divergent thinking
CEU CODE
•JK-28
Developing Divergent Thinking in the Mathematics Classroom
Betty Wood, Ph. D.
James Fetterly, Ph. D.
[email protected]
[email protected]
Arkansas iTunes U
Understanding and Developing Fractions
Alternative Algorithms
Problem Solving the Square-iest Situations
Author Unknown Good mathematics is NOT how many answers you know … but how you behave when you don’t know.
Mental Mathematics 64 -28
Creative Strategy 64 -28 -4 40 36
Van De Walle “If you have a solution, a good thing to do is see first why you think you are correct and try to articulate that. Then see if you can find a different way to solve the problem than the way you did it the first time.”
Which One Does Not Belong? Why?
2
6
5
10
DIVERGENT THINKING
FLUENCY FLEXIBILITY ORIGINALITY ELABORATION
BASIC ABILITIES INVOLVING DIVERGENT THINKING
Fluency
How many ideas can you come up with? Fluency is the number of responses which are relevant and not repeated in the list. Flexibility Can you think of another category or another way of looking at the ideas? Flexibility is the number of shifts to other ways of looking at the question.
BASIC ABILITIES INVOLVING DIVERGENT THINKING
Originality
Can you think of an idea that no one else has come up with? Originality is the uniqueness of a response in comparison with others of the group or a similar group. Elaboration Can you add to your ideas? Elaboration is the “fleshing out” of an idea by adding description details or relating it to other ideas.
TYPICAL QUESTIONING What is a fraction? If a boy goes to the store and buys $.35 worth of
candy and gives the man $.75, how much change will he get? Find the absurdity in this situation (A gentleman is writing check for more money than he has in his bank account.) Addition is to subtraction as multiplication is to _______.
DIVERGENT QUESTIONING List as many ways as possible to describe the
notion of a fraction? Think of different ways you can produce the number 7? Take the counting numbers and think of ways that they could be changed to make them more useful or interesting. Create a new mathematical operation by FLUENCY combining two or more operations.
FLEXIBILITY ORIGINALITY ELABORATION
Comparison What is a fraction? If a boy goes to the store and
buys $.35 worth of candy and gives the man $.75, how much change will he get? Find the absurdity in this situation (A gentleman is writing check for more money than he has in his bank account.) Addition is to subtraction as multiplication is to _______.
List as many ways as
possible to describe the notion of a fraction? Think of different ways you can produce the number 7? Take the counting numbers and think of ways that they could be changed to make them more useful or interesting. Create a new mathematical operation by combining two or more operations.
Lauren Resnick Children who are successful at making sense of mathematics are those who believe that mathematics make sense.
Geometry “The radius of a circle inscribed in a square is 6 inches. Find the area of the square.”
Geometry “The radius of a circle inscribed in a square is 6 inches. Find out all that you can about the square and circle.”
Which Ones Are The Same?
4
8
25 16
Number Theory “Show that the product of any four consecutive integers is divisible by 24.”
Number Theory “What conclusions can be drawn from the product of any four consecutive integers?”
Classroom Practices
• Challenge Assumptions • Change Behaviors • Contaminate Characteristics
Start And Jump Numbers: Searching For Patterns • You will need to make a list of numbers that begin with a “start number” and increase by a fixed amount we will call the “jump number.” First try 3 as the start number and 5 as the jump number. Write the start number at the top of your list, then, then 8, 13, and so on, “jumping” by 5 each time until your list extends to about 130.
Start And Jump Numbers: Searching For Patterns • Your task is to examine this list of numbers and find as many patterns as you possibly can. Share your ideas with the group, and write down every pattern you agree really is a pattern.
DIVERGENT THINKING Find and articulate as many patterns as possible in the list
of “Start Jump” numbers? Classify and label your patterns into groups. Name and describe them. Develop and describe a new problem or sequence derived from the original “Start Jump” problem. Create other sequences that are fundamentally different from the original. Describe how your new problem is different from the original “Start Jump” problem. How are the two similar? What conjectures or inferences can be made from this analysis? What other conclusions or relations might be stated?
DIVERGENT THINKING
FLUENCY FLEXIBILITY ORIGINALITY ELABORATION
Solve this problem. What was easy about it? Not so easy?
Create and solve a What’s one question problem similar to this someone should ask problem. themselves when they first look at this problem?
What’s a mistake that someone might make in trying to solve this problem? Why might they make that mistake?
What mathematical concepts or terms/vocabulary does this problem show? Be specific!
Write a step-by-step set of directions that tells someone who was absent today how to solve this problem.
H. Alang, A. Mironchuk, & J. Paul (Farragut Career Academy, Chicago IL
Consider the following tautology: Teachers
and students who think divergently with mathematics are those who believe that mathematics is a creative enterprise.
CCSS Mathematical Practices • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning Common Core Standards (CCSSO) 2010
Arkansas iTunes U
Understanding and Developing Fractions
Alternative Algorithms
Problem Solving the Square-iest Situations
Developing Divergent Thinking in the Mathematics Classroom
Betty Wood, Ph. D.
James Fetterly, Ph. D.
[email protected]
[email protected]
A Calculator Exploration Use a calculator to divide 1444 by 62. But you can’t use the “÷” symbol.
•JK-28
Developing Divergent Thinking in the Mathematics Classroom
Betty Wood, Ph. D.
James Fetterly, Ph. D.
[email protected]
[email protected]
Arkansas iTunes U
Understanding and Developing Fractions
Alternative Algorithms
Problem Solving the Square-iest Situations
Author Unknown Good mathematics is NOT how many answers you know … but how you behave when you don’t know.
Mental Mathematics 64 -28
Creative Strategy 64 -28 -4 40 36
Van De Walle “If you have a solution, a good thing to do is see first why you think you are correct and try to articulate that. Then see if you can find a different way to solve the problem than the way you did it the first time.”
Which One Does Not Belong? Why?
2
6
5
10
DIVERGENT THINKING
FLUENCY FLEXIBILITY ORIGINALITY ELABORATION
BASIC ABILITIES INVOLVING DIVERGENT THINKING
Fluency
How many ideas can you come up with? Fluency is the number of responses which are relevant and not repeated in the list. Flexibility Can you think of another category or another way of looking at the ideas? Flexibility is the number of shifts to other ways of looking at the question.
BASIC ABILITIES INVOLVING DIVERGENT THINKING
Originality
Can you think of an idea that no one else has come up with? Originality is the uniqueness of a response in comparison with others of the group or a similar group. Elaboration Can you add to your ideas? Elaboration is the “fleshing out” of an idea by adding description details or relating it to other ideas.
TYPICAL QUESTIONING What is a fraction? If a boy goes to the store and buys $.35 worth of
candy and gives the man $.75, how much change will he get? Find the absurdity in this situation (A gentleman is writing check for more money than he has in his bank account.) Addition is to subtraction as multiplication is to _______.
DIVERGENT QUESTIONING List as many ways as possible to describe the
notion of a fraction? Think of different ways you can produce the number 7? Take the counting numbers and think of ways that they could be changed to make them more useful or interesting. Create a new mathematical operation by FLUENCY combining two or more operations.
FLEXIBILITY ORIGINALITY ELABORATION
Comparison What is a fraction? If a boy goes to the store and
buys $.35 worth of candy and gives the man $.75, how much change will he get? Find the absurdity in this situation (A gentleman is writing check for more money than he has in his bank account.) Addition is to subtraction as multiplication is to _______.
List as many ways as
possible to describe the notion of a fraction? Think of different ways you can produce the number 7? Take the counting numbers and think of ways that they could be changed to make them more useful or interesting. Create a new mathematical operation by combining two or more operations.
Lauren Resnick Children who are successful at making sense of mathematics are those who believe that mathematics make sense.
Geometry “The radius of a circle inscribed in a square is 6 inches. Find the area of the square.”
Geometry “The radius of a circle inscribed in a square is 6 inches. Find out all that you can about the square and circle.”
Which Ones Are The Same?
4
8
25 16
Number Theory “Show that the product of any four consecutive integers is divisible by 24.”
Number Theory “What conclusions can be drawn from the product of any four consecutive integers?”
Classroom Practices
• Challenge Assumptions • Change Behaviors • Contaminate Characteristics
Start And Jump Numbers: Searching For Patterns • You will need to make a list of numbers that begin with a “start number” and increase by a fixed amount we will call the “jump number.” First try 3 as the start number and 5 as the jump number. Write the start number at the top of your list, then, then 8, 13, and so on, “jumping” by 5 each time until your list extends to about 130.
Start And Jump Numbers: Searching For Patterns • Your task is to examine this list of numbers and find as many patterns as you possibly can. Share your ideas with the group, and write down every pattern you agree really is a pattern.
DIVERGENT THINKING Find and articulate as many patterns as possible in the list
of “Start Jump” numbers? Classify and label your patterns into groups. Name and describe them. Develop and describe a new problem or sequence derived from the original “Start Jump” problem. Create other sequences that are fundamentally different from the original. Describe how your new problem is different from the original “Start Jump” problem. How are the two similar? What conjectures or inferences can be made from this analysis? What other conclusions or relations might be stated?
DIVERGENT THINKING
FLUENCY FLEXIBILITY ORIGINALITY ELABORATION
Solve this problem. What was easy about it? Not so easy?
Create and solve a What’s one question problem similar to this someone should ask problem. themselves when they first look at this problem?
What’s a mistake that someone might make in trying to solve this problem? Why might they make that mistake?
What mathematical concepts or terms/vocabulary does this problem show? Be specific!
Write a step-by-step set of directions that tells someone who was absent today how to solve this problem.
H. Alang, A. Mironchuk, & J. Paul (Farragut Career Academy, Chicago IL
Consider the following tautology: Teachers
and students who think divergently with mathematics are those who believe that mathematics is a creative enterprise.
CCSS Mathematical Practices • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning Common Core Standards (CCSSO) 2010
Arkansas iTunes U
Understanding and Developing Fractions
Alternative Algorithms
Problem Solving the Square-iest Situations
Developing Divergent Thinking in the Mathematics Classroom
Betty Wood, Ph. D.
James Fetterly, Ph. D.
[email protected]
[email protected]
A Calculator Exploration Use a calculator to divide 1444 by 62. But you can’t use the “÷” symbol.
