I am rereading Liping Ma’s Knowing and Teaching Elementary Mathematics for a class. It’s an exceptional book; I’d put it on the Mount Rushmore of books about math education (that might be a post for another day). For those who haven’t read it, the book is an exploration of elementary math instruction in the U.S. and China. It was published in 1999, but it doesn’t feel outdated whatsoever. And it laid the groundwork for a lot of the high quality routines that are emergent now.
I’m not going to recap the book, but I would like to tug on one thread: the necessity of more math for pre-service and practicing teachers. Not more math pedagogy, mind you, but more math.
Ma (1999) describes the differences in pedagogical approaches between groups of Chinese and U.S. Elementary teachers. One example she gives is the use of borrowing in subtraction problems versus regrouping in subtraction problems versus multiple ways of regrouping. She had teachers describe how to solve the subtraction problem 53-26. While most U.S. teachers focused on the algorithmic approach of stacking the numbers on top of one another, taking away a ten and adding it to the 3 in 53 and so on and so forth. Chinese teachers took the approach of regrouping 53 at 40+13. Then you can subtract 6 from 13 and 20 from 40 to get 27. Ma (1999) provides a couple other regrouping strategies offered by the Chinese teachers.
The reason for these different strategies lies, Ma (1999) contends, in the teachers’ mathematical content knowledge. Not, pedagogical content knowledge (PCK), necessarily. Ma (1999) notes that the knowledge gap between U.S. and Chinese elementary teachers mirrors the gap between U.S. and Chinese students. Ma (1999) notes that “most U.S. teacher preparation programs focus on how to teach mathematics rather than on the mathematics itself … [there is] no system in place to ensure that teachers get access to the knowledge they need” (p.145).
(As a quick aside, before we get to the crux of this blog post, this is why I always start my professional development, workshops, and coaching sessions with math. Recently, I spent a couple house virtually with some math friends in Michigan, the first hour of which was spent solving these two problems from this wonderful website, Daily Fractle created by Steven Greenstein (Twitter: @Psyclist).
The above quote by Ma (1999) is so meaty. There’s a lot to chew on there.
First off, as someone who aspires to teach “how to teach mathematics” some day, Ouch! But also, yeah. Campbell et al., (2014) correlates teacher mathematical knowledge with pedagogical content knowledge. In fact, that particular study suggests that teacher beliefs (i.e. teachers claim to know or desire to seek out students’ mathematical dispositions), matter… if and only if they also had a good amount of mathematical content knowledge.
Friends, we need teachers to acquire more math!
However, there are significant hurdles for both practicing and pre-service teachers.
Hurdle 1) Elementary teachers have negative self-perceptions about themselves as mathematicians.
This isn’t breaking new ground (Wilkins, 2009), although it is admittedly painting with a broad brush. We need to start by building teachers’ mathematical confidence and willingness to engage in mathematics beyond what their pre-service teaching program requires. No small endeavor, there. Even setting aside Elementary teachers’ understandable allergy to additional math classes, the idea of taking an additional, optional math class is unheard of in any discipline.
Hurdle 2) University Mathematics course offerings don’t allow future Elementary teachers to engage in math.
It’s not surprising that many elementary teachers’ post-secondary math coursework tops out at College Algebra. You want to take more math as a pre-service teacher? Your options are either boring, pseudo-remediative courses like finance or hyper-complex courses like Calculus. This is the more math that U.S. pre-service elementary teachers have access to. And neither are appealing.
Universities and university math departments need to get more creative with their offerings. Rather than striving for verticality (Calculus I → Calculus 2 → Vector Calculus! → DIFFERENTIAL EQUATIONS!), there ought to be more horizontal, elective-ish type offerings. There need to be classes where you play with decomposing numbers and shapes, courses where you find clever patterns, a whole course on fractions! Honestly, I think math majors might find a whole course devoted to fractions quite challenging (and quite possibly humbling).
There’s a whole playground to explore in these mathematical subjects, no less rigorous than Calculus, but certainly more accessible. You could teach these courses using Lockhart’s Measurement or Orlin’s Math With Bad Drawings. Lockhart in particular encourages doing math in creative and non-algorithmic ways. For practicing teachers, it would behoove University Math professors to host a summer workshop doing accessible mathematical tasks for a couple days. Just doing math together.
The two hurdles are very much a chicken-and-egg issue. Universities provide a course progression in math that communicates very loudly that only certain individuals have access to the discipline. Pre-service teachers internalize this message and stop taking math as soon as their program allows. Math departments are part of this problem, and they probably don’t realize it. My guess is, once given the scholarship motivating it, some math professors would jump at the chance to do accessible math with teachers and pre-service teachers., placing our discipline at the center of a teacher’s mathematical experience.
I know some of y’all who read this blog teach a course or run PD with mathematics as the central core of the action. In the give-a-penny, take-a-penny approach to education, consider sharing your course & description or workshop in the comments.
References
Campbell, P. F., Nishio, M., Smith, T. M., Clark, L. M., Conant, D. L., Rust, A. H., DePiper, J. N., Frank, T. J., Griffin, M. J., & Choi, Y. (2014). The Relationship Between Teachers’ Mathematical Content and Pedagogical Knowledge, Teachers’ Perceptions, and Student Achievement. Journal for Research in Mathematics Education, 45(4), 419–459. https://doi.org/10.5951/jresematheduc.45.4.0419
Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States (Studies in Mathematical Thinking and Learning Series).
Wilkins, J. L. M. (2009). Elementary School Teachers’ Attitudes Toward Different Subjects. The Teacher Educator, 45(1), 23–36. https://doi.org/10.1080/08878730903386856
emergent math 