Why Teach Mathematics?

a critical analysis

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Abstract


We discuss aspects of the role and practice of mathematics in education from elementary school to university.


 

                                  What will you be worth if you don´t know enough math? (Peter Tillberg)

Questions

This is a continuation of the knols Modern Mathematics Education Mathematics=Magics? and 


  • (Q1) Is knowledge of mathematics as important as reading and writing, for everyone?
  • (Q2) What is the net result of 12 years of school mathematics, for most people? 

The standard answer is YES to (Q1) and LITTLE to (Q2). A critical analysis [1] changes the answer of (Q1)
to NO but confirms the answer to (Q2), as presented in more detail below.
YES-LITTLE appears contradictory (while NO-LITTLE is not)  which reflects the central contradiction which has troubled school mathematics since it became a primary subject of study 150 years ago: 
  • Mathematics education is important to everyone, but most students learn very little and use less.


  

Standard Answers

  • 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 and lack of mathematical competence keeps those doors closed 
  • NCTM challenges the assumption that mathematics is only for the select few…attitudes that suggest only certain students are capable of mastering mathematics should be deprecated … everyone needs to understand mathematics 
  • all students, regardless of their personal characteristics, backgrounds, or physical challenges, must learn significant mathematics with depth and understanding…there is no conflict between equity and excellence
  • the mathematics taught in modern classrooms should be the skills that are most important to the students’ lives and careers
  • school mathematics curricula should focus on deepening students’ appreciation of mathematics as a discipline and as a human creation. 

These principles express: 

  • a circulatory definition of what mathematics to teach to what students
  • all students have to learn all the unspecified math skills required for life and career
  • appreciation of mathematics as a discipline should be indoctrinated.

The unfortunate net result of these principles is that a low level limit is put for all students including those interested in math, which is precisely the tragic contradiction of school mathematics:

  • mathematics is considered to be so important for everybody that knowledge has to be limited to practically zero. 

                                           Proof that math education works for everybody


To get an idea how math education can be presented to the public take a look at Cameron Heights Collegiate Institute presentation including the informative statement:


A snapshot from realities is given in the 2008 report by the National Mathematics Advisory Panel to the President:

  • prekindergarten-to-eighth-grade math curriculums should be streamlined and put focused attention on skills like the handling of whole numbers and fractions and certain aspects of geometry and measurement
  • by the end of the third grade, students should be proficient in adding and subtracting whole numbers and two years later, they should be proficient in multiplying and dividing them 
  • by the end of the sixth grade students should have mastered the multiplication and division of fractions and decimals
  • fractions are especially troublesome for Americans…the catchword for math teachers today should be “fractions” (Francis Fennell, president of the National Council of Teachers of Mathematics)
  • the current “talent-driven approach to math, that either you can do it or you can’t, like playing the violin” needs to be changed 
  • what should be avoided in mathematics is an approach that continually revisits topics year after year without closure…
Eight years of school to “handle” whole numbers (but not “troublesome”  fractions) without revisits?
  • US math textbooks are far too long – often 700-1000 pages…Excessive length makes our books unnecessarily expensive and tends to undermine coherence and focus.

Math books preventing math education?

Mathematics = Music


To get perspective one can compare with education in e.g. music following the same principles:

  • playing piano is necessary to succeed in life and career
  • all students are capable of mastering the piano. 

In a piano education based on these principles, all students would learn to play the C-scale on the piano, like “handling” whole numbers, but not much more.

Fortunately failure to play piano sonatas does not prevent reasonable lives and careers, which has allowed those who are interested to develop their talents and learn to play very well. Fortunately, there is no minimum standard to be met by everyone, which effectively would limit the maxium to the same low minimum. As a result there are fortunately lots of good piano players around to the enjoyment of many.

                                           Proof that piano teaching works for some students


In order to open to an equally productive mathematics teaching/learning, it is necessary to realize that

  • mathematics is not as important as reading and writing, for everybody 
  • acknowledging the relative unimportance of mathematics for everybody, the need of some who can understand and use mathematics in the information society, can be met    
as expressed in the debate article [2] and Dreams of Calculus causing an uproar from the Swedish Mathematical Society

A Critical Analysis


Mathematics education has not been subject to much of critical analysis, although it has been criticized a lot. An exception is the recent thesis [1] presenting the following analysis: 
  • school teaches mathematics in a way that makes it appear as something very powerful and everywhere present, but at the same time invisible and beyond comprehension for students and teachers. Using terminology from Slovene philosopher Slavoj Zizek, mathematics can thus be understood as the sublime object of an ideology sustained by school mathematics (similar to the God of a religion or the King in a monarchy).
  • school asks students/teachers to confess that mathematics is a universal and powerful tool, but admits that few students/teachers learn to use the tool
  • the failure of school mathematics is used to motivate more resources to school mathematics
  • school practices mathematics as an instrument for discipline, selection and sorting and to keep students busy.

In the spirit of Zizek it seems that the very fact that mathematics education is (more or less) meaningless for most students, by the school system is used as evidence that mathematics education is meaningful.  

                                                 Proof of the sublime nature of mathematics


In the same spirit, let us analyze the following main proposal by the 2003 Swedish Mathematics Delegation
(copying NCTM):

  • to develop activities increasing the interest/understanding of the value, role and importance of mathematics in everyday life, professional life, science and society.
The construction of this statement is revealing: It does not say that mathematics is important, and that
students should learn this important subject. It can be read as stating that the objective of mathematics education is to propagate the idea that mathematics is important (for its own sake or for society), whether it is or not (for the majority of people). In other words, it expresses that mathematics is a sublime object, worthy of worship, as a form of fundamentalism. With this objective mathematics teaching is very successful: even students failing miserably in mathematics become convinced of the value of mathematics, if not for them. 
After 5 years of incubation the proposal by the Delegation resulted in March 2009 in 525 million Swedish Crowns to mathematics education with the above objective. According to responsible Minister Jan Björklund the push to come up with the money came from detection of systematic errors in the subtraction of natural numbers committed by students (e.g. 51 – 49 = 18), which have to be corrected, at any cost. The system responsible for these systematic errors is under investigation.

What Mathematics Does Obama (or Bush) Need to Know?

We list some elements of mathematics which can be useful to large groups of students and which does not require 12 years of study, which a president of the US can be expected to understand and use:
  • counting: voters, income, expenses, budget deficits, soldiers, age, megabytes, pixels
  • performance: speed, miles/gallon, squaremeter/dollar, income/capita, bullets/second
  • listen to President Bush learns mathematics:


President Obama called for sweeping changes in American education on March 10 2009 following the advice of the Swedish Mathematics Delegation.

Observations

The facts that


  • modern information society is based on mathematics
  • mathematics education is in a (permanent) state of crisis
  • mathematics education has changed little the last 100 years

requires reform of mathematics education to meet the need of some people able to understand and use the tools of mathematics today and tomorrow, while acknowledging that the level of math skills of the President of the US can be satisfactory for life and career, for most people.
Proof that reform of math education is needed

Standard Views

Typical statements (of questionable truth value) about mathematics influencing opinions, are:


  • A mystery lurks beneath the magic carpet of science, something that scientists have not been telling, something too shocking to mention except in rather esoterically refined circles: that at the root of the success of twentieth-century science there lies a deeply ‘religious’ belief — a belief in an unseen and perfect transcendental world that controls us in an unexplained way, yet upon which we seem to exert no influence whatsoever. What this world is, where it is, and what it is to us is what this book is about. (John D. Barrow [3])
  • Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. (Bertrand Russell) 
  • The advancement and perfection of Mathematics are intimately connected with the prosperity of the State. (Napoleon I)
  • Nature’s great book is written in mathematical symbols. (Galileo)
  • All science requires Mathematics. The knowledge of mathematical things is almost innate in us… This is the easiest of sciences, a fact which is obvious in that no one’s brain rejects it; for laymen and people who are utterly illiterate know how to count and reckon. (Roger Bacon) 
  • Learning to think in mathematical terms is an essential part of becoming a liberally educated person. (Kenyon College Math Department Web Page
  • To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty of nature. … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. (Richard Feynman)

                                 Proof of the mathematical nature of Romanesque Cauliflower