The title page of the Principia Mathematica (shortened version), an important work of metamathematics
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics (and perhaps the creation of the term itself) owes itself to David Hilbert's attempt to secure the foundations of mathematics
in the early part of the 20th century. Metamathematics provides "a
rigorous mathematical technique for investigating a great variety of
foundation problems for mathematics and logic" (Kleene 1952, p.59).
An important feature of metamathematics is its emphasis on
differentiating between reasoning from inside a system and from outside a
system. An informal illustration of this is categorizing the
proposition "2+2=4" as belonging to mathematics while categorizing the
proposition "'2+2=4' is valid" as belonging to metamathematics.
History
Metamathematical metatheorems about mathematics itself were originally differentiated from ordinary mathematical theorems in the 19th century to focus on what was then called the foundational crisis of mathematics. Richard's paradox
(Richard 1905) concerning certain 'definitions' of real numbers in the
English language is an example of the sort of contradictions that can
easily occur if one fails to distinguish between mathematics and
metamathematics. Something similar can be said around the well-known Russell's paradox (Does the set of all those sets that do not contain themselves contain itself?).
Metamathematics was intimately connected to mathematical logic,
so that the early histories of the two fields, during the late 19th and
early 20th centuries, largely overlap. More recently, mathematical
logic has often included the study of new pure mathematics, such as set theory, category theory, recursion theory and pure model theory.
Serious metamathematical reflection began with the work of Gottlob Frege, especially his Begriffsschrift, published in 1879.
David Hilbert was the first to invoke the term "metamathematics" with regularity (see Hilbert's program), in the early 20th century. In his hands, it meant something akin to contemporary proof theory, in which finitary methods are used to study various axiomatized mathematical theorems (Kleene 1952, p.55).
The discovery of hyperbolic geometry had important philosophical
consequences for metamathematics. Before its discovery there was just
one geometry and mathematics; the idea that another geometry existed was
considered improbable.
When Gauss discovered hyperbolic geometry, it is said that he did not publish anything about it out of fear of the "uproar of the Boeotians", which would ruin his status as princeps mathematicorum (Latin, "the Prince of Mathematicians").[1]
The "uproar of the Boeotians" came and went, and gave an impetus to metamathematics and great improvements in mathematical rigour, analytical philosophy and logic.
Begriffsschrift is usually translated as concept writing or concept notation; the full title of the book identifies it as "a formulalanguage, modeled on that of arithmetic, of pure thought." Frege's motivation for developing his formal approach to logic resembled Leibniz's motivation for his calculus ratiocinator (despite that, in his Foreword
Frege clearly denies that he reached this aim, and also that his main
aim would be constructing an ideal language like Leibniz's, what Frege
declares to be quite hard and idealistic, however, not impossible task).
Frege went on to employ his logical calculus in his research on the foundations of mathematics, carried out over the next quarter century.
Principia Mathematica, or "PM" as it is often abbreviated, was an attempt to describe a set of axioms and inference rules in symbolic logic
from which all mathematical truths could in principle be proven. As
such, this ambitious project is of great importance in the history of
mathematics and philosophy, being one of the foremost products of the belief that such an undertaking may be achievable. However, in 1931, Gödel's incompleteness theorem
proved definitively that PM, and in fact any other attempt, could never
achieve this goal; that is, for any set of axioms and inference rules
proposed to encapsulate mathematics, there would in fact be some truths
of mathematics which could not be deduced from them.
One of the main inspirations and motivations for PM was the earlier work of Gottlob Frege on logic, which Russell discovered allowed for the construction of paradoxical sets. PM
sought to avoid this problem by ruling out the unrestricted creation of
arbitrary sets. This was achieved by replacing the notion of a general
set with notion of a hierarchy of sets of different 'types',
a set of a certain type only allowed to contain sets of strictly lower
types. Contemporary mathematics, however, typically avoids paradoxes
such as Russell's in less unwieldy ways, such as the system of Zermelo–Fraenkel set theory, though type theory still sees much use.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (e.g., a computer program, but it could be any sort of algorithm) is capable of proving all truths about the relations of the natural numbers (arithmetic).
For any such system, there will always be statements about the natural
numbers that are true, but that are unprovable within the system. The
second incompleteness theorem, an extension of the first, shows that
such a system cannot demonstrate its own consistency.
Tarski's definition of model-theoretic satisfaction
The Entscheidungsproblem (German for 'decision problem') is a challenge posed by David Hilbert in 1928. The Entscheidungsproblem asks for an algorithm that takes as input a statement of a first-order logic (possibly with a finite number of axioms beyond the usual axioms of first-order logic) and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem
can also be viewed as asking for an algorithm to decide whether a given
statement is provable from the axioms using the rules of logic.
In 1936, Alonzo Church and Alan Turing published independent papers showing that a general solution to the Entscheidungsproblem is impossible, assuming that the intuitive notation of "effectively calculable" is captured by the functions computable by a Turing machine (or equivalently, by those expressible in the lambda calculus). This assumption is now known as the Church–Turing thesis.
Although research into mathematics education is primarily
concerned with the tools, methods, and approaches that facilitate
practice or the study of practice, it also covers an extensive field of
study encompassing a variety of different concepts, theories and
methods. National and international organisations regularly hold conferences and publish literature in order to improve mathematics education.
Objectives
Boy doing sums, Guinea-Bissau, 1974
At different times and in different cultures and countries,
mathematics education has attempted to achieve a variety of different
objectives. These objectives have included:
The teaching and learning of basic numeracy skills to all students
The method or methods used in any particular context are
largely determined by the objectives that the relevant educational
system is trying to achieve. Methods of teaching mathematics include the
following:
Computer-based mathematics education:
involves the use of computers to teach mathematics. Mobile applications
have also been developed to help students learn mathematics.
Conventional approach: the gradual and systematic guiding through the hierarchy of mathematical notions, ideas and techniques. Starts with arithmetic and is followed by Euclidean geometry and elementary algebra taught concurrently. Requires the instructor to be well informed about elementary mathematics
since didactic and curriculum decisions are often dictated by the logic
of the subject rather than pedagogical considerations. Other methods
emerge by emphasizing some aspects of this approach.
Relational approach: uses class topics to solve everyday problems and relates the topic to current events. This approach focuses on the many uses of mathematics and helps
students understand why they need to know it as well as helps them to
apply mathematics to real-world situations outside of the classroom.
Historical method: teaching the development of mathematics within a historical, social, and cultural context. Proponents argue it provides more human interest than the conventional approach.
Discovery math: a constructivist method of teaching (discovery learning) mathematics which centres around problem-based or inquiry-based learning, with the use of open-ended questions and manipulative tools. This type of mathematics education was implemented in various parts of Canada beginning in 2005. Discovery-based mathematics is at the forefront of the Canadian "math wars" debate with many criticizing it for declining math scores.
New Math: a method of teaching mathematics which focuses on abstract concepts such as set theory, functions,
and bases other than ten. Adopted in the US as a response to the
challenge of early Soviet technical superiority in space, it began to be
challenged in the late 1960s. One of the most influential critiques of
the New Math was Morris Kline's 1973 book Why Johnny Can't Add. The New Math method was the topic of one of Tom Lehrer's
most popular parody songs, with his introductory remarks to the song:
"...in the new approach, as you know, the important thing is to
understand what you're doing, rather than to get the right answer."
Recreational mathematics: mathematical problems that are fun can motivate students to learn mathematics and can increase their enjoyment of mathematics.
Rote learning:
the teaching of mathematical results, definitions and concepts by
repetition and memorisation typically without meaning or supported by mathematical reasoning. A derisory term is drill and kill. In traditional education, rote learning is used to teach multiplication tables, definitions, formulas, and other aspects of mathematics.
Math walk: a walk where experience of perceived objects and scenes is translated into mathematical language.
Different levels of mathematics are taught at different
ages and in somewhat different sequences in different countries.
Sometimes a class may be taught at an earlier age than typical as a
special or honors class.
Elementary mathematics in most countries is taught
similarly, though there are differences. Most countries tend to cover
fewer topics in greater depth than in the United States. During the primary school years, children learn about whole numbers and
arithmetic, including addition, subtraction, multiplication, and
division. Comparisons and measurement are taught, in both numeric and pictorial form, as well as fractions and proportionality, patterns, and various topics related to geometry.
At high school level in most of the US, algebra, geometry, and analysis (pre-calculus and calculus)
are taught as separate courses in different years.
On the other hand, in most other countries (and in a few US states),
mathematics is taught as an integrated subject, with topics from all
branches of mathematics studied every year;
students thus follow a pre-defined course sequence – encompassing
different topics – rather than choosing courses à la carte
as in the United States.
Even in these cases, however, several "mathematics" options may be
offered, selected based on the student's intended studies post high
school.
(In South Africa, for example, the options are Mathematics, Mathematical Literacy and Technical Mathematics.)
Thus, a science-oriented curriculum typically overlaps the first year of university mathematics, and includes differential calculus and trigonometry at age 16–17 and integral calculus, complex numbers, analytic geometry, exponential and logarithmic functions, and infinite series in their final year of secondary school; Probability and statistics are similarly often taught.
Coverage in other programs will vary.
Business mathematics is usually limited to introductory calculus and (sometimes) matrix calculations; business majors often also take a parallel course in statistics and probability. Additional to these topics, economics programs typically cover optimization, often differential equations and linear algebra, and sometimes analysis; statistics is similarly extended, underpinning econometrics. The discipline of Management Science formalizes the use of mathematics and statistics in business.
Social science curricula require statistics also, often complemented by major-specific quantitative research courses. Students in the humanities and liberal arts
may be offered a course in "contemporary mathematics," "mathematics for
the liberal arts," or "quantitative reasoning," which may include
topics such as set theory and mathematical logic, and applications of mathematics to other fields. Mathematics courses have low rates of course success relative to other fields of study, and students at US colleges and universities may be required to retake high school mathematics courses through remedial education programs.
Standards
Throughout most of history, standards for mathematics
education were set locally, by individual schools or teachers, depending
on the levels of achievement that were relevant to, realistic for, and
considered socially appropriate for their pupils.
In modern times, there has been a move towards regional or
national standards, usually under the umbrella of a wider standard
school curriculum. In England, for example, standards for mathematics education are set as part of the National Curriculum for England, while Scotland
maintains its own educational system. Many other countries have
centralized ministries which set national standards or curricula, and
sometimes even textbooks.
Ma (2000) summarized the research of others who found,
based on nationwide data, that students with higher scores on
standardized mathematics tests had taken more mathematics courses in
high school. This led some states to require three years of mathematics
instead of two. But because this requirement was often met by taking
another lower-level mathematics course, the additional courses had a
"diluted" effect in raising achievement levels.
In North America, the National Council of Teachers of Mathematics (NCTM) published the Principles and Standards for School Mathematics in 2000 for the United States and Canada, which boosted the trend towards reform mathematics. In 2006, the NCTM released Curriculum Focal Points,
which recommend the most important mathematical topics for each grade
level through grade 8. However, these standards were guidelines to
implement as American states and Canadian provinces chose. In 2010, the
National Governors Association Center for Best Practices and the Council
of Chief State School Officers published the Common Core State Standards
for US states, which were subsequently adopted by most states. Adoption
of the Common Core State Standards in mathematics is at the discretion
of each state, and is not mandated by the federal government. "States routinely review their academic standards and may choose to change or add onto the standards to best meet the needs of their students." The NCTM has state affiliates that have different education standards at the state level. For example, Missouri
has the Missouri Council of Teachers of Mathematics (MCTM) which has
its pillars and standards of education listed on its website. The MCTM
also offers membership opportunities to teachers and future teachers so
that they can stay up to date on the changes in math educational
standards.
The Programme for International Student Assessment (PISA), created by the Organisation for the Economic Co-operation and Development (OECD), is a global program studying the reading, science, and mathematics abilities of 15-year-old students. The first assessment was conducted in the year 2000 with 43 countries participating. PISA has repeated this assessment every three years to provide
comparable data, helping to guide global education to better prepare
youth for future economies. There have been many ramifications following
the results of triennial PISA assessments due to implicit and explicit
responses of stakeholders, which have led to education reform and policy
change.
Research
Paul Ernest
proposed, in a tentative and incomplete way, some primary and secondary
objects of study of research in mathematics education. While the
secondary objects come after the primary objects, Ernest claimed that
they are "important too, and should not be neglected." These primary objects of study are
The nature of mathematics and school mathematical knowledge
The learning of mathematics
The aims and goals of mathematics teaching and schooling
The teaching of mathematics, including the methods and approaches involved
The full range of texts, materials, aids and electronic resources employed
The human and social contexts of mathematics learning/teaching in all their complexity
The interaction and relationships between all of the above factors.
The secondary objects of study are
The nature of mathematics education knowledge: its concepts, theories, results, literature, aims and function
The nature of mathematics education research: its
epistemology, theoretical bases, criteria, methodology, methods,
outcomes and goals
Mathematics education teaching and learning in teacher education, including practice, technique, theory and research
The social institutions of mathematics education: the
persons, locations, institutions (universities, colleges, research
centers), conferences, organizations, networks, journals, etc. and their
relationships with its overall social or societal contexts.
According to Hiebert and Grouws, "Robust, useful theories of classroom teaching do not yet exist." However, there are useful theories on how children learn mathematics,
and much research has been conducted in recent decades to explore how
these theories can be applied to teaching. The following results are
examples of some of the current findings in the field of mathematics
education.
Important results
One of the strongest results in recent
research is that the most important feature of effective teaching is
giving students "the opportunity to learn". Teachers can set
expectations, times, kinds of tasks, questions, acceptable answers, and
types of discussions that will influence students' opportunities to
learn. This must involve both skill efficiency and conceptual
understanding.
Conceptual understanding
Source:
Two of the most important features of
teaching in the promotion of conceptual understanding times are
attending explicitly to concepts and allowing students to struggle with
important mathematics. Both of these features have been confirmed
through a wide variety of studies. Explicit attention to concepts
involves making connections between facts, procedures, and ideas. (This
is often seen as one of the strong points in mathematics teaching in
East Asian countries, where teachers typically devote about half of
their time to making connections. At the other extreme is the US, where
essentially no connections are made in school classrooms.)
These connections can be made through explanation of the meaning of a
procedure, questions comparing strategies and solutions of problems,
noticing how one problem is a special case of another, reminding
students of the main point, discussing how lessons connect, and so on.
Deliberate, productive struggle with mathematical ideas
refers to the fact that when students exert effort with important
mathematical ideas, even if this struggle initially involves confusion
and errors, the result is greater learning. This is true whether the
struggle is due to intentionally challenging, well-implemented teaching,
or unintentionally confusing, faulty teaching.
Formative assessment
Formative assessment
is both the best and cheapest way to boost student achievement, student
engagement, and teacher professional satisfaction. Results surpass
those of reducing class size or increasing teachers' content knowledge.
Effective assessment is based on clarifying what students should know,
creating appropriate activities to obtain the evidence needed, giving
good feedback, encouraging students to take control of their learning
and letting students be resources for one another.
Homework
Homework
assignments which lead students to practice past lessons or prepare for
future lessons are more effective than those going over the current
lesson. Students benefit from feedback. Students with learning
disabilities or low motivation may profit from rewards. For younger
children, homework helps simple skills, but not broader measures of
achievement.
Students with difficulties
Students with genuine difficulties (unrelated to motivation or past instruction) struggle with basic facts, answer impulsively, struggle with mental representations, have poor number sense,
and have poor short-term memory. Techniques that have been found
productive for helping such students include peer-assisted learning,
explicit teaching with visual aids, instruction informed by formative assessment, and encouraging students to think aloud.
In particular, research surrounding students with
disabilities in a mathematics classroom is mostly done by special
education researchers. Some mathematics education researchers have
called for more collaboration across disciplines to better understand
supports that could be helpful to mathematics students with
disabilities.
Algebraic reasoning
Elementary school children need to spend a
long time learning to express algebraic properties without symbols
before learning algebraic notation. When learning symbols, many students
believe letters always represent unknowns and struggle with the concept
of variable. They prefer arithmetic reasoning to algebraic equations
for solving word problems. It takes time to move from arithmetic to
algebraic generalizations to describe patterns. Students often have
trouble with the minus sign and understand the equals sign to mean "the answer is...".
Cultural Equity
Despite the popular belief that mathematics is race neutral, some research suggests that effective mathematics teaching of culturally diverse students requires a culturally relevant pedagogy
that considers students' cultural backgrounds and experiences. The
three criteria for culturally relevant pedagogy are academic success,
cultural competence, and critical consciousness. More recent research proposes that culturally sustaining pedagogy explicitly aims to
perpetuate and foster cultural and linguistic pluralism within the
educational system, ensuring that students can thrive while retaining
their cultural identities.
Mathematics Teacher Education
Student teaching
is a crucial part of a teacher candidate's path to becoming a teacher.
Recommended reform in mathematics teacher education includes a focus on
learning to anticipate, elicit, and use students’ mathematical thinking
as the primary goal, as opposed to models with an over-emphasis on
classroom management and survival.
Methodology
As with other educational research (and the social sciences in general), mathematics education research depends on both quantitative and qualitative studies. Quantitative research includes studies that use inferential statistics to answer specific questions, such as whether a certain teaching method
gives significantly better results than the status quo. The best
quantitative studies involve randomized trials where students or classes
are randomly assigned different methods to test their effects. They
depend on large samples to obtain statistically significant results.
Qualitative research, such as case studies, action research, discourse analysis, and clinical interviews,
depend on small but focused samples in an attempt to understand student
learning and to look at how and why a given method gives the results it
does. Such studies cannot conclusively establish that one method is
better than another, as randomized trials can, but unless it is
understood why treatment X is better than treatment Y, application of results of quantitative studies will often lead to "lethal mutations" of the finding in actual classrooms. Exploratory qualitative research is also useful for suggesting new hypotheses,
which can eventually be tested by randomized experiments. Both
qualitative and quantitative studies, therefore, are considered
essential in education—just as in the other social sciences. Many studies are "mixed", simultaneously combining aspects of both quantitative and qualitative research, as appropriate.
Randomized trials
There has been some controversy over the relative
strengths of different types of research. Because of an opinion that
randomized trials provide clear, objective evidence on "what works",
policymakers often consider only those studies. Some scholars have
pushed for more random experiments in which teaching methods are
randomly assigned to classes. In other disciplines concerned with human subjects—like biomedicine, psychology, and policy evaluation—controlled, randomized experiments remain the preferred method of evaluating treatments. Educational statisticians and some mathematics educators have been
working to increase the use of randomized experiments to evaluate
teaching methods. On the other hand, many scholars in educational schools have argued
against increasing the number of randomized experiments, often because
of philosophical objections, such as the ethical difficulty of randomly
assigning students to various treatments when the effects of such
treatments are not yet known to be effective, or the difficulty of assuring rigid control of the independent variable in fluid, real school settings.
Elementary mathematics was a core part of education in many ancient civilisations, including ancient Egypt, ancient Babylonia, ancient Greece, ancient Rome, and VedicIndia. In most cases, formal education was only available to male children with sufficiently high status, wealth, or caste. The oldest known mathematics textbook is the Rhind papyrus, dated from circa 1650 BCE.
Pythagorean theorem
Historians of Mesopotamia have confirmed that use of the Pythagorean rule dates back to the Old Babylonian Empire (20th–16th centuries BC) and that it was being taught in scribal schools over one thousand years before the birth of Pythagoras.
In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium included the mathematical fields of arithmetic and geometry. This structure was continued in the structure of classical education that was developed in medieval Europe. The teaching of geometry was almost universally based on Euclid'sElements.
Apprentices to trades such as masons, merchants, and moneylenders could
expect to learn such practical mathematics as was relevant to their
profession.
Medieval and early modern
Illustration at the beginning of a 14th-century translation of Euclid's Elements
In the Middle Ages,
the academic status of mathematics declined, because it was strongly
associated with trade and commerce, and considered somewhat
un-Christian. Although it continued to be taught in European universities, it was seen as subservient to the study of natural, metaphysical, and moral philosophy. The first modern arithmetic curriculum (starting with addition, then subtraction, multiplication, and division) arose at reckoning schools in Italy in the 1300s. Spreading along trade routes, these methods were designed to be used in
commerce. They contrasted with Platonic math taught at universities,
which was more philosophical and concerned numbers as concepts rather
than calculating methods. They also contrasted with mathematical methods learned by artisan
apprentices, which were specific to the tasks and tools at hand. For
example, the division of a board into thirds can be accomplished with a
piece of string, instead of measuring the length and using the
arithmetic operation of division.
The first mathematics textbooks to be written in English and French were published by Robert Recorde, beginning with The Grounde of Artes
in 1543. However, there are many different writings on mathematics and
mathematics methodology that date back to 1800 BCE. These were mostly
located in Mesopotamia, where the Sumerians were practicing
multiplication and division. There are also artifacts demonstrating
their methodology for solving equations like the quadratic equation. After the Sumerians, some of the most famous ancient works on mathematics came from Egypt in the form of the Rhind Mathematical Papyrus and the Moscow Mathematical Papyrus. The more famous Rhind Papyrus
has been dated back to approximately 1650 BCE, but it is thought to be a
copy of an even older scroll. This papyrus was essentially an early
textbook for Egyptian students.
In the 18th and 19th centuries, the Industrial Revolution led to an enormous increase in urban populations. Basic numeracy skills, such as the ability to tell the time, count money, and carry out simple arithmetic, became essential in this new urban lifestyle. Within the new public education systems, mathematics became a central part of the curriculum from an early age.
By the twentieth century, mathematics was part of the core curriculum in all developed countries.
During the twentieth century, mathematics education was
established as an independent field of research. Main events in this
development include the following:
In 1893, a Chair in mathematics education was created at the University of Göttingen, under the administration of Felix Klein.
The professional periodical literature
on mathematics education in the United States had generated more than
4,000 articles after 1920, so in 1941 William L. Schaaf published a classified index, sorting them into their various subjects.
A renewed interest in mathematics education emerged in the 1960s, and the International Commission was revitalized.
Midway through the twentieth century, the cultural impact of the "electronic age" (McLuhan) was also taken up by educational theory and the teaching of mathematics. While the previous approach focused on "working with specialized 'problems' in arithmetic", the emerging structural approach to knowledge had "small children meditating about number theory and 'sets'." Since the 1980s, there have been a number of efforts to reform the
traditional curriculum, which focuses on continuous mathematics and
relegates even some basic discrete concepts to advanced study, to better
balance coverage of the continuous and discrete sides of the subject:
In the 1980s and early 1990s, there was a push to make discrete mathematics more available at the post-secondary level;
From the late 1980s into the new millennium, countries
like the US began to identify and standardize sets of discrete
mathematics topics for primary and secondary education;
Concurrently, academics began compiling practical advice on introducing discrete math topics into the classroom;
Researchers continued arguing the urgency of making the transition throughout the 2000s; and
In parallel, some textbook authors began working on materials explicitly designed to provide more balance.
Similar efforts are also underway to shift more focus to mathematical modeling as well as its relationship to discrete math.
.jpg)
.png)
