(Page 6 of 7) - Guiding Mathematical Discussions in High School Mathematics Classes authored by Staples, Megan.

to use the words IN and OUT as he explained. In this manner, she was pushing him to a new way
of thinking that (she believed) would support the development of his proficiency.

Similarly, later in the six-week sequence, Ms. Nelson explained to students that a rule such as

IN + OUT = 18 is good and correct, but that the best way to write rules is in the form “OUT =.”
Thus Ms. Nelson honored their present approach, but pushed them to recast the relationship they
saw in a manner that more closely aligned with how explicit functions are conceptualized.
Indeed, the difference between conceptualizing a pattern as IN + OUT = 18 and OUT = 18 – IN,
while trivial for those well versed with mathematics, was significant for these students. Ms.
Nelson recognized the different demands placed on the students and carefully attended to these
differences as she worked to develop new ways of thinking and proficiencies.

Discussion and Conclusion

The study provides a detailed analysis of how one teacher guided the mathematics during

lessons aligned with current calls for mathematics reform by centralizing students’ thinking,
implementing cognitively demanding tasks, and attending to a developmental map of algebra
learning. The results reported here echo some earlier findings, such as the importance of eliciting
students’ ideas. They also point to other aspects of teaching practice that may be critical
components of organizing productive, collaborative learning environments, such as using a small
inferential gap to guide task implementation. Further research is needed to examine the potential
value of these constructs in other settings.

The role of teacher knowledge also comes to the fore in this study. Ms. Nelson’s

developmental map of students’ algebra learning is a particularly interesting component of her
knowledge. This knowledge can be classified as pedagogical content knowledge (PCK)
(Shulman, 1986), and yet it seems to extend beyond this. The knowledge represented in this
developmental map in some respects is similar to what Ma (1999) has called profound
understanding of fundamental mathematics. Her focus was Chinese and American elementary
teachers’ knowledge of basic operations. Some of the Chinese teachers had an intricate and well-
connected conceptualization of the connections between mathematical ideas called knowledge
packets. The organization of the knowledge packet was based in part of mathematics and
relationship among ideas, but to large extent was informed by students’ thinking and how
students develop proficiency by progressing through a certain sequence of activities (e.g.,
addition of digits, addition within 20, etc.). Ms. Nelson’s map is similar in that respect. It relies
on a deep understanding of algebra, but is organized based on how students might learn,
including critical transitions in students’ ways of thinking. More research is needed to identify
the kinds of teacher knowledge needed to support classrooms aligned with NCTM’s vision of
good practice.

References

Ball, D. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school

mathematics. Elementary School Journal, 93, 373-97.

Boaler, J. (2003). Studying and capturing the complexity of practice: The case of the ‘Dance of

Agency’. Paper presented at the Psychology of Mathematics Education International
Conference, Honolulu, Hawaii.

Boaler, J. & Greeno, J. (2000). Identity, agency, and knowing in mathematics worlds. In J. Boaler

(Ed.), Multiple perspectives on mathematics teaching and learning (pp. 171-200). Westport,
CT: Ablex.