GeoGebra and the STEM agenda Mark Dawes University of Cambridge Comberton Village College GeoGebra Institute of Cambridge md437@cam.ac.uk www.geogebra.org

xkcd http://xkcd.com/435/

Every Child Matters (ECM) According to the British government, every child should: be healthy stay safe enjoy and achieve make a positive contribution achieve economic well-being

STEM Science Technology Engineering Mathematics achieve economic well-being Science Technology Engineering Mathematics

STEM Science Technology Engineering Mathematics Engineering Technology Maths Are the parts of mathematics that do not have applications to S, T and E considered to be worthless?

A better model? Science Technology Engineering Mathematics Maths

GeoGebra = Geometry + Algebra + Spreadsheet Algebra Window Geometry & Graphics Window

This talk GeoGebra and STEM How much I have learned from using GeoGebra: as a learner as a school-teacher from pupils in school as a lecturer from students at university

Some things I learned from my pupils

Question: “Make a conjecture about the triangle in a semi-circle Question: “Make a conjecture about the triangle in a semi-circle. Prove it.” I explained that there must always be a diameter involved in this triangle. I asked the class (of 14-year-olds) to draw this in GeoGebra and then to make a conjecture. I expected that they would focus on the size of the angle…

John’s Conjecture John drew this. I thought he would then use it to provide an algebraic proof/explanation of the angle CED = 90. This is what I expected …

John’s Conjecture y x y x This is what I expected: Label angle ACE = x Triangle ACE is isosceles, so angle AEC = x Label angle ADE = y Triangle ADE is isosceles, so angle AED = y In triangle DEC we have x+x+y+y = 180, so x+y = 90 and angle DEC is therefore a right angle. This is the proof I learned at school, and while it works and certainly proves that the angle is 90 degrees, it is not very satisfying because it doesn’t really explain what is going on.

What actually happened …

John’s Conjecture The two smaller triangles have the same area In fact, John had a different conjecture: The two smaller triangles have the same area. This is very interesting, because it was new to me and I didn’t immediately know whether it was true or not. John and the rest of the class didn’t know either. We tried a few things out. There are some special cases: when E is moved to C then the areas are both zero. When E is move to be vertically above A then there is symmetry and the areas are clearly the same. Some pupils used the area measurement tool to try this out, but we talked about this not providing a proof. I later asked a single question: “How do you work out the area of a triangle?” – and this was enough for them to prove that the conjecture was correct.

Maddie’s Diagram Maddie tried to make her own copy of John’s diagram, but instead of using “line segment” she drew a line. I used to do that all the time! It made me realise that I use the word “line” wrongly. If I ask pupils to “draw a line” I am asking them to do something impossible because a line should extend infinitely in both directions! What was particularly exciting here was that Maddie did not delete her line, but looked at the diagram differently.

Maddie’s Diagram She saw a rectangle! She then said “this is a rectangle, so the angle DEC must be a right angle”! How exciting is that! Again – I had never come across this, but the reason the angle in a semi-circle is a right angle is that the triangle is half a rectangle! Wow! It is a proof, but is even better than the algebraic proof because it is a proof that explain why it works too! But the proof is not complete. Someone else in the class pointed out we hadn’t proved that the shape really was a rectangle. Maddie said that the diagonals of the quadrilateral were the same length because they are both diameters, so it must be a rectangle. Someone else said that in an isosceles trapezium the diagonals are the same length (and in fact there are other non-special quadrilaterals where this is true too). Eventually we arrived at the diagonals being the same length and bisecting each other. This means it must be a rectangle.

Some things I learned as a teacher / as a learner

Maths Science

Maths Engineering

Interacting with important ideas

This talk GeoGebra and STEM How much I have learned from using GeoGebra: as a learner as a school-teacher from pupils in school as a lecturer from students at university

Tell me and I forget. Teach me and I remember. Involve me and I learn. Benjamin Franklin