Is teaching math to kids easy? Should any reasonably intelligent person who can do “plus, times, minus, and divide” be able to start teaching elementary math tomorrow, skipping college math and all those waste-of-time education classes? Could most smart people, if they decided to stop doing the sorts of jobs that smart people do (not elementary school teaching), walk in and immediately do a better job than most elementary teachers are currently doing?

Or, is teaching math to kids hard? Besides all the things that make teaching elementary school hard in general – planning engaging lessons for every subject, managing classroom behavior, addressing the needs of different children, creating partnerships with parents, preparing for tests, etc. – is there some special mathematical knowledge and pedagogical knowledge that teachers need in order to help students understand fractions and multiplication, or learn how to get started on a problem they’ve never seen before?

Last week I went to a big state meeting about the new math test that all prospective elementary and special education teachers in Massachusetts must take (MTEL test). The state instituted the test because many prospective teachers were missing most of the math questions on its general curriculum test, yet still passing and becoming teachers. At the first administration of the new test, only 27% of the teaching candidates passed; by the third administration, that figure was up to 37%. Candidates who almost passed can go on to teaching jobs, but must pass the test within three years. (I participated in the rather Byzantine process of setting the passing score, but that’s a post for another day).

Along with the test came new state guidelines for the mathematical preparation of elementary teachers , which were based on guidelines released in 2001 by a group of all the major math and math education organizations (I also was at a meeting a few years ago that led to some of the recommendations in the state guidelines).

The overall tone and structure of the meeting last week conveyed that math education for prospective teachers is a serious issue and a state priority. There were half a dozen or so high ranking state officials there, as well as representatives from each college or university in Massachusetts that prepares elementary or special education teachers (schools were encouraged to send deans, provosts, and VP’s). Most of what was said out loud fell squarely in the “teaching math to kids is hard” camp. We were there to work together, in an unprecedented way, to tackle this very difficult, yet critical challenge, without bashing or blaming teachers or professors. However, when it came to the details, the “teaching math to kids is easy” assumptions crept in, often unintentionally, albeit with the caveat, “as long as prospective teachers know enough math to pass the MTEL test.”

A national speaker discussed the complexity of the mathematical content involved in teaching kids, giving examples of mathematical issues that teachers have to resolve quickly, in a classroom setting, e.g. deciding whether a student’s method for doing multiplication works in general. She talked about the concept of Profound Understanding of ElementaryMathematics (PUFM), which articulates some of the ways that elementary teachers must understand elementary mathematics, and which people in other mathematically rich professions do not tend to have. This concept was neatly summed up by a few speakers as, “Elementary math is not elementary.” (I personally was shocked when I realized how difficult it is to teach fractions – I had a Ph.D. in math, but I had to completely rethink my understanding of fractions in order to learn to teach the subject, and I still find teaching fractions more difficult than teaching advanced classes for math majors).

Another speaker presented data on people who change careers to go into teaching. The majority of such people do not have a strong Science, Technology, Engineering, or Math (STEM) background, but those who do tend to have skills that are not aligned with what’s needed to teach elementary math, and like I did, need to learn to “unpack” their mathematical knowledge in order to make it accessible to novices.

As long as we were speaking generally, we all agreed: teaching math is hard and developing PUFM is different than doing well in ordinary math courses. As the meeting started getting more specific, that tone mostly continued. Several speakers talked about how they worked to change their undergraduate programs to meet the state guidelines that call for three math content courses for prospective teachers of grades 1-6. They focused on the internal politics at each institution, which made me appreciate my own school, which is smaller and a lot less bureaucratic than I often think it is. We were able to implement our three-course math content sequence about four years before the state guidelines came out, without pressure from the state.

The people speaking about undergraduate programs spoke more about courses than about the MTEL test. At Wheelock, we fear that the test will undermine our existing program, whereas at the other colleges and universities, their programs are new, are aligned with the test by design and with the assumption that such alignment means students will do well on the test. We don’t know yet, however. We don’t yet have enough data on how students do on the test after taking courses that meet state guidelines (at Wheelock, students have so far mostly avoided the new test, which they won’t be able to do for much longer). Some of the most important aspects of our courses at Wheelock are not represented on the test – for example, our courses emphasize speaking; we include many presentations and student-led discussions. Clearly speaking well is a critically important skill for future teachers. I don’t know whether the courses at other schools will emphasize speaking, but they certainly have incentive not to, if they are to be judged primarily by the MTEL test.

It’s when we got to talking about graduate programs that speakers at the state meeting started openly flirting with the “teaching math to kids is easy” framework. Unfortunately, graduate education programs are designed based on a “teaching is easy” framework, although I’m quite sure that no one associated with them believes that. Yet most programs are only about a year long. It takes three full years to get a law degree or a divinity degree, but only a little over a year to get an education masters.

If teaching math to kids is hard and most prospective teachers (including those with STEM backgrounds) need three math content courses, then these jammed full one-year masters programs need to either a) drop other courses to add math content courses or b) become longer programs. For a long time, however, many programs have operated under option c) that since these are* graduate students*, they learned all the math content they needed as undergraduates (even if the math content they had then was one semester of college algebra twenty years ago and they got a C). Now we have option d) if students pass the MTEL, then they know enough mathematics to teach it to kids.

Option d) works if the MTEL really tests the sorts of content that teachers need to teach math well. It doesn’t. It’s not a terrible test (I think it’s much better than the rest of the General Curriculum Test), but it’s mostly multiple choice, and much of what’s most important in testing PUFM is hard to test with multiple choice questions. Some topics are easier to test in this format, so they are overemphasized, while some of the most critical topics are barely tested at all (including the ubiquitous “plus, minus, times, divide” for whole numbers, what so many people think teaching elementary math is all about, and which requires a non-trivial amount of understanding to teach other than as a rote set of rules to follow). The test is also fairly easy for people with strong STEM backgrounds to pass, without taking any math content courses for elementary teachers – I’m quite sure I could have passed it when I was in high school – and it’s simply dishonest to assume that passing such a test means that a prospective teacher is well prepared in elementary math content.

I was also quite dismayed that it sounded like for the first three administrations, the state had used exactly the same test, which is a situation ripe for cheating.

The state guidelines are only recommendations, and it seems like neither the state nor the programs themselves are assuming they apply to graduate programs. I was pleased that one graduate program is requiring significant math content courses within the program. The other three programs that presented all had ways for students to avoid math content courses based on previous math courses that were not designed for teachers. One college was primarily doing test prep and not requiring any math content courses. With a few exceptions, however, the graduate population is no more likely to have a profound understanding of elementary math than the undergraduate population. Graduate students who don’t like math are often weaker than their undergraduate counterparts, as they’ve been away from the subject longer. Over half of all licensed teachers in the state come from post-baccalaureate programs.

The current of disrespect for teachers and for the jobs they do runs very deep. This disrespect is behind assumptions (stated or not) that people who have already proven themselves in more respected fields will have no trouble switching to teaching, and thus only need to pass tests, not to deeply engage with the content.

“Teaching math to kids is easy” is not a sentiment that comes from people involved in education, but it unfortunately reflects societal values and priorities, which help explain why graduate programs in education are so short. Surely we could prepare teachers better with a model more akin to medical school – several years of coursework, followed by several years of onsite supervised practice. Then we wouldn’t be tempted to pretend that a test could substitute for the course work. However, teaching is a relatively low paying profession that is often not very respected, and it’s a bad economy. It’s not clear that anyone could afford to come to such programs.

There were parts of the state meeting that I found very frustrating, particularly at the end when some of the state officials talked about some next steps, which seemed timid at best (e.g. require four years of high school math to get into state colleges, not three). What the state says they are demanding is a radical improvement in elementary mathematics teaching in Massachusetts. I am in complete agreement with such a demand, but I don’t see how it happens without radical changes in how teachers are prepared and supported and valued. A new test cannot make such big changes. Teaching math to kids is hard. Period

## Dot Gorenflo

Very clear and very well written. I totally agree that “teaching math is extremely hard” and that one key to overcoming the difficulty is to prepare teachers with a deep understanding of what is often seen as “simple” mathematical concepts. As prospective teachers begin to develop that depth of knowledge, the desire to continue learning math content is another aspect of teacher preparation. As teachers, we must continually in the mode of “teacher preparation”. The ongoing work of teachers requires continuous learning and support.

## dborkovitz

Thanks Dot! That’s my first ever comment on the whole website!

## Ellen F.

Debbie-

I finally had a chance to look at this!!! Thanks for sharing your thoughts on the meeting and math education in Massachusetts. I haven’t had a chance to look at all of the links-but your web site is amazing!

## dborkovitz

Thanks, Ellen. Sorry it took me a year to approve your comment! Illness…..sigh.

## Nancy

Deb,

I think we learn teaching math is hard whenever we put ourselves in the role of student. Once there, after communicating our thinking, we realize how different ways of thinking lead each participant to different methods of solutions. Reaching each one takes such time in the planning sessions. The joy of a class well taught……..is there anything better??

Thanks Deb for everything ,

Nancy