Knowledge of Algebra for Teaching: Framework, Item Development, and Pilot Results Joan Ferrini-Mundy...
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Transcript of 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
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
3
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 #0337595
NCTM Research Symposium April 25, 2006
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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?
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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?
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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1
Conceptual background
3
Pilot study 4
Next steps
2
Item development
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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1
Conceptual background
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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.
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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.
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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)
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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KAT ConceptualFramework
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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Tasks of Teaching
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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Categories of Knowledge
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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Core content knowledge Representation Content trajectories Applications and contexts Language and conventions Mathematical reasoning and proof
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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Overarching Categories Bridging Trimming Decompressing
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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2
Item development
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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AssessmentBlueprint
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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.
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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3
Pilot studies
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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Number of Participants in Pilot
Studies November 2004 - December 2005
Pre-service teachers 387
In-service teachers 423
Total 810
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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.
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
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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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
32
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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
33
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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
34
Results: Properties of Number Systems
Number of cases Difficulty
Pre-service teachers
86 0.151
In-service teachers
96 0.177
Total 186 0.161
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
35
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?
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
36
Results: Identifying Student’s Error
Number of cases Difficulty
Pre-service teachers
97 0.760
In-service teachers
14 0.946
Total 115 0.787
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
37
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?
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
38
Number of cases
Difficulty
Pre-service teachers
97 0.459
In-service teachers
14 0.757
Total 115 0.499
Results: Equivalent Expressions
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
39
4
Next steps
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
40
Validation Study: spring-summer, 2006 Status Study: 2006 -2007
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
41
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 #0337595
NCTM Research Symposium April 25, 2006
42
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 #0337595
NCTM Research Symposium April 25, 2006
43
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
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
44
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?
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
45
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 #0337595
NCTM Research Symposium April 25, 2006
46
Contacts for KAT Project Information
Xuhui Li (Project Manager) [email protected]
Joan Ferrini-Mundy (PI) [email protected] Senk (Co-PI) [email protected]
© 2006 Knowing Mathematics for Teaching Algebra NSF REC #0337595
NCTM Research Symposium April 25, 2006
47
Discussants
Nicole IceKennesaw State University
Cos FiThe University of North Carolina at Greensboro