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I don't mean to sound rude and denigrating to math educators but I always imagined that a science teacher (physics/chemistry), who has to know math, would be eligible to teach math?

Ignoring the pedagogical differences and the fact that there are far less science teachers than math teachers, content wise strictly speaking, isn't a science teacher thus then qualified to teach math? Is there something a math teacher can do that a science teacher cannot?

The only parts of math that science teachers probably cannot do as well are abstract math, number theory and statistics and probability.

Am I wrong in this thinking?

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    $\begingroup$ Why should we "[ignore] the pedagogical differences"? $\endgroup$
    – JRN
    May 28, 2022 at 2:05
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    $\begingroup$ A mathematics teacher has to know the language the students use. Does that make them qualified to teach that language? $\endgroup$
    – JRN
    May 28, 2022 at 2:07
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    $\begingroup$ Well, yes, as you already acknowledge, "up to a point". Just as high school math teachers can teach high school physics up to a point. These are the two closest things. In fact, the basic chemistry "stoichiometry" has a lot in common with "number theory", and little to do with calculus. $\endgroup$
    – paul garrett
    May 28, 2022 at 3:08
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    $\begingroup$ Which level of education are we talking about here? This is different in kindergarten and at university... $\endgroup$
    – Tommi
    May 28, 2022 at 6:58
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    $\begingroup$ Even many math majors graduate without much clue of what mathematics really is. They may not truly understand that mathematics is about inference, not executing algorithms. I would image that many science educators would earn their degree without ever being expected to produce a proof of a theorem. It would be disturbing for such a person to teach mathematics, because (while they may be proficient with using some of its results as tools) they do not know what the subject is even about. $\endgroup$
    – Steven Gubkin
    May 28, 2022 at 17:47

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That depends on the level. In general the rule is that to teach a subject coherently, one needs to know roughly speaking five times more than one is going to present to the students. If this condition is satisfied, then you should be just fine (the little quirks like what would be the best way/order to present this or that can be picked up on the way and there is no universal agreement on them anyway). If not, you'd rather leave it to other people. I acted as a tutor in elementary physics a couple of times (kinematics and dynamics: Newton laws, energy/momentum conservation and such) and I felt completely comfortable with that though I'm a pure math professor, and the results were satisfactory too, but I wouldn't go into, say, teaching quantum mechanics though I am not totally ignorant of it.

So just evaluate your knowledge and make the conclusion yourself. I have no doubt that you'll be fine teaching addition of fractions but I have my reservations about letting you teach a graduate university course in measure theory (though I don't know you well enough to declare that you are disqualified from it, of course). Just follow that 5 to 1 rule and you'll be well within your competence level, that's all.

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    $\begingroup$ Do you happen to have a reference for "In general the rule is that to teach a subject coherently, one needs to know roughly speaking five times more than one is going to present to the students"? I don't doubt its essence for a second (though I'm unsure about the precise factor), but I would love to show this to those of my students who intend to become math teachers and keep asking why they have to learn all those things which they believe they "won't need in school anyway". $\endgroup$
    – Jochen Glueck
    Jun 6, 2022 at 17:16
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    $\begingroup$ @JochenGlueck I wish I would but I don't at the moment. The factor is not precise, of course (after all, how exactly do you measure?), but it is roughly in that area. If you look at old university textbooks in various subjects, they contain the amount of material about 5 times more than one can possibly squeeze into a one semester course. I believe in the general principle though without any reference. If it is violated, we get homework problems asking to find the area of a triangle with sides 5,6,7 and altitude 4 to the side 6 (my daughter brought that masterpiece from school once). $\endgroup$
    – fedja
    Jun 6, 2022 at 17:34
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    $\begingroup$ @JochenGlueck Her geometry teacher was not stupid at all, but they were learning just the triangle area formula and the students were still confused about which side to multiply by the altitude, so he decided to give them all three choices to check understanding. Now, one can create a triangle with not too large integral sides and one integral altitude, but that requires some knowledge beyond the triangle area formula (though not too advanced). As to the (future) math teachers, they may meet some smart kids occasionally and those can ask questions of all sorts. $\endgroup$
    – fedja
    Jun 6, 2022 at 18:00
  • $\begingroup$ That's indeed a good anecdote; it illustrates the problem quite vividly. Regarding future highschool teachers: That they will sometimes have to teach smarts kids is of course a valid point - but I'm under the impression that many students are told this all the time as the reason why the have to learn so much advanced math. I understand why they don't find this, as a sole reason, completely convincing, and thus I very much prefer to explain the reasoning to them which you mentioned in your answer: in order to teach properly you need to understand much more than you teach. $\endgroup$
    – Jochen Glueck
    Jun 6, 2022 at 18:23
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    $\begingroup$ @JochenGlueck There is yet another way to look at it as far as the education of future teachers is concerned: in every subject, the students (except, perhaps, the brightest ones) fully comprehend and retain forever only a fraction of what they are exposed to, i.e., to have them to master simple things at the ace level, one needs to make them struggle for some time with more advanced ones. I have no reference for that either, but if you have ever taught some course that requires a prerequisite and that prerequisite was taken some time ago, you can easily decide whether I'm right or not :-) $\endgroup$
    – fedja
    Jun 6, 2022 at 18:53
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Is there something a math teacher can do that a science teacher cannot?

The essence of the mathematical discipline is rigorous deductive proof. Feel free to read that as "correct and understandable explanations and justifications" for mathematical principles.

Frequently, non-math science teachers have only taken computational math courses in college, like calculus, differential equations, and linear algebra (likely in their first two years or so).

But for college math majors, there is a radical shift around the end of the second year: courses suddenly shift to being about writing and understanding proofs as their central focus (generally talking U.S. system here; in other locales they wouldn't wait that long). It's common for math majors who are only good at mechanical calculations to find this very difficult or switch to another major as a result. Science majors usually never see these courses. (Conversely, some people who would be very good actual math majors never get through the mechanical-calculating prerequisites to get to this point.)

So it's quite likely that science teachers, given their background, could present a calculating algorithm, but be unable to explain why the method works. Not being trained and practiced in how to carefully attend to starting definitions and theorem-construction, it's quite likely that it would come out as a mishmash and be unclear on what the symbols, operations, starting assumptions, and sufficient and necessary conditions really are (in fact: I often experience this when I try to read posts on SE Physics, say).

Overall, such a mode of presentation represents cargo-cult (or faith-based) math, and leaves students unable to succeed at the next level, because all they're doing is memorizing a ton of separate facts instead of understanding more broadly-applicable principles.

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    $\begingroup$ could present a calculating algorithm, but be unable to explain why the method works -- A good example is being able to answer a question such as this MSE question (see my comment there), including being able to answer appropriate to the student's mathematical background level. $\endgroup$
    – Dave L Renfro
    May 29, 2022 at 7:17
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    $\begingroup$ In case that MSE question gets closed, and thus viewable only to those with sufficient reputation: Question Is there away we can prove say the division rule for indices with the same base? What is the intuition behind it? eg. 2^a/2^b=2^(a-b) My Comment What type of numbers are $a$ and $b$ -- positive integers with $a \geq b,$ integers, rational numbers, real numbers $\ldots$ ? For the intuition, try an example, such as $a=5$ and $b=2$, replacing exponents by repeated multiplication and then reducing the fraction by ordinary arithmetic properties of fractions: (continued) $\endgroup$
    – Dave L Renfro
    May 29, 2022 at 7:21
  • $\begingroup$ $\frac{2^5}{2^2} = \frac{2 \cdot 2 \cdot 2 \cdot 2 \cdot 2}{2 \cdot 2} = \frac{2 \cdot 2 \cdot 2}{1} \cdot \frac{2 \cdot 2}{2 \cdot 2} = \frac{2 \cdot 2 \cdot 2}{1} \cdot 1 = \frac{2 \cdot 2 \cdot 2}{1} = 2 \cdot 2 \cdot 2 = 2^3 = 2^{5-2}.$ $\endgroup$
    – Dave L Renfro
    May 29, 2022 at 7:21
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    $\begingroup$ I like this answer: a math teacher can explain more concretely about polynomials than a science teacher who just happens to know how to solve those. $\endgroup$
    – Lenny
    May 31, 2022 at 13:59
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    $\begingroup$ Replying to your comment "It's common for math majors who are only good at mechanical calculations to find this very difficult or switch to another major as a result. Science majors usually never see these courses." Indeed---I think that working physicists are often much better at, for example, integration than the average mathematician. Physicists are typically very good at computation, but often have no idea of why it is that their computations work (other than having some intuition about a physical process which, again, the mathematician may lack). $\endgroup$
    – Xander Henderson ♦
    Jun 5, 2022 at 16:55
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It's common in Madrid and mostly a failure, although there are of course individual exceptions. Most trained in something other than math don't know math well enough to teach it well. It's not so different from what happens when mathematicians have to teach computer programming. They can do it, those who don't have criteria by which to judge (students, parents, administrators, politicians) can't much tell the difference, but the result isn't what it could be (I speak from the experience of having taught C programming quite a few times).

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As long as the science teacher is appropriately motivated, I can't see why on earth he or she would not be able to teach mathematics to at least high-school level.

In order to reach a certain level of scientific sophistication as to be able to understand the science syllabus in order to teach it, there is bound to be at least a modicum of the "important" stuff in mathematics: namely, how to recognise a proof and how a proof is structured.

At the level of high school mathematics, it should not take a prohibitively long time to reach a level at which they can impart the knowledge and skill -- but as I say, the teacher has to be motivated.

If it's the level of: "we haven't got anyone to teach maths, we cast lots and you're it," is very probably not going to work well.

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    $\begingroup$ At my undergraduate institution, physics majors are required to take calculus, DEs, and PDEs (no proofs-based courses). At my PhD institution, physics majors must take calculus and DEs (PDEs is not required)---again, no proofs based classes. Hence your assertion that "there is bound to be at least a modicum of the "important" stuff in mathematics: namely, how to recognise a proof" does not mesh with my experience. $\endgroup$
    – Xander Henderson ♦
    Jun 3, 2022 at 14:29
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    $\begingroup$ @Thierry A teacher is expected to be expert enough in their field to help advise and guide students who may eventually do more in that field. I would expect that anyone teaching calculus should have an idea of what comes next. Where do the results from a high school calculus class show up again? How do we know that these results are even true? When and where were these results first understood? By whom? Someone whose highest mathematical education is a calculus class is (in my opinion) unqualified to teach calculus. Thus a typical physics major isn't going to cut it. $\endgroup$
    – Xander Henderson ♦
    Jun 4, 2022 at 15:22
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    $\begingroup$ @Thierry Okay, so consider examples from algebra. Why do we study polynomials? What is significant about the roots of a polynomial? How do you know that roots even exist? What is special about polynomials of degree less than four, and how do we know that? What is the significance of the analogy "integers are to rationals as polynomials are to rational expressions"? How far can one push that analogy? Circling back to the beginning, why do we care about polynomials? These are the kinds of questions that someone trained in mathematics ought to be able to answer, but others likely can't. $\endgroup$
    – Xander Henderson ♦
    Jun 4, 2022 at 22:08
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    $\begingroup$ Don't get me wrong---I think that the training that mathematics teachers in the US get is, very often, quite inadequate (I say that as someone who once held a a secondary ed teaching credential in mathematics---I know very well how my colleagues were trained). Still, someone trained to teach mathematics is far more likely to have engaged with the history and context of mathematics than someone trained to teach chemistry or physics. A chemist might know how to compute, but the mathematician will know why to compute. $\endgroup$
    – Xander Henderson ♦
    Jun 4, 2022 at 22:12
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    $\begingroup$ @XanderHenderson In some countries proof-based courses are standard from high school. For instance, when I was at high school I studied the limits with the $\epsilon-\delta$ definition, the Riemann integral, Rolle’s theorem (just to cite a few things) etc., all with the proofs. And my maths professor asked us to be able to reproduce all the proofs of the year. And when I studied engineering the analysis classes where the same of those for mathematicians, with proofs. So, certain generalizations about what someone with a physics or engineering background doesn’t know about math are excessive. $\endgroup$
    – Massimo Ortolano
    Jun 5, 2022 at 16:11
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In the USA, I'd say it's common, not ubiquitous, that from K-6, kids get their math instruction from generalist elementary teachers, lacking a math degree, or often even any non-ed degree. 7-12, you usually have dedicated instructors. So specialization is the norm. Most of the math teachers tend to have math degrees. But there's no reason for it to be 100%. As long as you know the content and have the teaching cert, it's not that hard to move (more likely "be moved") into teaching math versus science.

Usually, it seems like schools struggle harder to find the teaches for the science courses and for computer science. And there's competition for HS jobs. So having a subject matter degree is generally desirable. But I know of people teaching HS math with an undergrad in science.

I even know one with an English degree. They keep him away from AP Calculus, but he handles algebra and geometry fine. Not the norm really, but certainly rarely happens.

There's no conceptual reason you can't have an adjunct or even a tenure/track faculty in a university math department, who lacks a math degree. I'd say it's incredibly rare though. There's a big supply demand problem (for the jobseekers, it's a blessing for the colleges) in terms of the number of people looking for jobs versus the openings. So, competition makes it extremely unlikely.

All that said, I have reasonably frequently seen community colleges using non-math UGs to teach night school or summer sessions. Usually from a science or engineering background and often mid career with a day job.

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    $\begingroup$ It is worth noting that most high school math teachers in the US do not require a mathematics degree (or even a math minor). It is generally possible to get a secondary education credential in mathematics by taking 10-15 hours of upper division mathematics (along with one's courses on methodology, pedagogy, law, and so on). $\endgroup$
    – Xander Henderson ♦
    May 28, 2022 at 14:02
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    $\begingroup$ @XanderHenderson In my U.S. state it is possible for a teacher who has achieved baseline certification/preparation to get a credential in any subject (including secondary mathematics) by passing the state teacher certification examination for that subject. $\endgroup$
    – shoover
    May 31, 2022 at 21:56
  • $\begingroup$ In my limited experience with American grade school system, most teachers barely know their subject, and so-called science teachers are the worst. They cobble up their single-page leaflets from various internet sites and do not double-check the info, not even on Wikipedia. When questioned why they "teach" incorrect information they would say it is good enough and it will be taught the right way in HS anyway. This is American school system in nutshell: no system, no responsibility. $\endgroup$
    – Rusty Core
    Jun 1, 2022 at 18:49
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    $\begingroup$ Incidentally, we don't need to point our pointy fingers solely at the US. When I was aged 7 or 8 in (UK) school, and I questioned the teacher talking about "the four corners of the earth" (I think she was teaching us fairy stories, I can't remember her ever teaching us anything else) and told her I had been led to believe the Earth was spherical. She led the class in laughing at me, saying "But everybody would fall off!" What was worse was she was young, her husband was a cameraman for the BBC so sheltered and isolated she was NOT. $\endgroup$
    – Prime Mover
    Jun 4, 2022 at 15:33
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    $\begingroup$ @PrimeMover I don't see any contradiction. Can you clarify? $\endgroup$
    – Xander Henderson ♦
    Jun 4, 2022 at 17:21

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