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An infinite descent into pure mathematicsACOG seminar, University of Pittsburgh
Clive Newstead
Carnegie Mellon University
Thursday 1st March 2018
Pedagogy Motivation Development Preview Conclusion
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
1 What I learnt about learning
2 Why I wrote a textbook
3 Developing the book
4 Preview of the book
5 Concluding remarks
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
1 What I learnt about learning
2 Why I wrote a textbook
3 Developing the book
4 Preview of the book
5 Concluding remarks
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Traditional mathematics class
Course designChoose topics to be coveredChoose grade weightings for assessmentsChoose textbook & set schedule
ClassesLecturer presents contentMaybe some question-and-answer interaction
AssessmentProblem sheetsQuizzes and examinations
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Current research about teaching and learning
Good practices to maximise student learning:Proper alignment of learning objectives, teaching strategies andmethods of assessmentEngaging students actively in the learning processUse of inquiry-based strategies and assessmentsUsing a variety of teaching methods and assessments
Reference: How Learning Works (2010) by Susan Ambrose & co.
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Alternative teaching model
Course designChoose learning objectivesDesign classes around helping students achieve LOsChoose types of assessment that test these LOs
ClassesBefore: some content delivery (e.g. reading, video)During: activities targetting LOs, clarifications, some lecture
AssessmentProblem sheets, quizzes, examinationsProjects, presentations, group work, . . .
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
1 What I learnt about learning
2 Why I wrote a textbook
3 Developing the book
4 Preview of the book
5 Concluding remarks
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Concepts of Mathematics — descriptionFrom course catalogue:
“This course introduces the basic concepts, ideas and toolsinvolved in doing mathematics. As such, its main focus ison presenting informal logic, and the methods ofmathematical proof. [...list of topics...]”
From department website:“Truth values, connectives, truth tables, contrapositives. Quantifiers. Proofby contradiction. Sets, intersections, unions, differences, the empty set.Integers, divisibility. Proof by induction. Primes, sieve of Eratosthenes,prime factorization. Gcd and lcm, Euclid’s algorithm, solving ax + by = c.Congruences, modular arithmetic. Recursion. Linear recurrences.Functions and inverses. Permutations. Binomial coefficients, Catalannumber. Inclusion-exclusion. Infinite cardinalities. Binary operations.Groups. Binary relations, equivalence relations. Graphs. Eulercharacteristic, planar graphs, five color theorem, rationals, reals,polynomials, complex numbers.”
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Concepts of Mathematics — course design
Learning objectives
“Presenting informal logic” = communication“Methods of mathematical proof” = problem-solving
Mathematical topics
Symbolic logic, sets, functionsInduction on the natural numbersNumber theoryCombinatoricsOther topics (real numbers, probability theory, basic topology, ...)
Syllabus: X Next step: find a textbook.
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Concepts of Mathematics — textbook
Textbook criteria: a textbook for Concepts should. . .Be of an appropriate lengthCover enough mathematical topicsCover communication and problem-solving skillsPractise what it preachesBe as agnostic as possible
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Concepts of Mathematics — textbook
Solution: Write my own notes
Time frame: 51 days
Backup plan: Concede
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
1 What I learnt about learning
2 Why I wrote a textbook
3 Developing the book
4 Preview of the book
5 Concluding remarks
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Design considerationsLots of decisions to make
Mathematical areas to coverDefinitions and theorems to emphasiseInclude exercise solutions or not?Level of difficultyChoices of conventionLevel of verbosityLevel of detail in proofsGeneral tone of the bookHow to cover both skills and contentName of the bookLicensing and copyright issues
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Example dilemma #1
What is a function?
Possible definitions:(1) A set of ordered pairs such that . . .(2) A triple (X ,Y , f ) where f ⊆ X × Y such that . . .(3) A rule assigning to each x a unique y(4) An imaginary machine taking inputs and giving outputs(5) A primitive notion in terms of which all other mathematical notions
are defined
My choice: (3) because it is the most agnostic
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Example dilemma #2
Is zero a natural number?
Possible resolutions:(1) Yes(2) No(3) Choose your own adventure, make explicit when needed(4) Use N0 and N1 (or similar)
My choice: (1) for lots of reasons
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Example dilemma #3
What is the best domain of discoursefor number theory?
Possible answers:(1) Z(2) N (with or without zero?)(3) Sometimes N, sometimes Z
My choice: (1) because it generalises easily to more general rings
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Return to learning objectives
Communication skillsUsing notation accuratelyDeveloping mathematical fluencyEvaluating effectiveness of others’ proofsTypesetting in LATEX
Design principlesWrite accurately and clearlyInclude discussion exercisesProvide guidance on how to structure a proofProvide LATEX support
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Return to learning objectives
Problem-solving skillsIdentifying feasible proof strategiesIdentifying relevant definitions and theoremsCreativity in problem-solving approaches
Design principlesExamples and exercises galoreVary level of difficultyProvide problem-solving tipsDo not provide solutions to exercises
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
And so the writing began
Note: this is not actually me. . . I don’t have a Mac
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Growth of the book
Date Event # pages ∆May 9, 2015 Started writing 0 −Jun 29, 2015 Started teaching 21-127 134 134Aug 7, 2015 Finished teaching 21-127 183 49
May 24, 2016 Started preparing for 21-128 183 0Aug 29, 2016 Started teaching 21-128 204 21Dec 9, 2016 Finished teaching 21-128 348 144Mar 1, 2018 (today) 394 46
????? Finished writing < 500 ???
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
1 What I learnt about learning
2 Why I wrote a textbook
3 Developing the book
4 Preview of the book
5 Concluding remarks
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
1 What I learnt about learning
2 Why I wrote a textbook
3 Developing the book
4 Preview of the book
5 Concluding remarks
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Next steps
Remaining tasksFinish remaining chaptersAdd more examples, discussions & exercisesAdd more guidance for communication and proof-writingAdd more diagrams and graphicsInclude chapter introductions, reflections and summaries
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Reflection
What I have learntWriting a textbook takes a lot of time and effortWriting a textbook does not contribute towards PhD requirementsYou can’t make everyone happyLATEX is full of surprisesHaving a project to work on is funThis book might never be finished
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics
Pedagogy Motivation Development Preview Conclusion
Thanks for listening!Website
www.infinitedescent.xyz
These slidesmath.cmu.edu/˜cnewstea/talks/20180301.pdf
Clive Newstead ([email protected]) Carnegie Mellon University
An infinite descent into pure mathematics