Victoria Neumark reports
How much do people at work use tra-ditional maths learned in school? Not very much, to judge from a survey conducted recently by London University's Institute of Education. Twenty-eight nurses at a major London teaching hospital took part in a research awareness course aimed at finding out how they model the data they deal with every day. The results are surprising.
Nurses are the early warning system on a ward. They monitor and diagnose before doctors, which means they must be able to interpret graphs, charts and tables while they look for physical symptoms in patients. It can be crucial to know if a patient's temperature is "spiking" because she always gets hot in the evening or because she is reacting to a new drug; to know when it is necessary to change to a different drug regime; to grasp which lab results must be passed on to a doctor immediately and which can wait. Furthermore, these results come in different forms: in tables, graphs, words, charts. In fact, the recorded description of a patient's condition is partly mathematical.
In their nursing education, pre-registration students learn maths as part of pharmacology, but there are no mathematical statutory requirements. Is this desirable?
Nurses approach numbers from a clinical rather than a mathematical viewpoint; they are dealing with people, not sums. They may be unconfident mathematically, in contrast to doctors, who typically are better qualified in academic maths and science. Whether or not they have a sound grasp of "school maths", nurses tend to use intuitive and informal strategies which involve mental estimation and calculation rather than written algorithms.
Such strategies can work well enough in what Celia Hoyles, one of the authors of the study - funded by the Economic and Social Research Council - calls the "web of mathematics in life", but they can never be as reliable as a solid mathematical formula.
Professor Hoyles, Professor Richard Noss and Stefano Pozzi observed what they call the "ethnography of maths" over 80 hours on busy paediatric wards. They saw nurses juggling quantitative data with other knowledge effectively, but with little apparent awareness that what they were doing was mental maths.
In response to patients' questions, nurses often use the phrase "it depends" - referring to such factors as age, size, weight, state of nourishment and so on. Blood pressure and age, for instance, are well known to correlate; the rule of thumb for systolic pressure is Age+100. But what does a correlation mean, and how does it relate to real-life data? When the nurses learned how to present the data in a scattergram, they were disappointed by what they saw as "weak" relationships. In this form, they felt the variation dominated the overall relationship. They could not analyse the unfamiliar data model. Why?
From a statistical point of view, modelling any data should reveal variation; a set of identical values would be surprising. But, for nurses, who are thinking clinically, each substantial variation in a patient's vital signs has to be explained in order to make judgments about a patient's condition - and hence treatment.
For instance, a patient's blood pressure may rise as a result of overactivity. From the point of view of statistics, this one extreme reading will be an "outlier", and not representative. In daily ward rounds, though, a nurse has to distinguish different explanations, such as drug intolerance or a worsening illness, or even malfunctioning instruments.
Narrowing the mathematical focus of their survey, Hoyles, Noss and Pozzi used blood pressure charts of a fictional girl patient - Emma, aged eight - to discuss notions of the average with nurses. The results show a variety of informal strategies.
Only one of the nurses used the traditional school algorithm of adding all results and dividing by the number to get the mean or numerical average. The majority used the median (midpoint) or mode (most frequent) values to arrive at an "average" (guessed) reading.
The nurses said things such as: "I looked at the chart and judged which was the middle range"; "at a glance all the values are around that point" (median); "most of them are at 106" (mode). One gave garbled voice to confusion between algorithm and practice, switching her explanation from the mean - "You are asking me for the average aren't you, soIyou could add it all up and divide by the number of hours or whatever" - to a mixture of the median and mode..."perhaps I look to see how often she is being just underIhow often she is a little bit elevated".
A mathematical notion of the mean was co-opted into a broad clinical understanding invoking the "baseline" of a patient's physical condition from which blood pressure might be "under" or "elevated", adjusted for time of day and activity.
Such concepts of "normal" and "baseline" related only loosely to wider knowledge of expected readings for wider populations. As one nurse said, blood pressure of 100 over 50 could be normal for an eight-year-old if she had been running around, but if she was in bed all afternoon, then she would need more tests to establish how ill she was. Doctors, on the other hand, with different clinical and mathematical backgrounds, used a population average as a marker: is this reading normal for an eight-year-old? The nurses, with their hands-on experience, based judgments on their knowledge of the individual child: is this reading normal for this eight-year-old?
Nurses see their role as caring for patients, not populations. Their approaches to dealing with averages - to look at the midpoint, to focus on the mode - are not mathematically general, but can be clinically valid when supported by other knowledge.
Professor Hoyles says: "You don't go to the world of maths to make your calculations; you stay in the world of work and do your calculations there." Whether it be brick-laying or investment banking, number operations learned in the classroom will play second fiddle to other expertise: how much mortar for how many bricks depends on what kind of bricks and mortar; how far to venture capital depends on how bold the client is.
How can this practical approach connect to a greater mathematical understanding? There is no simple link between abstract and complex statistical concepts such as randomness, distribution and probability and the world of work. Yet real mathematics - for instance, analysing the significance of a distribution which is not symmetrical - could greatly enhance practice.
Since maths in the workplace is used idiosyncratically, vocational maths teaching has to be aimed at practice, however imperfect. Teachers must build bridges between the actual situation and general mathematical formulations (like how to calculate an average).
One method might be to check informal strategies, such as estimating modes and midpoints, within both class and work situations. One student would use intuition and another a formal procedure built into a calculator's software. Going through the checking process will make more impact than straight introduction of a formula.
Stefano Pozzi says: "I wouldn't necessarily undervalue algorithmic methods, only say that for safety purposes one would not want this as the only method available." Better a partial strategy which works correctly in a particular situation than an algorithm which is wrongly applied.
Above all, says Professor Hoyles, maths educators and policy makers "need to learn how maths is actually used and not assume it is according to instructions".
For more details of this unpublished research, contact Celia Hoyles at 0171 612 6651. E-mail: firstname.lastname@example.org