Fwd: course outline (pythonic math)
Here's something I originally posted to another list, but one that is whitespace insensitive. Hardly a venue for sharing anything Pythonic. Anyway, it's relevant here too. This is more of a sketch than something I've poured over for hours, just trying to give the gist of a possible Discrete Math course, one among hundreds or thousands or... not trying to write "the national standard" (blech). Kirby ==== DIGITAL MATH: Cardinality versus Ordinality Naming things Two different versus two the same Of zip codes and phone numbers Sorting things Equality, greater than, less than Hello World! A Biotum Class (Python) A Snake Class (Python) Numbers and Bases Positional Notation Lore: Algorithms liberate Europe (Liber Abacci) Lore: from ASCII to Unicode Decimal versus Hexadecimal Volume Bases: Tetrahedron versus Cube Functions and Relations (Part 1) Lore: the rise and fall of New Math Mappings Python's dictionary structure Caesar Codes Lore: from secret key to public key crypto Inverse Functions Injection, Surjection, Bijection What's a Relation? On Growth and Form Functions for Gnomon Growth Triangular Numbers Lore and Proof: Gauss summing 1..100 Square Numbers Proof: Sum of consecutive squares Tetrahedral Numbers Cubic Numbers Lore: Fibonaccis and Phi Generating Fibonaccis Generating Polyhedral Numbers Generating Pascal's Triangle Triangular and Tetrahedral Numbers Prime and Composite Numbers Euclid's Method for GCD Primes versus Composites Strangers Totatives and Totients Functions and Relations (Part 2) Permutations Polyhedral Rotations (dice in Casino Math) Composition of Functions (a kind of multiplying) Abstract Algebra I Multiplication: What is it Python and "Modulo Numbers" Vegetable Group Soup Cayley Tables Group Properties (CAIN and Abelian) Addition: What is it Rings and Fields Preview of Future Topics Supermarket Math SQL web frameworks Neolithic Math artifacts and encoded geography constellations Casino Math random number generators (Python) Deck and Card classes (Python) Martian Math Sphere packing and the Octet Truss Tetrahedral Accounting Pentagon Math Lore: geodesic spheres and domes, radomes Phi in Fuller's concentric hierarchy Notes for Teachers: Cardinality versus Ordinality  Before we order or sort, we need to recognize which things or objects are of which type. This course uses a type based mathematical logic known as the Python computer language, so awareness of types will be front and center from the get go. Exercises will include querying objects as to their types. Zaxis (depth dimension): if you've going through this in a spiral with plans to go deeper each time, then at some point your students may want to define their own classes and implement meanings for __lt__ __gt__ __eq__, Python's "ribs" (special names) for < , > and == respectively. However, this course outline does not make too many assumptions about which turn of the spiral one is in. Students will vary, as will teachers. Numbers and Bases  This should feel like fairly easy review. I recommend playing Tom Lehrer's 'New Math' from 'That Was the Year that Was' and making sure students get that it's mathematically correct. This is looking ahead to later lore, where we talk about the rise and fall of New Math. About Lore: this curriculum is premised on the notion that storytelling is integral to passing on a culture, and that too much time on a technical axis, to the exclusion of narrative context, is either counterproductive or is an intentionally applied filter aimed at testing student tolerance for "in the dark" learning. Functions and Relations (Part 1)  A lot of this is standard Algebra 1. New Math helped writers formalize their notion of function as distinct from a relation, using set theoretic constructs. This may not be the right place for a Python dictionary on a first pass. I was getting into Caesar Codes again recently, relating them to permutations and polyhedral rotations, and am freshly persuaded this is one of the better routes to elementary group theory, just ahead. http://mail.python.org/pipermail/edusig/2010March/009867.html http://mail.python.org/pipermail/edusig/2010March/009870.html On Growth and Form  Here's a way to connect the graphical and lexical without getting into XYZ coordinates or vectors right off. Gnomon studies and sphere packing keep the number sequences connected to the visualizations. Influences and valuable resources here would include 'The Book of Numbers' by Conway and Guy, 'Gnomon' by Midhat Gazale, and certain passages from 'Synergetics' by R.B. Fuller, with bolstering writing from H.S.M. Coxeter. Lore: getting a sense of Coxeter's outrage on finding nature's geometry had been patented by Fuller prefigures a generic distrust of the private ownership paradigm when applied to common natural heritage. In some classes, this might lead to a discussion of the free and open source software movement. I didn't include that in this particular segment though, as we're running low on time. Look in Pentagon Math under Future Topics. Abstract Algebra I  Because Python makes our exercises much more concrete and hands on, especially with scaffolding (prewritten / canned libraries or modules), it's new feasible to get more abstract. 'Concrete Mathematics' is an influence ('con' from continuous, 'crete' from discrete). With this kind of groundwork in place, it becomes easier to review topics such as dividing fractions, as we may now talk about division as "syntactic sugar" for "multiplying by the multiplicative inverse". Reviewing such basic concepts as the four operations with Q (Rational Numbers) would not be out of place in this segment. Preview of Future Topics  If this course or talk was used as a teaser or sampler, then here would be another chance to look ahead. I don't have anything on Pentagon Math at the Wikieducator site (Heuristics for Teachers). That's because you can easily fold it in with Neolithic and/or Martian Math. I split it out here because I wanted to dig into some of the lore in the Siobhan Roberts bio of Donald Coxeter. Remember: we think lore is very important, as are exercises and time alone with the Python interpreter (not all programming is pair programming in these initial stages, nor even later on  depends on the project). For further reading: http://www.4dsolutions.net/presentations/p4t_notes.pdf Kirby
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kirby urner