University of West Indies Problem solving workshop Dr
University of West Indies Problem solving workshop Dr Geoff Tennant g. d. tennant@reading. ac. uk Personal website: www. geofftennant. name © University of Reading 2007 www. reading. ac. uk
Approximate plan… What is a ‘problem’? Three contexts for problem-solving: - Olympiad format; - In class collaboratively; - ‘Real life’ problems. Support from ICT resources. I hope you’ve come prepared to work hard! 2
Opening example (1) At 0900 the hands on a traditional clock face are at 90° to each other. At what time exactly will the hands of the clock face again be at 90°? 3
Opening example (2) How did this make you feel? What did you do to start getting going with this problem? - Look at your own watch; - Change the hands on your watch; - Draw diagrams; - Write down equations; - Anything else? Crucial point: need to have the confidence to start when can’t see an end point in sight. 4
Opening example (3) My solution: Take t to be the time in minutes after 0900, and measure angles from 1200 clockwise (ie. using conventions from bearings). So at time t: the minute hand is at an angle of 6 t the hour hand is at an angle of 270 + t/2 So need 270 + t/2 -6 t = 90 5
Opening example (5) So need 270 + t/2 -6 t = 90 180 = 11 t/2 t = 360 / 11 = 32 8/11 ie. about approximately 0933. 6
Opening example (6) Brief discussion points: - Is this a problem we might want to do with children? - What might be the learning points arising, even if children don’t come to a full solution? - What might be the pitfalls of such an approach? - What would children already need to know before attempting this problem? 7
Opening example (7) Please note: • Setting out a solution like this implies that this is the ‘correct’ way to do this problem; • May be other approaches which may or may not be equivalent; • As teachers, need to ensure a ‘half full’ rather than ‘half empty’ approach. 8
Definition of a ‘problem’ A ‘problem’ is a task leading to a solution or set of solutions. These solutions are not immediately clear at the starting point, and these problems cannot be solved simply by the routine application of learnt routines. Problem-solving: - May be done individually (sometimes under examination conditions) or collaboratively; - May or may not be leading to one specific answer; - May or may not relate directly to curriculum material; - May or may not lead to some sense of ‘proof’; - May or may not be in some sense ‘real life’. 9
Why engage with problem-solving (1)? Part of the curriculum, eg. from MOEC (1998: 84): The mathematics curriculum developed for grades 7 - 9 has as its goals: 1. The development of the problem-solving approach to learning mathematics and the willingness to accept the challenges of new situations 2. The development of skills of creativity, enquiry, conjecturing , testing and generalizing. Ministry of Education and Culture (1998). Curriculum guide grades 7 -9 for career education, mathematics, language, arts, science and social studies. Kingston, Jamaica: MOEC. 10
Why engage with problem-solving (2)? - Fun, interesting, motivating (we hope!); - (If working in groups) facilitates social skills; - Enables pupils to learn to apply the mathematics they learn; - Develops transferable skills eg. to personal problem solving. 11
Why engage with problem-solving (3)? Warning! - May not look very good on paper; - Can reasonably expect children to use mathematics well below what they could be using in ‘closed’ contexts; - Is likely to be found difficult initially if pupils are used to working in a closed-ended manner; - Problems (eg. the 90 ° clock face problem) cease to be problems if we learn routines to solve them! - Need to think carefully about forms of assessment – can undermine the approach. 12
Olympiad type questions (1) For Jamaican Olympiad see: http: //myspot. mona. uwi. edu/mathematics/olympiad-resource-centre Sample question from the grade 9, 10, 11 2011 paper: 8) A certain number has five digits and their product is 100. Which of the following is possibly their sum? (a) 10 (b) 14 (c) 16 (d) 20 (e) 100 Previously seen: the probability that the next person you see will have an above average number of arms is: (a) 0 (b) nearly 0 c) ½ (d) nearly 1 (e) 1 See also United Kingdom Mathematics Trust http: //www. mathcomp. leeds. ac. uk/ 13
Olympiad type questions (2) What might we do as teachers to help children engage with these kinds of problems? - Practise on old papers: but beware, ‘problems’ are no longer ‘problems’ if we know a procedure for their solution! - Modelling of open ended approaches; - Giving pupils space to work things out for themselves, make mistakes; - Read question carefully; - Start make observations about the problem even if can’t see the end in sight. Try typing into Google: ‘problem solving mathematics olympiad’ See also United Kingdom Mathematics Trust http: //www. mathcomp. leeds. ac. uk/ 14
Problem solving in teams: the Zin Take one envelope per group. Each envelope contains 22 small pieces of paper. Share out the pieces of paper as equally as possible amongst the group. You may read out your card to the rest of your team BUT you may not show you cards to other members of the team. All the information you need to find out what the problem is and to be able to solve it are on the pieces of paper! Good luck! 15
The Zin: a ‘clean’ solution If there are 8 ponks in a schlib, then there are 16/2 = 2 schlib rest periods per day. So each worker works for 9 – 2 = 7 schlibs per day. There is one gang with 9 members, 8 of whom work, all of whom are making 150 blocks in a schlib. So the gang makes 8 x 7 x 150 = 8400 blocks per day. A Zin contains 100 x 50 x 10 = 50 000 blocks. So it takes 50000 / 8 400 = 5. 95 days to finish, therefore finish on the 6 th working day. Work starts on Aquaday, with 4 working days a week, so finishing day is second Neptimus. 16
The Zin: reflections Please note: the original activity includes several bits of irrelevant information, eg. “Green has special religious significance on Mermaidday. ” Did you like this activity? Why (not)? Would it work well with children? Why (not)? Could it be used with all attainments, possibly in carefully chosen groups? What might be problems in using this? How might these problems be overcome? 17
Problems directly relating to curriculum work (1) Choose from Crack It, Billiards, Jugs and Let’s Get Organised. Can you identify the key underlying mathematical concept in each of these problems? Key questions: -Might you want to use these problems with youngsters? Why (not)? -When would you be using these problems relative to introducing highest common factor formally?
Problems directly relating to curriculum work (2) The sheep pen problem A farmer has 100 m of fencing. What is the biggest area that can be made using this fencing? Investigate!
Problems leading to conjecture and proof Consider ‘simultaneously simultaneous’ and other problems on this sheet. Suggested approach: - Explore the problems, find some examples; - See if conjecture arise “Oh, so the answer seem to be”; - Can you express your conjectures algebraically? - Can you now justify your answers in the context of the original problem? Reflections: Are these problems you would want to use with children? Why (not)? How much time would you be wanting to give to these problems?
‘Real life’ problems: some examples • How far can you travel in 24 hours? • How could you fill a given space with a car park? • How would you go about spending a school council budget? • How much does it cost to be healthy? • In a class, decide somewhere you would like to go and work out what you would need to do to get there • How much does a baby cost? • Design your ideal bedroom. If doing these with children, firstly consider how you would address these problems yourself. Particularly, if pointing youngsters to getting information from the Internet, check you can do it first!
Support from ICT (1) Internet can be repository for problem solving idea, or give a platform for problem solving. Start at National Library of Virtual Manipulatives: http: //nlvm. usu. edu/ Starting points: Number and operations -> 9 -12 -> Circle 21 Algebra -> 9 -12 -> Coin Problem Algebra -> 9 -12 -> Peg Puzzle (also see http: //nrich. maths. org/1246 ) Algebra -> 9 -12 -> Towers of Hanoi (also see http: //www. cut-the-knot. org/recurrence/hanoi. shtml) 22
Support from ICT (2) Use of ICT may give immediate data and also error correction (eg. in Frogs problem). Need to consider how to tie up the problem in much the same kind of way. General point: there is nothing magical about ICT, underlying considerations about pupil engagement etc. all stay the same. 23
Closing reflections (from me) - Need to have safe place to make mistakes and explore ideas; - Need to develop confidence to start a problem when can’t see where it’s heading; - Need to experience success with challenging yet achievable problems; - When working individually, can be good to come away from the problem and start again; - Need to have a clear sense as to exactly why you are doing what you are doing and what the learning outcomes are intended to be; - If it’s difficult initially stick with it!
Closing reflections (from you) Discuss with the person next to you: - What can you take away from today for the rest of this week? - The rest of the year? - For next year? Thank you!
So here’s the problem! Think of a three digit number, with the first and last digits different. So 123 would be fine but 121 isn’t Reverse the digits – so 123 becomes 321 Subtract the smaller from the larger: in this case 321 – 123 If your number is less than 100 then put in 0 s to make it three digits – so 75 becomes 075 Reverse the digits – so 075 becomes 570 Add the two last numbers together: in this case 075 + 570 Now can my volunteer open the envelope and read out the contents!
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