RIMSE: Research Innovations in Mathematics and
Science Education
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
Arizona State University
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
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 they’re 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)
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
-PLC’s (“Effective PLC’s (FOP) (Carlson, Moore, et al.,
2007)
-Professional
support tools for teachers (student level
tasks, embedded research knowledge, conceptually
oriented assessments.) (Carlson, Oehrtman, Moore, Strom)
Developing the Pathways to Calculus
Professional Development Model
Design
Implementation
(Revise)
Review research
to provide the theoretical
grounding
--Cognitive Models of Learning
Key Ideas
-Implement
-Models of How Mathematical
Practices are Acquired
-Study Implementation
--Models for Effective Learning
Communities
-Models of how MKTP Develops
Revise
Frameworks
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)
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 don’t understand
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.
Student PCA Performance


PCA administered to 550 college algebra and 379 precalculus 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
Summary of student PCA pre- and
post-test gains in Pathways Precalculus
Courses
Pre-test Mean
Post-test Mean
8.2
15.2
(Previous Best Post Mean ScoreL 10.4)
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 tools—mini 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
What do we Mean by Mathematical Content
Knowledge for Teaching Proportionality (MKTP)?
How Does Mathematical Content Knowledge
for
Teaching Proportionality Interact with a Teacher’s
Pedagogical Actions?
W#3
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.
Copyright ©2010 Carlson and Oehrtman
13
TWO VARYING QUANTITIES ARE
RELATED BY A CONSTANT RATIO
W#3
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
w 10 5 25
= = =
h 6 3 x
x = 15
Copyright ©2010 Carlson and Oehrtman
14
W#3
CONSTANT MULTIPLE & SCALING
h is always the same
multiple of w.
h =6″ is 6/10
Why?
as long
as w1=10″
•w2=25″ is 25/10
Why?
as long as w1=10″
• 1
So h2 should be 6/10
as long as w2=25″
•
h = (6/10)25″
• 2
If we scale w by some
factor, we should also
scale h by the same factor.
•So h2 should be 25/10
as long as h1=6″
•w2=(25/10)6″
Copyright ©2010 Carlson and Oehrtman
15
W#3
PHOTO ENLARGEMENT – TABLE
Width in inches ×3/5 Height in inches
0
1
0
×3/5
2
6/5
3
×1/10
9/5
4
10
11
×5/2
25
3/5
12/5
×3/5
×1/10
6
33/5
×5/2
15
Copyright ©2010 Carlson and Oehrtman
16
What do we mean by MKT
Proportionality?
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.
A Solution
y = mx + b
8.3 = -2.2 × 3.35 + b
b = 15.67
A Second Solution
The candle’s 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.
Two Solutions?
y = mx + b
8.3 = -2.2 × 3.35 + b
b = 15.67
The candle’s 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.
Solution as a Consequence of PR.
y = mx + b
8.3 = -2.2 × 3.35 + b
b = 15.67
b = 8.3 + 2.2 × 3.35
The candle’s 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.
Why PR as a Focus?
Trigonometry?
Why PR as a Focus?
r
s
q=
r
q
s
r
Why PR as a Focus?
s
2=
r
Why PR as a Focus?
sin(2)?
Why PR as a Focus?
sin(2)
Why PR as a Focus?
Differential Equations?
Why PR as a Focus?
y' = cy
Why PR as a Focus?
Rate of change is proportional to amount…
y' = cy
Why PR as a Focus?
Rate of change is proportional to amount…
y' = cy
…is related to precalculus?
Why PR as a Focus?
Consider a mass of bacteria that grows continuously
at a rate of 25%.
Why PR as a Focus?
Consider a mass of bacteria that grows continuously
at a rate of 25%.
In precalculus, this means use “Pert”:
y = Pe
0.25t
Why PR as a Focus?
Consider a mass of bacteria that grows continuously
at a rate of 25%.
In precalculus, this means use “Pert”:
y = Pe
0.25t
But we are really saying:
y' = 0.25y
How do we support MKT
Proportionality?
Student PCA Performance


PCA administered to 550 college algebra and 379 precalculus 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
Summary of student PCA pre- and
post-test gains in Pathways Precalculus
Courses
Pre-test Mean
Post-test Mean
8.2
15.2
(Previous Best Post Mean ScoreL 10.4)