1.  Math and Technology Designing tasks that use DGS to support students’ high-level thinking

2. Technology Tasks  How do we, as teachers, select or design tasks that will use technology in support of high-level thinking?

3. Sinclair (2003) Design Principles  When a question aims to focus student attention, the sketch must provide the visual stimulus. It must draw attention through color, motion, and markings.  When a statement prompts action, such as asking students to drag, observe or deduce, the sketch must contain the necessary provisions. It must provide affordances so that the student can take the required steps.  Questions that invite exploration are open-ended. In order to explore uncharted territory, the student requires a sketch that allows options. Thus, when a question invites exploration, the sketch must provide alternate paths.  A question can surprise – which may lead to further exploration; how- ever, the teacher is not necessarily there to correct any misinterpretation. Thus, the sketch must support experimentation to unmask the confusion. It must be flexible enough to help students examine cases, yet constrained enough to prevent frustration.

4. Amplifier/Reorganizer  Amplifier: makes a task more accurate or efficient, but could be accomplished without it  Reorganizer: makes use of affordances of technology to accomplish a goal that would be practically impossible without it

5. Draw Attention GoalQuestionAmplifierReorganizer Foster connections across representations How does the sketch support connections between representations? Visual stimulus (e.g., color, motion, markings) used exclusively for aesthetic purposes. Visual stimulus linked with mathematical concepts to develop mathematical meaning, representations, and/or relationships.

6. Draw Attention: Amplifier

7. Draw Attention: Reorganizer

8. Provide Affordances GoalQuestionAmplifierReorganizer Make mathematically meaningful observations; look for invariant relationships. Do the sketch and prompts use DGS in a way that would be difficult to replicate without it? Students use dragging to create multiple static examples, and reason from those static examples. The sketch allows for continuous dragging, and students are guided to examine measurements or relationships dynamically.

9. Provide Affordances: Amplifier

10. Provide Affordances: Reorganizer

11. Provide Alternative Paths GoalQuestionAmplifierReorganizer Mathematical exploration; use appropriate tools strategically. How open-ended is the task? Sketch and prompts guide students to investigate the same examples to explore mathematical connections or invariances. Students are prompted to explore based on individual observations within the sketch. Prompts are open- ended or the sketch provides alternate paths.

12. Provide Alternative Paths: Amplifier

13. Provide Alternative Paths: Reorganizer

14. Support Experimentation GoalQuestionAmplifierReorganizer Foster curiosity and modify thinking; make and test conjectures Does the sketch provide feedback? Do the prompts encourage students to use feedback? Sketch does not provide feedback to allow students to explore their own conjectures; or prompts do not explicitly guide students to test conjectures. Sketch provides feedback and prompts encourage students to test and refine conjectures.

15. Support Experimentation: Amplifier

16. Support Experimentation: Reorganizer

17. Notes on this framework  Goals are essential  Foster connections across representations  Make mathematically meaningful observations; look for invariant relationships.  Mathematical exploration; use appropriate tools strategically.  Foster curiosity and modify thinking; make and test conjectures  Applies to tasks utilizing any dynamic software (e.g., GSP, GeoGebra, Tinkerplots, Fathom, etc.)  Not meant to be only evaluative  Can provide insight into how a sketch or prompts can be revised to better support students’ thinking