All in the mind

Peter Lacey on the muddle over keystage 3 pilot test in mental arithmetic

With, no doubt, the best of intentions, another assessment initiative in mathematics - mental arithmetic tests at key stage 3 - is set to deconstruct and distort the very nature of mathematics and consequently inhibit its learning.

This move follows the debacle of separate assessment of using and applying mathematics.

Mathematics is a mental universe. It exists in our minds and is a result of human endeavour. Whether a mathematical problem is presented in oral or written form, we need to use mathematical thinking and apply appropriate mathematical skills to solve it. These are mental activities.

The key stage 3 "written" tests require an understanding of which skills to apply, the recall of mathematical conventions, the successful application of those skills and the ability to use appropriate checking strategies. They demand the use of mental mathematical skills.

Oral assessment can provide critical and valuable information about pupils' knowledge and understanding, by which I mean the pupil speaks and the assessor listens. Alternative modes of assessment may be needed to ensure that instruments are not used to estimate a measure. But what makes these mental tests distinctive, and how do they improve the quality of assessment?

First, they are presented in oral mode, requiring a written response.

This raises several important issues: * Pupils whose first language is not English have an immediate problem. I was recently subjected to 10 mental arithmetic questions presented in French and, although I understand some French, I ran out of time and had to ask for the question to be repeated several times.

* Memorising the question, let alone calculating the result, is being tested. I have yet to find research showing that those with the best short-term memory make the best mathematicians. Identifying the salient points in the question is important, but this applies equally to written questions.

* On the pupils' response sheet, there is some "memory-jogging" information for some questions. Selected salient points are written down to help the pupil. In the question "Multiply three point nought six by one thousand', 3.06 is written in the right-hand column. But two questions later, "Multiply nought point two by thirty", nothing is written in the right-hand column. And two questions later still "What is nought point nine divided by nought point nought one?", both 0.9 and 0.01 are written.

* Special arrangements for pupils with hearing impairment allow flash cards to be used. Could these not be used for a wider range of pupils, say those who cannot retain the information presented orally? Or even as an additional aid for all pupils? Whatever is decided, preserving the flexibility afforded in the pilot to meet pupils' special educational needs is essential.

Each question also has a strict time limit. Again, this is ill thought-out: * The theory (whose theory?) is that five seconds is long enough to recall facts. I know the names of all my colleagues, but from time to time I forget, and need more than five seconds to recall what I know. And most questions in the five-second section of the tests are not recall.

In the question "What is four point seven multiplied by one hundred', pupils are not recalling the last time they multiplied 4.7 by 100, but quickly shuffle the digits 4 and 7 two hops to the left. This may take less time, if pupils know what to do, than calculating, say, 42 divided by six.

* Some calculations take longer than others, but categorising them as five, 10 or 15-second questions presupposes a science that does not exist. Cut out each question, drop them on the floor, then try putting them back into their categories.

* Even given that the categories are reasonable, for whom are they reasonable? The upper tier mental test is designed for those entering written tiers 4-6, 5-7 or 6-8. What distinguishes a pupil working at a particular level is not just level of knowledge, but confidence and flexibility in applying knowledge. Pupils at lower levels are more likely to be building methods from first principles, and this takes time. What they know and can do will not be credited.

Furthermore, one pupil may solve a problem using a quickly remembered trick, while another may take longer to build up and employ successful strategic thinking. One may argue that the second pupil's mathematics is more secure but, at the same time, it is more likely to be penalised.

These mental arithmetic tests are riddled with arbitrary decisions based on theories that have never been tested.

Working mentally is what mathematics is all about. Pupils need help to appreciate the perfect regularity and structure of our number system so they can develop confidence and competence in building and using calculating strategies. But this is a teaching matter.

Pupils' difficulties with calculating appear to centre around a dichotomy between the teaching of mental and written methods. The former are built on understanding the relationships between, and structure of, numbers. The latter are often recipes committed to memory.

There is a real danger here. Recipes committed to memory are prone to being forgotten. And written methods are sometimes taught in ways that contradict understanding acquired in working mentally. All calculations involve mental approaches, and how much needs to be written down relates to the complexity of the problem as well as what can be worked out in the head. Mental and written methods are inextricably linked. They are more helpfully understood as complements than compartments.

Having one assessment that presents calculations orally and disallows or discourages written work, and another that presents calculations in written form and often requires working to be shown, does little to promote this understanding. Separating the assessment of written and mental arithmetic will only compound difficulties in its teaching, with deleterious effects on efforts to improve performance.

Introduction of mental arithmetic tests into the statutory assessment arrangements should be postponed. Assessment of mental mathematics should be the subject of some rigorous and reliable research, and the possibility of incorporating such assessment into teacher assessment should be pursued with urgency.

Peter Lacey is senior education adviser to North East Lincolnshire Council

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