Mark Dow
"...we try to make predictions about Nature, to anticipate what
we’ll see
in places we have not yet looked. If additional observations
corroborate our expectations, then we’re on the right track.
Several skill sets are involved: one must know how to idealize the
world, and then how to work with that idealization. Remarkably enough,
our schools fail to teach either skill." -- Blake Stacey in The Necessity of Mathematics
Discrete math background
What math topics are not in pre-college cirriculum,
topics everyone with an interest in math, computer science, physics or
engineering should have seen?
Most of what is typically not covered involves
discrete math and related formalisms. This list of topics includes
mostly discrete mathematics, but also includes related connections with
continuous and toplological mathematics. There are many links to
Wikipedia topics
that you've heard about but nobody told you the formal name. I've found
that the Wikipedia math topics are very well written and consistent.
They go
into a lot of detail that I most often don't understand, but the first
paragraph of many of the math fields is usually very good, straight to
the point.
Mathematics involves the study of such concepts as
quantity, structure, space and change. We seek out patterns in numbers,
space, science, computers, imaginary abstractions, or elsewhere. The
patterns can be expressed and explored in a formal language unlike our
natural language.
How do we go from nothing to useful abstractions of
the real world using math?
Somehow we learn the language as we see concrete
examples of the abstraction. Most mathematical objects map onto space. For example, modular
numbers behave in the same way as symmetry rotations
of plane figures (like a square). This fact is generalized and applied across mathematics,
computer science, physical sciences, and increasingly in other fields
(linguistics, psychology, economics, etc.).
What is discrete math?
Discrete mathematics, also called finite mathematics,
is the study of mathematical structures that are fundamentally discrete
in the sense of not supporting or requiring the notion of continuity. Objects studied in discrete mathematics are largely countable sets such as integers, finite graphs, and formal languages. Discrete spaces are simple examples of a topological space or similar structure, one in which the points are "isolated" from each other.
Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages.
In some mathematics curricula, finite mathematics courses cover discrete mathematical concepts for business, whereas discrete
mathematics courses emphasize concepts for computer science majors, and
combinatorics and other specialized courses emphasize the mathematical
theory.
For contrast, see continuum, topology, and mathematical analysis.
Discrete mathematics includes the following topics:
Algebra
Algebra is a main branch of mathematics concerning the study of structure, relation, and quantity. Elementary algebra,
often taught in pre-college math classes, provides an introduction to
the basic ideas of algebra, including effects of adding and multiplying
numbers, the concept of variables, definition of polynomials, polynomial factorization and determining polynomial roots.
Algebra is much broader than elementary algebra and can be
generalized. In addition to working directly with numbers, algebra
covers working with symbols, variables, and set elements. Addition and multiplication are abstracted as general operations, and their precise definitions lead to structures such as groups, rings and fields.
complex numbers
Complex numbers are an extension of the
real numbers obtained by adjoining an
imaginary unit, denoted
i, which satisfies:
- i2 = -1
Any complex number z can be represented as a sum:
z = x + iy
where x and y are real numbers, called the real and imaginary parts of the complex number.
 |
Complex number notation
|
Complex numbers are a
field,
and thus have addition, subtraction, multiplication, and division
operations. These operations extend the corresponding operations on
real numbers, although with a number of additional elegant and useful
properties.
Often pre-college math classes introduce complex numbers as a sort of
curiosity. They are needed to express the solutions of some polynomial
equations like:
x2 - 1 = 0
where the solution can be expressed as:
x = 0 + i = i
x = 0 - i = -i
complex plane
The concept of the complex plane allows a
geometric interpretation of complex numbers. Under addition, they add like
vectors. [To Do: Diagram of vector addition.] The multiplication of two complex numbers can be expressed most easily in
polar coordinates – the magnitude or modulus of the product is the product of the two
absolute values, or moduli, and the angle or argument
of the product is the sum of the two angles, or arguments.
In
particular,
multiplication by a complex number of modulus 1 acts as a
rotation on the complex plane.
There are similarities and differences
between the real number plane (an infinite set) and the complex plane
(another infinite set). A point on the real plane is a 2-
tuple, or
coordinate pair. A point on the complex plane is a
single complex numbers,
z,
that has a real part (component) and an imaginary part (component).
Although complex numbers add just like 2-D vectors, complex number
multiplication is easily defined and well-behaved.
It is useful,
conceptually and practically, to
consider multiplication of real numbers as a "stretching", or scaling
operator. One real number is mapped to another by multiplication. [To
Do: Diagram.]. Many other operators can also be interpreted as this
kind
of spatial transformation. For example addition corresponds to a
translational transformation (moving sideways or up/down).
Multiplication
by -1 (negation) can be interpreted as a mirroring (with the mirror
point at 0) or as a half rotation (pi/2 radians) through the complex
plane.
What does it mean to multiply a
pair of points on the real plane? It doesn't mean anything unless you
can define what you mean by "multiply a pair of points"! [To Do:
Multiplication of points by scalars.]
[To Do: Argand diagram, corresponding Real plane, R2, C.]
If you multiply one complex number by another, it acts as a rotation and scaling
transform (an operator) mapping one complex number to a new compex number. [To Do: Argand diagram.]
Functions as mappings:
Functions are mappings of every member of the domain
of x to a new set of points within the range of the function. [To Do:
Show the formal notation of a concrete example, like f(x) = 2x, where x
is a member of R. In the imprecise shorthand of algebra class (but not math
papers) this is written f(x) = 2x, or just y = 2x.]
In programming, a function oftens acts as (is the same as, or
impements) an operator. Operators that act on a single object (like -x
= -1*x, negation) are called
unary operators. Operators that act on two objects (like multiplication of two numbers) are called
binary operators. Operators that act on more than two objects can always be broken down into sequences of unary or binary operations.
Note that the real number (
R2)
plane is not the same as the rational number plane; there are more
reals than rationals. Which set do we use in programming? We use
rational numbers that approximate real numbers. We call them "real
numbers", but they are necessarily of finite precision, and so can be
represented as the ratio of integers.
Wouldn't
it be nice to have a computing system that could represent real
numbers, instead of rationals that approximate real numbers. In one
sense quantum computers start to achieve this -- qubits are mixtures of
bits 1 and 0, and can represent aspects of real numbers better,
depending on how many qubits are interacting. But still not "perfectly"
-- no set of qubits can represent pi exactly, for example.
Remember for later:
ei(phi) is a complex number, and it is on the unit
circle (
|ei(phi)| = 1, or its modulus is one). If you multiply any
complex number by this value it acts as a rotation operator, mapping
the original complex number to a new compex number. This is the essence of
Euler's formula:
Richard Feynman called Euler's formula "our jewel" and "the most remarkable formula in mathematics".[2]
This form of complex number has many applications,
mostly for representing rotating or oscillating things. For example the
vibration of object can be (and most often is) modeled as
f(t) = kei( phi(t) ), where
f(t) describes the displacement (movement) over the time
t.
The real part of this complex number is the displacement, and the
imaginary part is the phase describing the part of the cycle of
vibration.
It could be written as
z(t) = a(t) + ib(t), where
sqrt( a(t)2 + b(t)2 )1/2 = |z| = 1, but there
is a
very nice reason for
writing it in this simple form. Here's the reason: if you want to multiply
two rotations (same as applying two rotation transforms in a row),
the math is dead simple --
ei(phi1) *
ei(phi2) = ei(phi1 + phi1). This is
easier, more clear, and the same as doing the calculation
(a +
bi)*(c + di) where
tan-1(a/b).
I haven't, and your teachers might never, tell you
why ei(phi) is a complex number with a modulus of 1, or even why
e = 2.71828... (the base of the natural logorithm, or
Euler's number) is so special. This takes some background in
trigonometry and
analysis,
in particular how irrational numbers can be represented with infinite
series of rational numbers . In short and without explanation,
e is the unique number
a, such that the value of the derivative (the slope of the tangent line) of the exponential function
f (x) = ax at the point
x = 0 is exactly 1.
Complex numbers whose real and
imaginary part are both integers form a lattice (a grid of points) on
the complex plane. This set of complex numbers, called the
Gaussian integers, is a generalization of the integers and is important in algebraic number theory.
quaternions, and other generalizations of complex numbers
Irish mathematician
Sir William Rowan Hamilton first described
quaternions,
recognizing that they formed a
mathematically consistent set analogous to complex numbers, and why
they are useful. For example, just as complex number multiplication can
be a model of 2-D rotation and scaling ("stretching" of
the "length"/modulus/magnitude/norm), multiplication of
quaternions models
rotation in
3-D space, just what's needed for applications like programming video
games.
"Hamilton knew that the complex numbers could be viewed as points in a plane, and he was looking for a way to do the same for points in space.
Points in space can be represented by their coordinates, which are
triples of numbers (3-tuples), and for many years Hamilton had known how to add
and multiply triples of numbers. But he had been stuck on the problem
of division: He did not know how to take the quotient of two points in
space.
On October 16, 1843, Hamilton and his wife took a walk along the Royal Canal in Dublin. While they walked across Brougham Bridge (now Broom Bridge), a solution suddenly occurred to him. He could not divide triples, but he could divide quadruples.
By using three of the numbers in the quadruple as the points of a
coordinate in space, Hamilton could represent points in space by his
new system of numbers. He then carved the basic rules for
multiplication into the bridge".
- i2 = j2 = k2 = ijk = − 1
- ℍ
The multiplication of quaternions is
non-commutative.
This corresponds to the fact that the order of rotation in three
dimensions (like "pitch then roll" and "roll then pitch") matters --
you don't end up at the same orientation if you swap the order of
rotation.
There are also eight dimensional
octanions! Octonions have limited applications in fields such as
string theory,
special relativity, and
quantum logic.
There are also corresponding 16-dimensional
sedenions.
However, is a price to pay for these systems: each increase
in dimensionality introduces new algebraic complications. Quaternion
multiplication is not
commutative anymore, octonion multiplication additionally is non-
associative, and sedenions do not form a
normed space with multiplicative norm.
There are various
hypercomplex number sets that aren't
fields -- they don't have division operators.
complex analysis
Complex analysis investigates functions of complex numbers/variables. It is useful in many branches of mathematics, including number theory and applied mathematics, and in physics.
Complex analysis is particularly concerned with the analytic functions of complex variables. Because the separable real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.
linear algebra
Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. A vector is a directed line segment,
characterized by both its magnitude, represented by length, and its
direction. Vectors can be used to represent physical entities such as
forces, and they can be added to each other and multiplied with scalars, thus forming the first example of a real vector space.
Modern linear algebra has been extended to consider spaces of arbitrary or infinite dimension. A vector space of dimension n is called an n-space.
Most of the useful results from 2- and 3-space can be extended to these
higher dimensional spaces. Although people cannot easily visualize
vectors in n-space, such vectors or n-tuples are useful in representing data. Since vectors, as n-tuples, are ordered lists of n components, it is possible to summarize and manipulate data efficiently in this framework. For example, in economics, one can create and use, say, 8-dimensional vectors or 8-tuples to represent the Gross National Productof
8 countries. One can decide to display the GNP of 8 countries for a
particular year, where the countries' order is specified, for example,
(United States, United Kingdom, France, Germany, Spain, India, Japan,
Australia), by using a vector (v1, v2, v3, v4, v5, v6, v7, v8) where each country's GNP is in its respective position.
Linear algebra is concerned with the study linear maps (also called linear transformations), and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by linear ones.
Linear maps
take elements from a linear space to another (or to itself), in a
manner that is compatible with the addition and scalar multiplication
given on the vector space(s). The set of all such transformations is
itself a vector space. If a basis for a vector space is fixed, every linear transform can be represented by a table of numbers called a matrix. The detailed study of the properties of and algorithms acting on matrices, including determinants and eigenvectors, is considered to be part of linear algebra.
solving simultaneous linear equations
One of the primary applications of linear algebra is the solution of simultaneous
linear equations.
The simplest case is when the the number of unknowns is equal to the
number of equations. Therefore, one could begin with the problem of
solving
n simultaneous linear equations in
n unknowns.
"It is remarkable that the deepest ideas of number theory reveal a
far-reaching resemblance to the ideas of modern theoretical physics.
... One would like to hope that this resemblance is no accident, and
that we are already hearing new words about the World in which we live,
but we do not yet understand their meaning."
Yuri I. Manin, pg. 99 in "Mathematics and Physics" (translation of "Matematika i fizika", Birkhauser, Boston 1981)
Number theory is
concerned with the properties of numbers in general, and integers in
particular, as well as the wider classes of problems that arise from
their study.
integers,
natural numbers, factors, primes, composites
ratios, and multiplication
rational numbers on the unit interval
The only
algebraic integers which are found in the set of
rational numbers are the ordinary integers. In other words, the intersection of
Q and
A is exactly
Z. The rational number
a/
b is not an algebraic integer unless
b divides
a. Note that the leading coefficient of the polynomial
bx −
a is the integer
b. As another special case, the square root √
n of a non-negative integer
n is an algebraic integer, and so is irrational unless
n is a
square number (perfect square).
Set theory
sets and ordered sets
universe, domain
The simplest version is that
any set can be a universe,
so long as the object of study is confined to that particular set. If
the object of study is formed by the
real numbers, then the
real line R, which is the real number set, could be the universe under consideration. Implicitly, this is the universe that
Georg Cantor was using when he first developed modern
naive set theory and
cardinality in the 1870s and 1880s in applications to
real analysis. The only sets that Cantor was originally interested in were
subsets of
R.
ordered pair
n-tuples (ordered lists of
n terms)
Let (a1, b1) and (a2, b2) be two ordered pairs. Then the characteristic or defining property of ordered pairs is
- (a1, b1) = (a2, b2) ↔ (a1 = a2 & b1 = b2).
Ordered pairs can have other ordered pairs as entries. Hence the ordered pair enables the recursive definition of ordered n-tuples (ordered lists of n terms). For example, the ordered triple (a,b,c) can be defined as (a, (b,c)
), as one pair nested in another. This approach is mirrored in computer
programming languages, where it is possible to construct a list of
elements from nested ordered pairs. For example, the list (1 2 3 4 5)
becomes (1, (2, (3, (4, (5, {} ))))). The Lisp programming language uses such lists as its primary data structure.
relations
binary relations
An example of binary relation is the "divides" relation between the set of prime numbers P and the set of integers Z, in which every prime p is associated with every integer z that is a multiple of p,
and no other. In this relation, for instance, the prime 2 is associated
with numbers that include −4, 0, 6, 10, but not 1 or 9; and the prime 3
is associated with numbers that include 0, 6, and 9, but not 4 or 13.
complement
types of numbers, unit division
rationals and integers, irrationals, reals etc.
one to one mapping
cover
covering and packing
sphere packing
Problems of arranging balls densely arise in many
situations, particularly in coding theory (the balls are formed by the
sets of inputs that the error-correction would map into a single
codeword).
Analysis (mathematical analysis)
"An infinite number of mathematicians walk into a
bar. The first one orders a beer. The second orders half a beer. The
third, a quarter of a beer. The bartender says "You're all idiots", and
pours two beers."
infinite series
recursion
fractal systems
rational numbers don't form a
complete metric space
but there is a completion if real numbers are included
Group theory
The ubiquity of groups in numerous areas—both within and outside
mathematics—makes them a central organizing principle of contemporary
mathematics.
Groups share a fundamental kinship with the notion of
symmetry. A
symmetry group encodes symmetry features of a
geometrical
object: it consists of the set of transformations that leave the object
unchanged, and the operation of combining two such transformations by
performing one after the other. Such symmetry groups, particularly the
continuous
Lie groups, play an important role in many academic disciplines.
Matrix groups, for example, can be used to understand fundamental
physical laws underlying
special relativity and symmetry phenomena in molecular
chemistry.
translation transforms
rotation transforms
reflection transforms
scaling transforms (http://scienceblogs.com/sunclipse/2008/10/rotation_matrices.php)
symmetry transforms
commutation
Arithmetic viewed as symmetry transforms on integers or reals
addition of integers as a discrete translational transform
addition of reals as a continuous translational transform
multiplication as a scaling transform
the complex plane
rotation as a product on the unit complex circle
Abelian groups
An abelian group satisfies the requirement that the product of elements does not depend on their order (the axiom of
commutativity). Abelian groups generalize the arithmetic of addition of integers.
- For the integers and the operation addition "+", denoted (Z,+), the operation + combines any two integers to form a third integer, addition is associative, zero is the additive identity, every integer n has an additive inverse, −n, and the addition operation is commutative since m + n = n + m for any two integers m and n.
- Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form an abelian group under addition, as do the integers modulo n, Z/nZ.
- Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
In general, matrices,
even invertible matrices, do not form an abelian group under
multiplication because matrix multiplication is generally not
commutative. However, some groups of matrices are abelian groups under
matrix multiplication - one example is the group of 2x2 rotation matrices.
Finite groups
The modular group is a fundamental object of study in
number theory,
geometry,
algebra,
and many other areas of advanced mathematics. Like spatial
transformations, the modular group can be represented as a group of
geometric transformations or as a group of
matrices.
of simple finite groups
Dihedral groups
Cyclic groups
modular numbers
The
nth roots of unity form a cyclic group of order
n under multiplication. e.g.,
0 = z3 − 1 = (z − s0)(z − s1)(z − s2) where
si = e2πi / 3 and a group of
{s0,s1,s2} under multiplication is cyclic.
Lie group examples
unit complex numbers
Structures that can be
represented as graphs are ubiquitous, and many problems of practical
interest can be represented by graphs. The link structure of a website
could be represented by a directed graph: the vertices are the web
pages available at the website and a directed edge from page A to page
B exists if and only if A contains a link to B. A similar approach can
be taken to problems in travel, biology, computer chip design, and many
other fields. The development of algorithms to handle graphs is
therefore of major interest in computer science. There, the
transformation of graphs is often formalized and represented by graph
rewrite systems. They are either directly used or properties of the
rewrite systems (e.g. confluence) are studied.
Visualization of diadic relationships
Properties of graphs
algorithms on graphs
Probability and statistics
permutations and combinatorics
discrete distributions
discrete states and equilibrium
error in measurement
representative sampling
Topology
connectedness
orientability
homeomorphisms
dimensions,
topological dimension
A topological space
X has topological dimension
m if every
covering
of
X has a refinement
in which every point of
X occurs in at most
m+1 sets in
, and
m is the smallest
such integer.
(Cantor's proof that there is a one-to-one correspondence between
and
).
(Peano's construction of a continuous map from
onto ).
Logic, algorithms, and computation
Truth tables, logic gates
bistable systems and state
transistors
bit-wise logic
Information theory
Shannon information
Complexity, Algorithmic information theory
Inductive Inference
A system with a short description is often described as having a low, or short,
Kolmogorov complexity",
or sometimes with terms like "low descriptive complexity,
Kolmogorov-Chaitin complexity, stochastic complexity, algorithmic
entropy, or program-size complexity". It is directly related (
1) to the principles of
Minimum Message Length (MML), a formal
information theory restatement of
Occam's Razor.
Big O notation (http://en.wikipedia.org/wiki/Big_O_notation)
linearity
Big
O notation has two main areas of application. In mathematics, it is
commonly used to describe how closely a finite series approximates a
given function, especially in the case of a truncated Taylor series or
asymptotic expansion. In computer science, it is useful in the analysis
of algorithms. In both of the applications, the function g(x) appearing
within the O(...) is typically chosen to be as simple as possible,
omitting constant factors and lower order terms.
It
is often useful to bound the running time of graph algorithms. Unlike
most other computational problems, for a graph G = (V, E) there are two
relevant parameters describing the size of the input: the number |V| of
vertices in the graph and the number |E| of edges in the graph.
An interesting example is
the graph isomorphism problem, the graph theory problem of determining
whether a graph isomorphism exists between two graphs. Two graphs are
isomorphic if one can be transformed into the other simply by renaming
vertices.
While a method for computing the
solutions to NP-complete problems using a reasonable amount of time
remains undiscovered, computer scientists and programmers still
frequently encounter NP-complete problems. An expert programmer should
be able to recognize an NP-complete problem so that he or she does not
unknowingly waste time trying to solve a problem which so far has
eluded generations of computer scientists. Instead, NP-complete
problems are often addressed by using approximation algorithms in
practice.
Determining whether or not
it is possible to solve these problems quickly is one of the principal
unsolved problems in Computer Science.
Turing machine
Cellular automata and rule systems
Computability
Alan Turing
In his momentous paper "On Computable Numbers, with an Application to the
Entscheidungsproblem"
[14] (submitted on 28 May 1936), Turing reformulated
Kurt Gödel's
1931 results on the limits of proof and computation, replacing Gödel's
universal arithmetic-based formal language with what are now called
Turing machines,
formal and simple devices. He proved that some such machine would be
capable of performing any conceivable mathematical problem if it were
representable as an
algorithm,
even if no actual Turing machine would be likely to have practical
applications, being much slower than practically realisable
alternatives.
Turing machines are to this day the central object of study in
theory of computation. He went on to prove that there was no solution to the
Entscheidungsproblem by first showing that the
halting problem for Turing machines is
undecidable:
it is not possible to decide, in general, algorithmically whether a
given Turing machine will ever halt. While his proof was published
subsequent to
Alonzo Church's equivalent proof in respect to his
lambda calculus,
Turing's work is considerably more accessible and intuitive. It was
also novel in its notion of a 'Universal (Turing) Machine', the idea
that such a machine could perform the tasks of any other machine. The
paper also introduces the notion of
definable numbers.
Maths notation
Background and references
Herbert S. Wilf
Discrete mathematics and proof in the high school
Aiso Heinze, Ian Anderson and Kristina Reiss
ZDM, The International Journal of Mathematics Education, Volume 36, Number 2 / April, 2004
"In the last 25 years, descrete mathematics has rapidly changed in
its methodologies, in the way in which it is viewed by mathematicians,
and in particular in the range of its applications. This is partly due
to an extended use of computer technology in the past decades.
Moreover, discrete mathematics has proved to be an important tool for
research and development for example in biology, chemistry, and
computer science. Discrete mathematics has its research roots in
different parts of mathematics most prominently in group theory,
geometry, number theory, algebraic combinatorics, graph theory, and
cryptography. Accordingly, it has been influenced by a variety of
mathematical results, methods, and representations. Their combination
and integration in a profound theory is essential for research in
discrete mathematics. In particular, the use of computer technology has
not only influenced mathematical results but nathematical methods as
well. It is necessary to discuss these methods, as they represent an
important development in mathematical argumentation."
"As an
active branch of contemporary mathematics that is widely used in
business and industry, it is clear that discrete mathematics should be
an integral part of the school mathematics curriculum, and in fact some
topics of discrete mathematics naturally occur in other areas of the
mathematics curriculum (National Council of Teachers of Mathematics,
2000). Combinatorics, iteration and recursion, and vertex-edge graphs,
for example, are mentioned explicitly as topics to be taught in all
grades from kindergarden to high school. ... but there is still a
substantial amount of research needed to identify in a more systematic
way those topics in descrete mathematics which are most relevant for
mathematics instruction. Some first steps have been taken by both
mathematicians and nathematics educators, but these attempts seem
sporadic and isolated (Kenney & Hirsch 1991; Rosenstein, Franzblau
& Roberts, 1997, DIMACS 2001)."
Kenney, M. J. & Hirsch, C. R. (eds.) (1991). Discrete Mathematics across the Curriculum, K-12, 1991 Yearbook. Reston, VA:
NCTM.
Rosenstein, J. G.; Franzblau, D. S. & Roberts, F. S. (1997). Discrete Mathematics in the Schools. Notices of the AMS, 47(6),
641–649.
DIMACS (2001) Center for Discrete Mathematics and Theoretical Computer
Science: Educational Program. http://dimacs.rutgers.edu/Education/.
National Council of Teachers of Mathematics (Ed.) (2000). Principles and Standards for School Mathematics, Reston, VA: NCTM.
Blake Stacey argues that "... we
disable ourselves if our “explanations” of science do not
include mathematics". While his examples are from geometry and calculus
(continuum mechanics), it includes references to group theoretical
concepts and the gist of it applies to all of discrete mathematics.
"Witnessing this kind of mathematics in action — the
establishment of equivalences between ideas — gives us a certain
perspective on what we should be putting in schoolbooks and the lessons
through which we must put teachers. The tools we’ve used in this
essay are not elaborate or abstruse: areas of triangles, properties of
parallel lines and so forth. I was taught this kind of geometry,
officially, in high school; in Alabama, it was a graduation
requirement, although I’m genuinely baffled why it took until
high school to get there.
But pacing is not the only problem.
All too often, discussions of what should be done with American
mathematics education polarize into a debate between, essentially,
pushing pencils by rote and punching calculator buttons by rote. The
issue touches all parents and students, yet our attempts to figure it
out get nowhere. To put the matter bluntly, we leave students
completely unprepared for the type of reasoning we have seen is
critical for natural science — yet we sell mathematics as part of
the curriculum partly because it’s important for understanding
how the world works. We can change what we teach in a great many
different ways, without ever delivering on that promise. To use
mathematics in the natural sciences, we first decide how we wish to
represent some aspect of the world in mathematical form. We then take
the diagrams and equations we’ve written and manipulate them
according to logical rules, and in so doing, we try to make predictions
about Nature, to anticipate what we’ll see in places we have not
yet looked. If additional observations corroborate our expectations,
then we’re on the right track. (It’s rarely so clean-cut as
that — the process can spread across thousands of people and
multiple generations of activity — but that’s the gist of
it.) Several skill sets are involved: one must know how to idealize the
world, and then how to work with that idealization. Remarkably enough,
our schools fail to teach either skill."
"Number Theory in Science and
Communication", M.R. Schroeder, Springer, Third edition 1997
