First there was mathematics. Then, 13 years ago, came national curriculum maths, followed swiftly by nationally tested maths. Two years ago the numeracy strategy for key stages 1 and 2 was introduced, and now we see the KS3 strategy.

Initiatives bring with them the risk of the “opposite effect” - they may end up achieving the opposite of that for which they are designed. The national curriculum for maths was conceived as an initiative to bring more young people into the subject, but there is an argument that suggests the distance between maths and its learning has been increased.

For example: there is maths, a subset of which is the national curriculum, a subset of which is tested, a subset of which is taught, and finally, a subset of which is learned.

Each of these steps distances the learner from the subject. Characteristic of maths is its inter-connectedness, which is a consequence of its logic and structure.

However, the national curriculum and numeracy frameworks express selections of maths in lists. It is analogous to describing a meal in terms of only its ingredients or a topological space in terms of only its nodes.

There is a danger that feeding ingredients to learners misses that essential matter of a recipe which blends the ingredients together. Changing to the topology metaphor, the territory of maths is described both in terms of its nodes and the arcs that connect the nodes.

I find the nodes and arcs metaphor helps to clarify my thinking on acceleration and enrichment in maths. A view of acceleration as simply accumulating nodes - bits of maths - does not develop better mathematicians. However, a view of enrichment as building arcs between existing nodes establishes a clearer picture of the territory of maths. Learners who develop habits of establishing arcs are likely to chart paths beyond the “taught” curriculum and encounter new nodes.

Enriching the curriculum is about “arc making” - establishing connections - which requires the application of mathematical thinking and reasoning. The nodes defined in the national curriculum and numeracy frameworks are necessary, but not sufficient.

The much-criticised strand of “using and applying maths” attempts to specify the skills needed to join the bits together - providing the dynamic of travelling between the bits of maths specified elsewhere. Presenting Ma1 as a separate attainment target (for assessment purposes) and integrating it in the programmes of study (for teaching purposes) is the complete reverse of what should be.

Enriching maths should be a feature of all maths lessons. The rigour of defining what will be learned by the end of a lesson has its place, but this has now become an orthodoxy.

Most lessons end on a predetermined node. What a pleasant change it is to observe a lesson that is defined by an interesting starting point and encourages pupils to develop their own learning trails.

Some years ago, in Newcastle, I worked with some sixth-formers on what I described as “off-piste” maths. It comprised a selection of highly challenging starting points which required a degree of mental acrobatics, but were unrelated to the examination syllabus.

I am saddened to see so many pupils restricted to the mathematical territory defined by the end-of-key stage assessment. It is particularly noticeable at KS4, where groups of pupils who are “targeted” for, say, the middle tier are not given the opportunity to encounter maths from the higher tier.

I have heard of cases at KS2 where pupils “targeted” for level 4 never have the opportunity to work beyond it. The assessed curriculum should reflect the taught curriculum, not determine it.

The national curriculum was designed as a minimum entitlement for pupils, but what was designed as a floor has become a ceiling. National assessment has distorted what is mathematical success: it is easier to count the nodes than to measure the arcs. The OFSTED inspection regime has left teachers with perceptions that every lesson should have a nodal end point called a learning objective (which must be written on the board) and every lesson must map directly on to a statement from either the framework or the national curriculum.

There are clear indications that the “opposite effect” exists and we may be approaching the time when the benefits of these national initiatives become outweighed by disadvantages.

It would be churlish not to recognise the benefits. Primary pupils are now more confident in their understanding of the structure of the number system and consequently more confident in the manipulation of numbers. In this sense they are better prepared for their next stages of schooling. The impending danger is a taught curriculum totally defined by external assessment and predetermined expectations of what pupils will attain in this external assessment. The territory of maths is then hidden and there is a loss to its learners.

We need to restore a taught curriculum which reflects the nature of maths, not as something done to pupils, but as something that engages them in creative thinking and mathematical reasoning. Perhaps we should recognise the mathematical potential in all of us and develop a mathematicians’, rather than maths, curriculum. This might take us beyond the frameworks and bring refreshed thinking to assessment.

Peter Lacey is chair of the general council of the Association of Teachers of Mathematics, 7 Shaftesbury Street, Derby DE23 8YB. Tel: 01332 346599Web: www.atm.org.uk