1. 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, :30 am
National Council of Teachers of Mathematics Annual Meeting,
America's Center 240
St. Louis, MO

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

3. 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

4. Assessing the Curriculum Content Needs of the Students

5. 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)

6. 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

15. One way to assist students in problem solving, communicating, reasoning, making connections, and showing representations in mathematics

17. 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

20. NCTM Process Standards
Communications
Problem Solving
Reasoning and Proof
Representation
Connections

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

22. 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

23. 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

24. an extended
response
example

25. MARTY’S SANDWICHES

26. 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!

27. 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!

28. 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.

29. Helping students use multiple representations to solve extended response problems

30. TEAM Alice: Always works a problem using algebra

31. 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

32. 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

33. 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

34. 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

35. 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

37. Before

39. Same
student
after!

41. Before
Before

43. Same student
after!

45. Before

47. Same student
after!

49. Before

51. Same student
after!

53. student favorites

55. later in the semester …

58. Assessing for mathematical knowledge, strategy, and explanation in problem solving

60. 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”

61. One-class sample results

62. 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

63. 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).
“Four Corners Graphic Organizer for Open-Ended Mathematical Problem Solving” Alan Zollman, Raising The MEANs Mathematics-Science Partnership. (2005).
“Four Corners is Better Than Four Squares in Math” Alan Zollman, ICTM 57th Annual Meeting, Springfield, IL. (Oct. 14, 2005).
Illinois Assessment web site:
Council of Chief State School Officers;Surveys of EnactedCurr.(SEC)
“Simmons Middle School Results: East Aurora District 131” Karen Lopez, MEANs Partnership Showcase, DeKalb, IL. (Apr. 6, 2006).

64. closure

65. The growing of good corn

66. 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.

67. 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.

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

69. 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.

70. 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.

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

72. dr. alan zollman dept. of mathematical sciences northern illinois university dekalb, il /