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Math Aint Kitty Litter: Thinking Outside the Box with Nonlinear Problem Solving Alan Zollman Northern Illinois University National Council of Teachers of Mathematics Annual Meeting April 11, 2008 Salt Lake City, Utah

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Log Rolling Apparently 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?

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What did you think of first?

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

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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.

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

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The Graphic Organizers 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. Thinking Outside the Box

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Relationship to the NCTM Process Standards Communications Problem Solving Reasoning and Proof Representation Connections

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What do I know?Brainstorm ways to solve this. 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 What do I want to find?

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What do I know?Brainstorm ways to solve this. 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 What do I want to find?

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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?

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The Driveway Problem Sarah 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?

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Switch Hitter or Second Best Is Not A Contradiction 2006 2007 Career MUTT 5/50 37/5042/100 JEFF 6/50 30/4036/90

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Switch Hitter or Second Best Is Not A Contradiction 2006 2007 Career MUTT 5/50 37/5042/100 JEFF 6/50 30/4036/90 2006 2007 Career MUTT0.100 0.740 0.420 JEFF0.120 0.750 0.400

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Switch Hitter or Second Best Is Not A Contradiction 2006 2007 Career MUTT 5/50 37/5042/100 JEFF 6/5030/4036/90 2006 2007 Career MUTT0.100 0.740 0.420 JEFF0.120 0.750 0.400 How can Jeff have both years better than Mutt, but Mutt have a better career? Problem from Bill Speer, UNLV

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Helping students use multiple representations to solve extended-response problems

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TEAM Alice: Always works a problem using algebra

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TEAM Alice: Always works a problem using algebra TEAM Cheryl: Always works a problem making a chart

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TEAM Alice: Always works a problem using algebra TEAM Cheryl: Always works a problem making a chart TEAM Darrell: Always works a problem drawing a picture

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TEAM Alice: Always works a problem using algebra TEAM Cheryl: Always works a problem making a chart TEAM Darrell: Always works a problem drawing a picture TEAM Thomas: Always works a problem guessing & testing

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TEAM Alice: Always works a problem using algebra TEAM Cheryl: Always works a problem making a chart TEAM Darrell: Always works a problem drawing a picture TEAM Thomas: Always works a problem guessing & testing TEAM Marvin: Always works a problem using manipulatives

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TEAM Alice: Always works a problem using algebra TEAM Cheryl: Always works a problem making a chart TEAM Darrell: Always works a problem drawing a picture TEAM Thomas: Always works a problem guessing & testing TEAM Marvin: Always works a problem using manipulatives TEAM Gwen: Always works a problem graphing it

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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 5X 5 patio is shown below. How many blue tiles do you need for: a)A 7x 7 patio; b)A 25 x 25 patio; c)An n x n patio? Source: NCTM Principles & Standards for School Mathematics

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TEAM Alice: Always works a problem using algebra TEAM Cheryl: Always works a problem making a chart TEAM Darrell: Always works a problem drawing a picture TEAM Thomas: Always works a problem guessing & testing TEAM Marvin: Always works a problem using manipulatives TEAM Gwen: Always works a problem graphing it

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What about Polyas four steps problem-solving hierarchy ? 1.Understand the problem. 2.Devise a plan 3.Carry out the plan. 4.Look back.

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

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

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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

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score Knowledge How well did they do on the problem? Strategy How well did they plan? Explanation How well did they describe it? 4 They got everything correct They got everything planned They explained why they did everything 3 They got most everything correct They got most everything planned They explained most of why they did things 2 They got some of it correct They got some of it planned They explained some of why they did things 1 They got a little of it correct They got a little of it planned They explained a little of why they did things 0 They did not tryThey did not try to plan They did not try to explain

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Before

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Same student after!

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Before

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Same student after!

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Results

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PRETEST SCORES 27% average N=186 Students POSTTEST SCORES 70% average N=183 Students OPEN-ENDED RESPONSE QUESTIONS Illinois Standards Achievement Test (ISAT) Scoring Rubric Grades 6-8

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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 Results for Grades 6-8

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Grades 3-5 Results n = 240

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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.

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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.

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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/ 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/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/assessmenthttp://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).

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Dr. Alan Zollman Dept. of Mathematical Sciences Northern Illinois University DeKalb, IL 60115 815/753-6750 zollman@math.niu.edu http://www.math.niu.edu/~zollman zollman@math.niu.edu http://www.math.niu.edu/~zollman zollman@math.niu.edu http://www.math.niu.edu/~zollman

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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

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