let’s enable students to make sense of mathematics: before and after algebra ii gail burrill...
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Life expectancy vs income Gapminder logTRANSCRIPT
Let’s Enable Students to Make Sense of
Mathematics: Before and After Algebra II
Gail BurrillMichigan State University
Napoleon’s 1812 march to Russia
Constructed by Charles Minard,1869
Life expectancy vs income
Gapminderlog
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 ProcessesStudents should engage inReasoning with definitions and theoremsConnecting conceptsImplementing algebraic/computational processesConnecting multiple representationsBuilding notational fluencyCommunicating
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) 0B) π/2C) πD) 3π/2E) 2π
Algebra and Precalculus Concept Readiness Alternate Test (APCR alternate) – August 2013
Analytic Expressions and Graphs a) f(0) = 2b) f(-3) = f(3) = f(9) =
0
c) f(2) = g(2)d) g(x) > f(x) for x > 2
Adapted from Illustrative Mathematics
(10, 4.55)
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
CCSS Progressions; Building Concepts: Ratios and Proportional Relationships
Building Procedural Understanding
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
A
-3
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
Advanced Placement AB, 2003, 23% correct
find
Intraocular Impact
Students learn when theyEngage in a concrete experienceObserve reflectivelyDevelop an abstract conceptualization based
upon the reflectionActively experiment/test based upon the
abstraction
An action/consequence An action/consequence principleprinciple
Zull, J. ( 2002). The Art of Changing the Brain: Enriching the Practice of Teaching by Exploring the Biology of Learning.
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
Just Do ItJust Do It