Affecting and Documenting Shifts in Secondary Precalculus
Teachers Instructional Effectiveness and Students Learning Marilyn
P. Carlson Arizona State University Kevin C. Moore University of
Georgia Katheryn Underwood Katheryn Underwood Arizona State
University RIMSE: Research Innovations in Mathematics and Science
Education
Slide 2
Pathways Phase I Goals: Greater student learning Understandings
of Key Ideas AND achievement on standardized exams Increased
student continuation in STEM course taking Improved mathematical
practices
Slide 3
Major Findings from Phase I: 5 key variables that impacted
teaching and student learning are teachers who have strong and
connected mathematical content knowledge of key ideas of the
content theyre teaching (e.g., precalculus level mathematics)
(Carlson & Rasmussen, 2009) believe that students can solve
novel applied problems and understand the mental processes involved
in doing so. (Moore & Carlson, 2012) implement pedagogy that
supports students construction of mathematical practices (sense
making, explaining thinking, persistence) (Clark, Moore, Carlson;
2008) participation in Pathways Professional Learning Communities
(PLCs) (Carlson, Moore, et al.; 2007). Use of inquiry-based and
conceptually oriented instruction (includes supports for teachers)
(Carlson, Oehrtman; 2011)
Slide 4
Pathways Phase II is scaling (and studying the scaling) of the
Pathways Professional Development Model for Teaching Secondary
Mathematics -Workshops/graduate courses focused on developing
mathematical content knowledge for teaching key ideas and reasoning
abilities of algebra through precalculus -PLCs (Effective PLCs
(FOP) (Carlson, Moore, et al., 2007) - Professional support tools
for teachers (student level tasks, embedded research knowledge,
conceptually oriented assessments.) (Carlson, Oehrtman, Moore,
Strom)
Slide 5
Developing the Pathways to Calculus Professional Development
Model Review research to provide the theoretical grounding
--Cognitive Models of Learning Key Ideas -Models of How
Mathematical Practices are Acquired --Models for Effective Learning
Communities -Models of how MKTP Develops -Implement -Study
Implementation Revise Frameworks Design Implementation
(Revise)
Slide 6
Foundational Understandings and Reasoning Abilities Needed to
Understand Calculus Covariation and Quantification Proportionality
Function (Composition and Inverse) Linear Functions Exponential
Functions Polynomial Functions Rational Functions Trigonometric
Functions (Precalculus)
Slide 7
Acquiring Productive Mathematical Practices: (Rules of
Engagement) The teacher expects (and acts on the expectation that)
students: Engage in Meaning Making Conceptualize quantities and
their relations Persist in making sense Attempt to make logical
connections Base conjectures on a logical foundation Students are
expected to express their thinking and speak with meaning about
their problem solutions pose meaningful questions when they dont
understand
Slide 8
The Precalculus Concept Assessment Instrument (PCA) Assesses
understanding of key ideas of precalculus that are foundational for
learning calculus 25 Item Multiple Choice Item choices are based on
common responses that have been identified in students (clinical
interviews) Validated to correlate with success in calculus
Tracking 277 students in beginning calculus, 80% of students who
scored 13 or above on PCA at the beginning of the semester received
an A, B, or C in calculus. 85% of the 277 students who scores 11 or
below received a D, F or withdrew.
Slide 9
Student PCA Performance PCA administered to 550 college algebra
and 379 pre- calculus students at a large southwestern university
Also administered to 267 pre-calculus students at a nearby
community college Mean score for college algebra: 6.8/25 Mean score
for pre-calculus: 9.1/25
Slide 10
Summary of student PCA pre- and post-test gains in Pathways
Precalculus Courses Pre-test MeanPost-test Mean 8.215.2 (Previous
Best Post Mean ScoreL 10.4)
Slide 11
How can teachers be supported to realize shifts in their
teaching that affect the above described shifts in students?
Workshops and PLCs that support teachers development of
Mathematical content knowledge for teaching precalculus (Silverman
& Thompson) Understanding of key connecting ideas and reasoning
abilities Image of how students acquire these understandings and
reasoning abilities Instructional toolsmini learning theory
Pedagogical Practices and Pedagogical Choices What they do in the
classroom to promote student thinking How they react to their
students expressed thinking
Slide 12
What do we Mean by Mathematical Content Knowledge for Teaching
Proportionality (MKTP)? How Does Mathematical Content Knowledge for
Teaching Proportionality Interact with a Teachers Pedagogical
Actions?
Slide 13
Copyright 2010 Carlson and Oehrtman 13 PHOTO ENLARGEMENT TASK
1. A photographer has an original photo that is 6 inches high and
10 inches wide and wants to make different-sized copies of the
photos so that new photos are not distorted. a. If the photographer
wants to enlarge the original photo so that the new photo has a
width of 25 inches, what will the height of the new photo need to
be so the image is not distorted? Explain the reasoning you used to
determine your answer. W#3
Slide 14
TWO VARYING QUANTITIES ARE RELATED BY A CONSTANT RATIO Let h =
the height of the new photo (in inches) Let w = the width of the
new photo (in inches) Then, as h and w vary together, their ratio
stays fixed Copyright 2010 Carlson and Oehrtman 14 W#3
Slide 15
CONSTANT MULTIPLE & SCALING h is always the same multiple
of w. h 1 =6 is 6/10 as long as w 1 =10 So h 2 should be 6/10 as
long as w 2 =25 h 2 = (6/10) 25 Copyright 2010 Carlson and Oehrtman
15 If we scale w by some factor, we should also scale h by the same
factor. w 2 =25 is 25/10 as long as w 1 =10 So h 2 should be 25/10
as long as h 1 =6 w 2 =(25/10) 6 Why? W#3
Analogous Problem: Given that a line contains the point (3.35,
8.3) and has a rate of change (slope) of -2.2, determine the
formula for the line.
Slide 19
A Solution
Slide 20
A Second Solution The candles length decreases at a constant
rate of 2.2 inches per hour. 3.35 hours have passed, which is 3.35
times as large as 1 hour. The length that has burned in this time
is 3.35 times as large as 2.2 inches (7.37 inches). The original
length was this amount plus the 8.3 inches left.
Slide 21
Two Solutions? The candles length decreases at a constant rate
of 2.2 inches per hour. 3.35 hours have passed, which is 3.35 times
as large as 1 hour. The length that has burned in this time is 3.35
times as large as 2.2 inches (7.37 inches). The original length was
this amount plus the 8.3 inches left.
Slide 22
Solution as a Consequence of PR. The candles length decreases
at a constant rate of 2.2 inches per hour. 3.35 hours have passed,
which is 3.35 times as large as 1 hour. The length that has burned
in this time is 3.35 times as large as 2.2 inches (7.37 inches).
The original length was this amount plus the 8.3 inches left.
Slide 23
Why PR as a Focus? Trigonometry?
Slide 24
Why PR as a Focus? s r r
Slide 25
Slide 26
Slide 27
Slide 28
Differential Equations?
Slide 29
Why PR as a Focus?
Slide 30
Rate of change is proportional to amount
Slide 31
Why PR as a Focus? Rate of change is proportional to amount is
related to precalculus?
Slide 32
Why PR as a Focus? Consider a mass of bacteria that grows
continuously at a rate of 25%.
Slide 33
Why PR as a Focus? Consider a mass of bacteria that grows
continuously at a rate of 25%. In precalculus, this means use
Pert:
Slide 34
Why PR as a Focus? Consider a mass of bacteria that grows
continuously at a rate of 25%. In precalculus, this means use Pert:
But we are really saying:
Slide 35
How do we support MKT Proportionality?
Slide 36
Student PCA Performance PCA administered to 550 college algebra
and 379 pre- calculus students at a large southwestern university
Also administered to 267 pre-calculus students at a nearby
community college Mean score for college algebra: 6.8/25 Mean score
for pre-calculus: 9.1/25
Slide 37
Summary of student PCA pre- and post-test gains in Pathways
Precalculus Courses Pre-test MeanPost-test Mean 8.215.2 (Previous
Best Post Mean ScoreL 10.4)