Mathematical habits of mind
and ways of thinking for
prospective teachers
Gail Burrill
Michigan State University
Background
• Association of Mathematics Teacher Educators
(AMTE)
• Preliminary Teacher Education Development Study
(PTEDS) (NCES)
• Teacher Education Development Study (TEDS) (NCES)
• PROM/SE Michigan State Mathematics Science
Partnership (NSF)
• Knowledge of Algebra for Teaching (KAT) (NSF)
• Teachers for a New Era (Carnegie Grant)
• Capstone Courses (MSU)
Association of Mathematics
Teacher Educators (AMTE)
Goals for the Preservice Education of Prospective
Secondary Teachers
1. Analyze and purposefully transform their beliefs and
dispositions about what mathematics is and what it
means to learn, do and teach mathematics
Conceptual Aspects Informing Teacher Preparation
PTEDS
© 2006 Michigan State University, Center for the Study of Curriculum
Supported by NSF Grant REC-0231886
X38: Algebra Knowledge Item
© 2005 MSU P-TEDS
Supported by NSF Grant REC-0231886
Doing mathematics
In general, a mathematical approach
involves defining a problem through
conjecturing in an established
mathematical area. Conjecturing may be
supported by technology, by compelling
ideas based on past work, computation,
or pattern exploration.
(Teachers for a New Era, 2003)
Doing mathematics
Mathematics involves representing a
mathematical concept concretely; a single
concept can have multiple representations.
One of the beauties of mathematics as a
whole is the interplay between various areas
of the subject. A particular way of
representing a problem may lead to an
especially efficient or enlightening result.
(Teachers for a New Era, 2003)
Capstone Course - MSU Senior
Math Majors:
• Deepen understanding of the mathematics
needed for teaching in secondary schools.
• Prepare students to
1. describe connections in
mathematics;
2. figure things out on their own.
High school math from an
advanced perspective
• Analyses of alternative definitions,
language and approaches to mathematical
ideas;
• Extensions and generalizations of familiar
theorems;
• Discussions of historical contexts in which
concepts arose and evolved;
• Applications of the mathematics in a
variety of settings;
High school math from an
advanced perspective
• Demonstrations of alternate ways of
approaching problems, with and without
technology;
• Discussions of relations between topics
studied in this course and contemporary
high school curricula.
Habits of mind: Mathematics
should make sense
• Know that all mathematics is not equal
• Understand that mathematics can be done
different ways
• Be willing to work hard on a challenging
problem for a long time
• Recognize that mathematics is about how
mathematical results are obtained not how
they are presented
Mathematically confident
enough to take risks
•
•
•
•
•
Using technology
Asking for explanations
Doing problems with students
Trying something new
Celebrating mistakes
A habit of mind

i
e  1
a b c
2
2
2
opposite
sin A 
hypotenuse
Polya’s Ten Commandments
Read faces of students
Give students “know how”, attitudes of
mind, habit of methodical work
Let students guess before you tell them
Suggest it; do not force it down their
throats (Polya, 1965, p. 116)
Polya’s Ten Commandments
Be interested in the subject
Know the subject
Know about ways of learning
Let students learn guessing
Let students learn proving
Look at features of problems that
suggest solution methods (Polya, 1965,p. 116)