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Learner-centred Education in Mathematics If you want to build higher, dig deeper

This article explores the concept of learner-centred education in mathematics and provides five key ingredients to consider. It emphasizes the importance of starting with rich challenges, valuing mathematical thinking, purposeful activity and discussion, and building a community of mathematicians. The article also suggests various activities and resources that promote deep learning and reflection in mathematics education.

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Learner-centred Education in Mathematics If you want to build higher, dig deeper

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  1. Learner-centred Education in MathematicsIf you want to build higher,dig deeper Charlie Gilderdale cfg21@cam.ac.uk

  2. Initial thoughts • Thoughts about Mathematics • Thoughts about teaching and learning Mathematics

  3. Five ingredients to consider • Starting with a rich challenge: low threshold, high ceiling activities • Valuing mathematical thinking • Purposeful activity and discussion • Building a community of mathematicians • Reviewing and reflecting

  4. Starting with a rich challenge:Low Threshold, High Ceiling activity To introduce new ideas and develop understanding of new curriculum content

  5. Making use of aGeoboard environment

  6. Why might a teacher choose to use this activity in this way?

  7. Some underlying principles Mathematics is a creative discipline, not a spectator sport Exploring → Noticing Patterns → Conjecturing → Generalising → Explaining → Justifying → Proving

  8. Tilted Squares The video in the Teachers' Notes shows how the problem was introduced to a group of 14 year old students: http://nrich.maths.org/2293/note

  9. Some underlying principles Teacher’s role • To choose tasks that allow students to explore new mathematics • To give students the time and space for that exploration • To bring students together to share ideas and understanding, and draw together key mathematical insights

  10. Give the learners something to do, not something to learn; and if the doing is of such a nature as to demand thinking; learning naturally results. John Dewey

  11. The most exciting phrase to hear in science, the one that heralds new discoveries, is not Eureka!, but rather, “hmmm… that’s funny…” Isaac Asimov mathematics

  12. There are many more NRICH tasks that make excellent starting points…

  13. Number and Algebra Summing Consecutive Numbers Number Pyramids What’s Possible? What’s It Worth? Perimeter Expressions Seven Squares Attractive Tablecloths

  14. Geometry and Measures Painted Cube Changing Areas, Changing Perimeters Cyclic Quadrilaterals Semi-regular Tessellations Tilted Squares Vector Journeys

  15. Handling Data Statistical Shorts Odds and Evens Which Spinners?

  16. …and for even more, see the highlighted problems on theCurriculum Mapping Document

  17. Time for reflection • Thoughts about Mathematics • Thoughts about teaching and learning Mathematics

  18. Morning Break

  19. Valuing Mathematical Thinking What behaviours do we value in mathematics and how can we encourage them in our classrooms?

  20. As a teacher, do I value students for being… • curious – looking for explanations – looking for generality – looking for proof • persistent and self-reliant • willing to speak up even when they are uncertain • honest about their difficulties • willing to treat ‘failure’ as a springboard to new learning

  21. … and do I offer students sufficient opportunities to develop these “habits for success” when I set tasks • to consolidate/deepen understanding • to develop fluency • to build connections

  22. We could ask… We could ask: 6cm Area = ? Perimeter = ? or we could ask … 4cm

  23. Perimeter = 20 cm Area = 24 cm² = 22 cm = 28 cm = 50 cm = 97 cm = 35 cm and we could ask …

  24. …students to make up their own questions • Think of a rectangle • Calculate its area and perimeter • Swap with a friend – can they work out the length and breadth of your rectangle? or we could ask …

  25. Can you find rectangles where the value of the area is the same as the value of the perimeter?

  26. Why might a teacher choose to use these activities in this way?

  27. We could ask students to find… (x + 2) (x + 5) (x + 4) (x - 3) … or we could introduce them to…

  28. Pair Products Choose four consecutive whole numbers, for example, 4, 5, 6 and 7. Multiply the first and last numbers together. Multiply the middle pair together… What might a mathematician do next?

  29. We could ask students to… Identify coordinates and straight line graphs or we could introduce them to…

  30. Will the route passthrough (18,17)? Which point will it visit next? How many points will it pass through before (9,4)? Route to Infinity Route to Infinity

  31. We could ask students to… List the numbers between 50 and 70 that are (a) multiples of 2 (b) multiples of 3 (c) multiples of 4 (d) multiples of 5 (e) multiples of 6 or we could ask students to play…

  32. The Factors and Multiples Game A game for two players. You will need a 100 square grid. Take it in turns to cross out numbers, always choosing a number that is a factor or multiple of the previous number that has just been crossed out. The first person who is unable to cross out a number loses. Each number can only be crossed out once.

  33. Why might a teacher choose to use these activities?

  34. Some underlying principles Consolidation should address both content and process skills. Rich tasks can replace routine textbook tasks, they are not just an add-on for students who finish first.

  35. There are many more NRICH tasks that offer opportunities for consolidation…

  36. What Numbers Can We Make? Factors and Multiples Game Factors and Multiples Puzzle Dicey Operations American Billions Keep It Simple Temperature Number and Algebra Painted Cube Arithmagons Pair Products What’s Possible? Attractive Tablecloths How Old Am I?

  37. Geometry and Measures Isosceles Triangles Can They Be Equal? Translating Lines Opposite Vertices Coordinate Patterns Route to Infinity Pick’s Theorem Cuboid Challenge Semi-regular Tessellations Warmsnug Double Glazing

  38. Handling Data M, M and M Which List is Which? Odds and Evens Which Spinners?

  39. …and for even more, see the highlighted problems on theCurriculum Mapping Document

  40. Time for reflection • Thoughts about Mathematics • Thoughts about teaching and learning Mathematics

  41. Lunch

  42. Promoting purposeful activity and discussion ‘Hands-on’ doesn’t mean ‘brains-off’

  43. The Factors and Multiples Challenge You will need a 100 square grid. Cross out numbers, always choosing a number that is a factor or multiple of the previous number that has just been crossed out. Try to find the longest sequence of numbers that can be crossed out. Each number can only appear once in a sequence.

  44. We could ask… 3, 5, 6, 3, 3 Mean = ? Mode = ? Median = ? or we could ask…

  45. M, M and M There are several sets of five positive whole numbers with the following properties: Mean = 4 Median = 3 Mode = 3 Can you find all the different sets of five positive whole numbers that satisfy these conditions?

  46. Possible extension How many sets of five positive whole numbers are there with the following property? Mean = Median = Mode = Range = a single digit number

  47. 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?

  48. Translating Lines Each translation links a pair of parallel lines. Can you match them up?

  49. Why might a teacher choose to use these activities?

  50. Rules for Effective Group Work • All students must contribute:no one member says too much or too little • Every contribution treated with respect:listen thoughtfully • Group must achieve consensus:work at resolving differences • Every suggestion/assertion has to be justified:arguments must include reasons Neil Mercer

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