Key Mathematical Concepts in the Transition from
Secondary to University Background
Changing mathematics curricula and their emphases have provoked some international concern about the ability of students entering university with regard to their apparently decreasing levels of competence (Smith, 2004). This has been particularly apparent with regard to essential technical facility, analytical powers, and perceptions of the place of precision and proof in mathematics. Such mathematical under-preparedness of students entering university has been seen as an issue (Hourigan & O’Donoghue, 2007; Kajander & Lovric, 2005; Luk, 2005; Selden, 2005), and one that may impact on students’ success in university mathematics (Anthony, 2000; D’Souza & Wood, 2003), although not all studies agree on the extent of the problem (Engelbrecht & Harding, 2008). Further, concern has been expressed about the levels of student enrolments in undergraduate mathematics programmes (Barton & Sheryn, 2009; the ICMI Pipeline Project) and the implications for the future of the subject.
While recent research has specifically addressed these issues with regard to the transition from school (Brandell, Hemmi & Thunberg, 2008; Engelbrecht & Harding, 2008; James, Montelle & Williams, 2008; Jennings, 2009), overall the volume of research in tertiary mathematics education has, until recently, been relatively modest (Selden & Selden, 2001).
MORE TO COME…
Several researchers studied the problems of transition between secondary school and university concerning the learning of Calculus. Some of theses studies focused on specific topics: real numbers (Bloch, Chiocca, Job & Scheider, M, 2006; Ghedamsi, 2008); functions (Dias, Artigue, Jahn & Campos, 2008; Vandebrouk, 2010); limits (Bloch et al. 2006; Bloch & Ghedamsi, 2005); sequences and series (González-Martin, 2009; Gyöngyösi, Solovej & Winsløw, 2010). They are located in several countries (Brazil, Canada, Denmark, France, Israel, Tunisia) and use different frameworks (Anthropological theory of didactics – ATD; textbooks analysis, analysis of students’ productions; use of CAS or innovative teaching and assessment methods).
I present a summary of the main articles reviewed.
Smida & Ghedamsi (2006) study the teaching practices of real analysis in the first year of mathematics/informatics courses in Tunisian university. They distinguish two kinds of teaching projects leading to two different models of teaching practices:
The projects where axiomatic, structures and formalism are the discourse which justify and generate the expected knowledge and know-how; this model only follows a mathematical logic;
The projects where the variety of choices for proving, illustrating, applying or deepening the mathematical results highlights a declared intent – by teachers – to enrol in a constructivist setting; this model combines the logic of mathematics and cognitive demands.
A questionnaire applied to 57 lecturers from 4 universities highlights 3 groups of lecturers: the lecturers with a logico-theoretical profile, who do not take into account cognitive demands (more or less 40%); the lecturers with a logico-constructivist profile, who have some cognitive concern (more or less 35%); lecturers who take into account cognitive demands (more or less 25%). However 80% of the lecturers report hardly or never give students tasks that lead them to formulate a conjecture. More than 90% of the lecturers do not consider the proof in Analysis as a mean to convince students of the validity of mathematical statements. Almost 60% do nor consider proof as a priority, as a logico-theoretical tool for validation.
Dias, Artigue, Jahn & Campos (2008) conduce a comparative study of the secondary-tertiary transition in Brazil and France, using the concept of functions as a filter. With ATD as a theoretical framework, the research is based on different approaches: institutional approach, an approach of the personal relationships developed by students with the concept of function; and an approach of continuities and discontinuities between teaching practices in secondary and tertiary institutions in the two countries. This paper focus on the analysis of institutional relationships through the analysis of evaluations used for the selection of students at university entrance or developed by specific universities (vestibular in Brasil; baccalauréat in France). They compare two tasks, considered as typical. A typical task in Brazil is the determination of terms and reason of arithmetic and geometric sequences, the associated praxeologies being based on algebraic techniques and technology. A typical task in France is the study of the convergence of such sequences both qualitatively and quantitatively, the associated praxeologies being the use of analytic techniques and technology. Furthermore, the study also shows a higher level of students’ guidance through hints and intermediate questions in France than in Brazil. They conclude that “becoming aware of such contextual influences which tend to remain invisible to those who stay inside a given educational system seems to us crucial for envisaging productive collaborative work, and also for envisaging evolutions inside a given system”.
Leviatan (2008) argues that while school mathematics tends to concentrate on problem solving skills, tertiary mathematics is more abstract and emphasises the inquisitive as well as the rigorous nature of mathematics. He presents a transition course aimed at bridging the gap for students of four-year secondary/high school teacher training programme. The objectives of this transition programme are: to identify and reinforce previous “core school mathematics”; to deepen and enrich the existing knowledge by adopting a more mature perspective to school mathematics; to introduce mathematical “culture” ( language, rules of logic, etc.); to get acquaintance with typical mathematical activities (generalizations, deductions, definitions, proofs, etc.); to re-introduce central mathematical concepts and tools; and to provide a rigorous, yet only semiformal, exposure to selected new topics in advanced mathematics. Students’ evaluations of the programme report increasing self-confidence, enjoying the sessions about misconceptions and playing the role of a reviewer. He concludes that a more systematic investigation is required and gives a list of possible follow-up.
Ghedamsi (2008) analyses the contributions of situations of pragmatic/formal mix control in a didactic engineering at Tunis University. The aim is to develop a form of conceptualisation deriving from a double thinking, natural and formal, more specifically: to resume the work with real numbers and enlarge the experimental field of students concerning the nature of real numbers and their appearance; leading students to an heuristic work which allows the development of pragmatic proofs; leading students to understand the link between the pragmatic experience and the existence of objects of real analysis through a go-between pragmatic proofs and formal proofs. Two approximation methods were used as experimental situations: the construction of the better rational approximation of and, if possible, its generalisation to other irrationals; the co sinus fix point. She concludes that the scope of students’ work on the experimental situations concerns 3 main points: a conceptual skip on numbers (students consider real numbers as mathematical objects); the problematic existence/accessibility (emergence of the link between research procedure and established proof); the link between micro-didactic and macro-didactic variables. She concludes that the irrational numbers situation gives a status to numbers that students have only consider as “notations”, while the co sinus fix point situation gives access to real numbers that we cannot explicit, and consequently requires the implementation of formal procedures.
Winslow (2008) studies the transition from concrete to abstraction in real analysis, in particular the study of real functions and of the operations on these functions associated with the limit process. He considers the concrete analysis as the part of this study centred on the study of specific functions while the abstract analysis focuses on the axiomatic systems. In secondary schools the focus is on the practical-theoretical blocs of concrete analysis, while at university level the focus is on more complex praxeologies of concrete analysis and on abstract analysis. He considers two kinds of transitions in the student’s mathematical activity.
The first kind is the transition from activity mainly centred on practical blocks, more or less independent, to working with more comprehensive and structured mathematical organisations. The second kind is the transition to tasks on theoretical objects. This second kind of transition presupposes the first one. As a consequence, an incomplete achievement of the first transition produces an obstacle for the second one by turning inaccessible the tasks to be worked on.
To help Danish students perform the two kinds of transition, Winslow give the students thematic projects, where they have to work on rather theoretical issues that are not presented in the textbook. The work is given by instructions for a several weeks’ theoretical work and done in groups. It leads to an oral individual presentation of the members of the group. This device provides a practical setting to undertake a didactical engineering work aiming at the second kind of transition.
González-Martin (2009) identifies two obstacles regarding the concept of infinite sum: the intuitive and natural idea that the sum of infinity of terms should also be infinite, and the conception that an infinite process must go through each step, one after the other and without any stop, which leads to the potential infinity concept. He argues that the concept of series is usually reduced to its algorithmic aspects. This leads to misconceptions of the integral concept. The aim of the research is to combine 3 dimensions (epistemological, cognitive and didactic) to study the learning of the series concept through textbooks analysis. He concludes that series represents 10% or more of the content, but that the textbooks do not foster the links between visual and algebraic representations. He did not find a consensus in the way to introduce series. However there are very few tasks showing the application of series in the real life and very few historical references. Furthermore, the textbooks do not explicitly take into account the difficulties highlighted by the literature. He concludes that the approach is very “traditional” and that textbooks use almost exclusively the algebraic register, with very few graphical representations. They do not seem to take into account results and recommendations from research.
Oehrtman (2009) argues that students’ reasoning about limit concepts appears to be influenced by metaphorical application of experiential conceptual domains. He analyses written assignment of 120 students from an introductory calculus course and interviews 9 of them to identify students’ strong metaphors for limit concepts. He identifies the following metaphors: Collapse metaphor (for the definition of the derivative, the volume of solids of revolution, definite integrals and the fundamental theorem of calculus; approximation metaphors (for infinite series, the definition of the derivative); proximity metaphors (for the limit of function and continuity, infinite series, the definition of the derivative); infinity as number metaphors; physical limitation metaphors (for a volume of revolution, the limit of a sequence of sets). He argues that the only metaphor cluster that demonstrated a consistent detriment to students’ understanding was physical limitation metaphors.
Although he observed several strong metaphors, he also observed students’ inability to apply abstract criteria for adopting, evaluating, or modifying particular metaphors. He concludes that many of students’ nonstandard interpretations are fertile sites for positive discussions. Recognising this potential for development of scientific reasoning requires an effort on the part of curriculum developers and instructors to see beyond students’ errors.
Gyöngyösi, E., Solovej, J & Winsløw C. (2010) report an experiment aiming at using CAS based work to ease the transition from calculus to real analysis in Denmark. Using ATD as a framework, they give examples of praxeologies to be developed by students and teachers and analyse them according to their pragmatic value (efficiency of solving tasks) and epistemic value (insight they provide into the mathematical objects and theories to be studied). The tasks are de signed for using Maple. They conclude that students with an overall lower performance also commit more errors in using instrumented techniques. The course evaluation done via the course website with student replies being anonymous was not concluding as students appear to have diverging voices.
Vandebrouk (2010) studies student’s conceptions of functions at the transition between secondary school and university. He claims that the transition between secondary school and university can be interpreted, in some sense, as a way to move from the conceptual-embodied world to the formal axiomatic one, embedding a higher level of conceptualization of the notions related to the domain of analysis. He points out three points of view on the object function: the point-wide point of view, the global point of view, and the local point of view. In the point-wide point of view, functions are considered as correspondence between two sets of numbers, an element of the first set being associated with a unique element of the second set. The main representations are arithmetic formula and tables of values. In the global point of view, the representations are tables of variation. Algebraic expression and graphical representation can exploit and can be exploited from a point-wide point of view as well as global point of view.
He claims that working at university level on functions implies that students can adopt a local point of view on functions whereas only the point-wide and global points of view are constructed at secondary school. However, a large algebraisation of tasks at the end of the secondary school tends to erase the point-wise and global points of view and doesn’t permit to reach the local point of view.
He presents a task for which algebraic techniques were not sufficient to solve the task. Two versions of the task were produced, one for secondary school level, and another for university level. The study of students’ productions shows that they face difficulties to solve tasks when algebraic techniques are not sufficient. He explains this by the non ability of students to consider functions as complex objects with point-wide as well as global properties; these difficulties are increased by the current practice of teaching in secondary schools in France, which reinforces tasks belonging to the algebraic frame only. Consequently students face difficulties to enter in the formal axiomatic world and to develop the local abilities which are necessary at the beginning of university.
As was the case for linear algebra, this area has been extensively investigated within the Francophone community, and the research developed obeys more or less the same characteristics, a diversity of approaches and themes but a shared vision of the importance to be attached to epistemological and mathematical analyses, the joint development of fundamental research and didactical engineering, the increasing influence of ATD.
Epistemological and mathematical perspectives
Several publications show this epistemological interest: (Robert, 2010) shows that the FUG perspective can be useful for approaching the teaching and learning of some notions in Analysis and Bridoux en her doctoral thesis on topological notions introduced in a first university course in Belgium shows the FUG character of these notions (Bridoux, 2011); (Rogalski, 2008) addresses the relationships between local and global perspectives which are crucial in Analisis; (Artigue, 2008) crosses epistemological and didactical perspectives regarding the concept of continuity; (Chorlay, 2009) formulates a series of hypotheses as to the long-term development of functional thinking, throughout upper-secondary and tertiary education at the light of a historical study of the differentiation of viewpoints on functions in 19th century elementary and non-elementary mathematics. Artigue (2009),Vandebrouck (2011) also address the evolution of functional thinking from secondary to university combining different perspectives: process / object duality, transition from the conceptual embodied and proceptual worlds to the formal axiomatic one, relationships between punctual, local and global perspectives, without forgetting the changes in the mathematical habitat of this concept (see also Najar above).
Didactical engineering research is also a long term tradition in the Francophone community linked to the theoretical influence played in this community by the Theory of didactical situations (TDS) due to Brousseau. Relying on epistemological and mathematical analysis as those mentioned above, didactical engineering has been designed and experimented in the last decade for supporting the transition between secondary and tertiary in Analysis. We mention some of these realizations below:
In Ghedamsi’s doctoral thesis (2009) regarding the concept of limit, two situations articulating knowledge on the nature and properties of real numbers and the notion of limit are designed. Through the development and use of approximation methods, these situations allow the students to productively connect the intuitive, perceptual and formal dimensions of this concept (see also (Bloch & Ghedamsi, 2010)).
In Bridoux’s doctoral thesis mentioned above, the author designs and experiments a succession of situations for introducing the notions of interior and closure of a set, of open and closed set, after identifying the FUG characteristics of these notions. This didactical engineering uses graphical representations for allowing the students to develop an intuitive vision of these notions on which the teacher will then rely for characterizing these notions in a formal language. As is the case in didactical engineering attached to FUG notions, a meta-mathematical discourse is also used. (see also Bridoux, 2011b).
As already mentioned, Praslon’s doctoral thesis in 2000 regarding the concept of derivative was a pioneering work in that area. Combining the affordances of ATD and of educational research already developed in the area of Calculus and Analysis, he showed that, in France, at the end of high school a substantial institutional relationship with the concept of derivative was already established, and that regarding this concept and its environment the transition secondary-tertiary was not a transition between intuitive and proceptual perspectives towards formal perspectives, but something more complex involving an accumulation of micro-breaches and changes in balance according several dimensions (tool/object dimensions, particular/general objects, autonomy given in the solving process, role of proofs…) to which university academics were poorly sensitive. He also developed and tested a series of didactic resources for making teachers and students more aware of these changes and allow specific work on these. In continuity with this analysis, Bloch identified 9 factors of discontinuity between high school and university in Analysis (Bloch, 2004). In the last decade, ATD has been used for the study of the secondary-tertiary transition by several researchers as already pointed out in the algebra part. One typical research work in Analysis is that developed by Bosch, Fonseca and Gascón in Spain (2004) regarding the concept of limit, in which they show the existence of strong discontinuities in the praxeological organization between high school and university, and build specific tools for qualifying and quantifying these. Another interesting contribution is that of Bergé (Bergé, 2008) investigating the evolution of students’ relationships with real numbers and the idea of completeness, and linking these relationships with the characteristics of the different courses where students meet these notions and work with them.
Generalised forms of arithmetic and Abstract Algebra: The case of Group Theory
Students’ encounter with Group Theory marks a significant point in the transition to advanced mathematical formalism and abstraction. This is a topic that is characterised by ‘deeper levels of insight and sophistication’ (Barbeau, 1995, p139) and asks of students commitment to what is often a fast-paced first encounter in lectures (Clark et al, 1997). Key to this encounter is the realisation of the need to ‘think selectively about its entities, paying attention to those aspects consistent with the context and ignoring those that are irrelevant.” (Barbeau, 1995, p140). As Hazzan (1999) the students’ difficulty with Abstract Algebra can be attributed to the novelty of dealing with concepts which are introduced abstractly, namely ‘defined and presented by their properties and by an examination of what facts can be determined just from their properties alone’ (Hazzan, 1999, p73). Furthermore the way that students approach proof writing, the type of practices and beliefs that they bring to the task often exacerbates some of this difficulty (Powers, 2010; Weber, 2001).
Below I summarise results from a few studies that focus on the difficulties students experience in their first encounters with key concepts in Abstract Algebra – and a few that touch on pedagogical insights emerging from our understanding of these difficulties. The text originates largely in (Nardi, 2000; 1996) and the preparations for the literature review in the thesis (Ioannou, in preparation) of my doctoral student, Marios Ioannou.
As mentioned above students’ skills in proof production are central to the quality of their first encounter with Group Theory. According to Hart (1991):
Students’ conceptual schemas is the key element in the success of problem solving in Group Theory;
Students’ overreliance upon concrete examples of groups often causes operation confusion;
The ability to translate concrete representations is critical in the students’ proof production (as is the overreliance on concrete representations)
Students need to learn how to apply domain-specific proving strategies
Dubinsky et al (1994) was the first comprehensive attempt to explore student encounters with fundamental concepts of Group Theory (group, subgroup, coset, normality and quotient group). Written largely in the language of APOS, the study marked the importance of students’ understanding of the process-object duality of mathematical concepts as a prerequisite for understanding in Group Theory; highlighted the importance of the concept of function in building group-theoretical understanding; and, identified specific issues of difficulty such as confusing normality with commutativity. Cosets and normality are also identified as major stumbling blocks in the early stages of students’ learning.
As a particular, and important, form of the concept of function, the concept of group isomorphism has attracted attention in several studies. For example, Leron et al (1995) distinguished between students’ naive and formal conceptualisations of isomorphism through an elaborate discussion of student attempts to distinguish isomorphic relations between two groups and isomorphism; to prove that a certain function is, or not, an isomorphism; to work with isomorphisms in the abstract or in concrete cases. The results highlighted students’ difficulties to link isomorphic relations with group orders; to distinguish between properties of group elements and properties of groups’; and, to construct isomorphisms between certain groups.
Lagrange’s theorem is another topic from the introductory parts of Group Theory that has attracted attention in several studies. For example, Hazzan and Leron (1996) noted that: students may use theorems such as this (particularly those with recognisable names to them) as slogans-style references in their proofs (in their data students use Lagrange’s theorem or some version of its converse where not appropriate or relevant to the problem and use the theorem and its converse, or ‘naïve’ versions of the converse, indistinguishably).
Other important introductory elements of Group Theory were treated by two papers published in the late 1990s in the Journal of Mathematical Behaviour: Brown et al (1997) focused on binary operations, groups and subgroups; and, Asiala et al (1997) focused on cosets, normality and quotient groups. Soon after, an analogous report by Asiala et al (1998) focused on permutations of a finite set and symmetries of a regular polygon. Once again emphasis, in the context of groups of symmetries and dihedral groups, was on the need to facilitate students’ transition to object understandings of key notions in Group Theory.
Group and group elements as encapsulated objects.
In addition to a focus on fundamental Group Theory concepts, some studies have focused on issues such as the relationship between visual and analytic thinking, and, largely, the need for both (Zazkis and Dubinsky, 1996). In these authors’ VA model, whether external or internal, visual representations are in a constant interplay with analytical ones. Eventually it is of little concern whether the emerging complex construct is visual or analytic as the elements of both types of thinking have merged into it effectively.
In resonance with the VA proposition by Zazkis and Dubisnky, Hazzan (1999) explored how students attempt to cope in Group Theory through reducing its high levels of abstraction. In a related paper Hazzan (2001) examines these attempts at reducing levels of abstraction in the context of a problem asking students to construct the operation table for a group of order four.
Mirroring many of the difficulties outlined generally in the above in her analyses of student responses to introductory Group Theory problem sheets, Nardi (2000) identified students’:
difficulties with the static and operational duality within the concept of order of an element as well as the semantic abbreviation contained in ;
often problematic use of ‘times’ and ‘powers of’ in association with the group operation;
ambivalent use of geometric images as part of meaning bestowing processes with regard to the notion of coset;
problematic conceptualization of multi-level abstractions embedded in the concept of isomorphism.
The duality underlying the concept of group, and the role of binary operation in this concept, were also discussed a little later by Iannone and Nardi (2002) who offered evidence of the students’ tendency
to consider a group as a special kind of set, often ignoring the binary operation that is fundamental to its entity
to consider the axioms in the definition of a group as properties of the group elements rather than the binary operation
to omit checking those axioms that they perceive as obvious (e.g. in some cases associativity)
Finally, I would like to mention a doctoral thesis (Ioannou, in preparation) which is expected to offer further evidence on some of the issues addressed above through drawing on a range of data collected during a semester of the students’ first encounter with Abstract Algebra (lecture notes and recordings; seminar notes and recordings; multiple interviews with 13 of the 78 students in that cohort; coursework and exam papers from all students; interviews with the lecturer, seminar leaders and markers). Examples of the mathematical topics that are currently the focus of the analysis include: the Subgroup Test, symmetries of a cube, equivalence relations, employing the notions of kerner and image in the First Isomorphism Theorem. At the time of writing it seems likely that the thesis will conclude that the students’ overall problematic experience of the transition to Abstract Algebra is characterised by the strong interplay between strictly conceptual matters, such as the ones addressed above, affective issues and issues that are germane to the wider study skills and coping strategies that students arrive at university with. A preliminary flavour of these results can be found in (Ioannou & Nardi, 2009a, b; 2010; Ioannou & Iannone, 2011). The thesis is due to be completed in 2011-12.
Most of the above studies offer some pedagogical insight into how teaching can facilitate students’ transition to this most abstract and formal topic in mathematics that students have met. As elaborating this insight may be outside the remit of this report, I list brief references to the pedagogical recommendations made in some of the above, and some other, studies:
Leron and Dubinsky (1995) propose an interactive approach that involves computer-based experimentation with group structures, followed by a more formal introduction to the topic.
Burn (1996; 1998) proposes reversing the order of presentation with examples and applications stimulating the discovery of definitions and theorems. At the heart of this new emphasis are the concepts of permutation and symmetry, historical precedents and platforms for the development of Group-Theoretical concepts.
Cnop and Grandsard (1998) propose a set of group-work activities that take the place of the formal introduction in the lectures and stimulate students’ taking over for the responsibility of their first encounters with the abstraction of Group Theory.
Alcock et al (2008) also encourage independent study of proofs in Group Theory through carefully prepared workbooks.
In a similar vein to the above emphasis of concretisation of abstract ideas, Thrash and Walls (1991) suggest activities that invite students to constructing multiplication tables for groups of small order as a stepping stone for an understanding of the general notion of group structure.
Larsen (2009) makes a similar case through presenting a series of tasks that explore the symmetries of an equilateral triangle and culminate in negotiating preliminary understandings of order of a group and isomorphism.