math aint kitty litter: thinking outside the box with nonlinear problem solving alan zollman...
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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/[email protected]@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