Same Surface Different Deep

1. Same Surface Different Deep

2. Planning schemes of learning • Some difficulties include: • Covering enough content for students to pass the exams whilst leaving sufficient time for revision • Balancing the need to get through content vs giving sufficient time for students to apply their skills to solve problems • Some topics might be too difficult for your students • How to order the topics and taking breaks in the year into account

3. Teaching problem-solving • Plan a strategy • What is the question asking? • What do I need to know? • How can I get that information? • Put the plan into action • Show all stages of working out • Check the answer makes sense in the context of the question

4. Teaching problem-solving • Plan a strategy • Possible strategies include… • Set out cases systematically • Work backwards from a value given in the problem • Find one or more examples that fit a condition for the answer • Look for and represent relationships between elements of the situation • Find features of the situation that can be acted on mathematically (AQA)

5. Teaching problem-solving • More strategies… • Guess and check • Look for a pattern • Make an orderly list • Draw a picture • Eliminate possibilities • Solve a simpler problem • Use symmetry • Use a model • Work backwards • Use a formula • Solve an equation • Be ingenious (!!!) • (Polya, How to Solve It, 1945)

6. Find one or more examples that fit a condition for the answer • Eliminate possibilities

7. Guess and check • Solve an equation • Work backwards from a value given in the problem • Find features of the situation that can be acted on mathematically

8. Set out cases systematically • Make an orderly list • Use symmetry

9. Look for and represent relationships between elements of the situation • Look for a pattern

10. Teaching problem-solving • Put the plan into action • Show all stages of working out • Check the answer makes sense in the context of the question • Good problem-solvers can reflect on their progress. • Ask these monitoring/planning questions: • What exactly are you doing? (Can you describe it precisely?) • Why are you doing it? (How does it fit into the solution?) • How does it help you? (What will you do with the outcome when you obtain it?)

11. Teaching problem-solving is hard • Students often don’t have the strong subject-specific knowledge required • There is no generic strategy for solving problems that we can teach • Students struggle to reflect on the effectiveness of their approaches

12. Is there a place for block practice? • This type of practice exposes students to lots of subject-specific examples and is therefore necessary for problem-solving but on its own is probably an ineffective revision technique.

13. Pitfalls to avoid • Teach a particular topic and then give them questions on that topic involving different contexts. • Students know what topic they are doing so they just look at the question to see how to apply the skills they have just been taught. • This approach robs our students of the opportunity to recognise the deep structure of a problem on a particular topic. • We inadvertently teach students to overlook the cues in a given problem that would tell them what the deep structure is. • Students often say I didn’t read the question fully

14. SSDD Problems https://ssddproblems.com/ • Gives students the chance to revisit topics they might not have seen for a while (spaced practice) • Creates natural interleaving (mixed practice) • Force students to consider each problem’s deep structure instead of working on autopilot • Gives students an opportunity to take charge of their own learning

15. SSDD Examples • Trapeziums 3

16. SSDD Examples • Trapeziums 3

17. SSDD Examples • The radius is 4

18. SSDD Examples • The radius is 4

19. How we could use SSDD • Starter – to see what students already know about particular topics and to identify “high fliers” and “low achievers” for the beginning of the lesson • Plenary – similar to using as a starter but will help identify the amount of progress students have made in a particular lesson • Break in a lesson • Part of lesson structure from the start of the year (group work) • Homework (useful since the questions are differentiated and there are worked solutions so students could peer mark) • Revision

20. Developing the “perfect” scheme of learning Things to consider: • Building in revision naturally into lessons (using things like Corbett Maths 5-a-day and SSDD problems) – can be used to revise areas of weakness for your cohort, high frequency topics or topics with common misconceptions • Don’t try to teach too much, leave enough time for past paper practice and revision at the end of the course • Topics which build naturally from other topics can be separated to create an opportunity to revise previous topics (eg after teaching Pythagoras leave a gap in your scheme of learning before teaching Trigonometry)

21. Optimal interval for revisiting topics

22. Developing the “perfect” scheme of learning • You have been given a cut out copy of Edexcel’s order of learning • In groups I’d like you to try and come up with your own order of learning bearing in mind what we have discussed today

23. Bringing our ideas together • What topics did we feel should be taught early on in the year? • What topics have strong links with one another and hence are good candidates to split up in our order of learning? • Are there any topics you feel are worth splitting rather than teaching them in one go? (for example Edexcel’s scheme of learning has split up mixed fractions from the rest of the work on fractions) • What topics are potentially too difficult for our GCSE Foundation students?

24. What topics should be taught early in the year? • Rounding • Written calculation methods • Order of operations • Negative numbers • Basic algebra (simplify, expand, factorise, substitution) • Solving equations • Fractions • Percentages • Ratios • Powers, cubes and roots • Factors, multiples and primes

25. Potential interleaved topics • Solving equations and basic algebra; inequalities; area, perimeter, volume and angles (algebraic context) • Substitution; sequences; volume and surface area of spheres and cones • Fractions; probability; pie charts; tree diagrams • Ratio; speed, distance and time; probability • Averages and range; bar charts; stem and leaf diagrams • Indices, powers and roots; simplifying algebra using index laws; standard form • Factors, multiples and primes; Venn diagrams and set notation • Constructions; loci • Area and perimeter; Pythagoras; Trigonometry • Enlargements; similarity

26. What topics could be split up? • Fractions(mixed fractions and reverse problems) • Percentages (repeated percentage change and reverse problems) • Ratios (reverse problems and double ratio questions) • Area, perimeter and volume (algebraic problems; spheres and cones) • Angles (algebraic problems)

27. What topics are potentially too difficult? • Rearranging equations • Simultaneous equations • Quadratic equations – factorising • Loci (is constructions worth teaching if loci is potentially too difficult?) • Trigonometry • Straight line graphs (using y = mx + c) • Angles in polygons • Algebraic proof

28. Our next event • Revision conference on key topics (carousel style) • Participating schools / colleges will bring along a group of students to sit in revision style workshops on a particular topic and will rotate until they have covered each topic • Hoping to do this in Brunel University • Date to be confirmed (October 2018) • If you’re interested in attending the conference or running one of the workshops please leave your details with me or send me an email: sde@woking.ac.uk

29. Evaluation • Please complete the evaluation before leaving. • Thank you for coming.