Task design in a technology-rich mathematics classroom: the case of dynamic geometry

1. Task design ina technology-rich mathematicsclassroom: the case of dynamicgeometry Allen Leung Associate Professor Department of Education Studies Hong Kong Baptist University GeoGebra Institute of Hong Kong Launching Seminar 9 November 2012

2. A Technology-rich Mathematics Classroom Technology in a mathematics classroom serves as a pedagogical tool like ruler and compasses. It facilitates teachers and students to teach and to learn. Technology is at the same time consequence and driving force of human intellect. Thus, technology can be a means to acquire knowledge and may itself become part of knowledge.

3. Utilizing Information Communication Technology (ICT) in a mathematics classroom is not merely about presenting traditional school mathematics via new ICT media like Powerpoint; Internet or even sophisticated virtual environment like the dynamic geometry environment GeoGebra; it is abouthow to harvest the power of the technology to create a new way of teaching, learning, and even thinking about mathematics.

4. In a technology-rich teaching and learning environment , the role of a teacher in the tradition sense must be cast-off in order to give room for students to discover and even to create knowledge. Teacher is to guide rather than to instruct, to suggest rather than to transmit. In this way, students could have ownership of the knowledge gained. ICT should open a new space of learning that is broader in scope than the traditional classroom.

5. Mathematical Experience Suitable ICT environments for mathematics learning have the power to allow students to conveniently make visible the different variations in a mathematical situationand to re-produce cognitive mental pictures that guide the development of mathematical concept (for example, in a dynamic geometry environment).

6. Leung, A. (2010). Empowering learning with rich mathematical experience: reflections on a primary lesson on area and perimeter, International Journal for Mathematics Teaching and Learning [e-Journal]. Retrieved April 1, 2010, from http://www.cimt.plymouth.ac.uk/journal/leung.pdf A mathematical experience can be seen as “the discernment of invariant pattern concerning numbers and/or shapes and the re-production or re-presentation of that pattern.” (Leung, 2010)

7. Brown, J. (2005). Identification of affordance of a technology rich teaching and learning environment (TRTLE). In Chick, H. L. & Vincent, J. L. (Eds.). Proceedings of the 29th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 185-192. Melbourne: PME. • An affordance in technology-rich teaching and learning environment is “the opportunity for interactivity between the user and the technology for some specific purpose” (Brown, 2005). • Task design must consider how the affordance of a chosen ICT environment can facilitate or impede mathematical learning and how to capitalize it to enhance students’ ability to experience mathematics under an inquiry mode.

8. Environmentally-situated o contextually oriented pedagogic task design • Technology-rich pedagogical environments: teaching and learning environments that are enhanced by the use of ICT (Information Communication Technology) to carry out the teaching and learning process.

9. Teaching and learning in a technology-rich pedagogical environment is a process where routines, procedures and actions are transformed to reasoning and creativity

10. Task Design Principles for (Technology-rich)Mathematics Classroom • Construct mathematical object using the technology involved • Interact with the technology involved • Observe and record • Explain or prove • Generalize findings into mathematical concepts • Hypothesize and make conjecture

11. Mason, J., & Johnston-Wilder, S. (2006). Designing and Using Mathematical Tasks. St. Albans: Tarquin Publications. “The purpose of a task is to initiate mathematically fruitful activity that leads to a transformation in what learners are sensitsed to notice and competent to carry out” (Mason & Johnston-Wilder, 2006, p.25) “The point of setting tasks for learners is to get them actively making sense of phenomena and exercising their powers and their emerging skills” (Mason & Johnston-Wilder, 2006, p.69)

12. Techno-pedagogic Task Design inMathematics Leung, A. (2011). An epistemic model of task design in dynamic geometry environment. ZDM - The International Journal on Mathematics Education, 43, 325-336. Task design that focuses on pedagogical processes in which learners are empowered with amplified abilities to explore, reconstruct (or re-invent) and explain mathematical concepts using tools embedded in a technology-rich environment.

13. Three nested epistemic task modes are put forward to guide the design of a techno-pedagogic task. They are: Gradual Evolution • Establishing Practices Mode • Critical Discernment Mode • Situated Discourse Mode

14. Establishing Practices Mode (PM) PM1 Construct mathematical objects or manipulate predesigned mathematical objects using tools embedded in a technology-rich environment PM2 Interact with the tools in a technology-rich environment to develop (a) skill-based routines (b) modalities of behaviour (c) modes of situated dialogue

15. Critical Discernment Mode (CDM) Observe Discover Record Re-present Re-construct Patterns of Variation and Invariant

16. Establishing Situated Discourses Mode (SDM) SD1 Develop inductive reasoning leading to making generalized conjecture SD2 Develop discourses and modes of reasoning to explain or prove

17. A Generic Nested Pedagogical Sequence 《創造》Situated Discourse 《審判、識別、領悟》Critical Discernment 《機械性操作》Practices

18. Dynamic Geometry Task Designs:Exploring Cyclic Quadrilateral

19. Design One 1. Construct a circle and four points on it 2. Join the four points with line segments to form a quadrilateral 3. Measure the four interior angles of the quadrilateral 4. Drag the four points to different positions on the circle 5. Investigate and make conjecture on the relationship among the angles 6. Explain (or prove) why the conjecture is true

20. Design Two 1. Construct a general quadrilateral ABCD 2. Measure two opposite interior angles, say ∠ABC and ∠CDA 3. Calculate ∠ABC + ∠CDA 4. Turn the Trace function on for point C 5. Drag point C continuously to keep ∠ABC + ∠CDA as close to 180° as possible 6. Observe the shape of the path that point C traces out 7. Make a conjecture on the shape of the path 8. Explain (or prove) why the conjecture is true

21. Design Three Task 1 1.Construct two points A and B 2.Explore how to construct a circle that passes through A and B 3. Investigate how many such circles can be constructed 4.Explain why the construction procedure works

22. Task 2 1.Construct three non-collinear points A, B and C 2.Join A, B and C with line segments to form a triangle ABC 3.Explore how to construct a circle that passes through the vertices of ABC 4.Explain why the construction procedure works

23. Task 3 1.Use the construction in Task 2 to explore how to construct a circle that passes through all the vertices of a general quadrilateral 2.Make a conjecture on the condition under which a quadrilateral can be circumscribed by a circle 3.Explain (or prove) why the conjecture is true

24. Possible Conjectures Design One: Given a cyclic quadrilateral, a pair of interior opposite angles always adds up to 180o. Design Two: For a quadrilateral to satisfy the condition “a pair of interior opposite angles adds up to 180o ”, the vertices of the quadrilateral must lie on a circle. Design Three: If the four perpendicular bisectors of the sides of a quadrilateral are concurrent, then the quadrilateral can be inscribed in a circle.

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