Let’s Enable Students to Make Sense of Mathematics: Before and After Algebra II Gail Burrill Michigan State University

Napoleon’s 1812 march to Russia Constructed by Charles Minard,1869

Life expectancy vs income Gapminder log

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 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 Relationshi ps Building Procedural Understanding

y = log 2 (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 -3-3 Compare the graphs of y=x 2 +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 they Engage in a concrete experience Observe reflectively Develop an abstract conceptualization based upon the reflection Actively experiment/test based upon the abstraction An action/consequence principle 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 It