1 Eric Mann, Ph.D. Purdue University [email protected] The Essence of Mathematics of Mathematics

11

Eric Mann, Ph.D.Eric Mann, Ph.D.Purdue UniversityPurdue University

[email protected]@purdue.edu

The Essence

The Essence

of Mathematics

of Mathematics

22

An Old RhymeAn Old Rhyme

Multiplication is vexation,Multiplication is vexation,

Division is just as bad;Division is just as bad;

The Rule of Three perplexes me,The Rule of Three perplexes me,

And Practice drives me mad.And Practice drives me mad.

33

Negative NumbersNegative Numbers A controversial topic for centuriesA controversial topic for centuries Initially used only as a computational toolInitially used only as a computational tool Existence as a real quantity challengedExistence as a real quantity challenged

• Descartes (1637): False, fictious numbersDescartes (1637): False, fictious numbers• Carnot (1803) to obtain an isolate negative Carnot (1803) to obtain an isolate negative

quantity, it would be necessary to cut off an quantity, it would be necessary to cut off an effective quantity from zero, to remove effective quantity from zero, to remove something from nothing: impossible operation. something from nothing: impossible operation.

• Busset (1843) Busset (1843) attributed the “failure of the teaching of mathematics in attributed the “failure of the teaching of mathematics in

France to the admission of negative quantities.” France to the admission of negative quantities.” compelled to declare that such mental aberrations could

prevent gifted minds from studying mathematics

44

Why is the product of twoWhy is the product of two negative numbers positive? negative numbers positive?

A fundamental question dealing A fundamental question dealing with the properties of operations on with the properties of operations on numbers numbers

• Obvious that the product is ab but is it Obvious that the product is ab but is it + or – ?+ or – ?

• Cannot be – as (-a by +b) gives –ab Cannot be – as (-a by +b) gives –ab and it can not have the same resultand it can not have the same result

• So, (-a by -b) must be positiveSo, (-a by -b) must be positive

Euler,1770Euler,1770

55

The Product of NegativesThe Product of Negatives

The Rule of Signs: To multiply a pair The Rule of Signs: To multiply a pair of numbers if both numbers have the of numbers if both numbers have the same sign, their product is the same sign, their product is the product of their absolute values product of their absolute values (their product is positive). (their product is positive).

TaughtTaught• Direct instructionDirect instruction• Rules and RhymesRules and Rhymes• Models (number lines, two-color chips)Models (number lines, two-color chips)

66

Does it make sense?Does it make sense?

PatternPattern

-2 x 3 = -6-2 x 3 = -6

-2 x 2 = -4-2 x 2 = -4

-2 x 1 = -2-2 x 1 = -2

-2 x 0 = 0-2 x 0 = 0

-2 x -1 = 2-2 x -1 = 2

-2 x -2 = 4-2 x -2 = 4

-2 x -3 = 6-2 x -3 = 6

Inductive ReasoningInductive Reasoning

If -1 x -1 = -1 thenIf -1 x -1 = -1 then

-1(1 + -1) = -1 x 1 + -1 x -1-1(1 + -1) = -1 x 1 + -1 x -1

-1(0) = -1 + -1-1(0) = -1 + -1

0 = -20 = -2

• Distributive Property (IN 5.3.3)• Add, subtract, multiply and divide

positive and negative rational numbers IN 6.2.1 & 6.2.2)

77

Don't Ask Why, Don't Ask Why, Just Invert and Multiply Just Invert and Multiply

2

11

2

3

1

3

2

1

3

1

2

1

88

Why?Why?

AlgebraicallyAlgebraically

VisuallyVisually

11

3

2

1

1

3

3

1

1

3

2

1

3

1

2

1

99

“ “We have known for some years We have known for some years

now…that most children’s now…that most children’s

mathematical journeys are in vain mathematical journeys are in vain

because they never arrive anywhere, because they never arrive anywhere,

and what is perhaps worse is that and what is perhaps worse is that

they do not even enjoy the journey.”they do not even enjoy the journey.”

(Whitcombe, 1988)(Whitcombe, 1988)

1010

Today’s Mathematics ClassroomToday’s Mathematics Classroom

Vision:Vision: a classroom where “students a classroom where “students confidently confidently

engageengage in complex mathematical tasks…draw in complex mathematical tasks…draw

on knowledgeon knowledge from a wide from a wide varietyvariety of of

mathematical topicsmathematical topics, sometimes , sometimes approaching approaching

the same problem from different the same problem from different

mathematical perspectives or representing mathematical perspectives or representing

the mathematics in different waysthe mathematics in different ways until they until they

find methods that enable them to make progress”find methods that enable them to make progress”

(NCTM, 2000)(NCTM, 2000)

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Today’s Mathematics ClassroomToday’s Mathematics ClassroomReality for many:Reality for many: time spent watchingtime spent watching as mathematical methods were as mathematical methods were demonstrated; where time was spent demonstrated; where time was spent committing to memory facts and committing to memory facts and algorithmsalgorithms (Pehkonen, 1997),(Pehkonen, 1997), conceiving conceiving mathematics as “a digestive processmathematics as “a digestive process rather than a creative one” rather than a creative one” (Dreyfus and Eisenberg, 1996),(Dreyfus and Eisenberg, 1996), conveying instead conveying instead the beliefthe belief that math is that math is divided into right and wrong and divided into right and wrong and that “the that “the essence of math is getting the right essence of math is getting the right onesones”” (Ginsburg, 1996, Balka, 1974).(Ginsburg, 1996, Balka, 1974).

1212

What our math classrooms might be…What our math classrooms might be…

1313

Essence of MathematicsEssence of Mathematics

““The essence of mathematics is not producing The essence of mathematics is not producing the correct answers, but thinking creatively” the correct answers, but thinking creatively” (Ginsburg, 1996). (Ginsburg, 1996).

Accuracy is important as the students’ responses Accuracy is important as the students’ responses must fit the context of the problem and be must fit the context of the problem and be mathematically correct mathematically correct

BUT, “if we reject original or clever applications BUT, “if we reject original or clever applications with small errors in application we discourage with small errors in application we discourage students from risk taking” (Poincaré, 1913). students from risk taking” (Poincaré, 1913).

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Creativity’s Vital Role in Creativity’s Vital Role in MathematicsMathematics

““Modern day commerce has no use for pupils graduating Modern day commerce has no use for pupils graduating

from school who have been from school who have been trained to mechanically trained to mechanically

solve problems in exactly one pre-given way, i.e. like solve problems in exactly one pre-given way, i.e. like

a machinea machine””

(Köhler, 1997).(Köhler, 1997).

For individuals to use mathematics in ways that For individuals to use mathematics in ways that go well go well beyond what they are taught creative mathematical beyond what they are taught creative mathematical thinking is keythinking is key..

(Sternberg, (Sternberg,

1996)1996)

1515

Mathematical Creativity DefinedMathematical Creativity Defined One view “includes the ability to One view “includes the ability to see new relationshipssee new relationships

between techniques and areas of application, and to between techniques and areas of application, and to makemake

associations between possibly unrelated ideas”associations between possibly unrelated ideas” (Tammadge as cited in Haylock, 1987). (Tammadge as cited in Haylock, 1987).

Krutetskii approaches creativity from a problem-solving Krutetskii approaches creativity from a problem-solving perspective characterizing creativity in the context of perspective characterizing creativity in the context of problem-formation (problem-finding), invention, problem-formation (problem-finding), invention, independence and originalityindependence and originality (Krutetskii, 1976; Haylock, 1987). (Krutetskii, 1976; Haylock, 1987).

Others have applied the Others have applied the concepts of fluency, flexibility concepts of fluency, flexibility and originalityand originality to mathematics to mathematics (Tuli, 1980; Haylock, 1997; Kim, Cho, Ahn, 2003). (Tuli, 1980; Haylock, 1997; Kim, Cho, Ahn, 2003).

1616

The ability toThe ability to Formulate mathematical hypothesesFormulate mathematical hypotheses Determine patternsDetermine patterns Break from established mind sets to obtain Break from established mind sets to obtain

solutions in a mathematical situation solutions in a mathematical situation Sense what is missing and ask questions Sense what is missing and ask questions Consider and evaluateConsider and evaluate unusual mathematical unusual mathematical

ideas, to ideas, to think through the consequencesthink through the consequences from a from a mathematical situation (divergent)mathematical situation (divergent)

Criteria for Measuring Creative Criteria for Measuring Creative Ability in Mathematics Ability in Mathematics (Balka, 1974)(Balka, 1974)

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A Criteria for Measuring Creative A Criteria for Measuring Creative Ability in Mathematics Ability in Mathematics (Balka, 1974)(Balka, 1974)

The ability to:The ability to: Formulate mathematical hypothesesFormulate mathematical hypotheses

concerning cause and effect in a mathematical concerning cause and effect in a mathematical situation (divergent)situation (divergent)

Determine patternsDetermine patterns in mathematical in mathematical

situations (convergent)situations (convergent)

1818

What is your hypotheses?What is your hypotheses?

Sample ProblemSample Problem: Is there a connection between the : Is there a connection between the number of sides of a polygon and the number of number of sides of a polygon and the number of diagonals you can make?diagonals you can make?

Problem is loosely structured – a research questionProblem is loosely structured – a research question

Cause:Cause: change in number of sideschange in number of sidesEffect: Effect: change in number of diagonalschange in number of diagonals

1919

Emma’s Work (9 years old) Emma’s Work (9 years old) (Koshy, 2001, p. 43-44)(Koshy, 2001, p. 43-44)

2020

Another ChallengeAnother Challenge

You have three jugs A, B, and CYou have three jugs A, B, and C The problem is to find the best way of The problem is to find the best way of

measuring out a given quantity of water, measuring out a given quantity of water,

using just three jugs.using just three jugs. You are told how much each jug holds.You are told how much each jug holds. There are no marks on the jugs so the only There are no marks on the jugs so the only

way to make an accurate measurement is way to make an accurate measurement is

to fill a jug to the brim.to fill a jug to the brim.

2121

An ExampleAn Example

Measure out 55 units if Jug A holds Measure out 55 units if Jug A holds

10 units, Jug B holds 63 units and Jug 10 units, Jug B holds 63 units and Jug

C hold 2 units.C hold 2 units. Solution: Fill B, pour off A, Fill C add Solution: Fill B, pour off A, Fill C add

to B orto B or• B – A + C = 63 – 10 + 2 B – A + C = 63 – 10 + 2

2222

Your TurnYour Turn

Measure Measure outout

Jug A Jug A holdsholds

Jug BJug B

holdsholdsJug C Jug C

holdsholdsSolutionSolution

11 52 units52 units 1010 6464 11

22 14 units14 units 100100 124124 55

33 3 units3 units 1010 1717 22

44 100 units100 units 2121 127127 33

55 20 units20 units 2323 4949 33

66 5 units5 units 5050 6565 55

2323

Your TurnYour Turn

Measure Measure outout

Jug A Jug A holdsholds

Jug BJug B

holdsholdsJug C Jug C

holdsholdsSolutionSolution

11 52 units52 units 1010 6464 11 B-A-2CB-A-2C




55 20 units20 units 2323 4949 33 A-CA-C

66 5 units5 units 5050 6565 55 CC

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A Criteria for Measuring Creative A Criteria for Measuring Creative Ability in Mathematics Ability in Mathematics (Balka, 1974)(Balka, 1974)

The ability to:The ability to: Formulate mathematical hypothesesFormulate mathematical hypotheses Determine patternsDetermine patterns Break from established mindBreak from established mind sets sets to obtain to obtain

solutions in a mathematical situation solutions in a mathematical situation (convergent)(convergent)

2525

The Einstellung EffectThe Einstellung Effect

The Einstellung effect: set successful procedure The Einstellung effect: set successful procedure

applied consistently even when it is less than applied consistently even when it is less than

efficient or inappropriateefficient or inappropriate The jug problem with 250, 11-12 year oldsThe jug problem with 250, 11-12 year olds

• 70% used the same procedure on all 6 problems70% used the same procedure on all 6 problems

• 11% used a different procedure on 1 problem11% used a different procedure on 1 problem

• 11% used a different procedure on 2 problems11% used a different procedure on 2 problems

• 8% couldn’t do the problems 8% couldn’t do the problems

(Haylock, 1985)(Haylock, 1985)

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Try This One Try This One

One acre of good soil usually contains about three One acre of good soil usually contains about three

million worms.million worms.

How many worms can Asa and David expect to find How many worms can Asa and David expect to find

between each 10-yard line as they dig up the between each 10-yard line as they dig up the

perfectly manicured grass on the junior high perfectly manicured grass on the junior high

school football field?school football field?

(Note: 1 acre = 4, 840 square yards)(Note: 1 acre = 4, 840 square yards)(Kleiman and Washington, 1996)(Kleiman and Washington, 1996)

2727

The Answer is…The Answer is… NOT ENOUGH INFORMATIONNOT ENOUGH INFORMATION You need to know the area of the football field You need to know the area of the football field

but the problem doesn’t give you its widthbut the problem doesn’t give you its width Divide area in square yards by yard per acre then Divide area in square yards by yard per acre then

find 1/10 of that, and then multiply by 3 million find 1/10 of that, and then multiply by 3 million worms per acreworms per acre

If you knew a football field was 55 yards wide the If you knew a football field was 55 yards wide the answer is 340,909.1 wormsanswer is 340,909.1 worms

Now the question is what is 0.1 worm?Now the question is what is 0.1 worm?

2828


The ability toThe ability to

Formulate mathematical hypothesesFormulate mathematical hypotheses Determine patternsDetermine patterns Break from established mind sets to obtain Break from established mind sets to obtain

solutions in a mathematical situation solutions in a mathematical situation Sense what is missingSense what is missing from a given from a given

mathematical situation andmathematical situation and to ask questionsto ask questions that that

will enable one to fill in the missing mathematical will enable one to fill in the missing mathematical

information (divergent)information (divergent)

2929


solutions in a mathematical situation solutions in a mathematical situation Sense what is missing ask questions Sense what is missing ask questions Consider and evaluateConsider and evaluate unusual mathematical unusual mathematical

ideas, to ideas, to think through the consequencesthink through the consequences from a mathematical situation (divergent)from a mathematical situation (divergent)


3030

How would you measure a puddle?How would you measure a puddle? (Westly, 1994,1995) (Westly, 1994,1995)

• Record all the Record all the

different ways different ways

you can think of.you can think of.

• Make sketches to Make sketches to

show your work.show your work.

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Here are some ways by aHere are some ways by a22ndnd grader…&…a 7 grader…&…a 7thth grader grader

(Westly, 1994,1995)(Westly, 1994,1995)

3232



solutions in a mathematical situation solutions in a mathematical situation Sense what is missing ask questions Sense what is missing ask questions Consider and evaluate unusual mathematical Consider and evaluate unusual mathematical

ideas, to think through the consequencesideas, to think through the consequences Split general mathematicalSplit general mathematical problems into problems into

specific sub-problemsspecific sub-problems (divergent)(divergent)

3333

Math in Strange Places Math in Strange Places At the local grocery store bulletin board, Asa and At the local grocery store bulletin board, Asa and

David found these 8 signs. Create three David found these 8 signs. Create three problems of your own based on these signs.problems of your own based on these signs.Are you Illiterate? If so, call

1-800-CANT-READ

Time Travelers’ Meeting: 7:45 p.m. last Tuesday

PSYCHICS’ MEETING: You Know Where, You know When

Weight Watchers: $14 All You Can Eat Hot Fudge Sundae and

Cheesecake Dinner

Janitors Wanted, $42,000 starting salary, Ph.D. required

Antique Collectors Display: Pentium Computers, Then and Now. See our collection dating all the way back to

1995

TIME MANAGEMENT WORKSHOP for people who find themselves just too busy. Meeting 6:30 – 9:30 weekly,

attendance mandatory

Central New York Computer Club: “Learning to Use Spell Check”Meat in the Lobby of Supper 8 Motel, 8-9 p.m., Wednesday

3434

Math in Strange PlacesMath in Strange PlacesIf you gain 1/2 pound for every Hot Fudge Sundae and 1/4 pound for each piece of cheesecake what are the possible combinations of sundaes and pieces of cheesecake that you ate if you gained 6 pounds? You have 4 friends and $97. Tickets for the roller coaster are $2.50. Do you have enough money to treat everyone to the Weight Watchers dinner and a ride on the roller coaster?

Let's suppose today is Friday. If the Psychic’s meetings are always 96.5 hours after the Time Traveler’s meeting when is (or when was) their next meeting?

In preparing for the auction at the antique display, Jeff wants to know how much the price of Pentium computers has changed since 1995. How can you help him?

What information would you need to decide if the job as a janitor is a good choice for someone with a Ph.D.?

3535

The Beauty of MathematicsThe Beauty of Mathematics

An understanding of mathematics is necessary to introduce your students to the beauty it holds yet…

3636

AlgorithmsAlgorithms Algorithms areAlgorithms are the focus in most mathematical the focus in most mathematical

classrooms and classrooms and the least importantthe least important as machines can as machines can

do it better and faster, they are boring, and they do it better and faster, they are boring, and they

offer the student no sense of the structure of offer the student no sense of the structure of

mathematics.” mathematics.” ““The human mathematical mind operates most The human mathematical mind operates most

efficiently when the following faculties are involved: efficiently when the following faculties are involved: LOGICAL - ALGORITHMICLOGICAL - ALGORITHMIC INTUITIVE - CREATIVEINTUITIVE - CREATIVE ASTHETIC - SPECULATIVEASTHETIC - SPECULATIVE

(Whitcombe, (Whitcombe, 1988)1988)

3737

AlgorithmsAlgorithms There is converging evidence that students often emerge There is converging evidence that students often emerge

from K-12 mathematics education with adequate problem from K-12 mathematics education with adequate problem

execution skills –– but with inadequate problem execution skills –– but with inadequate problem

representation skills representation skills (Mayer and Heagarty,1996) (Mayer and Heagarty,1996)

TIMSS Study: But can we use the math?TIMSS Study: But can we use the math?

3838

Whitcombe’s Model of the Whitcombe’s Model of the Mathematical MindMathematical Mind

Algorithms Beauty

Creativity

3939

AlgorithmsAlgorithms AdditionAddition SubtractionSubtraction MultiplicationMultiplication DivisionDivision PercentagesPercentages RatioRatio PythagorasPythagoras QuadraticsQuadratics Simple Simple

InterestInterest MeansMeans FormulasFormulas


4040

Algorithms Addition Subtraction Multiplication Division Percentages Ratio Pythagoras Quadratics Simple Interest Means Formulas

CreativityCreativity

Problem Problem SolvingSolving

InvestigationsInvestigations Pattern MakingPattern Making OriginalityOriginality SpeculationSpeculation ThinkingThinking ConceptsConcepts StrategiesStrategies


4141

Algorithms

Addition Subtraction Multiplication Division Percentages Ratio Pythagoras Quadratics Simple Interest Means Formulas

Beauty

Structure Form Relations Visualizations Economy Simplicity Elegance Order

Creativity

Problem Solving Investigations Pattern Making Originality Speculation Thinking Concepts Strategies


4242

Assessing Mathematical Reasoning Assessing Mathematical Reasoning

4343

The Fun in Math The Fun in Math

Can you read this Math Poem?Can you read this Math Poem?

((12 + 144 + 20 +(3 * 4((12 + 144 + 20 +(3 * 4(1/2)(1/2)))) ÷ 7) + (5 x 11)= 9))) ÷ 7) + (5 x 11)= 92 2 + 0+ 0

Jon SaxtonJon Saxton

4444

((12 + 144 + 20 +(3 * 4((12 + 144 + 20 +(3 * 4(1/2)(1/2)))) ÷ 7) + (5 x 11)= 9))) ÷ 7) + (5 x 11)= 92 2 + 0+ 0

A Dozen, a Gross, and a Score,A Dozen, a Gross, and a Score,

plus three times the square root of four,plus three times the square root of four,

divided by seven,divided by seven,

plus five times eleven,plus five times eleven,

equals nine squared and not a bit more.equals nine squared and not a bit more.

Found on-line at http://ctl.unbc.ca/CMS/poems.htmlFound on-line at http://ctl.unbc.ca/CMS/poems.html

4545

Problem Posing & Creativity RubricProblem Posing & Creativity RubricSheffield, 2003Sheffield, 2003

AssessmenAssessment Criteriat Criteria 11 22 33 44

Depth of Depth of UnderstandingUnderstanding

Little or no Little or no understandinunderstandingg

Partial Partial understanding; understanding; minor minor mathematical mathematical errorserrors

Good Good understanding, understanding, mathematically mathematically correctcorrect

In-depth In-depth understandingunderstanding; well ; well developed developed ideasideas

FluencyFluency

One One incomplete incomplete or or unworkable unworkable approachapproach

At least one At least one appropriate appropriate approach or approach or related questionrelated question

At least two At least two appropriate appropriate approaches or approaches or good related good related questionsquestions

Several Several appropriate appropriate approaches or approaches or new related new related questionsquestions

FlexibilityFlexibility

All approaches All approaches use the same use the same method (e.g., all method (e.g., all graphs, all graphs, all algebraic algebraic equations, etc.)equations, etc.)

At least two At least two methods of methods of solution (e.g. solution (e.g. geometric, geometric, graphical, graphical, algebraic, algebraic, physical physical modeling, etc.)modeling, etc.)

Several Several methods of methods of solution.solution.

4646


Assessment Assessment CriteriaCriteria 11 22 33 44

OriginalityOriginality

Method may Method may be different be different but does not but does not lead to a lead to a solutionsolution

Method will lead Method will lead to a solution but to a solution but is fairly commonis fairly common

Unusual, Unusual, workable workable method used method used by only a few by only a few studentsstudents

Unique, Unique, insightful insightful method use by method use by only one or two only one or two studentsstudents

Elaboration or Elaboration or EleganceElegance

Little or no Little or no appropriate appropriate explanation explanation givengiven

Explanation is Explanation is understandable understandable but may be but may be unclear in placesunclear in places

Clear Clear explanation explanation using correct using correct mathematical mathematical termsterms

Clear, concise, Clear, concise, precise precise explanations explanations making good making good use of graphs, use of graphs, charts, models charts, models or equationsor equations

4747


Assessment Assessment CriteriaCriteria 11 22 33 44

GeneralizationsGeneralizations

No No generalizationgeneralizations made, or s made, or they are they are incorrect and incorrect and the reasoning the reasoning is unclearis unclear

At least one At least one correct correct generalization generalization made; may not be made; may not be well supported well supported with clear with clear reasoningreasoning

At least one At least one well-made, well-made, supported supported generalization, generalization, or more than or more than one correct but one correct but unsupported unsupported generalizationgeneralization

Several well Several well supported supported generalizations; generalizations; clear reasoningclear reasoning

ExtensionsExtensions

None included, None included, or extensions or extensions are not are not mathematicalmathematical

At least one At least one related related mathematical mathematical question question appropriately appropriately exploredexplored

One related One related question question explored in explored in depth, or more depth, or more than one than one appropriately appropriately exploredexplored

More than one More than one question question explored in explored in depth.depth.

4848

Parting ThoughtsParting Thoughts ““Any” fruit of human endeavor shows creativity, if you Any” fruit of human endeavor shows creativity, if you

think about it. The interesting question to me is this: think about it. The interesting question to me is this: Why is it that a student who is only playing other Why is it that a student who is only playing other people's music instinctively understands that those people's music instinctively understands that those composers were creative, and that s/he might aspire to composers were creative, and that s/he might aspire to the same kind of creativity -- or, in English class, the same kind of creativity -- or, in English class, instinctively understands that those writers were instinctively understands that those writers were creative, even when s/he is just reading their creations creative, even when s/he is just reading their creations and answering quiz questions about them -- but doesn't and answering quiz questions about them -- but doesn't have the same instinctive understanding that Euclid and have the same instinctive understanding that Euclid and Newton and Pascal and Gauss and Euler were creative Newton and Pascal and Gauss and Euler were creative mathematicians? The most obvious answer has to do mathematicians? The most obvious answer has to do with the way these disciplines are taught.with the way these disciplines are taught.

(Bogomolny, 2000)

4949

References CitedReferences CitedBalka, D. S., (1974) Creative ability in mathematics, Balka, D. S., (1974) Creative ability in mathematics, Arithmetic Teacher,Arithmetic Teacher, 21, 633-636 21, 633-636

Dreyfus, T. & Eisenberg (1996) On different facets of mathematical thinking, In Sternberg, R., J & Ben-Zeev, Dreyfus, T. & Eisenberg (1996) On different facets of mathematical thinking, In Sternberg, R., J & Ben-Zeev, T. (Eds) T. (Eds) The Nature of Mathematical Thinking,The Nature of Mathematical Thinking, (pp. 253 - 284), (pp. 253 - 284), Mahwah, NJ: Lawrence Erlbaum Mahwah, NJ: Lawrence Erlbaum AssociatesAssociates

Haylock, D. W. (1987), A framework for assessing mathematical creativity in school children, Haylock, D. W. (1987), A framework for assessing mathematical creativity in school children, Education Education Studies in Mathematics, Studies in Mathematics, 18 (1), 59-74.18 (1), 59-74.

Haylock, D. W. (1985), Conflicts in the assessment and encouragement of mathematical creativity in Haylock, D. W. (1985), Conflicts in the assessment and encouragement of mathematical creativity in schoolchildren, schoolchildren, International Journal of Mathematical Education in Science and Technology, International Journal of Mathematical Education in Science and Technology, 16 (4) 16 (4) 547-553547-553

Ginsburg, H. P. (1996) Toby’s math, In Sternberg, R., J & Ben-Zeev, T. (Eds) Ginsburg, H. P. (1996) Toby’s math, In Sternberg, R., J & Ben-Zeev, T. (Eds) The Nature of Mathematical The Nature of Mathematical Thinking,Thinking, (pp. 175-282), (pp. 175-282), Mahwah, NJ: Lawrence Erlbaum Associates.Mahwah, NJ: Lawrence Erlbaum Associates.

Kim, H., Cho, S., & Ahn, D. (2003) Development of mathematical creative problem solving ability test for Kim, H., Cho, S., & Ahn, D. (2003) Development of mathematical creative problem solving ability test for identification of gifted in math. identification of gifted in math. Gifted Education International Gifted Education International 18, 184-174.18, 184-174.

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Kleilman, A., Washington, D. & Washington, M., (1996) Kleilman, A., Washington, D. & Washington, M., (1996) It’s alive! Math like you’ve never know it before, It’s alive! Math like you’ve never know it before, and may never know it again…,and may never know it again…,Waco, TX: Prufrock PressWaco, TX: Prufrock Press

Koshy, V. (2001), Koshy, V. (2001), Teaching mathematics to able children, Teaching mathematics to able children, London: David Fulton PublishersLondon: David Fulton Publishers

Knutenskii, V. A. (1976), The psychology of mathematical abilities in school children, Chicago, University Knutenskii, V. A. (1976), The psychology of mathematical abilities in school children, Chicago, University of Chicago Press.of Chicago Press.

Köhler, H. (1997) Acting artist-like in the classroom, Köhler, H. (1997) Acting artist-like in the classroom, International Reviews on Mathematical Education,International Reviews on Mathematical Education, 29 29 (3) 88-93. Electronic Edition (3) 88-93. Electronic Edition http://www.fiz-karlsruhe.de/fix/publications/zdm/adm97http://www.fiz-karlsruhe.de/fix/publications/zdm/adm97, accessed , accessed March 10, 2003.March 10, 2003.

Poincaré, H. (1913) Poincaré, H. (1913) The foundations of scienceThe foundations of science, New York: The Science Press., New York: The Science Press.

Pehkonen, E (1997) The state-of-art in mathematical creativity, Pehkonen, E (1997) The state-of-art in mathematical creativity, International Reviews on Mathematical International Reviews on Mathematical Education,Education, 29 (3) 63-66. Electronic Edition 29 (3) 63-66. Electronic Edition http://www.fiz-karlsruhe.de/fix/publicationshttp://www.fiz-karlsruhe.de/fix/publications/ zdm/adm97/ zdm/adm97, , accessed March 10, 2003.accessed March 10, 2003.

Principles and standards for school mathematics, Principles and standards for school mathematics, (2000) Reston, VA: The National Council of Teachers of (2000) Reston, VA: The National Council of Teachers of Mathematics, Inc.Mathematics, Inc.

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Sheffield, L. J. (2003) Sheffield, L. J. (2003) Extending the challenge in mathematics, developing mathematical promise Extending the challenge in mathematics, developing mathematical promise in K-8 students, in K-8 students, Thousand Oaks, CA: Corwin Press, Inc.Thousand Oaks, CA: Corwin Press, Inc.

Sternberg, R. J., (1996) What is mathematical thinking, In Sternberg, R., J & Ben-Zeev, T. (Eds) Sternberg, R. J., (1996) What is mathematical thinking, In Sternberg, R., J & Ben-Zeev, T. (Eds) The The Nature of Mathematical Thinking,Nature of Mathematical Thinking, (pp. 303-318), (pp. 303-318), Mahwah, NJ: Lawrence Erlbaum AssociatesMahwah, NJ: Lawrence Erlbaum Associates

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Documents

1 Eric Mann, Ph.D. Purdue University [email protected] The Essence of Mathematics of Mathematics