Greg Hargreaves

1. Greg Hargreaves

2. Teachers have a difficult role to perform in schools and we all need support in the execution of our job

3. Part of the role of a teacher is instruction – the teaching of content knowledge

4. Schools also have an obligation to prepare students for State testing

5. We must also look for ways of providing students with opportunities for success

6. We want to be able to build trust with our classes. To be able to build classrooms that reflect respect and the exchange of ideas.

7. We want students to be comfortable to explore, to think and be willing to share in a safe environment. We don’t want to produce robots.

8. Oh, how we long to run with a wilder herd. Alas we cannot.

9. We want students to explore their world and relate to what they are learning - not just learn unrelated facts. Solving problems & looking for creative ways of doing things are all part of good teaching practice. Results from the State testing would suggest that the open-ended questions are poorly done. Students demonstrate difficulty or an unwillingness to solve problems.

11. The Leverhulme Numeracy Research Project (1997-2000) has distilled a list of key characteristics of effective teaching for numeracy. (What we should be doing in the classroom. These teachers were those who: • challenge children to think mathematically • expose and relate to children’s existing knowledge • develop significant mathematics eg strategies and generalizations • develop connections between mathematical ideas and between mathematics and the real world

12. • stimulate children’s interest, curiosity, excitement and sustain engagement • permit access to mathematics and tasks for all pupils • • use symbols, diagrams and apparatus, not for window dressing or for objects in themselves but to communicate, represent and or provide good models of thinking • • involve a range of models of expression • • encourage the development of more sophisticated strategies • • focus on the mathematics not just on work or getting answers

13. The middle years numeracy research project Siemon (2001) has investigated effective teaching in the middle years and concluded that effective classroom teachingstrategies in the middle years would include: • • regular and systematic use of open ended questions, games, authentic problems and extended investigations to enhance students’ mathematical understanding and capacity to apply what they know • teaching strategies focused on connections and strategies for making connections

14. • students actively engaged in conversations and texts that encourage them to reflect on their learning and justify and explain their thinking. • learning activities designed or chosen appropriate to learners’ needs and interests and use a balance of teacher-directed and student-directed approaches • • opportunities provided for meaningful and enjoyable practice of essential knowledge and skills

15. Life is full of problems that we solve everyday! We must teach students survival skills in Math. Teach them ways of solving problems and teach them thinking strategies. These are skills that are transferable and will benefit them in their lives – set them up for life-long learning.

16. 100 Problem • Separate 100 into FOUR numbers, so that by • Adding 4 to the first • Subtracting 4 from the second • Multiplying the third by 4and • Dividing the fourth by 4 • ALL THE RESULTS WILL BE THE SAME

17. Not all problems have to be difficult or solved alone

18. What is a Problem? • A problem is a question that motivates you to search for a solution. • This implies first that you want or need to solve the problem and second that you have to search for a way to find a solution. • Whether a question is a problemor an exercise depends on the prior knowledge of the problem solver.

19. The main reason for learning all about math is to become better problem solvers in all aspects of life. Many problems are multi step and require some type of systematic approach. Most of all, there are a couple of things you need to do when solving problems:

20. Ask yourself exactly what type of information is being asked for. • Then determine all the information that is being given to you in the question i.e. recognize the important information • When you clearly understand the answers to those two questions, you are then ready to devise your plan

21. Understand the problem • Recognize important information • Select a strategy • Solve and check

22. Problem Solving Strategies MAKE SURE YOU UNDERSTAND THE PROBLEM.This may seem obvious but it is easy to jump straight into solving a problem before you really understand it.So, sure, have a bit of a play around with it at first, if you like, but then read the problem carefully two or three times if necessary. MAKING MISTAKES!!It's true. Good problem solvers make plenty of mistakes.You have heard the expression: learn from your mistakes.Well this statement is true. Try things out, make mistakes, then try some other way of attacking the problem.

23. Problem solvers must be organized and willing to make mistakes and learn from their mistakes.

24. KEEP A RECORD.If you do not keep a record of what you have done (that is all your rough working & notes), you might end up repeating some of your earlier work without realizing it.This is particularly true if you are going to leave your problem for a while before coming back to it. Make sure you write down exactly where you are up to, so it will be easy to get back into at that later date. MAKE A LIST, then LOOK FOR A PATTERN.Often in mathematical problems, there are patterns to be looked for that will help in their solution. (Consider the lifetime work of Leonardo Fibonacci)

25. As teachers, we must be flexible. There are often many ways to solve a problem. As teachers we must be willing to consider anything the students comes up with – there are often little gems waiting for us.

26. START WITH THE EASIER PARTS OF THE PROBLEM or MAKE THE PROBLEM SIMPLER. Understanding a simple version of a problem often is the first step to understanding a whole lot more of it. So don't be scared of looking at a simple version of a problem, then gradually extending your investigation to the more complicated parts.

27. Is the exact answer important? Can you estimate the answer? This is such an important skill. How often do we see students come up with answers that are obviously wrong – but that is what the calculator tells them!

28. The answer is 72.443

29. CHECK YOUR ANSWERS. You might think you've got it but you'd better check, just in case.This is the vital and final (hopefully) part of solving a problem.And if it is not the final part, it is just as well you checked - isn't it?So always check your answers.

30. Do your answers make sense? After all, anything is possible: If a = b Then a2 = ab So a2 – b2 = ab – b2 And (a+b)(a-b) = b(a-b) Eliminating (a-b) from both sides leaves a+b = b But a =b So 2b = b Dividing by ‘b’ leaves us with 2 = 1

31. Now it’s time for you to perform

32. Find the numerical values of the letters A, B, C, D & E if the following is true: A B C D E x4 E D C B A

35. Aunt Alice and the Silver Dollars Aunt Alice gave each of her three nieces a number of silver dollars equal to their ages. The youngest felt that this was unfair so they agreed to redistribute the money as follows: The youngest would split half of her silver dollars evenly with the other two sisters. The middle sister would then give each of the others four silver dollars. Finally the oldest sister splits half of her silver dollars evenly between the two younger sisters. After exchanging the coins, each girl had sixteen silver dollars. How old are the sisters?

36. Aunt Alice Solution: Let x = initial number of dollars for the oldest sister Let y = initial number of dollars for the middle sister Let z = initial the number of dollars for the youngest sister x + y + z = 48

37. Aunt Alice Solution: Let x = initial number of dollars for the oldest sister Let y = initial number of dollars for the middle sister Let z = initial the number of dollars for the youngest sister x + y + z = 48 Youngest sister Middle sister Oldest sister Youngest splits half her dollars z/2 y + z/4 x + z/4 Middle sister gives each sister 4 dollars z/2 + 4 y + z/4 -8 x + z/4 + 4 Oldest sister splits half her dollars with her sisters ¼(x + z/4 + 4) + z/2 + 4 ¼(x + z/4 + 4) + y + z/4 - 8 ½ (x + z/4 + 4) Number of coins each has at finish 16 16 16 a) So ½ (x + z/4 + 4) = 16 and (x + z/4 + 4) = 32 b) ¼(x + z/4 + 4) + z/2 + 4 = 16 ¼ (32) + z/2 + 4 = 16 8 + z/2 + 4 = 16 this means that z/2 = 4 and so z = 8 c) ¼(x + z/4 + 4) + y + z/4 – 8 = 16 ¼(32) + y + 8/4 – 8 = 16 8 + y + 2 – 8 = 16 this means that y + 2 = 16 and so y = 14 d) ½ (x + z/4 + 4) = 16 ½ (x + 8/4 + 4) = 16 ½ (x + 6) = 16 this means that x + 6 = 32 so x = 26 • ½ (x + z/4 + 4) = 16 • ½ (x + 8/4 + 4) = 16 • ½ (x + 6) = 16 • x + 6 = 32 so x = 26

38. At the end: 16 16 16 gave half her coins away 8 8 32 gave 4 to each sister 4 16 28 gave half evenly to each sister 8 14 26

39. Sliced Watermelon A 100 pound watermelon is 99% water. After being sliced and left uncovered it is 98% water. What weight of water has been lost?

40. Watermelon problem solution: • Watermelon problem solution: • 100lb watermelon contains 99lb of water and hence 1lb of flesh • After cutting and drying it is 98% water and hence 2% flesh • Now, we know that only the water has gone and the amount of flesh at the start and the finish is the same • Therefore 1lb is equivalent to 2% of the final weight. • This means the final weight is 50lb (1lb is 2%, how much is 100%) • So the weight loss is 100lb – 50lb = 50lb!

41. Weight of watermelon Weight of water Weight of flesh % water Initially 100 99 1 99 . 100 = 99% 100 After drying 100 - x 99 - x 1 (99 – x) . 100 = 98% (100 – x) OR Let x equal the amount of water lost: (99 – x) . 100 = 98 (100 – x) 9900 – 100x = 9800 –98x 100 = 2x and so x = 50lb equals the amount of water lost

42. SIX PACK On each line, using only the three numbers provided and any mathematical operations or symbols you choose make the total equal six. 1 1 1 = 6 2 2 2 = 6 3 3 3 = 6 4 4 4 = 6 5 5 5 = 6 6 6 6 = 6 7 7 7 = 6 8 8 8 = 6 9 9 9 = 6

43. Math problem solving websites: http://www.rhlschool.com/math.htm http://mathforum.org/library/topics/problem_solving/ http://www.abcteach.com/directory/basics/math/problem_solving/ http://www.hawaii.edu/suremath/intro_algebra.html http://www.fi.edu/school/math2/index.html http://intranet.cps.k12.il.us/assessments/Ideas_and_Rubrics/Rubric_Bank/MathRubrics.pdf (rubrics) http://www.readwritethink.org/lessons/lesson_view.asp?id=298 http://www.mav.vic.edu.au/PSTC/ http://www.rhlschool.com/mathv1-3.htm

44. Problem solving strategies http://www.mathstories.com/strategies.htm http://math.about.com/cs/testprep/a/ps.htm http://math.about.com/library/weekly/aa041503a.htm http://www.mathcounts.org/webarticles/anmviewer.asp?a=155&z=29 http://www.gamequarium.com/problemsolving.html http://www.kidinfo.com/Mathematics/Mathematics.html http://www.k111.k12.il.us/king/math.htm

45. “Mr. Hargreaves, may I pleased be excused? My brain is full!”