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Abstract
In Clark and Lovric (Suggestion for a theoretical model for secondary–tertiary transition in mathematics, Math. Educ. Res. J. 20(2) (2008), pp. 25–37) we began developing a model for the secondary–tertiary transition in mathematics, based on the anthropological notion of a rite of passage. We articulated several reasons why we believe that the educational transition from school to university mathematics should be viewed (and is) a rite of passage, and then examined certain aspects of the process of transition. Present article is a continuation of our study, resulting in an enhanced version of the model. In order to properly address all aspects of transition (such as a number of cognitive and pedagogical issues) we enrich our model with the notions of cognitive conflict (conceptual change) and culture shock (although defined and used in contexts that differ from the transition context, nevertheless, we found these notions highly relevant). After providing further justification for the application of our model to transition in mathematics, we discuss its many implications in detail. By critically examining current practices, we enhance our understanding of the many issues involved in the transition. The core section ‘Messages and implications of the model’ is divided into subsections that were determined by the model (role of community, discontinuity of the transition process, shock of the new, role of time in transition, universality of transition, expectations and responsibilities, transition as a real event). Before making final conclusions, we examine certain aspects of remedial efforts.
Notes
Notes
1. This is not the first time an anthropological model has been used in a mathematical context. See Citation11 for such an approach to number and time.
2. Such as contributions by Boland, Laird, and Taylor.
3. Biennial Southern Hemisphere conferences in mathematics educations, whose primary focus is tertiary education.
4. One could argue that there is a large difference in a successful (it is not at all clear what ‘successful’ really means) outcome between a cross-cultural sojourn and transition: the adoption, and creation of the balance between one's own and ‘new’ cultures for the former, and expectations that students reject their ‘high school’ culture (say, in terms of various parameters of their cognitive environment), and accept the new, ‘university’ culture for the latter.
5. The quote is from the Summary section of the report.
6. Third paragraph.
7. Common phrase used to critique US mathematics curricula (K-12) that tend to include too many topics; sometimes blamed for students’ poor achievement in mathematics.
8. Introduced in January 2001. Available at: http://www.ed.gov/nclb/landing.jhtml.
9. Raising the floor: Progress and setbacks in the struggle for quality mathematics education for all, workshop, May 7–10, 2006. Available at: http://www.msri.org/calendar/workshops/WorkshopInfo/388/show_workshop.
10. The individualism in certain cultures is fed by (or feeds) competitiveness - strong negative force, working against community forming. There are cases where students refuse to lend their class notes to a peer who missed the class; or (anecdotal evidence from McMaster University) a student would knowingly give an assignment containing wrong solutions to her/his peer.
11. McMaster University is by no means unique in this way. For instance, University of Auckland in New Zealand has been running a similar initiative for many years; see Citation27.
12. They also get paid for their job as teaching assistants.
13. Thus, on average, in a country with individualism index of 30 people believe and rely much more on group/collective values than people living in a country with the index of 70.
14. One could comment that this seems to be counter to contemporary practices in teaching mathematics (predominantly in the Western world) that foster group work and peer communication. However, anecdotal evidence (such as students opting out of group work if given the opportunity, as witnessed by a co-author of this article, M. Lovric in his problem-solving course at McMaster University) as well as research indicate that group work is not readily embraced by students (i.e. they need to be stimulated to engage in group work), nor are its benefits evident and/or present. For instance, examining practices in teaching in high school in Ontario, Canada, the author demonstrates that ‘Students perceived as low achievers never become legitimate participants in the peer discourse, despite their efforts.’ Citation31, p. 470].
15. China was not surveyed.
16. Of course, students’ experiences in high school (as well as in university) mathematics are strongly teacher-dependent. Here we refer to the rigidity of a high school curriculum, as compared to a more flexible coverage of the material in a university course.
17. Even more than that: material covered (i.e. choice of topics, as well as decisions about what theorems will be proved, or what applications will be presented) in a given course might depend on the course instructor.
18. We will not go into discussing reasons for it here. Probably the best evidence of this common practice is the fact that high school students in Ontario (and perhaps elsewhere) are told (by guidance counselors, older peers, or others), to expect that in university their grades will ‘drop by about 30%’.
19. The year in which this effect was noticed was 2001, with consequent dramatic lowering of first year calculus enrolments of up to 20%. At the time, concerns to the Qualifications Authority were expressed by both Victoria University and Auckland University.
20. Related to students who live on the Northern hemisphere, i.e. whose school vacations take place between June and August.
21. About the Center > Know the Facts.
22. ‘Going to school after school’.
23. Anecdotal evidence from McMaster University.
24. Sample of items from the inventory: assess prior knowledge, set goals, identify resources, assess progress, develop critical attitude, construct knowledge, scheduling time, etc.
25. Miroslav Lovric, co-author of this article.
26. By ‘remedial’ we mean courses that prepare students ‘for the academic concepts, skills and attitudes of their degree programmes as opposed to orientation programmes that prepare students for university life’ Citation10, p. 89]. Such courses are also called bridge or bridging courses, foundation courses, access courses, pre-calculus courses, etc. This is sometimes confusing, as some other types of courses, which are not remedial in nature (e.g. courses that introduce students to formal mathematical reasoning, or proof), are occasionally called foundation or bridge courses.
27. In this section, we discuss several important contributions in that direction. Our discussion does not aim to be comprehensive.
28. Auckland, New Zealand. The examination marks system used in Department of Mathematics at the University of Auckland allows for comparisons of results from one course to another, as well as looking at students’ performance as a group.
29. Wellington, New Zealand.
30. Students who took the Foundation course (along with many other first year university students) had an interview with a mathematics department member in order to identify the most suitable course for them from the list of available first-year mathematics courses. The issue was that the Foundation course grades were so unreliable as predictors that it became very difficult to give sound advice. On the contrary, the grades students received in secondary school mathematics turned out to be a fairly good indicator of what first-year university math course is most appropriate for them.
31. Large financial gains aside, for the moment.
32. Students need to complete 13 years of elementary and high school education before coming to university.
33. As mentioned earlier in the text, the course–although referred to as ‘foundation’ course–is actually remedial in nature.
34. Paper is a course.