# An Algebra-then-Digital-Math track

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## An Algebra-then-Digital-Math track

 What should we cover in Algebra to have students be ready for their next math courses, which use Python or some other computer language as a tool? We've already discussed the concept of Function quite a bit, which has its mathematical "Dolciani" meaning plus other meanings in English, such as you'll find in any dictionary, plus the meaning "subroutine" as Hansen calls it, though in Python is the formal type name (FunctionType), another type of object, passable as such to other functions as arguments. What about the concept of set?  Of course.  Even after the ebbing of the New Math (SMSG) tide, set notation was left high and dry on the shore, like driftwood or a whale skeleton:  intersection, union, subset... all still in Singapore Mathematics and Saxon, a basis for spiraling, as that notation lives on into higher math books, most definitely. Python's set is a lot like its dict, a mapping, but without the values, only keys.  dicts are lookup structures, like arrays but with whatever immutable object for a subscript, one might say.  Zoo sub "lion" i.e. Zoo["lion"] might retrieve the webcam object you need to look through, to make sure "lion" is OK.  So definitely we need sets combined with the formal Dolciani notion of function (with "into", "onto" "bi-jective" and all that).Next:  coordinate systems and vectors, with polyhedrons made of edges as defined by two vectors (which always originate at the origin).  Rotation matrix from linear algebra.  This is going more off IB (International Baccalaureate).  A lot of USA students never get to Vectors in algebra, yet are expected to work with them in physics.  It gets to be messy.  In Python, we'll be defining a Vector and Edge object (I have the source code out there in many venues, as do many digital math teachers), a Polyhedron object and so forth.  The nuts and bolts of Vector addition and subtraction, scaling, will be baked right in to the code.So that's a good little Algebra course:===Preview / OverviewData Structures with Lists and Mappings (set, dict...) Functions (many non-numeric examples, many OEIS sequences previewing geometric ideas)Types of Object (categorizing mathematical objects, taxonomy of)Sets (spiraling back to data structures, looking at one in particular) Functions in terms of sets (the more formal "ordered pair" idea -- no rules needed)Sequences using Functions (e.g. Fibonacci, power series, convergence / divergence / chaos)Geometric Objects:  Vectors, Edges and Polyhedrons (uses vZome by Vorthmann) Coordinate Systems:  XYZ, Spherical, sidebar into Quadray (exotic, check Wikipedia)Coding Types 101: an introduction to type definitions in Python (leverages Functions)Types coded:  Fraction, Number with Modulus, Vector, Edge, Polyhedron Summary / Overview===Obviously I'm jumping the gun a bit in already starting to use Python right inside the Algebra course, but this is exactly the kind of prep you need for Digital Math track work, which will include some calculus if you keep going, also some cryptography (e.g. RSA, per the Litvins' text). Kirby
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## Re: An Algebra-then-Digital-Math track

 On Jun 28, 2014, at 12:53 PM, kirby urner <[hidden email]> wrote: > A lot of USA students never get to Vectors in algebra, I would have said practically all of them don’t get to vectors, and most of them don’t even get algebra. Vectors generally come in physics and require the student be successful with algebra, geometry and trig first. And if your are going to start doing rotation matrices, then they will need some background in linear algebra. Baring a physics course, vectors and linear algebra would have to wait for algebra 3 or precalculus. Functions and sets would have already been pretty well baked by then, and they will have had sufficient geometry and if you are lucky, trig, which frees you to concentrate on vectors, transformations, linear algebra and programming. Bob Hansen ------- End of Forwarded Message
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## Re: An Algebra-then-Digital-Math track

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## Re: An Algebra-then-Digital-Math track

 In reply to this post by kirby urner-4 Working on the assumption that math is highly sequential (certain topics need be mastered as prerequisites for subsequent topics) then such a list does indeed seem reasonable. Though I tend to think we usually jump into the middle & back-track where necessary. But generally, I feel the major concern for students of math may lie in the affective domain. The ability to benefit from mistakes, the ability to concentrate for extended periods - not to simply give up. I tend to think of mathematics as something to be "figured out" rather than simply "learned". Students might well benefit from trying the chapter end exercises before studying the chapter itself. Students may well be able to 'complete the square' after being taught the method - but should be encouraged to figure out why it works. Math is possibly the aptest subject  area wherein a student can 'reinvent the wheel'. As for prerequisites for a math/programming course: early exposure to a 'turtle graphics' package would be well worth considering. It is visual, correlates closely to physical reality and is capable of producing dramatic images. Reinforces concepts of distance & motion including orientation. But best of all - it can provide enjoyment & the satisfaction of succeeding in achieving a goal. And most programming languages, including Python, have a turtle module. Gary Tupper, Terrace BC On 6/28/2014 9:53 AM, kirby urner wrote: What should we cover in Algebra to have students be ready for their next math courses, which use Python or some other computer language as a tool? We've already discussed the concept of Function quite a bit, which has its mathematical "Dolciani" meaning plus other meanings in English, such as you'll find in any dictionary, plus the meaning "subroutine" as Hansen calls it, though in Python is the formal type name (FunctionType), another type of object, passable as such to other functions as arguments. What about the concept of set?  Of course.  Even after the ebbing of the New Math (SMSG) tide, set notation was left high and dry on the shore, like driftwood or a whale skeleton:  intersection, union, subset... all still in Singapore Mathematics and Saxon, a basis for spiraling, as that notation lives on into higher math books, most definitely. Python's set is a lot like its dict, a mapping, but without the values, only keys.  dicts are lookup structures, like arrays but with whatever immutable object for a subscript, one might say.  Zoo sub "lion" i.e. Zoo["lion"] might retrieve the webcam object you need to look through, to make sure "lion" is OK.  So definitely we need sets combined with the formal Dolciani notion of function (with "into", "onto" "bi-jective" and all that). Next:  coordinate systems and vectors, with polyhedrons made of edges as defined by two vectors (which always originate at the origin).  Rotation matrix from linear algebra.  This is going more off IB (International Baccalaureate).  A lot of USA students never get to Vectors in algebra, yet are expected to work with them in physics.  It gets to be messy.  In Python, we'll be defining a Vector and Edge object (I have the source code out there in many venues, as do many digital math teachers), a Polyhedron object and so forth.  The nuts and bolts of Vector addition and subtraction, scaling, will be baked right in to the code. So that's a good little Algebra course: === Preview / Overview Data Structures with Lists and Mappings (set, dict...) Functions (many non-numeric examples, many OEIS sequences previewing geometric ideas) Types of Object (categorizing mathematical objects, taxonomy of) Sets (spiraling back to data structures, looking at one in particular) Functions in terms of sets (the more formal "ordered pair" idea -- no rules needed) Sequences using Functions (e.g. Fibonacci, power series, convergence / divergence / chaos) Geometric Objects:  Vectors, Edges and Polyhedrons (uses vZome by Vorthmann) Coordinate Systems:  XYZ, Spherical, sidebar into Quadray (exotic, check Wikipedia) Coding Types 101: an introduction to type definitions in Python (leverages Functions) Types coded:  Fraction, Number with Modulus, Vector, Edge, Polyhedron Summary / Overview === Obviously I'm jumping the gun a bit in already starting to use Python right inside the Algebra course, but this is exactly the kind of prep you need for Digital Math track work, which will include some calculus if you keep going, also some cryptography (e.g. RSA, per the Litvins' text). Kirby
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## Re: An Algebra-then-Digital-Math track

 On Sun, Jun 29, 2014 at 11:25 AM, Gary Tupper wrote:   As for prerequisites for a math/programming course: early exposure to a 'turtle graphics' package would be well worth considering. It is visual, correlates closely to physical reality and is capable of producing dramatic images. Reinforces concepts of distance & motion including orientation. But best of all - it can provide enjoyment & the satisfaction of succeeding in achieving a goal. And most programming languages, including Python, have a turtle module. Gary Tupper, Terrace BCYes, the Turtle Art movement, spearheaded by Ed Cherlin, OLPC spin-off SugarLabs, our own Gregor Lingl, Vienna (maintains Pythons turtle module) -- such a rich subculture. http://wiki.sugarlabs.org/go/Activities/Turtle_ArtIn my own coursework I'm sort of playing off Turtle Art with a branch / fork I call Tractor Art, where my turtle is imagined to be a tractor in a field, with added methods like plow and plant.  I've got this exotic blend spelled out in more detail in Pycon circles (some Python conference goers already know of my tractor obsession). :-Dhttp://www.slideshare.net/kirbyu/pycon-2012-proposed-lightning-talk (it says "proposed" but I've given this talk quite a few times already, including at Portland State University, Systems Science Program, by invitation).I'm not suggesting a "new standard" based on what I happen to do for Saturday Academy or O'Reilly School or whatever.  I'm a drop in the bucket.  Just saying, there's room for mentors / teachers / coaches / scout leaders to get creative and build on an already tremendous heritage of free (and not so free) software. Kirby
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