# Why do you need to study a subject at a university level to teach it at a high school?

If you want to teach mathematics at a high school for example, then why do you need to study it at a university? You will learn new stuff, but you won't teach anything of that at a high school.

• Comments are not for extended discussion; this conversation has been moved to chat. – StrongBad Feb 20 '18 at 15:21
• Because one day you get a student who knows all high-school mathematics, and wants to learn more. I did rotation-volumes, some linear algebra and Laurenz-transforms in middle-school! (I was lucky to have a teacher that could teach this), so you could imagine what would have happened if I did not go to a math-specialized high-school... – Per Alexandersson Feb 21 '18 at 10:24
• First of all, the pretext of this question is not true! In many states/countries this is not a requirement. Some municipalities just require sufficient knowledge of the subject which can be demonstrated in different ways [usually exams or degrees]. You certaintly do not need a math degree to teach math in high school in most states in the USA for example. Many of my math teachers had engineering, physics or other related [even unrelated] degrees. – WetlabStudent Feb 22 '18 at 4:26
• See the (virtually identical) question at MESE: matheducators.stackexchange.com/q/130/29. – mweiss Feb 22 '18 at 17:57
• "You will learn new stuff, but you won't teach anything of that at a high school." Of course you do. – pmf Feb 23 '18 at 9:25

On one hand, when I myself was in high school I was wondering about the same thing - especially that I learnt the whole high school material (in maths and physics, I mean) in the first year, so felt that I could have taught as well as a teacher. But indeed, as pointed out by Xander Henderson and Massimo Ortolano, only at the university (in fact, only during my PhD studies; well, in fact only as a post doc; well, in fact... hm, you get the idea) I learnt the bigger picture.

So, studies are so that if a student asks "what will I need the quadratic equation for?" you could give a better answer than "because it will be on the test".

What is a better answer in this case is a whole different issue. I have lots of friends who indeed, once they finished high school, never solved a quadratic equation again. But on the other hand, they also didn't need the knowledge about Hamlet, genetics, WW II history, etc., in everyday life. That doesn't mean that they, we, you shouldn't learn about those things. I have lots of knowledge that I don't need in my life - but I enjoy obtaining it, possessing and connecting it with other areas.

• "Because it's on the test" might be a bad answer, but its possible to do worse :) smbc-comics.com/comic/why-i-couldn39t-be-a-math-teacher – Clumsy cat Feb 19 '18 at 8:24
• @TheoreticalPerson That one's actually true ;) – corey979 Feb 19 '18 at 10:20
• This answer neglects to weigh the benefits of teacher erudition against its costs. It costs students: for a low-income school district in the US, following this policy increases the costs of math so corners are cut elsewhere, which might offset any advantages. (Also those students may especially want practical applications, which makes teacher's invocations to things not taught there seem pretentious.) It costs teachers years of debt: while in debt, they're meeker about, and thus more vulnerable to, administrative errors and systemic problems. – agc Feb 19 '18 at 18:51
• Regarding whenever you get a "When will I use...". Say that it is the wrong question and answer with a counter question "which of the things I have learned can I use in this situation?". If you haven't learned enough stuff in a situation like that then you won't be able to answer that question and then you will get stuck. – mathreadler Feb 19 '18 at 21:01
• Agreed, but I note that in my opinion we DO use "Hamlet, genetics, WW II history" in everyday life, because everyday life is more than work. – Rolazaro Azeveires Feb 20 '18 at 11:40

History and experience shows that those who do not have advanced training in, e.g., mathematics, and try to teach or write about it, give presentations that are somewhere between hilariously and atrociously wrong, broken, mangled, incorrect, damaging, and misleading. Secondary institutions can partially mask this by teaching to standardized tests (esp., multiple-choice ones), but the damage to actual understanding is still done, and people suffer from it later.

For an example attempt at remediating some of these effects, consider Hung-Hsi Wu's papers on "Teaching School Mathematics".

• In an ideal world one would think that those interested enough in mathematics to make a career out of teaching mathematics would jump at the chance to take a few mathematics courses in college rather than spend all their time in college taking courses in classroom management, educational policy, school psychology, educational measurement, cognitive science, human relations, along with the usual assortment of required (for most U.S. degrees) general electives in literature, history, sociology, etc. – Dave L Renfro Feb 19 '18 at 9:10
• Indeed I think multiple choice test were invented for that! Tough I had a excellent and passionate teacher that, based on his pedagogical research work, was sponsoring them. But he made oral examination too. – Alchimista Feb 19 '18 at 9:20
• I would add that the curriculum is not set in stone - it is easily possible for topics to be added that the teacher may not have studied in secondary school. It is still possible that a teacher could miss out on the particular topic at uni, but at least by studying to a higher level, they will be able to pick up the additional topic easily and thoroughly. Also, in my home of Victoria, Australia 3 possible maths subjects in the last year of high school. Someone taking the two math combination (M & S) needed to study maths at uni won't see the topics in the other subject (F) but would at uni. – Robert de Graaf Feb 22 '18 at 23:30
• @DaveLRenfro (et. al.): I might speculate that some people aim to become high school math teachers precisely because of the misconception in the question here -- belief that it's a trivially easy job on cruise control all the time. (I've personally seen cases of this.) – Daniel R. Collins Feb 24 '18 at 17:29
• Those interested in high school teaching should try substitute teaching first. Daniel, I suspect you've seen these before (cited in Mathematics Educators Stack Exchange), but for substitute teaching see this post, then this post, followed by this post, and then this post, and finally the posts I cite here. – Dave L Renfro Feb 24 '18 at 20:17

In short: If somebody doesn't like the idea of studying their subject at a university for a few years, I really don't like the idea of that person teaching children (in addition to the pragmatic reasons given by others).

Personally, what I remember most about math in high school was that my teachers were enthusiastic about the what they did (despite going into math, I don't recall learning any particular thing in high school; just that I got the standard fare).

I certainly would prefer students learn from somebody who at least enjoys their subject. While "study subject X for Y additional years" certainly doesn't guarantee that the person likes subject X, I'd hope that it discourages those who don't like the subject from pursuing this path.

I say this because I find few thoughts more depressing than thinking about the experience of a child learning from somebody who fears or resents the subject. And yet, it happens -- I've seen education majors that don't like or don't understand basic mathematics which they may some day teach...

• And sadly, little children learn math all the time "from somebody who fears or resents the subject". Grade school teachers often did not study math in-depth but just had a general university education, if at all. They became grade school teachers because they love little children, not because they are passionate about science. Math can be hard for anybody (me most certainly included) when leaving the intellectual comfort zone. For some grade school teachers it was an unloved subject, and that sentiment is inadvertently passed on to their students. – Peter A. Schneider Feb 21 '18 at 15:10
• @PeterA.Schneider It's true. It's a tough problem: It's understandable (but still unfortunate) that this would happen for primary/elementary schools, since one teacher teaches a variety of subjects. Still, later in an education, when the teachers usually "specialize", it's easy to imagine smaller (especially rural) schools where the teachers need to teach outside their comfort zone for some subjects. So, my answer really applies to some ideal version of the world, I suppose. – pjs36 Feb 22 '18 at 4:51
• What's more, math anxiety among female teachers leads to math anxiety among female students, which tends to perpetuate the underrepresentation of women in math-intensive subjects. – Michael Seifert Feb 22 '18 at 22:49
• "If somebody doesn't like the idea of studying their subject at a university for a few years, I really don't like the idea of that person teaching children" So you would turn away a renown business leader, if he doesn't have a degree, or a master carpenter (for wood working class) if the doesn't have a degree? – NPSF3000 Feb 25 '18 at 15:22
• @NPSF3000 This is a fair point. I had historically academic subjects like math, science, "English", history, etc, in mind. – pjs36 Feb 25 '18 at 18:50

If high school instruction was mere recitation of the information contained inside the textbook, I dare say that having a teacher at all would be completely unnecessary. The role of a teacher is to explain the material in an easily digestible manner, and more importantly, to be able to answer questions and correct mistakes. A mere high school education is not adequate to achieve these tasks. Consider this: a student who receives a 95% average in a high school course would certainly be considered to have a commanding grasp of the material, and would likely be near the very top of the class. If teachers were not required to gain further education in their subject area, then certainly a student with a 95% average would be an ideal candidate for a teaching position. But that individual doesn't actually understand the material completely! The fact that they didn't get 100% in the course means that they were tripped up by some homework or exam questions, so their own knowledge base is imperfect. Without further work and education they will have great difficulty correcting their misunderstandings and lack of knowledge.

Alternatively, in the interest of making sure that the teacher does not make fundamental mistakes or errors in their teaching, it is of paramount importance that they have further background in the material. Otherwise, they might be unable to answer difficult questions asked by curious students, or even worse, they might answer incorrectly; such a thing would only serve to confuse the students and slow down their education. Such incidents can truly leave a lasting negative impression; as reference, consider the following quote from the introduction to Jiří Lebl's book Basic Analysis: Introduction to Real Analysis

Let us use an analogy. An auto mechanic that has learned to change the oil, fix broken headlights, and charge the battery, will only be able to do those simple tasks. He will be unable to work independently to diagnose and fix problems. A high school teacher that does not understand the definition of the Riemann integral or the derivative may not be able to properly answer all the student’s questions. To this day I remember several nonsensical statements I heard from my calculus teacher in high school, who simply did not understand the concept of the limit, though he could “do” all problems in calculus.

While the above quote is in reference to high school mathematics teachers, the general idea certainly applies to instructors from all fields. Would you really want an English teacher who hasn't written a paper since high school? Certainly such an individual would be unlikely to be able to adequately offer writing advice to students, because they are at a comparable level of skill themselves. What about a physics teacher who only understands physics from an "intuitive" perspective, but is not able to explain anything more advanced? All in all, in most cases a high school teacher without considerable further education in their subject matter would not be able to offer instruction at the same level of quality as someone who has a bachelor's degree or more.

• I want to emphasize this part of your answer: "The role of a teacher is to explain the material in an easily digestible manner ...." A big part of teaching well is figuring out how to order parts of the presentation, how to conceptualize material, what analogies to use, and sometimes exactly what words to use in a particular sentence to best allow material that was initially foreign to a student to gradually slip into and re-form their thinking. Learning advanced material uses basic material, but in new ways, and with new connections to other parts of math, thus allowing more ways to teach. – Mars Feb 23 '18 at 5:26
• A small comment: even if the students gets 100%, that doesn't mean that such student has the maturity and/or depth to explain the topics to others, to assess progress, and to address questions. – Martin Argerami Feb 24 '18 at 16:52

I'm going to beg-the-premise of the question. It is not universally true that to teach a subject in high-school you need to major in it at university.

In Australia it is very common for high-school teachers to only have a teaching degree, or to teach subjects outside their original qualifications. (For example, the best drama teacher I had, had never formally studied drama. He was primarily a wood-working teacher, who happened to do semi-pro productions on the side, and was shoe horned into the position when our original teacher quit.)

My recollection was that the teachers who didn't have degrees majoring in their subject did not/could not teach it for the the university entrance exam level, but handled it at the introductory level. This makes sense -- one does not have to complete even high-school level calculus before one can complete a general teaching degree.

Conversely, the fancy private schools like to boast that all of their teachers have PhDs in the fields they are teaching (as well as a Grad. Dip. in teaching). No doubt this helps justify term fees that are as high as for an university education.

• In the USA, it's common that someone who is already licensed in one subject can become licensed in another subject by passing a standardized exam in that subject. E.g. if you studied English and became an English teacher, you can become a Math teacher by taking and passing your jurisdiction's Math Test for Teachers. Whether or not any particular person with an English degree is likely to know enough math to pass such a test, one can only speculate wildly. – Robert Columbia Feb 19 '18 at 13:03
• @RobertColumbia: A buddy in my year at university did a double major in Physics and Math, with a minor in English. Four night-school courses would have had him capable of teaching in any of the thee subjects, but he makes more money leading a cryptographic team at M.I.T. – Pieter Geerkens Feb 20 '18 at 12:40
• Having a Phd in anything doesn't mean one can be a successful teacher in that subject. Passing exams and writing papers isn't the same as teaching! – Tim Feb 25 '18 at 17:51
• @Tim Correct. And I never said otherwise. – Lyndon White Feb 26 '18 at 2:36
• I never said you did !! Just underlining what you did state, as it emphasises what you said about private schools. I taught for 40 yrs in state schools, no degree necessary, and my point is that it doesn't make a good and effective teacher. All it proves is you know a lot about a subject ! – Tim Feb 26 '18 at 7:50

The main point is summarized in the Pólya's rules of teaching:

The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach. (How to Solve It, p. 173).

But why we have to know "a little more"? Well, Steven G. Krantz has a reasonable answer:

One of the best arguments for even elementary college mathematics courses to be taught by people with advanced degrees is this: Because the material is all trivial and obvious to the professor, he can maintain a broad sense of perspective, he will not be thrown by questions, and he can concentrate on the act of teaching. (How to Teach Mathematics, p. 2).

• Too bad for his misuse of the word trivial. – AlexHeuman Feb 19 '18 at 20:20
• @AlexHeuman In general, the said word "is used to describe any result which requires little or no effort to derive or prove" (see this). In this sense it seems that, for people with advanced degrees, the material of elementary college mathematics courses is indeed trivial. So, why did you say misuse? – Pedro Feb 19 '18 at 22:47
• I suspect because in mathematics "trivial" has also the technical meaning of vacuously true, that is, true by lack of counterexamples, but I don't think that the common usage is a misuse. – Massimo Ortolano Feb 20 '18 at 0:03
• "Trivial" here is used in the exact historical sense: material covered in the first years of a university education. – Stephan Kolassa Feb 22 '18 at 12:49
• @MassimoOrtolano 'Vacuously' means 'true because the hypotheses are never true', whereas 'trivial' means 'true because the conclusion is never false'. – Jessica B Feb 22 '18 at 15:31

Some comments mention that the reason of “because you need to know MORE than what you need to teach” cannot explain why elementary or middle school teachers need to get a university degree.

At least in math, the “more” that you need to know might be at an incredibly high level, even for teaching elementary school. My university requires math education majors to take modern algebra, which they probably will never use directly. When they ask me why they have to do this, I tell them “So that when a fourth grader asks you why a negative times a negative is positive, you can appreciate the complexity of this question, and just maybe give a satisfactory answer.”

• +1: I would add things like "and why $3^{-1}=1/3$". – Martin Argerami Feb 20 '18 at 4:58

When I was in high school, I had a math teacher who did not have a degree. One day, we were learning about pi. The never ending digits of pi fascinated me. How could you ever know that there would not, somewhere over the next horizon, be an end to them? So I asked, in class, how people knew that they were infinite. My teacher gave me an annoyed look, said: "They just are.", and went on with the lesson. From the way she said it, it was clear that she also had no idea.

Because of things like these, I lost all respect for her, and I started to dislike math. I also felt like there was no real reason to listen to her teach, because she was just reading from the book. I also stopped asking questions that were not immediately relevant to exercises. None of this is good for learning.

In addition to this, getting a passing grade is not nearly mastering all of the material. With just a passing grade in high school math, you can solve the problems mechanically, if they are asked in a standard way, most of the time. It is better if a teacher is beyond that level, because they will see where someone is struggling more quickly, both during classes and while grading.

That said, all degrees will have some things that are useless to some students, and that is just the nature of teaching and learning. Education is expensive, highly specialized education prohibitively so, and so people often get a general range of subjects, instead of just the things they need. Making a specialized math course for every program that has math in the curriculum is impossible, so you will end up with math teaching majors learning math with everybody else. It's not a complete loss, though, because you can often use things in unexpected places.

• That's a sad story; however, I'm not convinced there is a correlation between the curtness of a short answer and the education level of a teacher. I would not be surprised to hear many anecdotes where highly-educated teachers might have stifled curiosity, or less-educated teachers might have fostered it. – J.R. Feb 22 '18 at 14:55
• @J.R. A teacher who actually know an answer has at least an opportunity to answer. A teacher without sufficient background is cornered by every question slightly beyond curriculum. – Benoît Kloeckner Feb 22 '18 at 21:17
• @BenoîtK - I know plenty of good teachers who, instead of feeling “cornered,” simply said, “That’s a good question; let me get back to you on that.” Interestingly enough, the pinnacle of our profession is research – guiding students toward answers to questions that no one yet knows. – J.R. Feb 23 '18 at 11:50
• @J.R.: Sure, good teacher can be asked question they don't know the answer of yet. But without a good mastery of the field (I am really meaning mastery, not universal knowledge) it is very difficult to find the answer to some questions, even with time ahead. You would maybe be surprised at the level of non-mastery displayed in the selective examination to hire math teachers in France -including by applicant who are ultimately hired. More to the point, knowing very well the high-school curriculum does not give the mastery and perspective to answer some questions. – Benoît Kloeckner Feb 23 '18 at 15:56
• @J.R.: I would add that given the question, one should think of setting expectations that shall give good results overall, in a system which needs to hire a large number of teachers. Then it makes a lot of sense to ask them to have some margin compare to what they teach, because it should help more of them to handle such questions appropriately. – Benoît Kloeckner Feb 23 '18 at 15:58

It's not the advanced "stuff" that is so important. What is important is the mastery. You want the teacher to have not just learned the content, but made it their own. That kind of mastery can be had without a university degree, but a degree is the simplest proof that one can posses that indicates their intellectual depth.

For me, the pattern is more obvious in the martial arts than it is in academia. In martial arts, in theory all one needs in order to instruct a white belt in the techniques they need to attain a yellow belt is a yellow belt. At that point, you've memorized all the things there are to memorize about these techniques. But we don't rely on a yellow belt to teach. The role of teaching goes to a higher belt. Most often this is a black belt, but sometimes a purple belt will do the teaching (as they are being taught how to teach). You want someone who not only has memorized the techniques, but someone who has internalized them and can express them fluidly. You want someone who won't teach you their own mistakes, because they have had enough time to work those mistakes out.

In the martial arts belt system, one view is that achieving the blackbelt is achieving "minimal competency in the material." It is the first point where you are truly expected to lead your art, and have others follow you. In our current academic system, a university degree is similar in nature to the black belt.

Is it truly required? Probably not. You can develop this breadth and depth of understanding on your own, without a university. But history has shown that it is an effective line in the sand to draw, so we draw it.

Teaching is, surprisingly, harder than it seems. At first glance it would seem that teaching is just repeating a set of materials, rinse and repeat. And to some extent there are a sub tasks where this might be true. However, there are quite a lot of mathematics, or any other skills that you just dont learn at high school level except very superficially.

Now do you want to be taught by a person who is just slightly better than you? No offcourse not. Why? Well simply a good teacher can teach the same thing in several ways. Different learners need different strategies, and it is hard to convert something you know how to do into one let alone three or four different explanations and approaches. Also subjects on pedagogy on sufficient level may not be offered at high school level.

There is also a second reason for having a higher education. Things change, and university level education is primarily a education into: How to find out stuff and learn new things rather than learning methods. Eventually something might change, maybe it is decided that you need this new subject matter (ok so its Math) in math. So a teacher needs to have a good enough palette in learning to go and learn that new thing, or at least the employer and parents want this.

• OMG! "teached" .. My graduated professor will say "taught " .. :) – ingconti Feb 25 '18 at 14:42

Another reason that has not been mentioned is that studying the type of math that deeply challenges your mathematical abilities is essential for developing humility and empathy for your students.

If every piece of math you ever learned was easy for you, it will be difficult to relate to a student struggling in your class. If you, however, for example struggle through a few upper [or even graduate] level math classes, you will learn to empathize with students who don't understand the more basic things you teach. We all reach what we think is our mathematical wall at some point, we must experience the process of pushing that wall further out and struggling to gain new knowledge.

My experience is only from German maths, but summarised I would say the following:

1. It is really helpful to study background material to the "school maths", like geometry, calculus, linear algebra because you begin to "really" understand what is happening there.
2. At university, prospective teachers are usually put into courses that "already exist". The result: The first year is useful, after that they often study very advanced topics with no connection to school maths at all (like topology, Galois theory, partial differential equations,...).
3. Many students who want to become teachers lack motivation for learning these advanced topics.
4. So the standard way of educating teachers may be cheap, but is not very efficient.
• Galois theory has a connection to school mathematics (extending the quadratic formula to higher-degree polynomials and usually failing in degree 5 and higher). – KCd Feb 21 '18 at 6:54
• @KCd I expected a comment like this. My opinion: I school, you solve quadratic equations. It is good to know that equations of order 3 or 4 have formulas, higher degrees don't. When I studied, this was one lecture in a one semester course. The rest was about rings, fields, extending fields etc. I have done a lot of work for maths olympics. There, you find a lot of interesting extensions or generalisations of school maths. Unfortunately, next to nothing is taught to prospective teachers. – J. Fabian Meier Feb 21 '18 at 8:18
• When teaching Galois theory I also show how to rationalize cubic or higher degree denominators (like rewriting the inverse of 5+2^(1/3) + 4^(1/3) as a sum with rational coefficients), which is connected to the high school topic of rationalizing degree-2 denominators and uses linear algebra in place of the trick of multiplying by a single "conjugate" term. I agree that few future teachers take a second semester of abstract algebra (although in my experience it is not an empty set), but there are definitely several connections to school math topics. – KCd Feb 21 '18 at 14:37

Since many good answer were already given, let me list a few reason very shortly: to teach high-school, one should have learned one's field to a much higher level and breadth than high-school curriculum because:

• one must be able to adjust to curriculum changes,
• one need to confidently spot errors in textbooks and correct them for students,
• one need to construct exercises for the class,
• one need to answer questions that go beyond the curriculum,
• one need to know where some simplifications happens for the sake of teaching, and know the unsimplified version to safeguard against possible mistake coming from the simplification.

I could give a striking example in my field, mathematics, showing that to teach polynomials it is a good idea to know about finite fields, but I said I would be brief.

If you want to teach mathematics at a high school for example, then why do you need to study it at a university?

Why questions can be answered at many many levels, and all of the answers I skimmed are stuck on only one side of one level of analysis, merely trying to present rationalizations for current law/practice.

I don't know the 'real' answer to your question, but I want to present:

## Potential answers, unmentioned in other responses, at different levels of analysis

• The law/policy says so.

This prompts Second Why do these laws/policies exist?

• Teacher's organizations don't want competition. If just anyone is teaching they have to compete more in the education market, which means it's harder to charge a high price. The size of the educational market is limited to people interested in being students, so on some level it's a zero sum game. Their organizations therefore do things like lobby in government, lobby school boards, create accreditation boards all to require new teachers to spend more time training. They justify it by claiming it's required for quality. Obviously these things can improve quality somewhat by removing charlatans, but they also eliminate all good/potential teachers who won't or can't get through their red tape.

Further, this is a ratchet that creates ever tightening requirements. Story example: 1st generation of teachers with an AS try to cut out high school grads by demanding a rule that you must have an AS. They eventually succeed. 2nd generation of teachers all have AS due to new rule, so some get a BS to show they're 'better qualified'. Teachers with BS in second generation generation try to cut out teachers with only an AS by demanding a rule that you must have a BS. Currently you need an MS, but don't be surprised if, in a decade or three, you need a PhD to teach high school.

Disclaimer 1: I want to stress I'm not taking a political position here. I'm only pointing out that requiring education beyond the material a teacher will teach is a trade-off. That trade-off may very well be worth it to most people, but the existing answers to your question show people aren't taking the time to seriously think about whether there are any downsides.

Disclaimer 2: Further I think most all teachers really do care about delivering good a educational product. Many if not most teachers I know would never consciously want to do anything specifically to limit competition. But their position as paid educators gives them a completely understandable bias towards the regulation side of the above tradeoff.

• People conflate depth of knowledge with mastery of knowledge People believe teachers should know more depth than just the information they teach. However while excess depth isn't a bad thing, it's never required. The grain of truth is that the extra study required to get more depth likely also confer more mastery of foundational topics. But it's also possible to build mastery of a foundational topic without studying deeper topics as you do in a graduate level program. Learning calculus may give you enough practice in algebra to make you very good at algebra, but you could also master algebra by practicing it directly.

• There is an over-supply of qualified teachers. This ties into both the anti-competition and depth of knowledge points. If you have more people who are interested in teaching than you have teaching positions, you can start being more choose-y about who you want to teach. Even without lobbying, you get high requirements as a de facto standard in job openings. This happens because employers need a way to filter out the sea of applicants, and they know there are plenty of applicants available even after adding the high requirement.

One major piece of evidence for this is the relatively low salary of teachers relative to other professions with similar education requirements, and relative to tuition. Over supply typically leads to low prices (in this case "prices" being wages). In the US at least this is heavily distorted by the fact that most education is government run and heavily regulated, not market based. There's no certainty that tuition would remain so high if all tuition went directly to teachers who then determined how to best use it to teach. But there's a market in some segments, and teacher salaries are still low there relative to some other master/PhD based careers.

## TL;DR: The idea that a master's or a PhD is important to teach it is not without merit, but it's certainly not the only potential answer to a your question.

A math teacher who does not know that

• the real numbers are an algebraic field;
• set theory and boolean algebra are isomorphic (and what an isomorphism is);
• modern math constructs the natural numbers from nested empty sets,

just to name a few, cannot competently teach 1st grade math. The reason lies in what I consider one goal of education in general, and here in math particularly:

Education should not be limited to conveying some degree of useful technical proficiency.

That is, the goal of teaching English1 should go beyond the ability to read a manual or write a shopping list. It should teach about poetry, the human condition, the beauty of language and rhythm; it should instill wonder and curiosity and open up perspectives which are perhaps not available at home or on TV.

The same thing goes for math. Math education1 should not be constricted to counting correct change or buying the right amount of wallpaper. Mathematics is about beauty and symmetry. It is about the power of abstraction. It is about beholding different infinities. It is about the realization that the mind has no limits. Math is like mental CGI; if you can think it, it is there. Math is as creative as sculpting: What you carve out of reality has always been there, you just made it visible. But it is about discipline and correctness, too. Like literature, math shows us what's underneath the surface.