MATH20009: Perspectives in Mathematics Teaching blocks 1 & 2 2017/18 This unit will be comprised of three periods of activity, in each of which there will be a choice between a task which uses the students’ maths background to develop communication or career skills, and one which gives scope for independent exploration of new mathematics.
The unit will also involve aspects of group work, peer review and support, peer assessment, as well as presenting work orally and in written form. As these skills are important in almost any career, whether professional or academic, it is suggested that all 2nd year students seriously consider taking this course.
This unit will be comprised of three periods of activity, in each of which there will be a choice between a task which uses the students’ maths background to develop communication or career skills, and one which gives scope for independent exploration of new mathematics. The unit will also involve aspects of group work, peer review and support, peer assessment, as well as presenting work orally and in written form. As these skills are important in almost any career, whether professional or academic, it is suggested that all 2nd year students seriously consider taking this course.

Recommended reading 0 items

Further Reading 2 items

How to think like a mathematician: a companion to undergraduate Mathematics. By Kevin Houston 2 itemsA copy of the full book is available in the library, but in addition Chapters 3 and 4 on how to write mathematics are freely available electronically and are linked below.

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T1B 10 itemsThese articles cover a variety of areas of mathematics and are reasonably short. However if you follow the link to one of these articles, you will be on the Mathematics Magazine website and can look around to find your own article on a topic that interests you.

The Abundancy Ratio, a Measure of Perfection  01/10/2000
Article

Compounding Evidence from Multiple DNATests  1999
Article

Disentangling Topological Puzzles by Using Knot Theory  01/12/2006
Article

The Lengthening Shadow: The Story of Related Rates  , , 01/02/2000
Article

Series That Probably Converge to One  2009
Article

Dinner, Dancing, and Tennis, Anyone?  01/02/2000
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Seeing Dots: Visibility of Lattice Points  2007
Article

The Geometry behind Paradoxes of Voting Power  2009
Article

On Lexell’s Theorem  , 2017
Article

How Long Until a Random Sequence Decreases?  2010
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T3B 21 itemsBelow are some suggestions for T3B topics. These are just suggestions  for the computing and maths exploration projects you are encouraged to come up with your own ideas of areas you want to research. I've included an example of some reading material, just to demonstrate that there is literature available. This may not be the most appropriate reference for your topic and may not end up being your main reference.

Education: Presenting maths in a school 2 itemsA maximum of one group will be selected for this type of project. Around week 19 you will work with a group of secondary school students for about an hour (exact time depends on the length of classes in the school) to introduce them to some exciting mathematical ideas. The school will provide us with some suggestions in January as to which year group would benefit from this and what topics would fit in with the students’ learning. All T3 projects require you do so some scholarly learning yourself, so just preparing and delivering the presentation is not enough for this project. You are also expected to read some maths education literature: for example, you could inform yourself on how to maximize understanding or engagement from students of the age you are working with. Your presentation and your written report will detail the learning you did, how you used it in your work with the school children and how it was received by them.

Education: Preparing a display for the new mathematics building 2 itemsThe School of Mathematics is in need of some engaging, permanent displays relaying some mathematical ideas to visitors to the School. Your audience is the general public: think of the secondary school students and their parents who visit on open days as they decide which university to go to. You will prepare an engaging display board, and possibly some interactive demonstrations to go with it. A selection of these projects will be chosen for display in the new building. All T3 projects require you to do some scholarly learning yourself, so just creating the display is not enough for this project. Either the mathematics presented should contain significant content that is not familiar to 2nd year students, or else you are expected to read some literature on information visualization or about how to actively engage an audience with a mathematical display. Your presentation and your written report will detail the learning you did, how you used it in creating your display, and whether you think it was effective.

Information visualization: perception for design  , 2013
Book

Visual explanations: images and quantities, evidence and narrative  c19970961392126
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Computing: Optimisation algorithm for placing Perspectives in Maths students in groups 2 itemsYou will be given a sample data set of Perspectives in Mathematics students with their selections for the type of T3 project, their preference of a couple of people to work with, and the groups they have already worked with in T1 and T2. Your job is to write an algorithm, using readily available software, to optimize T3 group allocation. All T3 projects require you to do some scholarly learning yourself, so just programming the code is not enough for this project. You will need to research optimization literature to educate yourself about what is known about this type of algorithm. Your presentation and written report will detail the learning you did and how you used it in creating your code.

Combinatorial Optimization  , October 6, 2005 (Hardcover)
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Combinatorial optimization for undergraduates  1984038790977X
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Computing: Fractal sets 1 itemFractal geometry concerns the mathematics of sets and functions that are not smooth. There is scope in this topic for mathematical exploration as well as gaining programming experience. Chapters 1 and 2 of the book below are an introduction to the mathematics of fractals.

Computing: Netflix prize problem 1 itemMany online salesbased businesses employ ``recommender systems” to improve the service they deliver or to tempt users into spending more money on their products. Imagine that you have viewed a set of items or, perhaps, bought a set of items from a website. A recommender system would attempt to determine which of the other items for sale on that web site would be most likely to attract your attention. Then they would advertise these items at online checkouts or in ad boxes. How would a recommender system work? The online retailer has access to a complete database of users and their habits and the system will work best if it can somehow infer information about your preferences based on the preferences of others. Such an approach is called collaborative filtering. The notes below are extracted from the first year Computational Mathematics unit and are just to give you an idea of the problem  you would need to find further resources about this type of algorithm

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Maths exploration: Counterexamples in Analysis 1 itemExplore some of the counterintuitive phenomena in analysis: Can an uncountable set have measure zero? Could a continuous function be differentiable nowhere? The reference below is a collection of examples of this type.

Counterexamples in analysis  , c1964
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Maths exploration: Special functions 1 itemSpecial functions such as hypergeometric functions or the gamma function surface in many areas of mathematics such as mathematical physics and analysis, and find applications in astronomy and other sciences. There are very old results as well as very current research on their properties. A discussion of Gamma and Beta functions are in Chapter 1 of:

Special Functions  , , 1999
Book


Maths exploration: the isoperimetric theorem 1 itemThe problem of maximizing the area of a region with a given perimeter goes back to Ancient Greeks dividing up plots of land. Two approaches (one geometric and one using Fourier analysis) to this problem are given in Chapters 35 and 36 of

Fourier Analysis  1988
Book


Maths exploration: Crystallography 1 itemParticles in a crystal arrange themselves in an orderly threedimensional configuration, resulting in a high degree of symmetry that lends itself to mathematical study. An introduction to crystallography can be found in Chapters 2 and 3 of

Maths exploration: Knots 1 itemThe mathematical theory of the classification of knots is related to fields such as topology, graph theory and abstract algebra. There are applications of this to DNA if a group wished to take it in this direction. Below is an excerpt from a book that gives a fairly easy introduction to the subject.

Maths exploration: Structural Rigidity 1 itemRigidity theory is an area of mathematics useful to the design of robust structures and involves graph theory and combinatorics. An introduction is given in the book below.

Counting on frameworks: mathematics to aid the design of rigid structures  , c20010883853310,9780883853313
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Maths exploration: Irrational numbers 1 itemIn twelve short pages, in Chapter 2 of the book below, you will find proofs that pi is irrational, as is ln2, cos(1) and the cube root of 17 and that e is transcendental.

Irrational Numbers  , 2014 (electronic resource)9781614440116
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Maths exploration: Partitions 1 itemThe number 4 can be partitioned in 5 ways: 4=3+1=2+2=2+1+1=1+1+1+1. Partitions can be studied combinatorially, analytically or geometrically. Hardy and Ramanujan found an asymptotic formula for the number of partitions of n. You can get a flavor of this area in Chapter 14 of the book below.

Introduction to analytic number theory  1976
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Maths exploration: Elliptic curves 1 itemElliptic curves were used by Wiles in the solution of Fermat’s Last Theorem. They are of interest to physicists, geometers and cryptographers. You can get an introduction in Chapter 1 of the book below.

Rational points on elliptic curves  , 2015
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Maths exploration: extending Probability 2 1 itemFor those who enjoyed the 2nd year unit Probability 2, Chapters 9 (Stationary Processes) and 10 (Renewals) of the book below extend ideas from that unit. It would be easier if some of your group had taken Probability 2.

Probability and random processes  , 20010198572239,0198572220,9780198572220
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Maths Exploration: Statistical Genetics 1 itemA little challenging, but very interesting material on statistical genetics can be found in chapters 1 through 4 in the reference below. It would be a good idea of some of your group has taken Probability 2

Maths exploration: Frieze and wallpaper groups 1 itemJust as in art or architecture, in mathematics a frieze is a twodimensional pattern that is repeated in one direction. In this topic you can investigate the set of symmetries of such a pattern. Chapter 2 in the reference below is a place to start.

Mathematics and technology  , , 2008 (electronic resource)
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Maths exploration: Random walks and electric networks 1 itemThis is a very readable piece on connecting Markov chains with electric networks. It is based on the observation that certain quantities like hitting probabilities satisfy a harmonic equation much similar to the ones found in elementary descriptions of electric resistor networks. This simple fact gives rise to the application of powerful physical ideas in Markov chain theory. One can push these as far as proving the celebrated Pólya theorem on recurrence and transience of simple random walk. This project might be especially suitable for students interested, or having some background, in (secondary school) physics. It is also useful if some of your group have taken Probability 2.

Random Walks and Electric Networks  2000
Article

