Linking algebra and geometry

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1 Future Curricular Trends in School Algebra and Geometry Linking algebra and geometry Keith Jones UoS School of Education

2 Future Curricular Trends in School Algebra and Geometry Linking geometry and algebra Keith Jones UoS School of Education

3 Linking geometry and algebra My research foci Emphasis and treatment of algebra and geometry in the UK Research influences on curriculum decisions in the UK Impact of technology Development of mathematical reasoning and proof in students Pre-service and inservice education of mathematics teachers Thoughts on changes over the next 10 years CSMC conference, 2-4 May

4 Sources of inspiration. Donald Coxeter ( the man who saved geometry ) I think that, by being careful, we could probably do the same amount of calculus and linear algebra in less time, and have some time left over for nice geometry. Eric Bell "With a literature much vaster than those of algebra and arithmetic combined, and at least as extensive as that of analysis, geometry is a richer treasure house of more interesting and half-forgotten things, which a hurried generation has no leisure to enjoy, than any other division of mathematics. Michael Atiyah..spatial intuition or spatial perception is an enormously powerful tool and that is why geometry is actually such a powerful part of mathematics - not only for things that are obviously geometrical, but even for things that are not. We try to put them into geometrical form because that enables us to use our intuition. Our intuition is our most powerful tool. CSMC conference, 2-4 May

5 UK context statutory integrated curriculum; national testing; increasing accountability (eg league tables ); impact of international comparisons (eg TIMSS); increasing focus on numeracy emerging concerns about proof CSMC conference, 2-4 May

6 Government advice on teaching the Grade 8 curriculum CSMC conference, 2-4 May

7 Research foci ICMI study on geometry Deductive and intuitive approaches to solving geometrical problems Can a deductive and an intuitive approach be mutually reinforcing when solving geometrical problems? Proof in dynamic geometry contexts Will the use of DGS improve the teaching and learning of proof, or will computer use be seen to replace any need for proof? CSMC conference, 2-4 May

8 UK context revised statutory curriculum; more explicit stipulations regarding proof; some encouragement for links within mathematics and across subjects continuing national testing; entrenched accountability (eg league tables ); overwhelming focus on numeracy ; Government Numeracy Task Force reports; national numeracy strategy Royal Society working group on geometry Government algebra and geometry initiative CSMC conference, 2-4 May

9 Research foci Royal Society report on geometry Conjecturing and proving Spatial thinking Visualising Linking geometry with other areas of mathematics CSMC conference, 2-4 May

10 Research foci Government algebra and geometry initiative Six projects One overall report Making connections between different mathematical concepts is important for developing understanding. One way of linking algebra and geometry is to exploit the capacity of dynamic geometry to provide novel ways of visualising algebraic relationships. A second approach to linking algebra and geometry is to use different approaches to tackle the same problem. CSMC conference, 2-4 May

11 UK context 2005-onwards revised statutory curriculum; less emphasis on specifying the curriculum in terms of subjects continuing national testing; entrenched accountability (eg league tables ); ICMI study on technology CSMC conference, 2-4 May

12 Research projects teaching 3D geometry with small software applications teaching calculus with dynamic geometry software CSMC conference, 2-4 May

13 Linking geometry and algebra 120 page report Role for geometry? Mention of proof? half a page zero CSMC conference, 2-4 May

14 Linking geometry and algebra Making connections between different mathematical concepts is important for developing understanding. One way of linking geometry and algebra is to exploit the capacity of dynamic geometry to provide novel ways of visualising algebraic relationships. A second approach to linking geometry and algebra is to use different approaches to tackle the same problem. CSMC conference, 2-4 May

15 The capabilities of dynamic geometry Visualising algebraic relationships dragon fountain, Singapore CSMC conference, 2-4 May

16 The capabilities of dynamic geometry Visualising algebraic relationships Clifton Bridge, Bristol CSMC conference, 2-4 May

17 The capabilities of dynamic geometry Visualising algebraic relationships Students use dynamic geometry software to plot quadratic functions that match the flight of the ball, providing a feel for how the various coefficients affect the shape of the graph. CSMC conference, 2-4 May

18 The visual world CSMC conference, 2-4 May

19 The visual world CSMC conference, 2-4 May

20 The visual world CSMC conference, 2-4 May

21 Different approaches to problems One of the oldest problems in number theory is no doubt the problem about Pythagorean triples, i.e. triples of whole numbers (a, b, c) which fulfil the Pythagorean relation: a 2 + b 2 = c 2 Yet there is a link between Pythagorean triples and the size of the radius of the incircle of a triangle CSMC conference, 2-4 May

22 Contemporary mathematics double bubble A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. It had been conjectured that two equal partial spheres sharing a boundary of a flat disk separate two volumes of air use a total surface area that is less than any other boundary. This equal-volume case was proved by Hass et al. (1995). In the case of two unequal partial spheres The corresponding triple bubble conjecture remains open CSMC conference, 2-4 May

23 Contemporary mathematics black holes In 2006, NASA scientists reached a breakthrough in computer modeling that allowed them to simulate what gravitational waves from merging black holes look like. The three-dimensional simulations, are the largest astrophysical calculations ever performed on a NASA supercomputer. CSMC conference, 2-4 May

24 Contemporary mathematics - flags The analytic, asymptotic and numerical solution of non-linear partial singular integro-differential equations with Cauchy Kernels. An unsteady movement of a 2D flag with bending stiffness. CSMC conference, 2-4 May

25 New UK curriculum from Sept 2008 Emphasis on essential skills and processes in mathematics Key concepts Competence Creativity Applications and implications of mathematics Critical understanding Key processes Representing Analysing Interpreting and evaluating Communicating and reflecting CSMC conference, 2-4 May

26 Teacher Education CSMC conference, 2-4 May

27 Concluding comments.. From an epistemological perspective the algebraic-geometric contrast, so far from being a dichotomy, represents something more like a spectrum. CSMC conference, 2-4 May

28 Concluding comments.. Donald Coxeter ( the man who saved geometry ) do the same amount of calculus and linear algebra in less time, and have some time left over for nice geometry. Eric Bell "geometry is a richer treasure house.. than any other division of mathematics. Michael Atiyah geometry is actually such a powerful part of mathematics - not only for things that are obviously geometrical, but even for things that are not. CSMC conference, 2-4 May

29 Concluding comments.. The soul of mathematics may lie in geometry, but algebra is its heart. [Usiskin, 2004] Without geometry, life is pointless. [Anon] Thank you for listening Keith Jones University of Southampton United Kingdom CSMC conference, 2-4 May