GeoGebra – The Mathematician’s Tool

1. GeoGebra – The Mathematician’s Tool Mark Dawes md437@cam.ac.uk [University of Cambridge, UK]

2. What do I do ? • I teach the new mathematics teachers at Cambridge University • I work with teachers in the East of England to help them improve their teaching • I teach mathematics in a high school in Cambridge

3. This talk • How GeoGebra is used in school • How GeoGebra could be used in school I will use my experience of schools in England. There will be lots of examples!

4. “most of the mathematics taught in schools does not accurately represent the practices of adult mathematicians” – Watson (2008) Journal for maths teachers: “Mathematics Teaching”

5. Teachers learning to use GeoGebra • The first file they make … Circle 3  • As the first file • Still using the file after a year? 

6. “How can I use GeoGebra?” • This is the question teachers ask me • In reality this is three questions

7. “How can I use GeoGebra?” Q1] “How does GeoGebra work?” For example: • How to draw a circle • How to create a Pythagorean diagram • How to label an angle • How to draw the graph of a function Using GeoGebra tools Using Euclidean tools Graph with slider Photograph Time

8. “How can I use GeoGebra?” Q2] “How can I use GeoGebra in the curriculum?” For example: • Which topics can I use it in? • Which ages of students? Reflect and test y = 2x + c

9. “How can I use GeoGebra?” Q3] “What is the pedagogical rationale behind using GeoGebra?” • This question is more difficult. • Teachers don’t usually ask this question. • It is very important!

10. Seymour Papert – creator of “Logo” • “In my vision, the child programs the computer.” (Papert, 1980)

11. In 2010 The are rooms of computers that are used by every subject. There is a projector in every classroom. The teacher explains the mathematics at the front of the class. In English schools: In 1985 There are computers only in mathematics classrooms. There are no projectors. The students use the computers to explore mathematics for themselves.

13. Three modes of using GeoGebra Teacher demonstration Pupils interact with teacher created files Pupils create their own files

14. Three modes of using GeoGebra Teacher demonstration GeoGebra is projected and the teacher (or a student) makes some changes. • Comments from teachers about this mode: • The teacher can guide the students • The teacher is in control of the lesson and what the students are doing • The teacher can focus on questioning/thinking • The students have to make conjectures Enlargement

15. Three modes of using GeoGebra The teacher makes a file for the students to interact with. The teacher makes sure the students will “discover” the new mathematics they want them to. Pupils interact with teacher created files • Comments from teachers about this mode: • Students don’t waste time • The teacher can control what the students will do • The students can start exploring the mathematics immediately Bridge Face

16. Three modes of using GeoGebra Pupils create their own files The students have a problem to solve and can choose to use GeoGebra to help them, starting with an empty file.

17. Question: “Make a conjecture about the triangle in a semi-circle. Prove it.”

18. John’s Conjecture The two smaller triangles have the same area

19. Maddie’s Diagram

20. Different ideas

21. Homework • “Draw any quadrilateral. Join the midpoints of the sides in order to make a new quadrilateral. What can you tell me about the new quadrilateral?” GeoGebra

22. Homework • “Draw any quadrilateral. Join the midpoints of the sides in order to make a new quadrilateral. What can you tell me about the new quadrilateral?”

23. Homework • “Draw any quadrilateral. Join the midpoints of the sides in order to make a new quadrilateral. What can you tell me about the new quadrilateral?”

25. How these students used GeoGebra • To visualise • To make a conjecture • To convince • To help provide the elements of the proof • Who is doing the mathematics? • The student!

26. Three modes of using GeoGebra Pupils create their own files • Comments from teachers about this mode: • It takes a long time • The pupils have to learn how to use GeoGebra • They might not do what the teacher wants them to do • They will behave like mathematicians • They might have an idea the teacher didn’t think of • They might be confused about what ‘proof’ means • They can work independently • They can create their own “What if … ?” questions

27. Pupils create their own files Pupils interact with teacher created files Teacher demonstration Are the modes connected? Teacher demonstration Pupils interact with teacher created files Pupils create their own files

28. Teacher demonstration Pupils create their own files Pupils interact with teacher created files The three modes Good teaching using GeoGebra

29. GeoGebra value-added Trapezium Circle again Triangle

30. Teacher demonstration Pupils create their own files Pupils interact with teacher created files GeoGebra can add value Good teaching using GeoGebra

31. GeoGebra – A tool for turning Children into Mathematicians Mark Dawes md437@cam.ac.uk [University of Cambridge, UK]