1. GeoGebra Conference First International July 14, 2009
Dr. Pamela J. Buffington

2. The Power of Convergence
Formative Assessment
Assessment Principle: Assessment should support the learning of important mathematics and furnish useful information to both instructors and learners. (NCTM, 2000, p 22)
Interactive Technology
Technology Principle: Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances learning. (NCTM, 2000, p 24)
Formative assessment can provide critical information about existing conceptions / misconceptions
Technology can provide accurate, interactive, visual images of mathematical concepts and ideas
Instructors can provide a targeted instructional intervention for learners’ specific needs and remediate potential barriers to learning

3. Students come to the classroom with preconceptions about how the world works. If their initial understanding is not engaged, they may fail to grasp the new concepts and information that are taught, or they may learn them for purposes of a test but revert to their preconceptions outside the classroom
-How People Learn, National Research Council
Note FA is about eliciting these preconceptions

4. Foundational Resources
Diagnostic Assessment Using Probes
Targeted Interventions Using Interactive Technologies
First will share briefly about the diagnostic assessment

5. Diagnostic Prompts: Example Errors
Overgeneralizations:
Applying incorrect rules such as “two negatives always make a positive”
Conceptual Misunderstandings
Viewing the equals sign as “place answer here”
Common Errors (Procedural)
Incorrect Decimal Placement when multiplying
Ask audience to think about preconceptions from their own experiences

6. Research Base Example #1
Direct translation of the words into an algebraic equation without understanding the relationship described.
Using variables to represent words

7. Example #2 – In GeoGebra

8. Research Base Example #2:
Failure to see the difference in the x and y scales applying an “up one over one” rule to misinterpret a rate of change of one.
Although the visual aspect of graphs makes them a powerful medium for generating meaning, it is also the source of many of the incorrect responses
Note they will see more examples

9. Foundational Resources
Diagnostic Assessment Using Probes
Targeted Interventions Using Interactive Technologies
Use this slide to transition to geogebra overview

10. Riding the Ferris Wheel
100 m
height
0 m
Loading dock
Ferris wheel
time
The distance from the loading dock at the bottom of the Ferris wheel to the very top is 100 meters. You are in a seat on the Ferris wheel at the 3:00 position and are being asked to graph the height of your seat as you move around the ride. The Ferris wheel is moving in a counterclockwise rotation.

11. Resources Research Brief around a Given Topic
Diagnostic Probe to Elicit Understanding and Misunderstandings
Instructional Materials:
Applet
Supporting Resources
Overview
Student Exploration
Comparing Fraction Example 11

12. Research Example: Rational Numbers
Research has verified that learners continue to use properties they learned from operating with whole numbers even though many whole number properties do not apply to rational numbers (Adding It Up, NRC )
Examples:
Reason that 1/8 is larger than 1/7 because 8 is larger than 7.
May believe that 3/4 equals 4/5 because in both fractions the difference between the numerator and denominator is 1.

13. Diagnostic Probe: Comparing Rational Numbers

14. Applet Example: Comparing Rational Numbers
- Many virtual manipulatives include dynamic numerals that change as the representation on the screen changes. This direct and immediate connection to numeric representation is impossible with physical models.
-VandeWalle

15. Applet Example: Comparing Rational Numbers
Directions to url
Learning Tools List # 5b Comparing Rational Numbers (Fractions) - 2

16. Applet Example: Comparing Rational Numbers
Explore the Applet
- What specific questions would you ask? What specific examples should be explored?

17. Research Example: Equivalent Expressions
In general, if students engage extensively in symbolic manipulation before they develop a solid conceptual foundation for their work, they will be unable to do more than mechanical manipulation. The foundation for meaningful work with symbolic notation should be laid over a long time (PSSM)
Examples:
Some students simply ignore bracketing symbols such as parentheses , as in 4(n+5) = 4n (Algebraic Thinking)
Some students make errors writing totals and products They wrote h10 , meaning h plus 10 or x4 meaning “x times 4” (Algebraic Thinking)
Transition to Pam 17

18. Diagnostic Probe: Equivalent Expressions

19. Applet Example: Equivalent Expressions
Learning Tools List # 12 Equivalent Expressions – Distributive Property (2)

20. Applet Example: Equivalent Expressions
Explore the Applet
How would you use this applet?
Which checkboxes would you turn on first? Explain why. Would you show all three expressions at once? Why or why not?
What examples would you use?
What questions would you ask students?

21. Supporting Resources: Equivalent Expressions
Applet Overview
Student Explorations

22. Diagnostic Probe: Equivalent Expressions
yes

23. Associated Applet: Equivalent Expressions
Learning Tools List # 6 Equivalent Expressions – Difference of Squares 1

24. Real Life Explorations: Comparing DVD Plans
yes
Interactive Mathematics Tools for Assessment & Instruction # 10 DVD Rental Plans Problem

25. Interactive Resources
Accessible online at
For Additional Information Contact:
Pam Buffington