By Diana White, Contributing Editor, University of Colorado Denver
Mathematics departments have long provided the bulk of the mathematics content training for both practicing teachers and those studying to be teachers. This is a tremendous responsibility, and one that presents a variety of challenges and opportunities. In this post, we start early in the mathematical spectrum – with elementary teachers and how mathematics departments impact their mathematical preparation.
Until fairly recently, at many higher education institutions students preparing to be elementary teachers would take one or more general education courses such as college algebra, math for liberal arts, or a version of calculus. It was expected that this would both meet some type of “quantitative reasoning” or “general education” requirement at their institution as well as prepare them with sufficient mathematics to teach elementary school. While there were exceptions, a prevailing thought was that elementary school mathematics was, well, taught in elementary school, so someone enrolled in college should have sufficient mathematics background already.
We have learned that this is far from a truism. While a great many researchers and practitioners have contributed to the development of knowledge in this area, we single out work by Deborah Ball and her many colleagues. Their work on mathematical knowledge for teaching (MKT), that is, the mathematics that teachers actually need to engage in the practice of teaching mathematics, has profoundly impacted courses and programs across the country.
In the practice of teaching mathematics, teachers engage in mathematical tasks such as responding to students’ mathematical statements, addressing students misconceptions, and providing multiple representations of concepts. The following document contains 35 problems covering a broad spectrum of elementary math topics that serve to illustrate the diversity of mathematical knowledge needed by elementary teachers: http://sitemaker.umich.edu/lmt/files/LMT_sample_items.pdf
These skills have both mathematical and pedagogical components and cannot neatly be separated into “content” and “methods” courses. In particular, addressing student misconceptions often crosses into both areas. To address student misconceptions, teachers must recognize the misconception and understand deeply the mathematics behind the topic. However, they must also have sufficient knowledge of student development and student thinking to respond productively to the student to help them grow in their mathematical understandings.
With the changing landscape of mathematics education, it is now well-accepted in the mathematics education community, and increasingly in mathematics departments, that elementary teachers need specialized content courses in mathematics. The latest Conference Board of the Mathematical Sciences (CBMS) recommendations in their Mathematical Education of Teachers II document suggest four such content courses. As a mathematical community, we remain far from this suggested standard in our typical course offerings.
A perhaps surprising challenge is that such courses usually contain content that is not typically familiar to mathematicians. For example, many of us are not familiar with a non-algebraic explanation of why the traditional “invert and multiply” rule for dividing fractions holds, one based only on an elementary understanding of the meaning of fractions, the meaning of multiplication, and the meaning of whole number division. However, building from definitions is solidly in our area of expertise, and we are well qualified to help elementary teachers learn to base their mathematical reasoning on age-appropriate definitions. After all, if the teachers do not have this skill set, then they will not be able to develop it in their students.
Even further removed from our knowledge base may be things like the whole associated with a fraction, unit rates, base 10 blocks or unifix cubes, fraction bars, double number lines, and the partial product or scaffolding algorithms for multiplication and division. Again, mathematicians are certainly capable of jumping in and learning these, but it is specialized mathematical knowledge that we do not just have by virtue of our advanced mathematics degrees.
So, what can departments and individuals do to contribute further to the mathematical education of elementary teachers?
First and foremost, we can increase our collective awareness of the importance of our role in preparing future elementary teachers to teach mathematics. At an individual level, we can stay abreast of key documents like the aforementioned CBMS recommendations, we can read articles in the AMS Notices, and we can attend a session or panel related to elementary mathematics education at the Joint Math Meetings or at Mathfest.
As part of our collective awareness, we can ensure that the importance of our role is emphasized by both formal and informal leaders within our departments, discussed or at least given genuine recognition at appropriate times during department meetings, and that a culture of respect for this part of our mission is established among faculty.
Going beyond the awareness level, departments can increase their participation and reward faculty participation. Likely there are a few mathematicians in each department who would enjoy and excel at becoming more actively involved in courses for elementary teachers. Encourage, support, and reward them for their efforts. Most importantly, respect their efforts and genuinely accept that it is important work and a much needed contribution to the mathematical spectrum.
Some mathematics departments, for example at the University of Nebraska and the University of Northern Colorado, go beyond the aforementioned faculty involvement, providing opportunities and training for their graduate students to teach courses for elementary teachers. These graduate students then enter the profession as faculty members who already have a basic knowledge base and skill set in this area, able to share their knowledge and contribute their skills to other departments.
Finally, reach out to our partners in education. Find out what the preservice teacher curriculum is at your local institution, volunteer to teach a content course for elementary teachers and put in the effort to learn the specialized knowledge, and consider volunteering at a local school or in some setting where you have direct mathematical interaction with elementary age students. A basic familiarity with where students are in their mathematical thinking can be invaluable as baseline knowledge to being involved in this type of work.
Thus, our level of involvement both individually and collectively can come at many levels, from simply increasing awareness to jumping in and becoming so involved that it becomes a major part of one’s career. Those looking to read further might check out some of the references at the end of this post.
We bear primary responsibility for the content preparation of elementary teachers, and I propose that we should take our responsibility in this area seriously and endeavor to excel at this crucial aspect of our mission. Elementary teachers are providing the initial mathematical training to our future scientists, engineers, and mathematicians. Perhaps more importantly, though, they provide the initial mathematical training to the future adults in our society, including our own children.
Conference Board of the Mathematical Sciences (CBMS). (2001). The Mathematical Education of Teachers. Providence, RI: American Mathematical Society.
Conference Board of the Mathematical Sciences (CBMS). (2012). The Mathematical Education of Teachers II. Providence, RI: American Mathematical Society.
Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers Understanding of Fundamental Mathematics in China and the United States, Mahwah, NJ: Lawrence Erlbaum.
Saul, M. (Ed.) (2011). Special Issue on Education, Notices of the American Mathematical Society, 58(3).