Four Corners is Better than Four Squares: Assessment Connections Dr. Alan Zollman Northern Illinois...

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).

closure

The growing of The growing of good corngood corn

I was raised on a farm in Indiana.I was raised on a farm in Indiana.

We had chickens, pigs, dairy cows. We had chickens, pigs, dairy cows. We grew alfalfa, clover, We grew alfalfa, clover,

soybeans, tobacco, oats, wheat,soybeans, tobacco, oats, wheat,

and especially corn. and especially corn.

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 60115815/753-6750zollman@math.niu.edu

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