Enhancing Problem-Solving in KS2 Teaching: Strategies and Techniques

1. Embedding problem solving in teaching and learning atKS2 Dr Ray Huntley Brunel University, London September 2, 2013

2. How to be good • Most learners make good progress because of the good teaching they receive • Behaviour overall is good and learners are well motivated • They work in a safe, secure and friendly environment • Teaching is based on secure subject knowledge with a well-structured range of stimulating tasks that engage the learners • The work is well matched to the full range of learners’ needs, so that most are suitably challenged • Teaching methods are effectively related to the lesson objectives and the needs of the learners

3. Assessment for Learning • Ensure that every learner succeeds: set high expectations • Build on what learners already know: structure and pace teaching so that they can understand what is to be learned, how and why • Make learning of subjects and the curriculum real and vivid • Make learning enjoyable and challenging: stimulate learning through matching teaching techniques and strategies to a range of learning needs • Develop learning skills, thinking skills and personal qualities across the curriculum, inside and outside of the classroom • Use assessment for learning to make individuals partners in their learning

4. Personalisation • Teaching is focused and structured • Teaching concentrates on the misconceptions, gaps or weaknesses that learners have had with earlier work • Lessons or sessions are designed around a structure emphasising what needs to be learnt • Learners are motivated with pace, dialogue and stimulating activities • Learners’ progress is assessed regularly (various methods) • Teachers have high expectations • Teachers create a settled and purposeful atmosphere for learning

5. Main part of a lesson • Introduce a new topic, consolidate previous work or develop it • Develop vocabulary, use correct notation and terms and learn new ones • Use and apply concepts and skills • Assess and review pupils’ progress • This part of the lesson is more effective if you… • Make clear to the class what they will learn • Make links to previous lessons, or to work in other subjects • Give pupils deadlines for completing activities, tasks or exercises • Maintain pace, making sure that this part of the lesson does not over-run and that there is enough time for the plenary • When you are teaching the whole class, it helps if you: • Demonstrate and explain using a board, flipchart, computer or OHP • Highlight the meaning of any new vocabulary, notation or terms, and encourage pupils to repeat these and use them in their discussions and written work • Involve pupils interactively through carefully planned and challenging questioning • Ask pupils to offer their methods and solutions to the whole class for discussion • Identify and correct any misunderstandings or forgotten ideas, using mistakes as positive teaching points • Ensure that pupils with particular needs are supported effectively • When pupils are working on tasks in pairs, groups or individuals it helps if you… • Keep the whole class busy working actively on problems, exercises or activities related to the theme of the lesson • Encourage discussion and cooperation between pupils • Where you want to differentiate, manage this by providing work at no more than three or four levels of difficulty across the class • Target a small number of pairs, groups or individuals for particular questioning and support, rather than monitoring them all • Make sure that pupils working independently know where to find resources, what to do before asking for help and what to do if they finish early • Brief any supporting adults about their role, making sure that they have plenty to do with the pupils they are assisting

6. Fishy Problem • A fish has a head that is 9cm long. • Its tail is the same length as its head plus half its body. • Its body is the same length as its head and tail together. • How long is the fish?

7. A Fishy Solution Head = 9cm Tail = Head + ½ Body, so ½ Body = Tail – 9 Body = Head + Tail, so Body = Tail + 9 ½ Body = Tail – 9, so Body = 2 Tails – 18 So Tail + 9 = 2 Tails – 18, so Tail = 27 And Body = 36, so fish is 72cm long

8. Equal Sets • Take a set of digits, say 1 to 8. Can you split them into 2 sets with the same total? • What about other sets of digits, say 1 to 5? 1 to 7? 1 to 20? • How can you decide whether it can be done?

9. Equal Sets Solution Need total of the set to be an even number. Each set totals to a triangle number. So it can be done for any set that totals to an even triangle number. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, … These are the 3rd, 4th, 7th, 8th, 11th, 12th, etc.. So it can be done if the number in the set is a multiple of 4, or 1 less than a multiple of 4.

10. Quadrilaterals • Start with a circle marked with 8 points evenly spaced on the perimeter. • What different quadrilaterals can you draw by joining 4 points? • How many are there? How do you know you have them all? • Can you sort the quadrilaterals by some criteria?

11. Quads solution There are 8 different shapes that can be made. Denoting each shape by the number of spaces between the points round the circle, they are: 1,1,1,5 - trapezium 1,1,2,4 – trapezium (different one) 1,1,3,3 - kite 1,2,1,4 – another trapezium 1,2,2,3 - irregular 1,2,3,2 – another trapezium 2,2,2,2 - square 1,3,3,1 - rectangle

12. Houses • There are some houses arranged around a square green. • There is the same number of houses on each side of the square. • Number 9 is opposite number 47. • How many houses are there around the square?

13. Houses solution • With 9 houses in each row, 9 is opposite 19 • With 10 in each row, 9 is opposite 22 • With 11 in each row, 9 is opposite 25 • Continuing this pattern, • With 18 in a row, 9 is opposite 46 • With 19 in a row, 9 is opposite 49 • So 9 can never be opposite 47!! (Sorry!)

14. Thank you! Please try these activities with your children and your teaching colleagues! Any feedback is always welcome! rayhuntley61@gmail.com

15. Coordinate shapes • Start with 2 points on a rectangular grid, marked by coordinates, say (2,1) and (4,3). • Can you find 2 more points to make a square (rhombus, trapezium, etc) ? • Can you do it in different ways? How many can you find?

16. Halftimes • If a sporting match (football, hockey) has a final score of 3-1, what are the possible half time scores? • If the fulltime score is a-b, what is the connection between a and b and the number of halftime scores possible?

17. Fractions • Can you find a fraction between ½ and ¾? • Can you find a fraction between any two fractions? • Can you devise a rule that will always do this? • How can you show why it works (not algebraic proof!)

18. 9s to make 1000 • Use a dice to generate 9 digits, each in the range 1-6. • Arrange them into three 3-digit numbers. • Add them. • Largest/Smallest/Closest to 1000 wins!

19. Magic square patterns • Draw a 3x3 magic square, where each row, column and diagonal adds to 15 using 1-9. • Find pairs of numbers in the square that have the same totals. Record this on a blank square colouring the positions of one pair in one colour and the other pair in another. • How many different coloured relationships can you find?

20. 5 presents • There are 5 presents, labeled A, B, C, D, E. • A and B together cost £6 • B and C cost £10 • C and D cost £7 • D and E cost £9. • All 5 presents together cost £21. • How much is each present?

21. Digit Sums • Without 0, write down as many 2-digit numbers as you can where the digits add to 6. • Now do the same for 3-digit numbers. • How many 4-digit numbers do you think there might be? Try it. • Now 5-digits, and finally 6-digits.

22. 15 cards • From a set of cards numbered 1 to 15, put down 7 cards in a row, face down. • Cards 1&2 add to 15, 2&3 add to 20, 3&4 add to 23, 4&5 add to 16, 5&6 add to 18 and cards 6&7 add to 21. • From this information can you work out what numbers are on the cards? • How many solutions are there?

23. Numbers of triangles • Take an integer number for the perimeter of a triangle, say 12. • What integer sides are possible? • Find all permutations. • How many possible permutations are there? • Is it always a triangle number?

24. IRATs • Start with an isosceles right-angled triangle (IRAT). Fold along the line of symmetry. • Open and cut along the fold line, what do you get? (Predict first!) • Now start with a new IRAT, fold along the line of symmetry and then again. • Open just the last fold and cut along the fold line. What do you get? (Predict first!)

25. IRATs • Now do 3 folds along lines of symmetry and cut along the 2nd fold line… what do you get? • What about 4 folds and cut along the 3rd fold line? • How does the pattern continue?