Four Corners is Better than Four Corners is Better than Four Squares: Four Squares: Assessment Connections Assessment Connections Dr. Alan Zollman Dr. Alan Zollman Northern Illinois University Northern Illinois University Successfully adapting the Graphic Organizer "Four Squares" Successfully adapting the Graphic Organizer "Four Squares" writing method to improve students’ extended responses in writing method to improve students’ extended responses in mathematical knowledge, strategy, and explanation in problem mathematical knowledge, strategy, and explanation in problem solving. solving. Friday, April 28, 2006 9:30 am Friday, April 28, 2006 9:30 am National Council of Teachers of Mathematics Annual Meeting, National Council of Teachers of Mathematics Annual Meeting, America's Center 240 America's Center 240 St. Louis, MO St. Louis, MO
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Four Corners is Better than Four Squares: Assessment Connections Dr. Alan Zollman Northern Illinois University Successfully adapting the Graphic Organizer.
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Four Corners is Better than Four Four Corners is Better than Four Squares: Squares:
Assessment ConnectionsAssessment Connections
Dr. Alan ZollmanDr. Alan ZollmanNorthern Illinois UniversityNorthern Illinois University
Four Corners is Better than Four Four Corners is Better than Four Squares: Squares:
Assessment ConnectionsAssessment Connections
Dr. Alan ZollmanDr. Alan ZollmanNorthern Illinois UniversityNorthern Illinois University
Successfully adapting the Graphic Organizer "Four Squares" writing method Successfully adapting the Graphic Organizer "Four Squares" writing method to improve students’ extended responses in mathematical knowledge, to improve students’ extended responses in mathematical knowledge,
strategy, and explanation in problem solving.strategy, and explanation in problem solving.
Friday, April 28, 2006 9:30 am Friday, April 28, 2006 9:30 am
National Council of Teachers of Mathematics Annual Meeting,National Council of Teachers of Mathematics Annual Meeting,
America's Center 240America's Center 240
St. Louis, MOSt. Louis, MO
question:question:
What’s the new national law that states What’s the new national law that states corporal punishment should only be corporal punishment should only be applied to the right side of a child’s applied to the right side of a child’s buttocks?buttocks?
question:question:
What’s the new national law that states What’s the new national law that states corporal punishment should only be corporal punishment should only be applied to the right side of a child’s applied to the right side of a child’s buttocks?buttocks?
No Child’s Left BehindNo Child’s Left Behind
Assessing the Curriculum
Content Needs of the Students
SURVEYS OF ENACTED CURRICULUM (SEC)
Council of Chief State School Officers (CCSSO)
• Surveys of Enacted CurriculumTools for Aligning Instruction, Standards, and Assessments
• Correlations of teacher-reported enacted curriculum vs. the Illinois State Achievement Test (ISAT)
MSTD Teachers vs. ISAT Exam Correlations
Left side slidesMSTD Partnership Teachers’Enacted Curriculum in Math
Right side slidesIllinois ISAT Exam2003 Grade 5 Math
One way to assist students in problem solving,
communicating, reasoning, making connections, and
showing representations in mathematics
Four Squares Graphic Organizer
1 + 4 + 3 + T + V
• ONE TOPIC
• FOUR MAIN POINTS (LAST IS CLOSURE)
• THREE SUPPORTING IDEAS FOR EACH POINT
• TRANSITION STATEMENTS BETWEEN MAIN POINTS
• VIVID DESCRIPTORS OF SUPPORTING IDEAS
NCTM Process Standards• Communications
• Problem Solving
• Reasoning and Proof
• Representation
• Connections
Adaptingfor
Extended Response Questions
Thinking like a Mathematician
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 Extended Response
First ParagraphWrite what you are to find.
“First, what I want to find is ..”
Second Paragraph
Write what you know from the problem.
“What I know from the problem is …”
Third Paragraph
Show your strategies you will try.
“So what I first tried to do was …”
Fourth Paragraph
Show your solution.
“Using numbers this is …” Fifth Paragraph
Explain your answer.“Therefore, what I did was …”
Dr. Alan Zollman, Northern Illinois UniversityFour-Corners Extended Response
an extendedresponseexample
MARTY’S SANDWICHESMARTY’S SANDWICHES
MARTY’S SANDWICHESMARTY’S SANDWICHES
Marty loves sandwiches. He likes hamburgers, Marty loves sandwiches. He likes hamburgers, hot dogs, and fish sandwiches. He loves hot dogs, and fish sandwiches. He loves
sandwiches with sandwiches with anythinganything on them! on them!
MARTY’S SANDWICHESMARTY’S SANDWICHES
Marty loves sandwiches. He likes hamburgers, Marty loves sandwiches. He likes hamburgers, hot dogs, and fish sandwiches. He loves hot dogs, and fish sandwiches. He loves
sandwiches with sandwiches with anythinganything on them! on them!
He always puts exactly one type of topping on He always puts exactly one type of topping on his sandwich: either ketchup, mustard, mayo, or his sandwich: either ketchup, mustard, mayo, or
even taco sauce!even taco sauce!
MARTY’S SANDWICHESMARTY’S SANDWICHES
Marty loves sandwiches. He likes hamburgers, Marty loves sandwiches. He likes hamburgers, hot dogs, and fish sandwiches. He loves hot dogs, and fish sandwiches. He loves
sandwiches with sandwiches with anythinganything on them! on them!
He always puts exactly one type of topping on He always puts exactly one type of topping on his sandwich: either ketchup, mustard, mayo, or his sandwich: either ketchup, mustard, mayo, or
even taco sauce.even taco sauce.
Marty says that for lunch he can eat a different Marty says that for lunch he can eat a different one-topping sandwich everyday for 2 weeks one-topping sandwich everyday for 2 weeks
without having the same thing. without having the same thing.
Can this be true? Can this be true? Explain why.Explain why.
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
Before
Same student
after!
Before
Before
Same student
after!
Before
Same student
after!
Before
Same student
after!
student favorites
later in the semester …
Assessing for mathematical knowledge, strategy, and explanation
in problem solving
Helping students self-reflect
In cooperative groups have students:
Design an abbreviated rubric that includes mathematical knowledge,
strategy, and explanation Assess “student” work using rubric Give recommendations to the “student”
One-class sample results
Here is the overall data:Category % of students who
scored an exceeds
% of students who scored a meets
% of students who scored less than
a meets
Squares and vertices problem
Pre Post Pre Post Pre Post
Mathematic knowledge
0 31 25 38 75 31
Strategy score 0 44 0 37 100 19
Explanation score 0 31 6 38 94 31
Tiling the patio problem
Pre Post Pre Post Pre Post
Mathematic knowledge
0 10 5 38 95 52
Strategy score 0 38 5 38 95 23
Explanation score 0 10 0 38 100 52
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).
Everyone who plants corn wants to Everyone who plants corn wants to have a good crop of corn. have a good crop of corn.
Initially, it requires money for good Initially, it requires money for good seed, fertilizers, insecticides, seed, fertilizers, insecticides, herbicides, and then a lot of herbicides, and then a lot of
time and effort.time and effort.
No one can make corn grow.No one can make corn grow.
You nurture the corn You nurture the corn
to allow it to grow.to allow it to grow.
You can’t plant one lone seed of You can’t plant one lone seed of corn to get corn. You can’t plant corn to get corn. You can’t plant
one row of corn to get corn. one row of corn to get corn.
Corn cross-pollinates. Corn cross-pollinates.
Fields of corn are needed. Fields of corn are needed.
You get the best corn when you You get the best corn when you and all your neighbors all grow and all your neighbors all grow
good corn.good corn.
You need your neighbors to grow You need your neighbors to grow good corn if you want good corn.good corn if you want good corn.
Thank you for helping me grow Thank you for helping me grow good corn by being good good corn by being good
neighbors! neighbors!
- Alan - Alan
dr. alan zollmandept. of mathematical sciences northern illinois universitydekalb, il 60115815/[email protected]