Errors and misconceptions
Teachers, especially of very young children, have a real aversion to telling pupils they've come up with wrong answers. It's a trait you hear satirised in staffrooms. "Two plus two, Camilla? Five? Yeeees, well, OK, that's a really good try... Anyone else now?"
It's easy to see that the intention is to be positive - to avoid stifling early enthusiasm for a subject full of traps and tricks, one where 0.25 turns out to be smaller than 0.5. There's probably another imperative at work: the teacher quickly seeks out a child who has the right answer for fear of the error becoming embedded.
However, as researchers have known for a long time, the mistakes that children make in maths can be important. Properly observed and studied, they provide starting points for further learning. The difficulty for teachers in the flow of classroom life is to understand what the various errors are saying to them.
There are errors and errors, of course. Everyone miscalculates under pressure, or out of boredom. The important ones for the teacher are those that signal misconceptions - the assumption, for example, that a rule that's seemed to work well up to now will continue to operate in a new set of circumstances: 25 has always been bigger than 5, so why should it be different just because the digits are now to the right of a decimal point?
Important here is the idea that a child is not the open and all-welcoming tabula rasa of Locke and Rousseau, but rather a thoughtful being who brings to each new proposition a set of theories constructed from experience.
These are often entrenched, committed beliefs - rules that the pupil has invented and therefore owns. That's why they're difficult to shift, and also why academics now dislike the negative word misconception. Julian Williams, professor of maths education at Manchester university, uses the phrase "potential windows into the child's mathematics" to describe them.
Nevertheless, this is where the most effective sort of teaching begins: patiently engaging with the child's existing ideas, perhaps over a considerable period. All of 10 years ago, an Ofsted summary of research into maths teaching recommended a teaching style that constantly exposes and discusses pupils' misconceptions in order to limit their extent.
If that doesn't happen - if a problem is ignored, or simply corrected by a one sentence exhortation, it leaves the child knowing something is wrong but lacking the necessary understanding to put it right. Hand on heart, any teacher knows that this scenario is very common. "Look, love, trust me - just remember that 0.5 is bigger, OK?" Consequently it's not difficult to see how, and why, so many people grow up not only with gaps in their maths knowledge, but with an attitude to the subject that varies between suspicion and outright hatred.
"Errors and misconceptions are often intelligent generalisations," says Julian Williams, "We really worry about teaching that simply ignores them."
To tackle misconceptions, then, they first need to be identified and understood. This means looking in detail at a child's mistakes and seeing within them the patterns that betray the underlying assumptions. This is where diagnostic assessment comes in.
A team at Manchester university, led by Professor Williams, has produced a comprehensive set of maths tests that can be used to identify and analyse the patterns of mathematical misunderstandings. Armed with that knowledge, teachers, rather than passing over children's errors, can bring them out to be dwelt upon and discussed. And it's here that Professor Williams and his team have been focusing their attention. Their project has produced materials that make up Mathematics Assessment for Learning and Teaching (MaLT), published by Hodder Murray.
At its heart are test questions, divided up into 10 stages covering the years from Reception to Year 9. The tests are able to do all the conventional summative functions - they're matched to the primary and secondary strategies, and calibrated to establish national curriculum levels subdivided into upper, middle and lower. They will also provide externally norm-referenced, age-standardised scores. There's much more to MaLT than that, though. It's the programme's formative, diagnostic capabilities that Julian Williams sees as highly significant.
"What I hope personally is that this is a tool that can take teachers on from where they are," he says. As the test manual puts it: "We attempt to diagnose a child's mathematical misconception from their erroneous response to a question, rather as a doctor might diagnose an infection from the symptom of a high temperature."
In this case, the temperature-taking lies in the way the test questions - and the mark scheme and record sheets - are devised so that they reveal patterns of errors that can then be tackled in class.
There's a clear role for ICT here, and MaLT offers two possibilities.
First, with the paper and pencil test package comes a CD-Rom with Scorer-Profiler software into which test results can be entered to produce analyses and summaries. Beyond that, the whole of MaLT also exists in an interactive computer-adaptive version as well as on paper. Given the increased flexibility and ease of testing, marking and data-handling it's difficult to see why an ICT-competent school would choose not to use it.
Putting diagnostic assessment online or on a school network opens up possibilities for the teacher, not least in terms of immediacy - the ability to discover error patterns in individuals and groups, analyse them and devise ways of tackling them all within a very small timescale.
Good diagnosis is only part of the story. The big question is whether, and how, a teacher will be able and equipped to deal with the consequent teaching and learning demands.
It's no small issue this. Julian Williams says, "In all the videos of UK classroom mathematics teaching we have seen in the public domain in recent years, we have been overwhelmed by the way they show teachers correcting errors.
"We take a contrary view, that because misconceptions are often supported by a child's generalised reasoning from experience, they require a different treatment which respects their intelligent behaviour."
Colin McCarthy, a consultant who's worked extensively on assessment issues with the Qualifications and Curriculum Authority and with Goal, an online exam body, believes that if teachers are to make proper use of the power of good diagnostic assessment, they will have to make a culture shift.
"There has to be a willingness to alter the teaching and learning agenda to match the needs of the learner," he says. "The people who most need to be consulted are the ones who aren't consulted, namely the learners.
Historically, teachers don't operate in that way very much."
How MaLT works diagnostically
MaLT provides a battery of test questions that are written - and standardised nationally - to highlight those particular errors and misunderstandings that signal key learning needs. If a number of test items taken together show that a child can count a row of objects, can also combine two groups of objects, counting them all together, but cannot say how many more objects are in one group than another, then there's good evidence of where the child is in terms of number concepts.
In MaLT 12, for Year 7, there's a question that asks, "A film on television starts at 8.30pm and lasts for 1hr 40ms. At what time will the film finish?" The mark scheme indicates that this question addresses the ability to "problem-solve in contexts involving time". It gives, as well as the correct answer (10.10 or 22.10), two sample wrong answers with comments about the underlying error. So we have "9.10 - made a carrying error in adding time." And "9.70 - added times as if 1hr = 100 minutes." Each type of wrong answer carries a code number, so you can discern patterns of errors showing areas that need to be tackled either with individual children, with groups or with the whole class.
First Steps in Mathematics: An australian approach
An alternative - or complementary approach - to the paper- and-pencil test is to give children diagnostic tasks. The teacher observes the children at work, discusses the content with them and then plans the learning activities that will help them to move on. That's the approach being used in Western Australia, after a five-year research programme showed that more than 60 per cent of 14-year-olds were producing the right answers but for the wrong reasons - a surefire formula for later frustration and failure.
The teaching programme that emerged - First Steps in Mathematics - is now published in this country by Steps Professional Development. The emphasis is on children talking about mathematical tasks - with partners, in groups and with their teacher. In one illustrative example in the Steps handbook, children are asked to measure a desk, using pens, and then to find out whether the desk would fit through the classroom doorway. The way they tackle this - whether or not they fit the pens end to end, for example, or remember to count the pens, or simply make a guess - provides insights for the teacher, who is then given appropriate teaching strategies.
Sue Dean, manager of Steps Professional Development UK, says, "We've found if you can help children to talk through their mistakes, they start to see for themselves where they were misunderstanding, and then begin to rectify it for themselves."
It's early days for First Steps in the UK. Professional development for teachers - it offers a three-day course - is an important component of the programme. Spreading the word takes time, especially when teachers are so preoccupied with the literacy and numeracy strategies, but Jeff Darby, head of Kingshurst junior school in Solihull, one of the UK research schools, is enthusiastic. "I'd been looking for something that would boost and support teachers' subject knowledge and would also cover the gaps in the national strategy. This is a very powerful resource, a commonsense guide to teaching maths, strong and easy to use. It supports the national strategy. It will support any curriculum in fact - it underpins it."
WHAT PUPILS MISUNDERSTAND
These are examples of children's misconceptions. All can be identified by diagnostic testing or by careful discussion.
Perhaps the commonest misunderstandings are associated with this idea, which involves taking in several concepts at once - that there are 10 digits, that the position of each digit tells its value, that zero is a place holder, that once 10 is reached in a column it is replaced by one in the next column.
Failure to understand it means pupils might make the following mistakes:
* Read 302 as 32 ("30, 2 = thirty-two") .
* Over-generalise the rule "to multiply by ten, add a nought". So using it on decimals: 4.5 times 10 becomes 4.50
* Treat the digits each side of the decimal point as separate whole numbers. So 3.4 plus 2.7 becomes 5.11
Orientation of shapes
Failing to recognise shapes that aren't printed as children have usually seen them - triangles with the apex towards the bottom of the page and the baseline at the top, triangles that aren't isosceles or equilateral, squares that are presented as diamonds.
Believing that a tall slim container holds more fluid than a short fat one, even if the child has seen the fluid poured from one to the other.
Failing to see the relationship between the hands on an analogue clock.
Reading a digital display in the way we commonly state time. So 10.05 becomes "ten past five".
What's to be done?
Whether you use MaLT, or First Steps (see page 10) or any other programme:
* Brush up your own subject knowledge. Read the support material for whatever scheme you use, and read professional books about your subject.
* Regard errors as windows into the child's learning.
* Take time discussing errors. Don't just correct them and move on. Be prepared to let them linger on the board as you and the children talk about them.
* Tease out misconceptions. Do the children understand place value? The way shapes are presented?
* Although maths progresses naturally through a hierarchy of concepts, be aware that children will extend early knowledge inappropriately into later learning - hence the many misunderstandings about decimals.
* When it becomes clear what's going wrong, your professional task is to devise learning that tackles the underlying concept. As Jeff Darby, head of Kingshurst juniors in Solihull, one of the UK research schools for First Steps, says, "A page of sums won't do it."
First Steps www.steps-pd.co.uk. There is no maths info on the website, so email email@example.com or Tel 01793 787930
Children's Errors in Mathematics: Understanding Common Misconceptions in Primary Schools Edited by Alice Hansen Learning Matters: Errors and Misconceptions in Maths at KS2 By Mike Spooner David Fulton, pound;13
KEYBOARD CONTROL MAKES DIAGNOSIS EASIER
Moving diagnostic assessment on to a computer has overwhelming advantages.
Pupils are more motivated, questions are more interesting, marking and analysing results is easier by several orders of magnitude. Small wonder that not only are paper tests migrating to CD-Rom andor the internet, but new on-screen tests are appearing all the time. Here's just a selection of what's on offer in addition to the material we've already covered in these pages.
Achieve from Harcourt Education is an interactive assessment for learning package devised in conjunction with Cambridge Assessment (formerly Cambridge Local Examinations Syndicate). Maths, science and English packages were launched at the BETT education technology show last month.
Jelly James is a simple and very easy to use assessment package in maths and spelling for Year 2 and Year 6 that's highly spoken of in the primaries that have used it. For Evelyn Donnelly, head of Shaftesbury primary in Shaftesbury, Dorset, the main attraction is that it provides children with instant feedback on their performance, spurring them on to make progress.
"The results were exceptional," says Ms Donnelly.
An assessment package from NFER is always going to be interesting. The number of e-assessments it is launching for key stages 1 and 2 shows evidence of its experience and academic expertise. The maths package has an impressive facility for analysing errors and misconceptions, not just at the level of what mistakes were made, but showing, for example, whether a child made a mistake before getting the answer right, or whether the help screen was used. The result is to give the teacher a huge amount of information about individual children's starting and sticking points.
At Granada Learning. NFER Nelson is making its widely used paper tests available in digital form - hence Progress in Maths 6 - 14 digital.
Although clearly based on the paper test, this is more attractive to look at it, and mathematically more sound.(For example, in this - and in other on-screen tests - angles can be truly shown as a measurement of rotation rather than as a static and ambiguous pair of lines.) www.testwise.net
Hodder Murray has published e-assessment materials for science and English.
Hodder Science Interactive Assessment, which complements the Hodder Science for KS3, is designed to help pupils explore difficult concepts and misconceptions. (There's another version for non-Hodder Science users.) The diagnostic Edinburgh Reading Test 4 - interactive is for secondary pupils and adults with reading difficulties.