Rejecting Platonism: Recovering Humanity in
Frederick A. Peck ID
Department of Mathematical Sciences, University of Montana, Missoula, MT 59812, USA;
email@example.com; Tel.: +1-406-243-4053
Received: 23 February 2018; Accepted: 25 March 2018; Published: 29 March 2018
In this paper, I consider a pervasive myth in mathematics education, that of Plato-formalism.
I show that this myth is ahistorical, acultural, and harmful, both for mathematics and for society.
I argue that, as teachers, we should reject the myth of Plato-formalism and instead understand
mathematics as a human activity. This philosophy humanizes mathematics and implies that math
education should be active, cultural, historical, social, and critical—helping students learn formal
mathematics, while also learning that mathematics shapes their lives, that this shaping is a result of
human work and choices, and that students are empowered to shape those choices.
Keywords: Platonism; formalism; philosophy of mathematics; sociocultural; critical theory
As a high-school math teacher, my ﬁrst activity for every class was a discussion exploring the
classic question: “was mathematics invented or discovered?” Although I professed to be agnostic in
this discussion, I was not. I believed that mathematics existed “out there”—independent of time, space,
and culture—and hence, that mathematics was discovered. Such a position is neatly summarized by
Martin Gardner :
[W]hen two dinosaurs met two dinosaurs there were four dinosaurs. In this prehistoric
tableau “2 + 2 = 4” was accurately modeled by the beasts, even though they were too stupid
to know it and even though no humans were there to observe it.
This position feels safe to me. Comforting. Yet, I am about to argue that this position is based
on a myth, one which is ahistorical, acultural, and ultimately, harmful. In taking this stance, I take
up a minority position within Western culture, where the Platonic ideals expressed by Gardner reign
]. In this paper, I will review the literature that supports this conclusion, and I will present
an alternative vision for mathematics and mathematics education.
First, however, I would like to frame this paper a bit. This is a paper about a mathematical
myth, written by a math teacher, for math teachers. It is also a paper about mathematical philosophy.
Why should teachers care about philosophy? Because it turns out that what teachers believe about
mathematics is more consequential for their teaching than what they believe about learning [
(but see [
] for a counter-argument). However, while literature about mathematical philosophy
abounds, very little of it is written for teachers and teacher educators [
] (at least within the last
30 years, but see [
]). In this paper, I aim to change that by arguing for a philosophy that is more
powerful and more faithful to the historical record—and ultimately more human—than the dominant
mathematical philosophy in the west.
To begin, I review this dominant mathematical philosophy, which I call the Plato-formalist
philosophy. I argue that this philosophy is a myth by showing it is both ahistorical and acultural.
I then describe a different approach: mathematics as a human—and therefore a social, cultural, and
Educ. Sci. 2018,8, 43; doi:10.3390/educsci8020043 www.mdpi.com/journal/education
Educ. Sci. 2018,8, 43 2 of 13
historical—activity, and I explain how this approach overcomes the shortcomings of Plato-formalism.
Finally, I discuss the signiﬁcance for math education of adopting this approach.
2. The Plato-Formalist Philosophy
Mathematical Platonism is the belief that mathematics exists as a complete structure, somewhere
“out there”, just waiting to be discovered. More formally, this belief holds the following three theses:
(1) mathematical objects exist; (2) they are neither physical nor mental, and they exist outside of space
and time; and (3) they exist independent of any sentient being and the culture thereof [
]. Such a
view is pervasive, both in popular culture and among mathematicians [2,3].
Alongside this philosophy is another belief about mathematics, called “formalism”. Formalism
was a movement in the late 19th and early 20th century to show that mathematics was a self-contained
system without any relation to the physical world [
]. The key consideration is that the system is
consistent (no statement is simultaneously true and false) and complete (every possible statement
can be proven to be either true or false). Even though Gödel [
] (translation in [
that such a program was impossible for any system that is complicated enough to include arithmetic,
formalist logic continues to dominate mathematics .
Platonism and formalism are not the same, and indeed one could argue that they are ideologically
opposed. For our present purposes it sufﬁces to consider them as forming an axis—the Plato-formalist
axis—upon which mathematics is often positioned in popular culture, professional work, and
mathematics education .
3. Plato-Formalism Is a Myth
Plato-formalism is popular, but it is a myth. To understand why, let us ﬁrst take a short detour
to Western science. In the West, we have folk notions of science as a process of pure discovery, a
dispassionate study of the objective reality that surrounds us. However, as sociologists and historians
of science [
] have shown, science is much more than observing and reporting. First, observations
are theory-laden [
] and are culturally-conditioned [
]. Additionally, the job of the scientist is to
produce facts that ﬁt the world through observation, but also to organize systems of relations in the
world such that those facts are valued. As Bruno Latour explains, “Scientiﬁc facts are like trains,
they do not work off their rails. You can extend the rails and connect them but you cannot drive a
locomotive through a ﬁeld” [
] (p. 155). The work of “extending and connecting the rails” is the
human work that is required to produce a world where facts can be accepted as “true”. It is the work
of turning an artifact (something produced by humans) into a fact (something that appears to have an
objective existence). This is a constructive and social process (documented extensively in [
belies folk notions of science as dispassionate discovery.
Just as the history of science is rife with examples of scientists doing work to create a world in
which their facts are accepted as true, so too is the history of mathematics. For example, concepts
including zero, negative numbers, complex numbers, inﬁnity, and the calculus (to name but a few) all
experienced turbulent introductions into Western mathematics, and their ultimate acceptance was a
product of human work to create the conditions under which these ideas could be accepted as true.
Today of course, these objects have become so naturalized that the human work required to make them
and sustain them as naturalized objects has become largely invisible [
] (for accessible historical
treatments that reveal these objects in all of their anthropological strangeness, see: zero: [
numbers: ; complex numbers: ; inﬁnity: ; the calculus: ).
As I write this, I can hear the objections. ‘Okay, maybe inﬁnity required human work to make it
true, but 2 + 2 = 4 does not. As the Gardner quote in the beginning of this paper demonstrated, this
fact was true for the dinosaurs. It always has been true and always will be true’. But the Gardner quote
is deceptive. To understand why, we need to examine the grammatical role of 2.
When Gardner discusses 2 dinosaurs, “2” is an adjective. It is being used to describe dinosaurs.
Thus, Gardner’s statement isn not a statement about numbers, it is a statement about dinosaurs.
Educ. Sci. 2018,8, 43 3 of 13
Substitute any object you want for the dinosaurs, and you are left with a statement about those objects:
a statement, in other words, about the physical world. In the equation, 2 + 2 = 4, however, things are
very different. Here, “2” is a noun. It is not describing an object, it is the object [27,28].
Of course, this is not just an argument about grammar. It’s an argument about the difference
between concrete quantity (i.e., 2 dinosaurs) and abstract number (i.e., 2). The difference is profound.
Somewhere, “2” became decontextualized and thingiﬁed as an object in its own right [29,30]. This is a
massive human achievement. Indeed, the eminent Russian psychologist Lev Vygotsky viewed such
decontextualization as the principle measure of human sociocultural evolution .
Like many human achievements, the thingiﬁcation of number in the West is an innovation that
was driven by material necessity—probably the demands of commerce. In fact, commerce drove
many mathematical achievements. For example, surviving evidence suggests that coordinate systems
developed from a need to parcel land in ancient Egypt [
]. For communities that have different
demands, we would expect to see different innovations in the ways that people interact with quantity
and space, and indeed we do [
]. For example, Pinxten, Van Dooren, & Harvey [
] studied a Navajo
community, and found that rather than parceling land via boundaries, members of the community
“systemically represent the world and every discrete entity as a dynamic and continually changing
] (p. 36). In this dynamic and relational view of space, “notions of boundary cannot easily
be grasped with the Western perspectives; again, the essentially dynamic nature of anything existing
has to be taken into account so that boundaries are recognized as extreme variations in process, rather
than static positions”  (p. 36).
Thus, both the history of Western mathematics as well as contemporary cross-cultural studies belie
the folk notions of Plato-formalism. Historically, the introduction of new concepts into mathematics
was almost always turbulent and required human work to create the conditions under which they could
be accepted. Contemporaneously, when we examine cross-cultural mathematics, we ﬁnd “various
types of mathematics, irreducible to each other” [
] (p. 457). All of this contradicts the Platonic
notion that a single mathematics exists in complete form, just waiting to be discovered. Similarly, the
tight coupling of mathematics to local practices belies formalist notions that mathematics is a “game
Despite this evidence, widespread belief in the myth of Plato-formalism still underscores many
educational practices and this has consequences, both for mathematics education and for society.
4. The Consequences of Plato-Formalism on Mathematics Education and Society
One result of the widespread belief in the myth of Plato-formalism in the West is that mathematics
is seen as value-neutral, and it is taught as such. However, mathematics has values. When we teach
mathematics as if it were neutral, we are teaching these values [
], and this has consequences [
One such value is objectism: the systemic decomposition of the environment into discrete objects,
to be categorized and abstracted. We can trace the evolution of objectism in mathematics back to Euclid,
who built his Elements on three geometric objects (the point, the line, and the plane). But Euclid did
not have to follow such a program. During his lifetime, there were two competing philosophies on the
nature of the world, embodied by Heraclitus on the one hand, who saw the world in terms of change
and ﬂux, and Democritus on the other, who saw the world in terms of ‘atoms’ and objects. In the
modern-day West, Heraclitus’s world-view is largely forgotten, and Western mathematics, science,
and society have all been structured in the mold of Democritus [34,37].
The consequence of this choice has been the reiﬁcation of abstraction as the gold standard in
mathematics. This has led to staggering progress in developing mathematics as a discipline. However,
it has also had the effect of restricting school mathematics for generations of students, who learn that
“mathematics” means memorizing formal algorithms and procedures for abstract symbol-manipulation.
These procedures are revered for their perceived generality; they can potentially be applied and used in
many different situations, often very different from those in which they were learned. However, such
transfer is problematic at best [
]. Often, people use a variety of situationally-relevant methods
Educ. Sci. 2018,8, 43 4 of 13
for computation—strategies that recruit features of the situation into the computation, rather than
strategies that abstract out those features [38,40–45]. Mathematics adheres in the relationship between
people and setting.
We do not have to look far to ﬁnd people that see the world in terms of relationships rather than
objects. For example, recall how the Navajo community described above represented the world in
terms of change and relationships. My point here is not to advocate for such a worldview, but to
re-present it so that those of us in the West can see that objectism is a cultural way of perceiving the
world , rather than “the way the world exists”.
Understanding objectism as a product of human work allows us to explore the consequences
for math education and society. For math education, one consequence is the fetishization of abstract
procedures, as I discussed above. For Western society, one likely consequence of objectism is our
insistence on rigid categorization systems, formalized in the Aristotelian “law of the excluded middle:”
the logic proposition that states “that which is not true is false”. It is surely the case that categorization
is an inevitable human activity [
]. However, the rigidity with which we categorize in the West is not
inevitable; in many cases dialectic notions are more appropriate than Aristotelian rigidity. However,
rigid categorization persists, and is highly consequential:
Each standard and each category valorizes some point of view and silences another. This is
not inherently a bad thing—indeed it is inescapable. But it is an ethical choice, and as such it
is dangerous—not bad, but dangerous  (pp. 5–6).
The above discussion stands in sharp contrast to the neutrality and objectivity that is often
associated with mathematics. But mathematics is neither objective nor neutral: it is imbued with
values including objectism. If we teach mathematics as if it were neutral we reify these values and
make them invisible, beyond the gaze of critical study [
]. In the remainder of this paper, I will
sketch an alternate vision that humanizes mathematics and math education, a vision that accepts
the notion that mathematics is not neutral, and that makes visible the values that have thus far been
5. Mathematics Is a Human Activity
By now, my general claim should be clear: mathematics is a human activity [
]. It happens as
humans mathematize the word, and it leads to the creation of mathematical objects, including concepts,
models, tools, strategies, symbols, and algorithms. These objects become reiﬁed—thingiﬁed [
culture, and thus can become tools that enable new forms of mathematical activity. This set of objects,
which is created by humans as they engage in joint activity and reiﬁed in culture, is what we in the
West now call mathematics.
Let us consider the nature of these mathematical objects. When I say that mathematical objects
are cultural, I mean that they are the collected product of human mathematical activity [
]. In this
way, mathematical objects exist in the same way that languages exist, or symphonies, or literature.
They are neither mental nor physical, but neither do they exist Platonically, independent of humans.
As cultural objects, mathematical objects can be manipulated and extended by humans to create
new culture. Just as human authors manipulate existing language to create new works (consider a
Shakesperian play), so too do humans manipulate existing mathematics to produce new mathematics.
This new mathematics is then subjected to social review, judged by how well it ﬁts with the existing
physical and cultural worlds. This review is a social process and the standards for review are simply
social agreements [
]. For example, the notion of what constitutes a “proof” in mathematics is not
universal, neither across social groups , nor across time .
It has been useful up to this point to draw a parallel between mathematical objects and other
cultural objects such as language, music, and literature. However, there is something that separates
mathematical objects from these other cultural forms, namely, the stunning regularity that mathematical
objects exhibit. To explore this further, let us return to the trope of 2 + 2 = 4.
Educ. Sci. 2018,8, 43 5 of 13
As I summarized earlier, “2” as a mathematical object is very different from the physical notion
of “2 dinosaurs”. Mathematically, 2 is a cultural object, one that exists in Western culture by shared
agreement. For example, in certain versions of set theory, 2 is understood as a very particular set that is
built recursively from the null set [
]. “2” also exists in the counting sequence, by agreement, between
1 and 3. From this perspective, we might say that 2 is “counted into existence” by humans following
an agreed-upon convention .
However, 2 exists as more than simply an abstract set, or a symbol in an abstract sequence, it is
intertwined with the physical world. Humans most likely invented 2 as an abstraction of the physical
], many of us understand 2 in terms of the physical world such as a collection of objects or
a distance [
], and we impose 2 on the physical world via counting, measurement, and coordinate
]. Thus, although 2 is a human creation, we are not free to make our own decisions about
what 2 is. It is constrained by social agreement to conform to certain aspects of the physical and
mathematical world. So invented and constrained, 2 is now very different than other cultural objects:
it has a life of its own, with consequences that became inevitable immediately upon its invention and
constraint [27,28,56,57]. One of those consequences is that 2 + 2 = 4.
Much mathematical work involves the “discovery” of these consequences. Even here, however,
mathematics is more than deductive discovery, it still retains constructive and social elements [
For example, as described by Bloor [
], we might explore the consequences of 2 and + in two contexts
in which we have agreed upon rules for their application: physical measurement and counting.
A common way to conceptualize the coupling of these applications is a number line, where “0”
represents the starting point in our measurement and our counting sequence, and increases in length
on the number line correspond to successions in the counting sequence (Figure 1). On the number line,
Figure 1. Measuring and counting along a line.
However, what if I told you that we were doing this on a wheel, as in Figure 2? Notice that on this
wheel, when we move forward one unit from 2, we arrive back at our starting point, which we have
agreed is “0”. Thus, following our agreed convention on the wheel, we ﬁnd that 2 + 1 = 0, and not 3 as
shown on the number line. We have a contradiction! Which is it?
Figure 2. On this wheel, 2 + 1 = 0.
Educ. Sci. 2018,8, 43 6 of 13
First of all, we should ask, why does it have to be one or the other? Why cannot 2 + 1 equal 0
and 3? I admit that this makes me uncomfortable, but I recognize that this is because I am wedded
to a mathematics that is built on the Aristotelian law of the excluded middle—that is, the notion that
“that which is not true is false.” My purpose is not to argue against the desirability of keeping this
notion in Western mathematics (although some have explored this, see [
] for a brief summary), I just
want to point out that it is a choice. My discomfort with the notion that 2 + 1 might have two different
answers has nothing to do with the “true” existence of 2 + 1, and is instead a historically-contingent
consequence of prior human work.
Okay, but let us say that we agree to keep the law of the excluded middle. We are then left with
a contradiction. How shall we resolve it? The answer is that we must come to a social agreement
about which one we want to keep. There is no “true” value of 2 + 1, just agreed upon meanings and
behaviors. If those behaviors come into conﬂict we must make decisions, and decision-making is a
social process. In the West we have decided that “everyday” arithmetic should remain consistent with
the number line, and therefore it is excluded from the wheel. However, in doing so-called “wheel”
arithmetic, we have invented a whole new arithmetic. Western mathematicians call this “modular”
arithmetic, and it exists right alongside everyday arithmetic both in mathematics and in our everyday
lives (for example, in the U.S. we use wheel arithmetic—that is, arithmetic modulo 12—to keep track
of time during the day, where adding an hour follows the counting sequence 8:00, 9:00, 10:00
. . .
until 12:00, at which point adding another hour takes us to 1:00, not 13:00; in many other societies,
the same principle is used but with 24 as the modulus). Examples of such inventions abound in
mathematics, from non-Euclidean geometries that spring from a rethinking of parallel lines, to p-adic
number systems that can be understood by rethinking the notion of distance.
When we understand that mathematics is a human activity, the historical controversies that have
surrounded the introduction of new mathematics are not difﬁcult to explain. They are the result
of situations like wheel arithmetic, where established principles come into conﬂict. Similarly, the
wheel arithmetic example shows us how it is perfectly possible that multiple mathematics can exist
across cultures and time: mathematics happens as humans solve problems in their environment, and
different problems lead to different mathematics. Hence, mathematics is not a Platonic structure
that exists “out there”, waiting to be discovered. Instead, mathematics is a human activity, and
what we in the West know today as mathematics is a human—and therefore a social, cultural,
6. Signiﬁcance for Math Educators
Rejecting the myth of Plato-formalism and adopting the perspective that mathematics is a human
activity implies that math education should be: active, cultural, historical, critical, and social.
The Plato-formalist view of mathematics has led to a view of math education as a two-step process.
First, students learn formal skills and algorithms, and then they apply those skills in exercises. When
we understand mathematics as a human activity, we see that this is an “anti-didactic inversion,” [
of teaching the results of an activity rather than the activity itself. Instead, students should engage in
mathematical activity ﬁrst, and through this activity they should invent mathematical objects. In other
words, rather than starting with the structure of mathematics, math education should engage students
in structuring activities [
]. These structuring activities should be “whole activities” [
intertwine multiple strands of mathematics [
]. In this way, skills are not separated from the practices
that give them meaning [45,63].
None of this is to suggest that students should not learn formal mathematics. Rather, it is a
suggestion for how students should learn formal mathematics. The job of the teacher is to engage
students in guided reinvention [
], such that students invent mathematical objects by mathematizing
their world—including the mathematical world.
Educ. Sci. 2018,8, 43 7 of 13
Over the past three decades, hundreds of studies have been conducted on active learning.
Overwhelmingly, the evidence suggests that students who experience active learning learn more
mathematics and develop more positive mathematical identities, as compared to students who
experience more passive forms of instruction such as lectures (for reviews and meta-analyses,
]). The evidence is so overwhelming that, after reviewing 225 studies, Freeman and
colleagues explained, “If the experiments analyzed here had been conducted as randomized controlled
trials of medical interventions, they may have been stopped for beneﬁt—meaning that enrolling
patients in the control condition might be discontinued because the treatment being tested was clearly
more beneﬁcial”  (p. 8413).
An amazing thing happens when mathematical objects are incorporated into activity: they enable
new forms of activity and transform mental functioning in the process [
]. An example will help to
make this point. Figure 3shows two representations of the height of an individual on a Ferris Wheel
as a function of time. Each representation is a mathematical object. Now, imagine that we want to
predict the height of the individual after riding the wheel for 30 s. The task is different depending on
the object used.
Figure 3. Two representations of the height of a Ferris Wheel as a function of time.
Analytically, we have to perform algebraic, arithmetic, and trigonometric operations. Graphically,
we have to coordinate horizontal and vertical distances. Thus we might ask, where did the algebra
go? The arithmetic? The trigonometry? The answer is that these operations were absorbed into the
graph. The graphical representation thus transforms a computation task into a spatial coordination
task (cf., ).
More generally, different mathematical objects transform the mental functioning necessary to solve
a problem. Consequently, mental operations cannot exist separately from these objects. The implication
is profound: mathematical cognition does not happen solely “between the ears” [
], but rather is
distributed across systems of persons and objects [77–79]:
If we ascribe to individual minds in isolation the properties of systems that are actually
composed of individuals manipulating systems of cultural artifacts, then we have attributed
to individual minds a process that they do not necessarily have. 
The key consideration for teachers is the classroom environment. Often, classrooms are organized
such that they restrict access to the resources that create cognition (consider, for example, the barren
conditions under which students take tests). This might make sense from a perspective in which
learning is seen as acquisition of knowledge (e.g., ), but it is counter-productive from the cultural
perspective that I have outlined here. From a cultural perspective, “humans create their cognitive
powers by creating the environments in which they exercise those powers” [
]. Thus, the way to
Educ. Sci. 2018,8, 43 8 of 13
make classrooms powerful places for cognition is to saturate them with cognitive resources and give
students the power to manipulate that environment [63,81,82].
Engaging students in the history of mathematics helps to make mathematics less mystical by
making visible the role of humans, showing students that “mathematics exists and evolves in time
and space, [and] human beings have taken part in the evolution” [
]. The goal is to help students
see mathematics as historically contingent [
], and to see themselves and their ancestors as historical
actors, actively playing a role in the colossal human achievement that is Western mathematics.
Western mathematics and Western society have grown together and inﬂuenced each other through
human decisions. The Plato-formalist view reiﬁes the resulting mathematical and societal structures
as natural. However, when we understand mathematics as a historically-contingent result of human
work, we can critically examine its consequences.
For instance, imagine a math class that engages in the wheel arithmetic example that I described
earlier. As the class struggles with the idea that 2 + 1 might equal both 0 and 3, it would be a great time
for the teacher to question why it is such a struggle to accept both. This might lead to a discussion of
rigid categorizations, and the real consequences of these categorizations. For example, a class might
discuss the consequences of rigid gender categorization in a society that insists on categorizing people
before they can use the restroom.
Furthermore, when we see how mathematics is intertwined with society, we can see how Western
mathematics can be used as a hegemonic tool. Such has been the case throughout colonial history, as the
supposedly “neutral” mathematical practices of the West have been used as a tool of domination and
cultural genocide [
]. Even in modern United States (U.S.) schools, mathematics continues to operate
as an instrument of colonization [
] that reproduces historical systems of privilege and inequality [
all while normalizing and obscuring these operations [
]. Students deserve to engage with these
], and understanding mathematics as a human activity can help to expose mathematics to
critical review within the math classroom.
It is not all bad of course. Mathematics is one of humanity’s greatest achievements, and it has
been used to shape society in an uncountable number of positive ways. Even if mathematics has
had and continues to have pernicious effects, so too can it be used as a tool for social justice [
As students engage in critical study of mathematics and its effect on society, they should do so with an
understanding that the effects of mathematics are human effects, and that they, as historical actors,
have the power to shape those effects (Gutstein  calls this critical agency).
Mathematics requires social interaction. Social interaction facilitates problem-solving [
and it is through social interaction that mathematical objects are invented and thingiﬁed—both
historically  and in classrooms [92,93].
But social interaction is more than a means to help students do mathematics. It is the requisite
background against which the previous four features (active, cultural, historical, and critical) are
enacted, and it is the mechanism by which students are produced as cultural people. For Radford [
this sort of production is more important than technical competencies:
What is important in teaching–learning mathematics is not really to become a good problem
solver. Although knowing how to solve problems in a technical sense may be an important
goal, more important, I think, is the range of possibilities that mathematics offers to our
students to live it as a social, historical, cultural, and esthetic experience. But to be truly
meaningful, this experience has to occur in the public space of words, deeds and actions—in
Educ. Sci. 2018,8, 43 9 of 13
the polis, that is to say, the organized space of the people “as it arises out of acting and
speaking together” (Arendt, 1958b, p. 198)  (p. 111).
Twenty ﬁve years ago, Barbeau  summarized the popular perception of mathematics:
Most of the population perceive mathematics as a ﬁxed body of knowledge long set into
ﬁnal form. Its subject matter is the manipulation of numbers and the proving of geometrical
deductions. It is a cold and austere discipline which provides no scope for judgment
or creativity (quoted in , p. 432).
Barbeau describes the popular perception of mathematics as a Plato-formalist structure. But
Plato-formalism is a myth. That this myth continues to exact such a strong hold on popular perceptions
is a tragedy, for our students, for our society, and for our mathematics. As teachers, we can make
things right for our students by rejecting Plato-formalism and embracing mathematics as a human
activity, engaging students in mathematical experiences that are active, cultural, historical, critical, and
social. This will help students to learn formal Western mathematics, as well as empower students to
shape and use mathematics in their daily lives, and to see, challenge, and inﬂuence the effects that
mathematics has on society.
In the past 25 years, mathematics education has made astonishing progress as a ﬁeld, and this
has had material effects for countless students (see e.g., the studies on active learning referenced in
Section 6of this paper). Yet, the situation that Barbeau described is still far too pervasive. We can
change that. Rejecting the myth of Plato-formalism is the ﬁrst step.
I thank Carrie Allen, Rubén Donato, Sara Heredia, Molly Shea, Bharath Sriraman, Joanna
Weidler-Lewis, and David Webb for helpful comments and feedback.
Conﬂicts of Interest: The author declares no conﬂict of interest.
Gardner, M. Is mathematics for real? The New York Review of Books, 13 August 1981. Available online:
http://www.nybooks.com/articles/1981/08/13/is-mathematics-for-real/ (accessed on 26 March 2018).
Radford, L. Culture and cognition: Towards an anthropology of mathematical thinking. In Handbook of
International Research in Mathematics Education; English, L.D., Kirshner, D., Eds.; Routledge: New York, NY,
USA, 2008; pp. 439–464.
Brown, J.R. Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures, 2nd ed.;
Routledge: New York, NY, USA, 2008; ISBN 9780415960489.
Philipp, R.A. Mathematics teachers’ beliefs and affect. In Second Handbook of Research on Mathematics Teaching
and Learning; Lester, F.K., Ed.; Information Age Publishing: Charlotte, NC, USA, 2007; pp. 257–315.
5. Ernest, P. What is our ﬁrst philosophy in mathematics education? Learn. Math. 2012,32, 8–14.
Pimm, D. Why the history and philosophy of mathematics should not be rated X. Learn. Math.
7. Wolfson, P. Philosophy enters the mathematics classroom. Learn. Math. 1981,2, 22–26.
8. Hawkins, D. The edge of Platonism. Learn. Math. 1985,5, 2–6.
Linnebo, Ø. Platonism in the philosophy of mathematics. In Stanford Encyclopedia of Philosophy; Zalta, E.N.,
Ed.; Stanford University: Stanford, CA, USA, 2011.
10. Balaguer, M. Fictionalism, theft, and the story of mathematics. Philos. Math. 2008,17, 131–162. [CrossRef]
11. Hanna, G. Some pedagogical aspects of proof. Interchange 1990,21, 6–13. [CrossRef]
Gödel, K. Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I.
Monatshefte für Mathematik und Physics 1931,38, 173–198. [CrossRef]
Van Heijenoort, J. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931; Harvard University
Press: Cambridge, MA, USA, 1967; ISBN 0674324498.
Ernest, P. Social Constructivism as a Philosophy of Mathematics; State University of New York Press: Albany, NY,
Educ. Sci. 2018,8, 43 10 of 13
Davis, P.J.; Hersh, R.; Marchisotto, E.A. The Mathematical Experience, Study Edition; Birkhäuser: Boston, MA,
Kuhn, T.S. The Structure of Scientiﬁc Revolutions; University of Chicago Press: Chicago, IL, USA, 1970;
Volume 2, ISBN 0226458083.
Latour, B. Give me a laboratory and I will raise the world. In Science Observed: Perspectives on the Social Study
of Science; Knorr-Cetina, K., Mulkay, M., Eds.; SAGE Publications: London, UK, 1983; pp. 141–169.
Latour, B.; Woolgar, S. Laboratory Life: The Construction of Scientiﬁc Facts, 2nd ed.; Princeton University Press:
Princeton, NJ, USA, 1987.
Wartofsky, M.W. Perception, representation, and the forms of action: Towards an historical epistemology.
In Models: Representation and the Scientiﬁc Understanding; Wartofsky, M.W., Ed.; D. Reidel: Dordrecht,
The Netherlands, 1979.
Pickering, A. The Mangle of Practice: Time, Agency, and Science; University of Chicago Press: Chicago, IL, USA,
1995; ISBN 0226668258.
Bowker, G.C.; Star, S.L. Sorting Things Out: Classiﬁcation and Its Consequences; MIT Press: Cambridge, MA,
Kaplan, R. The Nothing That Is: A Natural History of Zero; Oxford University Press, USA: New York, NY, USA,
1999; ISBN 0195128427.
Martínez, A.A. Negative Math: How Mathematical Rules Can Be Positively Bent; Princeton University Press:
Princeton, NJ, USA, 2006; ISBN 0691123098.
Derbyshire, J. Unknown Quantity: A Real and Imaginary History of Algebra; Plume: New York, NY, USA, 2006;
Maor, E. To Inﬁnity and Beyond: A Cultural History of the Inﬁnite; Princeton University Press: Princeton, NJ,
USA, 1991; ISBN 0691025118.
Bardi, J.S. The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of all Time; Basic Books:
New York, NY, USA, 2007.
27. Hersh, R. What Is Mathematics, Really? Oxford University Press: Oxford, UK, 1997.
Hersh, R. Fresh breezes in the philosophy of mathematics. In Mathematics, Education, and Philosophy:
An International Perspective; Ernest, P., Ed.; The Falmer Press: Bristol, PA, USA, 1994; pp. 11–20.
Sfard, A. On the dual nature of mathematical conceptions: Reﬂections on processes and objects as different
sides of the same coin. Educ. Stud. Math. 1991,22, 1–36. [CrossRef]
Stewart, I. Nature’s Numbers: The Unreal Reality of Mathematics; Basic Books: New York, NY, USA, 1995;
Wertsch, J.V. Vygotsky and the Social Formation of Mind; Harvard University Press: Cambridge, MA, USA,
1985; ISBN 0674943511.
Kinard, J.T.; Kozulin, A. Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom;
Cambridge University Press: Cambridge, UK, 2008; ISBN 0521876850.
Pinxten, R.; Van Dooren, I.; Harvey, F. The Anthropology of Space: Explorations into the Natural Philosophy and
Semantics of the Navajo; University of Pennsylvania Press: Philadelphia, PA, USA, 1983.
Bishop, A.J. Mathematical Enculturation: A Cultural Perspective of Mathematics Education; Kluwer Academic
Publishers: Dordrecht, The Netherlands, 1991.
Freire, P. Pedagogy of Hope: Reliving Pedagogy of the Oppressed; Continuum: New York, NY, USA, 1994;
36. Davis, P.J. Applied mathematics as a social contract. Math. Mag. 1988,61, 139–147. [CrossRef]
Shulman, B. What if we change our axioms? A feminist inquiry into the foundations of mathematics.
Conﬁgurations 1996,4, 427–451. [CrossRef]
Lave, J. Cognition in Practice: Mind, Mathematics, and Culture in Everyday Life; Cambridge University Press:
Cambridge, UK, 1988.
Packer, M.J. The problem of transfer, and the sociocultural critique of schooling. J. Learn. Sci.
Nunes, T.; Schliemann, A.D.; Carraher, D.W. Street Mathematics and School Mathematics; Cambridge University
Press: Cambridge, UK, 1993; ISBN 0521388139.
Carraher, T.N.; Carraher, D.W.; Schliemann, A.D. Mathematics in the streets and in schools. Br. J. Dev. Psychol.
1985,3, 21–29. [CrossRef]
Educ. Sci. 2018,8, 43 11 of 13
42. Scribner, S. Knowledge at work. Anthropol. Educ. Q. 1985,16, 199–206. [CrossRef]
Lave, J.; Murtaugh, M.; de la Rocha, O. The dialectic of arithmetic in grocery shopping. In Everyday Cognition:
Its Development in Social Context; Rogoff, B., Lave, J., Eds.; Harvard University Press: Cambridge, MA, USA,
1984; pp. 67–94.
Taylor, E.V. The purchasing practice of low-income students: The relationship to mathematical development.
J. Learn. Sci. 2009,18, 370–415. [CrossRef]
Nasir, N.S.; Hand, V.M. From the court to the classroom: Opportunities for engagement, learning, and
identity in basketball and classroom mathematics. J. Learn. Sci. 2008,17, 143–179. [CrossRef]
46. Freudenthal, H. Mathematics as an Educational Task; D. Reidel: Dordrecht, The Netherlands, 1973.
White, L.A. The locus of mathematical reality: An anthropological footnote. Philos. Sci.
Cole, M. What’s culture got to do with it? Educational research as a necessarily interdisciplinary enterprise.
Educ. Res. 2010,39, 461–470. [CrossRef]
Kleiner, I.; Movshovitz-Hadar, N. Aspects of the pluralistic nature of mathematics. Interchange
Joseph, G.G. Different ways of knowing: Contrasting styles of argument in Indian and Greek mathematical
traditions. In Mathematics, Education, and Philosophy: An International Perspective; Ernest, P., Ed.; The Falmer
Press: Bristol, PA, USA, 1994; pp. 194–204.
Jaffe, A.; Quinn, F. “Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical
physics. Bull. Am. Math. Soc. 1993,29, 1–13. [CrossRef]
52. Jech, T. Set Theory, 3rd ed.; Springer: Berlin/Heidelberg, Germany, 2006; ISBN 3540440852.
Rotman, B. Mathematics as Sign: Writing, Imagining, Counting; Stanford University Press: Stanford, CA, USA, 2000.
Lakoff, G.; Núñez, R.E. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being;
Basic Books: New York, NY, USA, 2001; ISBN 0465037712.
Latour, B. Circulating reference: Sampling soil in the Amazon Rainforest. In Pandora’s Hope; Harvard
University Press: Cambridge, MA, USA, 1999; pp. 24–79.
Glas, E. Mathematics as objective knowledge and as human practice. In 18 Unconventional Essays on the
Nature of Mathematics; Hersh, R., Ed.; Springer: New York, NY, USA, 2006; pp. 289–303.
57. Lockhart, J. A Mathematician’s Lament; Bellevue Literary Press: New York, NY, USA, 2009.
Lakatos, I. Proofs and Refutations: The Logic of Mathematical Discovery; Cambridge University Press: Cambridge,
Bloor, D. What can the sociologist of knowledge say about 2 + 2 = 4? In Mathematics, Education, and Philosophy:
An International Perspective; Ernest, P., Ed.; The Falmer Press: Bristol, PA, USA, 1994; pp. 21–32.
Fosnot, C.T.; Jacob, B. Young Mathematicians at Work: Constructing Algebra; Heinemenn: Portsmouth, NH,
Van den Heuvel-Panhuizen, M.; Wijers, M. Mathematics standards and curricula in The Netherlands. Zdm
2005,37, 287–307. [CrossRef]
Cole, M.; Grifﬁn, P. A socio-historical approach to re-mediation. Q. Newsl. Lab. Comp. Hum. Cogn.
Gutiérrez, K.D.; Hunter, J.D.; Arzubiaga, A. Re-mediating the university: Learning through sociocritical
literacies. Pedagogies 2009,4, 1–23. [CrossRef]
Gravemeijer, K.; Doorman, M. Context problems in realistic mathematics education: A calculus course as an
example. Educ. Stud. Math. 1999,39, 111–129. [CrossRef]
Freeman, S.; Eddy, S.L.; McDonough, M.; Smith, M.K.; Okoroafor, N.; Jordt, H.; Wenderoth, M.P. Active
learning increases student performance in science, engineering, and mathematics. Proc. Natl. Acad. Sci. USA
2014,111, 8410–8415. [CrossRef] [PubMed]
Laursen, S.L.; Hassi, M.; Kogan, M.; Weston, T.J. Beneﬁts for women and men of Inquiry-Based Learning in
college mathematics: A multi-institution study. J. Res. Math. Educ. 2014,45, 406–418. [CrossRef]
Barron, B.; Schwartz, D.L.; Vye, N.J.; Moore, A.; Petrosino, A.; Zech, L.; Bransford, J.D. The cognition
and technology group at Vanderbilt. Doing with understanding: Lessons from research on problem- and
project-based learning. J. Learn. Sci. 1998,7, 271–311. [CrossRef]
Educ. Sci. 2018,8, 43 12 of 13
Barron, B.; Chen, M. Teaching for meaningful learning: A review of research on inquiry-based and
cooperative learning. In Powerful Learning: What We Know about Teaching for Understanding; Edutopia:
San Rafael, CA, USA, 2008; ISBN 978-0-470-2766-9.
Savelsbergh, E.R.; Prins, G.T.; Rietbergen, C.; Fechner, S.; Vaessen, B.E.; Draijer, J.M.; Bakker, A. Effects of
innovative science and mathematics teaching on student attitudes and achievement: A meta-analytic study.
Educ. Res. Rev. 2016,19, 158–172. [CrossRef]
Stein, M.K.; Boaler, J.; Silver, E.A. Teaching mathematics through problem solving: Research perspectives.
In Teaching Mathematics through Problem Solving: Grades 6–12; Schoen, H.L., Ed.; NCTM: Reston, VA, USA,
2003; pp. 245–256.
Cai, J. What research tells us about teaching mathematics through problem solving. In Teaching Mathematics
through Problem Solving: Prekindergarten–Grade 6; Lester, F.K., Ed.; NCTM: Reston, VA, USA, 2003; pp. 241–254.
Kilpatrick, J. What Works? In Standards-Based School Mathematics Curricula: What Are They? What Do Students
Learn? Senk, S.L., Thompson, D.R., Eds.; Lawrence Erlbaum: Mahwah, NJ, USA, 2003; pp. 471–488.
Schoenfeld, A.H. What makes for powerful classrooms, and how can we support teachers in creating them?
A story of research and practice, productively intertwined. Educ. Res. 2014,43, 404–412. [CrossRef]
Vygotsky, L.S. Mind in Society; Cole, M., John-Steiner, V., Scribner,S., Souberman, E., Eds.; Harvard University
Press: Cambridge, MA, USA, 1978.
75. Hutchins, E. How a cockpit remembers its speeds. Cogn. Sci. 1995,19, 265–288. [CrossRef]
Mehan, H. Beneath the skin and between the ears: A case study in the politics of representation.
In Understanding Practice: Perspectives on Activity and Context; Chaiklin, S., Lave, J., Eds.; Cambridge University
Press: Cambridge, UK, 1996; pp. 241–268.
Hall, R.; Wieckert, K.; Wright, K. How does cognition get distributed? Case studies of making concepts
general in technical and scientiﬁc work. In Generalization of Knowledge: Multidisciplinary Perspectives;
Caccamise, D., Ed.; Routledge: New York, NY, USA, 2010; pp. 225–246.
78. Hutchins, E. Cognition in the Wild; MIT Press: Cambridge, MA, USA, 1995.
Cole, M.; Engeström, Y. A cultural-historical approach to distributed cognition. In Distributed Cognitions:
Psychological and Educational Considerations; Salomon, G., Ed.; Cambridge University Press: Cambridge, UK,
1993; pp. 1–46.
Anderson, J.R.; Schunn, C.D. Implications of the ACT-R learning theory: No magic bullets. In Advances in
Instructional Psychology; Glaser, R., Ed.; Erlbaum: Mahwah, NJ, USA, 2000; Volume 5, pp. 1–34.
Cole, M.; Hood, L.; McDermott, R.P. Ecological Niche Picking: Ecological Invalidity as an Axiom of Experimental
Cognitive Psychology; Rockefeller University, Laboratory of Comparative Cognition: New York, NY, USA, 1978.
Engeström, Y. Non scole sed vitae discimus: Toward overcoming the encaptulation of school learning.
Learn. Instr. 1991,1, 243–259. [CrossRef]
Jankvist, U.T. An empirical study of using history as a “goal”. Educ. Stud. Math.
,74, 53–74. [CrossRef]
Bishop, A.J. Western mathematics: The secret weapon of cultural imperialism. Race Class
Gutiérrez, R. (Re)deﬁning equity: The importance of a critical perspective. In Improving Access to Mathematics:
Diversity and Equity in the Classroom; Nasir, N.S., Cobb, P., Eds.; Teachers College Press: New York, NY, USA,
2006; pp. 37–50.
Secada, W.G. Agenda setting, enlightened self-interest, and equity in mathematics education. Peabody J. Educ.
1989,66, 22–56. [CrossRef]
87. Gutstein, E. Reading and Writing the World with Mathematics; Routledge: New York, NY, USA, 2006.
Esmonde, I.; Caswell, B. Teaching mathematics for social justice in multicultural, multilingual elementary
classrooms. Can. J. Sci. Math. Technol. Educ. 2010,10, 244–254. [CrossRef]
Gutiérrez, R. Embracing the inherent tensions in teaching mathematics from an equity stance. Democr. Educ.
90. Barron, B. When smart groups fail. J. Learn. Sci. 2003,12, 307–359. [CrossRef]
Barron, B. Achieving coordination in collaborative problem-solving groups. J. Learn. Sci.
Cobb, P.; Stephan, M.; McClain, K.; Gravemeijer, K. Participating in classroom mathematical practices.
J. Learn. Sci. 2001,10, 113–163. [CrossRef]
Educ. Sci. 2018,8, 43 13 of 13
Peck, F.A.; Matassa, M. Reinventing fractions and division as they are used in algebra: The power of
preformal productions. Educ. Stud. Math. 2016,92, 245–278. [CrossRef]
94. Radford, L. Education and the illusions of emancipation. Educ. Stud. Math. 2012,80, 101–118. [CrossRef]
Barbeau, E.J. Mathematics for the Public. In Proceedings of the Meeting of the International Commission on
Mathematical Instruction, Leeds, UK, September 1989.
Romberg, T.A. Further thought on the standards: A reaction to Apple. J. Res. Math. Educ.
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