5. Learning mathematics <ul><li>Learning involves learning to use new ‘mathematical’ tools </li></ul><ul><li>All learning is influenced by previous learning </li></ul><ul><li>Teachers are central </li></ul>
6. A person-plus view of learning <ul><li>The environments in which humans live are thick with invented artefacts that are in constant use for structuring activity , for saving mental work, or for avoiding errors or they are adapted creatively almost without notice . These ubiquitous mediating structures that both organise and constrain activity include not only designed objects such as tools, control instruments, and symbolic representations like graphs, diagrams, text, plans and pictures, but people in social relations , as well as features and landmarks in the physical environment” </li></ul><ul><li>Pea, 1993 p 48 </li></ul>
7. Mathematical tools <ul><li>Mathematical tools can be: </li></ul><ul><li>digital (e.g. dynamic geometry software, spreadsheets, graphic calculators) </li></ul><ul><li>non-digital (e.g.book, paper and pencil, ruler, compass, graph). </li></ul>
8. <ul><li>“ Einstein famously said that his pencil was more intelligent than he was - meaning, that he could achieve far more using his pencil as an aid to thinking than he could unaided. There is a need to recognise that mathematical digital technologies are the ‘pencils of today’ and that we will only fully exploit the benefits of digital technologies in teaching, learning and doing mathematics when it becomes unthinkable for a student to solve a complex problem without ready access to digital technological tools”. </li></ul><ul><li>Digital Technologies and Mathematics Education, Report for Joint Mathematical Council of UK </li></ul>
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10. Mathematical tools
11. Feynman diagrams <ul><li>A Feynman diagram is a tool invented by American physicist Richard Feynman for performing scattering calculations in quantum field theory . Particles are represented by lines, which can be drawn in various ways depending on the type of particle being depicted. A point where lines connect to other lines is called an interaction vertex, or vertex for short. Lines fall into three categories: internal lines (which connect two vertices), incoming lines (which extend from "the past" to a vertex and represent the initial noninteracting state) and outgoing lines (which extend from a vertex to "the future" and represent the final noninteracting state). Most commonly the bottom of the diagram represents the past and the top of the diagram represents the future. </li></ul>
12. All learning is influenced by previous learning — at school <ul><ul><ul><li>Int: Does L have to be a larger number than A? </li></ul></ul></ul><ul><ul><ul><li>Eloise: Yes because A starts off as 1 or something. </li></ul></ul></ul><ul><ul><ul><li>Int: What made you think that [L has to be a larger number than A?] </li></ul></ul></ul><ul><ul><ul><li>Eloise: Because when we were little we used to do a code like that...in junior school...A would equal 1, B equals 2, C equals 3.....there were possibilities of A being 5 and B being 10 and that lot.....but it would come up too high a number to do it.........it was always in some order... </li></ul></ul></ul>
13. and at home………. <ul><li>Int: Do either of you use Excel at home (Alan shakes head)? </li></ul><ul><li>Ray: Sometimes. My Dad uses it for his paper work </li></ul><ul><li>Int: And when you use it what do you use it for? </li></ul><ul><li>Ray: Umm, he uses it, cos when he’s got paper calculations and some are hard like for him, he puts it in Excel and then he puts, he circles it and then presses the equal button and it tells him what the sums are. </li></ul><ul><li>Int: What do you use it for? </li></ul><ul><li>Ray: Maths homework. </li></ul><ul><li>Alan: Cheat. </li></ul>
14. The teacher is central <ul><li>“ No educational reform can get off the ground without an adult actively and honestly participating — a teacher willing and prepared to give and share aid to comfort and to scaffold. Learning in its full complexity involves the creation and negotiation of meaning in a larger culture and the teacher is the vicar of the culture at large. You cannot teacher-proof a curriculum any more than you can parent-proof a family”. </li></ul><ul><li>Bruner 1996 p 84 </li></ul>
15. Using ICT in the classroom To start using ICT in the classroom requires a huge amount of support.
16. Proof & Geometry
17. What is proof? Students’ informal knowledge <ul><ul><ul><ul><ul><li>Is it kind of like a survey? </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Is it just like testing? </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>When you get school photographs you get those little ones that say ‘proof’’ on them. Is it like a sample or something? </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>Show your workings </li></ul></ul></ul></ul></ul><ul><ul><ul><ul><ul><li>It’s how to make sure…to prove it, to say that is right because… </li></ul></ul></ul></ul></ul>
18. In the classroom Marnie distributing laptops at the beginning of each session
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20. In the classroom
21. The teacher focuses attention on <ul><li>geometrical properties </li></ul><ul><li>Proof </li></ul><ul><ul><ul><li>“ just drawing lines will not work…you need to use your knowledge of shapes…….you need to use commands like the perpendicular bisectors, like parallel lines, that’s what you need to do” </li></ul></ul></ul>
22. Proving there is 180 degrees in a Triangle By Pema Seely and Billie Alder
23. To help us we will use the shape below A B C D E Line 1 Line 2 Lines 1 and 2 are parallel
24. We know that there is 180 degrees on a straight line. This means C+D+E=180 degrees C D E
25. Alternate angles are equal so angles B and D are equal. B D
26. Corresponding angles are equal, so A and E are equal. A E A E
27. So C+D+E=180 degrees D=B(alternate angles) E=A(corresponding angles) So C+B(D)+A(E)=180 degrees.
28. We know that there is 180 degrees on a straight line. This means C+D+E=180 degrees C D E
29. Alternate angles are equal so angles B and D are equal. B D
30. Corresponding angles are equal, so A and E are equal. A E A E
31. So C+D+E=180 degrees D=B(alternate angles) E=A(corresponding angles) So C+B(D)+A(E)=180 degrees.
32. The teacher is central <ul><li>The teacher is a gateway to the world of mathematical language and the world of mathematical tools. </li></ul>
33. Why ICT in initial teacher education <ul><li>ICTs are the ‘tools of the trade’ of the modern scientists, technologists, engineers and mathematicians. </li></ul><ul><li>Young people use ICTs in their everyday lives, but not usually for mathematical and scientific purposes </li></ul><ul><li>young people need to become proficient in using ICTs for mathematical and scientific purposes in school </li></ul>
34. Why ICT in initial teacher education <ul><li>Teachers are central and initial teacher education is key </li></ul>
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