Let’s Enable Students to Make Sense of

Mathematics: Before and After Algebra II

Gail BurrillMichigan State University

burrill@msu.edu

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