Using Research to Shape Placement Test and Curriculum
MAA NSF Grant – DUE 1122965
Marilyn P. Carlson, Professor School of Mathematical and Statistical Sciences
Arizona State University
Caren Diefenderfer, Professor Department of Mathematics, Statistics, and
Computer Science, Chair Hollins University
RIMSE: Research Innovations in Mathematics and Science Education
The Problem: Percent of 2012 College Graduates in 4 Countries
Earning STEM Degrees
5.6% USA
28.1% Germany
46.7% China
32.8% Korea
The Problem
US ranking on the Program for International Student Assessment (PISA)
n 30th in Math n 23rd in Science n 7th in Reading
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Calculus success rates of
high performing US students n Only about 25% of students who start beginning
calculus and intend to take 3 semesters of calculus go on to complete the calculus sequence (Thompson et al., 2007)
q Literature in Undergraduate Mathematics Education Research has revealed severe weakness in calculus students’ foundational understanding and reasoning abilities
q 87% of 17 year olds in Japan and 13% of secondary students in the US have completed calculus.
Addressing the Problem: Aligning Curriculum and
Assessments with Research Findings (instead of popular views)
What foundational mathematical understandings and reasoning abilities are needed to succeed in
STEM courses and STEM degrees?
A Research Based Model for Developing Curriculum and Assessment
Review research to provide the theoretical grounding
--Cognitive Models of Learning Key Ideas
-Models of How Mathematical Practices are Acquired -Assessment Frameworks
-Implement -Study Implementation
Revise
Frameworks
Design Assessments and
Curriculum: Implementation
(Revise)
A More Concrete Example: Identifying the Reasoning Abilities and
Understandings this Items Assesses Two cars that are originally 216 miles apart start traveling toward one another at 9 am. One car travels 62 miles per hour and the other travels 68 miles per hour, both with their cruise control on. Which formula represents d, the distance between the two cars t hours after 9 am.
Key Ideas and Reasoning Abilities
n Conceptualizing quantities and how they are related
n Variables are used to represent varying values of quantities
n Operations are used to define quantitative relationships
n Formulas, Graphs, Tables represent how two quantities change together
APCR Item Two cars that are originally 216 miles apart start traveling toward one another at 9 am. One car travels 62 miles per hour and the other travels 68 miles per hour, both with their cruise control on. Which formula represents d, the distance between the two cars t hours after 9 am.
(a) d = 62 t + 68 t (b) d = 216 – 62 t (c) d = 68 t – 62 t (d) d = 216 – (68 t + 62 t) (e) d = 216 + (68 t + 62 t)
Comparison: Standard Curriculum 32% Pathways: 79%
Calculus Concept Readiness Item: The model that describes the number of bacteria in a culture after t days has just been updated from to What implications can you draw from this information? (a) The final number of bacteria is 3 times as much as the initial value instead of 2 times as much. (b) The initial number of bacteria is 3 instead of 2. (c) The number of bacteria triples every day instead of doubling every day. (d) The growth rate of the bacteria in the culture is 30% per day instead of 20% per day. (e) None of the above.
Comparison: Standard Curriculum Used in 40 Colleges vs. Pathways 10 Year Research Project Standard Curriculum 29% Pathways: 73%
P(t)= 7(3)t( ) 7(2)tP t =
Other Foundational Concepts and Current Misconceptions
n Variable: q Something to “solve for” (Schoenfeld, 1989; Jacobs, 1999) Function:
Two expressions separated by an equal sign (Dubinsky, 1992; Thompson, 1994; Carlson, 1995)
Function Graph: Picture of an event (Monk, 1992; Kaput: 1992; Carlson 1998)
Average Rate of Change on an interval of a function’s domain: Add them up and divide (Carlson, 1998)
What understandings of key precalculus ideas are common among precalcus students at the end of the course?
Given f is defined by determine f(x+a)? What does f(g(x)) mean? f times g
What does the phrase “write a function that expresses s in terms of t” mean:
“there must be an s and a t and an equal sign” (Carlson, 1998)
f (x) = 2x2 + 3xf (x + a) = 2x2 + 3x + a
Studies have revealed that quantitative and covariational reasoning are foundational for
understanding key ideas of calculus
n Derivative: Carlson, 1998; Zandieh, 2002
n Limit: Williams: 2001; Oehrtman, 2008
n Related Rate Problems: Kaput, 1992; Carlson, 1998; Engelke; 2007
n Accumulation: Thompson, 1994a, 1994b; Smith 2006; Sealey, 2007
n The Fundamental Theorem of Calculus: Thompson, 1994; Carlson, 1998 ; Smith, 2007
Studies have consistently revealed that: n Memorizing procedures without students understanding
why the procedure works: q does not result in students evoking the right procedure
at the right time. q leads students to think that mathematics is about
meaningless memorization
q References on next slide
Studies have consistently revealed that: -continued References
n Carpenter, T. P. (1986). Conceptual knowledge as a foundation for procedural understanding. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 113-132). Hillsdale, NJ: Erlbaum.
n Cobb, P., Wood, T., & Yackel, E. (1993). Mathematics as procedural instructions and mathematics as meaningful activity: The reality of teaching for understanding. In C. Maher & R. Davis (Eds.), Relating schools to the reality of mathematics learning (pp. 119-133). Englewood Cliffs, NJ: Erlbaum.
n Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 3–20). Hillsdale, NJ: Erlbaum.
n Rittle-Johnson, B., Siegler, R. S., & Alabali, M. W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346-362.
Foundational Understandings and Reasoning Abilities Needed to Understand Calculus
n Quantities and Variables n Constant Rate of Change and AROC n Function (Composition and Inverse) n Function composition Exponenti n Function Inverse n Polynomial Functions n Rational Functions n Trigonometric Functions (Precalculus)
The Calculus Concept Readiness (CCR) & the Algebra Precalculus Concept Readiness (APCR)
The CCR and APCR were developed by MAA task forces of mathematicians and mathematics education researchers
n 25 Items n Validated to assess key concepts in taxonomy n Students who select correct answer demonstrate
understanding q Clinical interviews use to validate item stem and
distractors n Incorrect Item choices are aligned with common
incorrect ways of thinking
The Calculus Concept Readiness (CCR) & the Algebra Precalculus Concept Readiness
(APCR) - continued
Design Principles: n Based on Research on Student Learning
q Framework of Key Understandings and Reasoning Abilities
q Items are paired with key understandings in taxonomy n One item assesses multiple items in taxonomy
The CCR Taxonomy
Reasoning abilities n R1 Proportional reasoning: Observe that two quantities
that are changing together are related by a constant multiple and that as the two quantities change together the ratio of one quantity to the other remains constant; then use this knowledge to determine new values of one quantity for specific values of the other quantity.
n R2 Process View of Function: View a function as a process that maps input values in a function’s domain to output values in a function’s range.
n R3 Quantitative and Covariational Reasoning: Conceptualize quantities in situations and reason about how two quantities in a situation change together.
The CCR Taxonomy - continued
n Understand, represent and interpret function growth patterns n F1 Linear n F2 Exponential n F3 Non-linear polynomial n F4 Rational n F5 Periodic n Understand and use the following concepts or ideas n U1 Quantity n U2 Variable n U3 Slope/Constant rate of change n U4 Average rate of change n U5 Function composition n U6 Function inverse n U7 Function translations (horizontal and vertical shifts)
The CCR Taxonomy -continued
Understand central ideas of trigonometry n T1 Angle measure n T2 Radian as a unit of measure n T3 Sine and cosine functions as the covariation of an
arc’s length (measured in units of the circle’s radius) and the horizontal or vertical coordinate of the arc’s terminus (measured in units of the arc’s radius)
n T4 Sine and cosine functions as a representation of the relationship between an angle measure and sides of a right triangle
The CCR Taxonomy -continued
Other abilities n A1 Solve equations n A2 Represent and interpret inequalities n A4 Use and solve systems of equations n A5 Understand and use function notation to express
one quantity in terms of another
CCR Scores Compared with Course Grade
CCR, ACT and Course Grade Comparison
Role of CCR, PCA and APCR
n Provides formative and summative information about student learning relative to key ideas articulated in the taxonomy
n Facilitates comparisons in new approaches to teaching precalculus (e.g., Pathways Precalculus) q Pre- Post data shows gains
n Scores can be used to advise students
Instructional goals (CCR Taxonomy)
Assessment Curriculum Materials
The Didactic Triad: A Framework For Supporting Shifts in Teaching and
Student Learning
The Nature of Assessments Matter
n In terms of what teachers value and focus on n In terms of what you learn about student
thinking and understanding n Most importantly:
In terms of what students learn
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n In terms of ideas teachers must deal with
n In terms of coherence among ideas and how the ideas build off each other
n In terms of opportunities for students to think independently of the teacher
n In terms of the focus on developing students’ reasoning abilities and connections among idea
The Nature of the Curriculum Matters
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The Importance of Curriculum and Assessment in Teachers’ Development of
Mathematical Knowledge for Teaching
n Creates occasions for teachers to develop personal mathematical understandings
n Creates opportunities for students to think about mathematical ideas independently of the teacher
n Creates occasions where the teacher must deal with expressions of students’ thinking
A Research Based Curriculum: Project Pathways (Based on CCR and PCA Taxonomies)
q Investigations provide a scaffold to engage students in making meaning and to support students in developing understandings and reasoning abilities articulated in CCR Taxonomy n Research based learning progressions guide
curriculum design
q Assessments (similar to CCR) reveal student learning of key ideas and skills needed for calculus
Pathways Support Materials for Students and Teachers
n In-class investigations -detailed teacher notes and solutions
n Homework that is aligned to each investigation
n PowerPoints with dynamic animations that are tied to each investigation
n Other student resources -videos, solutions for some homework, etc.
Additional Supports for Teachers
n Professional development workshops for teachers q Intense Summer Workshops q Weekly/Daily (during first year of using Pathways)
n Exams for each module
n High quality instructional videos that can be viewed by both teachers and students
Major Findings • Pathways materials facilitate student development
of key concepts and ability to apply them in novel settings.
• Pathways materials facilitate student thinking and student development of mathematical practices described in the CCSSM.
Major Findings - continued • Pathways teacher support materials and workshops
are necessary to achieve dramatic improvements in teachers’ mathematical content knowledge of key ideas of the content they’re teaching (e.g., precalculus level mathematics)
• Leads to teachers becoming more effective in posing quality questions and leading discussions to support student learning
Procedures and Fluency: More Research Results
n Given the function f, defined by n f(x) = 3x2 + 2x – 4, find f(x + a).
n 34% of Precalculus students provided the correct answer (Over 2000 students at 40 different universities
n Pathways: 82% of Precalculus students provided the correct answer
Documented Problems In US Mathematics Teaching and Learning
n Below are drawings of a wide and a narrow cylinder. The cylinders have equally spaced marks on them. The same amount of water is poured into the two cylinders. The water reaches the 4th mark of the wide cylinder (see A) and the same amount of water reaches the 6th mark when poured into the narrow cylinder (see B).
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A B
Documented Problems In US Mathematics Teaching and Learning - continued
The water in the wide cylinder is poured into the narrow cylinder and the combined water rises to the 11th mark in the narrow cylinder. If we pour this combined amount into the empty wide cylinder, the water will rise to which mark? n To the 7 ½ mark n To the 9th mark n To the 8th mark n To the 7 1/3 mark n To the 11th mark
?
Documented Problems In US Mathematics Teaching and Learning - continued
n To the 7 ½ mark n To the 9th mark n To the 8th mark n To the 7 1/3 mark n To the 11th mark Only 22% of precalculus students selected the correct answer and 78% of Pathways Precalculus students selected the correct answer
?
The Precalculus Concept Assessment Instrument (PCA)
Reveals Gains in Student Learning n PCA assesses understanding of key ideas of
precalculus that are foundational for learning calculus q 25 Item Multiple Choice
n Validated to correlate with success in calculus q 85% of the 277 students who scores 11 or below
received a D, F or withdrew. q 87% of students who scored 13 or above received an
A, B, or C in calculus
Student PCA Performance
n Previous best PCA score when administered to students in over 40 colleges and universities in 2005-2012: q 9.1 out of 25
n Pathways students mean scores on PCA (2009 – 2013)
q (14.8 – 19.5) out of 25 (A score of 13 signals readiness for calculus)
Current Focus of Pathways Research
To understand how to adapt the Pathways model to support teachers in being “highly effective”
q What are the processes by which teachers transition to be “highly effective”?
q To what degree is each component of the Pathways
model effective?
Questions?
Contact Information Marilyn Carlson [email protected] Caren Diefenderfer [email protected]
