1. 1 Fraction Actions: Working with Fractions as Operators John Mason Calgary Oct 2014 The Open University Maths Dept University of Oxford Dept of Education Promoting Mathematical Thinking

2. 2 What Does it Mean? 3 divided by 5 Divide 3 by 5 The answer on dividing 3 by 5 The action of ‘three fifth-ing’ The result of ‘three fifth-ing’ of 1 on the numberline The value of the ratio of 3 to 5 The equivalence class of all fractions with value three fifth’s (a number) Place on the number line (number) …

3. 3 Different Perspectives  What is the relation between the numbers of squares of the two colours?  Difference of 2, one is 2 more: additive thinking  Ratio of 3 to 5; one is five thirds the other etc.: multiplicative thinking

4. 4 Raise your hand when you can see  Something which is 2/5 of something  Something which is 3/5 of something  Something which is 2/3 of something How did your attention shift? Flexibility in choice of unit  Something which is 1/3 of 3/5 of something  Something which is 3/5 of 1/3 of something  Something which is 2/5 of 5/2 of something  Something which is 1 ÷ 2/5 of something  Something which is 5/2 of something  Something which is 5/3 of something  Something which is 3/2 of something What others can you see?

5. 5 Doing & Undoing  What operation undoes ‘adding 3’?  What operation undoes ‘subtracting 4’?  What operation undoes ‘subtracting from 7’?  What are the analogues for multiplication?  What undoes ‘multiplying by 3’?  What undoes ‘dividing by 4’?  What undoes ‘multiplying by ¾ ’?  Two different expressions!  What operation undoes ‘dividing into 24’?

6. 6 SWYS Find things that are,,,,, of something Find something that is of of something What is the same, and what is different?

7. 7 Presenting Fractions as Actions

8. 8 Raise your hand when you can see … Something that is 1/4 – 1/5 of something else What did you have to do with your attention? What do you do with your attention in order to generalise? Did you look for something that is 1/4 of something else and for something that is 1/5 of the same thing? Commo n Measur e

9. 9 Stepping Stones … … R R+1 What needs to change so as to ‘see’ that

10. 10 Elastic Multiplication

11. 11 Two Journeys  Which journey over the same distance at two different speeds takes longer: –One in which both halves of the distance are done at the specified speeds? –One in which both halves of the time taken are done at the specified speeds? distance time

12. 12 Frameworks Doing – Talking – Recording (DTR) Enactive – Iconic – Symbolic (EIS) See – Experience – Master (SEM) (MGA) Specialise … in order to locate structural relationships … then re-Generalise for yourself What do I know? What do I want? Stuck?

13. 13 Mathematical Thinking  How describe the mathematical thinking you have done so far today?  How could you incorporate that into students’ learning?

14. 14 Possibilities for Action  Trying small things and making small progress; telling colleagues  Pedagogic strategies used today  Provoking mathematical thinking as happened today  Question & Prompts for Mathematical Thinking (ATM)  Group work and Individual work

15. 15 Human Psyche Imagery Awareness (cognition) Will Body (enaction) Emotions (affect) Habits Practices

16. 16 Three Only’s Language Patterns & prior Skills Imagery/Sense- of/Awareness; Connections Different Contexts in which likely to arise; dispositions Techniques & Incantations Root Questions predispositions Standard Confusions & Obstacles Only Behaviour is Trainable Only Emotion is Harnessable Only Awareness is Educable Behaviour Emotion Awareness

17. 17 Follow Up  j.h.mason @ open.ac.uk  mcs.open.ac.uk/jhm3  Presentations  Questions & Prompts (ATM)  Key ideas in Mathematics (OUP)  Learning & Doing Mathematics (Tarquin)  Thinking Mathematically (Pearson)  Developing Thinking in Algebra (Sage)