Improving Word Problems through Realistic Problem-Solving:

2. Improving Word Problems through Realistic Problem-Solving: Model-Eliciting Activities as a Formative Assessment and Learning Tool 2012 FCR-STEM Conference Dr. Melissa Dyehouse, CPALMS MEA Specialist Mr. Robert Lengacher, CPALMS K-8 Mathematics Coordinator

3. Agenda • MEA overview • MEA activity • Examining some word problems • Discussion and ideas • MEAs on CPALMS & the MEA Innovative Initiative

4. Realistic, open-ended problems with a client • Team-based • Product is the processfor solving the problem • A model the client can use • Integrated across subjects • Mathematics • Science • English Language/Arts • Other context areas • Model-Eliciting Activities

5. 6 Principles for Developing MEAs • Proposed, tested, and refined by hundreds of teachers, parents, and community leaders who worked with researchers. • The goals of the project that led to developing the 6 principles were: • Solutions to problems should involve important mathematical ideas. • Tasks should emphasize the kinds of problem understandings, characteristics, and abilities needed for success in real-life situations – not just in school settings. • Teachers should be able to gather useful information about their students’ conceptual strengths and weaknesses for more effective teaching. From Lesh et al. (2000)

6. Research • Learn about students’ developing knowledge • Formative assessment • Model eliciting = thought-revealing • Learning content • Example: Plants vs Pollutants MEA • Support 21st century skills • Problem-solving, communication, teamwork • How are MEAs used?

8. Students read a problem statement (e.g., letter, RFP) from a fictional “client” stating a problem. • Students work in teams to develop a procedure, or model, to give to the client. • Student teams write a letter back to the client explaining their models/procedures. • Students receive a 2ndproblem statement, or “twist”. • Teams test and modify (if needed) their models and write another client letter. • The teams present their work. • Basic steps of an MEA

9. Toothpaste MEA (grades 2-4 and 3-5 versions): this MEA is based on a children’s book. Students must choose among criteria such as cost and taste to decide the best toothpaste recipes. MEA Examples Turning Tires MEA: 9-12 grade students design a procedure to select the best tire material for certain situations by applying geometric concepts through modeling. Water Filter MEA (grades 2-4 and 3-5 versions): students are asked to develop a procedure for ranking water filters to clean the water for a turtle. Energy Sources MEA (grades 9-11): Students must select the most promising, sustainable, and helpful energy sources to invest in for the future.

10. Example Dataset at the Elementary Level • Toothpaste MEA • Using the data provided, students must write a letter back to the client explaining their procedure for selecting the best toothpaste recipe.

11. Example Dataset at the High-School Level • Turning Tires MEA • Using the data provided, students must write a letter back to the client explaining their procedure for selecting the best tire material to use in a given situation.

12. Imagine Improve Create Test Model Design Process What could be some solutions? Who is the intended user? What does the intended user need? Brainstorm ideas. Choose the best one. Go to User Model satisfies the goal Ask Plan Revise the Working model to make it even better. Test it out! Express the working model. Test the working model.

13. The following table shows the results of a poll which asked drivers how many accidents they had had over the previous 5 years. What is the median number of accidents per driver? • Word Problem CCSS Math.6.SP: Summarize numerical data sets in relation to their context, such as by: Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. Mathematics Assessment Resource Service (2012)

14. Read the letter from the client. • In small groups or with a shoulder partner, develop a procedure for determining how to rank the most dangerous intersections. • Activity – Traffic Light MEA

15. Determine whether your procedure still works with the new data. • If the procedure works, determine if it can be improved upon. • If the procedure does not work, determine if it needs to be tweaked or if additional steps need to be added. Make these changes. • Test your procedure

16. In what ways is the MEA different from a typical word problem? • What big mathematical ideas, or concepts, are present in this MEA? • At what point would your students need more support? Where would you step in? • What is something you would change in the Traffic Light MEA to modify for your students? • Debrief

17. Sheila works 8 hours per day on Monday, Wednesday and Friday, and 6 hours per day on Tuesday and Thursday. She does not work on Saturday and Sunday. She earns $324 per week. How much does she earn in dollars per hour? • Wordproblem example Mathematics Assessment Resource Service (2012)

18. How would you use one of the principles to modify the problem?

19. Dear engineering team, I am the production manager for the Winmac Factory, which makes computer hardware. Our factory has recently received a large order from a major computer company that is preparing to launch a new model of computer and has ordered our hardware. Because of this, we need to make lots of hardware in a short amount of time and we need your help. Please help us determine the two employees we should hire to best supplement our work force in filling this order. Our main goal is to minimize the cost of hardware but we also want to hire the best overall employee that we can. The Temp Agency has sent us a list of employees with previous computer hard drive assembly experience, along with their availability, work outputs, and error rates. Using this table of data provided, please write us a letter ranking the employees in the order that we should hire them from highest to lowest, the total cost for hiring the top two employees as well as the total hours they will work, the total number of hard drives each potential employee will make, the cost of hardware in terms of employee hours, and your procedure that you used to make your decisions so that we make similar decisions about hiring temporary employees in the future. Please note that employees cannot work more than eight hours in a single shift. Thank you for helping us at this critical time in our business. Sincerely, Bill Jobs, Production ManagerWinmac Factory • Production Problem MEA – Client Letter 1

20. Data set 1 * Number of working days per year = 250, number of working days per week = 5, number of working hours per day = 8. ** Tardiness refers to a record of employees’ tardiness at their previous job. This measure of tardiness is the average number of minutes per day that an employee was tardy during the surveyed week.

21. Dear engineering team, Thank you very much for your letter. The Winmac Factory was able to meet our demands for computer hardware in a short time period. However, the same computer company that recently purchased our hardware has experienced a huge success in selling their new computers and therefore has ordered even more hardware from us. That means that this time we need to hire two more temporary employees for another week of production and we would like to use the same procedure that you gave us to select these employees. We received updated information from the Temp Agency and they have added two additional employees to the list. We need you to see if your procedure still works in selecting the employees for us. If not please modify it to work with the additional data.Please help us determine the two employees we should hire to best supplement our work force in filling this order. Using the second data table, please write a letter telling me how you ranked the employees in the order in which we should hire them. If your team had to make any changes from your first procedure, please tell me what you changed and why each change was needed. Finally, be sure to include the total cost for hiring the top two employees as well as the total hours they will work and the cost of hardware in terms of employee hours. We expect to have similar situations in the future with companies who request our hardware so your procedure will be a great help for the company. Sincerely, Bill Jobs, Production ManagerWinmac Factory • Client Letter 2

22. Data set 2 * Number of working days per year = 250, number of working days per week = 5, number of working hours per day = 8. ** Tardiness refers to a record of employees’ tardiness at their previous job. This measure of tardiness is the average number of minutes per day that an employee was tardy during the surveyed week.

23. Is this an improvement on the original word problem…why or why not? • What big mathematical ideas, or concepts, are present in this MEA? • Discussion

24. Multiple-choice question • A board 7 feet 9 inches long is divided into three equal parts. What is the length of each part? • 2 ft. 6 1/3 in. • 2 ft. 7 in. • 2 ft. 8 1/3 in. • 2 ft. 8 in. • 2 ft. 9 in. • An assessment example From Lesh et al. (2000)

25. A board 7 feet 9 inches long is divided into three equal parts. What is the length of each part? Answer: ____________________ Explain your answer: • First improvement attempt

26. How would you use one of the principles to modify the problem?

27. Materials: 1,000 or more fake dollar bills, calculators • Tell students this story: Just as you decide to go to bed one night, the phone rings and a friend offers you a chance to be a millionaire. He tells you he won $2,000,000 in a contest. The money was sent to him in two suitcases, each containing $1,000,000 in $1 bills. He will give you one suitcase of money if your mom or dad will drive him to the airport to pick it up. Could your friend be telling you the truth? Could he make you a millionaire? • Involve students in formulating and exploring questions to investigate the truth of this claim. E.g., a) Can $1,000,000 in $1 bills fit in a standard-size suitcase? If not, what is the smallest denomination of bills that could fit? b) Could you lift the suitcase if it contained $1,000,000 in $1 bills? Estimate its weight. • Note: The dimensions of a $1 bill are approximately 6 in. by 2.5 in. • Example – The Million Dollar Problem From Lesh et al. (2000)

28. How well does the Million Dollar Problem Meet these principles?

29. “The stated questions (Could your friend be telling you the truth?...) do not create the need for the mathematics that the teacher intends to elicit. Students are being set up not by a phone friend, but by their own teacher, who leads them along a predetermined path toward a forgone conclusion, and the whole point of the leading questions seemed to be to try to guarantee that all students would interpret the situation as the teacher intended. In other words, what the teacher had in mind was for students to carry out a particular computation; it was not for students to develop a sensible interpretation of the situation that would allow a sensible decision to be made. Smart children would know that they were being set up.” • A teacher’s thoughts on The Million Dollar Problem From Lesh et al. (2000)

30. Teachers who helped to develop the six principles also found that the majority of problems asked of students in textbooks and texts violated allof the principles (Lesh et al., 2000) • Open-ended questions or those asking students to explain their answers can still elicit lower-level, procedural knowledge. • Students need to reason, predict, describe, experiment, connect, or interpret. • Form vs substance of the assessment • Writing good assessment problems take time and effort. • Payoff is that one good problem can assess and document learning better than many low-level questions can. • Final reflections

31. What are some things you like about MEAs? • Would you use an MEA like the Traffic Light MEA with your students? Why or why not? • Reflecting on MEAs

32. Navigate to www.cpalms.org • Under the “Resource Center” tab, do a keyword search for “MEA” • Find MEAs on CPALMS

33. Get involved with the CPALMS MEA Innovative Initiatives and get paid for writing MEAs: Email Melissa Dyehouse for more information: mdyehouse@lsi.fsu.edu • How to get involved