JILL NEWTONPURDUE UNIVERSITY

Preparing to Teach Algebra (PTA)

Today’s Talk

Description of algebra projectsPTA journey/project teamRationale for the PTA studyResearch questions/Methodology/TimelineWhat have we learned so far? What challenges

have we faced? 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: AlgebraAlgebra teachingIssues 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 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 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

Survey 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 physicsBetween 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?

Opportunity to Learn

Great extent

Some extent

Little extent

No extent

Do not know

Algebra □ □ □ □ □

Algebra teaching □ □ □ □ □

Issues in achieving equity in algebra learning

□ □ □ □ □

Algebra as described in the Common Core State Standards in Mathematics (CCSSM)

□ □ □ □ □

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). For example, the Structure of Algebra instructor described mathematical

modeling as the process of using mathematics to represent phenomena that one seeks to understand.

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. 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

A seminar course instructor stated that PSTs were encouraged to consider relationships among concepts in different chapters in the textbook.

With other mathematical fields 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 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 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.”

COMMENTS/QUESTIONS?

Thank you!