Putting It All Together
Hamburgers and High Ceilings KNOWLEDGE FOR TEACHING AND LEARNING by Joshua A. Taton and Cathryn Anderson
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o grow professionally, teachers need consistent, high-quality learning opportunities that stretch both their mathematical and pedagogical muscles. The Philadelphia Area Math Teachers’ Circle (PAMTC) takes a research-based approach to support teachers in doing both. In a provocatively titled piece, “Why Do Americans Stink at Math?”, Elizabeth Green (2014) relates a vexing anecdote about Americans’ struggles with numerical reasoning. She explains that in the 1980’s, the A&W Restaurants wanted to compete with the McDonald’s Quarter Pounder, and so they introduced a burger with a third-pound of beef at a comparable price. On taste tests, the A&W burger won, handsdown, but failed to sell. A&W eventually learned why: Americans thought that the Quarter Pounder was a better deal, because—you guessed it—they incorrectly reasoned that 1/4 was larger than 1⁄3. Concerns about Americans’ math ability are nothing new. According to Green, these concerns persist, despite changes in educational policies, curriculum, and technologies, largely because our style of teaching has remained stagnant. In particular, as Jo Boaler (2013) explains, American teaching is embedded within a set of faulty beliefs about who can do math and how math is learned, such as the flawed idea that mathematical dexterity requires speed. Another popular misconception, Green notes, is the idea that simply improving teachers’ knowledge of math results in improved math teaching. Decades of research, however, have proven otherwise. Mathematical Knowledge for Teaching So, what is it about our style of teaching that needs to change? Insight comes from a series of studies by Deborah Ball and Hyman Bass (2003; see also
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Ball, Hill, & Bass, 2005), who demonstrated that professional mathematicians and math teachers think differently. This should be obvious to anyone who has struggled to understand an otherwise brilliant math professor. Effective math teaching requires not only knowledge of math content, but also how to communicate—to show useful representations, to anticipate and decipher students’ errors, and so on. Ball and Bass call this broader knowledge “mathematical knowledge for teaching” (MKT). An example of MKT in action is explaining why some students obtain incorrect products such as 245 or 1,055 when trying to multiply 35 and 25. Mathematics professors have a difficult time explaining these errors, perhaps due to their flexible understanding of number systems. On the other hand, experienced math teachers are able to explain them easily, drawing on both strong numerical reasoning and a well-developed understanding of the relationships between concepts and algorithms. Building Mathematical Knowledge for Teaching... and our Circle According to Helen Doerr and her colleagues (Doerr, Goldsmith, Lewis, 2010), high-quality professional learning opportunities involve a number of factors, such as: teachers’ engagement in ongoing, collaborative activities; frequent opportunities to notice, analyze, document, and respond to students’ thinking; and developing key mathematical and pedagogical habits of mind. As an example of habits of mind, effective teachers manage classroom discussions in ways that involve students and support their reasoning (Chapin, O’Conner, & Anderson, 2003). The key here is that professional learning opportunities must engage both mathematical and pedagogical thinking. Doing
MTCircular · Summer/Autumn 2015 · American Institute of Mathematics