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# How Much Math Does a Teacher Need to Know to Teach Math?

I once asked a colleague if he’d read a particular book. “Read it?” he replied incredulously. “I haven’t even taught it!” A former college English professor, he came by the joke honestly. The first time I taught a course that I had never taken myself, I acknowledged the absurdity, at least to myself. I stayed about a week ahead of my students. Out-of-field teaching? Not exactly. I was teaching a course that was in my field, but outside of my immediate area of expertise. The teaching assignment was justified on the grounds that, as a Ph.D.-holder, I was deeply grounded in the core theoretical perspectives and research traditions in my discipline, and that I could therefore pick up the literature in a subfield quickly and accurately, and teach that literature competently. (At the time, no one was concerned with *pedagogical content knowledge*, the idea that there is practical knowledge of how to teach a subject that differs from mastery of the subject itself.)

Last week, the National Council on Teacher Quality released a report on the mathematics preparation of elementary school teachers who teach mathematics. The report indicts education schools for failing to select and prepare elementary teachers who have an adequate mastery of mathematics. Singling out algebra as a topic that is shortchanged in preparation programs, the authors offer a number of sensible recommendations for states, education schools, textbook publishers, and institutions of higher education.

The Teacher Education and Development Study in Mathematics (TEDS-M), a comparative study of how 18 countries, including the U.S., prepare mathematics teachers at the primary and lower secondary grades, is currently underway under the auspices of the International Association for the Evaluation of Educational Achievement. We’ll learn a great deal from this study that will complement the NCTQ recommendations.

It seems obvious that teachers must have knowledge of the subject matter they will actually teach. But how much *more* knowledge should a teacher have than what she or he is seeking to assist students in learning? The case of secondary school mathematics is instructive. Is it enough for a high school trigonometry teacher to know trigonometry cold – but not, say, real analysis, or ordinary differential equations?

In the US, many states have content specialty tests that prospective teachers must pass prior to assuming full-time teaching positions; presumably these tests tell us something about the mathematical content that states think is important for teachers to master. The four-hour Massachusetts test covers number sense and operations; pattern relations, and algebra; geometry and measurement; data analysis, statistics, and probability; trigonometry, calculus, and discrete mathematics; and integration of knowledge and understanding. Approximately 23% of the test is devoted to patterns, relations, and algebra, and there are 100 multiple-choice items and two constructed-response items. From tests such as these, we can infer that some states do not demand that high school math teachers have an extensive understanding of the discipline of mathematics.

One of the reasons I was unhappy with much of the press reporting on the Urban Institute’s study of Teach for America teachers’ effects on end-of-course tests in Algebra I, Algebra II, and Geometry (among other subjects) in North Carolina is that it shifted the locus of policy discussion to whether to expand alternate routes to teacher certification, without addressing the more challenging questions about what knowledge about subject matter and about how to teach it is optimal for student learning in particular subjects in high school. The reality is that even if we could count on the incremental achievement observed in the Urban Institute study, lots of other countries would still be kicking our butts in international assessments of mathematics and other subjects. I think we’d be better off examining how these countries prepare secondary math teachers – and teachers in other subjects – to see if there are approaches that we can adapt to the U.S. context. One thing that we might learn is that other countries demand much higher levels of subject matter competence from their elementary and secondary school teachers than we do.

Without excusing current conditions, it is worth understanding them.

Many teachers go into teaching at the secondary level for reasons that include love of or mastery of content.

This is rarely the case at the elementary level, and where there is content motivation, it is far, far more likely to be related to early reading than early mathematics.

Jonathan, I was hoping you'd comment on this. Content motivation is one of the issues to consider as part of a comparison of U.S. practices with those of other countries. Not every practice that we observe among the countries with high math achievement will necessarily transfer to the U.S. It's possible, for example, that recruitment into elementary and secondary teaching differs in countries that elevate teaching to a highly prestigious occupation, so that content motivation is more central, even at the elementary level. (Have you ever heard a math major say, "I want to be the best teacher of place value I can possibly be"?) Even in the absence of that motivation, we do need to develop ways to expand the mathematical knowledge of elementary teachers.

I think it is definitely important for the teacher to know more than the students he or she is teaching, and to know how to teach the material in the course. However, I see no reason for schools to demand that a teacher of middle school math be skilled in pre-calculus or calculus.

Example: I taught English to low-income high school students with extremely low reading and writing skills. I did not need to have a PhD in English Literature to teach them at their level: basic writing skills, sentence construction, parts of speech. However, at another school, I taught honors 11th grade English. At that school, it was imperative that I have a good background knowledge of literature and literary analysis. Just knowing how to construct a coherent paragraph was not enough.

Therefore, the type of student/level of study is integral to the type of preparation a teacher should have in his or her subject matter.

In fact, as a student, I found that very intelligent and well educated math and science teachers often were WORSE instructors of basic courses, because it was so difficult for them to relate to their students' lack of core knowledge about basic principles.

"In fact, as a student, I found that very intelligent and well educated math and science teachers often were WORSE instructors of basic courses, because it was so difficult for them to relate to their students' lack of core knowledge about basic principles."

This myth has so perpetuated the education field that principals at high-poverty, low-performing schools pass over teachers with stellar academic credentials so that they can hire someone who can relate to the students. This is why so many of these schools never make any progress in educating their students.

Between the engineer who can't teach, and the person with little knowledge of mathematics, but who can control a class, both are a problem, but the latter is far more dangerous.

The engineer who cannot adjust to the kids' lower level of knowledge and ability to work with abstractions, who cannot control a class, that engineer will leave sooner rather than later.

But the person who can run the class, but teaches things that are wrong, they can last a long time, and do damage year after year.

In any case, some engineers become good teachers, some do not. Knowledge of mathematics is a prerequisite, but does not guarantee that the person can become a good teacher.

I think when you go to other countries, you will find a far more sensible approach to training teachers at all levels than we have here. One example: Japanese math teachers attend "normal school" after high school, then apprentice with a master teacher for years. Most classrooms feature two teachers at any time. Keep in mind, they currently eat our lunch in math ed. We have math majors, who get masters degrees in math ed, and I probably don't need to go into the kind of results we get. And yet we persist in re-inventing the wheel with hairbrained schemes every 10 years.

We can't pin our hopes on finding enough math majors to teach our children. First, we don't need them if we train our teachers right. Second, we can't afford them. The best practices are out there. What we are lacking is leadership.

Jonathan: Of course intelligent and well educated teachers can teach basic math and science subjects. However, I believe that once the teacher has a core level of knowledge, expert knowledge in a subject area (e.g., PhD in physics) is only truly useful when teaching high level courses. How would a PhD in physics be particularly helpful in teaching 8th grade general science?

Which is not to say that NO knowledge of the subject area is necessary. A middle school math teacher should be well-versed in middle school math - but that's not the same as having college coursework in multivariable calculus.

I'm curious about the instances you referenced, in which schools choose not to hire very well-qualified teachers due to their "over-qualification." Did the schools specifically reject the applicants because they were TOO qualified, or did the school select another applicant with weaker academic qualifications, but with (for example) more teaching experience or stronger classroom management skills? I could see a principal picking a teacher with a bachelors in math education and 10 years experience over a teacher with a masters in math and 1 year of experience, for example.

Correction: My last post should have been addressed to "Bill" not Jonathan. Bill's the poster who commented on the problems with schools passing over highly qualified teachers.

Multivariate calculus sounds important and difficult. But Abstract Algebra (aka Modern Algebra)? These are undergraduate level courses, and yes, we should expect the equivalent of a bachelors in mathematics.

Unfortunately, there are far too few available. The example you gave:

"a teacher with a bachelors in math education and 10 years experience" or "a teacher with a masters in math and 1 year of experience"

would be an embarrassment of riches. We rarely get such choices, and when we do, we suspect that there is an underlying problem that made these candidates available.

Jonathan: It's unfortunate that schools (particularly low-income schools) can't get math/science majors as teachers. I suppose it's because there are so many higher paying jobs available in those fields, or perhaps science/math majors aren't generally the personality types who enjoy teachings?

In any event, thanks for your insights. Off to lunch...

Just FYI - the link in the text of your article for the UI study does not work - the study is at

http://www.urban.org/UploadedPDF/411642_Teach_America.pdf

Thanks

Doug Murray

Elementary teachers need to know math. Here are two reasons why. First - there is a growing movement of people (including myself) who feel that the Singapore Math program should be adopted by school systems. The program itself is not the point here. Many elementary teachers are against it only because the curriculum does not come with teacher's solution manuals. This is a K-5 grade mathematics program - anyone with a high school diploma should not need a solution manual to teach and grade it.

My second point is much more important. Algebra is a brick wall for many students, and there's no reason for it. In elementary school, students learn all the basic knowledge they need to solve arithmetic and algebra problems, but those skills are unavailable by the time they reach algebra. Reread that sentence - they learn the knowledge, but the skills are unavailable. It's the difference between learning what a hammer looks like, and learning how to use a hammer. The skills needed for solving algebra problems are dropped on the floor by elementary teachers who don't understand or appreciate the intimate connections between the arithmetic they teach in K - 5 and the algebra that follows in middle or high school.

I found an excellent report on this topic (I think this is different than the one mentioned in the blog) at http://www.nctq.org/p/publications/docs/nctq_ttmath_exec_summ.pdf.

Here's a quote from page 8 of the Executive Summary: "While elementary teachers do not deal explicitly with algebra in their instruction, they need to understand algebra as the generalization of the arithmetic they address while studying numbers

and operations, as well as algebra’s connection to many of the patterns, properties, relationships, rules, and models that will occupy their elementary students. They should learn that a large variety of word problems can be solved with either arithmetic or algebra and should understand the relationship between the two approaches."

Oops - that excellent report IS the one cited in this blog! Thanks for the link!

I agree with Jonathan in his characterization of elementary teachng candidates; I think it’s a fair generalization to acknowledge that college undergraduates who are strong in math do not gravitate towards elementary education. We tend to get a lot of people who are competent in math, but not obsessed by it.

While I think it’s reasonable to expect teacher education programs to ensure that their graduates know how to do college-level math. Most people would agree that math is a “use-it-or-lose-it” skill. The minute you stop solving algebraic equations is the minute you stop knowing how to solve them. I teach third grade. I was once very good at algebra; I’m still very good at subtraction and multiplication.

In their report, NCTQ includes five different math problems that they feel elementary teachers should be able to solve. Although there was a time in my life when I could have easily solved each of these problems, I have to admit that now is not one of those times. Why not? Simple: I do not work with this material on a day-to-day basis. It is not currently critical that I know how to do this type of math. That’s how people forget how to do things. (By the way, this is exactly why the American Red Cross makes us take CPR classes every two years to retain our certification. They assume we haven’t had to save anyone’s life lately, and they know we’ll forget how to do so.)

The NCTQ and its panel of experts (none of whom teach elementary school) believe that grade-school teachers should have more rigorous coursework in math content. Perhaps. But my concern is that a large portion of this “rigor” will be left behind once these teachers begin working with their students. I think their time in college would be better spent on methods courses and content-level math classes in which they focus on developing concrete understanding of the math with which they will have day-to-day interactions.

Hi, just wanted to comment on this excellent post. When I taught middle I had a math prep and a social studies prep. I had very poor, ill equipped social studies teachers that were boring and disengaged. I didn't learn geography, skills, causes of world events, etc. I can relate to being about a week ahead of my students and having to really work to prepare. When students would ask me questions I presented it as a learning experience for all of us as I didn't know the answers. I heavily relied on internet lessons with background information already compiled and used streaming videos or clips of videos to present the social studies content I was teaching. I used sites with newspaper and primary sources and created small webquests for students to use. Since social studies is not my forte I spent double the time that I spent planning for math. It was an learning experienc for all of us as I truly became a facilitator guiding students to find the answers because I didn't know the answers myself. We learned together and brainstormed resources that we would need to consult to complete an activity, answer a question, or take us to a new activity when the learning branched into a different direction that I wasn't quite expecting. I struggled a bit at first to find effective resources but I truly value those two years as I really grew professionally when I was able to step back and let the students run the show.

As a middle school and elementary teacher, I have seen first hand how crucial it is for elementary teachers to be skilled reading and math teachers. That is rarely the case. The depth and the quality of teaching for rigor and relevance that is needed isn't present in most teachers, even less so in new teachers until they find their niche. The students gap from where they currently perform and where they need to be widens each year that a student has ineffective teachers in critical content areas. I find this topic fascinating but don't see an easy solution on the horizon. I just blogged a similar post on my blog if you would like to review it and/or comment: http://kcaise.edublogs.org/2008/07/06/teachers-are-missing-the-mark-in-math/