1

What is Mathematics? A Pedagogical Answer with a Particular Focus on Proving

Guershon Harel
University of California, San Diego
harel@math.ucsd.edu- Guershon Harel 1-

2

0 Introduction
Why do we teach the long division algorithm, the quadratic formula, techniquesof integration, and so on when one can perform arithmetic operations, solve many
complicated equations, and integrate complex functions quickly and accurately using
electronic technologies? Typical answers teachers give to these questions include “thesematerials appear on standardized tests,” “one should be able to solve problems
independently in case a suitable calculator is not present,” “such topics are needed tosolve real-world problems and to learn more advanced topics.” From a social point ofview, there is nothing inadequate about these answers. Teachers must prepare students
for tests mandated by their educational system, they must educate students to carry out
elementary calculations independent of computer technologies, especially calculations
one might encounter in daily life, and they must prepare students to take advanced
courses where certain computational skills might be assumed by the instructors of thesecourses. These answers, however, are external to mathematics as a discipline, in that they
offer justifications that are neither cognitive (about thought processes) nor
epistemological (regarding the philosophical theory of knowledge) but mainly social. For
example, nothing in these answers suggests the role of computational skills in one’s
conceptual development of mathematics; nor do these answers reflect the role ofcomputations in the development of mathematics. A related question is: why teachproofs? The most typical answer given by teachers to this question was, “so that students
can be certain that the theorems we present to them are true.” While this is an adequateanswer—both cognitively and (by inference) epistemologically—it is incomplete. The
teachers who were asked this question had little to say when skeptically confronted about
their answers by being asked: Do you or your students doubt the truth of theorems thatappear in textbooks? Is certainty the only goal of proofs? The theorems in Euclidean
geometry, for example, have been proven and re-proven for millennia. We are certain of
their truth, so why do we continue to prove them again and again?
Overall, these teachers’ answers do not address the question of what intellectual
tools one should acquire when learning a particular mathematical topic. Such tools, I
argue, define the nature of mathematical practice. Judging from current textbooks andteaching practices, teachers at all grade levels, including college instructors, tend to view
mathematics in terms of subject matter, such as definitions, theorems, proofs, problems
and their solutions, and so on, not in terms of the conceptual tools that are necessary toconstruct such mathematical objects. While knowledge of and focus on subject matter is
indispensable for quality teaching, I argue it is not sufficient. Teachers should also
concentrate on conceptual tools such as problem-solving approaches, which, I argue,
constitute an important category of knowledge different from the subject matter category,as I will explain shortly.
What exactly are these two categories of knowledge, subject matter andconceptual tools? And what is the basis for the argument that both categories areneeded? Initially, pedagogical considerations, not philosophical ones, engendered thetwo questions. However, my inquiries into these questions, especially in relation to
students’ conceptions of proof, have led me into historical and philosophical analyses notinitially intended. These analyses have shed considerable light on my understanding of
cognitive processes of learning. For example, the philosophical debate during the- Guershon Harel 2-

3
Renaissance as to whether mathematics conforms to the Aristotelian definition of science
helped me understand certain difficulties able students have with a particular kind of
proof (see Harel, 1999). The juxtaposition of such epistemological and cognitive
analyses compelled me to look deeply into the nature of mathematical knowledge and its
implications for curriculum development and instruction. Thus, my answers to the above
two questions—the main concern of this paper—draw upon epistemological, cognitive,
and pedagogical considerations. These answers are situated within a broader theoreticalframework called DNR-based instruction in mathematics (DNR for short). The initials,
D, N, and R, stand for three leading principles in the framework—duality, necessity, andrepeated-reasoning—to be presented in Section 4. DNR stipulates conditions forachieving critical goals such as provoking students’ intellectual need to learn
mathematics, helping them acquire mathematical ideas and practices, and assuring thatthey internalize, organize, and retain the mathematics they learn.
The paper consists of five sections: Section 1 discusses a triad of key DNR
constructs: “mental act,” “way of understanding,” and “way of thinking.” On the basis ofthese constructs, a definition according to which mathematics consists of two categories
of knowledge is offered in Section 2. Epistemological considerations and pedagogicalconsequences of this definition are discussed in Section 2 and Section 3. Section 3
focuses mainly on long term curricular and research goals, with particular attention tolessons from history. The three foundational principles of DNR along with examples ofother DNR constructs are briefly presented in Section 4. However, DNR concepts andthemes are on every page of the paper. The paper concludes with a summary in Section
5.
1 Mental Act, Way of Understanding, and Way of Thinking
1.1 Mental Act
Humans’ reasoning involves numerous mental acts such as interpreting,conjecturing, inferring, proving, explaining, structuring, generalizing, applying,predicting, classifying, searching, and problem solving. 1 These are examples of mental
acts as opposed to physical acts. “Lifting” and “pulling” an object are examples of thelatter. However, many terms may refer to either physical acts or mental acts. For
example, searching may refer to the act of physically looking for a missing material
object—such as when one searches for missing keys—or to the act of mentally looking
for an abstract object—such as when one searches for the value of an equation’s
unknown. The distinction between “mental act” and “physical act” is not without
difficulty, as one can learn from the work of Lakeoff and Johnson (2003) and Johnson(1987), who argue that meaning, imagination, and reason have a bodily basis. Thisdebate, however, is beyond the scope and goals of this paper.
Humans perform mental acts, and they perform them in every domain of life, notjust in science and mathematics. Although all the aforementioned examples of mental
acts are important in the learning and creation of mathematics, they are not unique tomathematics—people interpret, conjecture, justify, abstract, solve problems, etc. in every
area of their everyday and professional life. Professionals from different disciplines are
1

The notion of “mental act” is taken as undefined in this paper.

- Guershon Harel 3-

4likely to differ in the extent they carry out certain mental acts; for example, a painter islikely to abstract more often than a carpenter, a chemist to model more often than a puremathematician, and the latter to conjecture and justify more often than a pianist. But a
more interesting and critical difference among these professionals is in the nature, thecharacteristics, of the mental acts they perform. A biologist, chemist, physicist, andmathematician all carry out problem-solving acts in every step in their professional
activities, and they may even produce similar solutions to problems their fields have incommon. The four, however, are likely to differ in the nature of the problem-solving act
and other related mental acts they perform while solving problems. Mental acts are basic
elements of human cognition. To describe, analyze, and communicate about humans’
intellectual activities, one must attend to their mental acts.
1.2 Way of Understanding Versus Way of Thinking
It was suggested earlier that teachers at all grade levels tend to view mathematics
in terms of “subject matter” (e.g., definitions, theorems, proofs, problems and theirsolutions), not in terms of the conceptual tools that are necessary to construct suchmathematical objects, and that cognitively, pedagogically, and epistemologically bothcategories are needed. In this section, these two categories of knowledge will be defined
more precisely in terms of the triad, mental act, way of understanding and way of
thinking. 2
Mental acts can be studied by observing peoples’ statements and actions. A
person’s statements and actions may signify cognitive products of a mental act carriedout by the person. Such a product is the person’s way of understanding associated withthat mental act. Repeated observations of one’s ways of understanding associated with agiven mental act may reveal certain cognitive characteristics of the act. Such acharacteristic is referred to as a way of thinking associated with that act. In the rest of thissection, these definitions will be explained and illustrated.
Again, a way of understanding is a particular cognitive product of a mental act
carried out by an individual. For example, upon seeing the symbol 3/ 4 one may carryout the interpreting act to produce a meaning for this symbol. The interpretation the
person produces is her or his way of understanding the symbol. Such a way ofunderstanding may vary with context, and when judged by an observer, it can be deemedright or wrong. For example, in one context a person may produce the meaning “ 3
objects out of 4 objects,” and in another the meaning “the sum 1/ 4 + 1/ 4 + 1/ 4 .” Oneperson may produce a mathematically sophisticated way of understanding, such as “theequivalence class {3n / 4n n is an integer different from zero} ,” and another a naive wayof understanding, such as “two numbers with a bar between them.” Likewise, aparticular solution to a problem and a particular proof of an assertion are products of the
problem-solving act and proving act, respectively; hence, each is a way of understanding.
A way of thinking, on the other hand, is a cognitive characteristic of a mental act.
Such a characteristic is always inferred from observations of ways of understanding—cognitive products of a mental act. For example, a teacher following her student’s
2

Until the terms, way of understanding and way of thinking, are defined and illustrated, the reader mightfind it helpful to associate this pair of terms with the pair, subject matter and conceptual tools,
respectively. The two pairs of terms, however, are not synonymous (see also Footnote 5).

- Guershon Harel 4-5mathematical behavior may infer that the student’s interpretation of mathematicalsymbols is characteristically inflexible, devoid of quantitative referents, or, alternatively,
flexible and connected to other concepts. Likewise, the teacher may infer that a student’s
justifications of mathematical assertions are typically based on empirical evidence, or,alternatively, based on rules of deduction.
Methodologically, when analyzing students’ mathematical behavior in terms ofways of understanding and ways of thinking, one begins with, and fix, a mental act underconsideration, looks at a class of its products (i.e., ways of understanding associated withit), and attempts to determine common cognitive properties among these ways ofunderstanding. Any property found is a way of thinking associated with the mental act.
To further illustrate the distinction between ways of understanding and ways ofthinking, consider the three mental acts of “interpreting,” “problem solving” and“proving.”
1.2.1 The Interpreting Act
The actual interpretation one gives to a term or a string of symbols is a way ofunderstanding because it is a particular cognitive product of her or his act of interpreting.
For example, one may interpret the string of symbols y = 6 x − 5 in different ways: asan equation (a condition on the variables x and y ), as a number-valued function (foreach number x , there corresponds the number 6x − 5 ), or as a proposition-valued
function (for each ordered pair ( x, y ) there corresponds the value “true” or the value“false.”). These ways of understanding manifest certain characteristics of the interpreting
act—for example, that “symbols in mathematics represent quantities and quantitative
relationships.” A person who holds more than one such way of understanding is likely toposses, in addition, the way of thinking that “mathematical symbols can have multiple
interpretations.” And a person who is able to vary the interpretation of symbols
according to the problem at hand is likely to possess the way of thinking that “it isadvantageous to attribute different interpretations to a mathematical symbol in theprocess of solving problems.” These are examples of mature ways of ways ofunderstanding and ways of thinking, which are absent for many high school and college
students. For example, when a class of calculus students was asked what y = 6 x − 5meant to them, many were unable to say more than what one of their classmates said: “Itis a thing where what you do on the left you do on the right.” For many students the actof interpreting algebraic symbols can be characterized as being free of quantitative
meaning.
It is not uncommon that students manipulate symbols without meaningful basis in
the context (as in (log a + log b) / log c = (a + b) / c ). Matz (1980) connects this(erroneous) way of understanding and a wide range of algebra errors to an
overgeneralization of the distributive property. Students factor out the symbol log fromthe numerator and cancel it, without attending to the quantitative meaning of their action.
The behavior of operating on symbols as if they possess a life of their own, not as
representations of entities in a coherent reality, is referred to as the non-referential
symbolic way of thinking. With this way of thinking, one does not attempt to attend tomeaning. For example, one does not ask questions such as “What is the definition of- Guershon Harel 5-

6

log a ?” “Does log⋅ a (multiplication) have a quantitative meaning?” “Islog a + log b = log(a + b) ?” and so on, for symbols are not conceived as representations ofa coherent mathematical reality. Of course, one may produce correct results and still
operate with the non-referential symbolic way of thinking. For example, we haveobserved students correctly solve systems of equations without attaching meaning to theoperations they apply or to the solution they obtain. Our current work (Harel, Fuller, &Rabin, submitted) points to an expected source of the phenomenon of non-attendance tomeaning by students: the way algebra is taught in school. We demonstrate how current
teaching practices of algebra teachers tend to deemphasize, ignore, or misuse
mathematical meaning.
Before proceeding with the other two mental acts, a remark on the non-referential
symbolic way of thinking, which I have just mentioned, is needed. The characterizationof this way of thinking may have evoked with the reader a different image from the one
portrayed here since relative to the reader's practice of mathematics it is not uncommon
that symbols are treated as if they possess a life of their own, and, accordingly, aremanipulated without (necessarily) examining their meaning. I will return to discuss thispoint in Section 3.2.
1.2.2 The Problem Solving Act
As to the mental act of problem solving, the actual solution—correct orerroneous—one provides to a problem is a way of understanding because it is a particularcognitive product of the person’s problem-solving act. A problem-solving approach, on
the other hand, is a way of thinking. For example, problem-solving approaches such as“look for a simpler problem,” “consider alternative possibilities while attempting to solvea problem,” and “just look for key words in the problem statement” characterize, at least
partially, the problem-solving act; hence, they are instances of ways of thinking.
The problem-solving act is not of the same status as the other mental acts listed
above, in that any of these acts is, in essence, a problem-solving act. The acts ofinterpreting, generalizing, and proving, for example, are essentially acts of problem
solving. Despite this, the distinction among the different mental acts is cognitively andpedagogically important, for it enables us to better understand the nature of mathematicalpractice by individuals and communities throughout history, and, accordingly, set explicit
instructional objectives for instruction. This viewpoint will be demonstrated in Sections
2 and 3.
1.2.3 The Proving Act
While problem-solving approaches are instances of ways of thinking associatedwith the problem-solving act, proof schemes are ways of thinking associated with theproving act. Proving is defined in Harel and Sowder (1998) as the act employed by aperson to remove or instill doubts about the truth of an assertion. Any assertion can be
self-conceived either as a conjecture or as a fact: A conjecture is an assertion made by aperson who has doubts about its truth. A person ceases to consider an assertion to be aconjecture and views it to be a fact once the person becomes certain of its truth. In Hareland Sowder (1998), a distinction was made between two variations of the proving act:
ascertaining and persuading. Ascertaining is the act one employs to remove one’s own
doubts about the truth of an assertion (or its negation), whereas persuading is the act one- Guershon Harel 6-

7
employs to remove others’ doubts about the truth of an assertion (or its negation). A
proof scheme characterizes one’s collective acts of ascertaining and persuading; hence, it
is a way of thinking.
A common proof scheme among students is the authoritative proof scheme, ascheme by which proving depends mainly on the authority of the teacher or textbook.
Another common proof scheme among students is the empirical proof scheme—a scheme
marked by its reliance on evidence from examples or visual perceptions. Against these
proof schemes stands the deductive proof scheme, a scheme by which one proves anassertion with a finite sequence of steps, where each step consists of a conclusion which
follows from premises (and previous conclusions) through the application of rules ofinference. 3 Note that while a proof scheme is a way of thinking, a proof—a particularstatement one offers to ascertain for oneself or convince others—is, by definition, a wayof understanding.
Mathematical reasoning centers on the deductive proof scheme. In contrast, theauthoritative proof scheme and the empirical proof scheme are examples of undesirable
ways of thinking. While undesirable, a dash of the authoritarian proof scheme is notcompletely harmful and is unavoidable; people may use this scheme to some extent when
they are sampling an area outside their specialties. In two of its worst forms, however,
either the student is helpless without an authority at hand, or the student regards ajustification of a result as valueless and unnecessary. As with the authoritarian proof
scheme, the empirical proof scheme does have value. Examples and nonexamples canhelp to generate ideas or to give insights. The problem arises in contexts in which a
deductive proof is expected, and yet all that is necessary or desirable in the eyes of thestudent is a verification by one or more examples.
Based on students’ work and historical development, Harel and Sowder (1998)
offered a taxonomy of proof schemes consisting of three classes: External Conviction,
Empirical, and Deductive. A brief description of the essential elements of this taxonomy
follows (For a more complete discussion of the taxonomy, see Harel & Sowder (1998),
Harel (2006), and Harel & Sowder (2007).):
Proving within the external conviction proof scheme class depends on (a) an
authority such as a teacher or a book (the authoritative proof scheme), (b) strictly theappearance of the argument (the ritual proof scheme, e.g., proofs in geometry must havea two-column format), or (c) symbol manipulations, with the symbols or the
manipulations having no coherent system of referents in the eyes of the student (the nonreferential symbolic proof scheme).
The empirical proof scheme class is marked by reliance on either (a) evidence
from examples of direct measurements of quantities, substitutions of specific numbers inalgebraic expressions, and so forth (the inductive proof scheme), or (b) visual perceptions(the perceptual proof scheme).
The deductive proof scheme class consists of two categories: the transformational
proof scheme category and the modern axiomatic proof scheme category. The essential
characteristics of the former are: generality, operational thought, and logical inference.
The generality characteristic has to do with an individual’s understanding that the goal is3

For an extensive taxonomy of proof schemes drawn from students’ mathematical behaviors and thehistorical development of proof, see Harel & Sowder (1998).

- Guershon Harel 7-

8
to justify a “for all” argument, not isolated cases, and that no exception is accepted. The
operational thought characteristic manifests itself when an individual forms goals andsubgoals and attempts to anticipate their outcomes during the proving process. Finally,
when an individual understands that justifying in mathematics must ultimately be basedon logical inference rules, the logical inference characteristic is being employed. The
modern axiomatic proof scheme category has all these characteristics but also includes
the attribute that one understands that in principle any proving process must start from
undefined terms and axioms and that these axioms determines one’s mathematical reality.
This is fundamentally different from Euclid’s Elements, where humans’ idealized special
reality determines the axioms (see Harel & Sowder, 2007). This partial taxonomy issummarized in Table 1.
Table 1A Partial Taxonomy of Proof Schemes
External Conviction
Empirical
Deductive
Authoritative
Inductive
Transformational
RitualPerceptual
Modern axiomatic
Non-referential________________________________________________________________________
1.2.4 Terminology
Two remarks on terminology are in order: The first remark concerns theadjective “cognitive” in the definitions of “way of understanding” and “way of thinking:”
a way of understanding is a cognitive product of a mental act, and a way of thinking is acognitive characteristic of a mental act. This is to indicate that the focus here is on
cognition rather than affect or physiology. For example, the product of feeling confusion
or frustration as one attempts to interpret a statement, prove an assertion, or solve a
problem, is not dealt with in the conceptual framework offered here. Nor does thisframework deal with physiological characteristics of mental acts—those that include, forexample, certain neurological activities in the brain. Thus, the adjective “cognitive” inthe above definitions intends to single out one type of products and characteristics—that
which signifies cognition. The focus on cognition rather than affect and physiology isalso evident in the examples discussed to illustrate the definitions.
The second remark concerns ease of terminology. It may not be easy to get
accustomed to the technical distinction between the terms “way of understanding” and“way of thinking” as is made here. This is partly because in communication among
educators and in the literature on learning and teaching the two terms are often usedinterchangeably (and without exact definitions). Also, the phrase “way of” seems toconnote a sort of a process and, hence, a dynamic image, whereas the definition of “wayof understanding” as a product of a mental act may connote an outcome, a static image.
My intention in using the phrase “way of” is to insinuate “one of several possible ways,”which suggests that a mental act in mathematics can, and should, have multiple products
and characteristics—an implied view in the DNR perspective, as we will see. The verbs“to understand” and “to think” are used in this paper in accordance with the definitions of
the corresponding terms: “to understand” means to “have a way of understanding,” and- Guershon Harel 8-

9“to think” means to “apply a way of thinking.” In DNR, and throughout this paper,“ways of understanding” and “ways of thinking” are distinguished from their values. For
example, one’s way of understanding can be judged as correct or wrong, useful or
impractical in a given context, etc.
2 A Definition of Mathematics: Epistemological Considerations and Pedagogical
Implications
The notions of “ways of understanding” and “ways of thinking” as defined here
are key constructs in the definition of mathematics I will now state. Mathematicians, thepractitioners of the discipline of mathematics, practice mathematics by carrying out
mental acts with particular characteristics (ways of thinking) to produce particularconstructs (ways of understanding). Accordingly, mathematics consists of these twocategories of knowledge. Specifically:
Definition: Mathematics consists of two complementary subsets:• The first subset is a collection, or structure, of structuresconsisting of particular axioms, definitions, theorems, proofs,
problems, and solutions. This subset consists of all theinstitutionalized 4 ways of understanding in mathematics
throughout history. It is denoted by WoU.
• The second subset consists of all the ways of thinking, which
are characteristics of the mental acts whose products comprisethe first set. It is denoted by WoT.

By this definition, mathematics is like a living organism. It grows continually asmathematicians carry out mental acts and their mathematical communities assimilate theways of understanding and ways of thinking associated with the mathematicians’ mental
acts. The assimilation is attained when new ways of understandings are integrated into
an existing mathematical edifice and ways of thinking are adopted in subsequent
mathematical practices. As one can learn from the history of mathematics, theassimilation process is gradual and often not without conceptual struggle. Some ways ofunderstanding and ways of thinking are regarded as inaccurate or faulty—sometimes longafter they have been institutionalized. They, too, are part of mathematics according tothis definition, as I explain later in this paper. In the rest of this section, I shall discuss
several epistemological issues concerning this definition and examine their pedagogicalconsequences.
2.1 Listability
Mathematics as a union of WoU and WoT is not listable—capable of being
completely listed. WoU contains more than the collection of all the statements appearing
in mathematical publications, and the members of WoT are largely unidentified. I
explain:4

Institutionalized ways of understanding are those the mathematics community at large accepts as correct
and useful in solving mathematical and scientific problems. A subject matter of particular field may beviewed as a structure of institutionalized ways of understanding.

- Guershon Harel 9-

10Consider a statement—say, a new theorem—that has appeared in a mathematical
publication, such as a book or research paper. Its publication indicates recognition by acommunity that a new way of understanding has been accepted. Individual
mathematicians might believe and act as if the published theorem represents a way ofunderstanding shared by the community at large, whereas, in fact, each individual
mathematician possesses an idiosyncratic way of understanding the theorem. Of all thelatter “private” ways of understanding, consider only the subset of those that areconsistent with the former “public” ways of understanding. These, too, are considered
institutionalized, since it is assumed by the mathematics community that any way ofunderstanding that is consistent with a “public” way of understanding is acceptable.
Thus, WoU contains all the statements that have appeared in mathematical publications—which the mathematics community views as representations of shared ways ofunderstanding. WoU also contains individual mathematicians' ideas which are consistent
with the published assertions. While statements in the first set are listable, those in thelatter are not. The reason they are not listable is this: Let S be a statement that has been
published, and let S’ be a particular person’s way of understanding S. Once this person
has expressed S’ to the community, S’ moves to the domain of those ways ofunderstanding assumed-to-be shared by the community. But the members of thecommunity, including this person, possess idiosyncratic ways of understanding S’.
A pedagogical implication of this analysis is that a way of understanding, such asa definition, theorem, proof, or solution to a problem, cannot and should not be treated byteachers as an absolute universal entity shared by all students. Any statement a teacher(or a classmate) utters or puts on the board will be translated by each individual student
into a way of understanding that depends on her or his experience and background. The
goal of the teacher is then that these necessarily different individual mental constructs arecompatible with each other. A classroom environment that promotes discussion anddebate among students is both necessary for and instrumental in achieving this goal.
As to the WoT subset of mathematics, its members are not formally recognized by
the mathematics community. They are neither explicitly targeted as instructional
objectives by mathematicians nor investigated and reported in formal publications.
Occasionally, however, they are informal parts of communications between collaborators.
Polya’s (1957) book “How to Solve It” is a rare attempt by a professional mathematician
to explicate desirable problem-solving approaches, which, as was explained earlier, areways of thinking. [For a discussion on Polya’s pedagogical and epistemologicalassumptions on mathematical heuristics, see Schoenfeld, 1992]. It is much more difficult
to reflect on and express in precise words ways of thinking than ways of understanding.
In DNR-based instruction, considerations of ways of thinking are central; they are an
essential part of curriculum development and instruction, as we will see.
2.2 Boundaries
A consequence of my definition of mathematics is that mathematics must include
ways of understanding and ways of thinking that from the vantage point of contemporary
mathematicians are imperfect or erroneous; Euclid’s Elements is an example. This leads
to the following question: Should ways of understanding and ways of thinking used or
produced by individuals (students, for example) while they are engaged in amathematical activity be considered mathematical even if they are narrow or faulty? My- Guershon Harel 10-

11answer to this question is affirmative in so far as the individual has utilized—with orwithout the help of an expert—such ways of understanding and ways of thinking for theconstruction of institutionalized knowledge—knowledge accepted at the time by themathematics community at large.
This position is consistent with the definitions of “way of understanding” and“way of thinking.” As can be seen from the examples discussed in the previous section,these terms do not imply correct knowledge. The terms only indicate the knowledge
currently held by a person, which may be correct or erroneous, useful or impractical.
Having said this, it must be emphasized that the ultimate goal of instruction must be
unambiguous: to help students develop ways of understanding and ways of thinking thatare compatible with those that are currently accepted by the mathematics community atlarge. From a pedagogical point of view this goal is meaningless without realizing thatthe process of learning necessarily involves the construction of imperfect and evenerroneous ways of understanding and deficient, or even faulty, ways of thinking.
Teachers must be aware of this phenomenon when working toward an instructional goal,
and their teaching actions must be consonant with this awareness. In particular, theymust attempt to identify students’ current ways of understanding and ways of thinking,regardless of their quality, and help students gradually refine and modify them toward
those that have been institutionalized—those the mathematics community at large accepts
as correct and useful in solving mathematical and scientific problems.
The repeated use of the term “institutionalized” here raises the question: what
about creativity—the discovery of new, not necessarily institutionalized, ways ofunderstanding and ways of thinking? Are such discoveries mathematical? By mydefinition of mathematics they are not. This position is based on the premise thatmathematics is a human endeavor, not a predetermined reality. As such, it is thecommunity of the creators of mathematics who makes decisions about the inclusion of
new discoveries in the existing edifice of mathematics. Such decisions may never bemade by the community at large, and the new discoveries may be forever lost as a result.
The work of Ramanujan would have likely been lost hadn’t G. H. Hardy recognized theprecious mathematical discoveries in the letter Ramanujan sent to him around 1913.
Other decisions may be delayed; the work of Grassman (19th century) and the work of
Cantor (19th century) are examples. Grassman’s work was ignored for many years but
became later the basis for vector and tensor analysis and associative algebras. Cantor’s
set theory, too, was ignored or boycotted for some time, but was later recognized as oneof the most important discoveries of the twentieth century mathematics. At the time of
their discoveries, prior to their institutionalization, these works did not belong tomathematics, according to the definition of mathematics I propose in this paper.
2.3 Relation to Ontology
There is a danger of confounding the above definition of mathematics with aparticular philosophical stance with which I vehemently disagree. Mathematics,
according to this definition, consists of ways of understanding and ways of thinking thathave evolved throughout history. Inevitably, some of these constructs are narrow andeven faulty if judged from a contemporary perspective. This does not entail thatparticular mathematical statements could be true for some people and false for others—aview that is implied by an extreme form of post-modernism, which asserts that

- Guershon Harel 11-12mathematical truth depends on the culture or bias of the mathematician (Buss, 2005).
Such disputed statements cannot be part of mathematics according to my definition, forthey have never been institutionalized by any mathematics community in the history ofmathematics. That is, no mathematics community, as far as I know, has ever acceptedthat a statement A and its negation, ~A, can both be true within the same system of rulesof inference. A statement such as “Every function on the real numbers is continuous” istrue for intuitionists but false for the rest of us because the two communities are
considering the statement within different systems of rules of inference. Also the term
“function” has different meanings for the two communities. Thus, a statement must not
be considered in isolation but within a context that constitutes its meaning.
What is disputed among philosophers, and to a lesser extent among
mathematicians, is the answer to an ontological question: What is the nature of the being
and existence of mathematics? For example, is mathematical practice an act of discoveryof eternal objects and ideas that are independent of human existence, an intuition-freegame in which symbols are manipulated according to a fixed set of rules, or a product ofconstructions from primitive intuitive objects, most notably the integers? The threepositions expressed in this question correspond, respectively, to the three major schoolsof thought, Platonism, Formalism, and Constructivism. Since Constructivism insists thatmathematical objects must be computable in a finite number of steps, it does not admit
many results accepted by the other schools as true. The basis for this rejection is not“cultural difference” or “personal bias,” as the extreme forms of post-modernism imply;
rather, the basis for the rejection is philosophical: nothing can be asserted unless there is a
proof—a constructivist proof—for it.
It is an open, empirical question whether mathematicians’ ontological stances on
the nature of mathematical practice have any bearing on their views of how mathematics
is learned and, consequently, how it should be taught. I conjecture that teachers’
approach to the learning and teaching of mathematics is not determined by their
ontological stance on the being and existence of mathematics. Dieudonné, a prominent
member of the Bourbaki group, calls in the following statement for an uncompromising
Formalist view:
Hence the absolute necessity from now on for every mathematician concernedwith intellectual probity to present his reasoning in axiomatic form, i.e., in a form
where propositions are limited by virtue of rules of logic only, all intuitive“evidence” which may suggest expressions to the mind being deliberately
disregarded. (Dieudonné, 1971, p. 253).
Yet, he cautions his reader:
We are saying that this is a form imposed on the presentation of the results; but
this does not lessen in any way the role of intuition in their discovery. Among the
majority of researchers the role of intuition is considerable, and no matter how
confused it may be, an intuition about the mathematical phenomena being studied
often puts them on the track leading to their goal. (Emphases added; Dieudonné,
1971, p. 253).

- Guershon Harel 12-

13
One can reasonably infer from these statements that Dieudonné’s approach to teaching is
to emphasize intuition despite his adherence to the Formalist school.
What, then, does determine one’s approach to learning and teaching ofmathematics?
2.4 Quality of Teacher’s Knowledge Base
Quality of instruction is determined largely by what teachers know. Building on
Shulman’s (1986, 1987) work and consistent with current views (Brousseau, 1997;
Cohen & Ball, 1999, 2000), teacher’s knowledge base was defined in Harel (1993) interms of three components: knowledge of mathematics, knowledge of student learning,and knowledge of pedagogy. Here I present a refined definition of these components thatis aligned with the definition of mathematics I have just discussed:• Knowledge of mathematics refers to a teacher’s ways of understanding andways of thinking. It is the quality of this knowledge that is the cornerstone ofteaching for it affects both what the teachers teach and how they teach it.
• Knowledge of student learning refers to the teacher’s understanding offundamental psychological principles of learning, such as how students learn
and the impact of their previous and existing knowledge on the acquisition of
new knowledge.
• Knowledge of pedagogy refers to teachers’ understanding of how to teach inaccordance with these principles. This includes an understanding of how toassess students’ knowledge, how to utilize assessment to pose problems that
stimulate students’ intellectual curiosity, and how to help students solidify andretain knowledge they have acquired.
Thus, while mathematical knowledge is indispensable for quality teaching, it is
not sufficient. Teachers must also know how to address students as learners. In DNR,
however, teacher’s knowledge of student learning and pedagogy rests on the teacher’s
knowledge of mathematics. That is to say, although each of the three components ofknowledge is indispensable for quality teaching, they are not symmetric: the developmentof teachers’ knowledge of student learning and of pedagogy depends on and isconditioned by their knowledge of mathematics. A brief example to illustrate this claim
follows. The example is from an on-site professional development study, currently
underway, aimed at investigating the evolution of teachers’ knowledge base. One of thefindings of this study is that teachers’ appreciation for students’ struggle with a particularconcept is a function of the quality of the teachers’ way of understanding that concept.
For example, Lisa, one of the teacher participants in this study, developed andenthusiastically implemented an instructional activity where her tenth-grade class
gradually discerned the formula for the sum of the interior angles in a convex polygon
along with a mathematically acceptable justification for it. In one of the interviews with
Lisa, she pointed out, with great satisfaction and a sense of accomplishment, that theclass understood well the proof of the formula and some students even developed it on
their own. On the other hand, Lisa, who had insufficient understanding of graphicalrepresentation of solutions to systems of linear inequalities, struggled to see the benefit ofa multi-stage instructional activity that was designed to involve students in developing asolid understanding of how to solve and graph the solution of such systems. She
inclined, instead, to provide the students with a prescribed procedure of how to solve- Guershon Harel 13-

14these systems. Thus, Lisa’s lack of a deep understanding of systems of linear inequality
prevented her from pursuing good teaching of this topic. Overall, Lisa’s knowledge ofpedagogy and of student learning seems to evolve hand in hand with the growth of andself reflection on her knowledge of mathematics, not out of institutional demand to
improve her students’ mathematical performance.
3 Long-Term Curricular and Research Goals
In the opening of this paper, it was argued that the instructional objectives
teachers set for their classes correspond merely to subject matter in terms of products ofmental acts—ways of understanding, such as particular definitions, procedures,
techniques, theorems, and proofs. Neither the actions of the teachers nor thejustifications they provide for their objectives indicate attention to the characteristics ofmental acts—to the ways of thinking that students are to develop by learning particularsubject matter. Objectives formulated in terms of ways of understanding are essential, as
it is asserted in one of the DNR principles, to be presented in Section 4, but without
targeting ways of thinking, students are unlikely to become independent thinkers when
doing mathematics. This brings up the question, when should we start targeting ways ofthinking with students?
3.1 Elementary Mathematics
The formation of ways of thinking is extremely difficult and those that have been
established are hard to alter. This is one of the main findings of our research (see forexample, Harel & Sowder, 1998). Hence, the development of desirable ways of thinking
should not wait until students take advanced mathematics courses; rather, students must
begin to construct them in elementary mathematics, which is rich in opportunities to help
students begin acquiring crucial ways of thinking. Consider, for example, the concept offraction. In current mathematics teaching, even when students learn mathematics
symbolism in context, the context is usually limited. For example, the most common
way of understanding the concept of fraction among elementary school students is what
is known in the literature as the part-whole interpretation: m / n (where m and n are
positive integers) means “ m out of n objects.” Many students never move beyond thislimited way of understanding fraction and encounter, as a result, difficulties indeveloping meaningful knowledge of fraction arithmetic (Lamon, 2001) and beyond(Pustejovsky, 1999). Seldom do students get accustomed to other alternative ways ofunderstanding such as m / n means “the sum 1/ n + " + 1/ n , m times” or “the quantitythat results from m units being divided into n equal parts” or “the measure of a segment
m -inches long in terms of a ruler whose unit is n inches” or “the solution to the equationnx = m ” or “the ratio m : n ; namely, m objects for each n objects.” This range of waysof understanding a fraction makes the area of fractions a powerful elementary
mathematics topic—one that can offer young students a concrete context to constructdesirable—indeed, crucial—ways of thinking, such as: mathematical concepts can beunderstood in different ways, mathematical concepts should be understood in different
ways, and it is advantageous to change ways of understanding of a mathematical concept
in the process of solving problems. These ways of thinking will be needed in thedevelopment of future ways of understanding. Indeed, without the above cluster of waysof thinking students are bound to encounter difficulties in other parts of mathematics. In- Guershon Harel 14-

15
calculus, for example, depending upon the problem at hand, one would need to interpretthe phrase “derivative of a function at a ,” or the symbol f ′(a ) , as “the slope of a line
tangent to the graph of a function at a ” or “the lim ( f (a + h ) − f (a ) ) / h ” or “theh →0instantaneous rate of change at a ” or “the slope of the best linear approximation to afunction near a .” Likewise, in solving linear algebra problems it is often necessary—orat least advantageous—to convert one way of understanding into another way ofunderstanding by using the equivalence among problems on systems of linear equations,matrices, and linear transformations.
The history of mathematics can provide a guide to ways of thinking worth
pursuing—in the classroom and in mathematics education research. In the rest of thissection, I will illustrate this claim with examples from the history of algebra and proof.
3.2 Algebra
According to Klein (1968) the revival and assimilation of Greek mathematics
during the 16th century resulted in a conceptual transformation that culminated in Vieta’s
development of symbolic algebra. Until then, mathematics had evolved for at least threemillennia with hardly any symbols. The following is an example to illustrate the colossal
role symbolic algebra played in defining modern mathematics. The work of Vieta that
led to the creation of algebra and that of Descartes and Fermat that led to the creation ofanalytic geometry constituted the conceptual foundation for the critical shift from “resultsof operations” as the object of study to the operations themselves as the object of study.
While the Greeks restricted their attention to attributes of spatial configurations and paid
no attention to the operations underlying them, 19th century mathematics investigated theoperations, their algebraic representations, and their structures. In particular, Euclidean
constructions using only a compass and straightedge were translated into statements
about the constructability of real numbers, which, in turn, led to observations about thestructure of constructible numbers. A deeper investigation into the theory of fields led tothe understanding of why certain constructions are possible whereas others are not. The
Greeks had no means to build such an understanding, since they did not attend to thenature of the operations underlying Euclidean construction. Thus, by means of analytic
geometry, mathematicians realized that all Euclidean geometry problems can be solved
by a single approach, that of reducing the problems into equations and applying algebraic
techniques to solve them. Euclidean straightedge-and-compass constructions were
understood to be equivalent to equations, and hence the solvability of a Euclidean
problem became equivalent to the solvability of the corresponding equation(s) in theconstructible field.
The monumental role that symbolic algebra played in defining modern
mathematics might be obvious to many, but it is worth pointing out in debates on the
future direction of school mathematics, particularly when attempts are made to
deemphasize symbolic manipulation skills. Often the rationale behind these attempts isthe availability of electronic technologies equipped with computer algebra programs thatcan carry out complex computations of all kinds and in all areas of mathematics. Whilethese technologies can have a positive role in the teaching of mathematics (see, Kaput &Hegedus, 2003) they can, if not used wisely, deprive the students of the experience
necessary for developing critical mathematical ways of thinking. In particular, they can- Guershon Harel 15-

16
deprive students of the opportunity to develop one of the most crucial mathematical waysof thinking, that of algebraic invariance.
Algebraic invariance refers to the way of thinking by which one recognizes thatalgebraic expressions are manipulated not haphazardly but with the purpose of arriving at
a desired form and maintaining certain properties of the expression invariant. If this wayof thinking were set as an instructional objective, elementary algebra—especially symbol
manipulation skills—would be taught differently and more meaningfully. The method ofcompleting the square, for example, would have an added value, not just as a method forsolving quadratic equations but as an activity to advance students toward acquiring thealgebraic invariance way of thinking. Assuming the students have already learned how tosolve equations of the form ( x + T ) 2 = L , the teacher’s action would be geared toward
helping them manipulate the quadratic equation ax 2 + bx + c = 0 with a goal in mind—
that of transforming the latter equation form into the former known equation form but
maintaining the solution set unchanged. The intellectual gain is that students learn thatalgebraic expressions are re-formed for a reason and would, accordingly, develop a senseof the actions needed in order to reach a desired algebraic form. Without this ability,
symbol manipulation is largely a mysterious activity for students—an activity they carryout according to prescribed rules but without a goal in sight. With this ability, on theother hand, symbol manipulation is not a matter of magic tricks performed by the teacher
but goal-directed operations learnable by all students. Of course, one reason symbolic
manipulation is being deemphasized is that this is not how it’s being taught!
With the algebraic invariance way of thinking as an instructional objective,
teaching techniques of integration, for example, will have an added value: would one
teach such techniques not only so that students know how to determine antiderivatives offunctions and values of integrals, but also to help students develop a critical way of
thinking in mathematics—that of utilizing the power of mathematical symbolism to solve
problems and make and prove conjectures. Techniques of integration provide an
excellent context to advance students toward this goal, which is why I believe this topic
should be maintained as part of the calculus curriculum. Take, for example, the simplest
technique of Reduction to Standard Formulas. In solving an integral such as ∫ tan θ dθ ,students in freshman calculus learn to set a goal of transforming this unknown integral
into an equivalent form that is familiar. Even if the students do not note that the symbolic
representation tan θ = sin θ / cos θ suggests the substitution u = cos θ , they would learn
to appreciate such a representation when they see how it is utilized to change the form of
the integral without changing its value through a sequence of symbolic transformations(e.g., ∫ tan θ dθ = − ∫ du / u = − ln u + C = ln u −1 + C = ln sec θ + C ). Likewise, thealgebraic invariance way of thinking is the basis for the concept of “equivalent systems;”
that is, for manipulating a system of equations but maintaining its solution set.
Algebraic invariance is of course a special case of the more general way of
thinking where one attempts to reduce a given problem into a familiar problem, represent
a mathematical entity by another, more useful one, etc. The algebraic invariance way of
thinking is not learned at once—one constructs it gradually by applying it in different
contexts, such as techniques of integration, systems of linear equations, matrix
factorization, etc. Students can start acquiring it in elementary mathematics, for example- Guershon Harel 16-

17when transforming fractions into decimals, and vice versa. It is crucial, however, thatsuch transformations are carried out meaningfully. Often students are taught to carry out
symbolic transformations without adequate emphasis on their justification. For example,
we have seen students learn to solve division problems involving decimal numbers (e.g.,
0.14 12.91 ) by transforming them into division problems involving whole numbers (e.g.,

14 1291 ) without ever attending to the mathematical basis for the transformation. Such
exercises—devoid of meaning—have no value in advancing the algebraic invariance wayof thinking among students, and they deprive the students of the opportunity to develop
other critical ways of thinking. When justifying the equivalence of 0.14 12.91 and14 1291 , for example, students reason proportionally (e.g., when justifying that

0.14
0.14 × 100=), attend to the nature of the number system (e.g., when justifying
12.91 12.91 × 100
that 0.14 × 100 = 14 and 12.91 × 100 = 1291 ), and begin to develop algorithmic way of
thinking (when dividing 1291 into 14 by using the long-division algorithm). Obviously,
such opportunities will not occur if the non-referential way of thinking dominates
students’ actions or if the students obtain the answer to 0.14 12.91 by using a calculator.
It should be clear that in applying the algebraic invariance way of thinking, it isnever the case that every single symbol is referential. It is only in critical stages—viewed
as such by the person who carries the symbol manipulations—that one forms, or attempts
to form, referential meanings. One does not usually attend to interpretation in the middleof symbol manipulations unless one encounters a barrier or recognizes a symbolic form
that is of interest to the problem at hand; thus, for most of the process the symbols aretreated as if they have a life of their own. It is in this sense that symbol manipulation
skills should be understood and, accordingly, be taught.
One might ask, what is then the difference between the algebraic invariance wayof thinking and the non-referential symbolic way of thinking? The answer is that theformer includes the ability to pause at will to probe into a referential meaning for thesymbols involved, whereas the latter does not. In applying the algebraic invariance wayof thinking, the attempt to form a referential meaning does not have to occur, and even if
it occurs it does not have to succeed. It is only that the person who carries out the
manipulation has the ability to investigate the referential meaning of any symbol andtransformation involved. In the non-referential symbolic way of thinking this ability islargely absent.
It is worth pointing out that the practice of manipulating symbols without
necessarily examining their meaning played a significant role in the development ofmathematics. For example, during the nineteenth century significant work was done indifferential and difference calculus using a technique called the “operational method,” amethod whose results are obtained by symbol manipulations without understanding their
meaning, and in many cases in violation of well-established mathematical rules. (See, forexample, the derivation of the Euler-MacLaurin summation formula for approximating
integrals by sums, in Friedman, 1991.) Mathematicians sought meaning for theoperational method, and with the aid of functional analysis, which emerged early in thetwentieth century, they were able to justify many of its techniques. Hence, the- Guershon Harel 17-

18
operational method technique is a manifestation of the algebraic invariance way of
thinking, not the non-referential symbolic way of thinking.
In sum, with the algebraic invariance way of thinking, teachers would recognizethat the goals of teaching manipulation skills include both learning how to compute
solutions to particular problems and constructing conceptual tools that are an essential
part of mathematical practice. The goal of teaching techniques of integration, forexample, is not just to obtain an antiderivative for a given function, but also to help
students acquire an important way of thinking—that of manipulating symbols with a goalof changing the form of an entity without changing a certain property of the entity, a wayof thinking that is ubiquitous and essential in mathematical practice. The role ofsymbolic algebra in the reconceptualization of mathematics raises a critical question
about the role of symbolic manipulation skills in students’ conceptual development ofmathematics. In response to increasing use of electronic technologies in schools,
particularly computer algebra systems, educators should ask: Might these tools deprive
students of the opportunity to develop algebraic manipulation skills which are needed forthe development of the algebraic invariance way of thinking?
3.3 Proof
Certain obstacles students encounter with the concept of proof seem to parallel
obstacles in the development of mathematics. I discuss here two related observations.
The first involves the transition from Greek mathematics to modern mathematics and thesecond the notion of Aristotelian causality.
3.3.1 Transition from Greek mathematics to Modern Mathematics
The deductive mode of thought was conceived by the Greeks more than 20
centuries ago and is still dominant in the mathematics of our day. The mathematicians of
the civilizations that preceded the Greeks established their observations on the basis of
empirical measurements; hence, they mainly possessed and employed empirical proof
schemes—schemes marked by their reliance on evidence from examples or visual
perceptions. In Greek mathematics, logical deduction is central in the reasoning process,and it was necessary for the geometric edifice they created. This edifice, however,represents a single model—that of idealized physical reality. This ultimate bond to areal-world context had an impact on the Greeks’ deductive proof scheme, in that Euclid
often uses arguments that are not logical consequences of his initial assumptions but arerooted in humans’ intuitive physical experience. While Euclid’s Elements is restricted toa single interpretation—namely that its content is a presumed description of humanspatial realization—Hilbert’s Grundlagen is open to different possible realizations, suchas Euclidean space, the surface of a half-sphere, ordered pairs and triples of real numbers,
etc., including the interpretation that the axioms are meaningless formulas. In otherwords, the Grundlagen characterizes a structure that fits different models. To reflect thisfundamental conceptual difference, I refer to the Greeks’ method of proving as the Greekaxiomatic proof scheme and to the modern mathematics’ method of proving as themodern axiomatic proof scheme. The transition between these two proof schemes isrevolutionary: It marks a monumental conceptual change in humans’ mathematical waysof thinking. Understanding this transition may shed light on epistemological obstacles
students encounter upon moving from concrete models of their quantitative or spatial- Guershon Harel 18-

19reality—such as the ones held by the Greeks—to a more abstract setting—such as thatoffered by Hilbert. As a historian might ask what events—social, cultural, andintellectual—necessitated the transition from one way of thinking to another (e.g. frompre-Greek mathematics to Greek mathematics to the mathematics of the Renaissance andto modern mathematics), a mathematics educator should ask what is the nature of the
instructional interventions that can bring students to refine and alter an existing way of
thinking to a more desirable one?
3.3.2 Causality
According to the definition of “proof scheme” presented in Section 1, certainty isachieved when an individual determines—by whatever means he or she deems
appropriate—that an assertion is true. Truth alone, however, may not be the only aim of
an individual, and he or she may also desire to know why the observation is true—thecause that makes it true. An individual may be certain of the truth of an observation andstill strive to understand what in that truth liberates her or him from doubt. “Proofs reallyaren’t there to convince you that something is true—they’re there to show you why it istrue,” said Gleason, one of the solver of Hilbert’s Fifth Problem (Yandell, 2002, p. 150).
Two millennia before him, Aristotle, in his Posterior Analytic, asserted: “To grasp thewhy of a thing is to grasp its primary cause.”
The 16-18th century conception of mathematics reflects global epistemologicalpositions that can be traced back to this position of Aristotle, according to which
explanations in science must be causal. According to the philosophers of the time, thisposition entails the rejection of proof by contradiction, for when a theorem “A implies B”
is proved by showing how not B (under the assumption of A) leads logically to an
absurdity, a person does not learn anything about the causality relationship between A
and B, nor—one might add—does one gain insight into how the theorem was—or might
have been—conjectured. Some students’ behavior with proof can be explained in termsof this epistemological position, in that many able students search for causal relationships
in proofs and dislike indirect proofs (see Harel, 1999). Likewise, for the decisive
majority of mathematicians the purpose of a proof is not only demonstrating that theassertion is true, but also explaining why it is true. Proofs by contradiction, while
accepted in modern mathematics, usually lack the explanatory power direct proofs canhave. As an example, it is worth mentioning the controversy that Hilbert’s proof of
Gordan’s Conjecture 5 raised. Hilbert didn’t find a basis that everyone had searched forbut merely proved that if we accept Aristotle’s law of the excluded middle (“Anystatement is either true or its negation is true”) then such a basis had to exist, whether we
could produce it or not. Why was Hilbert’s use of proofs-by-contradiction so
controversial—after all, he was not the first to use this method of argument? According
to Yandell (2002), previous uses had not dealt with a subject of such obvious
calculational complexity. A pure existence proof does not produce a specific object thatcan be checked—one had to trust the logical consistency of the growing body ofmathematics to trust the proof. The presence of an actual object that can be evaluated
5

The conjecture states: There is a finite basis from which all algebraic invariants of a given polynomial
form could be constructed by applying a specified set of additions and multiplications.

- Guershon Harel 19-

20
provides more than mere certainty; it can constitute a cause—in the Aristotelian sense—
for the observed phenomenon. The philosophers of the Renaissance rejected proof bycontradiction, and the practice of many mathematicians of that period, such as Cavalieri,
Guldin, Descartes, and Wallis, reflected this position by explicitly avoiding proofs bycontradiction in order to conform to the Aristotelian position on what constitutes perfectscience (Mancosu, 1996).
The implication of this history is not to avoid proofs by contradiction inmathematics curricula. On the contrary, proofs by contradiction represent an important,institutionalized way of thinking, which students should acquire. The point of thishistory is that modern proof schemes were born out of an intellectual struggle—a struggle
in which Aristotelian causality seems to have played a significant role. It is an open
question whether the development of students’ proof schemes necessarily includes someof these epistemological obstacles. The fact that even able students encounter theseobstacles makes this question even more relevant to the matter at hand. An answer to thisquestion may shed light on some of the roots of the obstacles students encounter withcertain kinds of proof, such as proof by contradiction. Accordingly, appropriate
instructional interventions can be devised to help students develop desirable proof
schemes as they encounter these obstacles, which, perhaps, are unavoidable.
3.3.3 Definitional Reasoning
The definition of mathematics presented earlier is consistent withmathematicians’ practice of mathematics but not necessarily with their perception of it.
There is a fundamental difference between the way mathematicians perceive mathematics
and the way they practice it in their research. Mathematicians’ perception of mathematics
manifests itself in how they communicate it in formal professional forums, such asconference presentations and journal publications, and in educational forums, when theyteach it or write textbooks on its topics. The main, explicit focus in both forums is largely
on ways of understanding—the content of axioms, definitions, theorems, proofs, etc., andtheir logical structure. Mathematicians’ research practice, which includes learning anddiscovering such contents, utilizes numerous ways of thinking; these usually are not partof their perception of mathematics. This is not surprising. In order for one’s practice of adiscipline to be part of one’s conscious image of that discipline, one must be aware of
how one comes to know and discover things in the discipline. But, attention to andreflection on the processes of knowing and discovering for the purpose of creating
conditions that others come to know and discover are not integral part of themathematician’s training.
This incomplete perception of mathematics is one source of a simplistic
conceptualization of mathematics learning and of teaching. Viewing mathematics merely
in terms of ways of understanding reduces curriculum development to a matter ofsequencing curricular content according to a logical structure of that content, paying noor scant attention to critical ways of thinking and to the complexity of the process
involved in acquiring and internalizing them. The effective trajectory for learning—and
therefore for teaching—a particular mathematical topic, according to this perception, is
one that parallels a logical structure of that topic. How should, for example, the conceptof fraction be introduced in the fifth grade? The answer according to thisconceptualization is simple: start with a mathematical definition (e.g., a fraction as a

- Guershon Harel 20-

21
point on the number line) and proceed through logical deduction to build the curricular
content related to this concept (e.g., the four arithmetic operations on fractions). Thisview, however, is irreconcilable with research findings, which reveal a different and more
complex terrain for the learning of fractions. More generally, reasoning in terms ofmathematical definition and justifying assertion deductively are ways of thinking notavailable to fifth graders. They are developmental and do not become integral part of therepertoire of students’ ways of thinking until advanced grades.
Students’ definitions of a concept and the way the concept is defined in acurriculum may or may not be mathematical. Generally speaking, a definition ismathematical if it is a description that applies to all objects to be defined, and applies
only to them. Typically, students’ definitions of concepts are not mathematical, even in if
the concepts were defined to them mathematically. Understanding the notion ofmathematical definition and appreciating the role and value of mathematical definitionsin proving is a developmental process, which is not achieved for most students until
adulthood. I refer to the way of thinking by which one defines objects and proves
assertions in terms of mathematical definitions as definitional reasoning. A crucial
feature of this way of thinking, which belongs to the transformational proof schemes
category, if not to the modern axiomatic proof schemes category, is that with it one iscompelled to conclude logically that there can be only one mathematical definition for aconcept within a given theory; namely, if D1 and D2 are such definitions for a concept C,then D1 is a logical consequence of D2, or vice versa; otherwise, C is not well defined.
Many students even in advanced grade do not possess definitional reasoning. For
example, Van Hieles’ stages imply that only in the highest stage of geometric reasoning
are students’ definitions of Euclidean objects mathematical. The work by the Van Hieles
is commonly applied to secondary school students. However, this way of thinking islargely absent among college students as well, even among undergraduate mathematics
and engineering majors. Undoubtedly, definitional reasoning must be a target formathematics instruction, but it cannot be assumed—definitely not among fifth-graders.
The lack of recognition by teachers, researchers, and curriculum developers of these andother ways of thinking and the nature of their development with learners is, I believe, oneof the reasons for the failure of instruction to help students acquire them.
4 DNR Based Instruction in Mathematics
DNR-based instruction in mathematics (DNR, for short) is a theoreticalframework for the learning and teaching of mathematics—a framework that provides alanguage and tools to formulate and address critical curricular and instructional concerns.
In this framework the mathematical integrity of the content taught and the intellectual
need of the student are at the center of the instructional effort. DNR has been developed
from a long series of teaching experiments in elementary, secondary, and undergraduate
mathematics courses, as well as teaching experiments in professional development
courses for teachers at each of these levels. Briefly, DNR can be thought of as a system
consisting of three categories of constructs:
1. Premises—explicit assumptions underlying the DNR concepts and claims.
2. Concepts—referred to as DNR determinants.
3. Instructional principles—claims about the potential effect of teaching actions
on student learning.

- Guershon Harel 21-

22
It goes beyond the scope of this paper to do more than present a brief outline of these
constructs. For more about DNR, see Harel (1989, 2001, in press, in preparation).
Premises. One of the DNR premises is the conceptualization premise:
Humans—all humans—possess the ability to develop a desire to be puzzled andto learn to carry out physical and mental acts to fulfill their desire to be puzzled
and to solve the puzzles they create.
This premise, which follows from Aristotle, is one of eight DNR premises. Note that it
assumes not only humans’ desire to solve puzzles but also humans’ desire to be puzzled.
It serves as a basis for many themes in DNR—the necessity principle, to be stated shortly,is one of them. It is also the basis of DNR’s interpretation of equity: All students arecapable of learning if they are given the opportunity to be puzzled, create puzzles, andsolve puzzles.
Concepts. “Mental act,” “way of understanding,” and “way of thinking” areexamples of DNR determinants.
Instructional Principles. Not every DNR instructional principle is explicitlylabeled as such. The system states only three foundational principles: the duality
principle, the necessity principle, and the repeated-reasoning principle; hence, theacronym DNR. The other principles in the system are derivable from and organizedaround these three principles.
Recall, according to my definition, mathematical knowledge consists of ways ofunderstanding and ways of thinking. The duality principle concerns the developmental
interdependency between these two constructs:
The Duality Principle: Students develop ways of thinking only through theconstruction of ways of understanding, and the ways ofunderstanding they produce are determined by the waysof thinking they possess.

The reciprocity between ways of understanding and ways of thinking claimed in
the duality principle is of mutual effect: a change in ways of thinking brings about achange in ways of understanding, and vice versa. The claim intended is, in fact, stronger:
Not only do these two categories of knowledge affect each other but a change in one
cannot occur without a corresponding change in the other.
Implied from the duality principle is that preaching ways of thinking to students
would have no effect on the quality of the ways of understanding they would produce.
For example, talking to them about the nature of proof in mathematics or advising themto use particular heuristics would have minimal or no effect on the quality of the proofs
and solutions they would produce. Only by producing desirable ways of understanding—by way of carrying out mental acts of, for example, solving mathematical problems andproving mathematical assertions—can students construct desirable ways of thinking.
This seems obvious until one observes, for example, teachers teaching problem-solving
heuristics explicitly and students following them as if they were general rules rather thanrules of thumb for solving problems.

- Guershon Harel 22-

23
Attention to ways of thinking, on the other hand, is necessary—according to theduality principle—for they direct teachers as to which teaching actions to avoid andwhich to pursue. As we have discussed earlier, attention to desirable ways of thinking—such as algebraic invariance, proportional reasoning, and algorithmic reasoning—highlights the need to focus on particular ways of understanding certain concepts andprocesses (e.g., the solution process of quadratic functions, techniques of integration, anddivision of decimal numbers; see Section 3.2). In particular, teachers must take into
consideration students’ current ways of thinking. For example, a college instructor may
start a course in geometry with finite geometries as a preparation for non-Euclidean
geometries. We found (Harel & Sowder, 1998) that most undergraduate students taking
college geometry are not prepared for such an instructional treatment because they do notpossess the modern axiomatic proof scheme—which includes the way of thinking thatgeometric properties are not limited to spatial imageries.6 As was discussed earlier, thisway of thinking was born at the turn of the 20th century with the publication of Hilbert’s
Grundlagen and is considered revolutionary in the development of mathematics.
Of critical pedagogical importance is the question: What is the nature of
instructional treatments that can help students construct desirable ways of understanding
and ways of thinking? This is addressed by the other two DNR principles: the necessity
principle and the repeated reasoning principle.
The Necessity Principle: For students to learn what we intend to teach them, theymust have a need for it, where ‘need’ refers tointellectual need, not social or economic need.

Most students, even those who are eager to succeed in school, feel intellectuallyaimless in mathematics classes because we—teachers—fail to help them realize an
intellectual need for what we intend to teach them. The term intellectual need refers to a
behavior that manifests itself internally with learners when they encounter an intrinsic
problem—a problem they understand and appreciate. For example, students might
encounter a situation that is incompatible with, or presents a problem that is unsolvable
by, their existing knowledge. Such an encounter is intrinsic to the learners because it
stimulates a desire within them to search for a resolution or a solution, whereby theymight construct new knowledge. There is no guarantee that the learners construct theknowledge sought or any knowledge at all, but whatever knowledge they construct ismeaningful to them since it is integrated within their existing cognitive schemes as a
product of effort that stems from and is driven by their personal, intellectual need. While
one should not underestimate the importance of students’ social need (e.g., mathematicalknowledge can endow me with a respectable social status in my society) and economic
need (e.g., mathematical knowledge can help me obtain comfortable means of living) as
learning factors, teachers should not and cannot be expected to stimulate (let alone fulfill)these needs. Intellectual need, on the other hand, is a prime responsibility of teachers andcurriculum developers.
6

For example, students in our study encountered insurmountable difficulty interpreting the statement“Given a line and a point not on the line, there is a line which contains the given point and is parallel tothe given line” in a finite geometry.

- Guershon Harel 23-

24
Even if ways of understanding and ways of thinking are necessitated throughstudents’ intellectual need there is still the task of ensuring that students internalize,organize, and retain this knowledge. This concern is addressed by the repeatedreasoning principle:
The Repeated Reasoning Principle: Students must practice reasoning in order tointernalize, organize, and retain ways ofunderstanding and ways of thinking.

Research has shown that repeated experience, or practice, is a critical factor inthese cognitive processes (see, for example, Cooper, 1991). DNR-based instruction
emphasizes repeated reasoning that reinforces desirable ways of understanding and waysof thinking. Repeated reasoning, not mere drill and practice of routine problems, isessential to the process of internalization—a conceptual state where one is able to apply
knowledge autonomously and spontaneously. The sequence of problems must
continually call for reasoning through the situations and solutions, and they must respondto the students’ changing intellectual needs.
These instructional principles are the basis for many of the pedagogical positions
expressed in this paper, and they have been used to organize my instruction, in general,and teaching experiments, in particular. Consider the following unit taken from a recent
teaching experiment with secondary mathematics teachers with limited mathematics
background. The teachers worked on justifying the quadratic formula. Prior to thisproblem, they had repeatedly worked with many quadratic functions, finding their roots
by essentially completing the square. They abstracted this process to develop thequadratic formula. In doing so they repeatedly transformed particular equations of theform ax 2 + bx + c = 0 into an equivalent equation of the form ( x + T ) 2 = L for some terms

T and L , in order for them to solve for x (as −T + L and −T − L ). To get to thedesired equivalent form, they understood the reason and need for dividing through by a ,bringing c / a to the other side of the equation, and completing the square. For theseteachers, the symbolic manipulation process stems from an intellectual need—the need toarrive at a particular form in order to determine the equation’s unknown—andconditioned by quantitative considerations—to transform the algebraic expressions
without altering their quantitative value. In these activities, the teachers practiced thealgebraic invariance way of thinking, whose importance I have discussed in Section 3.2.
We see here the simultaneous implementation of the duality principle, the necessity
principle, and the repeated reasoning principle. In particular, the repeated application of
the invariance way of thinking helped the participant teachers internalize it, whereby they
become autonomous and spontaneous in applying it.
5 SummaryCurrent teaching practices tend to view mathematics in terms of subject matter,such as definitions, theorems, proofs, problems and their solutions, not in terms of theconceptual tools that are necessary to construct such mathematical objects. The tenet of
this paper is that instruction should focus on both categories of knowledge: subject matterand conceptual tools. The paper defines these two categories and explains why bothcategories are needed. The definitions and explanations are oriented within the DNR- Guershon Harel 24-

25framework. Central to DNR is the distinction between “way of understanding” and “wayof thinking.” “Way of understanding” refers to a cognitive product of a person’s mental
act, whereas “way of thinking” refers to its cognitive characteristic. Accordingly,
mathematics is defined as the union of two sets: the set WoU, which consists of all theinstitutionalized ways of understanding in mathematics throughout history, and the set
WoT, which consists of all the ways of thinking that characterize the mental acts whose
products comprise the first set.
The members of WoT are largely unidentified in the literature, though somesignificant work was done on the problem-solving act (e.g., Schoenfeld, 1985; Silver,
1985) and the proving act (see an extensive literature review in Harel & Sowder, inpress). The members of WoU include all the statements appearing in mathematical
publications, such as books and research papers, but it is not listable because individuals(e.g., mathematicians) have their idiosyncratic ways of understanding. A pedagogicalconsequence of this fact is that a way of understanding should not be treated by teachers
as an absolute universal entity shared by all students, for it is inevitable that eachindividual student is likely to possess an idiosyncratic way of understanding that depends
on her or his experience and background. Together with helping students develop
desirable ways of understanding, the goal of the teacher should be to promote interactions
among students so that their necessarily different ways of understanding becomecompatible with each other and with that of the mathematical community.
Since mathematics, according to the definition offered in this paper, includes
historical ways of understanding and ways of thinking, it must include ones that might be
judged as imperfect or even erroneous by contemporary mathematicians. Also included
in mathematics are imperfect ways of understanding and ways of thinking used or
produced by individuals in the process of constructing institutionalized knowledge. The
boundaries as to what is included in mathematics are in harmony with the nature of theprocess of learning, which necessarily involves the construction of imperfect anderroneous ways of understanding and deficient and faulty ways of thinking. Theseboundaries, however, are not to imply acceptance of the radical view that particularmathematical statements could be true for some people and false for others.
My definition of mathematics implies that an important goal of research inmathematics education is to identify desirable ways of understanding and ways ofthinking, recognize their development in the history of mathematics, and, accordingly,
develop and implement mathematics curricula that aim at helping students constructthem. This claim was illustrated in the contexts of algebra and proof. The discussion on
algebra highlights the need to promote the algebraic invariance way of thinking among
students. With it, students learn to manipulate symbols with a goal in mind—that ofchanging the form of an entity without changing a certain property of the entity. It also
points to the risk that the use of electronic technologies in schools, particularly computer
algebra systems, can deprive students of the opportunity to develop this crucial way of
thinking. The discussion on proof focuses on the transition from Greek mathematics tomodern mathematics and the role of Aristotelian causality in the development ofmathematics during the Renaissance. It raises the question of whether the developmentof students’ proof schemes parallels those of the mathematicians and philosophers ofthese periods. An answer to this question would likely have important curricular andinstructional implications.

- Guershon Harel 25-

26
Since the formation of desirable ways of thinking is difficult and undesirable
ways of thinking that have been formed are hard to relinquish, an effort must be made inearly grades to help students acquire desirable ways of thinking. The concept of fraction,for example, can be taught with multiple ways of understanding, and in a context wherestudents can develop ways of thinking necessary for the acquisition of advanced
mathematics. Similarly, arithmetic problems such as division of decimals can provide
invaluable opportunities to engage in proportional reasoning and algorithmic reasoning
and revisit the nature of the decimal-number system.
Pedagogically, the most critical question is how to achieve such a vital goal as
helping students construct desirable ways of understanding and ways of thinking. DNR
has been developed to achieve this very goal. As such, it is rooted in a perspective that
positions the mathematical integrity of the content taught and the intellectual need of thestudent at the center of the instructional effort. The mathematical integrity of a curricular
content is determined by the ways of understanding and ways of thinking that have
evolved over many centuries of mathematical practice and continue to be the ground forscientific advances. To address the need of the student as a learner, a subjective approach
to knowledge is necessary. For example, the definitions of the process of “proving” and“proof scheme” are deliberately student-centered (see Section 1). It is so because theconstruction of new knowledge does not take place in a vacuum but is shaped by one’s
current knowledge. What a learner knows now constitutes a basis for what he or she willknow in the future. This fundamental, well-documented fact has far-reaching
instructional implications. When applied to the concept of proof, for example, this fact
requires that instruction takes into account students’ current proof schemes, independentof their quality. Again, despite this subjective definition the goal of instruction must be
unambiguous—namely, to gradually refine current students’ proof schemes toward theproof scheme shared and practiced by contemporary mathematicians. This claim isbased on the premise that such a shared scheme exists and is part of the ground foradvances in mathematics.
Instruction concerns what mathematics to teach as well as how to teach it. Whilethe definition of mathematics offered in this paper dictates the kind of knowledge toteach—ways of understanding and ways of thinking—the three DNR principles stipulate
how to teach that knowledge:
The duality principle concerns the developmental interdependency between waysof understanding and ways of thinking. Students would be able to construct a way of
thinking associated with a certain mental act or refine or modify an existing one only ifthey are helped to construct suitable ways of understanding associated with that mental
act. Conversely, students would be able to construct a way of understanding associatedwith a certain mental act or refine or modify an existing one only if they are helped toconstruct suitable ways of thinking associated with that mental act in the form ofproblem-solving approaches or proof schemes.
According to the necessity principle, problem solving is not just a goal but also
the means—the only means—for learning mathematics. Learning grows only out ofproblems intrinsic to the students, those that pose an intellectual need for them. In
general, an intellectually-based activity is one where students’ actions are driven by adesire to solve intrinsic problems. In a socially-based activity, on the other hand,- Guershon Harel 26-

27
students’ actions are carried out merely to satisfy a teacher’s will. 7 In an intellectuallybased teaching environment, students are continually challenged with new problems fromwhich they elicit new concepts and ideas. Such an environment is necessary for learning,
and is conducive to creativity.
The repeated reasoning principle is complementary to the duality principle andthe necessity principle, in that its aim is for students to internalize what they have learned
through the application of these two principles. Through repeated reasoning in solving
intrinsic problems, the application of ways of understanding and ways of thinking
become autonomous and spontaneous.

7

The notion of intellectually, but not socially, based activity is similar to what Brousseau (1997) calls an
adidactical situation.

- Guershon Harel 27-

28
References
Buss, S. (2005). Nelson's work on logic and foundations and other reflections on
foundations of mathematics. In W. Faris (Ed.), Quantum Theory and Radically
Elementary Mathematics. Princeton University Press.
Brousseau, G. (1997). Theory of Didactical Situations in Mathematics. Dordrecht:
Kluwer Academic Publishers.
Cohen, D. K. & Ball, D. L. (1999). Instruction, Capacity, and Improvement (CPRE
Research Report No. RR-043). Philadelphia, PA: University of Pennsylvania,
Consortium for Policy Research in Education.
Cohen, D. K. & Ball, D. L. (2000). Instructional innovation: Reconsidering the story.
Paper presented at the Annual Meeting of the American Educational Research
Association, New Orleans.
Cooper, R. (1991). The role of mathematical transformations and practice inmathematical development. In L. Steffe (Ed.). Epistemological Foundations ofMathematical Experience. New York. Springer-Verlag.
Dieudonné (1971). Modern axiomatic methods and the foundations of mathematics. In F.
Le Lionnais (Ed.), Great currents of mathematical thought (Vol.1), Dover, NewYork.
Friedman, B. (1991). Lectures on Applications-Oriented Mathematics, New York, John
Wiley & Sons, 1991.
Harel, G. (1993). On teacher education programs in mathematics, International Journalfor Mathematics Education in Science and Technology, 25, 113-119.
Harel, G. (1998). Two Dual Assertions: The First on Learning and the Second on
Teaching (Or Vice Versa). The American Mathematical Monthly, 105, 497-507.
Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory
studies. In A. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate
mathematics education III (pp. 234-283). Providence, RI: American
Mathematical Society.
Harel, G. (1999). Students’ understanding of proofs: A historical analysis andimplications for the teaching of geometry and linear algebra. Linear Algebra and
Its Applications, 302-303, 601-613.
Harel, G. (2001). The development of mathematical induction as a proof scheme: A
model for DNR-based instruction. In S. Campbell & R. Zazkis (Eds.), Thelearning and teaching of number theory (pp. 185-212). Dordrecht, The
Netherlands: Kluwer.
Harel, G. (2007). The DNR System as a Conceptual Framework for Curriculum
Development and Instruction, In R. Lesh, E. Hamilton., J. Kaput, (Eds.),
Foundations for the Future in Mathematics Education, Erlbaum, 263-280.
Harel, G., & Sowder, L (2007). Toward a comprehensive perspective on proof, In F.
Lester (Ed.), Second Handbook of Research on Mathematics Teaching and
Learning, National Council of Teachers of Mathematics, 805-842.
Harel, G., Fuller, E., & Rabin, J. (submitted). Attention to meaning by algebra teachers.
Harel, G. (in preparation). DNR-Based Instruction in Mathematics (a book manuscript).
Harel, G., & Sowder, L (in press). Toward a comprehensive perspective on proof, In F.
Lester (Ed.), Handbook of Research on Teaching and Learning Mathematics (2ndedition). Greenwich, CT: Information Age Publishing.

- Guershon Harel 28-

29
Kaput, J., & Hegedus, S. (2003). The effect of SimCalc connected classrooms on
students' algebraic thinking. In N. A. Pateman, B. J. Dougherty & J. Zilliox(Eds.), Proceedings of the 27th Conference of the International Group for thePsychology of Mathematics Education (Vol. 3, pp. 47-54). Honolulu, Hawaii:College of Education, University of Hawaii.
Klein, J. (1968). Greek mathematical thought and the origin of algebra (E. Brann,
Trans.). Cambridge, MA: MIT Press. (Original work published 1934).
Kleiner, I. (1991). Rigor and proof in mathematics: A historical perspective.
Mathematics Magazine, 64(5), 291-314.
Johnson, M. (1987). The Body in the Mind. The University of Chicago Press, Chicago.
Lakoff, G., Johnson, M. (2003). Metaphors we live by. The University of Chicago Press,
Chicago.
Lamon, S. (2001). Presenting and representing: From fractions to rational numbers. In
Cuoco, A. (Ed.), The roles of representation in school mathematics (2001
Yearbook of the National Council of Teachers of Mathematics). Reston, VA:
NCTM.
Mancosu, P. (1996). Philosophy of mathematical practice in the 17th century. New York:
Oxford University Press.
Matz, M. (1982). Towards a process model for high school algebra errors. In D. Sleeman& J. S. Brown (Eds.), Intelligent Tutoring Systems (pp. 25-50). New York:
Academic Press.
Pustejovsky, P. (1999). Beginning Undergraduate Students’ Understanding of
Derivative: Three Case Studies. Doctoral Dissertation, Marquette University.
Polya, G. (1957). How to Solve It. Princeton University Press
Schoenfeld, A. (1985). Mathematical Problem Solving. Orlando, FL: Academic Press.
Schoenfeld, A. (1992). Learning to think mathematically: Problem solving, metacgnition,and sense making in mathematics. In D. Grouws (Ed.). Handbook for Research on
Mathematics Teaching and Learning. New York: Macmillan, 334-370.
Shulman, L. (1986). Those who understand: Knowledge growth in teaching. Educational
Researcher, 15, 4-14.
Shulman, L. (1987). Knowledge and teaching: Foundations of the new reform. Harvard
Educational Review, 57, 1-22.
Silver, E. (1985). Teaching and learning mathematical problem solving: Multiple
research perspective. Hillsdale, NJ: Lawrence Erlbaum.
Weber, K. (2001). Student difficulty in constructing proof: The need for strategic
knowledge. Educational Studies in Mathematics, 48, 101-119.
Yandell, B. (2003). The Honors Class: Hilbert's Problems and Their Solvers. A K Peters
Ltd.

- Guershon Harel 29-