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Tony Gardiner
HYPERLINK "mailto:A.D.Gardiner@bham.ac.uk" A.D.Gardiner@bham.ac.uk: University of Birmingham, UK
Freedom is the freedom to say that two plus two makes four.
If that is granted, all else follows. (George Orwell, 1984)
Mathematics is more permanent than almost any other human cultural activity: school mathematics has its roots in methods often going back several thousand years. In contrast, concern about mathematical literacy (ML) is of relatively recent origin. In trying to disentangle these two strands in our analysis, the central points may be summarized as:
mathematics is very different from (basic) numeracy or mathematical literacy
numeracy and mathematical literacy are best seen as planned by-products, rather than as central goals, of effective mathematics education
attempts to redesign school mathematics as in England during the period 1982-2007, to give it a tight initial focus on numeracy, leaving more abstract ideas and methods to follow later, have predictable negative consequences for all students; such attempts contradict both the character of elementary mathematics and the way human beings learn
nevertheless, it may be helpful to consider what humane consequences these desirable practical outcomes could have for school mathematics.
Mathematics teaching is less effective than most of us would like; but we should hesitate before embracing the idea that school mathematics would be more effective on a large scale if the curriculum were to focus first on numeracy (useful mathematics for all), leaving more formal, abstract mathematics to follow later for those whose interest survives this thoroughly misleading introduction. For example, basic arithmetic became accessible to all only when it adopted notation and written procedures (or algorithms) which combined the profound abstractions of powers of 10, the index laws, and place value. One can program machines to do arithmetic. But human beings are not machines: to operate mathematically (as human beings) they need to grasp the abstract structure that underpins arithmetic.
The experience of committed teachers and small-scale projects in England reflects evidence from other countries: namely that it should be possible to help many more students to achieve a useable mastery of elementary mathematics but only if one is willing to interpret the goal of ultimate numeracy in a mathematical spirit. If we were to take this evidence seriously, we might restrict the initial focus to truly basic material (integers, fractions, decimals, proportion, word problems, algebra and geometry), and teach this material in a way which encourages all students to use these ideas effectively and which also prepares large numbers of students to move on to more serious mathematics when they need to. But such a program may look strangely like what good mathematics teaching has always been!
The key to effective mathematics teaching is elusive. When mathematical literacy and numeracy are presented as though they were alternatives to traditional school mathematics, rather than by-products of effective instruction (like literacy or maturity), there is a danger (i) that politicians and employers may seize upon the idea that there is a pragmatic-sounding alternative to difficult mathematics, (ii) that bureaucrats may imagine that focusing on numeracy from the outset might deliver what they see as the required (utilitarian!) end-product more directly and more cheaply; and (iii) that some educationists may see this paradigm shift as an opportunity to further undermine the idea of mathematics as the archetypal objective discipline. In particular, we should avoid being carried along on a flood of rhetoric of which the following is one of many examples (Steen, 2001):
Unlike mathematics, which is primarily about a Platonic realm of abstract structures,
numeracy is often anchored in data derived from and attached to, the empirical world
[and] does not so much lead upward in an ascending pursuit of abstraction as it moves
outward toward an ever richer engagement with lifes diverse contexts and situations.
Rather we should heed the warning of Hyman Bass (quoted in (Steen, 2004)), whose words describe with uncanny accuracy what happened in England in the late 1980s and 1990s:
the main danger is the impulse to convert a major part of the curriculum to this
form of instruction. The resulting loss of learning of general (abstract) principles
may then deprive the learner of the foundation necessary for recognizing how the
same mathematics witnessed in one context in fact applies to many others.
Towards a conception of mathematical literacy
Numeracy and mathematical literacy are not alternatives to, but would seem to be desirable by-products of school mathematics. It may therefore make more sense to interpret numeracy, or quantitative literacy, as a basic
willingness to engage effectively
with quantitative information
in simple settings.
We could then preserve the term mathematical literacy to denote
a more subtle, long-term aspiration involving
simple insights into the nature of elementary mathematics.
The mental universe of mathematics: imagination, literacy and the three Rs
We have lost sight of the basic fact that
the world of mathematics is a mental universe.
The mental nature of this mathematical universe has one truly liberating consequence (provided one remains faithful to the elusive, but undeniably objective, character of the discipline) namely that
elementary mathematics is accessible to anyone with a mind.
To open the minds of ordinary students to the power and flexibility of genuine mathematics
we must rediscover the fact that mathematics is in some ways inescapably abstract from the very beginning, and that effective mathematics teaching has to reflect this fact (in a sensitive way), and
we must concentrate on identifying and achieving mastery of core techniques, which are routinely (and flexibly) linked into multi-step wholes to solve extended exercises and problems.
Our concern is to identify those key ingredients of this mental universe of mathematics
(i) which are needed to lay a flexible mathematical foundation for all at ages 5-15;
(ii) which are relevant on the simplest level of numeracy; and
(iii) which might represent a modest outline of the kind of features one could include
in a more appropriate interpretation of mathematical literacy as a desirable adult
by-product of school mathematics.
As suggested earlier, we do not try to define numeracy, but simply take it to mean:
a willingness to engage effectively with quantitative information in simple settings.
We then list four components which would seem to be necessary in any program to achieve such a state. While these include competence in handling certain basic techniques, numeracy cannot be reduced to a detailed syllabus. Hence the list included in the third of our four components is deliberately short, and the other three components are in many ways more important.
1. The (not-so-traditional) three Rs: The first component of our attempt to pin down what might be meant by basic numeracy is the deliberately provocative Trinity of Obligations:
(i) to remember (i.e. to learn - if necessary by rote)
(ii) to reckon (i.e. to calculate accurately)
(iii) to reason (i.e. to think mathematically).
2. Mathematics as the science of exact calculation: The second component - which is essential if the other three components are to be effective - is the suggestion that all students should absorb and appreciate the spirit of the statement that, in contrast to the messy real world,
mathematical calculations (whether numerical, symbolic, logical or geometrical)
are special, in that they concern ideal rather than real objects, and so are exact.
3. Techniques: The third component is that all students should be expected to achieve mastery of a limited core of basic techniques (some large subset of: multiplication tables, place value and decimals, measures, fractions and ratio, negative numbers, triangles and circles, Pythagoras theorem, basic trigonometry and similarity, coordinates, linear and quadratic equations, how to handle formulae, straight line graphs and linear functions), which techniques constitute the background material in terms of which the other three components can be interpreted.
4. Applications: Whatever list of content and techniques is adopted, these should be used regularly and routinely to handle problems and situations which systematically cultivate the notion that, despite its ideal character, one important aspect of elementary mathematics (integer and decimal arithmetic; simple and compound measures; fractions, ratio and simple proportion; the simplest geometrical representations; etc.) is that it is useful.
The real test of these four components is whether they can help us to devise strategies which achieve more than we do at present for the majority of students. But these components are more than merely utilitarian, and should be seen as part of a profoundly humane education.
References
Gardiner, A. 2008. What is mathematical literacy? Proceedings of ICME10 [This 20 page version of the present summary with full references will be available at ICME11.]
Steen, Lynn A. (ed.) 2001. Mathematics and democracy, the case for quantitative literacy. National Council for Education and the Disciplines.
Steen, Lynn A. (ed.) 2004. Achieving quantitative literacy. Mathematical Association of America.
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