Leading view papers – Days 1 to 7

Making mathematics relevant and removing student boredom

 

 

First, let me make it clear that the ramblings of an ‘old pro’, which are about to follow, are specifically aimed at the state of mathematics education in Britain, although some points may be equally relevant to a whole range of subjects, if not education in general.

The main problem in Great Britain is the weak leadership from the very top that is allowing the examination boards to dictate what has to be taught, and the desire for votes at election time that is affecting how it is taught.  The examinations boards can be thought of as the ‘tail that wags the dog’! The exams at key stage three and General Certificate of Secondary Education (GCSE) are limiting the curriculum, as teachers spend more and more time preparing students for exams, rather than concentrating on developing a fuller understanding and thinking skills. We are in danger of becoming exam coaches rather than teachers of mathematics. Parents, too, have become seduced by the currency of the grades that the exam boards have created. A few weeks ago a parent asked me if I could give her son some private lessons. I interviewed the boy and reported back to his mother the areas that I thought he needed to focus on to improve his mathematics. Her reply was disappointing: ‘I don’t care about what he can or cannot do, just as long as he gets a C’. It is a message that I hear increasingly often. (I referred the lady and her son to another colleague!)

League tables are also harming the cause. School leaders are putting pressure on teachers to get ‘better’ results, but they are judging ‘better’ purely by those same exam grades of which I am challenging the validity. How many teachers currently have performance management targets that involve numbers of pupils passing a specified grade? How many teachers have a target focused on increasing the mathematical understanding of their students? I suspect the answer to the first question is far higher than the answer to the second. Teachers then feel pressure, causing them to resort to coaching for the exams rather than teaching mathematics. My firm belief is that if we consistently, and I mean throughout the children’s entire educational experience, put the development of understanding and thinking skills first, decent exam results will follow. In short, we should devise he right curriculum first, including decisions about the content, teaching and learning styles and resources, and then decide on the best ways to assess students’ progress. Let the dog wag its tail!

Current SATs, GCSE exams and, to a lesser extent, A-level exams have two main faults. Firstly, they are only attempting to test one, or at best, a limited range of intelligences. If we accept only part of Howard Gardner’s multiple intelligence theories, then we must also accept that maths exams are inadequate. I challenge you to pick up any GCSE maths paper and find examples of how a student with strong linguistic, musical, kinaesthetic, naturalistic, intra-personal or interpersonal intelligence profiles has chance to reasonably demonstrate their mathematical ability. Even the questions apparently aimed to test ‘shape, space and measures’ often resort to testing how a student can apply an algebraic formula logically. Because the exam is so limited, so have become our schemes of work and resources. Teachers are becoming coaches with the sole purpose of increasing the percentage of those magical C grades.

The second failing of the exams is that they often seem to be finding out what the pupil does not know or, even worse, merely what the student cannot reproduce in the exam, in artificial, stressful and unfamiliar conditions. The whole examination practice seems archaically primitive and could even be considered barbaric. Too many questions contain a twist in them: I suspect a deliberate ploy by the examiner to thwart the efforts of the exam coach (or teacher). For example, what is the purpose of a question that asks a student to find the area of a right-angled triangle given the hypotenuse and one other side? Surely it is to test the student’s knowledge of Pythagoras? So why make the question appear harder than it is by disguising it as a question about area? Numerous questions like this appear in front of the students every year. I do not accept the argument that the examiner is trying to test how the students can apply their mathematical skills, as there are far better ways of testing this.

There exists a third fault with exam questions, which, sadly also pervades many of our teaching resources: the dreaded contextualised question! The author attempts to make the question relevant to the student but, sadly, the decision as to its relevance lies wholly with the student and not the author at all. The majority of these questions may be irrelevant to the student, some may be totally alien and a few downright confusing. Consider this question:

‘Dervla is making a metal alloy by mixing copper and tin in the ratio 2:7. How much copper is needed to make 36kg of the alloy?’

How many students will not even attempt this question because they do not know what the word ‘alloy’ means; and where you or I may recognise Dervla as a beautiful Gaelic name, I have even heard students ask: ‘What’s a Dervla?’

Early in my career, I used to spend considerable time planning ‘real life’ problems for my students in an attempt to make mathematics relevant for them. Sometimes I was successful and sometimes not but that experience has taught me what I believe is an accurate working definition of what constitutes ‘relevant mathematics’.

Relevant mathematics is any mathematical problem or activity that engages the student.

Despite the dismal picture I have so far presented, I believe the simplicity of this definition is the key to a much brighter future for maths education and developing a meaningful and exciting curriculum and raising standards in terms of the students’ understanding, skill development and knowledge. The grades awarded to students would then be far more meaningful in terms of indicating their ability and capability. In order to turn this discussion in a more positive direction, I will give some examples of engaging, and therefore relevant, mathematics that have worked for me for students of different ages, abilities and attitudes to learning. I will then attempt to offer suggestions as to how we may better assess students.

Pentewan is a small coastal village close to my home in Cornwall. It has a small harbour that has been cut off from the Atlantic Ocean for nearly 100 years because the narrow channel leading to it has been silted up. Before the channel became blocked, it was a busy industrial harbour accepting trains full of English China clay to the quayside, where it was loaded onto merchant ships for export to the continent. The village is strangely served by three relatively large man-made reservoirs, linked to each other, and the harbour, by a series of locks. I accompanied a group of able 15 year old students on a trip to explore the history of the village. They became engrossed by the local people’s accounts of how, over a period of time, the prevailing winds had blown the sand and spilt clay from the loading of the ships back up the channel, until it became impossible for the merchant ships to reach the harbour. A reservoir was built so that it could be used to flush the channel and keep the harbour working. Unfortunately, the channel became blocked faster than the reservoir could be refilled, so a second reservoir was built to top up the first. Again, the local people underestimated Mother Nature and the channel was blocked faster than either of the reservoirs could refill. A third reservoir was added with the same result. The local people gave up their battle with nature and the harbour was abandoned, leaving the three reservoirs for the use of the tiny community. The students set their own questions: how much water had been used to flush the harbour channel? How long did it take to fill the reservoirs? How long did it take for the channel to become blocked? How large should the reservoirs have been? The next five weeks were spent on mathematical activities based around these questions: measuring on site, including using a theodolite, changing units, using map scales, trigonometry and Pythagoras, statistics, and more. Students even set up simulations using drain-pipes, sand and precisely measured amounts of water in a science laboratory. I could not hold back the smile from my face when the students demanded that I teach them about standard deviation. Maths was no longer being imposed on, or done to them: mathematics suddenly belonged to the students.

This example was time consuming and a major piece of work, but finding mathematics to engage students can be much less elaborate. Just last week I was teaching a bottom set of 26 13 year olds, including nine students on the register of special needs for behavioural problems. The syllabus demanded that I taught them how to add and subtract fractions. The first lesson was fairly traditional, with lots of diagrams and a fraction wall. The students did some work without being particularly motivated and probably completed about 10 questions each, with varying levels of success. I was certainly not satisfied that the students had internalised, or even understood the concept. Those who had experienced some success did so by just repeating the algorithm that I had presented to them earlier in the lesson. Before the next lesson I made up a game for two players taking it in turns to flick a counter onto a target board (like tiddly-winks). The target board was just a piece of A4 paper with a number of areas marked out, each instructing the player to add or subtract a fraction such as a half, quarters, thirds, sixths and mixed numbers, the highest of which was 1¾. In the next lesson, I gave the instructions for playing the game; first one to score 10 or more wins; paired the students off (carefully!) and let them play. It was a very noisy lesson, but the noise included students checking each others additions and subtractions (albeit to make sure their opponent wasn’t cheating) and confident students explaining to the less confident what to do. With five minutes left of the lesson, I estimated that students had, on average, completed about 40 questions each. I restored calm to the classroom by setting two test questions for my plenary:

⅔ + ⅝ and 2¼ - ⅓

The questions were done in silence, students working on whiteboards and showing their answers in unison after three minutes. Two of the students made a mistake on the second question, otherwise all responses were correct. The game was relevant to the students so they had wanted to learn the mathematics necessary. I had previously used a version of this game with two 11 year old students with severe learning difficulties, which meant they were unable to add one to a number without using bricks or number lines. For these, the target areas contained the instructions to either add or subtract 1, 2, 10 or 20. I started by giving them a target of 50, increasing it slowly up to 100 as the students played the game repeatedly, gaining in confidence as they went on. By the end of the lesson, they could both mentally add to or subtract 10 or 20 to their total with confidence.

Other activities include students working together to build (amazingly accurately!) a scale model of the school; using graphical calculators with a motion detector; role plays to introduce problems; students devising their games, such as vector football, and strategy games using pentominoes; students preparing lessons to teach each other, and colouring by numbers to practice topics such as percentages, fractions, averages, equations, and so on.

Teachers do not need to spend hours searching for these creative ideas; there are plenty already in existence. The Task Maths series by Barbara and Derek Ball was rich in activities that have successfully engaged my students and Exeter University’s Maths Enhancement Project contained some wonderful ideas amongst its support materials. I believe, to genuinely raise standards and make mathematics relevant to our students, we need to build our schemes of work around these types of activities. This will mean breaking away from the restrictions of the current schemes of work and resources that, in many cases, are written by the same examiners whose exams are so flawed.

So, how should we assess students? Coursework! I can already hear your groans. I am not talking of the coursework that has been imposed on us since the advent of GCSE maths, which has failed to achieve its purpose at phenomenal cost. I mean coursework in the true sense of the word - no special tasks. Let’s just look at the student’s workbooks they have used every lesson throughout their course. I feel confident to judge what grade any student of mine is capable of, and deserves. I can base my judgement on a wide variety of different types of work that the student engages in, including written, oral and practical work over a long period of time: something no exam has ever managed. If we trusted teachers’ judgement, there would be no need for exams as we know them. Think of the money that could be saved and reinvested more positively in education. To support this strategy, we should also dispose of the current measures used in league tables for schools. Teachers’ opinions can be supported by test results, but these would be very different from today’s experience. I see the tests as possibly being just one question. For example, two sets of data could be given and the student may simply be asked to write a report comparing them. Credit would be given for the range and appropriateness of the techniques used to compare the data. Students with less ability might only draw a bar chart, some may work out averages, while the more able might include calculations for standard deviation. The important thing is that we would be assessing the student’s ability to think mathematically and we would not be concerned with what the student doesn’t know. The test should last a stated maximum time, say 90 minutes, with students free to leave whenever they were satisfied they had finished to the best of their ability. Another test may ask the student to estimate the height of a building or tree in the school grounds. The answer would be far less important than the process the student chooses to use. These tests could be administered and marked within the school, and the results used to support the teacher’s judgement.

My arguments and solutions are simple but probably not politically acceptable but if you have managed to reach the end of my ramblings, I thank you for giving me your time.

ABOUT THE AUTHOR

Mr Adrian Smith is a mathematics teacher at Penryn College, in Penryn, Cornwall, in the United Kingdom.

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