Opening Session - "Challenges in the Mathematical Education of Teachers: Why is the Preparation of Mathematics Teachers so Difficult?"

Remarks by Roger Howe

I am very happy to see so many of you turn out for this MET summit, because I think that improving the mathematical education of teachers is a cornerstone of improving mathematics education. There is a broad consensus that mathematics education in the U.S. today faces serious challenges.

International comparisons [McKnight et al., 1987],  [McKnight & Schmidt, 1998], results from our own NAEP and other assessments [Dossey, 1997], [Dossey, 2000], reports of college placement examinations and other data all testify that we are far below where we should be. There is much less consensus about what to do to improve matters. My own view is that any serious plan for improvement must concentrate on the increasing the capacity of the teaching corps to deal with mathematics. This will take a lot of work. Teacher preparation is obviously a big part of that work.

To understand what is needed from teacher preparation courses, we have to know what the job of teaching mathematics requires. The main point is that new needs, especially the need to interpret the steady stream of quantitative information with which everyone is confronted daily, requires a higher level of understanding of mathematics than was the case when we (or at least, when those of us as old as I am) were young. Unfortunately, mathematics education research so far has told us much less than one might like to know about the requirements for teaching under the new conditions. However, during the 1990s research produced some useful information about the current situation [Ball, 1990], [Ma, 1999], [Post et. al., 1991], [Stigler & Hiebert, 1999]. I would summarize this work by saying that it showed the dominant mode of teaching mathematics in the U.S. is "mathematics as magic:" This is the way you do it, and look! It works!

Reasons for the success of procedures even as simple as two-digit subtraction are not taught. They cannot be taught, because they are not known to the teacher.1  Mathematics becomes a collection of unrelated facts to be memorized. Without understanding of the underlying unity of mathematics, there is too much to know. Memory is overwhelmed, and even well-practiced procedures soon fade.2 It never becomes possible to solve problems that require combining several principles, because the principles themselves are invisible. Instead of magic, mathematics becomes mystery.

Teaching the principles requires a different kind of teacher. We need teachers who regard mathematics as something to think about, not something to memorize. They must be able to deal flexibly with mathematics. They need to know several ways to do a computation or approach a problem. (See Chapters 3 and 7 of the MET report for examples.) They must be able to make connections between mathematical ideas, in particular, different methods for solving a given problem. They need to be grounded in basic principles, so that when a student produces a novel solution to a problem, they can parse the work and decide whether it is correct, and if not, where it went wrong. Such teachers would think of the rules of algebra as their friends. They would have a considerable capacity for mental mathematics.

Producing this teacher, or at least starting future teachers on the road to becoming this kind of teacher, must be the goal of a new kind of teacher preparation in mathematics. I would summarize the recommendations of the MET report concerning the nature of teacher preparation as follows:

  1. Emphasis on broad principles and flexible thinking.
  2. Close connection to the K-12 classroom.
  3. Sufficient number of courses.

To keep the discussion simple, I am going to focus on the elementary grades in these remarks. For elementary teachers, the report recommends three courses. This is quite conservative. Under current circumstances, what these courses have to do is reverse the effects of 12 years of learning bad mathematical habits. It is a tall order to do it at all, and three courses is the bare minimum or perhaps less. Paul Sally, who has been running teacher development programs at the University of Chicago for many years, wants his teachers for 200 hours   more like 5 or 6 courses. But the report had to take into account current conditions. Some states ask for three mathematics courses (a few even more), but others have no specific mathematics requirement for elementary teachers, so a widespread requirement of 3 would be an improvement. States who want more are encouraged to add to this.

What should we do with these courses? The later chapters of the MET report provide some hints, but the fact is that, at the moment, we are far from having a clear idea of the best ways to run such courses. A lot of experimentation and sharing of ideas will be necessary. I would mention these design considerations.

  1. Lots of good mathematics, emphasizing principles and reasoning.
  2. Close attention to the absorption and retention of ideas by students.
  3. Connection with classroom use.

As to the subject matter, I would make the first course a problem-solving course, whose main goal is to get students thinking about mathematics, and realizing that they can do it. As Deborah Schifter emphasizes, this change of viewpoint is essential. Without this, no understanding is likely, for the future teachers or their eventual students. Without it, all other courses will very likely be defeated by the "Water off a duck's back" phenomenon: students may get exposed to very good material, but unless they have the mental tools (and the proper environment) to digest it, it will not stay with them.

My second and third courses would focus on the topics that primary teachers use the most in the classroom: arithmetic, the algebra that helps to understand arithmetic, and some geometry. Despite the current drumbeat for statistics and data analysis, and without denying the value of these topics for functioning adults, I would treat that very briefly if at all, and only in ways designed to support the primary goal of understanding arithmetic.

However, I emphasize that just as important as the course topics is continuing attention to engaging teachers in doing mathematics, and changing their ideas about what mathematics is. These courses must be rigorous and friendly, friendly and rigorous.

They should also, to the extent possible, bridge the usual gap between the university and the K-12 classroom. They should present issues in terms of hypothetical classroom situations. They should show how general principles shed light on standard material, such as the particularities of various algorithms and procedures. They should also try to show how thinking in terms of principles can help resolve sticky questions such as might arise spontaneously in the classroom, particularly an interactive classroom with a lot of student input.

In designing, and also in teaching, these courses, collaboration of mathematicians with mathematics educators is advisable. Mathematics educators can provide examples, such as in the later chapters of the MET report, of the state of mind of teacher candidates, and how and what mathematical issues arise in classroom situations. It will be important to keep these in mind both in planning and giving teacher preparation courses. It would also be a good idea for a mathematics faculty who teach these courses to visit classrooms of the relevant grade level if possible. I believe that critical study of videotaped lessons, both for faculty preparation, and in the courses, can potentially be very valuable.

It will not be an easy task to create successful courses of this type. They are a large departure from traditional mathematics courses. Their goal is not simply to present good material correctly, but to take students who are not mathematically inclined, perhaps math phobic, and turn them into flexible mathematical thinkers. They also take much more responsibility for connecting coursework with future tasks of teaching. Much effort and much experimentation will be needed.

In marshalling this effort, it will be important to get the incentives right. The people who are creating these courses will need adequate support and recognition from their departments. If it is needed to visit classes to find the appropriate ways of presenting material, the time for this should be taken into account in computing workloads. Similarly, the effort of creating new materials should be recognized.

In turn, departments will need to make the case to their university administration that adequate resources should be allocated to this effort without compromising other departmental activities. Outside funding agencies can also play a valuable role in making the needed resources available.

As well as needing funding for development, these courses will also have to be supported by policies of educational authorities. If the department or school of education on campus does not support and require these courses, they will have little chance for success. This is an additional reason for these courses to be cooperative ventures between mathematicians and mathematics educators. Also, these courses will not be easy for students    they will be attempting to drastically change attitudes and thinking habits which by college age are pretty well ingrained. Unless the value of such training is backed up by state requirements or district hiring practices, students may simply avoid them by enrolling in easier programs. Hence the success of these courses depends as much on appropriate educational policy as it does on development efforts. The case for the needed resources and policy changes should be made on the basis of cost effectiveness. Our current system is very wasteful, not only in terms of human potential, but also in terms of repeated teaching, culminating in the large remediation programs in colleges.

I should point out that development of new, more successful teacher preparation programs in mathematics has potential benefits for the mathematical community as well as for the larger society. Initially, introduction of these courses can lead to better relations between mathematics departments and their campus administrations, as well as the local school or department of education. Also, greater expertise and cooperation on educational matters could lead to greater input by mathematicians into more issues of K-12 education. If one is very optimistic, one can also imagine a long-run payoff in better prepared students, and less isolation of the mathematical community from the wider society.

I want to emphasize the possibility of greater involvement of university mathematicians in K-12 education. I believe that this could be very beneficial, if carried out appropriately. There are many points in K-12 mathematics education where mathematical expertise can helpfully support educational decisions. Besides teacher preparation and inservice development, examples include the analysis of standards and frameworks, of assessments, and textbooks. Review and analysis of the mathematical content of lessons in teacher portfolios, such as are required of beginning teachers in many states, is another potential area where mathematicians could contribute. However, for the potential of fruitful involvement of university mathematicians to be realized, they will have to learn more about the issues of K-12 education. Involvement in developing and implementing this new type of teacher preparation will promote such learning, especially if it includes observation of lessons, live in the classroom, and/or on videotape. Mathematicians can also learn from mathematics educators, although currently there are not many mechanisms in place for this to happen efficiently. I have benefited greatly from contacts with mathematics educators on many committees, but this has taken a large amount of time. It is not an efficient way to create the reservoir of expertise that we need. We should be able to do better.

I hope that educational policymakers will realize that it is also important to make this kind of educational involvement to make sense professionally, rather than something one has to sacrifice to do. Up to now, very often the opportunities offered to mathematicians for participation in K-12 education have been voluntary service on committees. Volunteerism has a place, but over reliance on it will prevent development of a critical mass of mathematicians with the understanding needed to contribute constructively.

We should be mindful of the downside of volunteerism: like any seemingly free good, it is subject to waste through indiscriminate use. Overused volunteers may become cynical, and others may have a tendency to fanaticism, since success of their agenda in the only reward they can expect.

The enthusiastic turnout for this MET Summit testifies to the desire in the mathematical community to improve U.S. mathematics education.

I hope that the momentum generated by the mathematics education reform movement will carry through to effective practices and policies that preserve and build this enthusiasm in a virtuous cycle.


Ball, D., (1990) The mathematical understandings that prospective teachers bring to teacher education.  Elementary School Journal, 90, 449-466.

Dossey, J.A (1997)  Essential skills in mathematics: A comparative analysis of American and Japanese assessments of eighth graders (NCES97-885). Washington D.C: National Center for Education Statistics.

Dossey, J.A. (2000) The state of NAEP mathematics findings. In E.A. Silver & P.A. Kenney (Eds.)  Results from the seventh mathematics assessment of the National Assessment of Educational Progress  (pp. 23 - 43) Reston, VA: National Council of Teachers of Mathematics.

Ma, L. (1999),  Knowing and Teaching Elementary Mathematics: Teachers' understanding of fundamental mathematics in China and the United States  Mahwah, NL; Erlbaum.

McKnight, C.C., Crosswhite, F.J., Dossey, J.A., Kifer, E., Swafford, J.O., Travers, K.T., & Cooney, T.J., (1987)  The Underachieving Curriculum: Assessing U.S. school mathematics from an international perspective.  Champaign, IL: Stipes Publishing.

McKnight, C.C., & Schmidt, W.H., (1998) Facing facts in U.S. mathematics education: Where we stand, where we want to go.  Journal of Science Education and Technology,  7 (1), 57 -76.

Post, T.R., Harel, G., Behr, M.J., & Lesh, R., (1991) Intermediate teachers' knowledge of rational number concepts. In E. Fennema, T.P.

Carpenter & S.J. Lamon (Eds.)  Integrating research on teaching and learning mathematics  (pp. 194 - 217). Albany: State University of New York Press.

Stigler, J.W., & Hiebert, J. (1999)  The Teaching Gap: Best Ideas from the world's teachers for improving education in the classroom.  New York: Free Press.


1 This discussion is of course grosso modo. There are many teachers who do understand and teach ideas; but also, the research suggests that there are many too few.

2 In emphasizing the importance of understanding principles and ideas, I do not wish to downgrade the importance of being able to do computations. Understanding and facility should support each other. I am arguing against computational facility without understanding.