European Conference on Educational Research University of Vienna, September 28-30, 2009

1. European Conference on Educational Research University of Vienna, September 28-30, 2009 Transition to post-compulsory education: the case of algebra as a boundary object between school and college Paul Hernandez-Martinez University of Manchester

2. What the paper is about… • Algebra is the doorway to advanced mathematics. • And it has been identified by teachers as particularly problematic during transition to college. • The central question to this paper is: • Is Algebra the same at school and college? Is it a boundary object? IntWhat in your opinion is the main reason why a student drops out (of maths)? T It’s the algebra skills that they can’t cope with. It’s mainly algebra, and we’ve had some with Grade A who’ve actually dropped out of maths, and an awful lot of Grade Bs, which is quite worrying. (…) if they can’t cope with the algebra they’re not going to cope with the A-Level. Tina A-level teacher at college A Learning Maths at school Learning Maths at college Algebra Boundary objects are objects that are both plastic enough to adapt to local needs and constraints of several parties employing them, yet robust enough to maintain a common identity across sites (Star 1989: 46)

3. Background • This is an ongoing project studying the transition from GCSE (school – 16 year olds) to A-level (college – post-compulsory – 16 to 18 year olds) in mathematics. • We have 4 case study schools and 2 case study colleges (2 schools are feeder schools of each of our colleges), where we are interviewing authorities, teachers and students, and observing lessons. • Interviews with students are biographical in the sense that we ask them to narrate their ‘story’ of their experiences at school and of their transition to college (in particular about mathematics). We are interviewing them once before the transition and once after it.

4. The cognitive perspective • We took one GCSE (higher tier) textbook used at one of our feeder schools and compared selected topics with an AS-level textbook used at the receiving college (same assessment board in both textbooks). • The topics were identified by teachers or students as problematic during the transition (first couple of months in college) . • We ask: Is the discourse in both texts providing common elements for students to use it as a boundary object? Are these objects (mathematical concepts) flexible enough to act as boundary objects?

5. The cognitive perspective • Surds appear only in chapter 28 (of 41) – “Working with exact numbers” • Surds appear in the second chapter after a brief review of linear equations. • Surds are immediatly used in equations and inequalities, for example: GCSE • Scary surds… A surd is a square root that does not have an exact value. For example, and are surds but and are not. IntDid you find a big difference between Mathematics at college and Mathematics at school? P A huge difference! Because firstly, the first module surds, we just did the basic surds which was surd 5, and then it was dividing and multiplying and then brackets were involved and if you forget one of them… (…) I came third and stuff like that in the higher paper and then came here and my first lesson was surds and I haven’t, I didn’t… for one I didn’t re-do it over the summer so I didn’t really know what was going on, and then I just struggled with the work and it sort of scared me in the first week, that maybe, ‘cos it’s your first week so you’re thinking ‘Oh this is going to continue straight away’. A Level Expressions with root signs involving irrational numbers such as or are called surds. Peter Student at college A The step up in formality and complexity is considerable. The discourse is quite different.

6. The cognitive perspective • Factorise or not factorise… • The school text is a series of instructions on how to solve a problem: Int Yeah, and talking in particular about maths, maths topics and maths concepts, what is the biggest problem in your point of view? D Algebraic processing by a mile. The ability to expand brackets and factorise. One way of solving a quadratic equation is by factorising the expression ax²+bx+c. For example, x²+5x-6 (x+6)(x-1) For (x+6)(x-1) to equal zero, either (x+6)=0 or (x-1)=0. This gives two solutions, x=6 and x=1. Dan A-level teacher at college B At college, it is expected that students are already familiar with this factorisation, are able to abstract it, and know exactly what does it mean: A quadratic equation that can be written in the form (x-p)(x-q)=0 has roots p and q. The graph of (x-p)(x-q) crosses the x-axis at the points (p,0) and (q,0). Students are asked to function at certain level of abstraction and to have a deep understanding, all in a short time. Students’ conceptual understanding might not be flexible enough.

7. The socio-cultural perspective We are governed by the fact that it’s an exam to be honest. There is a GCSE for them to pass and that’s what we aim for. But to be honest, because as I’ve been teaching for so long and it ends up and its only been teaching up to GCSE, I’m good at what’s on the GCSE. I kind of know and I have kept up to date with kind of what’s the AS but I know I couldn’t teach the AS, I know my own limitations, so I teach what I’m happy at, I’ll get it to the A* and I’ll go through all of that but going Into more detail it’s too much preparation when there’s so much other stuff to do. The Maths discourse at school is about ‘exchange’ value, which is influenced by the performativity system in which schools compete. Well I know Patricia at school C… the main emphasis is with the heads of maths at the high schools… is to get as many students as possible through maths GCSE by whatever means possible, because it affects the funds and (league) tables. Patricia, Head of Maths at school C Sara, A-level Maths teacher at college A And I’m not sure to be honest how many grade Cs are left, I know I haven’t got any in my class, there’s at least one student that I know of, but we’ve had a number of students dropping from the course. And a surprising number of students with grade B because when we’ve looked at it more closely, it’s possible to get grade B with 50%, so if they just scraped a B, 50% knowledge of the GCSE syllabus, it hardly surprising that they’re not coping with it.

8. The socio-cultural perspective The Maths discourse at college is about ‘use’ value. Students are asked for a certain level of abstraction and understanding of the mathematical concepts to be used, all in a relatively short period of time. Int: And where do you think that problem comes from or why does it originate?  Dan: Why do I think… I mean I don’t know, but I think that, I think you can be successful at GCSE maths without having to tackle a lot of the algebraic content, so you can still get a grade A, I mean you’re not just, the one thing I think is the GCSE grade tells you nothing about their mathematical ability. It’s a poor educator, full stop. (…) Dan: Success rates. Kids’ success rates. that’s what we’re in that particular game. You play according to the rules really, you know, and my game is to get as many kids through GCSE maths as possible. That’s this November. I can’t see that a mentality of a maths teacher in school that will be any different. You know with all the league tables and everything else. Presumably they (students) can get compensating and get good marks on other parts of the GCSE so they don’t have to do very well on the algebra part, so it suggests that the GCSE is just not fit for purpose. I know Patricia actually gets in, erm, one of the Edexcel people. He comes and does, erm, he’s one of the chief examiners, what’s his name? I can’t remember, I do know him actually. But he comes in on a regular basis to tutor the students and tells them the particular questions and says right if you just write down this you get a mark, kind of thing. Focuses on the questions where he thinks that they can gain marks and tells them what to do with it. Dan, A-level Maths teacher at college B Tine, A-level Maths teacher at college A

9. Conclusions Maths discourses at school and college are very different: At school, maths takes an exchange value and the object of learning maths is dominated by the motive of obtaining a qualification. Institutions that compete in the league tables have to ensure their survival by ‘passing as many students as possible through GCSE by whatever means available’. In most cases, this translates in ‘teaching to the test’. Algebra can be avoided or become a series of routines to be memorised, without too much conceptual understanding. At college, maths takes a use value and the object of learning maths is dominated by the motive of using previous knowledge to acquire new knowledge in a short period of time. This requires understanding and flexibility of the concepts (mainly algebraic concepts) previously acquired, something that many students lack of. At the moment, algebra does not constitute a boundary object for many students, and this translates into drop outs and failures. Algebraic concepts are not flexible enough and do not provide sufficient commonalities during transition.

10. Questions and feedback Correspondence: Paul.Hernandez-Martinez@manchester.ac.uk This research has been funded by the ESRC under the project “Mathematics Learning, Identity and Educational Practice: the Transition into Post-compulsory Education” (RES-000-22-2890).