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Hence we also reviewed literature analysing the learning of mathematics on one or both sides of the transition boundary. To achieve this we formed the somewhat arbitrary division of this mathematics into: calculus and analysis; abstract algebra; linear algebra; reasoning, argumentation and proof; and modelling, applications and applied mathematics, and report findings related to each of these fields. We were aware that other fields such as geometry and statistics and probability should have been included, but were not able to do so.

The Survey We considered it important to obtain data on transition from university mathematics departments. We wanted to know what topics are taught and how, if the faculty think the transition should be smooth, or not, their opinions on whether their students are well prepared mathematically, and what university departments do to assist those who are not.

Hence, we constructed an anonymous questionnaire on transition using an Adobe Acrobat pdf form and sent it internationally by email to members of mathematics departments. The 79 responses from 21 countries were collected electronically. The sample comprised 56 males and 23 females with a mean of Of these 45 were at the level of associate professor, reader or full professor, and 30 were assistant professors, lecturers or senior lecturers.

Clearly the experience for beginning university students varies considerably depending on the country and the university that they attend. For example, while the majority teaches pre-calculus 53, Most of the respondents 50, The mathematicians were specifically asked whether students were well prepared for calculus study. Those whose students did, rated secondary school calculus as preparation to study calculus at university at 2.

These results suggest that there is some room for improvement in school preparation for university study of calculus and analysis. Since the view has been expressed e. A few mentioned changes that could be made at the university, such as: better placement of students in classes; increasing the communication between secondary and tertiary teachers; and, addressing student expectations at each level.

This lack of communication between the two sectors was highlighted as a major area requiring attention by the two-year study led by Thomas Hong et al. Since one would expect that, seeing students with difficulties in transition, universities would respond in an appropriate manner see e. Some respondents said that they change the course content for the first year students based on a decision by an individual member of faculty who diagnoses student needs and background.

In some of the courses, students were encouraged to use tools for calculation and visualisation. In contrast, six departments increased the complexity and the rigour of their first year mathematics courses. The survey considered the notion of proof in several questions. Only 8 respondents replied that definitions are not important in first year mathematics. While some had separate courses e.

One respondent used the modified Moore method in interactive lectures. These responses appear to show a good level of agreement with employing the suggested approaches as components of a course on proof construction. It may be that these are ideas that the Mathematical modelling in universities was another topic our survey addressed.

Reasons given for choosing dedicated courses include: the majority of all mathematics students will end up doing something other than mathematics so applications are far more important to them than are detailed theoretical developments; most of the mathematics teaching is service teaching for students not majoring in mathematics so it is appropriate to provide a relevant course of modelling and applications that meets the needs of the target audience; if modelling is treated as an add-on then students may not learn mathematical modelling methods.

Those who chose integrated courses did so because students need to be equipped with a wide array of mathematical techniques and solid knowledge base. Hence, it is appropriate for earlier mathematics courses to contain some theory, proofs, concepts and skills, as well as applications. When asked for their opinion on how modelling should be taught in schools, most of the answers stated that it should be integrated into other mathematical courses.

The main reasons presented for this were: the many facets of mathematics; topics too specialised to form dedicated courses; to allow cross flow of ideas, avoid compartmentalization; and students need to see the connection between theory and practice, build meaning, appropriate knowledge.

The key differences pointed out by those answering this question were: at school, modelling is poor, too basic and mechanical, often close implementation of simple statistics tests; students have less understanding of application areas; university students are more independent; they have bigger range of mathematical tools, more techniques; they are concerned with rigour and proof.

One message for transition is to construct more realistic modeling applications for students to study in schools. The MLS has a drop-in help room, and runs a series of seminars on Maths skills. These are also available to students on the web. There is some evidence that bridging courses can assist in transition Varsavsky , by addressing skill deficiencies in basic mathematical topics Tempelaar et al. Other successful transition courses e.

Overall the survey confirmed that students do have some difficulties in transition and these are occasionally related to a deficit in student preparation or mathematical knowledge. Literature Review A number of different lenses have been used to analyse the mathematical transition from school to university.

Some have been summarised well elsewhere see e.

One theory that is in common use is the Anthropological Theory of Didactics ATD based on the ideas of Chevallard , with its concept of a praxeology comprising task, technique, technology, theory. ATD focuses on analysis of the organisation of praxeologies relative to institutions and the diachronic development of didactic systems.

A second common perspective is the Theory of Didactical Situations TDS of Brousseau , where didactical situations are constructed in which the teacher orchestrates elements of the didactical milieu under the constraints of a dynamic didactical contract.

Dubinsky and McDonald for studying learning. This describes how a process can be constructed from actions by reflective abstraction, and subsequently an object is formed by encapsulation of the process.

This describes thinking and learning as taking place in three worlds: the embodied; the symbolic; and the formal.

In the embodied world we build mental conceptions using visual and physical attributes of concepts and enactive sensual experiences. In the symbolic world symbolic representations of concepts are acted upon, or manipulated, and the formal world is where properties of objects are formalized as axioms, with learning comprising building and proving of theorems by logical deduction from these axioms. We use the acronyms above to refer to each of these frameworks in the text below. Calculus and Analysis A number of epistemological and mathematical obstacles have been identified in the study of the transition from calculus to analysis.

These include: Functions: Students have a limited understanding of the concept of function Junior and need to be able to switch between local and global perspectives Artigue ; Rogalski ; Vandebrouck Using a TWM lens Vandebrouck suggests a need to reconceptualise the concept of function in terms of its multiple registers and process-object duality.

The formal axiomatic world of university mathematics requires students to adopt a local perspective on functions, whereas only pointwise functions considered as a correspondence between two sets of numbers and global points of view representations are tables of variation are constructed at secondary school.

Limits: Students need to work with limits, especially of infinite sequences or series. Another, employing a TDS framework Ghedamsi developed situations that allowed students to connect productively the intuitive, perceptual and formal dimensions of the limit concept. Institutional factors: An aspect of transition highlighted by the ATD is that praxeologies exist in relation to institutions.

Employing the affordances of ATD, Praslon showed that by the end of high school in France a substantial institutional relationship with the concept of derivative is already established. Building on this work Bloch and Ghedamsi identified nine factors contributing to a discontinuity between high school and university in analysis and Bosch et al. Also employing an institutional approach, Dias et al.

They conclude that although contextual influences tend to remain invisible there is a need for those inside a given educational system to become aware of them in order to envisage productive collaborative work and evolution of the system. Other areas: One TDS-based research project examined a succession of situations for introducing the notions of interior and closure of a set and open and closed set Bridoux , using meta-mathematical discourse and graphical representations to assist students to develop an intuitive insight that allowed the teacher to characterise them in a formal language.

The conclusion was that many students have a weak understanding of ideas such as the suprema of bounded subsets, convergence of Cauchy sequences and the completeness of R.

Some possible ways to assist the calculus-analysis transition have been considered. A similar use of graphing calculator technology in consideration of the Fundamental Theorem of Calculus by Scucuglia made it possible for the students to become gradually engaged in deductive mathematical discussions based on results obtained from experiments.

In addition, Biehler et al. Abstract Algebra Understanding the constructs, principles, and eventually axioms, of the algebra of generalised arithmetic could be a way to assist students in the transition to study of more general algebraic structures.

This agrees with the observations of Godfrey and Thomas , who, using the TWM framework, provided evidence that many students have a surface structure view of equation and fail to integrate the properties of the object with that surface structure. The role of verbalisation in this process, as a semantic mediator between symbolic and visual mathematical expression, may require a level of verbalisation skills that Nardi , notes is often lacking in first year undergraduates.

As with other aspects of mathematics, counting requires combining a conceptual understanding of the nature of number with a fluent mastery of procedures that allow one to determine how many objects there are. When children can count consistently to figure out how many objects there are, they are ready to use counting to solve problems.

It also helps support their learning of conventional arithmetic procedures, such as those involved in computation with whole numbers. Preschool children bring a variety of procedures to the task of learning simple arithmetic.

Most of these procedures begin with strategic application of counting to arithmetic situations, and they are described in the next section. As with the distinction between conceptual understanding and procedural fluency, this categorization is somewhat arbitrary, but it provides a good example of how children can build on procedures such as counting in extending their mathematical competence to include new concepts and procedures.

Strategic Competence Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them. An important feature of mathematical development is the way in which situations that involve extended problem solving at one point can later be handled fluently with known procedures. Simple arithmetic tasks provide a good example.

Most preschoolers show that they can understand and perform simple addition and subtraction by at least 3 years of age, often by modeling with real objects or thinking about sets of objects.

In one study, children were presented with a set of objects of a given size that were then hidden in a box, followed by another set of objects that were also placed in the box. The majority of children around age 3 were able to solve such problems when they involved adding and subtracting a single item, although their performance decreased quickly as the size of the second set increased.

Much research has described the diversity of strategies that children show in performing simple arithmetic, from preschool well into elementary school. Some children will model the problem using available object or fingers; others will do it verbally. These strategies are discussed in detail in Chapter 6.

Kindergartners use all of these strategies, and second graders use all of them except for counting all. When 5-year-olds were given four individual sessions over 11 weeks in which they solved more than addition problems, most of them discovered the counting-on-from-larger strategy, which saves effort by requiring them to do less counting. They then were most likely to apply it to problems e.

The diversity of strategies that children show in early arithmetic is a feature of their later mathematical development as well. In some circumstances the number of different strategies children show predicts their later learning. Solving word problems. Young children are able to make sense of the relationships between quantities and to come up with appropriate counting strategies when asked to solve simple word, or story, problems.

Word problems are often thought to be more difficult than simple number sentences or equations. Young children, however, find them easier. If the problems pose simple relationships and are phrased clearly, preschool and kindergarten children can solve word problems involving addition, subtraction, multiplication, or division.

Will every bird get a worm? But they are less able to represent changes in sets or relationships between sets, in part because they fail to realize that the order of their actions is not automatically preserved on paper.

Adaptive Reasoning Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations and to justify and ultimately prove the correctness of a mathematical procedure or assertion. Adaptive reasoning also includes reasoning based on pattern, analogy, or metaphor. Research suggests that young children are able to display reasoning ability if they have a sufficient knowledge base, if the task is understandable and motivating, and if the context is familiar and comfortable.

Situations that require preschoolers to use their mathematical concepts and procedures in unconventional ways often cause them difficulty. For example, when preschool children are asked to count features of objects e. Most preschool children enter school with an initial understanding of procedures e.

A major challenge of formal education is to build on the initial and often fragile understanding that children bring to school and to make it more reliable, flexible, and general. They see mathematics as a meaningful, interesting, and worthwhile activity; believe that they are capable of learning it; and are motivated to put in the effort required to learn.