1. December 11, 2014 Presented by: Charlotte Thompson & Taryn Miley

2. Jumping Right In… Real World Application – Grade 4

3. Real World Application – Grade 4 72 total

4. Take it a Step Further… Explain your thinking Molly believes 1 bus, 1 van, and 6 cars can hold all of the students. Explain why she is incorrect.

5. What Must Students Do When Solving Problems? To solve even simple problems, students must: -understand the vocabulary and instructions contained within the problem -recall mathematical rules and formulas -recognize patterns -use sequential ordering to solve multi-step problems

6. 6 During this session, participants will: experience learning mathematics through problem solving solve problems in different ways develop strategies for teaching mathematics through problem solving Overall Learning Goals for Problem Solving

7. What is Problem Solving? “Problem solving means engaging in a task for which the solution method is not known in advance.” --Principles and Standards for School Mathematics It encompasses exploring, reasoning, strategizing, estimating, conjecturing, testing, explaining, and proving.

8. What’s so great about problem solving? Various strategies to solve the problems (pictures, numbers, equations, graphing) Could have more than 1 solution Encompasses multiple standards

9. Key Questions Are you learning? Are you learning it? Do you know? Did you get that answer?

10. What is a Problem? A problem is a task that requires the learner to reason through a situation that will be challenging but not impossible. Most often, the learner is working with a group of other students to meet the challenge. Problems can generally be defined as an obstacle, that remains perplexing until solved.

11. Problem or Exercise? An exercise is a set of number sentences intended for practice in the development of a skill. A problem is what we commonly refer to as a “word problem.” But beware! Problems can become exercises!!

12. It’s More Than This… I have read 134 of the 512 pages of my book. How many more pages must I read to reach the middle? Word problems have their place in mathematics, but it’s not necessarily problem solving!

13. Problem Solving Strategies Two Methods You Can Use with Students

14. Students can learn to become better problem solvers. Polya’s (1957) “How to Solve It” book presented four phases or areas of problem-solving, which have become the framework often recommended for teaching and assessing problem-solving skills. The four steps are: 1. understanding the problem, 2. devising a plan to solve the problem, 3. implementing the plan (Solve), and 4. reflecting on the problem (Check). Four Phases

15. UPS Strategy Understand- what is the problem asking you to do? Plan- how will you solve the problem? What strategy will you use? Solve & Check- Solve the problem and Check your answer to make sure it makes sense

16. MY PROBLEM I have 4 shirts one is red, one yellow, one white, and one blue. I have 2 pairs of pants that are black and khaki and one skirt that is dark blue. I can wear all these with all 4 shirts. How many different outfits do I have?

17. Step One- U Understand what the problem is asking: How many different outfits do I have?

18. Look for Clues Read the problem carefully. Underline clue words. Ask yourself if you've seen a problem similar to this one. If so, what is similar about it? What did you need to do? What facts are you given? What do you need to find out?

19. Step Two- P Plan how you will solve the problem. What strategy will you use?  Draw a picture  Use logical reasoning  Guess and check  Look for a pattern  Choose an operation  Use a formula  Write an equation  Solve a simpler problem  Work backwards  Make a list, table, graph, or diagram

20. Problem Solving Strategy The method I have chosen to solve my problem is to draw a picture to show how many different outfits I can make

22. Step Three- S Solve and Check- Solve the problem and Check your answer to make sure it makes sense – 3 outfits with the red shirt – 3 outfits with the yellow shirt – 3 outfits with the blue shirt – 3 outfits with the white shirt – I have a total of 12 outfits with the clothes that I have in my closet.

23. Other Ways to Solve? How else could you have solved this problem?  Use logical reasoning  Guess and check  Choose an operation  Write an equation  Make a list, table, graph, or diagram

24. Practice Problem solving requires practice! The more your practice, the better you get. Practice, practice, practice.

25. CUBES Strategy C- Circle the key numbers U- Underline the question B- Box in key words E- Evaluate and Eliminate S- Solve and Check

26. My Problem Camden has a wall in his room that measures 13 feet long and 8 ½ feet high and is freshly painted. He wants to hang his favorite posters on the wall. Each poster measures 3 feet long and 2 feet high. What is the greatest number of posters that he can hang on the wall so that the posters do not overlap?

27. Step One- C Circle Key Numbers Camden has a wall in his room that measures 13 feet long and 8 ½ feet high and is freshly painted. He wants to hang his favorite posters on the wall. Each poster measures 3 feet long and 2 feet high. What is the greatest number of posters that he can hang on the wall so that the posters do not overlap?

28. Step Two- U Underline the Question Camden has a wall in his room that measures 13 feet long and 8 ½ feet high and is freshly painted. He wants to hang his favorite posters on the wall. Each poster measures 3 feet long and 2 feet high. What is the greatest number of posters that he can hang on the wall so that the posters do not overlap?

29. Step Three- B Box in Key Terms Camden has a wall in his room that measures 13 feet long and 8 ½ feet high and is freshly painted. He wants to hang his favorite posters on the wall. Each poster measures 3 feet long and 2 feet high. What is the greatest number of posters that he can hang on the wall so that the posters do not overlap?

30. Step Four- E Evaluate and Eliminate Wall = 13 ft. x 8 ½ ft. Poster = 3 ft. x 2 ft. Question: What is the greatest number of posters that he can hang on the wall so that the posters do not overlap? What strategy could I use to solve?

31. Step Five- S Solve and Check Explain your thinking- How do you know your answer is correct?

32. Try It Out… 4 hungry teachers want to order burritos for lunch on Friday. Each burrito costs $4.12. The 4 teachers have $20 total. Will there be enough to buy 4 burritos? Show your answer two different ways.

33. Problem Solving Strategies What About Students with Disabilities

34. Concrete-Representational-Abstract (C-R-A) Phase of Instruction Instructional method incorporates hands-on materials and pictorial representations. Students first represent the problem with objects - manipulatives. Then advance to semi-concrete or representational phase and draw or use pictorial representations of the quantities Abstract phase of instruction involves numeric representations, instead of pictorial displays.

35. Interventions Found Effective for Students with Disabilities Reinforcement and corrective feedback for fluency Concrete-Representational-Abstract (C-R-A) Instruction Direct/Explicit Instruction Demonstration/Modeling Verbalization while problem solving Big Ideas Metacognitive strategies: Self-monitoring, Self-Instruction Computer-Assisted Instruction Monitoring student progress Teaching skills to mastery

36. Common Characteristics of a Good Problem It should be challenging to the learner. It should hold the learner’s interest. The learner should be able to connect the problem to her life and/or to other math problems or subjects. It should contain a range of challenges. It should be able to be solved in several ways.

37. Classroom Application Amping up the Rigor

38. M/C - one right answer (traditional format) M/C - more than one right answer (at least A-F) Short Answer - fill-in-the blank Open Ended - compare/contrast; explain/defend; how do you know? Create Different Question Formats: Middle School

39. Choosing Problem-Solving Tasks The problem must be meaningful to the students. The teacher must sometimes adapt the problem to make it more meaningful. The teacher must work the problem to anticipate mathematical ideas and possible questions that problem might raise.

40. Presenting the Problem It must be interesting and engaging. It must be presented so that all children believe that it’s possible to solve the problem, but that they will be challenged. The teacher has to decide whether students will work individually or in groups.

41. Group Work or Individual Work? In groups, students don’t give up as quickly. Students have greater confidence in their abilities to solve problems when working in groups. When in a group, students hear a broad range of strategies from others. Kids enjoy working in groups! Students remember what they learn better when they assist each other. If students are less productive, arrangements can be made for them to work alone. There will be a heightened noise level—but conversation is an important part of the learning process.

42. Once the Kids Are Working… Allow students to “wrestle” with the problem without just telling them the answer! If we are just telling them what to do, the students are not engaged in the process. Finally, teachers have to determine how to assess what the students are learning and what they need to learn next.

43. Learning Mathematics through Problem Solving Students learn to apply the mathematics as they are learning it. They can make connections within mathematics and to other areas of the curriculum. Students can understand what they have learned. Students can explain their thinking to others