TEACHER INTERVIEWS, STUDENT INTERVIEWS, AND CLASSROOM OBSERVATIONS IN COMBINATORICS: FOUR ANALYSES.
Abstract: This research consists of teacher interviews, student interviews,
and classroom observations, all based around the mathematical content area of
combinatorics. Combinatorics is a part of discrete mathematics concerning the ordering and
grouping of distinct elements. The data are used in four separate analyses. The first
provides evidence that student interviews can be a useful source ... read moreof data when considering
the qualities of instruction. The case analysis shows that the teacher's instruction
shifted. During interviews, the student responses showed indications of the shifts. The
student interviews allowed us to see things we would not have seen through classroom
observations or written assessments, and these things reflected the qualities of the
instruction. The second analysis explores a framework of types of teacher knowledge in a
novel way. The analysis assigns knowledge types to statements made during interviews. Not
all teachers showed the same relative frequency of the different types. The implication is
that with more teachers and in connection with classroom data, we may understand what these
profiles suggest about a teacher's work and the types of supports that would help them. The
third analysis examines the connections between students solving problems involving the
multiplication principle and solving problems involving permutations. Analysis of
interviews showed that on problems involving permutations, students often incorrectly
overextended the multiplication principle. Students are struggling to make the transition
from multiplication principle problems to permutation problems. This suggests that they
need support to understand of how the two types of problems differ. The fourth analysis
looks at students' representations in combinatorics. Both interviews and classroom
observations showed novel student representations. The analysis shows that students
generate useful non-canonical representations and that we can benefit from utilizing these.
The four analyses connect to different areas of research. The first two papers consider the
complex characteristics of teacher knowledge. They aim to become part of the ongoing
conversation about how to prepare, evaluate, and support math teachers. The third and
fourth papers focus on elements of student thinking in combinatorics. These provide
examples to indicate that there is still much we do not know about this
Thesis (Ph.D.)--Tufts University, 2012.
Submitted to the Dept. of Education.
Advisor: Bárbara Brizuela.
Committee: Analúcia Schliemann, Montserrat Teixidor i Bigas, Heather Hill, and Maria Blanton.
Keywords: Mathematics education, Teacher education, and Secondary education.read less