Student perceptions of decision mathematics and legitimate mathematical practice

Student perceptions of decision mathematics and legitimate mathematical practice Abstract Decision mathematics is at present the least established and arguably the most contested strand of applied mathematics within the British A-level mathematics and further mathematics qualifications. This article presents data from a study comprising 10 A-level further mathematics candidates who chose as a cohort to study a second module of decision mathematics. The students described their developing attitudes towards, and perceptions of decision mathematics in two sets of questionnaires and one round of follow-up interviews. Together they reported different levels of challenge, mostly positive affective profiles, and a high level of appreciation for the utility of decision mathematics. Using the concept of figured worlds, this article explores further how the students understood the activity contained within the decision mathematics modules, and the extent to which they positioned it as legitimate mathematical practice. The participants’ accounts show that the study of decision mathematics not only augmented the students’ knowledge of how mathematics might be applied in the world, but for some also worked to extend and reconfigure their perceptions of mathematics as a curriculum discipline. 1. Introduction Students who choose to study mathematics in England, Wales and Northern Ireland beyond the age of 16 typically opt to work towards an A- or advanced level in mathematics. The syllabuses for this qualification, as well as those for the succeeding A-levels in further mathematics and additional further mathematics, currently comprise a mixture of six ‘pure’ and ‘applied’ modules (Porkess, 2003). Almost all applied modules fall into three strands: mechanics, statistics and decision mathematics. Decision mathematics is an area of study that draws on material and applications from the fields of discrete mathematics and operational research. Whilst there are some differences between syllabuses, A-level decision mathematics modules characteristically focus on the use of algorithms and graphs to model and solve a range of context-based problems. For instance, in an examination a student might be required to select and apply an appropriate algorithm to find the shortest path between two given points on a road network. They could then be asked to adapt this learnt algorithm to find the shortest path which also goes through a given intermediate point, to comment on the possible limitations of the model, or to discuss the complexity of the algorithm. A skeleton outline of the content of the decision mathematics modules D1 and D2, as devised by the group Mathematics in Education and Industry (MEI) and administered by the examining board OCR, is given below in Table 1 by way of example (MEI/OCR, 2013).1 Table 1. Summary of OCR (MEI) Decision Mathematics Modules D1 and D2 Decision mathematics 1 (D1)  Decision mathematics 2 (D2)  Introduction to algorithms  Linear programming using the simplex algorithm  Packing and sorting algorithms  Algorithms on networks: Floyd’s algorithm, the travelling salesperson problem and the Chinese postperson problem  Modelling with graphs and networks  Logic, combinatorial circuits and Boolean algebra  Algorithms on networks: Prim’s, Kruskal’s and Dijkstra’s algorithms  Decision trees and using networks in decision analysis  Linear programming using a graphical method    Critical path analysis    Simulation    Decision mathematics 1 (D1)  Decision mathematics 2 (D2)  Introduction to algorithms  Linear programming using the simplex algorithm  Packing and sorting algorithms  Algorithms on networks: Floyd’s algorithm, the travelling salesperson problem and the Chinese postperson problem  Modelling with graphs and networks  Logic, combinatorial circuits and Boolean algebra  Algorithms on networks: Prim’s, Kruskal’s and Dijkstra’s algorithms  Decision trees and using networks in decision analysis  Linear programming using a graphical method    Critical path analysis    Simulation    Although decision mathematics modules have been taught in schools for two decades now, it remains the least established and least prevalent strand of applied mathematics within the A-levels. The popular examination board Edexcel offers its candidates five mechanics modules and four examining statistical knowledge, but only two in decision mathematics (EdExcel, 2013), whilst smaller boards such as WJEC do not offer decision modules at all. Although it is hard to draw transparent comparisons when many modules can count towards either A-level mathematics or A-level further mathematics, the evidence suggests that decision mathematics modules are offered less frequently and less widely by schools than the analogous units in mechanics and statistics, such that decision mathematics is often ignored, relegated to further mathematics only or assigned as independent study (Ward-Penny et al., 2013). The relative place of decision mathematics within A-level mathematics was apparent in the evaluation of participation carried out by the QCA (2007), which reported notional results for the examination board Edexcel by unit combination (p. 11). Only 1.0% of candidates had studied both D1 and D2, compared to 21.8% who had studied two mechanics modules and 20.2% who had studied two statistics modules. Similarly, just 12.9% of combinations had included some decision mathematics, whereas 71.6% included at least one unit of mechanics and 72.5% some study of statistics. Participation in decision mathematics is likely to decrease from current levels in the immediate future, as it is soon to be excluded fully from the newly linear A-level in mathematics in the next round of qualification reforms, for first teaching in 2017 (DfE, 2015, p. 26). Nonetheless, decision mathematics does continue to feature in some draft specifications for further mathematics. For instance, EdExcel (2016) are planning to retain two decision modules, whilst the OCR (MEI) draft specification has replaced them with a new minor option, ‘Modelling with Algorithms’ (OCR, 2016). Decision mathematics also appears to be the most contentious area of applied mathematics. After observing some D1 lessons and interviewing staff, Hernandez-Martinez and Williams (2008) noted cases where decision mathematics ‘is taught in a procedural way’, and questioned its relative level of challenge: ‘in some colleges (it) has become a way to achieve a good pass rate’ (p. 47). Similarly, the QCA (2007) evaluation reported on an ‘informal straw poll’ of teachers and experts that placed D1 amongst the easiest options (p. 10), and the survey of teachers’ perceptions of applied strands carried out by Ward-Penny et al. (2013) found that many teachers viewed decision mathematics as easier overall (pp. 6–7). One of the teachers interviewed questioned the intrinsic worth of decision mathematics: ‘I think we would probably say that we regard D1 as being the easiest one … one reason for that is that it doesn’t contain any maths! Or very little … there’s almost no connection with anything algebraic …’ (p. 13). Conversely, other teachers in the same article commended decision mathematics for being relevant and accessible, and one claimed that their students in fact found decision mathematics ‘hard because they have not seen anything like it’ (p. 9). Another said that it was divisive amongst their students, characterizing it as ‘a ‘‘love it or hate it’’ subject. Those that can work logically and systematically tend to do well and find it easy. Those that can’t get over it ‘‘not being proper maths’’ (their words, not mine) struggle and tend to get low grades …’ (p. 13). The contention that decision mathematics is an easier option for students has also been disputed by awarding bodies and the regulatory agency Ofqual on statistical grounds (ALCAB, 2014, p. 8). Finally, decision mathematics is arguably the most unfamiliar area of applied mathematics for the A-level students themselves: although mechanics can be located against the background of physics, and statistics has been met before within the mathematics curricula prescribed for the UK nations, the content and approaches of the decision mathematics modules are largely novel. In spite of these distinctions there is limited data reporting students’ own considerations of decision mathematics or studying how decision mathematics impacts students’ conceptions of mathematics as a whole. The report of the QCA (2006, p. 61) notes instances of both positive and negative attitudes towards decision mathematics, but offers limited detail; Hernandez-Martinez and Williams (2008) offer greater depth but focus on teachers and only the first decision mathematics module. Between 2013 and 2015 the author was one of the two further mathematics teachers of a small group of students who had studied D1 and who, after some reflection, consciously elected to study D2. It was recognized that these students were in an exceptional position. Not only were they in the middle of studying both decision modules, but the process of choosing between modules for themselves as a class had led them to consider the worth, content and relative difficulty of different areas of mathematics. They therefore constituted an especially rich and accessible source of data regarding students’ perceptions of decision mathematics, and presented an opening for an exploratory study. This article reports on how these learners’ perceptions of decision mathematics evolved as they came to know more about the field. Moreover, it takes advantage of this circumstance to look for evidence regarding how students construct and re-construct their perceptions of mathematics as a curriculum discipline. The students’ accounts relate how the topics taught within decision mathematics might continue to have a particular place in the wide spectrum of the A-level mathematics and further mathematics curricula, and suggest that the attendance of decision mathematics topics within the A-level qualifications can extend learners’ appreciation of the potency and scope of applied mathematics. 2. The students and their context The learners who make up this study were studying the OCR MEI form of A-level further mathematics over two years in a partially selective, inner city London academy. Ten students, eight male and two female, had both successfully passed the first year of the course and opted to continue into the second year in order to gain the full qualification. In line with the school’s predictions and the students’ own aspirations, the majority were expecting to attain one of the top grades, although a small number were working towards a lower passing mark. The author was one of two teachers who shared the class, and was responsible for two of five further mathematics lessons each week. All 10 students had already studied the first decision mathematics module, D1, as part of their first- year work, and received the results of all of their first-year examinations. Their second-year curriculum was expected to consist of: the pure mathematics module FP2; the mechanics module M2; and a module on differential equations, DE. However, my colleague and I were concerned about the balance of topics and techniques covered in these three modules. It was felt that, having done well with D1, the students attaining at a lower level might benefit from swapping DE for the second decision mathematics module, D2. Conversely, some of the higher performing students might particularly enjoy the content and challenges of DE. After deliberation both teachers felt it was reasonable, and perhaps fairest, to put this choice to the students themselves. The teachers together sketched out both modules to the students, attempting to outline them in a balanced way that focused on the strengths of each module, and to award genuine agency to the students in this matter. In brief, DE was presented as being more algebraic, complementary to the mechanics work, related to engineering and containing a coursework component, whereas D2 was summarized as being more algorithmic, building on D1, related to computer science and free of coursework. The students were given this choice over a matter of weeks and some did research into their options, so that this was a reasonably informed and considered choice. It was also repeatedly made clear to the students that both teachers were happy to teach both modules and that the teachers considered each to have its own particular value. The final result was six votes to four in favour of D2, and so the second decision module was chosen to be taught mid-way through the students’ second year of study. Shortly after this ballot the author recognized the unusual potential for researching students’ conceptions of decision mathematics and constructed the research questions of this study. 3. Research questions and theoretical framework The leading research question for this project was predominantly descriptive: how did these students view decision mathematics, and how did their perceptions evolve after working through the second decision mathematics module? The associated second research question was more exploratory: in what ways might these learners’ developing perceptions of decision mathematics indicate how they appraised and located mathematical activity? Both of the research questions were predicated on a broad concept of mathematical practice, as well as the idea that students can and do read validity and worth into classroom activity. The study also assumed the position that the students’ notions of mathematical practice were related to, and reflective of, their own mathematical identities and self-positioning as practitioners of mathematics. Whilst a full discussion of the place of identity in mathematics lies outside the scope of this article, these students’ identities were considered to be developing and related to both practice and agency (Boaler and Greeno, 2000; Black et al. 2009). In particular, these A-level students were in the midst of selecting and securing places in higher education; they would have therefore been questioning the extent to which they wanted to continue to align themselves outwardly with the field of mathematics. In light of the students’ context, the research questions and the relevance of the students’ own identities, the concept of ‘figured worlds’ (Holland et al., 1998) was selected as an analytic frame for use throughout this study. The term ‘figured world’ references a ‘constructed realm of interpretation in which particular characters and actors are recognized, significance is assigned to certain acts, and particular outcomes are valued over others’ (p. 52). Figured worlds are intersubjective, ‘socially produced and culturally constructed activities’ (pp. 40–41) which provide a context for both meaning and identity work such that ‘identities are formed in the process of participating in activities organised by figured worlds’ (p. 57). Urietta (2007) likewise describes figured worlds as ‘processes or traditions of apprehension that give people shape and form as their lives intersect with them’ (p. 108). Finally, this study was conducted with recognition of the idea of fixed and growth mindsets as presented by Dweck (2000). A student with a fixed mindset typically believes that mathematical ability is an innate quality, sees academic difficulties as a sign that things are getting ‘too hard’ and perceives grades or marks in reference to themselves as a person. A student with a growth mindset instead considers mathematical ability to be malleable, sees challenge as a natural part of the learning process and understands grades as being reflective of their own performance, not their person. This distinction was considered pertinent, not only because of its relationship to mathematical identity, but also because concerns about accomplishment and formative assessments would likely have been paramount in the final year of the A-level course as the students applied to university and received offers dependent on grades. Darlington’s (2015) review also reinforced the particular relevance of performance to further mathematics students. For instance, 80.9% of students surveyed therein said that they had chosen further mathematics as they were ‘better at Maths than at other subjects’ (p. 15). 4. Methodology The study took place in two phases and adopted a mixed methods approach. In each phase the central research instrument was a questionnaire chiefly consisting of 16 items, presented with five-point Likert-type scales. The wording of these questions drew on previous research into teachers’ perceptions of applied modules (Ward-Penny et al., 2013) as well as wider instruments measuring learner affect (such as Tapia and Marsh, 2004) but it was adapted and bolstered in light of the particular context. The questions were phrased both positively and negatively, and together were considered at the design stage to encompass three non-orthogonal dimensions of perception related to outcomes, affect and value. Cronbach’s alpha calculations using Microsoft Excel with the first set of questionnaires (n = 10) yielded results of 0.919, 0.879 and 0.810, respectively; these results shifted to 0.844, 0.877 and 0.734, respectively when both sets were combined (n = 20). In this way, although the size of the study limits any proposal of well-defined constructs, reliability analysis supports these groupings as a typology for the presentation of the results. The first set of questionnaires was distributed in the second year of the two-year further mathematics course, hence well after D1 but before D2 was taught; the second set was distributed towards the end of the second year, both after the teaching for D2 had been finished and some time had passed to allow for past paper practice. The first questionnaire included an introductory item that asked each student to explain the reasoning behind their vote, and both versions concluded with two open-ended questions about decision mathematics that were intended to check content validity and further promote reflection; these are considered in more detail below. After a preliminary analysis of both sets of questionnaires, five students were purposively selected for a short interview using their questionnaire responses, so as to ensure a mix of genders, attainment levels and perspectives. The questions for the semi-structured interviews were designed to promote validity through retelling and clarification. After transcription these comments were combined with the qualitative questionnaire data for analysis. Some quotes presented herein include minor edits, such as the removal of spelling errors, misspeaking and spoken fillers such as ‘like’, for the purposes of clarity. The author’s dual role as teacher and researcher raised ethical concerns that were addressed throughout. Primarily, there was a conscious effort to manage both the timing and execution of the research so that it was clearly subordinate to the students’ learning. It was made immediately and repeatedly clear that student responses would not affect grades, reports or the conduct of either teacher. In light of the age and relative maturity of the students, the aims and methods of the research project were also shared openly, and all 10 agreed to the anonymized use of their data; in addition, gender neutral pronouns have been used throughout this article. As the research repeatedly involved the same students, a high premium was put on their time. This limited the number of open-ended items on the questionnaire and contributed to the decision to sample the students at the interview stage. It was considered that interviews with half of the students would provide enough data for analysis without presuming further on the students’ time. Further to this, the author’s dual role also resulted in some less obvious tensions which had to be navigated during the research (Atkinson, 1994). For example, both parties had to renegotiate their roles with regard to the act of questioning. As a teacher, the author would typically ask a question of a student from a position of knowledge, and there existed a mutual recognition from both parties that extended lines of questioning most often arose to address misconceptions, such that the student was expected to ultimately change their response. However, as a researcher the author was asking questions from a position of enquiry, and follow-up questions were intended to confirm, not challenge, the students’ contributions. To address this, each interview started with a short introduction that explicitly signalled a change in expectations, stating that there were ‘no right or wrong answers’ and stressed a genuine interest in each student’s personal opinions. Whilst the author’s influence was inevitable in both the design and exercise of this research, steps were also taken to minimize deliberate or subconscious prompting of the students beyond that which would have occurred naturally through unobserved teaching. For instance, the first questionnaires were not analysed at the level of the individual until both sets had been collected, so that the author could not purposely attempt to influence the perceptions of any specific student through targeted comments or action whilst teaching D2. Further, the author attempted to draw principally on discourses surrounding the value and worth of the curriculum content which were already present in some form in existing resources such as the MEI textbook. The incidence of a co-teacher also meant that the students were exposed to a second perspective from a person in a position of power. Nonetheless, it is apposite to note briefly here that the author holds a principally positive perception of decision mathematics, and that he personally considers all strands of applied mathematics to have utility and appeal. 5. Results 5.1 Reasons for voting The first questionnaire began by exploring the vote through which the students had chosen D2 over the differential equations module, DE. The opening item asked each student to identify which module they had voted for and then briefly offer their reasons. All six learners who voted for D2 referenced D1 in their answer: four stated that they had ‘liked’ or ‘enjoyed’ the first decision module and two said that they wanted to build on what they had already achieved. Three of those who voted for D2 expressed specific reservations about DE, regarding either the perceived relative difficulty of more calculus, or the presence of a coursework element. Conversely, some of the students who voted for DE said that they had been unimpressed with D1: ‘D2 looks okay but D1 was not particularly inspiring so much’; ‘I really really disliked D1 … also calculus is my type of maths.’ Thematically, the comments volunteered at this initial stage by the students who voted for D2 were clustered around outcomes and affective elements, whereas half of those who voted for DE expressed thoughts about future utility: ‘DE seems more useful in general and especially for continuing maths at uni’; ‘I thought it would be more useful for what I want to do later on.’ 5.2 Outcomes A total of 4 of the 16 Likert-type items explored the students’ perceptions of how the inclusion of decision mathematics modules might affect their achievement outcomes. These items were included in recognition of both the debate surrounding the relative difficulty of decision mathematics, and the import of the qualification to the students’ future plans. Two items explicitly concerned the students’ marks and are presented in Table 2; Q6 was worded as a reverse item. Table 2. Questionnaire results relating to outcomes - part one (n = 10); SD = strongly disagree, D = disagree, N = neutral, A = agree, SA = strongly agree   After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q2: I am expecting to get a high mark in my decision mathematics modules.  0  2  2  5  1  0  0  4  5  1  Q6: I find it difficult to get a good mark on decision mathematics examination papers.  0  6  2  1  1  0  4  5  1  0    After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q2: I am expecting to get a high mark in my decision mathematics modules.  0  2  2  5  1  0  0  4  5  1  Q6: I find it difficult to get a good mark on decision mathematics examination papers.  0  6  2  1  1  0  4  5  1  0  These results show that the students considered decision mathematics modules to be manageable both before and after studying D2. Only three students chose to select a different response to their original one when answering Q2 in the second iteration of the questionnaire; overall, the students had like expectations over time. This consistency is not contradicted by the slight shift over time in the responses to Q6, since the grade boundaries for the practised D2 examination papers were typically lower than those seen in other modules. Attitudes here may also have been steered by attainment in D1 and the other mathematics units as well as the students’ personal criteria for success. For instance, one of the academically weaker students stated in the open-ended questions that ‘if you practise a lot of questions it’s easier to get a good mark than other modules’, but one of the students aiming for a top grade offered that the make-up of the examination papers was an issue for them: ‘I do like aspects and can see how they’re relevant to real-life situations, but I dislike the unpredictability of the question paper.’ The results of the remaining two outcome-focused items are given in Table 3. Even in light of the small sample size, the responses to Q10 entail a notable spread, with the students’ attitudes appearing to polarize somewhat after the study of D2. Hence, whilst the students felt that they were able to achieve in this module, and despite the concerns in the extant literature about decision mathematics being a slighter option (see Section 1), these further mathematics students did not uniformly find decision mathematics easy. Table 3. Questionnaire results relating to outcomes - part two (n = 10); SD = strongly disagree, D = disagree, N = neutral, A = agree, SA = strongly agree   After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q10: Decision mathematics is easier than other mathematics modules.  2  2  2  4  0  1  4  1  3  1  Q16: I am expecting to meet my overall predicted grade in my decision mathematics modules.  0  2  1  5  2  0  0  3  6  1    After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q10: Decision mathematics is easier than other mathematics modules.  2  2  2  4  0  1  4  1  3  1  Q16: I am expecting to meet my overall predicted grade in my decision mathematics modules.  0  2  1  5  2  0  0  3  6  1  5.3 Affect Another six of the Likert-type items explored the affective dimension of the students’ study of decision mathematics, with two items (Q9 and Q15) worded negatively. All six items were constructed in light of the differing opinions present in the existing literature (see Section 1). The items are presented together in Table 4 in the order that they appeared on the instrument. Table 4. Questionnaire results relating to affect (n = 10); SD = strongly disagree, D = disagree, N = neutral, A = agree, SA = strongly agree   After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q1: I enjoy learning decision mathematics.  0  1  3  4  2  0  0  2  7  1  Q4: I have a lot of self-confidence when it comes to decision mathematics.  1  2  1  5  1  0  0  6  4  0  Q8: I prefer decision mathematics to my other mathematics modules.  1  3  4  2  0  1  4  4  0  1  Q9: I do not enjoy working with the ideas and techniques of decision mathematics.  2  5  1  2  0  2  6  1  1  0  Q13: I find decision mathematics interesting.  0  1  1  5  3  0  1  1  6  2  Q15: I do not feel comfortable solving problems using decision mathematics.  0  7  3  0  0  1  7  2  0  0    After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q1: I enjoy learning decision mathematics.  0  1  3  4  2  0  0  2  7  1  Q4: I have a lot of self-confidence when it comes to decision mathematics.  1  2  1  5  1  0  0  6  4  0  Q8: I prefer decision mathematics to my other mathematics modules.  1  3  4  2  0  1  4  4  0  1  Q9: I do not enjoy working with the ideas and techniques of decision mathematics.  2  5  1  2  0  2  6  1  1  0  Q13: I find decision mathematics interesting.  0  1  1  5  3  0  1  1  6  2  Q15: I do not feel comfortable solving problems using decision mathematics.  0  7  3  0  0  1  7  2  0  0  Even though it might be presumed that students working towards a full A-level in further mathematics are likely to have a positive baseline attitude towards mathematics, the results of Q1 and Q9 in particular endorse that these 10 students generally enjoyed decision mathematics as a cohort, and that they reported slightly higher levels of enjoyment after studying D2. A paired comparison of the two sets of questionnaires showed that nine and eight students, respectively disclosed an equal or more positive response to these two items in the second questionnaire. The item for which the most students altered their response was Q4, where only two students chose the same answer on both questionnaires. Although this degree of reporting is offered cautiously in light of factors such as central tendency bias, the table alone gives the impression of a positive shift in the self-confidence of the pupils who had at first struggled with D1, as well as an affective diminution at the top end, possibly influenced once more by the higher level of challenge and the shifted grade boundaries. Conversely, the negatively phrased Q15 was the most stable item on the entire questionnaire, with eight students offering the same response on both iterations. In Q13, 80% of the cohort agreed or agreed strongly that decision mathematics was interesting. In light of the broadly positive affective profile, the muted neutral and negative responses to Q8, ‘I prefer decision mathematics to my other mathematics modules’, were perhaps then a little surprising. This ambivalence was also present in the supporting qualitative data when pupils were asked to give their overall impression of decision mathematics. The mood of this ranged from the extremely positive ‘it’s great’ to the following oblique criticism from the single student who had chosen ‘strongly disagree’ for this item on both questionnaires: ‘not good, I prefer the other modules even statistics’. Nonetheless, when this student was asked the same question on the second questionnaire, they simply wrote that decision mathematics was ‘decent’. 5.4 Value The remaining six Likert-type items concerned various aspects of the value or utility of decision mathematics. These items were intended to address further some of the debates summarized in question one whilst also speaking to the second, wider research question regarding students’ appraisal of mathematical activity. Two items were again reversed to allow for internal reliability checks, specifically Q5 and Q11. The results are presented in Table 5. Table 5. Questionnaire results relating to value (n = 10); SD = strongly disagree, D = disagree, N = neutral, A = agree, SA = strongly agree   After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q3: I think of decision mathematics as a worthwhile part of mathematics/further mathematics.  0  1  3  4  2  0  0  1  7  2  Q5: I do not think that decision mathematics is very mathematical.  0  4  5  1  0  2  5  1  2  0  Q7: It is easy to see how decision mathematics relates to real-life situations.  0  0  0  4  6  0  0  2  3  5  Q11: I find it hard to see how decision mathematics might be used in jobs or careers.  4  3  3  0  0  6  3  1  0  0  Q12: I think that decision mathematics helps me develop my problem-solving and thinking skills.  0  0  2  5  3  0  0  2  4  4  Q14: I think that decision mathematics is a valuable branch of applied mathematics.  0  0  2  6  2  0  0  1  7  2    After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q3: I think of decision mathematics as a worthwhile part of mathematics/further mathematics.  0  1  3  4  2  0  0  1  7  2  Q5: I do not think that decision mathematics is very mathematical.  0  4  5  1  0  2  5  1  2  0  Q7: It is easy to see how decision mathematics relates to real-life situations.  0  0  0  4  6  0  0  2  3  5  Q11: I find it hard to see how decision mathematics might be used in jobs or careers.  4  3  3  0  0  6  3  1  0  0  Q12: I think that decision mathematics helps me develop my problem-solving and thinking skills.  0  0  2  5  3  0  0  2  4  4  Q14: I think that decision mathematics is a valuable branch of applied mathematics.  0  0  2  6  2  0  0  1  7  2  Of the three groups, these questions implicate the biggest change in opinion, with perhaps the largest shifts relating to the place of decision mathematics within the mathematical canon: Q3 and Q5 both show affirmative swings, and in both items five of the students altered their responses between the questionnaires in favour of decision mathematics. Smaller positive moves can be read into the overall results of Q11 and Q14. The results of Q7 seem to stand against this trend, with four students agreeing less strongly that ‘it is easy to see how decision mathematics relates to real-life situations’. This may be understood in light of the curriculum content of the OCR MEI syllabus, as outlined in Table 1. Whereas the bulk of D1 easily lends itself to context-based work, the applications of more abstract topics such as logic and Boolean algebra in D2 are perhaps less tangible and immediate for students. Notwithstanding, the qualitative data fully supports the idea of decision mathematics overall as being strongly anchored in context. In the first questionnaire one student reported that ‘D1 + 2 are very useful in the real-world. It is more applied than other modules. Very interesting’. The utility of decision mathematics was widely noted, though it was not always allied with either affect nor outcome: ‘It’s okay, useful, but not all that interesting’; ‘I haven’t found it easy at all, as this has taken time to adapt to, however I have found it valuable’. Another told that it was particularly relevant in preparing to study computer science. The results of Q12 suggest that the students concerned were further able to connect their decision mathematics learning with the development of wider problem-solving skills. This was returned to by some in the open-ended questions, with one student offering that ‘I like it because it requires more thinking in terms of the context, which would be useful for my future career as an engineer.’ When the students were asked in the first questionnaire to express what they thought of when they thought of decision mathematics, the most common response categories were problems/problem-solving (five learners) and practical/real-life contexts (four learners). Once again though, there was a significant undercurrent of negative comments in this regard; some felt that elements were laborious or tantamount to number crunching, with one learner stating that decision mathematics felt like ‘going through the motions (with) a little less room for creativity’. In summary, the questionnaire results show that the students taken as a whole judged decision mathematics to be manageable though not always easy; interesting and enjoyable in the main; and broadly valuable and related to real-life. Their attitudes changed slightly after studying D2, typically in a positive direction, with the largest observed changes relating to the value and utility of decision mathematics. When they were asked at the end of the second questionnaire whether they would recommend decision mathematics to a friend about to study A-level mathematics, seven said ‘yes’, and the other three gave a conditional ‘it depends’ style response. However, while some students offered positive comments without provision, there was a persistent tension implied by some of the remarks, such as this summative juxtaposition: ‘it’s alright but it’s not the most interesting branch of mathematics. It’s a pretty cool way of looking at problems though I guess.’ This was explored further in the interviews. 5.5 Figuring decision mathematics The interviews were semi-structured and based around a set of five questions designed to explore the students’ thoughts on decision mathematics, how they saw the progression from D1 to D2 and how they felt decision mathematics fit within the wider A-level qualification. Each of the five interviews lasted between 6 and 10 minutes, and the students’ accounts are discussed here in turn. The first interview question asked the participant to summarize their thoughts about decision mathematics in their own words. Student A said that ‘I think it’s formalising the logic you’d use in normal mathematical problems.’ This was the first of a number of comments that included the use of a linguistic marker which separated off decision mathematics, in this case the word ‘normal’. Elsewhere they variously described decision mathematics as a ‘brand of maths’, ‘a different take to maths … a different flavour of maths.’ Student A considered that D2 was a stronger module than D1 in some respects, but not all: D2 ‘took what D1 was and made it more powerful so simplex is like linear programming, but expanded so you have more variables. Floyd’s is just Dijkstra n times… I think it’s just expanding what we did on D1’. Indeed, ‘it’s very much, kind of, going through the process … maybe if there was something like on how to improve the algorithms some of that might be a bit more interesting …’. Student A had originally wanted to study DE instead of D2 and uniquely had gone on to study the differential equations module as an additional endeavour. They felt that the models they had used in DE were more sophisticated. For student A, decision mathematics was offered as part of A-level mathematics and further mathematics in order to diversify the modules available and to appeal to a wider variety of people: ‘I’m sure there are some people in this class who would hate DE but like D2 and vice versa.’ Nonetheless, the tenor of their comments implied some distance between their own preferences for practice and decision mathematics. Student B had also voted for the differential equations module. They had presented the most negative affective profile in the two questionnaires, and were the only student who had strongly disagreed with Q8, ‘I prefer decision mathematics to my other mathematics modules.’ When asked in the questionnaire whether they would advise a friend to take decision mathematics, they wrote ‘if you want a more full on maths module then decision is not for you.’ In a similar manner to student A, during the interview B repeatedly distanced themselves as an individual mathematician from decision mathematics: it was ‘not my cup of tea … just different thinking, that’s not really my type of thinking’. Later, when discussing its utility, they said that ‘I can see how it can be useful. Probably not useful for me, personally.’ Similarly, they admitted ‘I think it does make you think differently. I don’t know if that thinking differently will apply to me…’. Other resonances were observable when student B considered why decision mathematics was present in the curriculum: it was included because ‘everyone’s different’… it may be helpful for ‘people who want to do programming’ or set up a business. Their perception of decision mathematics as something meaningful but subordinate was particularly evidenced by their use of a metaphor: mathematics was a big tree, where core maths was the trunk of the tree, and ‘decision might be a thinner branch, like mechanics might be thick, slightly thicker…’ Unlike the preceding two participants, student C had voted for D2. This was a partly strategic choice, for although they had struggled a little with D1 they were retaking this earlier module and felt that the overlap would aid their revision. Ultimately, student C showed the greatest increase in affect towards decision mathematics between the two questionnaires. Student C described decision mathematics as ‘applying maths and logic to real world situations and interpreting them.’ They felt that moving from D1 to D2 had widened the relevance of decision mathematics. Beforehand they ‘knew why people did it for these very specific scenarios but I didn’t really understand at the time how that would, sort of, broaden…’. Later they confirmed that knowing of more applications helped them feel decision mathematics was more worthwhile. Further, whilst ‘D1 for me was very procedural’, D2 ‘asks more of you, asks more of your understanding’. In opposition to the comments of student A, student C felt that this qualitative shift was greater for decision mathematics than for other strands, citing that the second mechanics module was instead largely about ‘factoring in more things’. This perceived shift in the requisite depth of understanding was important to student C in a way they confessed was ‘weird’: ‘I found it harder but I’m sort of more aware of why I’m doing things … it’s sort of the way I learn, if I don’t really fully understand something I really, really struggle just to do it.’ When questioned about Q5, ‘I do not think that decision mathematics is very mathematical’, student C noted that they had responded with a neutral answer both times, but not for the same reason. After D1 they had not been convinced that decision mathematics was not as mathematical as their other work, ‘but when I put neutral now it’s because my understanding of “mathematical” changed’. Student C explicitly stated that studying more decision mathematics had overtly shifted their perception of what qualified as mathematics; this is discussed in further depth in the next section. Again in opposition to some of the preceding interviews, they shared that decision mathematics ‘doesn’t have that surface complexity, but … it has a complexity behind it that requires you to think mathematically…’. By way of contrast, student D strongly endorsed decision mathematics in both questionnaires, offering the most positive affective profile overall. When they were asked to summarise their thoughts about decision mathematics the leading feature of their response was the applicability of the strand, rather than its character, or any specific content. They praised the immediacy of the applications and compared the models of decision mathematics favourably with other strands, saying that in mechanics ‘it feels like they’re quite basic models and you can do so much more to sort of actually give a realistic thing … but with D2 it just feels like you’re already doing stuff that can be useful.’ As with student C, student D noted that their learning experiences had shifted their internal perspective of mathematics: ‘I feel maybe I changed my idea on what mathematical is between the two (questionnaires) … it’s not just, something needs to, be like pure maths …’. Another possible element contributing to this shift was the introduction of more ‘advanced’ mathematical tools, symbols and notation. Student D recounted that D1 primarily involved the application of The General Certificate of Secondary Education (GCSE) level skills, but felt that features such as the use of matrices had made D2 more mathematical. Student E was academically the weakest of the interviewees in terms of their overall results, but they had been relatively successful with decision mathematics; this is likely to have influenced their attitude. They had voted for D2, and said that ‘I like it. It seems quite nice, really logical. Yeah, it sort of makes sense to me.’ Indeed, although they conceded that this was a personal view, when they were asked to volunteer a less logical module, they unexpectedly proposed the further pure module FP2, which consists of topics such as polar co-ordinates and Maclaurin series, as well as more advanced work involving matrices and complex numbers. Student E’s perceptions had been challenged when moving from D1 to D2. They felt that D2 was ‘more complex’ than D1, ‘which goes against what I thought decision maths was in terms of mathematical stuff’. When they were asked to expand on this point they said that ‘I have it in my head that decision maths is much less complex in terms of like pure mathematics … D2 seemed to go against that more than D1 did.’ They had felt that the simplex tables and Boolean algebra of D2, arguably the most symbolic elements of the module, were more mathematical. On the questionnaires, student E had ‘agreed’ both times through Q5 that decision as a strand was not very mathematical. They were not entirely confident explaining their view however: ‘Not too sure to be honest … much less obvious number plugging, and rearranging equations and stuff, which seems to be more like mechanics or the other maths modules… .’ Regardless, they were certain about wanting decision mathematics to remain on offer to students; not only had they liked it, ‘it wouldn’t be under any other subject… I do think it’s useful and worth doing.’ 6. Discussion: positioning legitimate mathematical activity The narratives of positioning outlined in the five interviews above can be construed in terms of figured worlds and mathematical identities. An A-level student might react to the fact and practice of the decision mathematics modules by rationalizing decision mathematics as being somehow separate from what has come before, attaching less significance to the practice of decision mathematics, predominantly maintaining their sense of the figured world of mathematics and insulating their mathematical identity. Alternatively, the novel aspects of decision mathematics can drive a marginal renegotiation of a learner’s conception of the quality of being mathematical, alter their sense of the figured world of mathematics and inform their perception of themselves as a mathematician. Both the explicit comments and linguistic markers in the data are indices of these two contrasting cases. In the first case, the learner locates decision mathematics as an adjunct, and then positions themselves at some distance from this applied strand. This approach can be read into the comments of student A, who uses the labels ‘normal’ and ‘different’ and places more emphasis on the procedural aspects of decision mathematics, ostensibly devaluing its intellectual currency. A similar tactic can be inferred from the words of student B, who sites himself ‘personally’ aside from decision, thinking of it as other people’s mathematics and characterizing it as a ‘thinner branch’ on the tree of mathematics. Since both of these learners were high-achieving students, this restricted assimilation of decision mathematics might be construed as an ego defence strategy. It has already been noted above that decision mathematics assessments are in some ways less predictable, and this teaching of D2 in particular involved some expectation of high challenge, unexpected elements and lower grade boundaries. If a very able student possessed a fixed mindset inasmuch as their mathematical identity was bound up with perfection or high percentage scores, aspects of the decision mathematics modules and assessments could be perceived as threats. Related ego defence strategies have been noted in the accounts of slightly older undergraduate mathematicians who were forced to move away from the expectations of ‘one hundred percent’ that they inherited from school mathematics (Ward-Penny et al., 2011). Building on this reading, it is possible to interpret student E’s account as a supporting counterpoint. Student E was aiming for a lower grade than the previous two students, and likely saw themselves as a different sort of actor within the figured world of mathematics than the other interviewed students. However, student E was both more comfortable with, and achieving more highly in, decision mathematics than in the other modules. Student E’s ego defence thus works in the opposite direction to that of students A and B. Asserting the alterity of decision mathematics helps student E to rationalize their success without having to change their sense of themselves and the figured world of mathematics, by fencing off and validating an area of achievement on the periphery. Decision mathematics is awarded the title of ‘logical’ and detailed in opposition: in the questionnaire it ‘is nice, seems to be more about logic and clear thinking more than it relies on complicated maths like FP2 for example’, just as in the interview ‘it seems quite nice, really logical’. Learners who are perhaps less involved with measures of absolute success are arguably freer to extend their sense of the figured world of mathematics, by attaching significance to decision mathematics and positioning it as fully legitimate mathematical practice. This may have been the case for both student C and student D, who remarked explicitly in the preceding section about how their perceptions of mathematics had changed through studying the two decision modules. This willing accommodation is further hinted at in the words of student D, who liked starting the first year with D1, because it was ‘a nice introduction to what further maths is like’; decision mathematics had evidently featured in their sense of the figured world of (further) mathematics. This is not to say that students who are more ready to assign value to decision mathematics do not recognize differences in the demands of the decision modules; on their questionnaire student C conceded that ‘technically, it might not be as mathematically challenging but it changes your perspective on maths as a whole’. However, such distinctions are construed differently than before, such that these students seemed more likely to foreground the conceptual demands of decision mathematics. Two students who were not interviewed offered consonant comments on their questionnaires: ‘The maths isn’t as complex as other maths modules, but the overall concept is harder to understand’; ‘Still Maths but you need to think differently and actually understand what is going on more than just memorising the steps as you maybe would in other modules.’ Perhaps the most interesting narrative under this reading is that of student C, who appears to have moved from the first cluster of students to the second between the two decision mathematics modules. D1 for student C had been ‘very procedural’ with a narrow focus of application; hence, in the first questionnaire they exhibited some of the linguistic markers discussed above, saying ‘I think of logic when I think of decision mathematics.’ Contrariwise, they felt D2 ‘asks more of your understanding’, has a wider range of application, ‘required me to use a different way of thinking and has changed the way I approach problems, and view situations. I haven’t found it easy at all, as this has taken time to adapt to, however I have found it valuable.’ These descriptions are resonant not only with the two groups of interview accounts, but also with some of the trends in the overall questionnaire data, particularly those discussed in Section 5.4. Student C’s transition also gives rise to the question of what might have happened to the perceptions of the group as a whole if the students had gone on to study a third module of decision mathematics. In this way, the five interviews can be understood as being indicative of two contrasting ways in which students might perceive and position decision mathematics. Framing these accounts in terms of figured worlds and mathematical identities may also speak to the findings of the literature as summarized in Section 1, suggesting additional reasons why decision mathematics might be so divisive, and why some students cannot (or elect not to) get over it ‘not being proper maths’. 7. Conclusion Whilst the scale of this research prohibits generalization, the findings of this study do appear to mitigate in some measure some of the more general criticisms of decision mathematics cited in recent research and reports such as those listed in Section 1. The majority of these able further mathematics students found worth in, were challenged by, and enjoyed the experience of learning decision mathematics. Further, the students’ perception and appreciation of decision mathematics evolved after studying the second module. Such advances are arguably predictable, but it is nonetheless tendered that there is worth in mentioning them here; although debates will continue to be held regarding the comparative academic challenge and place of the applied strands, the expansion and elaboration present in D2 should not be overlooked because of the relative frequency of the present teaching of this module. This reminder is particularly pertinent in light of the present changes to the place of decision mathematics in future A-level portfolios. Since decision mathematics has the potential to be positioned both within and to one side of each learner’s personal mathematical canon, these students’ accounts have also exemplified how a learner might adjust their sense of the figured world of mathematics in light of novel curriculum elements. In particular, the narrative of student C strongly suggests that an individual can switch between positioning strategies, and serves as a reminder that learners’ perceptions of mathematics are open to renegotiation, such that the quality of being mathematical is clarified by both narrative and action. There is scope for subsequent, fuller research to map and gauge the incidence of such strategies in other contexts, for instance, by looking at how students of different attainment might locate mechanics with respect to mathematics, or considering whether and how positioning strategies might be related to issues of affect and performance in a fuller range of students. Finally, it has been argued that to meet the needs of the 21st century, curriculum divisions should be ecological, not bureaucratic, and unshackled from any ‘rigid disciplinary structure’ (Noddings, 2013, p. 146). The pressing global problems of today undoubtedly demand logical thinking and quantitative literacy, but their solutions are unlikely to fall precisely within the single curriculum area that is mathematics. Noddings (2013) further proposes a strategy of ‘stretching the disciplines from within’ (p. 62) to facilitate a broad and connected education. For at least some of the students in the study, decision mathematics was understood in this way: as an area of study which supplemented the more senior aspects of the A-level mathematics curriculum, extended their sense of what mathematics was, and served to demonstrate how mathematical thought might be applied in a wide range of settings. Hence, whilst decision mathematics might not serve as a direct prerequisite for higher education courses, as noted by ALCAB (2014), these stories submit that it can work to augment learners’ perceptions of mathematics as a discipline, what it allows for, and its power within the world. This is certainly a valuable intention for students who are about to leave the formal study of mathematics behind, and a positive motivating factor for those who might choose at this critical juncture to pursue it further. Footnotes 1 The current OCR (MEI) specification is unique in that it also offers a third related module, Decision Mathematics Computation (DC) which integrates technology into the teaching and assessment of decision mathematics. Robert Ward-Penny received his PhD in Education from the University of Warwick in 2013 and is currently working as a mathematics teacher. His interests include cross-curricular teaching and learning in secondary school mathematics, and the relationship between philosophy and practice in mathematics education. References ALCAB ( 2014) Report of the ALCAB panel on mathematics and further mathematics. A Level Content Advisory Board. (https://alevelcontent.files.wordpress.com/2014/07/alcab-report-on-mathematics-and-further-mathematics-july-2014.pdf) [accessed April 2017]. Atkinson S. ( 1994) Rethinking the principles and practice of action research: the tensions for the teacher-researcher. Educ. Act. Res ., 2, 383– 401. Google Scholar CrossRef Search ADS   Black L., Mendick H., Solomon Y. (eds.) ( 2009) Mathematical Relationships in Education: Identities and Participation . London: Routledge. Boaler J., Greeno J. G. ( 2000) Identity, agency and knowing in mathematics worlds. Multiple Perspectives on Mathematics Teaching and Learning  ( Boaler J. ed.). Westport, CT: Ablex, pp. 171– 200. Darlington E. ( 2015) Students’ perceptions of A-level Further Mathematics as preparation for undergraduate Mathematics. In Adams, G. (ed.) Proceedings of the British Society for Research into Learning Mathematics, 35, 13–18. DfE ( 2015) Reformed GCSE and A level subject content consultation. Government response. Department for Education. (https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/397672/Reformed_GCSE_and_A_level_subject_content_Government_Response.pdf). Dweck C. ( 2000) Self-theories: Their Role in Motivation, Personality and Development . Lillington, NC: Psychology Press. EdExcel ( 2013) Specification. GCE Mathematics. Issue 3. Pearson. (https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2013/Specification%20and%20sample%20assessments/UA035243_GCE_Lin_Maths_Issue_3.pdf). EdExcel ( 2016) Specification DRAFT. Pearson Edexcel Level 3 Advanced GCE in Further Mathematics. Pearson. (http://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-further-mathematics-specification.pdf). Hernandez-Martinez P., Williams J. ( 2008) Ethics, performativity and decision mathematics. In Joubert, M. (ed.) Proceedings of the British Society for Research into Learning Mathematics, 28, 43–48. Holland D., Lachicotte W.Jr., Skinner D., Cain C. ( 1998) Identity and Agency in Cultural Worlds . Cambridge, MA: Harvard University Press. MEI/OCR ( 2013) MEI Structured Mathematics. MEI/OCR. (http://www.ocr.org.uk/Images/75811-specification.pdf). Noddings N. ( 2013) Education and Democracy in the 21st Century . New York, NY: Teachers College Press. OCR ( 2016) OCR Level 3 Advanced GCE in Further Mathematics B (MEI) (H645) Specification (Draft). OCR. (http://www.ocr.org.uk/Images/308768-specification-draft-a-level-gce-further-mathematics-b-mei-h645.pdf). Porkess R. ( 2003) The new AS and A levels in mathematics. MSOR Connect , 3, 12– 16. Google Scholar CrossRef Search ADS   QCA ( 2006) Evaluation of participation in GCE mathematics. Interim report. Qualifications and Curriculum Authority Research Faculty. QCA ( 2007) Evaluation of participation in GCE mathematics. Final report. Qualifications and Curriculum Authority Research Faculty. Tapia M., Marsh G. E. ( 2004) An instrument to measure mathematics attitudes. Academic Exchange Quarterly , 8, 16– 21. Urietta L.Jr. ( 2007) Figured worlds and education: an introduction to the special issue. Urban Rev . 39, 107– 116. Google Scholar CrossRef Search ADS   Ward-Penny R., Johnston-Wilder S., Johnston-Wilder P. ( 2013) Discussing perception, determining provision: teachers’ perspectives on the applied options of A-level mathematics. Teach. Math. Appl ., 32, 1– 18. Ward-Penny R., Johnston-Wilder S., Lee C. ( 2011) Exit interviews: undergraduates who leave mathematics behind. For the Learning of Mathematics , 31, 21– 26. © The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Teaching Mathematics and Its Applications: International Journal of the IMA Oxford University Press

Student perceptions of decision mathematics and legitimate mathematical practice

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Institute of Mathematics and its Applications
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© The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com
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1471-6976
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Abstract

Abstract Decision mathematics is at present the least established and arguably the most contested strand of applied mathematics within the British A-level mathematics and further mathematics qualifications. This article presents data from a study comprising 10 A-level further mathematics candidates who chose as a cohort to study a second module of decision mathematics. The students described their developing attitudes towards, and perceptions of decision mathematics in two sets of questionnaires and one round of follow-up interviews. Together they reported different levels of challenge, mostly positive affective profiles, and a high level of appreciation for the utility of decision mathematics. Using the concept of figured worlds, this article explores further how the students understood the activity contained within the decision mathematics modules, and the extent to which they positioned it as legitimate mathematical practice. The participants’ accounts show that the study of decision mathematics not only augmented the students’ knowledge of how mathematics might be applied in the world, but for some also worked to extend and reconfigure their perceptions of mathematics as a curriculum discipline. 1. Introduction Students who choose to study mathematics in England, Wales and Northern Ireland beyond the age of 16 typically opt to work towards an A- or advanced level in mathematics. The syllabuses for this qualification, as well as those for the succeeding A-levels in further mathematics and additional further mathematics, currently comprise a mixture of six ‘pure’ and ‘applied’ modules (Porkess, 2003). Almost all applied modules fall into three strands: mechanics, statistics and decision mathematics. Decision mathematics is an area of study that draws on material and applications from the fields of discrete mathematics and operational research. Whilst there are some differences between syllabuses, A-level decision mathematics modules characteristically focus on the use of algorithms and graphs to model and solve a range of context-based problems. For instance, in an examination a student might be required to select and apply an appropriate algorithm to find the shortest path between two given points on a road network. They could then be asked to adapt this learnt algorithm to find the shortest path which also goes through a given intermediate point, to comment on the possible limitations of the model, or to discuss the complexity of the algorithm. A skeleton outline of the content of the decision mathematics modules D1 and D2, as devised by the group Mathematics in Education and Industry (MEI) and administered by the examining board OCR, is given below in Table 1 by way of example (MEI/OCR, 2013).1 Table 1. Summary of OCR (MEI) Decision Mathematics Modules D1 and D2 Decision mathematics 1 (D1)  Decision mathematics 2 (D2)  Introduction to algorithms  Linear programming using the simplex algorithm  Packing and sorting algorithms  Algorithms on networks: Floyd’s algorithm, the travelling salesperson problem and the Chinese postperson problem  Modelling with graphs and networks  Logic, combinatorial circuits and Boolean algebra  Algorithms on networks: Prim’s, Kruskal’s and Dijkstra’s algorithms  Decision trees and using networks in decision analysis  Linear programming using a graphical method    Critical path analysis    Simulation    Decision mathematics 1 (D1)  Decision mathematics 2 (D2)  Introduction to algorithms  Linear programming using the simplex algorithm  Packing and sorting algorithms  Algorithms on networks: Floyd’s algorithm, the travelling salesperson problem and the Chinese postperson problem  Modelling with graphs and networks  Logic, combinatorial circuits and Boolean algebra  Algorithms on networks: Prim’s, Kruskal’s and Dijkstra’s algorithms  Decision trees and using networks in decision analysis  Linear programming using a graphical method    Critical path analysis    Simulation    Although decision mathematics modules have been taught in schools for two decades now, it remains the least established and least prevalent strand of applied mathematics within the A-levels. The popular examination board Edexcel offers its candidates five mechanics modules and four examining statistical knowledge, but only two in decision mathematics (EdExcel, 2013), whilst smaller boards such as WJEC do not offer decision modules at all. Although it is hard to draw transparent comparisons when many modules can count towards either A-level mathematics or A-level further mathematics, the evidence suggests that decision mathematics modules are offered less frequently and less widely by schools than the analogous units in mechanics and statistics, such that decision mathematics is often ignored, relegated to further mathematics only or assigned as independent study (Ward-Penny et al., 2013). The relative place of decision mathematics within A-level mathematics was apparent in the evaluation of participation carried out by the QCA (2007), which reported notional results for the examination board Edexcel by unit combination (p. 11). Only 1.0% of candidates had studied both D1 and D2, compared to 21.8% who had studied two mechanics modules and 20.2% who had studied two statistics modules. Similarly, just 12.9% of combinations had included some decision mathematics, whereas 71.6% included at least one unit of mechanics and 72.5% some study of statistics. Participation in decision mathematics is likely to decrease from current levels in the immediate future, as it is soon to be excluded fully from the newly linear A-level in mathematics in the next round of qualification reforms, for first teaching in 2017 (DfE, 2015, p. 26). Nonetheless, decision mathematics does continue to feature in some draft specifications for further mathematics. For instance, EdExcel (2016) are planning to retain two decision modules, whilst the OCR (MEI) draft specification has replaced them with a new minor option, ‘Modelling with Algorithms’ (OCR, 2016). Decision mathematics also appears to be the most contentious area of applied mathematics. After observing some D1 lessons and interviewing staff, Hernandez-Martinez and Williams (2008) noted cases where decision mathematics ‘is taught in a procedural way’, and questioned its relative level of challenge: ‘in some colleges (it) has become a way to achieve a good pass rate’ (p. 47). Similarly, the QCA (2007) evaluation reported on an ‘informal straw poll’ of teachers and experts that placed D1 amongst the easiest options (p. 10), and the survey of teachers’ perceptions of applied strands carried out by Ward-Penny et al. (2013) found that many teachers viewed decision mathematics as easier overall (pp. 6–7). One of the teachers interviewed questioned the intrinsic worth of decision mathematics: ‘I think we would probably say that we regard D1 as being the easiest one … one reason for that is that it doesn’t contain any maths! Or very little … there’s almost no connection with anything algebraic …’ (p. 13). Conversely, other teachers in the same article commended decision mathematics for being relevant and accessible, and one claimed that their students in fact found decision mathematics ‘hard because they have not seen anything like it’ (p. 9). Another said that it was divisive amongst their students, characterizing it as ‘a ‘‘love it or hate it’’ subject. Those that can work logically and systematically tend to do well and find it easy. Those that can’t get over it ‘‘not being proper maths’’ (their words, not mine) struggle and tend to get low grades …’ (p. 13). The contention that decision mathematics is an easier option for students has also been disputed by awarding bodies and the regulatory agency Ofqual on statistical grounds (ALCAB, 2014, p. 8). Finally, decision mathematics is arguably the most unfamiliar area of applied mathematics for the A-level students themselves: although mechanics can be located against the background of physics, and statistics has been met before within the mathematics curricula prescribed for the UK nations, the content and approaches of the decision mathematics modules are largely novel. In spite of these distinctions there is limited data reporting students’ own considerations of decision mathematics or studying how decision mathematics impacts students’ conceptions of mathematics as a whole. The report of the QCA (2006, p. 61) notes instances of both positive and negative attitudes towards decision mathematics, but offers limited detail; Hernandez-Martinez and Williams (2008) offer greater depth but focus on teachers and only the first decision mathematics module. Between 2013 and 2015 the author was one of the two further mathematics teachers of a small group of students who had studied D1 and who, after some reflection, consciously elected to study D2. It was recognized that these students were in an exceptional position. Not only were they in the middle of studying both decision modules, but the process of choosing between modules for themselves as a class had led them to consider the worth, content and relative difficulty of different areas of mathematics. They therefore constituted an especially rich and accessible source of data regarding students’ perceptions of decision mathematics, and presented an opening for an exploratory study. This article reports on how these learners’ perceptions of decision mathematics evolved as they came to know more about the field. Moreover, it takes advantage of this circumstance to look for evidence regarding how students construct and re-construct their perceptions of mathematics as a curriculum discipline. The students’ accounts relate how the topics taught within decision mathematics might continue to have a particular place in the wide spectrum of the A-level mathematics and further mathematics curricula, and suggest that the attendance of decision mathematics topics within the A-level qualifications can extend learners’ appreciation of the potency and scope of applied mathematics. 2. The students and their context The learners who make up this study were studying the OCR MEI form of A-level further mathematics over two years in a partially selective, inner city London academy. Ten students, eight male and two female, had both successfully passed the first year of the course and opted to continue into the second year in order to gain the full qualification. In line with the school’s predictions and the students’ own aspirations, the majority were expecting to attain one of the top grades, although a small number were working towards a lower passing mark. The author was one of two teachers who shared the class, and was responsible for two of five further mathematics lessons each week. All 10 students had already studied the first decision mathematics module, D1, as part of their first- year work, and received the results of all of their first-year examinations. Their second-year curriculum was expected to consist of: the pure mathematics module FP2; the mechanics module M2; and a module on differential equations, DE. However, my colleague and I were concerned about the balance of topics and techniques covered in these three modules. It was felt that, having done well with D1, the students attaining at a lower level might benefit from swapping DE for the second decision mathematics module, D2. Conversely, some of the higher performing students might particularly enjoy the content and challenges of DE. After deliberation both teachers felt it was reasonable, and perhaps fairest, to put this choice to the students themselves. The teachers together sketched out both modules to the students, attempting to outline them in a balanced way that focused on the strengths of each module, and to award genuine agency to the students in this matter. In brief, DE was presented as being more algebraic, complementary to the mechanics work, related to engineering and containing a coursework component, whereas D2 was summarized as being more algorithmic, building on D1, related to computer science and free of coursework. The students were given this choice over a matter of weeks and some did research into their options, so that this was a reasonably informed and considered choice. It was also repeatedly made clear to the students that both teachers were happy to teach both modules and that the teachers considered each to have its own particular value. The final result was six votes to four in favour of D2, and so the second decision module was chosen to be taught mid-way through the students’ second year of study. Shortly after this ballot the author recognized the unusual potential for researching students’ conceptions of decision mathematics and constructed the research questions of this study. 3. Research questions and theoretical framework The leading research question for this project was predominantly descriptive: how did these students view decision mathematics, and how did their perceptions evolve after working through the second decision mathematics module? The associated second research question was more exploratory: in what ways might these learners’ developing perceptions of decision mathematics indicate how they appraised and located mathematical activity? Both of the research questions were predicated on a broad concept of mathematical practice, as well as the idea that students can and do read validity and worth into classroom activity. The study also assumed the position that the students’ notions of mathematical practice were related to, and reflective of, their own mathematical identities and self-positioning as practitioners of mathematics. Whilst a full discussion of the place of identity in mathematics lies outside the scope of this article, these students’ identities were considered to be developing and related to both practice and agency (Boaler and Greeno, 2000; Black et al. 2009). In particular, these A-level students were in the midst of selecting and securing places in higher education; they would have therefore been questioning the extent to which they wanted to continue to align themselves outwardly with the field of mathematics. In light of the students’ context, the research questions and the relevance of the students’ own identities, the concept of ‘figured worlds’ (Holland et al., 1998) was selected as an analytic frame for use throughout this study. The term ‘figured world’ references a ‘constructed realm of interpretation in which particular characters and actors are recognized, significance is assigned to certain acts, and particular outcomes are valued over others’ (p. 52). Figured worlds are intersubjective, ‘socially produced and culturally constructed activities’ (pp. 40–41) which provide a context for both meaning and identity work such that ‘identities are formed in the process of participating in activities organised by figured worlds’ (p. 57). Urietta (2007) likewise describes figured worlds as ‘processes or traditions of apprehension that give people shape and form as their lives intersect with them’ (p. 108). Finally, this study was conducted with recognition of the idea of fixed and growth mindsets as presented by Dweck (2000). A student with a fixed mindset typically believes that mathematical ability is an innate quality, sees academic difficulties as a sign that things are getting ‘too hard’ and perceives grades or marks in reference to themselves as a person. A student with a growth mindset instead considers mathematical ability to be malleable, sees challenge as a natural part of the learning process and understands grades as being reflective of their own performance, not their person. This distinction was considered pertinent, not only because of its relationship to mathematical identity, but also because concerns about accomplishment and formative assessments would likely have been paramount in the final year of the A-level course as the students applied to university and received offers dependent on grades. Darlington’s (2015) review also reinforced the particular relevance of performance to further mathematics students. For instance, 80.9% of students surveyed therein said that they had chosen further mathematics as they were ‘better at Maths than at other subjects’ (p. 15). 4. Methodology The study took place in two phases and adopted a mixed methods approach. In each phase the central research instrument was a questionnaire chiefly consisting of 16 items, presented with five-point Likert-type scales. The wording of these questions drew on previous research into teachers’ perceptions of applied modules (Ward-Penny et al., 2013) as well as wider instruments measuring learner affect (such as Tapia and Marsh, 2004) but it was adapted and bolstered in light of the particular context. The questions were phrased both positively and negatively, and together were considered at the design stage to encompass three non-orthogonal dimensions of perception related to outcomes, affect and value. Cronbach’s alpha calculations using Microsoft Excel with the first set of questionnaires (n = 10) yielded results of 0.919, 0.879 and 0.810, respectively; these results shifted to 0.844, 0.877 and 0.734, respectively when both sets were combined (n = 20). In this way, although the size of the study limits any proposal of well-defined constructs, reliability analysis supports these groupings as a typology for the presentation of the results. The first set of questionnaires was distributed in the second year of the two-year further mathematics course, hence well after D1 but before D2 was taught; the second set was distributed towards the end of the second year, both after the teaching for D2 had been finished and some time had passed to allow for past paper practice. The first questionnaire included an introductory item that asked each student to explain the reasoning behind their vote, and both versions concluded with two open-ended questions about decision mathematics that were intended to check content validity and further promote reflection; these are considered in more detail below. After a preliminary analysis of both sets of questionnaires, five students were purposively selected for a short interview using their questionnaire responses, so as to ensure a mix of genders, attainment levels and perspectives. The questions for the semi-structured interviews were designed to promote validity through retelling and clarification. After transcription these comments were combined with the qualitative questionnaire data for analysis. Some quotes presented herein include minor edits, such as the removal of spelling errors, misspeaking and spoken fillers such as ‘like’, for the purposes of clarity. The author’s dual role as teacher and researcher raised ethical concerns that were addressed throughout. Primarily, there was a conscious effort to manage both the timing and execution of the research so that it was clearly subordinate to the students’ learning. It was made immediately and repeatedly clear that student responses would not affect grades, reports or the conduct of either teacher. In light of the age and relative maturity of the students, the aims and methods of the research project were also shared openly, and all 10 agreed to the anonymized use of their data; in addition, gender neutral pronouns have been used throughout this article. As the research repeatedly involved the same students, a high premium was put on their time. This limited the number of open-ended items on the questionnaire and contributed to the decision to sample the students at the interview stage. It was considered that interviews with half of the students would provide enough data for analysis without presuming further on the students’ time. Further to this, the author’s dual role also resulted in some less obvious tensions which had to be navigated during the research (Atkinson, 1994). For example, both parties had to renegotiate their roles with regard to the act of questioning. As a teacher, the author would typically ask a question of a student from a position of knowledge, and there existed a mutual recognition from both parties that extended lines of questioning most often arose to address misconceptions, such that the student was expected to ultimately change their response. However, as a researcher the author was asking questions from a position of enquiry, and follow-up questions were intended to confirm, not challenge, the students’ contributions. To address this, each interview started with a short introduction that explicitly signalled a change in expectations, stating that there were ‘no right or wrong answers’ and stressed a genuine interest in each student’s personal opinions. Whilst the author’s influence was inevitable in both the design and exercise of this research, steps were also taken to minimize deliberate or subconscious prompting of the students beyond that which would have occurred naturally through unobserved teaching. For instance, the first questionnaires were not analysed at the level of the individual until both sets had been collected, so that the author could not purposely attempt to influence the perceptions of any specific student through targeted comments or action whilst teaching D2. Further, the author attempted to draw principally on discourses surrounding the value and worth of the curriculum content which were already present in some form in existing resources such as the MEI textbook. The incidence of a co-teacher also meant that the students were exposed to a second perspective from a person in a position of power. Nonetheless, it is apposite to note briefly here that the author holds a principally positive perception of decision mathematics, and that he personally considers all strands of applied mathematics to have utility and appeal. 5. Results 5.1 Reasons for voting The first questionnaire began by exploring the vote through which the students had chosen D2 over the differential equations module, DE. The opening item asked each student to identify which module they had voted for and then briefly offer their reasons. All six learners who voted for D2 referenced D1 in their answer: four stated that they had ‘liked’ or ‘enjoyed’ the first decision module and two said that they wanted to build on what they had already achieved. Three of those who voted for D2 expressed specific reservations about DE, regarding either the perceived relative difficulty of more calculus, or the presence of a coursework element. Conversely, some of the students who voted for DE said that they had been unimpressed with D1: ‘D2 looks okay but D1 was not particularly inspiring so much’; ‘I really really disliked D1 … also calculus is my type of maths.’ Thematically, the comments volunteered at this initial stage by the students who voted for D2 were clustered around outcomes and affective elements, whereas half of those who voted for DE expressed thoughts about future utility: ‘DE seems more useful in general and especially for continuing maths at uni’; ‘I thought it would be more useful for what I want to do later on.’ 5.2 Outcomes A total of 4 of the 16 Likert-type items explored the students’ perceptions of how the inclusion of decision mathematics modules might affect their achievement outcomes. These items were included in recognition of both the debate surrounding the relative difficulty of decision mathematics, and the import of the qualification to the students’ future plans. Two items explicitly concerned the students’ marks and are presented in Table 2; Q6 was worded as a reverse item. Table 2. Questionnaire results relating to outcomes - part one (n = 10); SD = strongly disagree, D = disagree, N = neutral, A = agree, SA = strongly agree   After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q2: I am expecting to get a high mark in my decision mathematics modules.  0  2  2  5  1  0  0  4  5  1  Q6: I find it difficult to get a good mark on decision mathematics examination papers.  0  6  2  1  1  0  4  5  1  0    After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q2: I am expecting to get a high mark in my decision mathematics modules.  0  2  2  5  1  0  0  4  5  1  Q6: I find it difficult to get a good mark on decision mathematics examination papers.  0  6  2  1  1  0  4  5  1  0  These results show that the students considered decision mathematics modules to be manageable both before and after studying D2. Only three students chose to select a different response to their original one when answering Q2 in the second iteration of the questionnaire; overall, the students had like expectations over time. This consistency is not contradicted by the slight shift over time in the responses to Q6, since the grade boundaries for the practised D2 examination papers were typically lower than those seen in other modules. Attitudes here may also have been steered by attainment in D1 and the other mathematics units as well as the students’ personal criteria for success. For instance, one of the academically weaker students stated in the open-ended questions that ‘if you practise a lot of questions it’s easier to get a good mark than other modules’, but one of the students aiming for a top grade offered that the make-up of the examination papers was an issue for them: ‘I do like aspects and can see how they’re relevant to real-life situations, but I dislike the unpredictability of the question paper.’ The results of the remaining two outcome-focused items are given in Table 3. Even in light of the small sample size, the responses to Q10 entail a notable spread, with the students’ attitudes appearing to polarize somewhat after the study of D2. Hence, whilst the students felt that they were able to achieve in this module, and despite the concerns in the extant literature about decision mathematics being a slighter option (see Section 1), these further mathematics students did not uniformly find decision mathematics easy. Table 3. Questionnaire results relating to outcomes - part two (n = 10); SD = strongly disagree, D = disagree, N = neutral, A = agree, SA = strongly agree   After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q10: Decision mathematics is easier than other mathematics modules.  2  2  2  4  0  1  4  1  3  1  Q16: I am expecting to meet my overall predicted grade in my decision mathematics modules.  0  2  1  5  2  0  0  3  6  1    After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q10: Decision mathematics is easier than other mathematics modules.  2  2  2  4  0  1  4  1  3  1  Q16: I am expecting to meet my overall predicted grade in my decision mathematics modules.  0  2  1  5  2  0  0  3  6  1  5.3 Affect Another six of the Likert-type items explored the affective dimension of the students’ study of decision mathematics, with two items (Q9 and Q15) worded negatively. All six items were constructed in light of the differing opinions present in the existing literature (see Section 1). The items are presented together in Table 4 in the order that they appeared on the instrument. Table 4. Questionnaire results relating to affect (n = 10); SD = strongly disagree, D = disagree, N = neutral, A = agree, SA = strongly agree   After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q1: I enjoy learning decision mathematics.  0  1  3  4  2  0  0  2  7  1  Q4: I have a lot of self-confidence when it comes to decision mathematics.  1  2  1  5  1  0  0  6  4  0  Q8: I prefer decision mathematics to my other mathematics modules.  1  3  4  2  0  1  4  4  0  1  Q9: I do not enjoy working with the ideas and techniques of decision mathematics.  2  5  1  2  0  2  6  1  1  0  Q13: I find decision mathematics interesting.  0  1  1  5  3  0  1  1  6  2  Q15: I do not feel comfortable solving problems using decision mathematics.  0  7  3  0  0  1  7  2  0  0    After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q1: I enjoy learning decision mathematics.  0  1  3  4  2  0  0  2  7  1  Q4: I have a lot of self-confidence when it comes to decision mathematics.  1  2  1  5  1  0  0  6  4  0  Q8: I prefer decision mathematics to my other mathematics modules.  1  3  4  2  0  1  4  4  0  1  Q9: I do not enjoy working with the ideas and techniques of decision mathematics.  2  5  1  2  0  2  6  1  1  0  Q13: I find decision mathematics interesting.  0  1  1  5  3  0  1  1  6  2  Q15: I do not feel comfortable solving problems using decision mathematics.  0  7  3  0  0  1  7  2  0  0  Even though it might be presumed that students working towards a full A-level in further mathematics are likely to have a positive baseline attitude towards mathematics, the results of Q1 and Q9 in particular endorse that these 10 students generally enjoyed decision mathematics as a cohort, and that they reported slightly higher levels of enjoyment after studying D2. A paired comparison of the two sets of questionnaires showed that nine and eight students, respectively disclosed an equal or more positive response to these two items in the second questionnaire. The item for which the most students altered their response was Q4, where only two students chose the same answer on both questionnaires. Although this degree of reporting is offered cautiously in light of factors such as central tendency bias, the table alone gives the impression of a positive shift in the self-confidence of the pupils who had at first struggled with D1, as well as an affective diminution at the top end, possibly influenced once more by the higher level of challenge and the shifted grade boundaries. Conversely, the negatively phrased Q15 was the most stable item on the entire questionnaire, with eight students offering the same response on both iterations. In Q13, 80% of the cohort agreed or agreed strongly that decision mathematics was interesting. In light of the broadly positive affective profile, the muted neutral and negative responses to Q8, ‘I prefer decision mathematics to my other mathematics modules’, were perhaps then a little surprising. This ambivalence was also present in the supporting qualitative data when pupils were asked to give their overall impression of decision mathematics. The mood of this ranged from the extremely positive ‘it’s great’ to the following oblique criticism from the single student who had chosen ‘strongly disagree’ for this item on both questionnaires: ‘not good, I prefer the other modules even statistics’. Nonetheless, when this student was asked the same question on the second questionnaire, they simply wrote that decision mathematics was ‘decent’. 5.4 Value The remaining six Likert-type items concerned various aspects of the value or utility of decision mathematics. These items were intended to address further some of the debates summarized in question one whilst also speaking to the second, wider research question regarding students’ appraisal of mathematical activity. Two items were again reversed to allow for internal reliability checks, specifically Q5 and Q11. The results are presented in Table 5. Table 5. Questionnaire results relating to value (n = 10); SD = strongly disagree, D = disagree, N = neutral, A = agree, SA = strongly agree   After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q3: I think of decision mathematics as a worthwhile part of mathematics/further mathematics.  0  1  3  4  2  0  0  1  7  2  Q5: I do not think that decision mathematics is very mathematical.  0  4  5  1  0  2  5  1  2  0  Q7: It is easy to see how decision mathematics relates to real-life situations.  0  0  0  4  6  0  0  2  3  5  Q11: I find it hard to see how decision mathematics might be used in jobs or careers.  4  3  3  0  0  6  3  1  0  0  Q12: I think that decision mathematics helps me develop my problem-solving and thinking skills.  0  0  2  5  3  0  0  2  4  4  Q14: I think that decision mathematics is a valuable branch of applied mathematics.  0  0  2  6  2  0  0  1  7  2    After D1 only   After D1 and D2   SD  D  N  A  SA  SD  D  N  A  SA  Q3: I think of decision mathematics as a worthwhile part of mathematics/further mathematics.  0  1  3  4  2  0  0  1  7  2  Q5: I do not think that decision mathematics is very mathematical.  0  4  5  1  0  2  5  1  2  0  Q7: It is easy to see how decision mathematics relates to real-life situations.  0  0  0  4  6  0  0  2  3  5  Q11: I find it hard to see how decision mathematics might be used in jobs or careers.  4  3  3  0  0  6  3  1  0  0  Q12: I think that decision mathematics helps me develop my problem-solving and thinking skills.  0  0  2  5  3  0  0  2  4  4  Q14: I think that decision mathematics is a valuable branch of applied mathematics.  0  0  2  6  2  0  0  1  7  2  Of the three groups, these questions implicate the biggest change in opinion, with perhaps the largest shifts relating to the place of decision mathematics within the mathematical canon: Q3 and Q5 both show affirmative swings, and in both items five of the students altered their responses between the questionnaires in favour of decision mathematics. Smaller positive moves can be read into the overall results of Q11 and Q14. The results of Q7 seem to stand against this trend, with four students agreeing less strongly that ‘it is easy to see how decision mathematics relates to real-life situations’. This may be understood in light of the curriculum content of the OCR MEI syllabus, as outlined in Table 1. Whereas the bulk of D1 easily lends itself to context-based work, the applications of more abstract topics such as logic and Boolean algebra in D2 are perhaps less tangible and immediate for students. Notwithstanding, the qualitative data fully supports the idea of decision mathematics overall as being strongly anchored in context. In the first questionnaire one student reported that ‘D1 + 2 are very useful in the real-world. It is more applied than other modules. Very interesting’. The utility of decision mathematics was widely noted, though it was not always allied with either affect nor outcome: ‘It’s okay, useful, but not all that interesting’; ‘I haven’t found it easy at all, as this has taken time to adapt to, however I have found it valuable’. Another told that it was particularly relevant in preparing to study computer science. The results of Q12 suggest that the students concerned were further able to connect their decision mathematics learning with the development of wider problem-solving skills. This was returned to by some in the open-ended questions, with one student offering that ‘I like it because it requires more thinking in terms of the context, which would be useful for my future career as an engineer.’ When the students were asked in the first questionnaire to express what they thought of when they thought of decision mathematics, the most common response categories were problems/problem-solving (five learners) and practical/real-life contexts (four learners). Once again though, there was a significant undercurrent of negative comments in this regard; some felt that elements were laborious or tantamount to number crunching, with one learner stating that decision mathematics felt like ‘going through the motions (with) a little less room for creativity’. In summary, the questionnaire results show that the students taken as a whole judged decision mathematics to be manageable though not always easy; interesting and enjoyable in the main; and broadly valuable and related to real-life. Their attitudes changed slightly after studying D2, typically in a positive direction, with the largest observed changes relating to the value and utility of decision mathematics. When they were asked at the end of the second questionnaire whether they would recommend decision mathematics to a friend about to study A-level mathematics, seven said ‘yes’, and the other three gave a conditional ‘it depends’ style response. However, while some students offered positive comments without provision, there was a persistent tension implied by some of the remarks, such as this summative juxtaposition: ‘it’s alright but it’s not the most interesting branch of mathematics. It’s a pretty cool way of looking at problems though I guess.’ This was explored further in the interviews. 5.5 Figuring decision mathematics The interviews were semi-structured and based around a set of five questions designed to explore the students’ thoughts on decision mathematics, how they saw the progression from D1 to D2 and how they felt decision mathematics fit within the wider A-level qualification. Each of the five interviews lasted between 6 and 10 minutes, and the students’ accounts are discussed here in turn. The first interview question asked the participant to summarize their thoughts about decision mathematics in their own words. Student A said that ‘I think it’s formalising the logic you’d use in normal mathematical problems.’ This was the first of a number of comments that included the use of a linguistic marker which separated off decision mathematics, in this case the word ‘normal’. Elsewhere they variously described decision mathematics as a ‘brand of maths’, ‘a different take to maths … a different flavour of maths.’ Student A considered that D2 was a stronger module than D1 in some respects, but not all: D2 ‘took what D1 was and made it more powerful so simplex is like linear programming, but expanded so you have more variables. Floyd’s is just Dijkstra n times… I think it’s just expanding what we did on D1’. Indeed, ‘it’s very much, kind of, going through the process … maybe if there was something like on how to improve the algorithms some of that might be a bit more interesting …’. Student A had originally wanted to study DE instead of D2 and uniquely had gone on to study the differential equations module as an additional endeavour. They felt that the models they had used in DE were more sophisticated. For student A, decision mathematics was offered as part of A-level mathematics and further mathematics in order to diversify the modules available and to appeal to a wider variety of people: ‘I’m sure there are some people in this class who would hate DE but like D2 and vice versa.’ Nonetheless, the tenor of their comments implied some distance between their own preferences for practice and decision mathematics. Student B had also voted for the differential equations module. They had presented the most negative affective profile in the two questionnaires, and were the only student who had strongly disagreed with Q8, ‘I prefer decision mathematics to my other mathematics modules.’ When asked in the questionnaire whether they would advise a friend to take decision mathematics, they wrote ‘if you want a more full on maths module then decision is not for you.’ In a similar manner to student A, during the interview B repeatedly distanced themselves as an individual mathematician from decision mathematics: it was ‘not my cup of tea … just different thinking, that’s not really my type of thinking’. Later, when discussing its utility, they said that ‘I can see how it can be useful. Probably not useful for me, personally.’ Similarly, they admitted ‘I think it does make you think differently. I don’t know if that thinking differently will apply to me…’. Other resonances were observable when student B considered why decision mathematics was present in the curriculum: it was included because ‘everyone’s different’… it may be helpful for ‘people who want to do programming’ or set up a business. Their perception of decision mathematics as something meaningful but subordinate was particularly evidenced by their use of a metaphor: mathematics was a big tree, where core maths was the trunk of the tree, and ‘decision might be a thinner branch, like mechanics might be thick, slightly thicker…’ Unlike the preceding two participants, student C had voted for D2. This was a partly strategic choice, for although they had struggled a little with D1 they were retaking this earlier module and felt that the overlap would aid their revision. Ultimately, student C showed the greatest increase in affect towards decision mathematics between the two questionnaires. Student C described decision mathematics as ‘applying maths and logic to real world situations and interpreting them.’ They felt that moving from D1 to D2 had widened the relevance of decision mathematics. Beforehand they ‘knew why people did it for these very specific scenarios but I didn’t really understand at the time how that would, sort of, broaden…’. Later they confirmed that knowing of more applications helped them feel decision mathematics was more worthwhile. Further, whilst ‘D1 for me was very procedural’, D2 ‘asks more of you, asks more of your understanding’. In opposition to the comments of student A, student C felt that this qualitative shift was greater for decision mathematics than for other strands, citing that the second mechanics module was instead largely about ‘factoring in more things’. This perceived shift in the requisite depth of understanding was important to student C in a way they confessed was ‘weird’: ‘I found it harder but I’m sort of more aware of why I’m doing things … it’s sort of the way I learn, if I don’t really fully understand something I really, really struggle just to do it.’ When questioned about Q5, ‘I do not think that decision mathematics is very mathematical’, student C noted that they had responded with a neutral answer both times, but not for the same reason. After D1 they had not been convinced that decision mathematics was not as mathematical as their other work, ‘but when I put neutral now it’s because my understanding of “mathematical” changed’. Student C explicitly stated that studying more decision mathematics had overtly shifted their perception of what qualified as mathematics; this is discussed in further depth in the next section. Again in opposition to some of the preceding interviews, they shared that decision mathematics ‘doesn’t have that surface complexity, but … it has a complexity behind it that requires you to think mathematically…’. By way of contrast, student D strongly endorsed decision mathematics in both questionnaires, offering the most positive affective profile overall. When they were asked to summarise their thoughts about decision mathematics the leading feature of their response was the applicability of the strand, rather than its character, or any specific content. They praised the immediacy of the applications and compared the models of decision mathematics favourably with other strands, saying that in mechanics ‘it feels like they’re quite basic models and you can do so much more to sort of actually give a realistic thing … but with D2 it just feels like you’re already doing stuff that can be useful.’ As with student C, student D noted that their learning experiences had shifted their internal perspective of mathematics: ‘I feel maybe I changed my idea on what mathematical is between the two (questionnaires) … it’s not just, something needs to, be like pure maths …’. Another possible element contributing to this shift was the introduction of more ‘advanced’ mathematical tools, symbols and notation. Student D recounted that D1 primarily involved the application of The General Certificate of Secondary Education (GCSE) level skills, but felt that features such as the use of matrices had made D2 more mathematical. Student E was academically the weakest of the interviewees in terms of their overall results, but they had been relatively successful with decision mathematics; this is likely to have influenced their attitude. They had voted for D2, and said that ‘I like it. It seems quite nice, really logical. Yeah, it sort of makes sense to me.’ Indeed, although they conceded that this was a personal view, when they were asked to volunteer a less logical module, they unexpectedly proposed the further pure module FP2, which consists of topics such as polar co-ordinates and Maclaurin series, as well as more advanced work involving matrices and complex numbers. Student E’s perceptions had been challenged when moving from D1 to D2. They felt that D2 was ‘more complex’ than D1, ‘which goes against what I thought decision maths was in terms of mathematical stuff’. When they were asked to expand on this point they said that ‘I have it in my head that decision maths is much less complex in terms of like pure mathematics … D2 seemed to go against that more than D1 did.’ They had felt that the simplex tables and Boolean algebra of D2, arguably the most symbolic elements of the module, were more mathematical. On the questionnaires, student E had ‘agreed’ both times through Q5 that decision as a strand was not very mathematical. They were not entirely confident explaining their view however: ‘Not too sure to be honest … much less obvious number plugging, and rearranging equations and stuff, which seems to be more like mechanics or the other maths modules… .’ Regardless, they were certain about wanting decision mathematics to remain on offer to students; not only had they liked it, ‘it wouldn’t be under any other subject… I do think it’s useful and worth doing.’ 6. Discussion: positioning legitimate mathematical activity The narratives of positioning outlined in the five interviews above can be construed in terms of figured worlds and mathematical identities. An A-level student might react to the fact and practice of the decision mathematics modules by rationalizing decision mathematics as being somehow separate from what has come before, attaching less significance to the practice of decision mathematics, predominantly maintaining their sense of the figured world of mathematics and insulating their mathematical identity. Alternatively, the novel aspects of decision mathematics can drive a marginal renegotiation of a learner’s conception of the quality of being mathematical, alter their sense of the figured world of mathematics and inform their perception of themselves as a mathematician. Both the explicit comments and linguistic markers in the data are indices of these two contrasting cases. In the first case, the learner locates decision mathematics as an adjunct, and then positions themselves at some distance from this applied strand. This approach can be read into the comments of student A, who uses the labels ‘normal’ and ‘different’ and places more emphasis on the procedural aspects of decision mathematics, ostensibly devaluing its intellectual currency. A similar tactic can be inferred from the words of student B, who sites himself ‘personally’ aside from decision, thinking of it as other people’s mathematics and characterizing it as a ‘thinner branch’ on the tree of mathematics. Since both of these learners were high-achieving students, this restricted assimilation of decision mathematics might be construed as an ego defence strategy. It has already been noted above that decision mathematics assessments are in some ways less predictable, and this teaching of D2 in particular involved some expectation of high challenge, unexpected elements and lower grade boundaries. If a very able student possessed a fixed mindset inasmuch as their mathematical identity was bound up with perfection or high percentage scores, aspects of the decision mathematics modules and assessments could be perceived as threats. Related ego defence strategies have been noted in the accounts of slightly older undergraduate mathematicians who were forced to move away from the expectations of ‘one hundred percent’ that they inherited from school mathematics (Ward-Penny et al., 2011). Building on this reading, it is possible to interpret student E’s account as a supporting counterpoint. Student E was aiming for a lower grade than the previous two students, and likely saw themselves as a different sort of actor within the figured world of mathematics than the other interviewed students. However, student E was both more comfortable with, and achieving more highly in, decision mathematics than in the other modules. Student E’s ego defence thus works in the opposite direction to that of students A and B. Asserting the alterity of decision mathematics helps student E to rationalize their success without having to change their sense of themselves and the figured world of mathematics, by fencing off and validating an area of achievement on the periphery. Decision mathematics is awarded the title of ‘logical’ and detailed in opposition: in the questionnaire it ‘is nice, seems to be more about logic and clear thinking more than it relies on complicated maths like FP2 for example’, just as in the interview ‘it seems quite nice, really logical’. Learners who are perhaps less involved with measures of absolute success are arguably freer to extend their sense of the figured world of mathematics, by attaching significance to decision mathematics and positioning it as fully legitimate mathematical practice. This may have been the case for both student C and student D, who remarked explicitly in the preceding section about how their perceptions of mathematics had changed through studying the two decision modules. This willing accommodation is further hinted at in the words of student D, who liked starting the first year with D1, because it was ‘a nice introduction to what further maths is like’; decision mathematics had evidently featured in their sense of the figured world of (further) mathematics. This is not to say that students who are more ready to assign value to decision mathematics do not recognize differences in the demands of the decision modules; on their questionnaire student C conceded that ‘technically, it might not be as mathematically challenging but it changes your perspective on maths as a whole’. However, such distinctions are construed differently than before, such that these students seemed more likely to foreground the conceptual demands of decision mathematics. Two students who were not interviewed offered consonant comments on their questionnaires: ‘The maths isn’t as complex as other maths modules, but the overall concept is harder to understand’; ‘Still Maths but you need to think differently and actually understand what is going on more than just memorising the steps as you maybe would in other modules.’ Perhaps the most interesting narrative under this reading is that of student C, who appears to have moved from the first cluster of students to the second between the two decision mathematics modules. D1 for student C had been ‘very procedural’ with a narrow focus of application; hence, in the first questionnaire they exhibited some of the linguistic markers discussed above, saying ‘I think of logic when I think of decision mathematics.’ Contrariwise, they felt D2 ‘asks more of your understanding’, has a wider range of application, ‘required me to use a different way of thinking and has changed the way I approach problems, and view situations. I haven’t found it easy at all, as this has taken time to adapt to, however I have found it valuable.’ These descriptions are resonant not only with the two groups of interview accounts, but also with some of the trends in the overall questionnaire data, particularly those discussed in Section 5.4. Student C’s transition also gives rise to the question of what might have happened to the perceptions of the group as a whole if the students had gone on to study a third module of decision mathematics. In this way, the five interviews can be understood as being indicative of two contrasting ways in which students might perceive and position decision mathematics. Framing these accounts in terms of figured worlds and mathematical identities may also speak to the findings of the literature as summarized in Section 1, suggesting additional reasons why decision mathematics might be so divisive, and why some students cannot (or elect not to) get over it ‘not being proper maths’. 7. Conclusion Whilst the scale of this research prohibits generalization, the findings of this study do appear to mitigate in some measure some of the more general criticisms of decision mathematics cited in recent research and reports such as those listed in Section 1. The majority of these able further mathematics students found worth in, were challenged by, and enjoyed the experience of learning decision mathematics. Further, the students’ perception and appreciation of decision mathematics evolved after studying the second module. Such advances are arguably predictable, but it is nonetheless tendered that there is worth in mentioning them here; although debates will continue to be held regarding the comparative academic challenge and place of the applied strands, the expansion and elaboration present in D2 should not be overlooked because of the relative frequency of the present teaching of this module. This reminder is particularly pertinent in light of the present changes to the place of decision mathematics in future A-level portfolios. Since decision mathematics has the potential to be positioned both within and to one side of each learner’s personal mathematical canon, these students’ accounts have also exemplified how a learner might adjust their sense of the figured world of mathematics in light of novel curriculum elements. In particular, the narrative of student C strongly suggests that an individual can switch between positioning strategies, and serves as a reminder that learners’ perceptions of mathematics are open to renegotiation, such that the quality of being mathematical is clarified by both narrative and action. There is scope for subsequent, fuller research to map and gauge the incidence of such strategies in other contexts, for instance, by looking at how students of different attainment might locate mechanics with respect to mathematics, or considering whether and how positioning strategies might be related to issues of affect and performance in a fuller range of students. Finally, it has been argued that to meet the needs of the 21st century, curriculum divisions should be ecological, not bureaucratic, and unshackled from any ‘rigid disciplinary structure’ (Noddings, 2013, p. 146). The pressing global problems of today undoubtedly demand logical thinking and quantitative literacy, but their solutions are unlikely to fall precisely within the single curriculum area that is mathematics. Noddings (2013) further proposes a strategy of ‘stretching the disciplines from within’ (p. 62) to facilitate a broad and connected education. For at least some of the students in the study, decision mathematics was understood in this way: as an area of study which supplemented the more senior aspects of the A-level mathematics curriculum, extended their sense of what mathematics was, and served to demonstrate how mathematical thought might be applied in a wide range of settings. Hence, whilst decision mathematics might not serve as a direct prerequisite for higher education courses, as noted by ALCAB (2014), these stories submit that it can work to augment learners’ perceptions of mathematics as a discipline, what it allows for, and its power within the world. This is certainly a valuable intention for students who are about to leave the formal study of mathematics behind, and a positive motivating factor for those who might choose at this critical juncture to pursue it further. Footnotes 1 The current OCR (MEI) specification is unique in that it also offers a third related module, Decision Mathematics Computation (DC) which integrates technology into the teaching and assessment of decision mathematics. Robert Ward-Penny received his PhD in Education from the University of Warwick in 2013 and is currently working as a mathematics teacher. His interests include cross-curricular teaching and learning in secondary school mathematics, and the relationship between philosophy and practice in mathematics education. References ALCAB ( 2014) Report of the ALCAB panel on mathematics and further mathematics. A Level Content Advisory Board. (https://alevelcontent.files.wordpress.com/2014/07/alcab-report-on-mathematics-and-further-mathematics-july-2014.pdf) [accessed April 2017]. Atkinson S. ( 1994) Rethinking the principles and practice of action research: the tensions for the teacher-researcher. Educ. Act. Res ., 2, 383– 401. Google Scholar CrossRef Search ADS   Black L., Mendick H., Solomon Y. (eds.) ( 2009) Mathematical Relationships in Education: Identities and Participation . London: Routledge. Boaler J., Greeno J. G. ( 2000) Identity, agency and knowing in mathematics worlds. Multiple Perspectives on Mathematics Teaching and Learning  ( Boaler J. ed.). Westport, CT: Ablex, pp. 171– 200. Darlington E. ( 2015) Students’ perceptions of A-level Further Mathematics as preparation for undergraduate Mathematics. In Adams, G. (ed.) Proceedings of the British Society for Research into Learning Mathematics, 35, 13–18. DfE ( 2015) Reformed GCSE and A level subject content consultation. Government response. Department for Education. (https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/397672/Reformed_GCSE_and_A_level_subject_content_Government_Response.pdf). Dweck C. ( 2000) Self-theories: Their Role in Motivation, Personality and Development . Lillington, NC: Psychology Press. EdExcel ( 2013) Specification. GCE Mathematics. Issue 3. Pearson. (https://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2013/Specification%20and%20sample%20assessments/UA035243_GCE_Lin_Maths_Issue_3.pdf). EdExcel ( 2016) Specification DRAFT. Pearson Edexcel Level 3 Advanced GCE in Further Mathematics. Pearson. (http://qualifications.pearson.com/content/dam/pdf/A%20Level/Mathematics/2017/specification-and-sample-assesment/a-level-l3-further-mathematics-specification.pdf). Hernandez-Martinez P., Williams J. ( 2008) Ethics, performativity and decision mathematics. In Joubert, M. (ed.) Proceedings of the British Society for Research into Learning Mathematics, 28, 43–48. Holland D., Lachicotte W.Jr., Skinner D., Cain C. ( 1998) Identity and Agency in Cultural Worlds . Cambridge, MA: Harvard University Press. MEI/OCR ( 2013) MEI Structured Mathematics. MEI/OCR. (http://www.ocr.org.uk/Images/75811-specification.pdf). Noddings N. ( 2013) Education and Democracy in the 21st Century . New York, NY: Teachers College Press. OCR ( 2016) OCR Level 3 Advanced GCE in Further Mathematics B (MEI) (H645) Specification (Draft). OCR. (http://www.ocr.org.uk/Images/308768-specification-draft-a-level-gce-further-mathematics-b-mei-h645.pdf). Porkess R. ( 2003) The new AS and A levels in mathematics. MSOR Connect , 3, 12– 16. Google Scholar CrossRef Search ADS   QCA ( 2006) Evaluation of participation in GCE mathematics. Interim report. Qualifications and Curriculum Authority Research Faculty. QCA ( 2007) Evaluation of participation in GCE mathematics. Final report. Qualifications and Curriculum Authority Research Faculty. Tapia M., Marsh G. E. ( 2004) An instrument to measure mathematics attitudes. Academic Exchange Quarterly , 8, 16– 21. Urietta L.Jr. ( 2007) Figured worlds and education: an introduction to the special issue. Urban Rev . 39, 107– 116. Google Scholar CrossRef Search ADS   Ward-Penny R., Johnston-Wilder S., Johnston-Wilder P. ( 2013) Discussing perception, determining provision: teachers’ perspectives on the applied options of A-level mathematics. Teach. Math. Appl ., 32, 1– 18. Ward-Penny R., Johnston-Wilder S., Lee C. ( 2011) Exit interviews: undergraduates who leave mathematics behind. For the Learning of Mathematics , 31, 21– 26. © The Author 2017. Published by Oxford University Press on behalf of The Institute of Mathematics and its Applications. All rights reserved. For permissions, please email: journals.permissions@oup.com This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

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Teaching Mathematics and Its Applications: International Journal of the IMAOxford University Press

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