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# Mathematical Skills I: Reasoning & Communication - MAT00011C

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• Department: Mathematics
• Module co-ordinator: Dr. Michael Bate
• Credit value: 10 credits
• Credit level: C
• Academic year of delivery: 2017-18

## Module will run

Occurrence Teaching cycle
A Autumn Term 2017-18 to Summer Term 2017-18

## Module aims

The first years of all mathematics programmes are designed to give students a thorough grounding in a wide spectrum of mathematical ideas, techniques and tools in order to equip them for the later stages of their course. During first year, as well as consolidating, broadening and extending core material from pre-University study, we initiate a cultural transition to the rigorous development of mathematics which is characteristic at University. Students will develop both their knowledge of mathematics as a subject and their reasoning and communication skills, through lectures, tutorials, seminars, guided self-study, independent learning and project work. This development is addressed in all of our first year modules, although different modules have a different emphasis.

In addition to the above broad aims of the first year, this module focusses on mathematical reasoning and communication, developing the basic ideas of proof, familiarity with mathematical language and notation, the ability to construct mathematical arguments and present them to others. They will also develop a skill which is essential for their future careers, namely, the ability to write a good CV.

## Module learning outcomes

Subject content

Reasoning component.

• Develop the ability to: follow a mathematical argument (proof), critically; analyse such for correctness and completeness; appreciate the meaning and correct use of mathematical language (including elementary set theory, standard logical symbols and standard phrases); be familiar with some standard methods of proof (contradiction, contrapositive, induction); construct correct mathematical arguments in both familiar and unfamiliar contexts.

• Mathematical reasoning as a refined version of everyday reasoning. The intention of proof.

• The language of mathematics: statements, connectives, implication (various forms), negation. How to read and write mathematics. Analysing the clarity and precision of a written argument.

• Elementary set theory (“naive set theory") and its applications to giving problems a mathematical setting, including intersection, union, products and the notation for standard sets of numbers. De Morgan's Laws and their logical meaning.

• What is a function? Relations versus functions. Injective, surjective and bijective functions (including a function is a bijection if and only if has an inverse). Comparing these notions between functions on finite sets and real-valued functions.

• Standard methods of proof: syllogism, necessary compared with sufficient, contradiction compared with contrapositive, “There exists" and “For all" type arguments. The role of counter-examples. Common errors.

• Counting arguments, Peano's Axiom and mathematical induction. Application e.g. to the uniqueness of prime factorisation.

• Addition modulo n, through equivalence modulo n, leading onto equivalence classes as elements of a set.

• Infinity: its meaning, uses and abuses. The perils of infinite sums.

• The use of axiomatising (example: the axioms for a group). Consequences.

• The role of intuition. Methods for “getting started". Building a hypothesis from examples, and tightening up informal arguments. Analysing fallacies and paradoxes. (N.B. this aspect will be introduced early and revisited throughout the module).

Communication component.

• To be able to communicate mathematical ideas clearly and precisely, using calculations, data and arguments.

• Syllabus: The principles of clear writing and presentation, developed by analysing examples of clear exposition. Developing the self-discipline of analysing one's own writing critically as a method for achieving better understanding and therefore greater clarity.

• Academic skills: this module concentrates on key skills which are essential for any mathematician to have: the importance of rigour and the ability to communicate mathematical ideas are cornerstones of what it means to do mathematics.

• Many of the techniques and ideas developed in this module are ones which graduates employed as mathematicians will use from day to day in their work. Mathematics graduates are prized by employers for their ability to think logically and precisely; when combined with a well-developed ability to communicate this makes students much more employable.

• Each student will develop basic CV writing skills, through the process of writing a CV and receiving summative feedback on this.

Other Learning Outcomes

• The skills thread through the programme is designed to help students integrate into the life of the department and find out what it means to be a mathematician early on in their time at York, and then maintain that idea of being part of the wider community throughout their time with us. This module will help us to identify what is really important for students to get to grips with straight away, and develop that with an emphasis on small groups and a collegiate atmosphere.

## Module content

The Reasoning component will be taught via 2 lectures per week in Autumn term, together with fortnightly small-group tutorials with fortnightly assessed coursework marked by the tutor, with detailed feedback (written and face-to-face in class). An exam in January will also test the learning outcomes.

The Communication component will begin with 4 lectures on the principles of clear writing and

presentation, including the role of LaTeX. The fortnightly tutorials from Autumn will continue, helping students to develop their group projects and presentations. Projects will be handed in at the end of Spring term, and presentations will follow in early Summer. Alternate fortnightly drop-in sessions for LATEX advice and presentation advice. The group presentation assessment includes the delivery itself and the material developed for the presentation (supporting notes from each individual and displayed material).

The intention is that the presentations will be held over a couple of days as a departmental event, open to all, similar to arrangements for final year presentations.

CV writing skills will be introduced through a one-hour practical, in a computer room, in the middle of Spring term. The CV will be due in Spring week 10 and assessed by one staff member from a small panel convened for this purpose.

## Assessment

Task Length % of module mark
Essay/coursework
Coursework (CV) Essay : Coursework (CV) Essay : Coursework (CV) Essay : Coursework (CV) Essay : Coursework (CV)
N/A 10
Essay/coursework
Essay : Coursework
N/A 20
Essay/coursework
Essay : Project Report
N/A 20
Oral presentation/seminar/exam
Presentation
N/A 10
University - closed examination
Mathematical Skills I: Reasoning & Communication
1.5 hours 40

### Special assessment rules

Non-compensatable

### Reassessment

Task Length % of module mark
Essay/coursework
Coursework (CV) Essay : Coursework (CV) Essay : Coursework (CV) Essay : Coursework (CV) Essay : Coursework (CV)
N/A 10
Essay/coursework
Essay : Coursework
N/A 20
Essay/coursework
Essay : Project Report
N/A 20
Oral presentation/seminar/exam
Presentation
N/A 10
University - closed examination
Mathematical Skills I: Reasoning & Communication
1.5 hours 40

## Module feedback

Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Numbers and proof, RBJT Allenby, Arnold 1997 (S 0.1 ALL).

Introduction to proofs in mathematics, J Franklin & A Daoud, Prentice Hall, 1988 (S 0.1 FRA, out of print but full pdf available online).

A Concise Introduction to Pure Mathematics, M Liebeck (S 0.2 LIE).

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.

## Coronavirus (COVID-19): changes to courses

The 2020/21 academic year will start in September. We aim to deliver as much face-to-face teaching as we can, supported by high quality online alternatives where we must.

Find details of the measures we're planning to protect our community.