What mathematics do teachers need to know ?

In the same way that engineers need mathematics applicable to problems of  design and construction and executives need mathematics useful in business applications, teachers need mathematics they can use in their work. The question, therefore, becomes one of identifying the mathematical knowledge, skill, and habits needed for teaching. This requires an understanding of the work of teaching, and in particular of the mathematical work of teaching.

Further, when identifying mathematical knowledge useful to teaching, it may be helpful, directly or indirectly, to connect that mathematics to classroom teaching. If we say teachers need to know the different meanings of subtraction, then it might be useful to identify some of the tasks where teachers are likely to use such knowledge: when sizing up the difficulty of word problems in a textbook, anticipating likely student thinking, or hearing the different interpretations children use to explain their thinking. This exercise can establish more definitively that teachers actually need this knowledge, can focus attention on what it is about this mathematics that is important, and can clarify when and how to use the knowledge. Over the past two decades, researchers have begun to explore where, when, and how teachers use mathematics. While more study is needed, some useful domains have begun to emerge: mathematical practices, or doing mathematics, and mathematical tasks of teaching. Taken together, they begin to identify a body of knowledge for an applied mathematics of teaching.

Many forces act against the changes proposed here — the size and scope of the problem, entrenched political stances, the institutional and philosophical divisions among mathematicians, mathematics educators, education researchers, and teachers, and our incomplete understanding of the body of mathematical knowledge entailed in teaching. What we do today, though, will shape the opportunities we create for the future. Whose responsibility is it and who cares enough to do something? The improvement of mathematics courses for teachers, and of the mathematical preparation of teachers, should concern mathematics departments across the country for several reasons.

First, teachers prepare the students who arrive in university classes. The mathematics that prospective teachers learn, or do not learn, in courses taught by mathematicians plays an important role in determining what the K-8 students of those teachers learn. Second, mathematics courses for prospective teachers help finance mathematics  departments. Elementary school teachers represent one of the largest career groups in the United States, with approximately two million active K-8 teachers nationwide. And all of them take mathematics courses to become certified. Yet, like other fields such as business, engineering, and medicine, service courses for teachers need to serve. They need to be useful to professional work if they are to to remain viable. Losing these courses would be a huge financial loss, but it would also be a shame because mathematicians have valuable insight and perspective to offer classroom teachers.

Finally, teachers represent mathematics to children. In doing so, teachers lay the foundation for our society’s

understanding of mathematics and provide the most visible face of the discipline. If mathematicians care whether children learn mathematics, if they care about our country’s economic and political vitality, if they care whether society holds them in high regard and funds them, then they should care about their largest public relations staff — teachers.