Assessing Pre-Service Secondary School Teachers’ Mathematical Knowledge for Teaching Algebra Sharon Senk Joan Ferrini-Mundy Michigan State University January 12, 2006

BACKGROUND Mathematical Knowledge for Teaching Mathematical Education of Teachers RAND Mathematics Study Panel Shulman Ma Ball & Bass Many other scholars

KNOWING MATHEMATICS FOR TEACHING ALGBRA (KAT) PROJECT (NSF REC No ) Joan Ferrini-Mundy, PI Robert Floden Raven McCrory Mark Reckase Sharon Senk Karen Allen & Xuhui Li

GOALS OF KAT PROJECT Instrument Design ( ) Develop theoretical constructs, items, test forms Framework Validation (2006) Do data support theoretical framework? Status Study ( ) How does knowledge for teaching algebra vary among pre-service and in-service teachers across the U.S.?

Item Development August October 2005 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

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.

Pilot Testing Volunteers were 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

Number of Participants in Pilot Studies November December 2005 Pre-service teachers387 In-service teachers423 Total810

KAT Pilot Sites Calvin College (MI) Cal. State Univ. - Fullerton George Mason University Grand Valley State Univ. Kennesaw State University Michigan State University Oregon State University St. Xavier University San Diego State University Syracuse University Texas State University - San Marcos University of Arizona UC Berkeley University of Delaware UNC Greensboro Univ. of South Florida Valparaiso University Western Michigan Univ. In-service teachers in CA,DE,IL, MI, OH

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.

Sample Item: Identify 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

Results: Identify Exponential Function Number of casesDifficulty Pre-service teachers In-service teachers Total

Sample Item: Number Systems (Advanced 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

Results: Properties of Number Systems (Advanced Knowledge) Number of casesDifficulty Pre-service teachers In-service teachers Total

Sample Item: 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?

Results: Identify Student’s Error Number of casesDifficulty Pre-service teachers In-service teachers Total

NEXT STEPS Validation Study (March - August, 2006) Need at least 1600 participants in-service secondary teachers pre-service secondary teachers math majors (juniors & seniors) math graduate students

Contacts for KAT Project Information Xuhui Li, Project Joan Ferrini-Mundy, PI, Sharon Senk,