The importance of Puzzles and Problem-Solving

1. The importance of Puzzles and Problem-Solving Kate Cowling Uplands College

2. Maths of the future • Cambridge are currently starting a project which embraces change in Maths Education – the course will be conceptually taught with applications in the real world. It will embrace technology as a tool to enhance conceptual understanding. • Conrad Wolfram

3. Kids vs Computers Computers – fast, accurate and stupid (but very cool to learn from!) Kids – slow, inaccurate but brilliant and creative!

5. 4.10 Different aspects of achievement … East Asian students overall outperform their Western counterparts in mathematics - …superior performance is more evident in certain aspects of mathematics achievement, such as using basic skills of computation and algebraic manipulation, solving routine problems and … applying algorithms. …Western students performed as well, or sometimes better … using visual and graphical representations and solving open-ended problems.

6. Outcomes from research into teaching with Technology • Kids are more prepared to: take risks work with realistic numbers “ There is a substantial amount of good evidence on calculators in schools, mainly from the US, and none of it shows their use is detrimental to pupils’ learning.” Anne Watson, Emeritus Professor of Mathematics Education at Oxford University

7. My ideal primary school “desk”

8. Adapt or Die!! We need people who are able to function well in situations for which they were not specifically trained.” Seymour Papert (MIT 1978) e.g. Civil Engineer build roads Then 4 days ago you read the paper and this is what you see?

10. Replacing existing methods with problem solving • Problem- solving isn’t just a Friday last lesson activity. What I would love to see in all classrooms is problem-centred teaching. • Objective is to engage the students and get them thinking like Mathematicians but still learn all the “language” of Maths at the same time.

11. Mathematical “Habits of Mind” (Cuoco; Goldenberg & Mark 1996)

12. Problem Solving - Getting it right Bosse Larson Bored Anxiety Challenge Good Skill

13. Conceptual Change Origins in Science Where we challenge existing concepts - Some examples: Fractions can be greater than 1 Discreteness of numbers Trigonometry – definition of ratios where angle greater than 900 The need for a malleable mind when concepts can be challenged, adjusted and expanded (or sometimes rejected!)

14. The journey of a learner and ¾ What is ¾ ? Well cut up a pizza into 4 equal parts and eat 1 part. What is left? Or I have 3 pizzas and I share it evenly between 4 people so they each get ¾ of a pizza

15. ¾ of a packet of 12 sweets How do we calculate this? Or

16. ¾ continued fraction divided by a fraction? ? ? 8 (NB: Equivalent fractions not just - “Tip and Times”!!!!)

17. ¾ continued Or When tossing two coins the probability of getting at least 1 tail is

18. Don’t rush When you see just some of the different representations of ¾ you can understand why the kids struggle with fractions. Don’t rush – spend time with all the representations exploring each –understanding how, when and why you use ¾

19. Bonds of 10 – Speed Tests • Anxiety ≠ fun • Slow processing ≠ less intelligent • Fluency ≠ understanding

20. Alternative problem Using the numbers 1 to 5 put a number in each square so that the sum of all the numbers in each row is 10. Questions to probe understanding: If the sum in each row no longer has to be 10 1. Can the number in the overlap square be odd? Or even? Is your solution unique? Does the sum change depending on what number is in the overlap square?

21. Extension Using the numbers 1 to 9 put a number in each square so that the sum of all the numbers in each row is the same. Using the numbers 1 to 9 put a number in each square so that the sum of all the numbers in each row is the same. (does it matter which number is left out?) Your turn!

22. Bonds of 300 You have 30 seconds to complete this. 197 + … 55 + … 284 + … 65 + … 279 + … 151 + … 242 + … 167 + … 245 + … 138 + … 129 + … 299 + … 65 + … 217 + … 281 + … 7 + … 144 + … 95 + … 212 + … 180 + … 384 + … 287 + … 65 + … 527 + …

23. Alternative problem Find a route that starts at Numbi gate and visits at least 2 camp sites and returns to Numbi which is at most 300 km long.

24. Don’t be afraid to extend their knowledge in problems e.g Area Estimate the area of the sea Covered by the oil slick in the Gulf of Mexico – May 2010. By refining their initial estimate from a rectangle to a series of smaller shapes where the area is still easy to calculate they are actually heading towards integration.

25. Gulf Disaster

26. Blue Fin Tuna? Weight of Blue Fin Tuna – 250 kg Cost of gold per ounce $1292.4 1 kg = 35,27 oz Is Tuna more expensive than gold?

27. Solution Tuna $ 1 760 000 for 250 kg Which is $ 7040 per kg Gold $ 1292 per oz Which is $ 45574 per kg So gold is about 6 ½ more expensive than Blue Fin Tuna. I am afraid Rhino horn is fetching more than double the price of gold!!!

28. Puzzles ( I haven’t forgotten!!) • This is pure fun – play lessons • Great for the bright kids when they are “finished” • Excellent for developing strategy and testing conjectures • All about those “habits or mind” for Mathematicians.