Preparing to Teach Algebra (PTA)
JILL NEWTON
PURDUE UNIVERSITY
Today’s Talk
 Description of algebra projects
 PTA journey/project team
 Rationale for the PTA study
 Research questions/Methodology/Timeline
 What have we learned so far? What challenges have
we faced?
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Framework development
Protocol development
Survey development
Preliminary findings
 Questions/Comments
Algebra as a Research Focus
 Justification and Argumentation: Growing
Understanding of Algebraic Reasoning (JAGUAR)
Megan Staples (University of Connecticut)
 Sean Larsen (Portland State University)

 Pre-service Secondary Teachers’ Mathematical
Knowledge for Teaching Equations & Inequalities
Rick Hudson (University of Southern Indiana)
 Aladar Horvath, Sarah Kasten, & Lorraine Males

 Preparing to Teach Algebra (PTA)
 Yukiko Maeda (Purdue University)
 Sharon Senk
PTA Journey
 EXCITE Conference (MSU, 2008)
 Two algebra groups (K-12, TE)
K-12 – Mary Kay Stein, Ed Silver, Glenda, Beth, etc.
 TE – Raven, Kristen, Mike, Sandra, Sharon, Jill, Betty, etc.

 NSF REESE, 2009
 Important concept
 Implementation challenges - participation
 Scale down
 Revised, NSF REESE, 2010 (Funded)
 Sharon Senk, Jill Newton, Yukiko Maeda
 Scale down more
PTA Team
 MSU
 PI - Sharon Senk
 GAs – Jia He & Eryn Stehr
 Purdue
 PIs – Yukiko Maeda, Jill Newton
 GAs – Vivian Alexander, Hyunyi Jung, Alexia Mintos, & Kari
Wortinger/Tuyin An
 Undergraduates – Adam Hakes, Jules McGee/Ali Brown
 Project Team Meetings
 MSU, Purdue, Advisory Board Meeting (MSU), South Haven
PTA Team
 Advisory Board
 Tom Hoffer, Joint Center for Education Research at NORC
 Eric Hsu, San Francisco State University
 Karen King, NCTM
 Vilma Mesa, University of Michigan
 MSU Internal Advisory Board
 Dorinda Carter Andrews
 Robert Floden
 Glenda Lappan
 Jean Wald
 Suzanne Wilson
Rationale for the Study
 Algebra as foundation for advanced mathematics
and gatekeeper for post-secondary opportunities

(e.g., Kilpatrick & Izsák, 2008; Moses & Cobb, 2001)
 Algebra course and/or end-of course exam
requirement in most states and increasing diversity
of population of students taking algebra

(e.g., Perie, Moran & Lutkus, 2005)
 Failure rates in algebra are high
 (e.g., Loveless, 2008)
 Debates about how algebra should be taught
 (e.g., Chazan, 2008; Kieran, 2007)
Rationale for the Study
 Common Core State Standards for School Mathematics
(CCSSM) includes both old and new visions of algebra in
three strands: (1) Algebra, (2) Functions, (3) Modeling

(CCSSM, 2010)
 Research base about teaching algebra is thin, and lacks
strong connection to the research on students’ learning
algebra.

(Kieran, 2007)
 Most extant mathematics teacher education literature
was written by researchers studying aspects of programs
offered by their own institutions.

(Adler et al., 2005)
Rationale for the Study
 “Both strong content knowledge (a body of
conceptual and factual knowledge) and pedagogical
content knowledge (understanding of how learners
acquire knowledge in a given subject) are important”

(NRC, 2010, p. 4)
 “Prospective high school teachers of mathematics
should be required to complete the equivalent of an
undergraduate major in mathematics, [including] a
6-hour capstone course connecting their college
mathematics courses with high school mathematics.”

(CBMS, 2001, p. 7)
Rationale
 Great variation exists across mathematics teacher
education in the US (less variation in other
countries); less mathematics than “A+” countries,
also more general pedagogy than math pedagogy.

(Schmidt, Cogan, & Houang, 2011)
 More students taking algebra earlier; underprepared
students admitted to algebra do not fare well
(without additional supports); different versions of
algebra are being created.

(Stein, Kaufman, Sherman, & Hillen, 2011)
Research Question
What opportunities do secondary mathematics
teacher preparation programs provide to learn
about:
 Algebra
 Algebra teaching
 Issues in achieving equity in algebra learning
 The algebra, functions, and modeling standards
and mathematical practices described in the
Common Core State Standards for Mathematics
(CCSSM)?
Methods
 National survey of a stratified random sample of at
least 200 secondary teacher preparation programs

Carnegie classification for stratification, oversampled 2x
 Case studies of learning opportunities in four
purposefully chosen secondary teacher preparation
programs

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Doctoral-granting university with very high research activity
Large master’s level universities (rural/suburban, urban)
 Focus groups with student teachers at each of the
case study programs
Timeline
 Year 1
 Develop, pilot, and revise instruments
Survey
 Frameworks
 Interview protocols (Instructor, Focus group)


Select sample

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Locate contact people at institutions in survey sample
Collect and analyze pilot case study data

Three institutions
 Survey think-aloud
 Five instructor interviews & one focus group interview
Timeline
 Year 2
 Administer revised survey


Analyze, Summarize, & Disseminate
Identify courses at three case study institutions
Collect course materials (with site coordinator)
 Interview instructors
 Conduct focus groups
 Administer survey
 Transcribe, Analyze, Summarize, & Disseminate


Identify fourth case study institution

Collect course materials (with site coordinator)
Timeline
 Year 3
 Identify courses at fourth case study institution
Collect course materials (with site coordinator)
 Interview instructors
 Conduct focus groups
 Administer survey
 Transcribe, Analyze, Summarize, & Disseminate

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Survey

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Continue analysis, summarizing, & dissemination
Three case study sites

Continue analysis, summarizing, & dissemination
What have we learned so far?
 Framework development
 Stay focused on Algebra
 Select a set of big ideas in Algebra
Nature of Algebra (CBMS, 2001; InTASC, 1995)
 Reasoning & Proof (InTASC, 1995; NCTM, 2009; TNE)
 Contexts & Modeling (CCSSM, 2010; NCATE, 2003)
 Algebra Connections (CBMS, 2001; NCTM, 2000)
 Tools & Technology (CBMS, 2001; CCSSM, 2010)
 Equity in Algebra Learning (NBMS, 2010; NCTM, 2000)
 Functions (InTASC, 1995; NBMS, 2010; NMP, 2008)
 History of Algebra (NBMS, 2010; NCATE, 2003)

Connections
 Within algebra
 Between algebra & other mathematical fields
 Example: symmetry groups of polygons <-> geometry of
transformations
 Between algebra & other non-mathematical fields
 Example: quadratic functions used to model motion of
projectile in physics
 Between college level algebra & school algebra
 Example: Rings, integral domains, and fields related to
the number systems used in high school algebra
What have we learned so far?
 Protocol development
 Investigated semi-structured and focus group protocols
 Developed series of parallel questions for interviews
Instructor interview
 What are the big ideas that you would like students to take away
from this course?
 To what extent does your course emphasize functions?
 Focus group interview
 Will you please give us some examples of experiences from this
list in which you had opportunities to either learn algebra or
learn how to teach algebra?
 What experiences have you had to learn about functions or to
learn to teach functions?

What have we learned so far?
 Protocol development
 Challenges
Typical interviewing challenges
 Probing
 Avoiding evaluative language
 Consistency across interviewers
 Parallel structures of instructor and focus group interviews
 Maintaining focus on algebra
 Sharing big ideas (When? How?)

What have we learned so far?
 Survey development
 Reviewed items from related surveys
The Mathematics Teaching in the 21st Century (MT21) Study
(Schmidt, et al., 2007)
 TEDS-M Institutional Program Survey (Tatto, et al., 2008)
 Secondary Mathematics Teacher Education Programs in Iowa
Survey (Murdock, 1999)
 2000 National Survey of Science and Mathematics Education
School Mathematics Program Questionnaire (Horizon, Inc.,
2000).


Developed items to collect information to answer our research
questions
What have we learned so far?
 Survey development
 Challenges
Locating program information and contact person
 Diversity and complexity of programs across diverse institutions
 4 year/5 year?
 Degree?
 Online?
 Licensure by exam?
 Variable knowledge of program contact person
 Department?
 Instructor/Coordinator?
 Limited previous studies on mathematics teacher education

Survey Item (Program characteristics)
What is the most common degree that pre-service secondary
mathematics teachers obtain upon completion of the secondary
mathematics teacher education program at your institution ?
 4-year Bachelor's degree
 5-year Bachelor's degree
 Post-baccalaureate(i.e., for licensure only, not for Master's degree)
 Master's degree and initial certification (e.g., M.A.T. not for
previously certified individuals)
 No degree
 Other: Please specify___________________
Survey Item (Research questions)
To what extent does the secondary mathematics teacher education
program provide opportunities to learn in the following areas?
Great
extent
Some
extent
Little
extent
No
extent
Do not
know
Algebra
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Algebra teaching
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Issues in achieving
equity in algebra
learning
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Algebra as described
in the Common Core
State Standards in
Mathematics (CCSSM)
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Opportunity to Learn
Survey Item (CCSSM)
How has the recent release of Common Core State Standards for
Mathematics (CCSSM) influenced your secondary mathematics teacher
education program? (Check the one answer that best describes the
situation at your institution.)
 Discussions about CCSSM have not begun in our program.
 Discussions about CCSSM are ongoing, but no programmatic changes
have been made.
 Minor changes have been made to the program as a result of CCSSM.
 Major changes have been made to the program as a result of CCSSM.
 I am not familiar with CCSSM.
Briefly describe the changes that have occurred in your program
based on CCSSM.
Preliminary Findings
 Contexts & Modeling
 Instructors seemed to hold a similar conception of the purpose of
modeling; that is, one which connects the use of mathematics to solve
real world problems (e.g., loan repayment schedules, population growth,
and security problems).

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The program provided a wide range of opportunities for PSTs to engage
with C&M related to algebra through course activities and assignments,
including using C&M to motivate course topics.

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For example, the Structure of Algebra instructor described mathematical
modeling as the process of using mathematics to represent phenomena that
one seeks to understand.
Two of the instructors interviewed alluded to the motivational quality of C&M
that could encourage student interest and persistence in mathematics.
PSTs reported that opportunities to engage in C&M arose in the
following courses in the program: Math Modeling, Differential
Equations, Math Software, Probability & Statistics, Geometry, Abstract
Algebra, Calculus, and Mathematics Methods.
Preliminary Findings
 Connections
 Within algebra
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With other mathematical fields

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Middle School Math Methods course instructor emphasized algebra
lessons including geometric or statistical concepts in the micro-teaching
and the Connected Mathematics Project lessons. In addition, a PST said,
“When you are in geometry, you just can't say geometry, because we need
algebra to complete the proofs.”
With non-mathematical fields

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A seminar course instructor stated that PSTs were encouraged to consider
relationships among concepts in different chapters in the textbook.
Differential Equations instructor explained that students develop
“functional models for certain physical situations that will often involve
exponential functions or sines and cosines.”
Between high school and university-level algebra.

Goal in the Abstract Algebra course: “Make a connection between what
they [PSTs] have already learnt through high school and to new
mathematical systems.”
Preliminary Findings
 Reasoning & Proof
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Instructors reported that R&P played a significant role in their courses. PSTs also
reported having opportunities to access content knowledge and pedagogical content
knowledge related to R&P.
PSTs’ opportunities in mathematics courses:
 Learn about axiomatic systems by examining what qualifies as a proof, differences
between definitions and theorems, and specific techniques of constructing proofs, and
proving the equivalence of two statements.
 Fewer chances to engage in making conjectures. The Capstone Course instructor said
that making conjectures did not arise often in classes although he thought
“conjectures are important…because they give the sense that mathematics is alive.”
PSTs’ opportunities in methods courses:
 Discuss conceptions of proof as related to K-12 mathematics, engage in activities such
as generalizing, formalizing and refining mathematical arguments.
 Engage in designing tasks and questions to support students’ learning of R&P. Both
instructors of methods courses and PSTs in the focus group mentioned class activities
related to one PST called “what qualifies as proof.”
Thank you!
C O M M E N T S / Q U E S TI O NS ?