Imagine a small box with an American flag painted on each side. This represents the box in which we American teachers think about education … how we view the role of the teacher, the nature of the learner and the purpose of school. I say this, because ongoing international research studies show that teaching is a cultural phenomenon. We do not teach the way we were trained to teach; we teach the way we were taught. While there are differences among us American teachers, there are glaring commonalities that we uniquely share. The same can be said for our counterparts abroad. We could make a similar box and cover it with French flags and have a conversation about how French teachers think about education. We could do the same for the Japanese or any country for that matter. The issue is that while our education system has a few great things to share with world, for the most part, countries which outperform us in academic math tests do so because their box is far superior to ours.

That lesson can be found in the Trends in International Mathematics and Science Study (TIMSS). In March of 1998, The Math Projects Journal was granted an interview with Dr. William Schmidt, the American Coordinator of TIMSS. He claims that among the top-performing countries in mathematics (no, the United States is not one of them) there is no common methodology, but there are common principles of instruction that all the top-performing countries share: teaching to conceptual understanding and teaching with mathematical substance. In his writings and public presentations he stresses two additional components: standards and accountability.

Therefore, I have consolidated these findings into what I have dubbed “The Four and a Half Principles of Quality Math Instruction,” or in the vernacular of the digital age, “Q.M.I. version 4.5.” The first four principles come from the research shown in the international comparisons of the TIMSS report:

1) STANDARDS: Focus on a the limited number of topics that your students need to know; don’t just cover the textbook.
2) CONCEPTS: Teach students to understand what they are doing, not just to mimic what you are doing.
3) SUBSTANCE: Intellectually challenge students; raise your level of questioning.
4) ACCOUNTABILITY: Hold students to knowledge and performance expectations that go beyond grades and unit credit.

The fifth principle comes from professional experience and opinion rather than research, and therefore, its emphasis is demoted to a half-principle.

½) RAPPORT: No philosophy, technique, methodology, instructional material or textbook can replace the student-teacher relationship. You must reach ‘em before you teach ‘em.

Since I don’t have a PhD after my name, I found backup from someone who does. I was reading a book titled Six Easy Pieces by Dr. Richard Feynman. It caught my eye because the subtitle of the book was, “Physics Taught by its Most Brilliant Teacher.”  The preface of Six Easy Pieces is full of insights into the teaching philosophies and methods of one of the finest teachers of arguably the most difficult subject in contemporary academia. Here are some quotes by Dr. Feynman regarding teaching:

Dr. Feynman on Standards:

First figure out why you want the students to learn the subject and what you want them to know, and the method will result more or less by common sense.

Dr. Feynman on Concepts:

I wanted to take care of the fellow who cannot be expected to learn most of the material in the lecture at all. I wanted there to be at least a central core or backbone of material which he could get…the central and most direct features.

Dr. Feynman on Rapport:

The best teaching can be done only when there is a direct individual relationship between a student and a good teacher — a situation in which the student discusses the ideas, thinks about the things, and talks about the things. It’s impossible to learn very much by simply sitting in a lecture, or even by simply doing problems that are assigned.

So where is “the use of math projects” in the list? Math projects are not on the list, because in-and-of themselves they are not critical to quality math instruction. Projects are effective tools of instruction only when they embody these four and a half basic principles of teaching discussed herein — in particular, teaching to conceptual understanding and with mathematical substance. To gain further verification of the potential effectiveness of math projects, though, I once again call upon Dr. Feynman.

Dr. Feynman on Projects:

I think one way we could help the students more would be by putting more hard work into developing a set of problems which would elucidate some of the ideas in the lectures. Problems give a good opportunity to fill out the material of the lectures and make more realistic, more complete, and more settled in the mind the ideas that have been exposed.

Thank you, Dr. Feynman, for the encouragement to keep creating and implementing quality math lessons and problems, and to persist in developing good student-teacher relationships. Thank you, Dr. Schmidt, for revealing to us the value in teaching to conceptual understanding and with mathematical substance, and for pressing us to see the need for standards and accountability. May we all keep in mind the most important lesson offered by these studies: The greatest determining factor in the quality of the education that a student receives is the decisions that a teacher makes on a daily basis!