knowledge of algebra for teaching: framework, item development, and pilot results joan ferrini-mundy...

Knowledge of Algebra for Teaching: Framework, Item Development, and

Pilot Results

Joan Ferrini-MundySharon Senk

Division of Science and Mathematics EducationMichigan State University

NCTM Research Symposium April 25, 2006

© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595


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Knowing Mathematics for Teaching Algebra (KAT) Project

NSF REC No. 0337595 (2004-2007)

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



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Fundamental Question:

What knowledge of algebra for teaching do secondary school teachers of algebra need to support their instruction?



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Fundamental Question, v.2

What knowledge of algebra for teaching “do/should/might” secondary school teachers of algebra “draw upon” to support their instruction?



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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?



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1

Conceptual background

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Pilot study 4

Next steps

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Item development



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1

Conceptual background



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Pedagogical Content Knowledge Lee Shulman, 1986, pp. 9-10

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



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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.



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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



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Mathematics for TeachingAl Cuoco, 2001

The vertical disconnect. Most teachers see very little connection between the mathematics they study as undergraduates and the mathematics they teach.



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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)



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KAT ConceptualFramework



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Tasks of Teaching



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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



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Categories of Knowledge



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Core content knowledge Representation Content trajectories Applications and contexts Language and conventions Mathematical reasoning and proof



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Overarching Categories Bridging Trimming Decompressing



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Item development



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AssessmentBlueprint



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Item Development and RefinementAugust 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



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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 knowledgeknowledge of typical errors, canonical uses of school math, curriculum trajectories, etc.



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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



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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



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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



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3

Pilot studies



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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



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Number of Participants in Pilot

Studies November 2004 - December 2005

Pre-service teachers 387

In-service teachers 423

Total 810



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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



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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.



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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 onlyB. ii onlyC. iii onlyD. i and ii onlyE. ii and iii only



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Results: Identifying an Exponential Function

Number of cases Difficulty

Pre-service teachers

138 0.297

In-service teachers

287 0.324

Total 431 0.313



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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 numbersi. the set of complex numbersiii. the set of integers mod 6iv. the set of integers mod 5v. the set of 2x2 matrices with real number entries

A. i only D. i, ii, iii and iv onlyB. i and ii only E. i, ii, iii, iv, and vC. i, ii and iv only



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Results: Properties of Number Systems



86 0.151

In-service teachers

96 0.177

Total 186 0.161



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Sample Item 3: Identifying Student’s Error in Solving a Linear Equation

(Teaching Knowledge)

A student solved the equation3(n - 7) = 4 - n

and obtained the solution n = 2.75.

What might the student have done wrong?



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Results: Identifying Student’s Error



97 0.760

In-service teachers

14 0.946

Total 115 0.787



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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 - s2

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?



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Number of cases

Difficulty


97 0.459

In-service teachers

14 0.757

Total 115 0.499

Results: Equivalent Expressions



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4

Next steps



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Validation Study: spring-summer, 2006 Status Study: 2006 -2007



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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.



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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.



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Participants Needed for Validation Study

March 20th – August 15th

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



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Status Study (2006-07): 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?



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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



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Contacts for KAT Project Information

Xuhui Li (Project Manager) [email protected]

Joan Ferrini-Mundy (PI) [email protected] Senk (Co-PI) [email protected]



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Discussants

Nicole IceKennesaw State University

Cos FiThe University of North Carolina at Greensboro

knowledge of algebra for teaching: framework, item development, and pilot results joan ferrini-mundy...

Documents

teaching algebra nsf

knowing mathematics

knowledge of algebra

understanding slide

basic mathematics inherent

conceptual background

item development slide

xuhui li slide