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A New Model of Mathematics Education: Flat Curriculum with Self-Contained Micro Topics

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Abstract

The traditional way of presenting mathematical knowledge is logical deduction, which implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent developments and methodical inventions in mathematics education, many curricula are still close in spirit to this hierarchical structure. However, this organisation of mathematical ideas may not be the most conducive way for learning mathematics. In this paper, we suggest that flattening curricula by developing self-contained micro topics and by providing multiple entry points to knowledge by making the dependency graph of notions and subfields as sparse as possible could improve the effectiveness of teaching mathematics. We argue that a less strictly hierarchical schedule in mathematics education can decrease mathematics anxiety and can prevent students from ‘losing the thread’ somewhere in the process. This proposal implies a radical re-evaluation of standard teaching methods. As such, it parallels philosophical deconstruction. We provide two examples of how the micro topics can be implemented and consider some possible criticisms of the method. A full-scale and instantaneous change in curricula is neither feasible nor desirable. Here, we aim to change the prevalent attitude of educators by starting a conversation about the flat curriculum alternative.
philosophies
Article
A New Model of Mathematics Education: Flat Curriculum with
Self-Contained Micro Topics
Miklós Hoffmann 1,2,† and Attila Egri-Nagy 3,*,†


Citation: Hoffmann, M.;
Egri-Nagy, A. A New Model of
Mathematics Education: Flat
Curriculum with Self-Contained
Micro Topics. Philosophies 2021,6, 76.
https://doi.org/10.3390/
philosophies6030076
Academic Editor: Rossella Lupacchini
Received: 6 August 2021
Accepted: 8 September 2021
Published: 13 September 2021
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4.0/).
1Department of Mathematics, Eszterházy Károly University, 3300 Eger, Hungary;
hoffmann.miklos@uni-eszterhazy.hu
2Faculty of Informatics, University of Debrecen, 4032 Debrecen, Hungary
3Department of Mathematics and Natural Sciences, Akita International University,
Yuwa, Akita 010-1292, Japan
*Correspondence: egri-nagy@aiu.ac.jp
These authors contributed equally to this work.
Abstract:
The traditional way of presenting mathematical knowledge is logical deduction, which
implies a monolithic structure with topics in a strict hierarchical relationship. Despite many recent
developments and methodical inventions in mathematics education, many curricula are still close in
spirit to this hierarchical structure. However, this organisation of mathematical ideas may not be the
most conducive way for learning mathematics. In this paper, we suggest that flattening curricula
by developing self-contained micro topics and by providing multiple entry points to knowledge
by making the dependency graph of notions and subfields as sparse as possible could improve
the effectiveness of teaching mathematics. We argue that a less strictly hierarchical schedule in
mathematics education can decrease mathematics anxiety and can prevent students from ‘losing the
thread’ somewhere in the process. This proposal implies a radical re-evaluation of standard teaching
methods. As such, it parallels philosophical deconstruction. We provide two examples of how the
micro topics can be implemented and consider some possible criticisms of the method. A full-scale
and instantaneous change in curricula is neither feasible nor desirable. Here, we aim to change the
prevalent attitude of educators by starting a conversation about the flat curriculum alternative.
Keywords:
mathematics; education; philosophy of education; flat curriculum; micro topics; decon-
struction; dependency graph
1. Introduction
Mathematics as a scientific discipline has been deductively described and presented in
the scientific community for centuries. Even if the presentation is not fully axiomatic, one
must possess a solid basis of the preliminary hierarchical system of notions and statements
to understand the actual mathematical topic. The theoretical aspects of these educational
hierarchical systems, also called learning hierarchies, have been widely studied from the
viewpoint of many disciplines for decades (cf. [
1
4
] and the references therein). However,
less attention is paid to the fundamental question of the extent to which it is necessary to
maintain this hierarchy in mathematics education from practical and theoretical points of
view. Although the deductive approach in mathematics education is becoming increasingly
backward, giving way to many modern methodological approaches, such as inquiry-
based learning, research-based learning, etc. We still do not seem to be able to go beyond
the approach arising from the abovementioned hierarchical structure of concepts and
statements in education. This insistence, in turn, makes the mathematical knowledge
provided over the school years similar to a monolithic, large structure with a highly
interconnected system of concepts, where there is little chance to understand a topic
without possessing knowledge from other, previously discussed topics in the curriculum.
Philosophies 2021,6, 76. https://doi.org/10.3390/philosophies6030076 https://www.mdpi.com/journal/philosophies
Philosophies 2021,6, 76 2 of 11
There is practically a single entry point into mathematics education. Many children and
pupils in school may be frightened when realising the weight, rigour and robustness of this
huge building-like structure. Mathematics anxiety, which involves fear, nervousness, or
even bodily symptoms related to mathematical activities, is an existing issue in education.
Classical experimental tools exist to measure math anxiety, e.g., Fennema–Sherman
Mathematics Attitudes Scales. The outcome of these tests shows that approximately 93% of
adult US students indicate that they experience some level of math anxiety [
5
], 59% of the
15–16-year-old students reported that they often worry that math classes will be difficult
for them, 33% reported that they become very tense when they have to complete math
homework and a further 31% stated that they become very nervous solving math problems.
Math anxiety correlates negatively with PISA math task achievement [
6
], and there is no
significant decrease in math anxiety throughout the years. It generally increases with age
among young people.
We aim to provide a significantly and structurally different mathematics education
approach: flattening the curriculum and reconsidering the dependencies of topics and
notions by introducing a set of self-contained micro topics instead of a single structure. Con-
sequently, our vision of a new system allows for multiple entry points and can contribute
to a decrease in math anxiety. As every mathematical and methodological approach has its
own philosophical background, our proposal for change also has its foundation inherently
embedded in Derrida’s deconstruction, which we discuss in detail in an upcoming section.
Ultimately, we can characterise our vision by describing mathematical knowledge as a
complex decentralised system, but we do not investigate this perspective here.
We claim that the underlying philosophy of mathematics influences how the subject
is taught. Curiously, the philosophical assumptions are less critical for actually doing
mathematics. In any case, it is fitting to state our assumptions upfront. As a simple
classification, we identify two ways of thinking about Mathematics, somewhat similar
to [7].
Mathematics is a beautiful, grand structure of ideas highly connected by powerful ab-
stractions, organised into a hierarchical structure along with the axiomatic-deductive
method.
Mathematics is a set of cognitive tools for efficient thinking. One can conceptualise
mathematical thinking as a way to delegate our cognitive load in problem solving to
mathematical symbolism.
Roughly speaking, these can be identified as the perspectives of mathematicians
and of scientists and engineers. These are often contrasted, especially in the context of
pure versus applied mathematics, but we do not see them as contradicting approaches.
An engineer can appreciate the beauty of mathematical coherence while simultaneously
using a specific method from a narrow branch of mathematics. Similarly, a mathematician
can appreciate real-world applications. Whatever professions are associated with these
approaches, mathematics education struggles with applying both of them. While trying to
demonstrate the importance of problem solving, which is related to the second approach, it
cannot break away from the hierarchical structure of the first approach. The result, however,
is that the daily struggle to understand pieces of the grand structure draws its breath away
from the joy of problem solving. Mathematics education, despite many efforts, is still
mostly about building and understanding the hierarchy and connectivity of theoretical
notions and statements.
Regarding the ontological status of mathematics, we endorse a noncommittal version
of realism. Mathematical objects exist in the sense that they demonstrate some resistance
to our inquiries (they ‘kick back’), and they have causal power (through the actions of
humans). Whether they really exist in some platonic realm or they are merely constructed
socially is irrelevant for our discussion. What is crucial for us is that people have access to
this reality and that they naturally enjoy thinking about mathematical ideas. Moreover, it
is beneficial to society if more people have access to mathematical experiences. The actual
mode of existence of mathematical structures can be separated from how we think about
Philosophies 2021,6, 76 3 of 11
them, as a set of tools or as a grand structure, as outlined above. However, the platonist view
tends to conflate these two: objective realism pairs with the axiomatic method. Therefore,
what we say will likely confront the platonist view. That is not our goal.
Our claim is that how we teach mathematics and how we present mathematics as a
whole depends on which ontological viewpoint we are closer to. Curricula at all levels of
education seem to be based mainly on the grand structure view. We argue that this is out
of balance. By thinking about a second perspective, we could make mathematical ideas
accessible for a wider audience (not just gifted pupils and students with math majors).
2. Metaphors for the Process of Learning Mathematics
A metaphor can also describe the gist of our argument. It rephrases the view of the
foundationalist description of the nature of mathematics as a solid building. This idea can
be traced all the way back to Aristotle. He considered science in his metaphysics as a series
of deductions from principles, where principles play a similar role as the foundation of
a house. Without putting the necessity and validity of this foundation into question, we
argue that the mathematical building of Aristotle is way too high today for a student, and
instead of a single giant skyscraper, perhaps we could try to build several smaller buildings
in education. However, let us also describe this using an alternative, geographic metaphor.
Climbing a high mountain peak is a dangerous activity. Many become stuck in the
middle, and in extreme cases, they may even fall down. The air is thin. However, those
few who can reach the summit have a complete view of the countryside, similar to an
all-encompassing map—a privileged view.
Walking the country from village to village visiting one hut at a time is a pleasant
activity. Spending some time in one place, talking to the people there, then moving to
another place is also enjoyable and meaningful. Even if the places wandered do not come
together to form a complete map of the countryside, the journey is a valuable life experience.
In addition, one can even start to see the big picture as a result of regular hiking.
As these metaphors imply, we suggest that experiencing a collection of small-sized,
relatively self-contained mathematical topics can increase the efficiency of mathematics
education. We propose this method since we think that the traditional hierarchical structure
of the curriculum not only fails to convey the grand structure of mathematical knowledge
but also harms most students in the process. Following from the abovementioned anxiety-
related problems, our main target readership consists of those teachers who feel that some
or most of their students are of mediocre level or labeled as underachievers. Even students
who cannot successfully conduct their mathematical studies, lost interest, or suffer from
math anxiety could benefit from the proposed method. It is well evidenced in the literature
that one of the main hindrances to teacher creativity and successful teaching practice is
the curriculum itself: the curricular restrictions [
8
]. We aim to soften these restrictions in a
revolutionary manner drastically. It is not the primary purpose, but we even suspect that
a flattened curriculum can lead to a coherent understanding of Mathematics with higher
probability. We rely on the natural pattern-matching abilities of the human brain and mind.
We claim that this method is even better than giving a ready-made structure, as that does
not entail personal knowledge [
9
]. It is a fundamental change of view, but it may be worth
trying after so many failures in mathematics education.
3. Philosophical Background—Deconstruction
Mathematical knowledge is organised as a hierarchical system by the deductive ax-
iomatic method. We take this as a fact, though it is conceivable that there might be other
principles of organisation. Concepts of the mainstream of 20th century philosophy (of sci-
ence), such as logicism, positivism, formalism, structuralism and even social constructivism,
consider to be mathematics a large, massive building with solid axiomatic foundations.
This view has deeply influenced didactical concepts in several countries, and for several
decades, the teaching of mathematics has been, and still is, somewhat similar to the con-
struction of this building through numerous school years. However, we challenge the
Philosophies 2021,6, 76 4 of 11
view that mathematics is best taught according to its inherent structure. In this section, we
discuss how the concept of Derrida’s deconstruction can change this view and what aspects
of thoughts and didactical works can be philosophically supported by deconstruction,
yielding a less frightening and less oppressive methodical approach for students. The mu-
tually beneficial relationship of mathematics education and philosophy has a long tradition
in European history, from the pivotal figures of the ancient Greek classical period—where,
in fact, several innovators represented both fields—to the eminent scientists of the 20th
century. From Plato to Husserl and from Kant to Bertrand Russell, many philosophers
have been influenced by their contemporary (and sometimes by their own) mathematical
thoughts and initiatives, while mathematicians have always reflected on the philosophical
movements of their period. For a good overview of this cross-fertilisation, see, e.g., [
10
].
Since the first half of the 20th century, well-known and popular(ised) concepts of the philo-
sophical mainstream have been, among others, logicism (Frege, Russell), logical positivism
(Vienna Circle) and formalism (Hilbert). Recently, structuralism has emerged as a view of
mathematics (Benacerraf, Saphiro), see, e.g., [
11
]. What is common in these—sometimes
rival—concepts is that they consider mathematics (or science in general) as a large, massive
structure, and the aim of our scientific and methodical community is nothing else than to
prepare foundations of this structure that are as solid as possible and to build this structure
as high as we can. Other recent approaches, such as social constructivism (
see [12]
) may
add critical aspects to the absolutism of mathematics, emphasising its human construction,
but the fundamental idea is still to construct something. This view has been and still
is the ultimate approach in our education, aiming to construct a large structure called
mathematics from preschool (sets and logic) to university. However, as it is well-known
since Gödel, the foundation of this building cannot be solid anymore. An alternative way of
considering and teaching mathematics can be based on the philosophical concept emerged
in the last decades, called deconstruction (or deconstructivism), introduced by Derrida [
13
]
and inspired by Saussure and Heidegger. Deconstruction, in its primal form, is an attempt
to criticise and put into question the fundamental concepts of methodical approaches and
forms of description. One can understand the approach of Derrida from his critical note on
classical philosophical (as well as mathematical) view: “we are not dealing with the peace-
ful coexistence of a vis-a-vis [of notions], but rather with a violent hierarchy. One of the
two terms governs the other (axiologically, logically, etc.), or has the upper hand” [
13
]. In
terms of mathematics, the revolutionary approach is to deconstruct this “violent hierarchy”
of definitions and theorems and to let teachers and students deal with coexistent notions
and concepts without being pressured by the weight of the large structure. We suggest that
the widely accepted methods, which organise textbooks and curricula in a quasi-linear or
circularly ascending (helix-like) fashion with strict dependencies, are fragile. It is prone
to losing students. One may miss a step for some reason, rendering subsequent topics
incomprehensible, leading to math anxiety. Instead of these structures, we propose a ‘flat
curriculum’ approach: a collection of self-contained, small size exploratory/constructive
problems in no particular order and with as few dependencies as possible. This means the
deconstruction of the building, cutting the helix of topics into pieces and preserving the
parts (and introducing new ones) that can be studied, understood and enjoyed without
constant references to other pieces.
This is, in some sense, indeed contradictory to what we usually think about mathemat-
ics. However, let us clarify here again: we are not questioning the structure of mathematics
as a scientific discipline. What is beautiful for many mathematicians is that everything
is connected to everything and organised into a vast hierarchy. Instead of doubting the
importance of this structure, we want to reconsider the way we deal with mathematics at
school. We argue that the teaching approach can follow a drastically different way, where
we want to separate the study of each sub-area as much as possible from the other areas. In
a classical textbook series, the entire mathematical system is built on a few fundamental
notions (e.g., function, set), highly unified and synthesised. However, failure and anxiety
are almost surely coded into this system: if you do not understand the first notions, you will
Philosophies 2021,6, 76 5 of 11
understand nothing further due to the continuous reference to these notions. Unfortunately,
it is particularly difficult to get the first notions right, since they are abstract (by definition).
Studying a set of ideas without constant referencing of their dependencies and without
the requirement of forming a basis of developing further ideas may lead to a more relaxed
pathway to the same skill set as planned to be achieved by regular curricula. Contrary
to the standard educational approach, we are not trying to find “the Truth” but to study
coexistent “truths”.
Considering this view, some further connections to and directions of philosophy
(of science) may come to our mind. Questioning the need for absolute foundations and
the rationale behind multiple, co-existing alternatives may lead us to fields such as anti-
foundationalism (for a good overview, see, e.g., [
7
,
14
]) and pluralism of
mathematics [15].
Perhaps meaning is not derived from a sound base but the interactions inside a network of
ideas? Could it be that mathematics is also non-foundational? Do we just have a persistent
illusion since Euclid? Here, we focus primarily on education; thus, the study of these
questions and connections is beyond the scope of this paper.
4. Implementation
Once we grounded our new approach in theory, the fundamental question nat-
urally arises. How to put it all into practice? How to implement a deconstructed,
flattened curriculum?
First, we need to produce a set of micro topics from the existing curriculum. These
should be independent monadic structures. Second, we need to bring in new topics,
possibly those thought to be inaccessible in school mathematics. The alleged lack of
reachability is due to the simple fact that recent developments, by definition, are at the end
of the dependency chains of concepts. Once we give up the hierarchical order, we have no
reason for excluding newer mathematics.
Let us consider the dependency graph of notions and concepts in the mathematics
curriculum to understand this approach better. It is similar to the graph in [
16
], where
vertices denote notions and concepts, and an edge connects two vertices if they are depen-
dent, meaning that the understanding of one notion/concept is required to understand
the other one. This is a directed graph, and hopefully, it is acyclic, although this latter
property is not entirely assured and evidenced in every textbook. Rephrased in this graph-
theoretical language, we aim to make this graph relatively sparse and to discuss as many
notions/concepts (i.e. vertices) as possible. We intend to identify and separate a more
extensive set of so-called micro topics, small subgraphs with as few incoming edges as
possible but with several potential outgoing edges.
4.1. Creating Micro Topics
Amicro topic is a piece of mathematical knowledge that can be presented and under-
stood in one session. The length of a session depends on the age group, so we cannot
specify it precisely. Moreover, a micro topic is self-contained. We claim that mathematics
can be presented in this monadic format, even though the subject is inherently hierarchical.
Of course, a significant amount of work is needed to create this presentation format. By
using the following algorithm, we can pick any mathematical idea, and then, we need to
take the following steps:
1. Listing dependencies;
2. Substituting dependencies, i.e., removing incoming edges; and
3. Installing hooks, i.e., indicators of outgoing edges.
First, we list the dependencies of the concept and construct a dependency graph. We
need to clarify what objects we need to mention when expressing the chosen idea. In other
words, we need to identify the incoming arrows in the graph. This is different from the
linear chain of dependencies of a traditional curriculum since we put the chosen topic
in the center and explore the connecting ideas. It is a standard feature of mathematical
Philosophies 2021,6, 76 6 of 11
monographs to show a dependency graph and to suggest several paths to traverse it. Our
approach simply takes this custom one step further.
The next step is substituting dependencies. Although possible in some cases, we cannot
simply erase all dependencies. That would result in unreasonable requests to believe
something without explanation, and understanding through explanations is what we are
trying to facilitate. However, human thinking has an enormous capacity for compressing
ideas with analogies, images, stories, metaphors and intuitive descriptions. A classic
example is defining continuity by saying that we can draw a curve without lifting the pen.
Once deep we are deep into calculus, we need the rigorous
ε
-
δ
definition, but the intuitive
definition serves as a great entry point. Another example would be a statistics course,
where calculus is not a prerequisite. Suppose we want to go from discrete probability
distributions to continuous ones. Traditionally, this assumes background knowledge in
integral calculus. However, the exact definition of the Riemann integral can be substituted
with the intuitive explanation of increasingly thinner bars in the histogram. With a bit of
squinting, we can see a continuous curve. In general, finding substitutes such as this may
require creativity.
Finally, we install hooks, the outgoing dependencies. The monadic nature of a micro
topic facilitates the sense of achievement but does not aim to limit the learner to cut off
the outgoing connections. Instead, we want to maintain their potential. These can be
represented as intriguing questions, without referring to other micro topics explicitly but
letting the pupils themselves discover these connections.
4.2. Prior Art
It is often remarked that teaching mathematics and conducting research may require a
different set of skills. Gian-Carlo Rota remarked that “gifted expositors” of mathematics
could be rarer than successful researchers [
17
]. He also added that “if you wish the reader
to follow rather than decipher, the linear deductive technique of exposition is the worst”,
but beyond pointing to the engineers, the alternatives were not explored.
Developing these types of micro topics is also not a new idea. It has been done
several times in popular science math books. The classical example is the works of Martin
Gardner [18],
but there are numerous other examples (for instance [
19
,
20
]). Sadly, these
works seldom make it into the classroom. Moreover, regular textbooks never take this
approach. Why is this the case? Is there a true educational, cognitive science-based reason,
or is it simply just a continuing misunderstanding that real work cannot be entertaining?
We believe in the latter, but this a future task for research. Here, we aim to change the
general attitude, so teachers will be more open to experiment with this approach.
In his seminal work entitled Proofs and Refutations, Imre Lakatos also developed some
problems of mathematics in the form of dialogues—these pieces can also be considered as
micro topics in our context [
21
]. In the form of small maths gems, micro topics often also
appear at public promotional presentations in science centers; math museums; and other
outreach events, such as Researchers’ Night. A crucial part of these events is to provide
the audience with understandable, exciting, one-of-a-kind lectures on some mathematical
topics without requiring specific preliminary knowledge or special training.
4.3. Examples of Micro Topics
Here, we provide some examples of micro topics. These already proved successful
from the practice of the authors. We emphasise that our goal here is not to process the
whole of mathematics and develop yet another entirely new curriculum but just to shed
light through a few examples of what we mean by a specific micro topic.
4.3.1. Group Theory in Primary School
As an example for bringing advanced mathematics into the classroom, we mention
a successful educational endeavor. Can we explain the basic concepts of group theory
to primary school pupils? We think the answer is yes, assuming enough thoughtful
Philosophies 2021,6, 76 7 of 11
preparation. Within the program ‘Mathematicians in Schools’ by CSIRO, Australia’s
science and research agency, we had a chance to trial this idea in an extracurricular session
for 15 students of mixed ages from 9 to 12. Naturally, the class format has to be different
from a university lecture for such an age range. The session has to include some hands-on
experiences.
The main idea of the session comes from an informal definition of symmetry for a
popular science mathematics book.
“You could think of the total symmetry of an object as all the moves that the
mathematician could make to trick you into thinking that he hadn’t touched it
at all”.
[
22
]
We used large-sized cut-out geometric shapes laid out on the floor to demonstrate
that a symmetry operation’s result is not distinguishable from doing nothing. Half of the
class was asked to turn away, while the other half observed the instructor’s action. Then,
the non-observing half had to decide whether the instructor touched the shape or not.
The recognition of symmetry operations led to counting their numbers for a given shape.
Again, we relied on a crisp formulation of the idea of a symmetry group.
“Numbers measure size, groups measure symmetry”.
[
23
]
The children were fascinated by the idea of the identity operation. They found it
amusing that doing nothing is a necessary transformation. The counting of symmetries
was facilitated by labeling the corners of the shapes. Finally, the symmetries were put
into a composition table to track how they combine. We intentionally avoided calling it a
multiplication table since that would have caused some confusion. The session ended with
the students recognising patterns in the composition table of the cyclic group.
The books’ quotes show that much work is needed to make a short session possible.
For both authors, the concise informal definitions are the distilled summaries of years of
work. Creating micro topics will require a large amount of work, but some of them might
have been completed already, as the example shows.
Group theory also allows for thinking about the limits of the micro topics method. Can
we summarise quickly the ideas involved in the Monster Group? A short book can
[24]
,
which is still far from being a micro topic.
4.3.2. Differential Geometry of Curves and Surfaces in Secondary School
Another example of bringing advanced micro topics into the classroom is introducing
some basic concepts of differential geometry of curves and surfaces. This micro topic
has also been proven successful in several extracurricular classroom events in various
secondary schools.
We need to avoid the application of exact calculus. Nevertheless, keeping our dis-
cussion almost exclusively on geometry, we can quickly introduce various fundamental
notions. The first notion must be the naive concept of differentiable parametric curve,
where differentiable simply means something that can be drawn smoothly, with no cusps.
The parameter can be the time as we travel along the curved path. Connecting two points
of this curve by a straight line and moving one point towards the other along the curve,
we obtain the notion of the tangent line, where students can discover the fundamental
difference between the elementary definition of the tangent of a circle and the general
notion. This finishes the substitution for the missing calculus.
The central notion is the curvature. Suppose we draw a couple of convex curves
touching the same line at a point with different (higher and higher) curvatures. In that case,
students can easily grasp the “feeling” of being less or more curved: it is soon discovered
that the more curved curves differ more radically from the original direction than the less
curved ones. Here, we can also refer to driving a bicycle on a curved road. Using the
concept of a parameterised path, the curvature can be jointly discovered and introduced
Philosophies 2021,6, 76 8 of 11
as the deviation of tangent lines at two neighboring points at the same distance along the
different curves, measured by the angle of the two tangent line.
The bicycle path analogy can also lead to the discovery of curves with constant
curvature (no need to move the handlebar at constant speed), inflection points (a moment
in an S-shape path where the handlebar is in its original position), etc.
After understanding the notion of curvature of curves, the Gaussian curvature of
surfaces can also be introduced. Students can imagine a hilly landscape, where at one
point, there are infinitely many directions paths, and in each direction, one can measure
the curvature of the actual path at that point (normal curvatures). It is evident that—if
the landscape is flat and not a perfect sphere—one can find the steepest path and, on the
contrary, the least sloping path and their curvatures (principal curvatures). Although it is
generally a big surprise for students that a Gaussian curvature is the product (and not the
average) of these principal curvatures, the further discovery of the meaning and geometric
consequences of this product (sign, being equal to zero) soon convinces them of the benefits
of multiplying the two extrema.
4.4. Creative Substitution
Substitutions can be made just for a single purpose, or they can be more systematical.
A successful example of popularising mathematics is [
25
], where the underlying metaphor,
mathematics-as-cooking, spans the whole book. This is a notable example, since it manages
to convey the basic ideas of category theory, which is often considered too abstract even by
mathematicians. Clearly, this is a result of years of thinking or rather a way of living life.
Powerful substitutions are not easy to find, but they are possible.
5. Flattening the Curriculum
From the viewpoint of the foundations of mathematics, nothing is possible without set
theory (or any other foundational part of mathematics). Furthermore, by definition, no step
in the reasoning process can be left out in logical deduction. We argue that the presentation
of a topic in education must not follow this logical order and the order of construction.
Set theory, being foundational, is the starting point of the standard curriculum. However,
should it be the first to present? Deductive logic says so, but the history of mathematics
shows that mathematical ideas could be discussed before set theory was worked out. One
might argue that it gives a unifying abstract language for the whole subject, but we also
know that abstraction does require mental effort. The Wason selection task demonstrates
that people reason better in familiar social contexts than with abstract objects [26,27]. The
question is, what is the right balance between the logical ideal and cognitive accessibility?
The fundamental problem is that we still insist that the strong interrelationship of
mathematical fields and notions must be present in mathematics education. Mathematics
is provided to students as a single massive logical structure, an unquestionable monolithic
building. Even if the pedagogical and methodological experiments of the last decades,
especially inquiry-based learning, try to soften this deductive approach, we cannot break
away from this epistemic viewpoint. Inquiry-based learning and similar methodological
initiatives try to show something new within a specific subfield to make mathematics more
“user-friendly”, but they do not question the strict interdependence of the subfields, the
global logical structure of mathematics and the necessity of presentation of this structure.
However, human thinking has several other modes of operation, e.g., metaphorical
thinking. There are different ways of thinking about the same idea [
28
]. While logical
formalism can precisely fix the meaning, it may make more nuances and the more tenuous
connections become less apparent. A monadic/flat curriculum can maintain these nuances,
as the different meanings can play a role in different topics.
We argue that flattening the curriculum can be an alternative educational approach to
mathematics. To flatten the curriculum, the key idea is to make the dependency graph of
the presented subfields as sparse as possible. The ultimate goal of flattening the curriculum
Philosophies 2021,6, 76 9 of 11
is to provide more than one entry point to make it more accessible for students who have
missed or understood less of the previous topics.
Programming education has a similar setup. The traditional approach starts at the
bottom of the computing stack, e.g., the binary representation of numbers, digital logic,
and computer architecture, and then proceeds to the level of the operating system and
eventually to application programming in a higher-level language. This is not a possible
approach in coding for kids. Additionally, on the undergraduate level, learning higher-level
programming languages for students not majoring in computer science may be obstructed
by the assumed background knowledge of the inner workings of computers. With a bit of
thinking and careful language choice, this can be avoided.
6. Possible Criticisms
Here, we are trying to answer some of the possible criticisms preemptively. We expect
resistance to the idea of a flat, multi-entry curriculum. Even if these criticisms become
straw man arguments, they might help understand the concept deeper.
6.1. Assessment
Grading and assessment are always an issue. Our proposal implies that each school
can have its own randomised order of topics (even when assuming a centrally curated
collection of topics). How can we then compare the performance characteristics of schools?
How do we evaluate the performance of individual students in the bite-size topics? Indeed,
switching to a flat curriculum while keeping the system of standardised tests would be a
disastrous combination. Developing assessment methods along the flat curriculum could
require even more work than creating micro topics. However, if we are not happy with the
learning efficiency of standardised tests, this work is unavoidable anyway.
6.2. Lack of the Complete Picture of the Subject
One might argue that having a coherent worldview is vital to dealing with information
overload in today’s world. Such a perspective should be the primary task of education,
but the proposed flat curriculum method fails to do that. Our rejoinder is that giving the
ready-made hierarchical view of mathematics also fails to provide that coherent picture
reliably. Moreover, a complete, hierarchically structured knowledge may not be the only
way to provide a helpful perspective. It can be in the process of approaching problems, in
the strategy, in the way of relating to challenges.
Are the micro topics just another pure discovery learning method [
29
]? No, we do not
measure the success of a flat curriculum by the students’ ability to reconstruct the whole
picture of mathematics. However, we assess success by the quality of the relationship with
mathematical thinking that they develop. Roughly speaking, the monolithic curriculum
frequently yields students feeling left out and developing math anxiety, while the grand
picture is not guaranteed even for students who can keep up. We think that a flat curriculum
with multiple entry points can lead to a better experience of mathematics. We do not claim
that micro topics automatically lead to a complete understanding. However, we suspect
that the human brain’s pattern-matching abilities will naturally spring into action once the
anxiety is released. Additionally, a flat curriculum has plenty of opportunities for guided
discovery. Of course, the proof of these claims requires future empirical investigations.
7. Conclusions
We described an alternative method for organising a mathematics curriculum. Instead
of the strictly hierarchical and quasi-linear or helix-like traditional school curriculum, we
promote micro topics, monadic pieces of mathematical knowledge to improve accessibility
to mathematics. From a philosophical point of view, this approach is based on the principle
of deconstruction.
Micro topics are proven successful in out-of-classroom events, such as the outreach
event of the Researchers’ Night. There, a topic was presented without the requirement
Philosophies 2021,6, 76 10 of 11
of specific preliminary knowledge. Why not apply this approach in the curriculum? We
argue that we made a mistake by not using this micro topic approach instead of the
“serious” (i.e., a hierarchically established large system of) mathematics in the classroom.
A set of largely independent micro topics provides a chance for multiple entry points to
mathematics while not eliminating the possibility for students to discover connections. The
directed dependency graph of notions and concepts of these micro topics is much sparser
in incoming edges than in a usual curriculum. However, outgoing directions, called hooks,
preserve the potential of discovery of connections. This way, the curriculum is flattened.
Instead of a large monolithic structure, it contains various, horizontally spreading smaller
subfields, decreasing the feeling of the monumentality of a hierarchical structure and
hopefully decreasing math anxiety.
We are fully aware of the radicality of this proposal, and it is clear that much work
is needed to create a flattened curriculum and to investigate its benefits and possible
disadvantages. Especially, for this reason, it is critical now that more researchers and
educators open their minds up to this way of thinking. This is what we try to achieve with
this paper.
Author Contributions:
The fundamental ideas presented in this paper were developed indepen-
dently by the two authors. At a chance meeting, M.H. described the climbing metaphor and suggested
writing it up in a more detailed way. A.E.-N. and M.H. contributed equally to writing the paper. All
authors have read and agreed to the published version of the manuscript.
Funding:
This research was partially funded by the internal research grant of Akita International
University and by the MTA KOZOKT2021-44 Public Education Development Research Program of
the Hungarian Academy of Sciences.
Conflicts of Interest:
The authors declare no conflicts of interest. The funders had no role in the
writing of the manuscript.
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