National Council of Teachers of Mathematics Annual Meeting, Four Corners is Better than Four Squares: Assessment Connections Dr. Alan Zollman Northern Illinois University Successfully adapting the Graphic Organizer "Four Squares" writing method to improve students’ extended responses in mathematical knowledge, strategy, and explanation in problem solving. Friday, April 28, 2006 9:30 am National Council of Teachers of Mathematics Annual Meeting, America's Center 240 St. Louis, MO

question: What’s the new national law that states corporal punishment should only be applied to the right side of a child’s buttocks?

question: What’s the new national law that states corporal punishment should only be applied to the right side of a child’s buttocks? No 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 slides MSTD Partnership Teachers’ Enacted Curriculum in Math Right side slides Illinois ISAT Exam 2003 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

Adapting for Extended Response Questions Thinking like a Mathematician Adapting for Extended Response Questions

Brainstorm ways to solve this. What do I know? Brainstorm ways to solve this. What do I want to find? Try it here. Things I need to include in my extended-response write up Dr. Alan Zollman, Northern Illinois University Four-Corners Extended Response

Write what you know from the problem. 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 …” First Paragraph Write what you are to find. “First, what I want to find is ..” Fifth Paragraph Explain your answer. “Therefore, what I did was …” Dr. Alan Zollman, Northern Illinois University Four-Corners Extended Response

an extended response example

MARTY’S SANDWICHES

MARTY’S SANDWICHES Marty loves sandwiches MARTY’S SANDWICHES Marty loves sandwiches. He likes hamburgers, hot dogs, and fish sandwiches. He loves sandwiches with anything on them!

MARTY’S SANDWICHES Marty loves sandwiches MARTY’S SANDWICHES Marty loves sandwiches. He likes hamburgers, hot dogs, and fish sandwiches. He loves sandwiches with anything on them! He always puts exactly one type of topping on his sandwich: either ketchup, mustard, mayo, or even taco sauce!

MARTY’S SANDWICHES Marty loves sandwiches MARTY’S SANDWICHES Marty loves sandwiches. He likes hamburgers, hot dogs, and fish sandwiches. He loves sandwiches with anything on them! He always puts exactly one type of topping on his sandwich: either ketchup, mustard, mayo, or even taco sauce. Marty says that for lunch he can eat a different one-topping sandwich everyday for 2 weeks without having the same thing. Can this be true? 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: 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: 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: 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: 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: 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 Mathematic knowledge 31 25 38 75 Strategy score 44 37 100 19 Explanation score 6 94 Tiling the patio problem 10 5 95 52 23

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/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).

closure

The growing of good corn

I was raised on a farm in Indiana. We had chickens, pigs, dairy cows. We grew alfalfa, clover, soybeans, tobacco, oats, wheat, and especially corn.

Everyone who plants corn wants to have a good crop of corn. Initially, it requires money for good seed, fertilizers, insecticides, herbicides, and then a lot of time and effort.

No one can make corn grow. You nurture the corn to allow it to grow.

Corn cross-pollinates. Fields of corn are needed. You can’t plant one lone seed of corn to get corn. You can’t plant one row of corn to get corn. Corn cross-pollinates. Fields of corn are needed.

You need your neighbors to grow good corn if you want good corn. You get the best corn when you and all your neighbors all grow good corn. You need your neighbors to grow good corn if you want good corn.

Thank you for helping me grow good corn by being good neighbors! - Alan

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