Mathematics
Education

Colloquium

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**PEDAGOGICAL AND COGNITIVE FLOW IN MATHEMATICS**

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November 27, 2007

Room 114 at 4:00

This
paper is based on partial results of an exploratory study that looked at the
potential pedagogical and cognitive flow in secondary school mathematics
classes as revealed by 36 public school mathematics teachersŐ choices of
activities and problems or exercises they plan to use in a standard mathematics
class. The teachers were given
three instruments to complete: a demographics questionnaire, a beliefs
questionnaire, and a curriculum questionnaire. Responses to the curriculum questionnaire, which provided
the main data for this part of the study, were analyzed using content analysis
methods. The teachers were asked
to select from a list of 10 problems and exercises on the content lesson __Equations
of a Line__ typical problems or exercises they would use for each of 6 parts
in a normal mathematics class, namely: motivation or introduction to the
lesson, demonstration of solving a sample problem, practice exercises, review,
assigned homework, in-class assessment.
Using the TIMSS 2003 framework, the researcher classified each problem
or exercise as developing low-level skills (*Knowing* and *Solving Routine Problems*), mid-level skills (*Using Concepts* and *Solving Routine Problems*) and high-level skills (*Reasoning*).
The cognitive sequence resulting from the choices of problems or
exercises for each part of the lesson was categorized based on the potential
cognitive response of students: rewarding, challenging, or frustrating. Descriptions of each category or
cognitive response are discussed in detail.

Results
show that 21 of the 36 teachers could potentially develop low-level, low to
mid-level, and mid-level skills at best while only 15 could potentially develop
mid to high-level and high-level skills at best. Students could potentially end up frustrated in 20 of the 36
teachersŐ classes while students could potentially feel cognitively rewarded in
only 7 of the 36 teachersŐ classes.
Selected teachersŐ explanations for their choices indicate lack of
understanding of the cognitive demands of a particular problem or exercise and
inadequate pedagogical and content training, among others. Implications for teacher training are
offered and these include the need for introducing a more systematic approach
to developing pedagogical content knowledge among mathematics teachers.