Dan Lortie's 1975 book

*Schoolteacher*clarified an idea that teachers already know: how we teach is greatly influenced by the way we've been taught. Lortie called the idea

*apprenticeship of observation*, and it specifically refers to how teachers, having spent 13,000+ hours in classrooms as students, take that experience as a lesson in how to be a teacher. What we often fail to deeply reflect on, however, is that we were only seeing the end product of teaching. We didn't see the lesson planning, go to summer conferences, attend professional development workshops, study the science of learning, or take part in the hundreds of decisions a teacher makes every day. Just observing isn't a proper apprenticeship, even after thousands of hours watching good teachers. I think of it this way: I watch a lot of baseball, and I can tell good baseball from bad. This hardly makes me ready to play, sadly, because I'm not spending hours taking batting practice, participating in fielding drills, studying video, digesting scouting reports, and working out in the offseason. Just as watching a lot of baseball doesn't

*really*prepare me to play baseball, watching a lot of teaching doesn't

*really*prepare someone to teach. Still, all those hours heavily influence our beliefs, both of teaching and of subject matter.

Deborah Ball (CC BY-NC-ND House Committee on Education and the Workforce Democrats) |

*Unlearning to Teach Mathematics*, she describes a project involving teaching permutations to her class of introductory preservice elementary teachers. The goal was not simply to teach her students about permutations, but also to learn more about their beliefs about the nature of mathematics and to develop strategies that might enlighten those beliefs and break the cycle of simply teaching how you were taught.

By selecting permutations as the topic, Ball hoped to expose these introductory teachers to a topic they'd never studied formally. By carefully observing how her students constructed their knowledge, Ball would be able to see how their prior understandings about mathematics influenced their learning. The unit lasted two weeks. In the first phase of the unit, Ball tried to engage the students in the sheer size and scope of permutations, like by thinking about how the 25 students could be sat in 1,551,121,000,000,000,000,000,000 different seating arrangements. Working back to the simplest cases, with 2, 3, and 4, students, students could think and talk about the patterns that emerge and understand how the permutation grows so quickly. For homework, Ball asked students to address two goals: increase their understanding of permutations, but also think about the role homework plays in their learning, including how they approach and feel about it and why. In the second phase of the unit, Ball has her students observe her teaching young children about permutations, paying attention to the teacher-student interactions, the selection of tasks, and what the child appears to be thinking. In the last phase of the unit, the students become teachers and try helping someone else explore the concept of permutations. After discussing this experience, students wrote a paper reflecting on the entire unit.

From other research, Ball knew that teacher educators often assumed their students had mastery of content knowledge. Even moreso, future elementary math teachers themselves assumed they had mastery over the mathematical content they'd be expected to teach. She knew, however, that there was something extra a teacher needed to teach that content. Citing Shulman's

*pedagogical content knowledge*, along with numerous others, Ball describes some ways we can think about what that special content knowledge for teaching is, but admits that her permutations project was too narrow to explore how teachers construct and organzie that knowledge. The project would, however, give insight to her students' ideas about mathematics, and assumptions they make about what it means to know mathematics. For example, a student named Cindy wrote:

I have always been a good math student so not understanding this concept was very frustrating to me. One thing I realized was that in high school we never learned the theories behind our arithmetic. We just used the formulas and carried out the problem solving. For instance, the way I learned permutations was just to use the factorial of the number and carry out the multiplication ... We never had to learn the concepts, we just did the problems with a formula. If you are only multiplying to get the answer every time, permutations could appear to be very easy. If you ask yourselfwhydo we multiply and really try to understand the concept, then it may be very confusing as it was to me. (p. 44)

Comments like this revealed that many of Ball's students relied on a procedural view of mathematics, one where the question "Why?" had been rarely asked. Ball also noticed a theme in her students' reflections about knowing math "for yourself" versus for teaching. Alison wrote:

I was trying to teach my mother permutations. But it turned out to be a disaster. I understood permutations enough formyself, but when it came time to teach it, I realized that I didn't understand it as well as I thought I did. Mom asked me questions I couldn't answer. Like the question about there being four times and four positions and why it wouldn't be 4 x 4 = 16. She threw me with that one and I think we lost it for good there.

From observing a young student learn about permutations in phase two, Ball noticed that some of her students started to challenge some of their assumptions they made about themselves as learners. Both from her experience and from the literature, Ball knew that elementary preservice teachers are often the most apprehensive about teaching mathematics. In some cases, these students choose to teach elementary in the hopes of avoiding any mathematical content they might find difficult. Changing these feelings about mathematics and about themselves is a difficult task for the teacher educator, but Ball did see progress. Christy, for example, said, "Most of all, I realized that I

*do*have the ability to learn mathematics when it is taught in a thoughtful way" (p. 45). Unfortunately, not all shared this experience, as Mandy said she "did not enjoy the permutations activities because I was transported in time back to junior high school, where I remember mathematics as confusing and aggravating. Then as now, the explanations seemed to fly by me in a whirl of disassociated numbers and words" (p. 45).

In her conclusion, Ball says activities like the permutations project can be used by teacher educators to expose students' "knowledge, beliefs, and attitudes" (p. 46) about math and teaching math. By understanding the ideas prospective teachers bring with them, teacher educators can better develop preparation programs that address those beliefs in ways that strengthen the positive ones while changing some negative ones. Also, by including these kinds of activities with introductory preservice teachers, this can raise their expectations for what they will encounter later in methods classes. Summarizing, Ball concludes:

How can teacher educators productively challenge, change, and extend what teacher education students bring? Knowing more about what teachers bring and what they learn from different components of and approaches to professional preparation is one more critical piece to the puzzle of improving the impact of mathematics teacher education on what goes on in elementary mathematics classrooms. (p. 46)

References

Ball, D. L. (1988). Unlearning to teach mathematics.

*For the Learning of Mathematics*,*8*(1), 40–48. Retrieved from http://www.jstor.org/stable/40248141