Posing and Pursuing
One’s Own Questions:
Experiences of Graduate Students
in Math Education and Math
Eden M. Badertscher, Institute for
Learning at University of Pittsburgh
Juliana Belding, Harvard University
Overview of the Course

Main Goal: Students develop ability and
desire to ask and investigate their own
mathematical questions

Other Goals: Mathematical communication,
Appreciation for process/struggle of math

Who: Graduate students in Math Ed (also
Physics Ed and Mathematics)
Structure of Course

In-Class Investigations
– Weekly 2 hr. classes
– Three topics (last 4-5 weeks each)
– Small groups, informal presentations

Individual Projects
– Student-designed, Outside of class
– Studio Times (1 hr., 5 during semester)
– Final written project

Journals (in-class and weekly project
updates)
In-Class Investigations:
Wrestling with…

1: Rational Numbers
Farey Sequences, Representation of Rationals

2: Geometry
Definitions of A Parabola, Taxi-cab Geometry

3: The Real Numbers
Cardinality, Representation of Reals
The Creative Process in
Mathematics
“Mathematics has two faces. Presented in finished form,
mathematics appears as a purely demonstrative
[deductive] science, but
mathematics in the making is
a sort of experimental science.
A correctly written mathematical paper is supposed to
contain strict demonstrations only, but the creative
work of the mathematician resembles the creative
work of the naturalist;
observation, analogy, and conjectural
generalizations, or mere guesses
play an essential role in both.” -Polya, 1952
Mathematical Themes
Through the creative process, students develop need for
and appreciation of




Definitions (their role, principled choices, sensemaking)
What constitutes a proof?
Multiple viewpoints (geometric, algebraic, etc.)
Precise mathematical language and notation
“What-If-Not”
From The Art of Problem Posing, Brown and
Walter, 1983
Given a mathematical object/situation/problem:
 List attributes
 Ask “what-if-not” (tweak attributes)
 Formulate new questions
Example: A Parabola

Definition: The locus of points equidistant
from a line (directrix) and a point not on the
line (focus)

What-if-not…
–
–
–
–
Not Equidistant (1/2 as far, 2x as far)
Directrix is another object (circle, a parabola)
Focus is on the line (degenerate conics)
Non-Euclidean distance (Taxi-cab geometry)
Other Resources
Habits of Mind: An Organizing Principle
of Mathematics Curricula, Cuoco,
Goldenberg and Mark. 1996
 The Roles of The Aesthetic in
Mathematical Inquiry, Sinclair, 2004

Instructors’ Role

In class:
– participant (“having new eyes”)
– translator (model communication)
– facilitator (restart, regroup and recap)

Outside of class:
– reframe questions
– restructure groups
according to interest, facility with formal math,
working style and new questions that arise…
The Students’ Experience
“It made me understand what it means to
explore math
as opposed to learn math and solve problems”
(from student’s reflection)
Freedom of exploration/“Playfulness”
 Increased confidence in math & validity of
own questions
 Improved communication of mathematics
(through journals/project write-ups)

What can a Mathematician
(in training) gain?

Challenges:
– Being the know-it-all “math guy”
– When to hold back, when to contribute
– Some topics already familiar
– At times, less rigorous than used to
What can a Mathematician
(in training) gain?

Benefits:
– More freedom to ask questions, make
guesses (than in graduate math classes)
– Exposure to more experimental, intuitive,
“naïve” approaches to math
– Practice for graduate research (formulating
a thesis question, trying multiple
approaches)
Why?
All learners (educators in particular) need
hands-on experience with..

how math is created (what do
mathematicians do?)
 how to communicate about math across
disciplines (eg: Hy Bass)
 how to learn math outside a classroom or
textbook
For More Information

CSSM Institute at Educational Development
Center, Newton, MA
http://cssm.edc.org/AboutCSSM.html

Learning to learn Mathematics: Voices of
doctoral students in mathematics education.
In M. Strutchens & W. Gary Martin (eds.) The Learning
of Mathematics. 69th Yearbook of the National
Council of Teachers of Mathematics. NCTM: Reston,
VA.
Appendix: Getting Into the Problems

How did we set up each of the
investigations?
Rationals
Geometry
For each definition,
1.
2.
3.
locus of points equidistant from focus
and directrix
locus made from cutting plane parallel
to side of cone
equation of form y = ax^2 + bx + c
Generate attributes, what-if-nots and
questions.
Real Numbers
Read Zeno’s Paradox
 Read Hotel Infinity
 Generate, categorize and refine
questions about real numbers

Individual Projects

Key Elements:
– Students choose/develop own question
– Instructors and peers give weekly feedback
– Structured communication (oral and written)
Some Examples
 Moving a Couch Through a Doorway
 Finding Geometric Proofs of Trig Identities
 Iterating Rational Functions