Let’s Enable Students to Make
Sense of Mathematics: Before
and After Algebra II
Gail Burrill
Michigan State University
burrill@msu.edu
Napoleon’s 1812 march to
Russia
Constructed by Charles Minard,1869
Life
expectancy
vs income
log
Gapminder
Interactive Dynamic Technology
 SimCalc (Roschelle et al., 2000) – real contexts linked to
graphical representations of those contexts; students
explore the mathematics of change and variation.
 Dynamic geometry software (Laborde, 2001) – students
interact directly with objects, their shapes and
measurements related to those shapes, looking for
consequences invariant with respect to a shape.
 Computer algebra systems (CAS) – students make
changes in variable values and parameters of
functions and see immediate consequences (Heid et al,
2002).
Mathematical Processes
Students should engage in
Reasoning with definitions and theorems
Connecting concepts
Implementing algebraic/computational processes
Connecting multiple representations
Building notational fluency
Communicating
Mathematical Practices for AP Calculus
Making Connections: Words,
graphs, numbers
Building Concepts: Fractions, What is a Fraction?
Words and Graphs
5. Given the following graph of f (θ) = cosθ, at what angle
measure do the output values of f change from increasing
at an increasing rate to increasing at a decreasing rate?
A) 0
B) π/2
C) π
D) 3π/2
E) 2π
Algebra and Precalculus Concept Readiness Alternate Test (APCR alternate) – August 2013
Analytic Expressions and Graphs
a) f(0) = 2
b) f(-3) = f(3) = f(9) = 0
(10, 4.55)
c) f(2) = g(2)
d) g(x) > f(x) for x > 2
Adapted from Illustrative Mathematics
Developing Understanding
Building Concepts: Expressions and Equations, What is a Variable?
Looking at rate of
change
Building Concepts: Ratios and Proportional Reasoning
As you drag the point, describe
what happens to:
 Average rate of change
 Instantaneous rate of change
 Average value
Random sampling
Building Concepts: Statistics and Probability, Samples and Means
Building Procedural
Understanding
CCSS Progressions; Building Concepts: Ratios and Proportional Relationships
y = log2 (8x) for each positive
real number x. Which of the
following is true if x doubles:
a) y increases by 3
b) y increases by 2
c) y increases by 1
d) y doubles
e) y triples
Algebra and Precalculus Concept Readiness Alternate Test (APCR alternate) – August 2013
Developing
Procedural Fluency
Building concepts: Expressions and Equations, Ratios and Proportional Relationships
Not just
Compare the graphs of y=x2 +1 and y=(x-2)2 +1,
But also:
Identify how the three graphs are
related and write functions
describing the relationship.
So given the graph
3
find
A
Advanced Placement AB, 2003,
23% correct
Intraocular Impact
An action/consequence principle
Students learn when they
 Engage in a concrete experience
 Observe reflectively
 Develop an abstract conceptualization based upon the
reflection
 Actively experiment/test based upon the abstraction
Zull, J. ( 2002). The Art of Changing the Brain: Enriching the Practice of
Teaching by Exploring the Biology of Learning.
Just Do It
NO EXCUSES (no money, too difficult to learn, too much
time off task)—make it happen because dynamic
interactive technology can make a real difference in
what students learn.
Give kids a chance to PLAY with mathematical ideas. Use
the technology to
 Connect concepts
 Develop conceptual understanding
 Gain a foundation for applying procedures
 Increase notational fluency
 Make sense of mathematics