The Application of Mathematics: Teachers Proclivities
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The application of mathematics: teachers’ proclivities and curriculum opportunities/constraints
John Monaghan & Louise Sheryn University of Leeds
Paper presented at the British Educational Research Association Annual Conference, University of Glamorgan, 14-17 September 2005
Please note that this is merely a working paper which supports a symposium presentation.
Abstract The application of school mathematics to everyday and work settings is one of the main rationales for the place of mathematics as a core subject in the National Curriculum but little is known about how mathematics teachers introduce applications in their classrooms. This paper reports on one aspect of a wider study which investigates teaching and learning issues arising when secondary mathematics teachers design learning activities so that links between school mathematics and out-of-school activities are made manifest. Two datasets are introduced. The first is questionnaire data on how secondary mathematics teachers view various issues surrounding the application of mathematics. The second dataset is a set of interviews with eight teachers who planned to link school mathematics to out-of-school mathematical activities. We explore tensions, constraints and opportunities with a focus on teacher planning.
Introduction We ‘set the scene’, consider how the application of mathematics may be taught and then introduce the project we gather data from.
It is, to us, a constant source of amazement that mathematics, the ultimate cerebral discipline, has such wide application in the world. This is rarely mentioned in articles or books but it is worth taking time to consider how is such a product of mind applicable to the material world? The answer undoubtedly lies in mathematics being a social product of people’s interaction with the world (which is material). We simply note this wonder and move on to educational matters. The application of mathematics is, to us, using mathematics to make sense of ‘real world’ phenomena: anything from checking your change in a shop to advanced physics. We briefly and selectively survey UK educational policy initiatives designed to ensure that school students meet such use in their studies.
Cockcroft (1982) saw the application of school mathematics to everyday and work settings as “sufficient reason for teaching mathematics” (ibid., paragraph 1). A famous proposal from this report, though not exclusively targetted at application issues, is its ‘foundation list of mathematical topics’ which all pupils should cover. Such ‘content’ solutions rest on the assumption that content learnt in mathematics classrooms ‘transfers’ to out-of-classroom settings.
The National Curriculum acknowledges that “Mathematics is important in everyday life, many forms of employment, ...” (DFEE, 1999, 14). Its solution is using and applying, a process curriculum. A product of this philosophy appears to be a perception that setting the
1 1 1 question ‘ 2 3 ?’ as ‘Louise has half an apple and John has a third of an apple. How much of an apple do they have between them?’ is somehow applying mathematics. Cooper & Dunne (2000) discuss problems that pupils experience in playing such ‘maths games’.
The now dead Tomlinson Report (DfES, 2004a), the Smith Report (DfES, 2004b) and the 14- 19 White Paper (DfES, 2005) propose ‘functional mathematics’ but no one really knows what this means and there are suggestions (see Roper et al., to appear) that a battle between functional mathematics as basic skills and functional mathematics as modelling is ongoing.
Turning from policy to research does not immediately help teachers as most research on the use of mathematics in out-of-school activities shows a strong discontinuity between school and out-of-school mathematical practice. According to the situated cognition paradigm, e.g. Lave (1988), this discontinuity is a consequence of the fact that learning in and out of school are two different social practices. Further to this, school mathematics is often ill-suited to out- of-school practices. In some cases the problems are only apparently similar to school mathematics problems, but in reality there is a range of explicit and implicit restrictions which makes school methods unsuitable.
Despite the evident discontinuity between school mathematics and out-of-school practices some authors have observed an interplay between them. Saxe (1991) found evidence that school mathematics and the mathematics of street children’s candy-selling practice in Brazil affect each other. Pozzi, Noss and Hoyles (1998) found cases of nurses looking for a mathematical explanation for the conceptually simple mathematical procedures used in their daily practice. Magajna & Monaghan (2003), in a study of technicians designing moulds for bottles, found evidence that in making sense of their practice the technicians resorted to a form of school mathematics.
It is against this ‘optimistic research’ background that our project is set. We basically wish to understand how learning activities can be designed so that links between school mathematics and out-of-school activities are made manifesti. An assumption is that such links are unlikely to arise without learning activities being designed to engender this links. Further to this learning activities cannot be separated from the participants (teachers and students) and resources involved in using them. This project thus has simultaneous but distinct foci on tasks, resources and participants, teachers and students.
Of the participants, teachers are central agents in making school mathematics relevant to out of school activities: they direct students activities and students ‘privilege’ the techniques and resources that are privileged by their teachers (Kendal & Stacey, 2001). Many mathematics teachers, however, do not perceive that there is a problem in the application of mathematics – and if no problem is perceived, then involvement in this proposed project is unlikely to affect their practice.
Secondary mathematics teachers in the wider Leeds area were invited to apply to be involved with the project and eight were selected. We make no claims about the ‘representativeness’ of these project teachers, all of whom teach in comprehensive schools. We restricted the number to eight simply because of the intensive nature of the work we expected to do with them and their classes. Teachers were free to pursue any link with out-of-school mathematics that wanted to do with any of their classes. The project had several stages. One stage was an initial survey of a wider sample of teachers views on making maths applicable. This is reported on
2 in the next section. The principle stage is working with these teachers over two years on classroom activities. At the time of writing the project has just started on the second year. Data collection is extensive but we do not detail this here as the next two sections deal with data collection relevant to this paper and further details may be obtained via http://www.education.leeds.ac.uk/research/cssme/outofschool.php
The questionnaire results Given our project with a small number of project teachers who would try to link school mathematics to out-of-school mathematical activities we wanted to get an idea of how other teachers did this. To this aim we designed a questionnaire which was sent to Heads of Mathematics Departments of the 43 LEA secondary schools in Leeds. We chose Leeds because that is where we live but Leeds does have some claim to representing the nation’s education: Ofsted (2000) states that Leeds is “in many ways a microcosm of national diversity” (para. 4) and that Leeds’ performance is about level with national statistics. We obtained returns from 25 questionnaires. This doesn’t make our sample perfect but it provides some indicators of how issues of linking school mathematics to out-of-school mathematical activities are approached in schools. We now describe the questions asked and summarise teachers’ responses.
1) Teachers were asked if school mathematics should attempt to make links with some real life or out-of school activities? (citizenship, leisure or the world of work). All said ‘yes’. They were then asked if this is a difficult thing to do. 15 said ‘yes’, 7 said ‘no’ and 3 were unsure. 6 of the 15 who thought it was difficult had suggestions for how to address the problem: (i) give teachers time to prepare (ii) give teachers time & money (iii) write better textbooks/materials (iv) bring in professionals to talk to pupils (v) arrange after school or lunchtime activities and invite a speaker to the school (vi) capture students’ imagination.
2) Teachers were asked if linking school mathematics to out-of-school mathematical activities is best achieved by doing work in lots of different out-of-school areas or picking just a few and working on these areas in depth? 10 said ‘a few activities’, 7 said ‘lots of different activities’, 1 said a bit of both, 3 did not respond and 4 gave sentence answers: try to relate all maths to real life situations; work on a few to start with but develop new ones to widen the range; linking cross-curricular always, reinforcement; whatever you can get.
3) Teachers were asked if their school offered vocational courses/modules, e.g. GNVQ, key skills and, if so, were asked to provide brief details. All but 1 school provided vocational courses or modules. 2 appeared to offer to only offer these at 16+. 1 offered an ICT GNVQ (the Thomas Telford model).
Teachers in schools which offered vocational courses/modules were asked if mathematics teachers teach on these vocational courses/modules and, if so, were asked to provide brief details. Of the 24 in schools with vocational courses/modules: 17 said that mathematics teachers were not involved, 6 said mathematics teachers taught mathematical components and 1 (the ICT GNVQ) stated that GCSE mathematics formed the mathematics component. Of the 6 who taught mathematics: 1 taught 16+ Key Skills Application of Mathematics, 1 taught 16+ Key Skills Numeracy, 1 taught 14-19 Key Skills Application of Number, 1 taught 16+ Application of Number and 2 taught GNVQ Business Studies.
3 4) We were interested in the amount of time teachers devoted to applications/links and whether this varied over age and type of course. Teachers were asked what proportion of class time, to the nearest 20%, is devoted to mathematics with links to out-of-school activities.
Key Stages 3 & 4 5 schools said that they did not devote any class time within KS 3 & 4 to mathematics with links to out-of-school activities. 10 schools said they devoted less than 20% to this type of mathematics at this level and a further 8 schools indicated they devoted 20% of their class time to these activities. 2 schools said they devoted more than 20% (30 & 40%) to applications of mathematics at this level.
Vocational courses/modules All but 1 school that responded taught vocational courses or modules and yet only 3 schools indicated that they devoted any class time to linking mathematics to out-of-school activities (50, 40 and 20%). This appears strange but it be may be that teachers within the mathematics department are not involved with vocational courses.
AS/A2 21 of the schools who responded had a sixth form and 8 of these said they devoted some AS/A2 class time (from 5 – 20%) to links to out-of-school activities. 1 who said they did not devote time to this commented that there was “not enough time” but that some practical work was included in Mechanics and Statistics.
The teachers were also asked if there were any groups of pupils for which links to out-of- school activities is particularly important or unimportant and any real life areas that should (or should not) be covered? 11 respondees believed that making links within mathematics classes to out-of-school mathematics activities is important for lower ability students and 7 who believed that all students should be taught with a focus on these links. 2 schools felt that ‘disaffected’ or work-related students would benefit most from this approach and 2 schools who felt gifted and talented students would benefit from this approach to mathematics. Real- life areas that teachers said should be covered included personal finance, time and timetables, measurement, percentages and ratios. Others referred to ‘the world of work’, ‘living skills’ and ‘practical numeracy’.
5) Teachers were asked a series of questions on resources and activities. a) Do you mainly use textbooks or worksheets with: Key stage 3 23 used textbooks, 2 used worksheets. Key stage 4 24 used textbooks, 1 used worksheets. Vocational courses/modules Only 5 teachers responded: 4 used worksheet, 1 used a textbook. AS/A2 All schools with sixth forms used textbooks. b) What written resources, if any, do you use when pupils do schoolwork linked to real life applications? 17 said they used teacher produced worksheets, 7 said they used their usual textbook and 1 said they used specially produced commercial material. c) Does your Department use any special resources for linking school mathematics to out-of- school mathematical activities? Anything from surveying equipment, to catalogues to computer software, please specify. 13 said they do not use any special resources. 4 had a
4 supply of holiday and Argos catalogues and 2 of these said they used them with lower ability students. Other resources mentioned were trundle wheels and clinometers. d) Does your Department arrange any special activities designed to help pupils appreciate the use of mathematics out of the classroom? e.g. joint work with other Departments, visits or field trips? 10 teachers said that they did. Activities included: visiting speakers for Yr 11; a day of mathematical activities organised by the RAF; maths trails; trip to a Science centre; joint work with the Geography department.
Initial interviews with project teachers Data relevant for the focus of this paper includes a roundtable discussion with seven of the project teachers and the authors in July, 2004 (one was ill on the day) and individual interviews with teachers in their schools in the period October – November, 2004. The roundtable discussion allowed teachers to outline and share their initial ideas. The discussion was audio recorded and transcribed. The individual interviews started with an open question “Tell me about your plans”. Follow up questions on the following areas were asked if these were not addressed in response to the open question: motives; perceptions of student gain; source of ideas; curriculum opportunities and constraints. Interviewers also referred teachers back to their July plans and asked them about reasons for changes in their plans.
The authors shared the following work: conducting, transcribing and repeated reading of the interviews (and making and discussing notes as we did this). This immersion in the verbal/written data generated four important themes: curriculum/assessment pressures; planning project work around mathematics content or around a theme; use of out-of-school people; the role of technology. We briefly address these in turn.
With regard to curriculum/assessment pressures, English High Schools students take high- stakes examinations and this can create very real tensions for teachers. We illustrate this with one teacher who wanted to do project work with a class of 14-15 year olds. The class varies in mathematical attainment but most are ‘disaffected’, i.e. do not like school. The students have chosen to follow a ‘vocational curriculum’ but the mathematics element of this curriculum is the same as other 14-15 year olds in England follow. The teacher feels that these children would be more motivated by application of mathematics project work but also has a strong moral sense that she must get the best examination results possible for her students. To compound this dilemma she is also aware that focusing on work directed at obtaining good examination results for her students is highly likely to ‘turn her students off’. She speaks of: … a real dilemma in terms of meeting the needs of the pupils, in terms of attainment when they leave school and delivering the curriculum in the way that they’re going to be examined …I don’t see that they match up… knowing all these pupils, knowing the set ups of their days, knowing the time scales we’ve got, knowing the disaffection that some of them are coming with, I’m not so sure that I can achieve both needs
With regard to planning project work around mathematics content or around a theme a spectrum of views exists. The poles of this spectrum are: plan around an out-of-school theme and see what mathematics it generates; plan around an area of mathematics and choose a theme (or themes) that appears appropriate. An illustration of the former is: I’d like to do the project and see what maths comes out of it rather than say, now, there’s going to be some work on … but I don’t really want to do that … what I’d like
5 to see is, you know, talk to designers… and find out what maths do they use … rather than my preconceptions An illustration of the latter is: I’d rather take a small sections of work and group that together around … a concept rather than take a project itself which is extended. So maybe look at ratio as a topic and then try and work into that quite a few different aspects maybe different work areas and see if we could have a look at that in that way.
With regard to use of out-of-school people all but one of the teachers planned to consult with workers in various fields and, if possible, to bring these people into school to speak to their project classes. In should be noted, however, that we, at the July meeting, suggested that this may be a fruitful activity. The range of people teachers were approaching varied from manual workers such as carpet layers to stockbrokers. In each case the teachers expressed their belief that their students would be interested in the work of these practitioners. The one exception was a teacher planning to use the music software ‘Cubasis’ for students to write their own music. This is his main hobby and he did not feel the need for external expert help.
With regard to technology teachers’ considerations about the use of technology in their project lessons varied considerably. Some teachers, like the ‘Cubasis teacher’ noted immediately above, placed technology at the heart of their project work plans. Another teacher who did this was looking into using a computer package ‘Prozone’ which is used by professional soccer trainers to analyse players’ performance. In contrast to this one teacher, who did not mention technology in the interview, was asked about what resources she might use. She responded: You see my resource is the motivation of the pupils. It’s very much driven, I personally think, to what I think will engage them. The actual hands on money or equipment resource is very peripheral. If necessary we get some pipes or tubing or all sorts of other things. It’s the holy grail of total engagement that I’m after.
Not all teachers were at these extremes of use/non-use of technology. One teacher was planning to get his class playing the stock market. He intended to use a website in which his students could set up dummy portfolios but this technology was not at the heart of his planning.
Discussion We discuss five issues that strike us as important: the peculiarities of the English education system; a teacher’s dilemma; does the mathematics determine the application (or is it the other way around?); resources; out-of-school people.
Peculiarities of the English education system We take it as given that any national (or even local) education system will have peculiarities which enable and constrain aspects of teaching and learning. With regard to the application/links focus of our study a peculiarity of the English system that appears noteworthy is a high stakes examination (GCSE), a perception of the need to teach to this examination and the fact that this examination appears to try to put everything into a pseudo- 1 1 context. We alluded to this earlier when we mentioned that ‘ 2 3 ?’ is likely to be set as ‘Louise has half an apple and John has a third of an apple. How much of an apple do they have between them?’ Another example, obtained through a quick skim through a popular GCSE textbook, is:
6 7 The costs, C pence, of an advertisement in the newspaper is worked out using the formula C 25n 40 where n is the number of words in the advertisement. a Joy puts in an advertisement of 16 words. Work out the cost. b The cost of Debbie’s advertisement is 540 pence. How many words were in Debbie’s advertisement? (MEG, Paper 2F, Q6, 1998)
We call it a pseudo-context because it is totally invented for the sake of doing some algebra – actual newspaper advertisement prices are based on pages or single column centimetres! So the scene is set: Teachers, you don’t need to address real applications of mathematics, just make up some, that’s what your students will find in the examinations.
This example, although from a specimen paper, was taken from a textbook. This textbook has sample examination questions (under topics) at the end of each chapter (quite a common feature of textbooks now). Some textbooks are even called as GCSE Higher Mathematics for
A teacher’s dilemma A dilemma is a no win situation. This appears to correctly describe the possibilities facing the teacher described above who wished to motivate her disaffected pupils. We emphasis the problem by reducing her comments to their bare bones: (1) I want to motivate my pupils. (2) I want to help my pupils to gain the best GCSE grade possible. (3) Teaching for the test will not motivate my pupils. (4) Teaching for the test is likely to help my pupils gain the best GCSE grade. (5) Realistic application work is likely to motivate my pupils. (6) Realistic application work is unlikely to help my pupils gain the best GCSE grade.
The dilemma, of course, is in the mind of the teacher. (1) and (2) we take as given. (3) and (4) are based on her teaching experiences with this class. (5) and (6), however, are assumptions. She has clearly questioned assumption (6) or she would not have volunteered to be involved in the project. She stands at the brink, her hesitancy to a large extent, we believe, created by a perceived need to teach for the test. A way out of this dilemma is to select project work that covers curriculum content, which brings us on to the next theme.
Does the mathematics determine the application or is it the other way around? As mentioned above, when planning classwork around an application one may plan around an application and see what mathematics it generates or plan around an area of mathematics and choose an application(s) that appears appropriateii. Actually we have found that the mathematics you expect may arise often does not and that other mathematics appears. For example a teacher in the project looked at a designing packaging to hold six bags of loose tea (a problem set by a local tea company). He included surface area and volume in his pre-lesson plans but these were unimportant to the pupils in practice (the net of the cardboard was given and once you arrange it to contain the six packets the surface area and volume are unimportant). Mathematics that pupils engaged in that he did not expect were producing curves and tessellations.
8 This decision is not really a dilemma. Our interviews with teachers show that just about every option was considered (the interview extracts above show the poles in this debate but another teacher used real NHS data; the data was very rich and he directed pupils to statistical analysis which was on the syllabus). Interestingly, although textbooks (which the survey showed are extensively used) tend to focus on ‘bits of applications’ that fit into content areas, most project teachers ended up letting the application determine the mathematics. If nothing else this speaks well of these teachers, that they feel able to stand up to the alienation created by high stakes examinations.
These considerations may bear on the debate over what functional mathematics should or could be. If functional mathematics is focused on basic skills, then it may be best to organise ‘bits of applications’ around curricula themes but if functional mathematics is focused on modelling, then the context that surrounds and weaves together (Cole, 1996) the threads of the model must be explored for pupils to make sense of the mathematics in context.
Resources We focus on tools and textbooks. The term ‘tools’ encompasses many things and may be subdivided into cultural, physical and semiotic tools or combinations of these. In the mathematics classroom pencil and paper tools, algorithms and ‘methods’, are privileged. In workplace mathematical activities physical tools are omnipresent, as Pozzi et al. (1998) note “workers rarely think mathematically without an artefact to help them organise or compute the data”. We have no problem with school mathematics doing what school mathematics does when the goal is the perpetuation of a cultural tradition (learning to solve, say, linear equations and paying attention to keeping the equals signs underneath each other). But when the goal of the mathematics is applications (workplace or leisure) it would appear to be sensible to pay attention to the tools used in these applications. Our survey, however, suggests that, apart from the odd trundle wheel or clinometer, this is rarely done. Interviews showed, however, that some of our project teachers were conceiving of their applications with tool-use in mind. We posit that a reconsideration of the place of tools in school mathematics is made when teachers plan to undertake significant applications work with their classes.
The survey shows that virtually all teachers make substantial use of textbooks. Only seven of the 25 survey respondees, however, used these textbooks when schoolwork was linked to real life applications, which suggests that teachers perceive a mismatch between ‘school mathematics’ and application work. We view this perception, and our project teachers’ reconsideration of the place of tools, as positive indicators that mathematics teachers are aware of tensions between school mathematics, in its current state in England, and real life applications of mathematics.
Out-of-school people As noted above, all but one of the project teachers planned to consult with workers in various fields in planning their application activities. This contrasts with the activities reported in part (5)(d) of the survey, where only two teachers mentioned using outside visitors. Although these uses of out-of-school people contrast both are understandable: if you are considering an significant out-of-school project, then it makes sense to consult and/or bring in an expert in this application; ‘normal’ school mathematics teaching is quite an insular occupation.
It is beyond the brief of this paper but we would like to say that in every case, when project teachers brought in outside speakersiii, the students were very responsive and motivated: they knew the speakers knew what they were talking about and they knew that the mathematics
9 problem they were being given was a real life one. We were also impressed by the goodwill of these outside speakers. We are optimistic that if there is a future for real life applications in school mathematics, that outside speakers can make a substantial positive contribution to students’ experiences.
Acknowledgements This work was supported by the Economic and Social Research Council, Award Ref. No. RES-000-22-0739.
Notes
References Cockcroft (1982) Mathematics Counts. London: HMSO. Cole, M. (1996) Cultural Psychology: A Once and Future Discipline. Cambridge, MA: Harvard University Press. Cooper, B. & Dunne, M. (2000) Assessing Children’s Mathematical Knowledge: Social Class, Sex and Problem Solving. Open University Press. DES (1982) Mathematics Counts. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools under the Chairmanship of Dr. W.H. Cockcroft. London: HMSO DfEE (1999) Mathematics in the National Curriculum. London: HMSO. DfES (2004a) Making Mathematics Count. The Report of Professor Adrian Smith’s Inquiry into Post- 14 Mathematics Education. London: HMSO DfES (2004b) 14-19 Curriculum and Qualifications Reform. Final Report of the Working Group on 14-19 Reform. London: HMSO DfES (2005) 14-19 Education and Skills. London: HMSO Kendal, M. & Stacey, K. (2001) ‘The Impact of Teacher Privileging on Learning Differentiation with Technology’. International Journal of Computers for Mathematical Learning, 6(2), 143-165. Lave, J. (1988) Cognition in Practice, CUP, Cambridge. Magajna, Z. & Monaghan, J. (2003) ‘Advanced Mathematical Thinking in a Technological Workplace’. Educational Studies in Mathematics, 52(2), 101-122 Ofsted (2000) Inspection of Leeds Local Education Authority. pdf file available on http://www.ofsted.gov.uk/inspect/index.htm Pozzi, S., Noss, R. and Hoyles, C. (1998) ‘Tools in practice, mathematics in use’. Educational Studies in Mathematics, 36(2), 105-122. Roper, T., Threlfall, J. & Monaghan, J. (to appear) ‘Functional Mathematics: What is it?’. Proceedings of the British Society for Research into Learning Mathematics. Saxe, G. B. (1991) Culture and Cognitive Development: Studies in Mathematical Understanding, Laurence Erlbaum Associates, Hillsdale NJ.
10 i See http://www.education.leeds.ac.uk/research/cssme/outofschool.php for a fuller project description. ii This is related to, but not identical to, the issues raised in the second question in the survey: lots of little bits of application or a few application in depth. We were quite surprised that 10 of the 25 respondees stated ‘a few in depth’. Given how GCSE examinations and textbooks flit from application to application, we expected that most would say ‘lots of applications’. iii Most did much more that just speak to the class they were working with and all made more than one visit.