Functioning Mathematically Charlie Gilderdale cfg21@cam.ac.uk 2 July 2012

Music teachers challenge students to listen and participate Music teachers challenge students to listen and participate. English and History teachers invite students to journey into other worlds. Art and Drama teachers offer students opportunities to explore. What are we to offer students if they are to function mathematically?

NRICH problems require students to work mathematically To experiment To be systematic To notice patterns / look for connections To conjecture To test ideas To generalise To explain, justify and prove

What’s it Worth? Each symbol has a numerical value. The total for the symbols is written at the end of each row and column. Can you find the missing total that should go where the question mark has been put? Once one solution has been found: “Can you find the solution in a different way?” then “Here are eight different starting points for solutions, can you finish them all off?” (worksheet in TN) Finally, discuss which approaches are most elegant, pleasing, whatever. Rationale for using What’s it Worth? The main focus is not the solution but the approaches taken. We are interested in reasoning and justification, and people being able to make sense of each other’s thinking.

Odds, Evens and More Evens Here are the first few sequences from a family of related sequences: A1 = 1, 3 ,5 ,7 ,9 ,11 ,13 ,15 ,17 ,19 ,21 ,23 ,25 ,27 ,29 ... A2 = 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42 ... A3 = 4, 12, 20, 28, 36, 44, 52, 60 ... A4 = 8, 24, 40, 56, 72, 88, 104 ... A5 = 16, 48, 80, 112, 144 ... A6 = 32, 96, 160 ... A7 = 64 ... A8 = ... Which sequences will contain the number 1000?

See the Curriculum Mapping Documents NRICH provides free mathematical resources designed to develop subject knowledge and higher order problem-solving skills. See the Curriculum Mapping Documents http://nrich.maths.org/curriculum

What next? Follow up materials on the NRICH site: https://nrich.maths.org/cpdfollowup

… a teacher of mathematics has a great opportunity … a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. Polya, G. (1945) How to Solve it