RIMSE: Research Innovations in Mathematics and
Science Education
Pathways to Achieving High Gains in Secondary
Mathematics Students’ Learning: The Role of
Curriculum, Teachers, and Instruction
Marilyn P. Carlson
marilyn.carlson@asu.edu
Professor, School of Mathematical and Statistical Sciences
Arizona State University
Framing the Problem:
The Percent of 2012 College Graduates
Earning STEM Degrees
5.6%
USA
28.1%
Germany
46.7%
China
32.8%
Korea
Where are we losing our STEM students?
Passing rates in first-semester calculus by students who had
passed the prerequisite precalculus ranged from 17%-37% at the
six colleges and universities (Schattschneider’s, 2006).
In another study, less than 20% of students who received an A, B, or C
) in precalc (and had a declared major that required calculus) went on
to complete beginning calc. and less than 3% of these students
completed calc II (Thompson, 2008)
Select research results reveal
sources of the problem

Precalculus level students are not developing the foundational
understandings and reasoning abilities that are necessary for
learning and using ideas of calculus
(e.g., Making the Connection: Research and Teaching in Undergraduate Mathematics
Education, Carlson & Rasmussen, 2009; Kaput, 1992; Dubinsky 2002; Monk, 1992;
Thompson 1994; Carlson, 1998; Jacobs, 2000; Carlson, Oehrtman, Engelke, 2010)

Teachers are not receiving the support they need to teach for
meaning (MKT)
(e.g., Carlson & Moore; in press; Thompson, 2011; Thompson, Carlson & Silverman:
2007; Thompson 2014)

Curriculum materials have a strong focus on methods for getting
answers (Oehrtman, 2009; Teuscher, 2010; Thompson, 2008)
Common Conceptions Held by Precalculus Level
Students at the End of the Course

Variable:


Function/Formula:


Something to “solve for” (Schoenfeld, 1989; Jacobs, 1999)
Two expressions separated by an equal sign (Thompson, 1994; Carlson,
1995)
Function Graph:

Picture of an event (Monk, 1992; Kaput: 1992; Carlson, 1998)

Constant Rate of Change: m or the speed stays the same

Average Rate of Change on an interval of a function’s
domain:
Add them up and divide (Carlson, 1998)
The speed the car is going most of the time (Carlson & Oehrtman)
Give an example of an exponential function?
f (x) = x 2
What does the phase, “write a function that
expresses s in terms of t” mean?
there must be an s and t and an equal sign
What does f(g(x)) mean?
f times g times x
What does f-1(x) represent? (1/x), 1/(f(x)), 1/f
(Carlson, 1998; Carlson, Frank, Fowler, in press)
Problem Solving Behaviors :
Classroom Rules of Engagement:
Individuals are expected to:
 Persist in sense making/meaning making
 Attempt to make logical connections
 Exhibit Mathematical Integrity
 Don’t pretend to understand when you don’t
 Base conjectures on a logical foundation
 Students are expected to express their thinking and
speak with meaning about their problem solutions
 Students are expected to listen to the meaning being
conveyed by their peers and ask questions when the
meaning is not clear
Research on calculus learning revealed that students’ weak
covariational reasoning abilities and weak conceptions of ideas of
quantity, variable, function, and rate of change are obstacles for
learning and understanding key ideas of calculus

Rate of Change, Changing Rate of Change:
(Thompson, 1992; Carlson, 1998, 2002; Zandieh, 2004)

Limit (Cottrill, Brown, Dubinsky, 1998; Jacobs, 1999; Oehrtman, 2009;
Williams, 2002)

Derivative (Tall, 1992; Zandieh, 2004)

Accumulation (Carlson, Oehrtman, & Thompson 2009; Smith, 2009;
Carlson, Larsen, Jacobs; 2007)

Related Rate Problems (Engelke, 2008; Carlson, 2007)

Fundamental Theorem (Thompson, 1994; Carlson, Oehrtman,
Thompson, 2009; Smith, 2009)
Covariational Reasoning:
Conceptualizing two quantities and how they
change togeter
Quantitative Reasoning:
Identifying attributes of an object that can
be imagined as being measured
(Thompson, 1989)
Variable:
A letter that represents the varying values
that a quantity may assume
Essential Ways of Thinking:
The Bottle Problem
Imagine this bottle filling with water. Sketch a
graph of the water’s height as a function of the
amount of water that’s in the bottle. Explain the
thinking you used to construct your graph.
Covariational Reasoning
(Carlson et al., 2002)

MA1: Imaging 2 quantities that are changing together

MA2: Imaging how the direction of the 2 quantities
are changing together

MA3: Imaging how one quantity is changing while
imagining changes in the other quantity

MA4: Imaging how the average rate of change of the
output quantity with respect to the input quantity is
changing while imaging uniform increments of
change in the input quantity.
A Research Based Response for Precalculus Level
Mathematics: Over-arching Ideas and Reasoning Abilities






Quantity, Variable, Formula, Function, Function
Composition, and Function Inverse
Constant Rate of Change & Average Rate of Change
Exponential Functions
Polynomial Functions
Rational Functions
Trigonometric Functions (Algebra II and Precalculus)
Pathways: Other Cross-Cutting Ideas/Methods



Evaluating
Representational equivalence
Solving
Investigation #1
QUANTITIES & CO-VARIATION OF QUANTITIES
The number of teachers in a school varies with the
number of students in the school so that there are
seven times as many students as teachers.
a. Write a formula that defines the number of
students s in terms of the number of teachers t.
Copyright © 2010 Carlson and Oehrtman
13
Investigation #1
QUANTITIES & CO-VARIATION OF QUANTITIES
The number of teachers in a school varies with the
number of students in the school so that there are
seven times as many students as teachers.
a. Write a formula that defines the number of
students s in terms of the number of teachers t.
Number of teachers in
the school
Number of students in
the school
1
7
2
14
3
21
Copyright © 2010 Carlson and Oehrtman
14
Investigation #4
CONSTANT RATE & LINEARITY
6. Alan and Alissa are attempting to find each other in a
crowded mall. They finally spot each other near the food
court, at which point they are 140 feet apart. They begin
walking towards each other, Alan at a constant rate of 5 feet
per second, Alissa at a constant rate of 4 feet per second.
a. Write a formula to represent the distance between Alan and Alissa
in terms of the number of seconds since they spotted each other.
140 ft.
5 ft/s
4 ft/s
Alan
Alissa
? ft.
? ft.
Constant Quantities
? ft.
Varying Quantities
? seconds
Copyright © 2010 Carlson and Oehrtman
15
Investigation #4
CONSTANT RATE & LINEARITY
Alan and Alissa are attempting to find each other in a
crowded mall. They finally spot each other near the food
court, at which point they are 140 feet apart. They begin
walking towards each other, Alan at a constant rate of 5
feet per second, Alissa at a rate of 4 feet per second.
Write a formula to represent the distance between Alan and Alissa in
terms of the number of seconds since they spotted each other.
140 ft.
5 ft/s
4 ft/s
Alan
d1 = 5t ft.
Alissa
d = 140 – 9t ft.
d2 = 4t ft.
t seconds
Copyright © 2010 Carlson and Oehrtman
16
Investigation #4
CONSTANT RATE & LINEARITY
6. Alan and Alissa are attempting to find each other in a
crowded mall. They finally spot each other near the food
court, at which point they are 140 feet apart. They begin
walking towards each other, Alan at a constant rate of 5
feet per second, Alissa at a rate of 4 feet per second.
How many seconds it will take for Alan and Alissa to reach one
another?
Alan and Alissa reach one another when the distance (in feet) between them is zero.
Thus, we let d = 0 and solve for t:
0 = 140 – 9t
–140 = –9t
t = 15.556.
Therefore it will take Alan and Alissa 15.556 seconds to reach each other.
Copyright © 2010 Carlson and Oehrtman
17
Constant Rate of Change Item
Buddy runs straight down the track at a
constant rate of change, traveling a distance of
15 meters every 2 seconds.
 How long will it take Buddy to run 5 meters?
10 meters? 60 meters?120 meters?
 How far did Buddy travel over any 3 second
interval? 1 second interval? 10 second
interval?
Understanding Rate of Change

What does it mean to understand constant
rate of change?
<see animations>
What does it mean for two quantities to change together
at a constant rate of change?
The changes in the two quantities are proportional
 As the change in distance and change in time
CHANGE TOGETER,


the ratio of these changes remains constant
The change in distance is always some constant times
as large as the change in time
Dd
m=
Dt
Dd = mDt
Given that the x and y axes have the same scale, what is
the approximate rate of change of y with respect to x for
the linear functions whose graphs are given?
W#1
MODELING THE CO-VARIATION OF
QUANTITIES IN FOUR
FUNCTION CONTEXTS:
Problem context (a box), a formula, a table (numeric) and a graph
1.
a. Create an open box (without a top) by: i) cutting four
equal-sized square corners from an 8.5 by 11-inch sheet of
paper and ii) folding up the sides and taping them together
at the edges. Define a function that expresses the volume of
the box in terms of the length of the side of the square
cutoout
22
Copyright © 2009 Carlson and Oehrtman
What are the dimensions of a box with the maximum volume
that can be formed from cutting equal squares from the corner
of an 11” X 13” piece of paper?
Occurred after he had
created a diagram and
determined V=13*11*x
 Travis appeared to focus on the object of the box, but
not the attributes of length and width of the box
 The length and the width of the box had fixed
measurements. The various references to length
were not distinct in Travis’ mind.
Shifting the Instructional Approach
After Instruction that supported
students in imagining
relevant quantities and
how they change together
…

 Travis appeared to conceive of the various quantities (attributes
of the box that could be measured) and how they change together
 Tavis defined and conceptualized variables and then constructed
a formula (for the box’s volume in terms of the cutout length)
that was meaningful to Travis.
Process of Developing and Refining Pathways
Interventions
Review research
to provide the theoretical
grounding
Design
Implementation
(Revise)
--Cognitive Models of Learning
Key Ideas
-Implement
--Develop Curriculum and Aligned
Assessments
-Study Implementation
-Models of how MKTP Develops
Revise
Frameworks
Variables that Impact Teachers and Teaching
Teachers’ MKT
for teaching
concept X
Teacher
Beliefs
Teacher
Goals for
Student
Learning
A Framework
For Supporting Shifts in Teaching and Student
Learning
Learning Goals
for Students:
(Influenced by
teachers’ MKT,
beliefs, etc.)
Materials
Teachers
Use/Design
Teaching &
Reflection
on Teaching
The Didactic Triad: A Framework for Supporting Shifts
in Teaching and Learning
Learning Goals
for Students
Materials
Teachers
Use/Design
Teaching &
Reflection
on Teaching
Looking at each in relation to the others
The Didactic Triad
Learning Goals
for Students
Materials
Teachers
Use/Design
Teaching &
Reflection
on Teaching
Looking at each in relation to the others
The Didactic Triad
Learning Goals
for Students:
(Influenced by
teachers’ MKT,
beliefs, etc.)
Instructional
Materials
Teaching &
Reflection on
Teaching
Looking at each in relation to the others
The Didactic Triad
Your
Your
understanding
Your
understanding
ofunderstanding
students’
of students’
mathematics
of students’
mathematics
mathematics
Materials
You Use/
Design
Your
understanding
of the content
Learning Goals
for Students:
(Influenced by
teachers’ MKT,
beliefs, etc.)
Your
understanding
of the content
Your
understanding
Teaching &
of the content
Reflection
on Teaching
Curriculum that is facilitating shifts in teachers’:
i) MKT, ii) teaching practices, iii) students’ learning

One Example—Project Pathways: Conceptually Oriented
Materials that facilitate teacher development of mathematical
and pedagogical knowledge, and ongoing reflection on
student learning
Pathways materials include instructional tools such as:
i) Carefully scaffold in-class activities
ii) Detailed teacher notes and videos;
v) Animations to support student learning of key concepts
iv) PPts to support in-class discussions
v) Formative assessments
Laying the Foundation for Success in
Calculus (and Beyond)

Tasks support students in developing foundational
reasoning abilities and understandings

Classroom tasks engage students in meaning making
and promote the development of students’ problem
solving abilities.

Teacher Support Tools Help Teachers Acquire “Content
Knowledge for Teaching”


PowerPoint Slides with Dynamic Animations and Links to Applets
Help Novice Teachers Engage Students in Conceptual
Conversations
Worksheets (with detailed solutions) help teachers promote
critical reasoning abilities and understandings in students
Classroom Rules of Engagement:
Individuals are expected to:
 Persist in sense making
 Attempt to make logical connections
 Exhibit Mathematical Integrity
 Don’t pretend to understand when you don’t
 Base conjectures on a logical foundation
 Students are expected to express their thinking and
speak with meaning about their problem solutions
 Students are expected to listen to the meaning being
conveyed by their peers and ask questions when the
meaning is not clear
Comparison Data: Precalculus Concept
Assessment Instrument

A 25 item multiple choice exam

Based on the body of literature related to knowing and
learning the concept of function (Carlson, 1998, 1999,
2003; Dubinsky et al., 1992; Monk,1992; Thompson and
Thompson, 1992; Thompson, 1994; Sierpinska, 1992)

Validated and refined over a 12 year period
Predictive Potential of PCA

How does PCA score relate to course grade
in calculus?

83% of 277 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 Responses

A ball is thrown into a lake, creating a circular ripple that travels
outward at a speed of 5 cm per second. Express the area A of the
circle in terms of the time t in seconds that have passed since the ball
hits the water.





a)
b)
c)
d)
e)
A = 25πt (21%)
A = πr2 (7%)
A = 25πt2 (17%)
A = 5πt2 (34%)
None of the above (20%)
WHY ARE WORD PROBLEMS SO DIFFICULT FOR STUDENTS?
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
Post Mean PCA Score at 5 universities Pre: 8.4; Post: 13.6
Precalculus: Pathways to Calculus
Summary of PCA pre- and post-test
Academic Yr.
Pre-test
Post-test
2009-10
8.2
15.2
2010-11
9.1
15.8
2011-12
9.7
15.9
2012 –13
10.2
16.3