What Is Mathematics?
A Pedagogical Answer with a Particular
Reference to Proving
Guershon Harel
University of California at San Diego
harel@math.ucsd.edu

http://www.math.ucsd.edu/~harel
1

Two Fundamental Questions
1. What mathematics should we teach inschool?
2. How should we teach It?

DNR’s stance: an overview

2

1

Presentation StructurePart I:DNR as a conceptual framework: a synopsis

Part II:
DNR’s stance on the 2nd question:
How should mathematics be taught?

Part III:
DNR’s stance on the 1nd question:
What mathematics should be taught?
3Part I: What is DNR?
DNR-based instruction in mathematics is a theoretical framework
consisting of a system of three categories of constructs:– Premises:
explicit assumptions, most of which are taken from or based on
existing theories.

– Concepts:
definitions oriented within the stated premises.

– Claims:
statements formulated in terms of the DNR concepts, entailed fromthe DNR premises, and supported by empirical studies.

•

Instructional principles: claims about effects of teachingpractices on student learning.

The term DNR refers to three foundational instructional principles:–The Duality Principle–The Necessity Principle–The Repeated Reasoning Principle
4

2

DNR Premises
Mathematics Premise1.
Mathematics: Knowledge of mathematics consists of all ways ofunderstanding and ways of thinking that have evolved throughout history.
Learning Premise2.
Epistemophilia: Humans—all humans—possess the capacity to develop
a desire to be puzzled and to learn to carry out mental acts to fulfill their
desire to be puzzled and to solve the puzzles they create. Aristotle
3.
Adaptation: Learning is adaptation; namely, learning is a developmental
process, which proceeds through a continual tension between assimilation
and accommodation, directed toward a (temporary) equilibrium—abalance between the structure of the mind and the environment. Piaget
4.
Content: Learning is context dependent. Cognitive Psychology
Teaching Premise6.
Teaching: Construction of scientific knowledge is not spontaneous.
There will always be a difference between what one can do under expert
guidance or in collaboration with more capable peers and what he or she
can do without guidance. Vygotsky
Ontology Premise7.
Subjectivity: Any observations humans claim to have made is due towhat their mental structure attributes to their environment. Piaget
8.
Interdependency: Humans’ actions are induced and governed by theirviews of the world, and, conversely, their views of the world are formed
5 by
their actions. Piaget

Part II: The 2st Fundamental Question
How should mathematics be taught?
In DNR, teaching effectively entails:•••

preserving the mathematical integrity of what
we teachaddressing the intellectual needs of the student
assuring that students•••

internalize,
Structure,retainthe mathematics they learn

63The Necessity Principle
For students to learn the mathematics we intend
to teach them, they must have a need for it,
where ‘need’ refers to intellectual need, notsocial or economic need.

7

Violation of the Necessity Principle:
Some Examples

84Premature introduction of algebraic symbolism
Tom and John are roommates. They decided to painttheir room. Tom can paint the room in 4 hours andJohn, a perfectionist, in 8 hours. How long would ittake them to paint their room if they work together?
Kate:… 4x + 8x …
Teacher:
What is x?
Kate:… x?... x is the house.
Teacher : You want to find x, … so you wantto find the house?
9

Tom and John are roommates; they decided to painttheir room. Tom can paint the room in 4 hours andJohn, a perfectionist, in 8 hours. How long would ittake them to paint their room if they work together?
Kate:
Tom paints ¼ of the house in 1 hour.

105Tom and John are roommates; they decided to painttheir room. Tom can paint the room in 4 hours andJohn, a perfectionist, in 8 hours. How long would ittake them to paint their room if they work together?
Kate:John paints 1/8 of the house in 1 hour.

11

Tom and John are roommates; they decided to painttheir room. Tom can paint the room in 4 hours andJohn, a perfectionist, in 8 hours. How long would ittake them to paint their room if they work together?
Kate:

1 hour

1 hour12

6

Tom and John are roommates; they decided to paint their room.
Tom can paint the room in 4 hours and John, a perfectionist,in 8 hours. How long would it take them to paint their room ifthey work together?
Kate:

1 hour

1 hour• ¾ of the house is painted in 2 hours, ¼ of the house will be
painted in 2/3 of an hour.
• The whole house will be painted in 2 hours and 40 minutes.
• …This can’t be right … there is no x …
13

intellectual need
response

social need
response

The house will be painted in 2 hours and 40 minutes…This can’t be right … there is no x …
147

Harriet’s solution to the pool problem:
A pool is connected to two pipes. One pipe can fillup the pool in 20 hours, and the other in 30 hours.
How long will it take the two pipes together to fill up
the pool?

20-h pipe

44430-h pipe

6

6

15

Linear Algebra Textbooks“So far we have defined a mathematical systemcalled a real vector space and noted some of its
properties ....
[In what follows], we show that each vector space
V studied here has a set composed of a finite
number of vectors that completely describe V. It
should be noted that, in general, there is morethan one such set describing V. We now turn toa formulation of these ideas.”

Following this, the text defines the concepts:
Linear independence
Span
Basis

and proves related theorems.

168

Is Students’ Intellectual Need Considered?
• Can this “motivation” help students see a needfor the pivotal concepts, linear independence,
span, and basis?
• Can it constitute a need for the concept finitely
generated vector space (alluded to in the“motivating paragraph)?
• Can it help students see how these threepivotal concepts contribute to thecharacterization of finitely generated vector
space?
17

Five Categories of intellectual need: The 5 C’s
1. Need for certainty
2. Need for causality
3. Need for computation
4. Need for communication
5. Need for connection

18

9Categories of intellectual need(a) Need for certainty
Is it true that the sum of the angles in any triangle is 180º?

(b) Need for causality(Need for enlightenment)

What makes the sum of the angles in any triangle 180º?

(c) Need for computation
I am looking for a number whose square is less than its cube.
Can I find all such numbers?

(d) Need for communication
How can I convince Jill that I am right?

(e) Need for connection
What does the negative solution of an equation mean in the19physical reality represented by the equation?

Need for certainty
versusNeed for causality

20

10

Aristotle’s Definition of Science“We do not think we understand something until wehave grasped the why of it. … To grasp the why ofa thing is to grasp its primary cause.” Aristotle,
Posterior Analytics.

21

Mathematics is not a perfect science, argued 1617th Century philosophers, because an"implication" is not just a logical consequence; itmust also demonstrate the cause of theconclusion.

22

11

Euclid’s Proof to Proposition 1.32
The sum of the three interior angles of a triangle
is equal to 1800.
Proof:
A

B

E

C

D

23

“Mathematics is not scientific:”
Examples of other arguments• Proof by contradiction is not a causal proof
since it does not provide sufficient insight of
how the result was obtained.
• If mathematical proof are scientific (i.e., causal),then equivalent statements (i.e., “A iff B”
statements) are an absurdity.
“A implies B” means “A causes B”“B implies A” means “B causes A”
Hence: “A causes A”—an absurdity.
24

12

Attempts to Conform to the Aristotelian
Theory of Science
Rejection of proof by contradiction and use
ostensive (causal) proofs– Descartes’ appeal to a priori proofs against proofs bycontradiction– Cavalieri’s geometry of indivisibles– Insistence on movement in geometrical proofs

25

Theorem: For every triangle, the sum of themeasures of its interior angles is 1800.
Proof 1

Proof 2

C

A

A

B

B

K

C

J

C

A

B

K

J
C

A

26
L

B

13

Students’ Conceptions of Proof:
Selected Results• Students justify mathematical assertions by
examples• Often students’ inductive verifications consistof one or two example, rather than a multitude
of examples.
• Students’ conviction in the truth of an assertionis particularly strong when they observe apattern.
27•

Students view a counterexample as an
exception—in their view it does notaffect the validity of the statement.

•

Confusion between empirical proofs andproofs by exhaustion.

•

Confusion between the admissibility of
proof by counterexample with theinadmissibility of proof by example.
28

14

Teaching Actions with Limited Effect– Raising skepticism as to whether theassertion is true beyond the cases
evaluated.
– Showing the limitations inherent in the useof examples through situations such as:
The conjecture “ 1141y + 1 is an integer” is false for 1 ≤ y ≤ 10 .
The first value for which the statement is true is:30,693,385,322,765,657,197,397,208
2

25

29

Why showing the limitations inherent in
the use of examples is not effective?
• Students do not seem to be impressedby situations such as:• The conjecture “ 1141 y + 1 is an integer” is false for 1 ≤ y ≤ 10 .
The first value for which the statement is true is:30,693,385,322,765,657,197,397,208
2

25• Students view a counterexample as an
exception—in their view it does not affectthe validity of the statement.
30

15

How can instruction facilitate the transition fromempirical reasoning to deductive reasoning?
The Role of the Need for Causality

31

The Necessity Principle
For students to learn what we intend to teachthem, they must have a need for it, where ‘need’refers to intellectual need, not social or
economic need.
To implement the necessity principle:
1. Recognize what constitutes an intellectual need fora particular population of students, relative to theconcept to be learned.
2. Present the students with a sequence problems
that correspond to their intellectual need, and from
whose solution the concept may be elicited.
3. Help students elicit the concept from the problem
solution.
32

16

33

Doris Solution
Dimension
1

Number ofWhite Squares
0(1-1)234(3-1)2516(5-1)2736(7-1)234

17

Doris Solution
Dimension
1

Number ofWhite Squares
0(1-1)234(3-1)2516(5-1)2736(7-1)2(x-1)2x

35

John’s Solutionx⎛ x −1⎞
⎜
⎟
⎝ 2 ⎠

2

⎛ x −1⎞
⎜
⎟
⎝ 2 ⎠

2

⎛ x −1⎞
⎜
⎟
⎝ 2 ⎠

2

⎛ x −1⎞
⎜
⎟
⎝ 2 ⎠

2

⎛ x −1⎞
4⎜
⎟
⎝ 2 ⎠

2

= ( x − 1)

2

36

18

Towns A and B are 300 miles apart. At 12:00 PM,
a car leaves A toward B, and a truck leaves B
toward A. The car drives at 80 m/h and the truckat 70 m/h. When and where will they meet?
Students’ reasoning:
After 1 hour, the car drives 80 miles and truck 70
miles.
Together they drive 150 miles.
In 2 hours they will together drive 300 miles.
Therefore,
They will meet at 2:00 PM.
37
They will meet 160 miles from A.

Towns A and B are 300 miles apart. At 12:00 PM,
a car leaves A toward B, and a truck leaves B
toward A. The car drives at 80 m/h and thetruck at 70 m/h. When and where will they
meet?

38

19

Towns A and B are 300 100 miles apart. At
12:00 PM, a car leaves A toward B, and a truck
leaves B toward A. The car drives at 80 m/h
and the truck at 70 m/h. When and where will
they meet?
Students’ reasoning:
It will take them less than one hour to meet.
It will take them more than 30 minutes to meet.
They will meet closer to B than to A.
Let’s try some numbers:
50

80 +

60
4 0

50

?
70 = 100

60
80 +

60

40

?
7 0 = 1 0 0 (Y e s !)60

39

Towns A and B are 118 miles apart. At 12:00 PM,
a car leaves A toward B, and a truck leaves B
toward A. The car drives at 80 m/h and thetruck at 70 m/h. When and where will they
meet?
Necessitating the concept of unknown

Students’ reasoning:
It will take them less than one hour to meet;
It will take them more than 30 minutes to meet;
They will meet closer to B than to A.
?
Let’s try some numbers: 5 0 8 0 + 5 0 7 0 = 1 1 860deate g
p
e
R
in
so n
a
e
R

40

60
80 +

60

?
70 = 118

60

30
80
60

40

+

30
60

?
70 = 118

40

20

Goal: To necessitate the e-N definition of limit
1lim
Teacher: What is
and why?
n →∞ nStudents:11
lim = 0 because the larger n gets the closer is to zero.
n →∞ nnTeacher:

4

2

5

10

f(x) = -1
-2

-4

1
1
lim = −1 because the larger n gets the closer is to -1.
n →∞ nn
41

Part III: The 1st Fundamental Question
What mathematics should we teach in school?
DNR’s stance:
Mathematics consists of two categories ofknowledge:
ways of understanding and ways of thinking
Hence, it is necessary to focus on bothcategories of knowledge

42

21

ways of understanding = subject matter
E.g., definitions, theorems, proofs, problems, solutions, etc.

ways of thinking = conceptual tools
E.g., problems solving approaches, beliefs aboutmathematics

43

Ed’s Background
1. Second grader
2. Reported by his teacher to be slightly above
average in mathematics and average inother subjects
3. Ed’s formal instruction had included
addition and subtraction without regrouping
4. Ed was informally exposed to the basic
notion of multiplication as repeated addition
and the meaning of division as sharing
equally
5. Ed was never taught any division strategy44

22

Ed’s solution to Division Problems(Ed is a 71/2-year-old child)

1. What is Ed doing?
2. How would I respond to Ed?
3. What value, if any, do I see in Ed’s solution?

45

Ed’s Strategy for Solving Division Problems
Interviewer:Ed:

How much is 42 divided by 7?
… That’s easy …
40 divided by 10 is 4
3 plus 3 plus 3 plus 3 is 12
12 plus 2 is 14
14 divided by 2 is 7
2 plus 4 is 6
The answer is 6! (triumphantly)
Interviewer (to himself):
Okay, miracles do happen … 46

23

Interviewer :Ed:

How about 56 divided by 8?
You do the same thing (Impatiently)
50 divided by 10 is 5
5 times 2 is 10
10 plus 6 is 16
16 divided by 2 is 8
5 plus 2 is 7
The answer is 7.

Interviewer :Ed:… And 72 divided by 9?
Are you going to ask me every single
problem?

47

Ed:… okay … 72 divided by 9 …
70 divided by 10 is 7
7+2 is 9
1 and 7 is 8
The answer is 8.

48

24

A survey of mathematics teachers• Why do we teach the long division algorithm,
the quadratic formula, techniques ofintegration, and so on when one can perform
arithmetic operations, solve many
complicated equations, and integrate complex
functions quickly and accurately using
electronic technologies?

49

Teachers’ Answers

Justifications• “These materials appear on
standardized tests.”• Teachers must prepare
students for tests mandated
by their districts.

• “One should be able to solve
problems independently incase a suitable calculator is
not present.”• Teachers must teachstudents to do calculations
without a calculator,
especially those needed indaily life.

• “Such topics are needed tosolve real-world problems
and to learn more advanced
topics.”• Teachers must prepare
students for more advanced
courses where certain
computational skills might beassumed.
50

25

Teachers’ answers are external tomathematics as a discipline
The justifications for these answers are:• neither cognitive– Role of computational skills in one’s conceptual
development of mathematics• nor epistemological– Role of computations in the development ofmathematics• mainly social– Role of computational skills in the context of social
expectations
51

Why teach proofs?
Typical Answer:• So that students can be certain that thetheorems we teach them are true.
While this is an adequate answer—both
cognitively and (by inference)
epistemologically—it is incomplete.

52

26

The teachers had little to say when skepticallyconfronted about their answers by being
asked:– Do you or your students doubt the truth oftheorems that appear in textbooks?
– Is certainty the only goal of proofs?
– The theorems in Euclidean geometry, forexample, have been proven and re-proven formillennia. We are certain of their truth, so why
do we continue to prove them again andagain?

53• Textbooks and teachers at all levels tend toview mathematics– in terms of subject matter such as definitions,
theorems, proofs, algorithms, and problems
and their solutions.
– not in terms of conceptual tools necessary toconstruct such entities.

• Knowledge of subject matter is indispensable,
but it is not sufficient.
• Conceptual tools constitute an important
category of knowledge different from thesubject matter category.
54

27

• What exactly are these two categories ofknowledge, subject matter and conceptual
tools?
• What is the basis for the argument that bothcategories are needed?

55

A way of understanding is a particular product ofa mental act carried out by an individual.
¾
Interpreting Act
Ways of Understanding (products)• Three objects out of four objects• The sum ¼ + ¼ + ¼• The measure of the quantity resulting from dividing 3 units into 4 equal parts• The measure of a 3 cm long segment with a ruler whose unit is 4 cm long
• The solution to the equation 4x=3
• The equivalence class {3n/4n| n≠0}
• Two numbers with a bar between them.
56

28

A way of thinking is a characteristic of a mental
actInterpreting act
Ways of Thinking (characteristics)
One’s interpretation of symbols might be characteristically

1. inflexible: a symbol has a single interpretation
2. flexible: symbols can have multiple interpretations
3. non-referential: devoid of referents (quantitative,
spatial, functional, etc.)
4. referential: a representation of an entity within acoherent reality
57

A way of understanding is a particular product ofa mental act carried out by an individual.
Show that 2 is an upper bound for:2, 2 + 2 , 2 + 2 + 2 ,K

Proving act
Proof 2

Proof 1
2 = 1.41421 < 2

2 is less than 2
Therefore

2 + 2 = 1. 8478 < 2
2 + 2 + 2 = 1. 9616 < 2

2 + 2 is less than 4
Therefore
2 + 2 is less than 2

Therefore
Every term in thesequence is less than 2.

Therefore
2 + 2 + 2 is less than 4
Therefore
2 + 2 + 2 is less than 2

58

29

A way of thinking is a characteristic of a mental
act
proving act
Ways of Thinking (characteristics)
One’s proving act might be characteristically

1. inductive: based on examples or measurements
2. perceptual: based on visual perceptions
3. authoritarian: based on an authority (e.g., teacher)
4. deductive: based on rules of deduction
59

Proof Scheme
A person’s proof scheme is a persistentcharacteristic of one’s act of proving.

60

30

Taxonomy of Proof Schemes
External Conviction Proof Schemes
RitualAuthoritarian

Non-Referential
Symbolic

Empirical Proof Schemes
Inductive

Perceptual
61

Deductive Proof Schemes
Transformational

Modern Axiomatic

Generic
Contextual

CausalAxiomatizing Structural

Greek Axiomatic Constructive
Arithmetical Symbolic

Structural Symbolic

Quantitative Symbolic
62

31

Mental Act
Humans’ construction of knowledge involves
numerous mental acts
such as:representing, interpreting, defining,
computing, conjecturing, inferring, proving,
structuring, symbolizing, transforming,
generalizing, applying, modeling, connecting,predicting, reifying, classifying, formulating,searching, anticipating, problem solving.
63

Interpreting act
productcharacteristic

particular interpretationcommon property

Proving act
productcharacteristic

Proof

Proof scheme

Mental act
product

way of understanding

characteristic

way of thinking
64

32

Mathematics consists of two complementary subsets:
The first subset is a collection, or structure, of
structures consisting of particular axioms, definitions,
theorems, proofs, problems, and solutions. This
subset consists of all the institutionalized ways ofunderstanding in mathematics throughout history.
It is denoted by WoU.
The second subset consists of all the ways of thinking,which characterize the mental acts whose products
comprise the first set. It is denoted by WoT.

M = WoU UWoT

65

Thinking in Terms Ways of Thinking

Pedagogical implications
Mathematics curricula at all grade level, including
curricula for teachers, should be thought of interms of the constituent elements of mathematics,
ways of understanding and ways of thinking.

66

33

Thinking in Terms Ways of Thinking

Why do we teach the long division algorithm, thequadratic formula, techniques of integration, andso on when one can perform arithmetic
operations, solve many complicated equations,and integrate complex functions quickly andaccurately using electronic technologies?

67

Algebraic Invariance
Algebraic invariance is the way of thinking
where one recognizes that algebraic expressions
are manipulated not haphazardly but with the
purpose of forming a desired structure while
maintaining certain properties of the expression
invariant.
computational skills
IS
algebraic invariance
68

34

Why Teach the Proof of the Quadratic Formula?

ax 2 + bx + c = 0 (a ≠ 0)

( x + T )2 = L

2 x + 5x + 3 = 0

4⎞
1⎛⎜x+ ⎟ =
4 ⎠ 16
⎝

2

2

5
3⎞
⎛
2 ⎜ x2 + x + ⎟ = 0
2
2⎠
⎝
2
⎡⎛
5 ⎞ 25 3 ⎤2 ⎢⎜ x + ⎟ − + ⎥ = 0
4 ⎠ 16 2 ⎦⎥
⎣⎢⎝
2
⎡⎛
4⎞
1⎤
2 ⎢⎜ x + ⎟ − ⎥ = 0
4 ⎠ 16 ⎦⎥
⎣⎢ ⎝

1 5
x=± −
4 4

2

4⎞
1
⎛
⎜x+ ⎟ − =0
4 ⎠ 16⎝
69

Why Teach the Long Division Algorithm?
0.14 12.91

14 1291

12.91 12.91 × 100 1291==
0.14
0.14 × 100
14
Proportional
reasoning as a wayof thinking

Algorithmic
way of thinking

A way of understanding thedecimal system
70

35

Why Should We Teach Techniques of Integration?
1−
tan θ dθ∫ u du∫u = cos θ∫ tan θ dθ =sin θ

∫ cosθ dθ =1

∫ − u du =− ln u + C =ln sec θ + C
71

Depriving Students the Opportunity to
Develop Ways of Thinking

72

36

Multiplication and Division(Junior-high school students)
A cheese weighs 5 pounds. 1 pound costs $12. Find out theprice of the cheese. Which operation would you have toperform?
(a) 12÷5(b) 5÷12(c) 5×12(d) 12+12+12+12+12Correct responses (c and d): 83%
A cheese weighs 0.823 pounds. 1 pound costs $10.50. Findout the price of the cheese. Which operation would you
have to perform?
(a) 10.50+0.823(b) 10.50×0.823(c) 10.50÷0.823(d) 10.50-0.82373
Correct responses: 29%

Teachers teach the following solution strategy:
A cheese weighs 0.823 pounds. 1 pound costs $10.50. Find out theprice of the cheese.

1. Replace the decimal numbers (0.823 & 10.50) by
any whole numbers (say, 8 & 10):
A cheese weighs 8 pounds. 1 pound costs $10. Find out theprice of the cheese.

2. Find the expression that solves the newproblem:
8 × 103. Replace the numbers in this expression with theoriginal decimal numbers:
0.823 × 10.50
74

37

A cheese weighs 0.823 pounds. 1 pound costs $10.50.
Find out the price of the cheese. Which operation
would you have to perform?
(a) 10.50 + 0.823(b) 10.50 × 0.823(c) 10.50 ÷ 0.823(d) 10.50 - 0.823
Rita: None [none of the given choices is asolution to the problem].
T: How would you solve the problem?
Rita: (A long pause)
One thousandth of a pound costs 10.50
divided by 1000. Then I times that by 823.
75

Teacher: One pound of candy cost $7. How
much would 3 pounds cost?
Tammy: Three times seven: 21.
Dan:
I agree, 3 times 7.
Teacher: How much would I pay if I buy only
0.31 of a pound?
Tammy: It is the same. You only changed thenumber. 0.31 times 7.
Dan:
No way! It isn’t the same. … Can’t be.
It isn’t times.
Teacher: How would you, Dan, solve theproblem?
Dan:
Divide 1 by 0.31. Take that number,whatever that number is, and divide 7
76
by it.

38

Developmental Interdependency• Ways of Understanding– Products of mental acts

• Ways of Thinking– Characteristics of mental acts
77

Developmental Interdependency• Ways of Understanding
definitions, problems and their solutions, algorithms,
theorems, proofs, and so on

The Duality Principle
Students develop ways of thinking through the
production of ways of understanding.
Conversely:
The ways of understanding students produce are
impacted by the ways of thinking they possess.

• Ways of Thinking
algebraic invariance, algorithmic reasoning, proportional
reasoning, and deductive reasoning
78

39

Duality Principle
Students develop ways of thinking through the productionof ways of understanding.
And:
The ways of understanding students produce are impacted
by the ways of thinking they possess.
Necessity Principle
For students to learn what we intend to teach them, they musthave a need for it, where ‘need’ refers to intellectual need, notsocial or economic need.
Repeated-Reasoning Principle
Students must practice reasoning in order to organize,internalize, and retain what they learn.

79

DNR Theoretical Framework
http://www.math.ucsd.edu/~harel

DNR-Based Instruction in
Mathematics

Duality

Repeated Reasoning

Necessity

80

40