The School of Babel of Mathematics

crisis of mathematics education

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Abstract


Mathematicians no longer understand mathematicians.


This knol is related to the knols Why School Mathematics? Modern Mathematics Education  Simulation Technology  and Mathematics=Magics? 

The eminence, safety, and well-being of nations have been entwined for centuries with the ability of their people to deal with sophisticated quantitative ideas (=mathematics). Leading societies have commanded mathematical skills that have brought them advantages in medicine and health, in technology and commerce, in navigation and exploration, in defense and finance, and in the ability to understand past failures and to forecast future developments. History is full of examples. (National Mathematics Advisory Panel 2008)

                           And the whole earth was of one language, one speech and one mathematics. 


Crisis = Mathematics Crisis?

The crisis in mathematics education is getting worse every year. Students absorb less and less of shrinking curricula, and leaders of Western industrial nations feel urged to take action:
  • President Obama issued a challenge [Video] to states to increase the quality of reading and math instruction to keep American students at pace with other countries: It is time to give all Americans a complete and competitive education from the cradle up through a career…We have accepted failure for too long – enough. America’s entire education system must once more be the envy of the world…What’s at stake is nothing less than the American Dream. (March 2009)
  • Business – Higher Education Forum: Mathematics and science education in this country is falling short of what is required to keep America productive, stable, and secure. 
  • Swedish Minister Jan Björklund: If Sweden is going to survive as an industrial nation, it is necessary to improve mathematics education. Systematic errors in the subtraction of natural numbers committed by students (e.g. 51 – 49 = 18) have been detected.  525 million Swedish Crowns is not allocated in a major push to come to grips with the system behind the errors. (March 2009)

But the present crisis is just a continuation of a constant crisis: The National Commission on Excellence in Education presented in 1983 its report to President Reagan entitled A Nation at Risk: the Imperative for Educational Reform:


  • the educational foundations of our society are presently being eroded by a rising tide of mediocrity that threatens our very future as a Nation and a people
  • only one-third can solve a mathematics problem requiring several steps.
Is this a crisis of
  • mathematics?
  • mathematics education?
  • information society?
  • democracy and market economy?

Or is there no crisis? 

                                                             Credibility gap?

Mathematicians Don’t Understand Mathematicians

Mathematics consists of the following main disciplines:

  • pure: symbolic – pen+paper
  • (classical) applied: symbolic/digital – pen+paper
  • computational: symbolic/digital – computer
  • didactic: symbolic/digital – pen+paper+calculator 

where we indicate the main tools of pen+paper and computer and the main forms of expression of symbolic (formulas) and digitial (numbers).
Observation indicate that 
  • pure (mathematicians) don’t understand applied/computational and vice-versa
  • classical applied don’t understand computational
  • didactic are supposed to understand pure/applied/computational but not vice-versa.

                               Unity of pure and applied mathematics in the School of Athens.


The split pure/applied/computational expresses a contradiction of mathematics as the Queen of science of abstraction combining unity and generality. The split away from unity occured in two steps during the 20th century:

  • pure from applied in the early 1900s driven by focus on the foundations of mathematics
  • computational from classical applied/pure in the 1950s driven by the computer and prepared by the constructivists in the 1930s. 

The big battle in the 1930s between the logicists/formalists on one side and the constructivists on the other, ended with Gödel’s incompleteness theorem giving the victory to the constructivists, but ironically the winners were then expelled from the math departments in control of pure logicists/formalists to departments of computer science as a form of constructive mathematics.
The fundamentally different mathematical techniques and standards used in pure and computational mathematics, and the related difficulty of reaching consensus, is illustrated in The Clay Navier-Stokes Millennium Problem. 
Today math departments are controled by pure math with an agenda different from applied/computational. 
The mathematician Hilbert observed in the 1930s that physics is becoming too difficult for the physicists, and would probably say today that:

  • mathematics is becoming too difficult for the mathematicians.

If pure/applied/computational cannot unite, didactic math is left to invent its own unity, or separate identity
of school mathematics with its own questions and answers: The new math reform of the 1970s can be seen as a self-induced didactics reaction to the logicist set theoretic bible  Principia Mathematica by Russell-Whitehead published in 1910-13, a reform which was a mystery to both pure and applied mathematicians as well as teachers, students and parents, and therefore ended in chaos.

                                                 Last supper of math education inside a dodecahedron?

National Council of Teachers of Mathematics (NCTM)

The vision of the  National Council of Teachers of Mathematics adopted by most states of the US, is grand (although criticized for valuing understanding before correctness and referred to as new-new-math):
  • the vision for mathematics education is highly ambitious…it requires solid mathematics curricula, competent and knowledgeable teachers who can integrate instruction with assessment, education policies that enhance and support learning, classrooms with ready access to technology, and a commitment to both equity and excellence…the challenge is enormous and meeting it is essential… our students deserve and need the best mathematics education possible, one that enables them to fulfill personal ambitions and career goals in an ever-changing world
  • in this changing world, those who understand and can do mathematics will have significantly enhanced opportunities and options for shaping their futures…mathematical competence opens doors to productive futures…lack of mathematical competence keeps those doors closed…everyone needs to understand mathematics…all students should learn significant mathematics with depth and understanding
  • from a young age, children are interested in mathematical ideas
  • assessment should reflect the mathematics that all students need to know and be able to do, and it should focus on students’ understanding as well as their procedural skills

based on the axioms that

  • mathematics is one of the greatest cultural and intellectual achievements of human-kind, and citizens should develop an appreciation and understanding of that achievement, including its aesthetic and even recreational aspects
  • there is no conflict between equity and excellence.
But the difficulties are considered to be enormous:
  • unfortunately, learning mathematics without understanding has long been a common outcome of school mathematics instruction…learning without understanding has been a persistent problem since at least the 1930s, and it has been the subject of much discussion and research by educators over the years…in the twenty-first century, all students should be expected to understand and be able to apply mathematics

and the demands on the teachers are monumental:
  • teachers must be information providers, planners, consultants, and explorers of uncharted mathematical territory
  • they must adjust their practices and extend their knowledge to reflect changing curricula and technologies
  • they must be able to describe and explain why they are aiming for particular goals
  • they need sustained, ongoing professional development in order to offer students a high-quality mathematics education
  • they must continue to learn new or additional mathematics content and use new materials and technology
  • they must develop their own professional knowledge using research as the knowledge base of the profession
  • they control the range of mathematical ideas made available to their students and have the responsibility to ensure that a full range of mathematical content and processes are taught into a coherent whole
despite the fact that
  • unfortunately, the preparation today’s teachers have received is in many instances inadequate for the needs of tomorrow…unless teachers are able to take part in ongoing, sustained professional development, they will be handicapped in providing high-quality mathematics education… the current practice of offering occasional workshops and in-service days does not and will not suffice.

In short, mathematics teachers are supposed without adequate training to bring unity, from pre-school to university level, into a science and practice of mathematics split by professional mathematicians separated from didactics. Is this really possible?

               Destiny of human civilization according to mathematical modeling of global climate = ??

National Mathematics Advisory Panel

The Final Report of the National Mathematics Advisory Panel presented to the President of the US in March 2008  makes the following (remarkable) statements:
  • mathematics literacy is a serious problem in the US: 78% of adults cannot explain how to compute the interest paid on a loan, 71% cannot calculate miles per gallon on a trip, and 58% cannot calculate a 10% tip for a lunch bill
  • a broad range of students and adults also have difficulties with fractions, a foundational skill essential to success in algebra. 
  • a conceptual understanding of fractions and decimals and the operational procedures for using them are mutually reinforcing. One key mechanism linking conceptual and procedural knowledge is the ability to represent fractions on a number line 
  • the curriculum should afford sufficient time on task to ensure acquisition of conceptual and procedural knowledge of fractions and of proportional reasoning 
  • instruction focusing on conceptual knowledge of fractions is likely to have the broadest and largest impact on problem-solving performance when it is directed toward the accurate solution of specific problems
  • when children believe that their efforts to learn make them “smarter,” they show greater persistence in mathematics learning
  • unfortunately, little is known from existing high-quality research about what effective teachers do to generate greater gains in student learning.

The school mathematics of the 19th century also concentrated the study to fractions motivated by a need of  transforming between different units of base other than 10 using pen and paper, which however today is diminishing…and what is meant by “represent fractions on a number line”? 

What Can Be Done?

The credibility gap between the grand vision formulated by the National Council of Mathematics Teachers and the realities of the Advisory Panel is huge, and as in any activity with eroding credibility, permanent crisis is the result: When the credibility of the King disappears, so does the Kingdom. Why School Mathematics? indicates a possible way out of the crisis. 
What is needed is that professional mathematicians take on responsibility of school mathematics and form a new synthesis of pure/applied/computational mathematics in the new age of the information society. But is such a unity possible? Or will the math wars continue and students and society continue to suffer?
Inspection shows this debate to be a form of scholasticism lacking contact with realities.

                                  Did Math Formula Cause the Financial Crisis? 

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