5.1: Comparing geometry and arithmetic

The opening quotations remind us that the mental universe of formal mathematics draws much of its initial inspiration from human perception and activity - activity which starts with infants observing, moving around, and operating with objects in time and in space. Many of our earliest pre-mathematical experiences are quintessentially proto-geometrical. We make sense of visual inputs; we learn to recognise faces and objects; we crawl around; we learn to look ‘behind’ and ‘underneath’ obstructions in search of hidden toys; we sort and we build; we draw and we make; etc.. However, for this experience to develop into mathematics, we then need to

• identify certain semi-formal “objects” (points, lines, angles, triangles),
• pinpoint the key relations between them (bisectors, congruence, parallels, similarity), and then
• develop the associated language that allows us to encapsulate insights from prior experience into a coherent framework for calculation and deduction.

Too little attention has been given to achieving a consensus as to how this transition (from informal experience, to formal reasoning) can best be established for beginners in elementary geometry. In contrast, number and arithmetic move much more naturally

• from our early experience of time and quantity
• to the notation, the operations, the calculational procedures, and the rules of formal arithmetic and algebra.

Counting is rooted in the idea of a repeated unit - a notion that may stem from the ever-present, regular heartbeat that envelops every embryo (where the beat is presumably felt long before it is heard). Later we encounter repeated units with longer time scales (such as the cycles of day and night, and the routines of feeding and sleeping). The first months and years of life are peppered with instances of numerosity, of continuous quantity, of systematic ordering, of sequences, of combinations and partitions, of grouping and replicating, and of relations between quantities and operations - experiences which provide the raw material for the mathematics of number, of place value, of arithmetic, and later of ‘internal structure’ (or algebra).

The need for political communities to construct a formal school curriculum linking early infant experience and elementary formal mathematics is a recent development. Nevertheless, in the domain of number, quantity, and arithmetic (and later algebra), there is a surprising level of agreement about the steps that need to be incorporated - even though the details may differ in different educational systems and in different classrooms. For example:

• One must somehow establish the idea of a unit, which can be replicated to produce larger numbers, or multiples.
• One must then group units relative to a chosen base (e.g. 10), iterate this grouping procedure (by taking “ten tens”, and then “ten hundreds”), and use position to create place value notation.
• One must introduce “0” - both as a number in its own right, and as a placeholder for expressing numbers using place value.
• One can then use combinations and differences, multiples and sharing (and partitions), to develop arithmetic.
• At some stage one introduces subunits (i.e. unit fractions) and submultiples (i.e. multiples of these subunits) to produce general fractions; one can then use equivalence and common submultiples to extend arithmetic to fractions.
• If we restrict to decimal fractions, then our ideas of place value for integers can be extended to the right of the decimal point to produce decimals.
• At every stage we need to

  — relate these ideas to quantities,

  — require pupils to interpret and solve word problems, and

  — cultivate both mental arithmetic and standard written algorithms

  .
• Towards the end of primary school, attention begins to move beyond bare hands computation, to consciously exploit internal structure in preparation for algebra.

Our early geometrical experience is just as natural as that relating to number; but it is more subtle. And there is as yet no comparable consensus about the path that needs to be followed if our primitive geometrical experience is to be formalised in a useable way.

The 1960s saw a drive to modernise school mathematics, and at the same time to make it accessible to all. Elementary geometry certainly needed a re-think. But the reformers in most countries simply dismissed the traditional mix (e.g. in England, where one found a blend of technical drawing, Euclidean, and coordinate geometry in different proportions for different groups of students) in favour of more modern-sounding alternatives. Some countries favoured a more abstract, deductive framework; some tried to exploit motion and transformations; some used matrices and groups; some used vectors and linear algebra; some even toyed with topology. More recently we have heard similarly ambitious claims on behalf of dynamic geometry software. And although each approach has its attractions,

none of the alternatives has succeeded in helping more students to visualise, to reason, and to calculate effectively in geometrical settings.

At a much more advanced level, geometry combines

• with abstract algebra (where the approach proposed by Felix Klein (1849-1925) shows how to identify each geometry with a group of transformations), and
• with analysis and linear algebra (where, following Gauss (1777-1855), Riemann (1826-1866) and Grassmann (1809-1877), calculus, vector spaces, and later topology can be used to analyse the geometry of surfaces and other spaces).

However, these subtle formalisms are totally irrelevant for beginners, who need an approach

• based on concepts which are relatively familiar (points, lines, triangles etc.), and
• whose basic properties can be formulated relatively simply.

The subtlety and flexibility of dynamic geometry software may be hugely impressive; but if students are to harness this power, they need prior mastery of some simple, semi-formal framework, together with the associated language and modes of reasoning. Despite the lack of an accepted consensus, the experience of the last 50 years would seem to suggest that the most relevant framework for beginners at secondary level involves some combination of:

• static, relatively traditional Euclidean geometry, and
• Cartesian, or coordinate (analytic) geometry.

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