Math Ain’t Kitty Litter: Math Ain’t Kitty Litter: Thinking Outside the BoxThinking Outside the Box

with Nonlinear Problem Solvingwith Nonlinear Problem Solving

Alan ZollmanAlan ZollmanNorthern Illinois UniversityNorthern Illinois University

National Council of Teachers of MathematicsNational Council of Teachers of MathematicsAnnual Meeting April 11, 2008Annual Meeting April 11, 2008

Salt Lake City, UtahSalt Lake City, Utah

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?

What did you think of first?

GRAPHIC ORGANIZER One way to assist students in

problem solving, communicating, reasoning, making connections, and

showing representations in mathematics

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.

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

Thinking Outside the BoxThinking Outside the Box

The Graphic Organizer’s pictorial The Graphic Organizer’s pictorial orientation allows students to put down orientation allows students to put down their ideas in their ideas in whateverwhatever order they occur. order they occur.

It bolsters students to “muck around” It bolsters students to “muck around” working on a problem. working on a problem.

Further, teachers quickly can identify Further, teachers quickly can identify where students are confused in solving a where students are confused in solving a problem.problem.

Thinking Outside the BoxThinking Outside the Box

Relationship to theNCTM Process StandardsRelationship to theNCTM Process Standards

Communications Problem Solving Reasoning and Proof Representation Connections

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 UniversityFour-Corners and a Diamond Graphic Organizer

What do I want to find?

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 UniversityFour-Corners and a Diamond Graphic Organizer

What do I want to find?

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?

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?

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?

Switch Hitter or“Second Best” Is Not A Contradiction

2006 2007 CareerMUTT 5/50 37/50 42/100JEFF 6/50 30/40 36/90

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

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

How can Jeff have both years better than Mutt, but Mutt have a better career? Problem from Bill Speer, UNLV

Helping students use multiple representations to

solve extended-response problems

TEAM Alice: Always works a problem using algebra

TEAM Alice: Always works a problem using algebra

TEAM Cheryl:Always works a problem making a

chart

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 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 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 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

Tiling the Patio ProblemYou 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) A 7’x 7’ patio;

b) A 25’ x 25’ patio;c) An n x n patio? Source: NCTM Principles & Standards for School Mathematics

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

What about Polya’s four steps problem-solving

hierarchy?

What about Polya’s four steps problem-solving

hierarchy?1. Understand the problem.2. Devise a plan3. Carry out the plan.4. Look back.

What about Polya’s four steps problem-solving

hierarchy?

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.

Assessing for mathematical knowledge, strategy, and explanation

in problem solving

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”

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 try They did not try to plan

They did not try to explain

Before

Same student after!

Before

Same student after!

Results

PRETESTPRETESTSCORESSCORES 27% average 27% averageN=186 StudentsN=186 Students

POSTTESTSCORES

70% averageN=183 Students

OPEN-ENDED RESPONSE QUESTIONS

Illinois Standards Achievement Test (ISAT) Scoring Rubric

Grades 6-8

PRETESTState meets or exceeds

4% in Math Knowledge, 19% in Strategic Knowledge 8% in Explanation

N=186 Students

POSTTESTState meets or exceeds

75% in Math Knowledge, 68% in Strategic Knowledge68% in Explanation

N=183 Students

Using ISAT Scoring Rubric

Results for Grades 6-8

Pre-Test Ave rage

Out of 12 Possible Points

Post-Test Ave rage

Out of 12 Possible Points

3rd Graders – Perimeter 6.90 9.50

3rd Graders – Area 6.30 9.10

4th Graders – Perimeter 4.95 6.65

4th Graders – Area 4.71 5.76

5th Graders – Perimeter 4.70 6.89

5th Graders – Area 4.81 7.90

Grades 3-5 Resultsn = 240

Teachers’ Comfort Levels Teaching Problem Solving and Using the State Scoring Rubric: Pre-Test and Post-Test Surveys

NC: Not at all Comfortable SU: Somewhat Uncomfortable n= 10

SC: Somewhat Comfortable VC: Ver y Comfortable

Number of Teachers Pre-Test Number of Teachers Post-Test

Teaching Problem Solving

NC SU SC VC

0 5 5 0

NC SU SC VC

0 0 2 8

Using the State Scoring Rubric

NC SU SC VC

2 4 3 1

NC SU SC VC

0 0 3 7

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.

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.

references: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/assessmentCouncil 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).

Dr. Alan ZollmanDr. Alan ZollmanDept. of Mathematical Sciences Dept. of Mathematical Sciences Northern Illinois UniversityNorthern Illinois UniversityDeKalb, IL 60115DeKalb, IL 60115815/753-6750815/753-6750zollman@math.niu.eduzollman@math.niu.edu

http://www.math.niu.edu/~zollmanhttp://www.math.niu.edu/~zollman

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