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In order to answer the question ‘is learning mathematics at university different to learning mathematics at school’, this essay will focus on the difference between A-level and undergraduate mathematics education. In particular, we will consider differences in teaching styles, environments and mathematical thinking and how this affects students’ approaches to learning. I also will look into the difference between examination questions of both A-level and undergraduate examinations using application of the Mathematical Assessment Task Hierarchy (MATH) taxonomy (Smith et al., 1996), an adjustment to Bloom’s taxonomy for mathematics (Bloom et al., 1956), and Galbreith and Haines’ (2000) taxonomy. Finally, I will look into the different types of discourse used at A-level and undergraduate level.
The greatest change I noticed when coming to university was the change in teaching style. The type of teaching of a lecture is frequently described as transmission teaching, where the students listen to the lecturer and take notes from the board with little to no interaction between the students and the lecturer (Jaworski, 2017). Trigwell et al. (1994) describe this as a ‘teacher-focused’ strategy and based on the passive transmission of information to students. On the other hand, an A-level class has a teaching strategy described as ‘teacher/student interaction’ where students are taught through interactions and active engagement (Trigwell et al., 1994). Another difference is the change from classroom to lecture theatre as a learning environment, in particular the difference in class size. The average mathematics A-level class size in 2015-2016 was 13.9 in year 12 and 15.5 in year 13 (Parish et al., 2017) compared to a university lecture with a class size of 290 which is the size of the module MATH1010 Mathematics 1 at the University of Leeds (2013) this current year. Therefore, it is likely that students will have a different learning experience. Personally, I came from a small Sixth Form college with only 8 people in the Mathematics A-level class and this, in contrast with a lecture theatre filled with over 200 people that I did not know, was extremely different and something I had not experienced before. This affected my inclination and willingness to ask questions. I did not want to ask what I thought might be deemed a ‘silly’ question by my peers particularly in the first few weeks of university when you feel the people on your course are formulating their opinions of the people around them. At Sixth Form, I had no problem with asking any question; which was most likely due to the fact I had known my peers since year 7 and there was significantly fewer of them, which meant I felt more comfortable doing so. A similarity of class size can however be made between an A-level class and a tutorial class. Tutorials are used to work through problems set by the lecturer, which means that there can be more teacher/student interaction than in a lecture. However, my own experience is that this is not necessarily the case, which is supported by Jaworski (2017) who observed teaching during tutorials and found that often students struggle with beginning to solve these problems and so the tutorial frequently adopts a transmission approach to learning. Another difference is that at university there are significantly fewer contact hours than at A-level. This puts a meaningful importance on independence and a key aim for students at university is to become a successful, independent learner (Baird, 1988, cited in Gow and Kember, 1990). At university there is no one checking that you have put in the extra hours of independent study or reminding you to. Therefore, Mathematics study at university requires a greater degree of self-motivation and independent study than at A level. These findings above, supported by own experiences, show that both the teaching style, especially the ‘teacher focused’ strategy, and learning environments, such as a lecture theatre, mean that learning mathematics at school if different to university.
Another difference between A level and undergraduate mathematical study is in the level of mathematical thinking, with students experiencing a step up in mathematical thinking from A-level to university. Tall (1991, p.20) labels this step as ‘from describing to defining, from convincing to proving in a logical manner based on definitions’ and says that at this higher-level, students find difficulty with proofs as they are not familiar with the new mathematical culture. This is supported by Hanna et al. (2008) who argue that one of major reasons that mathematics students find the transition to university difficult is the emphasis on proof writing along with the new definitions and formal language involved in the proving. My own experiences support these statements. Before starting university, I had not come across proofs in the A-level and they were a large focus of many of my first-year modules which I felt unprepared for and it was a new type of learning that I was experiencing. I had to have complete understanding of the mathematics behind the proof. A deep approach to learning involves understanding the meaning of the subject matter, such as involved in proofs, and a surface approach focuses on memorising only the essentials (Gow and Kember, 1990). The surface approach to learning therefore means there is no need for the student to analyse or draw their own conclusions, something that is necessary for the deep approach to learning needed at university. In a questionnaire given to students by Crawford et al. (1994) 76% of students take a surface approached to learning at A-level contrasting with the deep approach taught at university. This drastic change in the approach to learning can play a part in why many students, like myself, feel unprepared for their undergraduate Mathematics study. Research has investigated this, with Darlington and Bowyer (2017) conducting an online questionnaire with Mathematics undergraduates who had taken Further Mathematics and Mathematics A-levels. The survey revealed that students felt that they had had unsatisfactory preparation for a Mathematics degree due to both A-levels not going into adequate depth on core topics, especially in Pure Mathematics, or mathematical concepts and felt there was a gap between the content at A-level and university especially in the ‘lack of proof and rigorous formal argument at A-level’ (Darlington and Bowyer, 2017, p.8).
The Advisory Committee on Mathematics Education [ACME] (2011) writes that there are concerns about the deficiency of mathematical knowledge of new undergraduate students and this is supported by multiple reports that students attaining high grades at A-level are struggling to apply basic mathematical skills and solve multi-step problems at undergraduate level (Hawkes and Savage, 2000; Savage, 2003, cited in Darlington and Bowyer, 2017). To study mathematics at university the Maths A-level is normally required at grade A or B (Universities and Colleges Admissions Service, 2018). The fact that students who have succeeded well at A-level are ill-equipped and struggling at undergraduate level highlights the difference in learning mathematics between the two due to the level of mathematical thinking involved. The ACME (2011) also reports there were criticisms from lecturers that students were unable to apply their learning to new ideas at university due to having been taught such specific methods in order to pass school examinations This supports the claim that many students are unprepared for the mathematical thinking required and find it hard to adjust to this new type of learning at university.
Another difference in the learning of mathematics at school and university can be seen in the examination papers of the A-level and at undergraduate level. It is useful to look at examination papers as they give a clear indication of the level that students are expected to attain. Darlington (2014) applies the Mathematical Assessment Task Hierarchy (MATH) taxonomy (Smith et al., 1996), an adjustment to Bloom’s taxonomy for mathematics (Bloom et al., 1956), to examination papers from A level and undergraduate level. Bloom’s taxonomy is a hierarchal classification of educational objectives increasing in complexity (Bloom et al., 1956) and the MATH taxonomy by Smith et al. (1996) is a modification of this, categorising skills needed to complete a task. The MATH taxonomy classifies examinations questions into 3 groups based on the essence of the task rather than the difficulty. Group A consists of factual knowledge and understanding, Group B includes application in new situations and Group C covers justifying and evaluation. Darlington (2014) found that at A-level the majority of questions were from Group A (72.6%) with one paper having 90.7% from Group A. In comparison, the majority of marks awarded at undergraduate level were from Group C (54.1%) and only 31.6% being awarded from Group A. The difference between A level and undergraduate can also be seen when looking at the type of Group A question the marks are awarded for. At A level 88.7% of these were for ‘routine use of procedures’ and at undergraduate 90% of them were for ‘factual knowledge and fact systems’. Supporting this is an investigation by Tallman et al. (2016, cited in Thoma and Nardi, 2018). Questionnaires were given to lecturers to investigate the intended focus of their examination questions. In the questionnaires the lecturers state that they ask students to justify and explain their answers, an example of a Group C question. These findings therefore support the view that learning mathematics is different at university than at A-level as highlighted by the difference in the nature of examination questions, with the aim of learning moving from recall to justification. These findings support my own experiences where I have found undergraduate examinations ask me to explain and justify my answers unlike at A-level in which I found the examination questions to almost be predictable solvable by the same routines each time.
Additionally, when comparing examinations at A-level to university the questions that students face become more complex and involved and this can be related to what Polya (2004) described as ‘routine’ and ‘non-routine’ problems. Polya (2004, p.171) writes:
In general, a problem is a ‘’routine problem’ if it can be solved either by substituting special data into a formally solved general problem, or by following step by step, without any trace of originality, some well-worn conspicuous example.
Berry et al (1999, cited in Darlington, 2014) found routine questions to provide the bulk of the marks in the mathematics A-level papers.
Similar to this notion of ‘routine’ and ‘non-routine’ questions is the idea of conceptual and procedural questions (Tasara, 2018). Boaler (1997, cited in Tasara, 2018) defines procedural questions as “those questions that could be answered by a simplistic rehearsal of a rule, method or formula” comparable to Polya’s (1945) definition of a routine problem and conceptual questions seen as questions in which “the use of some thought and rules or methods committed to memory in lessons would not be of great help” like that of a non-routine question. Building on this, Galbreith and Haines (2000) propose a taxonomy with 3 categories in order of increasing mathematical difficulty based on the degree of procedural and conceptual knowledge needed to solve a task. The 3 categories are:
- Mechanical: students must use procedural knowledge guided by the wording of the question.
- Interpretive: students must use conceptual knowledge and apply it to the question. It does not encompass using mathematical procedures to solve the question.
- Constructive: students must use both procedural and conceptual knowledge initiated by themselves and not guided by the question.
This taxonomy’s 3 categories correspond with the MATH taxonomy developed by Smith et al. (1996) with mechanical tasks relating to Group A, interpretive tasks relating to Group B and constructive tasks relating to Group C. We can now relate this back to the findings of Darlington (2014) where the majority of marks at A-level were from Group A questions which can also be interpreted as mechanical tasks using only procedural knowledge like that of a ‘routine’ question which supports the findings of Berry et al. (1999) cited in Darlington (2014). This can be compared to undergraduate level where the majority of marks came from Group C questions which can be translated as constructive task using both procedural and conceptual knowledge. This shows the change from routine problems at A-level to non-routine problems at undergraduate level and supports the claim that learning mathematics at school is different to university as the efforts required to answer examination questions have a different focus.
Finally, I will examine the different discourses used at A-level and university to compare the difference in learning. Thoma and Nardi (2018) comment on how the different type of discourses used at A-level differ to that at university. Sfard (2008, p.93, cited in Thoma and Nardi, 2018) defines discourses as:
different types of communication, set apart by their objects, the kinds of mediators used, and the rules followed by participants and thus defining different communities of communicating actors.
Mathematical discourse is defined by its word use (the mathematical terminology used in the discourse), visual mediators (e.g. graphs and symbols), routines (e.g. proving and defining), and narratives (e.g. theorems) (Thoma and Nardi 2016; Güçler et al., 2015). Sfard (2000) describes university level discourse as one with rigorous and precisely defined rules and A-level mathematical discourse as more relaxed and less rigorous. She also comments on meta-discursive rules (meta-rules) which are unseen rules that control the movement of exchange between teacher and student. When comparing the different discourses Sfard (2000) comments on the individuality of each discourses’ set of meta-rules and suggests that there are clear differences in these meta-rules easily seen when comparing lecture at university and school mathematics lesson. Consequently, learning mathematics at school is different from learning mathematics at university due to these different sets of meta-rules. Sfard (2000) also comments on the change from school to university as the transfer into a meaningful university level discourse due to the consistent and coherent mathematics that provides justification rather than school where everything learnt can seem arbitrary. My own experiences support this; at university I no longer felt that I was learning something for the sake of it and just having to accept information without reasoning like I did at A-level. At university I gained a deeper understanding of the mathematics I was learning. Thoma and Nardi (2016) identify key concepts of the assessment discourse by analysing compulsory questions from an undergraduate examination paper. In this paper students are directed to justify their answers which is a key objective introduced at undergraduate level and needed to be successful at university (Darlington 2014). These directions given in the examination paper are aimed to help the students shift their mathematical discourse from what is required at A-level to what is required at university (Thoma and Nardi 2016) and again highlight the differences in learning at A-level compared to university.
From this essay it is clear that there is a large difference in learning mathematics at school compared to university. Whilst some similarities can be drawn between the teaching style and size of a university tutorial and an A-level class, beyond this there are many differences. Differences in learning can be seen between the ‘teacher-focused’ teaching strategy of a lecturer in a lecture theatre and the ‘student-teacher’ interaction of a classroom and also the content of undergraduate mathematics heavily based on proofs, something not covered at A-level which influences students’ approaches to learning. Through analysing examination questions, it is clear that A-level learning is based on routines but at undergraduate level one must begin to justify and explain too. The final difference can be pulled from examining the discourses inferring that learning will always be different between the two due to the meta-rules involved. It is not surprising that learning mathematics is somewhat different at the two levels, but it is interesting that it is due to more than just increasing the level of difficulty of the mathematics. It is also different due to how the student approaches the learning, how it is taught and the variances in discourse. All of these areas are supported by my own personal experience and I can conclude that learning mathematics at university is different to learning mathematics at school.
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