mathematicians do. Defining what a mathematician is is like defining what a point and a line are. It can be done, but not everyone is going to agree with you. $\endgroup$ Commented Jun 19, 2017 at 19:19
  • $\begingroup$ I totally agree with that sentiment, and to be perfectly clear, I have great respect for Douglas Hofstadter. I was just pointing out that he is not the best archetypical example of a mathematician. But maybe I misunderstood your point. $\endgroup$ Commented Jun 19, 2017 at 20:00
  • $\begingroup$ I was thinking you meant Hostetler: en.wikipedia.org/wiki/Jeff_Hostetler#Washington_Redskins $\endgroup$
    – guest
    Commented Dec 23, 2018 at 16:37

You could say "Well, there are high school level 'biologists,' 'mathematicians,' 'physicists,' etc. (just entertain their loose definitions). The people doing exciting work (and maybe describe what PhD graduates are doing) are PhDs, and studied years beyond what your teachers did."

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    $\begingroup$ This implies that work in education is not "exciting work." $\endgroup$
    – JRN
    Commented Apr 22, 2016 at 22:08
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    $\begingroup$ @Joel, I do not feel it implies education is not exciting. It's a reply to the quote "Being a mathematician seems a bit boring...." $\endgroup$
    – user1815
    Commented Apr 23, 2016 at 12:33

I would personally distinguish between 'professional mathematician', meeting the definition you seem to be using, and 'mathematician', as someone who thinks that way, but may or not do that as their profession. Someone might be a musician without being primarily employed as such.

I think I would focus on talking about what real mathematics is with the students (and maybe also the teachers), rather than convincing the teachers to stop identifying as mathematicians. They could be mathematicians who spend most of their time teaching arithmetic instead. Or its possible they've never really encountered nice maths for themselves. Generally (in my view) too much of the population believes that mathematics is what you do at school (and there's nothing better!). So with students I'd take the line of 'you have only seen one example of things you can do with a maths degree, there are many many others'.


It's impossible (like the joke about the Maine directions...you can't get there from here). What I mean is you can't really communicate this idea kindly to your colleagues. Furthermore, however nicely you get it across to your colleagues is not going to have the effect you want (to change their behavior).

In addition, there is a little bit of a muddle here. Do you need to "tell your colleagues" or talk to the students? Really the students are your objective (and you control communications in your class). So instead of "how do I tell my colleagues", the question is how do I tell my students.

In terms of your students, the easy way to communicate the difference in terms is by analogy to football players. Teachers of math are like coaches of sports. (Even if some coaches/teachers play, the activity of coaching/teaching is different than playing/researching.) You can even continue the analogy to pro football players (and note the small numbers of pro players versus school players...similar to small number of pro mathematicians.) Bottom line is teaching versus doing. In addition to research of course there are some "doing" aspects of math like calculating things, cracking codes, statistical modeling. I think if you just differentiate coaching versus playing, teaching versus doing (to your students), it makes the point. And make the conceptual point, but don't dwell on the "this is the only way to use the word 'mathematician' and your other teachers were wrongity wrong-wrong" picayune terminology aspect.

The other thing is to maybe give them some aspect/glimpse of what advanced math is about (and not terms like Sobolov spaces and manifolds and Lie group crap, but simpler concepts like "these are the courses after calculus in the engineering track", "these are the ones in the math track", "these are the major schools of research math (analysis, algebra)", "these are the major areas in applied math (stats, numerical methods, etc.)".

Realistically VERY FEW of your students will go on to get a BS in math, let alone a Ph.D. and then research activities. So don't spend a bunch of time on trying to inspire the few. Drop a few pearls, maybe, but limit the time/effort.

As far as some kid saying something wrong, don't let it get to you. There are even people wrong on the Internet. And sometimes I try to pin them down on the figurative high school wrestling mat of forum debates, but they never admit they were wrong. And they're probably not worth the effort either. ;-)

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    $\begingroup$ Lie groups and manifolds are not waste. $\endgroup$ Commented Dec 24, 2018 at 3:15

When I use the term "mathematician", I set a pretty high bar. By way of example, I have been working in mathematics for more than a decade now: I started as a secondary math teacher (after finishing a BA in mathematics), then went back for a masters (the local school district was undergoing a hiring freeze following the financial crisis of 2008, so it was return to academia, or continue moving from one long-term substituting gig to another for an indefinite period of time), and ended up pursuing a PhD (which will be done in another month or three). I have a paper in a good journal, have presented at a number of important conferences in my field, and have even been flown to the UK to give a talk. Yet I don't really consider myself to be a mathematician—I only have one good pub, and no PhD yet.

All of that being said, I would never presume to tell someone else that they are not a mathematician—it simply isn't my place. I (obviously?) have sympathy for the point of view that a middle school mathematics instructor is not a mathematician, but I am not going to attempt to impose my point of view on others. This just seems arrogant to me. If a middle school math instructor claims to be a mathematician, great.

On the other hand, I can see the problems that this might cause. I think, however, that the correct approach is to try to change the minds of your students, rather than to try to change he behaviour of your colleagues. For example, you suggest that students say things such as

Being a mathematician seems a bit boring and easy. My Grade 9 teacher is a mathematician and it really doesn't seem like an interesting or difficult occupation. There's only so much you can learn about fractions and trigonometry.

A reasonable response might be

Your grade 9 teaching is one kind of mathematician. Her job is to take very difficult ideas and explain them in an approachable manner. It is, perhaps, a good sign that her job seems easy and uninteresting, as this indicates that she is doing an excellent job of making the material seem easy and approachable.

Of course, the are different kinds of mathematicians. There are, for example, mathematicians who spend their time working on difficult mathematical puzzles which require a lot of heavy abstraction. These "pure mathematicians" typically work at universities (though not always). As a student, you might encounter them as an instructor in a classroom, but this is only a small part of their job. Mathematicians of this type spend the majority of their time engaged in research, which involves a lot of explaining their ideas to others in papers and conference talks.

Another type of mathematician might work in "industry". These mathematicians solve practical, real world problems. For example, these kinds of "applied mathematicians" might work for health organizations, where they may be tasked with understanding how diseases (such as the coronavirus) spread. Such mathematicians are also involved in understanding climate change, and in designing rocket systems which might one day take people to Mars (and beyond!). There are even applied mathematicians working in the CIA and NSA, where they seek design (and break) better methods for sending secret methods from one place to another.

If you don't like my examples of other kinds of mathematicians, pick your own examples. The main point is that a middle school teacher is just one kind of mathematician. If students don't find that kind of mathematics interesting, there are other ways of engaging with the field.


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