Examples why university education is important for future high school teachers

Question

At my university, the students in math are mixed up (1/3-1/2 are bachelor/master students, the rest are future high school teachers). A problem arising very often is the discussion dramatically summarized by "I don't need this in high school/I already know enough, don't bother me with this abstract stuff".

Are there good examples/showcases/questions/arguments/etc. illustrating how important math education at university is for their life as high school teachers?

(Expect from general discussion like "You need to know more than the school kids/What if they change the content of high school education/What will you tell the school kids when the say that they wanted to study something completely unrelated to math?" - If you want to, you can also add general issues, but I am more interetesed in concrete examples)

Answer 1

I'm not completely sure how well this addresses the question, but here is my best response. A few years ago I was teaching a Methods course for preservice secondary math teachers. Over the course of the semester there was a fair amount of griping, not about my course but about the other courses they were required to take, and in particular about their Abstract Algebra course, which they felt was completely useless for their future careers.

At a certain point in the semester we were doing some collaborative lesson planning for a hypothetical set of classes about exponential functions. Since our Methods class was structured in part around the "backwards design" approach of Wiggins & McTighe, we spent some time at the beginning discussing the "big ideas" of exponential functions. One of my students proposed as a "big idea" that there was an analogy between exponential growth and linear growth: in the functions $f(x)=mx+b$ and $g(x)=AB^x$, the coefficient $A$ is analogous to the constant term $b$ (both give the y-intercept or "starting value" for the function) and the exponential base $B$ is analogous to the slope $m$ (both capture the idea of a "constant growth rate", one additively and one multiplicatively).

After talking about this for a while I said something like: "You know, there is a precise name for 'analogies' like this. A mathematician would call it an isomorphism. In fact what you seem to be saying is that the additive group of real numbers is isomorphic to the multiplicative group of positive real numbers. In fact the properties of logarithms and exponents — like $e^{a+b} = e^a e^b$ and $\log(ab)=\log(a) + \log(b)$ — really don't say anything more than that the log and exp functions are homomorphisms."

One of my students got one of those "aha" looks and commented that I had just taught her more about abstract algebra in 2 minutes than she learned in an entire semester of the other course.

(An even more basic version of this: Students already know that the rules for multiplying positive and negative numbers are, in a sense, "just like" the rules for adding even and odd numbers. Abstract algebra provides a language for describing exactly what this "just like" sense means.)

My point is that there are ways in which higher-level undergraduate math courses (like abstract algebra) can be directly relevant for teaching secondary mathematics... But also that those applications are usually not made explicit. So students are probably not wrong in feeling that those courses are useless for their future careers. It is up to us to connect the dots and show them how and why these things matter.

Edited, November 2020:. I hope it is not considered shilling to add this here, but in the six years since I originally wrote this post I have written and published a mathematics textbook that is explicitly organized around the goal I articulated in the last paragraph: "to connect the dots [between advanced level mathematics and the secondary curriculum] and show them [i.e. preservice and/or inservice secondary teachers] how and why these things matter". The book is called Secondary Mathematics for Mathematicians and Educators, and is published by Routledge. (More information available at the link in the previous sentence.)

Answer 2

Here is some text from the paper "Teacher Characteristics and Student Achievement Gains: A Review" (Wayne and Youngs, 2003):

Three analyses take advantage of the detailed teacher data in NELS:88 on degrees. The analysis by Goldhaber and Brewer (1997a) illustrates the key finding most clearly. No differences were evident when the authors examined whether 10th-grade mathematics students scored better when their teachers had master’s degrees. However, introducing information about the subject of the teachers’ degrees produced significant results. Mathematics students whose teachers had master’s degrees in mathematics had higher achievement gains than those whose teachers had either no advanced degrees or advanced degrees in nonmathematics subjects. In addition, students whose teachers had bachelor’s degrees in mathematics learned more than students whose teachers had bachelor’s degrees in nonmathematics subjects. The contributions of these two indicators of subject preparation were independent of several other teacher characteristics; the student achievement model controlled for certification, mathematics certification, and years of high school teaching experience. The two other analyses of the NELS:88 data replicated this finding, again for mathematics, but with slight variations. Goldhaber and Brewer (2000) used the 12th- rather than 10th-grade students and again found that students learned more from teachers with mathematics majors and from teachers with master’s degrees in mathematics. Rowan, Chiang, and Miller (1997) used a single variable to indicate whether the teacher had an undergraduate or graduate degree in mathematics. In addition, as described earlier, they added a crude control for teachers’ test scores in mathematics knowledge.

Answer 3

The question is this: "why do you want to teach math ?"

I'll try to stay positive here, I think anyone who has had experience with potential teachers is well aware that bad teachers are probably first bad students. Moreover, as one of my colleagues said, difference between doctors and us: they bury theirs we just graduate them.

That aside, I would like for them to be motivated by genuine curiosity for math. If they don't have it already then comments ought to be made to shame them into caring. What are numbers? Why does algebra work like it does? Why do we need calculus? What if you did trigonometry with some shape other than a circle? I read in this book that you can cut a volume into two pieces with the same volume, what's the deal with that? What does it mean to be in 4-dimensions? Why are functions important, who thought of them first?

These are just some examples, but, I think we can agree that a student will be much more likely to respect you as a teacher if you can at least start down a path which tells them how they might want to think to answer such questions. In other words, they ought to study abstraction and origins with some fervor. If your best answer is for them to google it, why bother teaching? As teachers of math we need to know math. It's just that simple.

Answer 4

How can you hope to clearly explain something you yourself do not understand well?

I have found the following line of reasoning is quite convincing for many skeptics:

How hard did precalculus seem when you took it? After taking a semester of calculus, how did your perception change? How about after taking 3 semesters of calculus?

Mathematics is cumulative. The more you learn, the more you reinforce the basics.

By the time you finish the standard sequence of several semesters of calculus, differential equations and linear algebra, you start to find that algebra and the basic precalculus material is equally difficult with basic arithmetic. When first confronted with derivatives, maybe they seemed weird/difficult/tricky, but after a couple of semesters of material building on the basics, taking a derivative is like adding 2 and 2 or breathing.

The point is, the more you know, the more you will consider basic/easy/trivial. When teaching a subject, you want to be able to focus on how you are presenting the material - how to best communicate this to you students. You don't want to be fighting with understanding it yourself.

Along the same lines, the more you know, the better perspective you have as to why different topics are important. The more math you learn, the better you will be armed when confronted with a skeptical student of your own who asks, "Why do I need to learn this?"

Admittedly modern algebra as taught at most universities is a hard sell for you typical unmotivated future highschool teacher (why you would enter such a profession half-hearted, I do not know, but it seems common). If you focus on ring theory, you have a chance at selling the connections to factorization and the like. Groups are trickier because, honestly, that specific material may very well be completely unapplicable. However, students who spend a semester really engaging their modern algebra (or any other proof oriented) class should leave with at the very least bullet proof logic skills. This of course is of immeasurable value when teaching mathematics.

Anyway, that's how I've approached this topic with many students (and been successful as far as I can tell).

Answer 5

I have faced exactly the same issue. I often teach masters level math courses to students getting a masters in education with a focus in mathematics; so these are future high-school math teachers.

The way I look at it, my real task is to provide them with mathematical enrichment and sophistication, to show them a larger context and provide a deeper perspective from which they might look upon the more elementary material with a greater degree of mathematical insight.

But indeed, they will not directly be teaching this more sophisticated material in their high-school classes. Nevertheless, I do believe that the enrichment I offer is indeed enriching, and develops important parts of their mathematical thinking, such as a greater appreciation for mathematical proof and a greater facility with abstract thinking. And this I believe is in turn beneficial for their own students.

Faced with the occasional objection that they don't need that stuff (and it is usually only the weaker students who make such objections), I have found that I can almost always convince them simply by asking:

To teach 2nd grade reading well, does it suffice for the teacher to have only 3rd grade reading ability?

Everyone agrees, of course, that a 2nd grade (age 7) reading teacher should read at far better than 3rd grade (age 8) level. And similarly, for the same obvious reasons, a math teacher at any level should know far more than just the material they are teaching their students.

Answer 6

Of course, as already mentioned, it is important for a teacher to know more than her or his students. But one thing where this principle is especially important is exercise (and test) designing. Let me give a few simple examples.

1. How do teachers manage to chose the second degree polynoms that have not-too-complicated roots? They start from the factorization, which needs a slightly deeper understanding of polynoms than root finding methods.
2. How can a teacher improvise a $2\times 2$ affine system with a unique solution (or with no or multiple solutions) at the board, whenever needed (e.g. some students finished all exercises already)? They use determinant. Variant: finding $3$ affine equations on $2$ variables that have solutions by using row manipulations, which needs some understanding of independence in linear algebra.
3. We know how to compute sequences defined by rank-two induction by reducing the corresponding matrix. This can be used to design a problem (e.g. on Fibonacci sequence) where the relevant change of variable is given to the students.

There are obviously a whole lot of such examples, where deeper understanding of somehow more abstract mathematics than taught is needed for teachers simply to design good exercises.

Answer 7

While teaching polynomials to future high-school teachers, I realized one example of pretty advanced mathematics that is needed to understand why a certain important (but often kept implicit) result in high school mathematics is a result, and not something so obvious that it need not be even stated:

Two real polynomials taking the same value at each point are equal coefficient-wise.

It has proved hugely difficult to make my students realize there where something to state here, because they did not question the notion of equality, and did not realize that the definition of equality for functions and for polynomials might conflict each other.

Now, once one has seen finite fields, one can be given an example of a non-zero polynomial whose polynomial function vanishes identically. I don't see any better way to show that there is something to prove in the real case. There are some tricks, such as using trigonometric polynomials for comparison ($\cos^2+\sin^2$ is coefficient-wise different from $1$).

But the finite-field example made it also possible for me to convince my student that they would never, ever be able to prove that coefficient-wise equality and functional equality agree just by using algebraic manipulations, because they would be able to apply their proof to the finite field case. This helped them accepting that it is worth giving a proof, but also that this proof has to use something more than the field operations (I gave two flavor of proofs, one using polynomial algebra and the fact that there are infinitely many real numbers, and one using topological arguments).

Of course, as high-school teachers they will not teach finite fields, and will probably not insist on the distinction between equality as polynomial and functions. But they have learned that there is something they will be pushing under the rug, making them certainly better at presenting the material, understanding some possible misconceptions by students, and answering some questions if they turn out. These look like invaluable teaching skills.

Answer 8

To teach mathematics, you need to know, first, what mathematics is, no exceptions.

The problem of school-math being a whole different thing having little in common with mathematics, is caused, among others, by people who teach school-math, but have a bad idea about what mathematics is. However, it's a hard chance that someone will have contact with mathematics in high school, in fact, if some student does, with high probability he/she will go to college (perhaps with major other than math, but still).

This might change in the future, but right know, the best option to learn mathematics is to go to the academia. Of course, just attending university courses is not enough, but itself. However, there are opportunities, many times more than somewhere else.

To this end, I would gladly require any math teacher to do some research (I'm not suggesting that teachers should teach research in high school, only that knowing what research looks like in math gives you an appropriate perspective). It might be very small and insignificant, but still. On the other hand, I understand this is highly unfeasible, there aren't enough people that would be able to do it, after all, nowadays you can do a PhD without having an independent thought $\ddot\frown$

So, what's left is hope that going to some university will be enough. It's not encouraging and I wouldn't ever use such an argument to convince some future teacher (the effect might be just the opposite). Nevertheless, it is a real concern in present times.

To conclude: going to universities is the future teachers' best chance to get acquainted with mathematics.

I hope this helps $\ddot\smile$

Answer 9

When my high school offered the AP course in Calculus, the teacher chosen to teach the class was the one who had studied it most recently. She was rusty and it showed. She was continually making mistakes in her demonstrations and the students wound up frequently correcting her.

When I tutored college algebra, I frequently encountered prospective elementary teachers who did not like or understand mathematics, and suspected that their attitudes and lack of understanding would be reflected in their teaching. The same would be of course be true in high-school teachers.

Answer 10

Examples from Abstract Algebra

• The reason why the product of two negative numbers is positive boils down to a proof from abstract algebra. $(-3)(4)$ should be the opposite of $(3)(4)$ because you add the two to get zero. Then $(-3)(-4)$ should in turn be the opposite of $(-3)(4)$.

• Why negative numbers exist, and why subtraction is just adding a negative.

• Why fractions exist.

• Why zero behaves oddly with multiplication and division.

• Why $-1$ is not a prime number. Why $1$ is not a prime number.

• Why $\infty$ is often not considered a number.

• What we mean in general when we talk about a "number".

• $\cdots$

With all of the above questions, I think only a trained mathematician would know how to answer the question honestly if they needed to explain it to a child.

I'm sure there are many examples from analysis too, but abstract algebra in particular is practically the direct study of everything taught in basic elementary school math: integers, addition and multiplication, divisibility, fractions, and so on.

Answer 11

This is but a part of the bigger picture and certainly not the most important one: Speeding up the legwork.

Obviously you have to be better than your students in what you teach to, e.g., design and correct tests in a reasonable time. There are two ways of getting better at such things: Getting more routine with the techniques you are teaching or having a deeper understanding and knowing more advanced techniques. The former has its limits (and so does the latter) and thus you need the latter, at least to some extent.

One illustrative example (that is unfortunately far from the high-school level and also not fully realistic):

Is the following calculation correct? $$152345+10340+532045+139525+10530+123525=968313$$

You can decide about this having learnt only how to add by simply calculating the correct result and checking it. But most likely you will quickly notice that the sum of some numbers ending on 5 and 0 can never end on 3. But that already requires more knowledge than just being able to add numbers mechanically.

Answer 12

I don’t think that you can explain this to the students. Perhaps instead, you should better familiarize yourself with the high school standards and draw the connections.

A high school mathematics teacher will teach material that heavily overlaps with all undergraduate coursework, and is only a slight extension of most ideas.

While a high school mathematics teacher may spend their time teaching a bunch of 9th grade algebra, there is much to connect in their linear algebra and/or modern algebra courses.

They should also be prepared to teach through BC calculus or HL IB mathematics, which are not superficial. Further, AP stats requires a solid understanding of introductory statistics.

I would recommend these students work related to making some of these curricular connections and how they could be creative about teaching them. Incorporate the wealth of research on teaching and learning advanced mathematics, discuss math teams, research opportunities for high schoolers, etc.

High school teaching is a very demanding profession that requires deep understanding of formal mathematics at a high level of you are going to be a true professional and find continuing opportunities for yourself. There should be no argument about this, just look at what high school mathematics students are expected to do in terms of material content through precalc in the CCSS and the AP or IB exams.

Answer 13

I think the HS teachers have a point. It seems like you love the topic and want to find a justification for it (examples why the teachers are wrong) rather than really considering the question openly if maybe they are right. Maybe the reason you strain so hard for a rationale is the the math teachers have a point!

Consider: I like gymnastics as a sport, but it is a stretch to say that working on it will help a high school football player get better at football or be a sound use of his time. (Wrestling on the other hand, especially for defensive players has some noticeable benefit. But is still not a requirement.)