What did you think of first ?

1. Math Ain’t Kitty Litter: Thinking Outside the Boxwith Nonlinear Problem SolvingAlan ZollmanNorthern Illinois UniversityNational Council of Teachers of MathematicsAnnual Meeting April 11, 2008Salt Lake City, Utah

2. Log RollingApparently the ancient Egyptians moved the large stone blocks for the pyramids by rolling them on logs. If we were to attempt the same task with a large block, rolling it on 1-meter circumference logs, how far would the block travel for a single rotation of the logs?

3. What did you think of first?

4. GRAPHIC ORGANIZER One way to assist students in problem solving, communicating, reasoning, making connections, and showing representations in mathematics

5. Graphic organizers allow (even expect) the learner • to sort information as essential or non-essential; • structure information and concepts; • identify relationships between concepts; and • organize communication about an issue or problem.

6. Thinking Outside the Box Using a graphic organizer allows a student quickly to organize, analyze, and synthesize one’s knowledge, concepts, relationships, strategy, and communication.

7. Thinking Outside the Box The Graphic Organizer’s pictorial orientation allows students to put down their ideas in whatever order they occur. It bolsters students to “muck around” working on a problem. Further, teachers quickly can identify where students are confused in solving a problem.

8. Relationship to theNCTM Process Standards • Communications • Problem Solving • Reasoning and Proof • Representation • Connections

9. What do I know? Brainstorm ways to solve this. What do I want to find? Try it here. Things I need to include in my extended-response write up Dr. Alan Zollman, Northern Illinois University Four-Corners and a Diamond Graphic Organizer

10. What do I know? Brainstorm ways to solve this. What do I want to find? Try it here. Things I need to include in my extended-response write up Dr. Alan Zollman, Northern Illinois University Four-Corners and a Diamond Graphic Organizer

11. Pail Problem A small pail can be filled to 7/8 full using 2/3 of a gallon of water. How much will the pail hold if filled completely?

12. The Driveway ProblemSarah can sweep the driveway in 40 minutes.And Robert can sweep the driveway in 50 minutes.If Sarah begins 4 minutes before Robert joins her, how long will it take them both to finish the whole driveway?

13. Switch Hitter or“Second Best” Is Not A Contradiction 2006 2007 CareerMUTT 5/50 37/50 42/100JEFF 6/50 30/40 36/90

14. Switch Hitter or“Second Best” Is Not A Contradiction 2006 2007 CareerMUTT 5/50 37/50 42/100JEFF 6/50 30/40 36/90 2006 2007 CareerMUTT 0.100 0.740 0.420JEFF 0.120 0.750 0.400

15. Switch Hitter or“Second Best” Is Not A Contradiction 2006 2007 CareerMUTT 5/50 37/50 42/100JEFF 6/50 30/40 36/90 2006 2007 CareerMUTT 0.100 0.740 0.420JEFF 0.120 0.750 0.400How can Jeff have both years better than Mutt, but Mutt have a better career? Problem from Bill Speer, UNLV

16. Helping students use multiple representations to solve extended-response problems

17. TEAM Alice: Always works a problem using algebra

18. TEAM Alice: Always works a problem using algebraTEAM Cheryl: Always works a problem making a chart

19. TEAM Alice: Always works a problem using algebraTEAM Cheryl: Always works a problem making a chartTEAM Darrell: Always works a problem drawing a picture

20. TEAM Alice: Always works a problem using algebraTEAM Cheryl: Always works a problem making a chartTEAM Darrell: Always works a problem drawing a pictureTEAM Thomas: Always works a problem guessing & testing

21. TEAM Alice: Always works a problem using algebraTEAM Cheryl: Always works a problem making a chartTEAM Darrell: Always works a problem drawing a pictureTEAM Thomas: Always works a problem guessing & testingTEAM Marvin: Always works a problem using manipulatives

22. TEAM Alice: Always works a problem using algebraTEAM Cheryl: Always works a problem making a chartTEAM Darrell: Always works a problem drawing a pictureTEAM Thomas: Always works a problem guessing & testingTEAM Marvin: Always works a problem using manipulativesTEAM Gwen: Always works a problem graphing it

23. Tiling the Patio Problem You are to tile a patio. The patio will be a square, with the inside tiles always being blue and the border tiles always being white. A 5’X 5’ patio is shown below. How many blue tiles do you need for: A 7’x 7’ patio; A 25’ x 25’ patio; An n x n patio? Source: NCTM Principles & Standards for School Mathematics

24. TEAM Alice: Always works a problem using algebraTEAM Cheryl: Always works a problem making a chartTEAM Darrell: Always works a problem drawing a pictureTEAM Thomas: Always works a problem guessing & testingTEAM Marvin: Always works a problem using manipulativesTEAM Gwen: Always works a problem graphing it

25. What about Polya’s four steps problem-solving hierarchy? Understand the problem. Devise a plan Carry out the plan. Look back.

26. What about Polya’s four steps problem-solving hierarchy? Students mistakenly feel that Polya’s problem solving steps need to be accomplished in order. Our graphic organizer varies from Polya, not in intent, but in deployment.

27. Assessing for mathematical knowledge, strategy, and explanation in problem solving

28. Helping Students Self-Reflect In cooperative groups have students: • Students help design an abbreviated rubric that includes mathematical knowledge, strategy, and explanation • They assess “student” work using rubric • They give recommendations to the “student”

30. Before

32. Same student after!

34. Before

36. Same student after!

39. Results

40. OPEN-ENDED RESPONSE QUESTIONS PRETESTSCORES 27% averageN=186 Students Illinois Standards Achievement Test (ISAT) Scoring Rubric Grades 6-8 POSTTESTSCORES 70% averageN=183 Students

41. Results for Grades 6-8 PRETEST State meets or exceeds 4% in Math Knowledge, 19% in Strategic Knowledge 8% in Explanation N=186 Students POSTTEST State meets or exceeds 75% in Math Knowledge, 68% in Strategic Knowledge 68% in Explanation N=183 Students Using ISAT Scoring Rubric

42. Grades 3-5 Results n = 240

44. Reflections • From research (National Reading Panel 2000), we know graphic organizers work well with elementary students in the reading-writing process. • A good learning strategy for reading and writing is also a good teaching method in mathematics.

45. Reflections • For students, graphic organizers have overlapping effects in connecting, communicating, justifying, and solving mathematical problems. • For teachers, graphic organizers offer a quick, efficient diagnosis of the weaknesses and strengths in individual student’s problem-solving abilities and skills. • For teachers, graphic organizers a comfortable, familiar method to facilitate problem-solving instruction.

46. references: “Four Square Writing Method for Grades 1-3” written by Judith S. and Evan Jay Gould published by Teaching and Learning Company, Carthage, Illinois. (1999). “Four Corners Graphic Organizer for Open-Ended Mathematical Problem Solving” Alan Zollman, NIMS Mathematics-Science Partnership. (2004). “Four Corners Graphic Organizer for Open-Ended Mathematical Problem Solving” Alan Zollman, MSTD Mathematics-Science Partnership. (2005). http://www.mstd-d41-d4.niu.edu/ “Four Corners Graphic Organizer for Open-Ended Mathematical Problem Solving” Alan Zollman, Raising The MEANs Mathematics-Science Partnership. (2005). http://www.means-d131.niu.edu/ “Four Corners is Better Than Four Squares in Math” Alan Zollman, ICTM 57th Annual Meeting, Springfield, IL. (Oct. 14, 2005). Illinois Assessment web site: http://www.isbe.net/assessment Council of Chief State School Officers;Surveys of EnactedCurr.(SEC) http://www.ccsso.org/projects/surveys_of_enacted_curriculum/ “Simmons Middle School Results: East Aurora District 131” Karen Lopez, MEANs Partnership Showcase, DeKalb, IL. (Apr. 6, 2006).

47. Dr. Alan ZollmanDept. of Mathematical Sciences Northern Illinois UniversityDeKalb, IL 60115815/753-6750zollman@math.niu.eduhttp://www.math.niu.edu/~zollman

48. http://www.mstd-d41-d4.niu.edu/ http://www.means-d131.niu.edu/ partially supported by the Illinois Mathematics and Science Partnerships Program/ISBE/ US Department of Education, funded by NCLB, Title II, Part B, US DOE