What math professors and k-12 teachers think of each other

By Michael Pershan, St. Ann’s School

I.

What do primary/secondary math educators think of the teaching that happens in colleges? And — the other way around — what do mathematics professors think of primary and secondary math teaching?

I’m nearing my tenth year as a primary and secondary classroom math teacher, and every once in a while I end up in a conversation with a graduate student or professor who suggests (politely, almost always!) that math education before college is fundamentally broken. A few weeks ago, a mathematician told me that PhDs are needed to help redeem secondary teaching from its “sins.” Once, at the summer camp where I teach, a young graduate student told me that there is simply no real math happening in American schools.

Well — I disagree! But how widespread is that view? And why does it exist?

The flipside phenomenon is also interesting. When a mathematician criticizes primary/secondary math education, primary/secondary educators sometimes lash back. Often we point our collective finger at pure lecture. Primary/secondary educators tend to think of pure lecture as uniquely ineffective. It gives the teacher no knowledge of whether students understand the material, and students no chance to practice new ideas in class. It is rarely used in primary/secondary math classes. Still, pure lecture was the main teaching mode in my own college classes, across subjects. We therefore bristle at mathematicians critiquing our work; “Let those without pedagogical sin throw the first stone!” I’ve even said this before, or something not far from it.

I have heard this “anti-lecture” critique expressed by some primary/secondary educators, but I wonder how widely held this view is. Is it held even by some math professors? And, in general, do primary and secondary educators tend to see flaws in the way math is taught in colleges?

In short, I wanted to better understand how mathematics professors and educators of younger students relate to each other’s teaching.

Needing a way to approach the question, I created a survey and shared it on social media. I asked people to share their job, the highest degree they have earned, and their views on primary, secondary, and post-secondary education. So far, thirty-four people working in math or math education shared their reactions to these three statements:

• “Primary (K-8) math teaching is generally effective.”
• “Secondary (9-12) math teaching is generally effective.”
• “College (or University) math teaching is generally effective.”

They rated their agreement/disagreement on a scale of 1 through 5 and explained their response at whatever length they chose.

Disclaimer: since the survey made no effort to be representative, themes and patterns that emerged from the responses are at most suggestive. Nonetheless, suggestive themes and patterns did emerge, some of which surprised me.

There were three questions I wanted to more clearly understand:

1. What exactly are the issues people have with math teaching at the elementary, secondary and post-secondary levels?
2. Do people tend to have a rosy view of their own setting while leveling harsh critiques against what goes on in other settings?
3. Where do extreme views of different educational settings come from, and why are they sometimes so deeply held?

The background question, though, the one driving this whole project, is whether there is any chance of coming closer together. What are the best ways for us to learn from each other?

II.

One mathematics PhD who responded to the survey rated the effectiveness of primary math teaching as just 2 out of 5. This same person rated secondary and college math both at 4. Here is how they explained the ratings:

My experience with a couple of districts is that the primary teachers up through Grade 4 tend to feel uncomfortable with math, so they do their best to opt out of teaching it as much as they can.  I have some horror stories, including concerned parents of students who brought their math workbook home at the end of the year untouched.

Here is a college mathematics professor voicing similar views:

I think many elementary teachers are math phobic or have math anxiety, and this impacts how much time they spend on math. Also weak skills and knowledge of teachers could lead to misconceptions for students.

This view — that primary teachers tend to be uncomfortable with (or ignorant of) math, and therefore avoid it — showed up again and again in the survey responses.

I was expecting this, because it is maybe the most prominent critique of math education at any level. It’s the sort of thing that, every so often, pops up in the New York Times. A prominent version of this complaint, for instance, comes from Hung-Hsi Wu:

My own observation is that among teachers, especially elementary teachers, their prolonged immersion in textbook school mathematics has often rendered them incapable of routinely asking why, much less looking for the answer.

What surprised me, though, was how frequently this concern (or ones like it) was also voiced by those working in primary and secondary education. In fact, the math PhD quoted at the top of this section is actually a high school teacher. Meanwhile, a 5th Grade math teacher rated primary teaching’s effectiveness at 2 out of 5, and explained that what primary teachers need most of all is the help of math pedagogy specialists. They can’t handle the math on their own, it seems. Likewise, a high school department chair blamed “lack of content knowledge” for a “strong focus on algorithms” at the elementary level.

The second half of this department chair’s worry — the “strong focus on algorithms” — appeared often on the survey as well, though expressed in slightly different ways. Here is a sample, along with the respondent’s professional role and their rating of primary education’s effectiveness:

Too often students are taught processes instead of concepts (8th Grade math teacher, 3 out of 5)

In the memorization and skills-based way mathematics is taught in most K-8 classrooms, I do not think teaching is effective. (High school math teacher/community college adjunct, 2 out of 5)

Are there shades of distinction between “algorithms,” “processes” and “memorization”? Maybe. But I think there is more in common than not in these complaints, and they are probably trying to say something like what Wu said above — there is no thinking going on in too many primary classrooms. The reason? Because teachers fear or misunderstand mathematics. I would say that this picture is the major critical narrative facing primary education.

To put my cards on the table, I think this narrative probably overstates the problem, though it gets at something real. I’ve heard something like this story told by many primary teacher educators and coaches — people who really would know. I do think this critical story misses two important things: (1) what looks like a mindless call for a procedure to an adult is often a thought-provoking and interesting problem for children, when presented appropriately and (2) a lot of good schools do a great job of helping math-phobic teachers teach math at a high level. They provide training, coaching and strong curricular materials that can help teachers overcome their fears and become more mathematically confident. My wife, for one, worked through a lot of her math phobia when she used the TERC Investigations curriculum to teach multiplication to her 4th Grade class.

So much for primary teaching. What about math teaching at colleges and universities?

My own college teaching relied almost entirely on pure lecture as a classroom teaching technique; often the teacher would not even pause for questions. I was fully expecting to hear this come up in the survey. I thought I would hear it from my primary/secondary colleagues, but I was curious to know whether any of those working in higher education would raise the “pure lecture” critique themselves.

The answer was no. The “pure lecture” critique certainly did come up, as expected. It was almost entirely raised by those working in primary/secondary teaching. The high school department chair (quoted above) mentioned “a preponderance of lecture as the instructional strategy” in college classes as an issue. An instructional coach at a middle school bemoaned “lecture-based math classes.” Another response, from someone who works with middle school students, critiqued “lectures that are too difficult to follow, or very hard to be engaged in.”

This “pure lecture” critique was not raised at all by professors or graduate students. It came only from primary/secondary educators in my survey.

However, college mathematics teachers did raise other issues. One college teacher (“assistant professor of mathematics at a small private liberal arts college”) described college math teaching as “highly variable, and depends strongly on how much the institution and the individual instructor value teaching as part of the academic job.” Another (“visiting Professor of Math at a 4 year college”) wrote that “the biggest issue for post-secondary mathematics is the mindset of I know mathematics therefore I can teach it and I will have minimal pushback because I have a terminal degree in mathematics.”

This narrative is different than the “lecture” complaint. It alleges that some professors either do not value teaching highly or that they are too confident in their ability to teach well. Strong content knowledge can perhaps present its own challenges for a teacher of mathematics; deep knowledge can make it difficult to understand the point of view of the struggling student. Curiously, this was only brought up by college teachers; primary/secondary teachers didn’t mention it.

To sum up the situation, in this informal survey, the knock against primary teaching was that its teachers avoid or misunderstand math. This results in students being presented with a distorted picture of mathematics as a subject devoid of thinking but full of procedures to follow. You could even hear this critique from primary/secondary educators themselves. On the other hand, primary/secondary educators were somewhat apt to critique college teachers as relying too heavily on pure lecture. College teachers themselves did not bring up lecture, but did mention the relationship of professors to the work of teaching as an area of concern.

One last thing: I haven’t mentioned what people said about secondary teachers! This is for two reasons: (a) there’s plenty to talk about with primary/university, but also (b) secondary teachers seemed sort of stuck in the middle on the survey. The critiques they (we!) received seem to me best understood as watered-down versions of the concerns leveled at primary teachers. Secondary teachers are taken to have stronger content knowledge, but in the survey we were said to be too procedural, too dry, too focused on memorization and not enough attuned to the needs of the discipline. There were more strong opinions about primary and college than secondary teachers, who seemed to get a slightly different, but weaker, version of the critiques aimed at primary teachers.

III.

I had assumed that the responses to my survey would be super-skewed, with everyone defending their own turf but taking issue with the work in other educational settings. But for the most part this was not true. Without getting statistically precise about this, people’s ratings stuck pretty close to whatever their own baseline was. People who thought math education was basically working across the board didn’t distinguish a lot between primary, secondary and post-secondary schooling. Likewise, there were many others who thought that math education was across the board not getting the job done, and they didn’t distinguish very much between settings.

(Getting statistically precise: people’s ratings did not deviate much from the mean of the three ratings they provided. The absolute deviation from the mean of their three ratings was 0.493, on average.)

I was surprised by this. I had expected mathematicians would have much harsher things to say about primary/secondary education, as compared to their own work. But, at least on my little survey, asymmetrical harshness turned out to be the exception rather than the rule. For instance, one respondent with a Master’s in mathematics said, “The early grades K-3 tend to explore more and most kids are not left behind.” That doesn’t sound damning at all!

Echoing this was a PhD and college mathematics instructor: “Especially with better training nowadays, I generally think that elementary teachers do a pretty good job.” That’s also pretty positive; this person gave primary and secondary teaching a 4 and college teaching a 3. People who were overall cheery about math education tended to be so across the board, even across educational settings.

The flipside also tended to be true. People who were critical of math education overall often made significant criticisms of their own educational setting. The source of many strident critiques against primary education, for example, came from those who oversee primary math education. One of my favorite lines on the survey was from a mathematician who explained their middling evaluation of primary education: “I read about it in NCTM literature.”

(NCTM is the National Council of Teachers of Mathematics, the largest professional organization for primary and secondary math educators in the United States. They have been trying to reform primary/secondary math education for decades and are often highly critical of the way mathematics is typically taught to younger children.)

This is interesting! It suggests that we are not exactly a profession divided against itself, as much as a profession that can’t seem to agree as to whether things are fundamentally broken or not. (I tend to side with the “not”s, for the record.)

In general, things were both less critical and less adversarial than I was expecting. People tended to judge the entire math education system, from primary through university, as a whole. One high school teacher said, “I’m certain that experiences differ widely (as with primary and secondary), but I had good university teachers and I learned a lot.” One mathematics teacher educator repeated the same comment for primary, secondary and college education: “Too much focuses on procedural knowledge and less on higher-order thinking.”

Both of those views makes a lot of sense to me; whether good or bad, we’re all in this together.

IV.

To summarize:

• There wasn’t a great deal of quantitative polarization; people tended to be overall happy or overall unhappy with how things are going in math education.
• There was, however, a great qualitative difference in the issues critics recognized in primary, university and (to a lesser extent) secondary math teaching.

It’s worth dwelling on this, if only for a moment. Critics identify entirely different flaws in primary and college education. The “bad version” of primary teaching looks almost nothing like the “bad version” of college teaching. The stereotypes, to whatever extent they are believed, are completely unlike each other.

This is fertile ground for extreme views. Under these conditions, people can identify problems with other areas of math education and think to themselves, nothing like that is happening where I teach — and they would be largely correct. If you are a middle school teacher, none of your colleagues could ever be accused of talking for the whole period straight. And whatever problems exist in college teaching, nobody would ever accuse a professor of turning mathematics into nothing but mindless routines. Quite the opposite! Students are, absolutely, asked to think.

These differences, I feel confident in saying, result from deep differences between our teaching contexts. It is sometimes tempting to see the similarities between our educational settings — there are students, desks, whiteboards and textbooks — instead of uncovering the deeper structural differences. But the differences are vast! Here are just a few of the variables that differ in significant ways between primary/secondary and college classes: the number of students in a class; the frequency of the class’ meeting; whether we are accountable to a test or not; the ability of students to study the material independently; our control over the curriculum; etc. We could easily name more.

These contextual differences should make us slower to come up with educational solutions for other people’s problems. If our educational settings are different enough that bad teaching looks different, it seems to me that good teaching in primary, secondary and college settings ought to look very different as well. This means that we can’t simply assume that the techniques that are useful for teaching math in one setting will also be useful in another. Which means that we should be cautious before offering advice to those who teach in other contexts based on our own teaching experiences.

It reminds me of something Neil Gaiman says about writing:

When people tell you something’s wrong or doesn’t work for them, they are almost always right. When they tell you exactly what they think is wrong and how to fix it, they are almost always wrong.

When those outside our professional setting tell us that our teaching is not as successful as it should be, we should listen. But as the criticisms and prescriptions get more specific, they almost always grow less useful, at least to me. The culprit is context; we rarely truly understand other people’s constraints.

I don’t intend to sound pessimistic. I think we really can learn from each other’s perspectives. But specific solutions and criticisms can only be supported by a deep understanding of the teaching context. That’s why people who move between these educational settings are capable of doing such important work. These are the PhD graduates who become high school department chairs, the primary teachers who attend mathematics lectures, the secondary teachers who pursue graduate work in mathematics. These are the people who can not only hear the criticisms, but can turn them into something that really works.

In various little ways I have benefited from others who have done this work. When I began teaching math to 3rd Graders at my school, I was committed to sharing “real” mathematics with these students. What surprised me, though, was just how real the mathematics of the curriculum is for these students. Just in the past few weeks I have heard some pretty amazing mathematical conversations about fairly straightforward mathematical questions; things such as 120 + __ = 210 and 3 x 8 = __. But, as mathematical critics of primary teaching have said, there is more to mathematics than arithmetic, and I wanted to expose my students to more. How?

Joel David Hamkins is a professor of logic. At some point I came across his blog, and found materials he posted. For several years, Hamkins had gone in to his daughter’s elementary classroom as a guest math teacher. Each year he put together a pamphlet of problems for the children, and was generous enough to share them online.

For the last several years, my students have loved doing his “Graph Theory for Kids” for a few days each year. They learn about circuits, planar and non-planar graphs, and chromatic numbers. They color maps and hear, for the first time, of the four-color theorem. I feel so grateful that Hamkins was able to really be there, in person, to teach his daughter and her elementary classmates. Do other professional mathematicians do classroom visits? If not, couldn’t they? What if our professional organizations were to organize such things, at some sort of scale?

Primary and secondary teachers have a unique sort of expertise in pedagogy, but I see no way for us to share it unless we tangle in-depth with the context of college math teaching. I have never heard of primary/secondary educators visiting college courses as guest speakers, but why not? Surely there are some who work in primary/secondary settings who could be invited to give lessons or talks in university math courses. We could try to adapt our methods to the university setting, and work together with professors to design different styles of lessons. Could this be a way to learn what primary/secondary pedagogy looks like in a different setting?

The truth is that I am optimistic that something like this cross-pollination is already occurring, though very slowly. For the past few years I have taught at a summer math camp in New York City. The students are all entering the 7th Grade; the faculty are about evenly split between middle/high school teachers, graduate students and college teachers. Every summer, I’m surprised by the sorts of pedagogical discussions we end up having. Assumptions are frequently challenged as we all look for a common language to describe our teaching. For six weeks, we talk daily about how we are structuring our lessons and helping our students. We’ve seen each other teach, we’ve seen the problems each has shared with their students, and we’ve shared our successes and struggles.

When camp is over we say goodbye to each other, taking whatever ideas we have learned over the past few weeks, and help students learn mathematics in all sorts of different classrooms, wherever they happen to be.