Knowledge of Algebra for Teaching: Framework, Item Development, and Pilot Results Joan Ferrini-Mundy Sharon Senk Division of Science and Mathematics Education Michigan State University NCTM Research Symposium April 25, 2006

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Knowing Mathematics for Teaching Algebra (KAT) Project NSF REC No ( ) PI Joan Ferrini-Mundy Co-PIs Robert Floden, Raven McCrory, Sharon Senk Senior personnel Mark Reckase, Gail Burrill, William Schmidt Project Manager Karen Allen, Xuhui Li

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Fundamental Question: What knowledge of algebra for teaching do secondary school teachers of algebra need to support their instruction?

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Fundamental Question, v.2 What knowledge of algebra for teaching “do/should/might” secondary school teachers of algebra “draw upon” to support their instruction?

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Fundamental Question, v.3 What knowledge of algebra for teaching “do/should/might” secondary school teachers of algebra “draw upon/bring to bear” to support their instruction?

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Conceptual background 3 Pilot study 4 Next steps 2 Item development

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Conceptual background

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Pedagogical Content Knowledge Lee Shulman, 1986, pp For the most regularly taught topics in one’s subject area: The most useful representations of ideas The most powerful analogies, illustrations, examples and demonstrations Ways of representing and formulating the subject that make it comprehensible to others A veritable armamentarium of alternative forms of representation Understanding of why certain concepts are easy or difficult to learn Conceptions and preconceptions that students bring Strategies to help students reorganize their understanding

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Curricular Knowledge Lee Shulman, 1986, p. 10 Understandings about curricular alternatives available for instruction Familiarity with the under study by a teachers’ students in other subjects Familiarity with the topics that have been and will be taught in the subject during the preceding and later years in school, and the materials that embody them.

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Profound Understanding of Fundamental Mathematics Liping Ma, 1999, p. 122 Connectedness, multiple perspectives, basic ideas, longitudinal coherence Awareness of the conceptual structure and basic mathematics inherent in elementary mathematics Ability to provide a foundation for that conceptual structure for students

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Mathematics for Teaching Al Cuoco, 2001 The vertical disconnect. Most teachers see very little connection between the mathematics they study as undergraduates and the mathematics they teach.

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Mathematical Knowledge for Teaching Deborah Ball & Hyman Bass, 2000 “a kind of understanding [that] is not something a mathematician would have, but neither would be part of a high school social studies’ teacher’s knowledge” “teaching is a form of mathematical work… involves a steady stream of mathematical problems that teachers must solve” Features include: unpacked knowledge, connectedness across mathematical domains and over time (seeing the mathematical horizons)

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, KAT Conceptual Framework

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Tasks of Teaching

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Analyzing students’ mathematical work and thinking Designing, modifying, and selecting mathematical tasks Establishing and revising mathematical goals for students Accessing and using tools and resources for teaching Explaining mathematical ideas and solving mathematical problems Building and supporting mathematical community and discourse

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Categories of Knowledge

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Core content knowledge Representation Content trajectories Applications and contexts Language and conventions Mathematical reasoning and proof

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Overarching Categories Bridging Trimming Decompressing

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Item development

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Assessment Blueprint

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Item Development and Refinement August 2004 – October 05 Constructs defined Item writing workshops with mathematicians, math educators, secondary teachers Additional Items written by KAT faculty & GAs Items reviewed by mathematicians Items edited by KAT staff

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Mathematical Knowledge for Teaching Algebra (simplified for assessment design) Knowledge of school algebra algebra in middle and high school Advanced mathematical knowledge related college math, e.g. calculus, abstract algebra Teaching knowledge knowledge of typical errors, canonical uses of school math, curriculum trajectories, etc.

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Knowledge of School Algebra Knowledge of mathematics in the intended algebra curriculum for middle and high school The knowledge we expect of students in school algebra NCTM Principles and Standards “big ideas” NAEP and state standards and expectations Topics in textbooks and instructional materials

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Advanced Mathematical Knowledge Other mathematical knowledge, including college level mathematics Content that conveys the trajectory and growth of mathematical ideas beyond school algebra Mathematical Education of Teachers examples -- broader and deeper Calculus, linear algebra, number theory, abstract algebra, analysis, and modeling Alternate definitions, extensions and generalizations, applications

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Teaching Knowledge Knowledge specific to teaching algebra that might not be taught in advanced courses What makes a particular concept difficult to learn What misconceptions lead to specific mathematical errors Mathematics needed to identify mathematical goals for instruction, choose tasks, identify trajectories Aspects of pedagogical content knowledge Mathematics content applied in teaching

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Pilot studies

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Pilot Testing Volunteers recruited to administer forms of the item sets to target samples Students in mathematics teacher preparation Practicing teachers Interns/those in professional development Wanted variety in teaching experience and type of mathematics course work

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Number of Participants in Pilot Studies November December 2005 Pre-service teachers387 In-service teachers423 Total810

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Some Pilot Sites Calvin College (MI) Cal. State Univ. - Fullerton George Mason University Grand Valley State University Kennesaw State University Michigan State University Oregon State University St. Xavier University San Diego State University Syracuse University Texas State Univ.– San Marcos University of Arizona UC Berkeley University of Delaware Univ. of North Carolina-Greensboro University of South Florida Valparaiso University Western Michigan Univ. In-service teachers in CA,DE,IL, MI, OH

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Analysis of Items Items were analyzed to check a number of features. Difficulty: proportion correct/mean performance Spread of scores Items that were outside target difficulty range, showed little spread of scores, or showed negative discrimination were revised or eliminated. As of December 2005 we have about 100 items of which about 50 meet our criteria.

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Sample Item 1: Identifying an Exponential Function (School Knowledge) Which of the following situations can be modeled using an exponential function? i. The height h of a ball t seconds after it is thrown into the air. ii. The population P of a community after t years with an increase of n people annually. iii. The value V of a car after t years if it depreciates d% per year. A. i only B. ii only C. iii only D. i and ii only E. ii and iii only

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Results: Identifying an Exponential Function Number of casesDifficulty Pre-service teachers In-service teachers Total

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Sample Item 2: Properties of Number Systems (Advanced Mathematical Knowledge) For which of the following sets S is the following statement true? For all a and b in S, if ab = 0, then either a = 0 or b = 0. i. the set of real numbers i. the set of complex numbers iii. the set of integers mod 6 iv. the set of integers mod 5 v. the set of 2x2 matrices with real number entries A. i onlyD. i, ii, iii and iv only B. i and ii onlyE. i, ii, iii, iv, and v C. i, ii and iv only

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Results: Properties of Number Systems Number of casesDifficulty Pre-service teachers In-service teachers Total

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Sample Item 3: Identifying Student’s Error in Solving a Linear Equation (Teaching Knowledge) A student solved the equation 3(n - 7) = 4 - n and obtained the solution n = What might the student have done wrong?

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Results: Identifying Student’s Error Number of casesDifficulty Pre-service teachers In-service teachers Total

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Sample Item 4: Interpreting and Relating Equivalent Expressions (Teaching Knowledge) Hot tubs and swimming pools are sometimes surrounded by borders of tiles. The drawing at the right shows a square hot tub with sides of length s feet. This tub is surrounded by a border of 1 foot by 1 foot square tiles. s How many 1-foot square tiles will be needed for the border of this pool? a. Paul wrote the following expression: 2s + 2(s+2) Explain how Paul might have come up with his expression. b. Bill found the following expression: (s+2) 2 - s 2 Explain how Bill might have found his expression. c. How would you convince the students in your class that the two expressions above are equivalent?

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Number of casesDifficulty Pre-service teachers In-service teachers Total Results: Equivalent Expressions

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Next steps

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Validation Study: spring-summer, 2006 Status Study:

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Validation Study (2006) Constructs to be Assessed Components of KAT Knowledge of School Algebra Advanced Knowledge Teaching Knowledge The hypothesis is that it will be possible to distinguish among these components through analysis of item response data.

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Types of Validity Evidence Judgments of item content and cognitive levels – If judges agree on classification into categories, validity of inferences is supported. Statistical analysis of the structure of item response data – Analyses could show that sets of items define relatively unique constructs. Predictive analysis of performance of groups – Groups can be identified that should differ on the three components. If they do differ, it supports the validity of the inferences.

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Participants Needed for Validation Study March 20 th – August 15 th Paper-and-pencil assessment (up to one hour) Target participants: - preservice secondary math teachers - inservice secondary math teachers - undergraduate mathematics majors - math/math education graduate students Detailed descriptions on last page of the handout

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Status Study ( ): Original Goals How does knowledge for teaching algebra of preservice teachers compare with that of experienced teachers? What is the status of knowledge for teaching algebra among preservice teachers in different mathematics and mathematics education course settings? What is the status of knowledge for teaching algebra among secondary school mathematics teachers who have participated in various algebra-related professional development experiences?

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Populations of Interest Teacher groups likely to have varied algebra knowledge for teaching profiles: Preservice course settings: linear algebra, abstract algebra, math methods, capstone courses Inservice settings: mathematics master’s degree programs, algebra PD programs Secondary school teachers using various algebra curricula with varied approaches

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Contacts for KAT Project Information Xuhui Li (Project Manager) Joan Ferrini-Mundy (PI) Sharon Senk (Co-PI)

© 2006 Knowing Mathematics for Teaching Algebra NSF REC # NCTM Research Symposium April 25, Discussants Nicole Ice Kennesaw State University Cos Fi The University of North Carolina at Greensboro