Pure mathematics - Wikipedia, the free encyclopedia. An illustration of the Banach.
Although it is proven that it is possible to convert one sphere into two using nothing but cuts and rotations, the transformation involves objects that cannot exist in the physical world. Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts.
A-Level Maths Resources: For teaching or learning A-level Maths and Further Maths, this is a collection of the hundreds of worksheets, interactive spreadsheets and other resources designed for introducing new concepts, testing.
This was a recognizable category of mathematical activity from the nineteenth century onwards. On the other hand, many pure mathematicians draw on natural and social phenomena as inspiration for their abstract research. History. Plato helped to create the gap between .
Plato regarded logistic (arithmetic) as appropriate for businessmen and men of war who . The idea of a separate discipline of pure mathematics may have emerged at that time.
The generation of Gauss made no sweeping distinction of the kind, between pure and applied. In the following years, specialisation and professionalisation (particularly in the Weierstrass approach to mathematical analysis) started to make a rift more apparent. The logical formulation of pure mathematics suggested by Bertrand Russell in terms of a quantifier structure of propositions seemed more and more plausible, as large parts of mathematics became axiomatised and thus subject to the simple criteria of rigorous proof. In fact in an axiomatic setting rigorous adds nothing to the idea of proof. Pure mathematics, according to a view that can be ascribed to the Bourbaki group, is what is proved. Pure mathematician became a recognized vocation, achievable through training.
The case was made that pure mathematics is useful in engineering education. Category theory is one area of mathematics dedicated to exploring this commonality of structure as it plays out in some areas of math. Generality's impact on intuition is both dependent on the subject and a matter of personal preference or learning style. Often generality is seen as a hindrance to intuition, although it can certainly function as an aid to it, especially when it provides analogies to material for which one already has good intuition. As a prime example of generality, the Erlangen program involved an expansion of geometry to accommodate non- Euclidean geometries as well as the field of topology, and other forms of geometry, by viewing geometry as the study of a space together with a group of transformations. The study of numbers, called algebra at the beginning undergraduate level, extends to abstract algebra at a more advanced level; and the study of functions, called calculus at the college freshman level becomes mathematical analysis and functional analysis at a more advanced level. Each of these branches of more abstract mathematics have many sub- specialties, and there are in fact many connections between pure mathematics and applied mathematics disciplines.
Teach Yourself an A Level in Mathematics (Please tweet or Facebook the link out if you found it helpful!). Fellowships opportunities for people from the developing countries. The Faculty for the Future Scholarship: https://www.fftf.slb.com/ The L’Oreal - Unesco (.). This module explores how students learn, both generally and in the discipline of mathematics. At the end of this module you will be able to: explain different learning theories as they relate to mathematics. Mathematics (from Greek . There is a range of views among mathematicians and philosophers as to.
Explain how a square is. Broadly speaking, pure mathematics is mathematics that studies entirely abstract concepts. This was a recognizable category of mathematical activity from the nineteenth century onwards, at variance with the trend towards.
A steep rise in abstraction was seen mid 2. In practice, however, these developments led to a sharp divergence from physics, particularly from 1. Later this was criticised, for example by Vladimir Arnold, as too much Hilbert, not enough Poincar. The point does not yet seem to be settled, in that string theory pulls one way, while discrete mathematics pulls back towards proof as central. Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics. One of the most famous (but perhaps misunderstood) modern examples of this debate can be found in G. H. Hardy's A Mathematician's Apology.
1 Water use 1.1 Water as a resource. Water is arguably the most important physical resource as it is the only one that is essential for human survival; we would die very quickly without it. Other physical resources can make. Questions regarding J Zipped folders containing all AQA A-level Maths exam papers and mark schemes from Jan 2005 to Jun 2012.
It is widely believed that Hardy considered applied mathematics to be ugly and dull. Although it is true that Hardy preferred pure mathematics, which he often compared to painting and poetry, Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express physical truth in a mathematical framework, whereas pure mathematics expressed truths that were independent of the physical world. Hardy made a separate distinction in mathematics between what he called . Moreover, Hardy briefly admitted that. In that subject, one has the subareas of commutative ring theory and noncommutative ring theory. An uninformed observer might think that these represent a dichotomy, but in fact the latter subsumes the former: a noncommutative ring is a not necessarily commutative ring. If we use similar conventions, then we could refer to applied mathematics and nonapplied mathematics, where by the latter we mean not necessarily applied mathematics.
It deals with concepts such as continuity, limits, differentiation and integration, thus providing a rigorous foundation for the calculus of infinitesimals introduced by Newton and Leibniz in the 1. Real analysis studies functions of real numbers, while complex analysis extends the aforementioned concepts to functions of complex numbers. Functional analysis is a branch of analysis that studies infinite- dimensional vector spaces and views functions as points in these spaces. Abstract algebra is not to be confused with the manipulation of formulae that is covered in secondary education. It studies sets together with binary operations defined on them.
Sets and their binary operations may be classified according to their properties: for instance, if an operation is associative on a set that contains an identity element and inverses for each member of the set, the set and operation is considered to be a group. Other structures include rings, fields, vector spaces and lattices. Geometry is the study of shapes and space, in particular, groups of transformations that act on spaces. For example, projective geometry is about the group of projective transformations that act on the real projective plane, whereas inversive geometry is concerned with the group of inversive transformations acting on the extended complex plane.
Number theory is the theory of the positive integers. It is based on ideas such as divisibility and congruence. Its fundamental theorem states that each positive integer has a unique prime factorization.
In some ways it is the most accessible discipline in pure mathematics for the general public: for instance the Goldbach conjecture is easily stated (but is yet to be proved or disproved). In other ways it is the least accessible discipline; for example, Wiles' proof that Fermat's equation has no nontrivial solutions requires understanding automorphic forms, which though intrinsic to nature have not found a place in physics or the general public discourse. Topology is a modern extension of Geometry. Rather than focusing on the sizes of objects and their precise measurement, topology involves the properties of spaces or objects that are preserved under smooth operations such as bending or twisting (but not, for example, tearing or shearing). Topology's subfields interact with other branches of pure math: traditional topology uses ideas from analysis, such as metric spaces, and algebraic topology relies on ideas from combinatorics in addition to those of analysis. See also. Mac. Tutor History of Mathematics archive.
Retrieved 1. 2 July 2. A History of Mathematics (Second ed.).
John Wiley & Sons, Inc. Plato is important in the history of mathematics largely for his role as inspirer and director of others, and perhaps to him is due the sharp distinction in ancient Greece between arithmetic (in the sense of the theory of numbers) and logistic (the technique of computation). Plato regarded logistic as appropriate for the businessman and for the man of war, who . A History of Mathematics (Second ed.). John Wiley & Sons, Inc.
Evidently Euclid did not stress the practical aspects of his subject, for there is a tale told of him that when one of his students asked of what use was the study of geometry, Euclid asked his slave to give the student threepence, . A History of Mathematics (Second ed.). John Wiley & Sons, Inc.
It is in connection with the theorems in this book that Apollonius makes a statement implying that in his day, as in ours, there were narrow- minded opponents of pure mathematics who pejoratively inquired about the usefulness of such results. The author proudly asserted: .
Aust MS : M2 - Models of maths learning. Introduction. This second module will investigate how people learn mathematics and continue to investigate the nature and attributes of students. You will engage with literature and video elements to examine the major theories and research relevant to tertiary mathematics education. It is important to note that this is an open area of study and that there are a range of other theories and approaches that have not been discussed.
Learning Outcomes. This module explores how students learn, both generally and in the discipline of mathematics. At the end of this module you will be able to: explain different learning theories as they relate to mathematicsdescribe different approaches to mathematics learning and how they can inform your mathematics teachingoutline different learning styles and describe ways in which your teaching may be adjusted to cater to them. Module Structure. The module proceeds as follows: Perspectives of learning. There are a range of views of learning that you may consciously or inadvertently adopt in your mathematics teaching.
For instance: Behaviourism - learning and teaching is essentially about changing behaviours through stimulus materials and consequent rewards/punishments. Constructivism - learning and teaching is about designing activities that enable students to build upon prior knowledge to construct internally consistent mental models.
Socio- constructivism - learning is a collaborative, negotiated experience that is most effectively achieved through peer and group interaction. Authentic learning - learning is best facilitated through realistic, relevant, complex, and problematised tasks. Other perspectives of learning including cognitivism, constructionism and connectionism and connectivism.
Watch the following two videos that explore some of these perspectives on learning. This is expounded in the video Why We (Should) Teach the Way We (Should) Teach: How Learning Theory Should Impact the Design of Classroom Experiences. In this presentation Dr Nancy Casey, Associate Professor of education emphasises that . Bonaventure University, n. An abridged version of Dr Casey's presentation is provided below (IAm.
Bona, 2. 00. 9): Casey stresses that teachers at all levels need to acknowledge that the important thing is not the teaching, but the learning. For instance, it would be erroneous to assume that because you have transmitted the content to students in a lecture, they have learnt the material. Task 2. 1 Perspectives of learning for your classes Select three perspectives of learning that most interest you to investigate in more depth. Outline the key features of the three perspectives of learning that you have selected and compare and contrast the relative advantages and disadvantages of each. Which approach do you believe to be most important in mathematics learning? Why? Which approach/approaches do you see yourself adopting in your classes? Explain. Models of mathematics learning.
Several experienced mathematics education researchers have presented models and heuristics to guide mathematics teaching. It is an interesting and open question - what are the different ways in which students learn mathematics? Three models of mathematics learning that will be examined here are Schoenfeld's model of mathematical problem solving, Skemp's Instrumental versus Relational Understanding, and Threshold concepts. Schoenfeld's approach to mathematical problem solving. Alan Schoenfeld identifies four types of knowledge and skills that are required to be successful in mathematics.
They are: Resources - factual and procedural knowledge of mathematics. Heuristics - problem solving strategies and techniques (deductive reasoning, working backwards, drawing diagrams etc).
Control - the ability to make decisions about which strategies are used when and how. Beliefs - a mathematical epistemology or . When confronted with having to do a proof, the students need to be able to recognise what knowledge they need to apply and when to apply it.
I can help with this by showing them the thought processes behind the proofs that I do in class, pointing out how to start one, think about what you know about the topic, and how to put the knowledge together to get a proof. Further reading. Skemp's Instrumental versus Relational understanding of mathematics.
Richard Skemp draws the distinction between learning and teaching that takes a relational approach and that which takes an instrumental approach. Relational understanding refers to knowing both what to do and why - an understanding of all of the parts, how they relate, and why they are applied in the manner they are. On the other hand instrumental understanding refers merely to being able to apply a series of steps without knowing why they are being applied in that way or what they mean - 'rules without reasons'.
Quite often when students learn mathematics they experience (perhaps encouraged by the approaches of their teachers) temporary success by forming an instrumental understanding of the topic. However this is at the detriment of their long- term success compared to their potential if they had formed a relational understanding.
For instance, when learning differentiation students might easily learn 'the rules' of calculating derivatives but have little concept of when to use each rule and what the derivative means in practice. The consequence of forming this instrumental rather than relational understanding is that they cannot sense the utility of differential calculus and they are unable to solve problems with it. Previous participants in this module identified that to promote relational understanding, teachers need to consciously set tasks requiring relational understanding, and to reward it (or not reward instrumental understanding as highly).
The rewards of relational understanding suggested were not only marks(!) but also the satisfaction of the “Aha!” kind of experience, and the better recall that comes from relational understanding. Here is one of their suggestions: Try to change the assessment tasks to include tasks that require a relational understanding.
For example, “Write a paragraph that a peer could understand, including diagrams where appropriate, explaining ..”Further reading. Skemp, R. Relational Understanding and Instrumental Understanding, Mathematics Teaching, 7. Threshold concepts in mathematics. It is valuable when planning your unit and learning activities, to spend some time considering the .
This is a relatively new insight into learning that has been developed by Meyer and Land (2. It represents a transformed way of understanding, or interpreting, or viewing something without which the learner cannot progress. Understanding threshold concepts can be a kind of initiation into the predominant way of thinking, perceiving and practising in a particular discipline (Meyer & Land, 2.
Threshold concepts are: transformative (they trigger a shift in perception)irreversible (they cannot be easily unlearned or discarded)integrative (they expose previously hidden or unrecognised connections and interrelations)bounded (they open up a new conceptual space, and to move beyond it may take engagement with a further threshold concept)troublesome (appearing complex, alien, counterintuitive or incoherent).(Meyer & Land, 2. For instance, in mathematics, a complex number consisting of a . In pure mathematics the concept of a limit is another threshold concept. A previous participant identified understanding a scalar function of two variables as a surface to be a threshold concept in vector calculus. Well before the idea of threshold concepts gained acceptance across many disciplines, the mathematician David Tall had described the “difficult transition” to Advanced Mathematical Thinking that must take place during undergraduate mathematical education (Tall, 1.
Given the centrality of such concepts within sequences of learning and curricular structures, their troublesome nature for students assumes significant pedagogical importance (Meyer and Land, 2. Thus it is critical that teachers dedicate adequate time and focus to developing students' understanding of threshold concepts. Possible strategies include extra time dedicated to instruction (including provision of applied examples), time for students to experiment with problems, and peer explanation tasks. Further reading. Meyer, J., & Land, R. Threshold concepts and troublesome knowledge (2): Epistemological Considerations and a Conceptual Framework for Teaching and Learning.
Higher Education, 4. The Transition to Advanced Mathematical Thinking: Functions, Limits, Infinity and Proof.
A PDF of this article is available online. Task 2. 2: Ways your students learn mathematics. Post responses to the following questions on the discussion board: Which of the four areas outlined by Schoenfeld do you feel will cause or does cause students in your classes the most difficulty?
What strategies can you apply as a teacher to address this? You notice that some of the students in your class are only aiming for an instrumental rather than relational understanding of the mathematics being covered in your unit. What can you as the teacher do? What are the key threshold concepts in unit/s you are teaching? What can you do to support students to acquire these threshold concepts?
General education frameworks relevant to mathematics. As well as models of learning that are specific to mathematics, there are also general models of learning that can be applied to learning and teaching mathematics.
Two that will be examined here are Anderson and Krathwohls' (2. Taxonomy of Learning, Teaching and Assessing, and the Structure of Observed Learning Outcomes (SOLO) Taxonomy. Anderson and Krathwohl's taxonomy of learning, teaching and assessing.
Anderson and Krathwohl (2. Bloom's taxonomy of learning objectives to derive their comprehensive framework for Learning, Teaching and Assessing.