Math and Algorithms in the Information Age

Formatted with Python docutils and S5 [7]

My Background

• Math Ph.D. from Stanford
• 20 year teaching in the Math and Computer Science Department at Loyola teaching both subjects, including high school level math a number of times.
• Last 5 years in separated Computer Science Department
• My Hands-on Python Tutorial [8] is used in all sections of our Introduction to Computing - not math centered
• Presenting my Tutorial again at PyCon [5] in Chicago, Mar 25

Convergence of Factors

• Trends in hardware size and price
• Social/political changes in education
• Software options in school and beyond

Hardware Context

Newly developed sub-notebook computers, are under $200 and going lower:

• For a review of low-cost options from December: http://www.vitalwaveconsulting.com/publications/pdf/Landscape_Analysis_of_Low-cost_Computing_Devices_Dec08.pdf
• XO machine from http://laptop.org

One-to-One Computing in Education

• The curriculum in De La Salle High School in Chicago is strongly tied to the tablet computers each student has. http://www.dls.org
• Governor Quinn is a strong proponent, see http://www.standingupforillinois.org/connect/.
• Example of OLPC XO computers in US classroom use: http://www.olpcnews.com/countries/usa/conclusions_of_olpc_pilot_in_n.html

Software Beyond School

• Particularly in science & engineering numerical and symbolic manipulations are done routinely by computer:
• Computer allows modeling of large real-world problems that would be difficult or impossible by hand.

Software Options in School Now

• Numerical and graphical calculator applications (CALC) - useful, not controversial. There are even more choices on a real computer; not my focus.
• Computer Algebra Systems (CAS): computer symbolic manipulation:

Problem Application

• CALC: Can use big numbers, but end up using algebra of limited complexity, or it is very boring to grind through.
• CAS: Since solution is easy once a problem is set up, there is an emphasis on modeling, translation from real world - important

Symbol Sense

Symbol sense is the ability to manipulate symbols and more abstract quantities than just literal numbers.

• This is important for students.

• Hand symbolic calculations like simplifying a complex linear expression to solve a linear equation provide some symbol sense.

• A black-box CAS loses out here - jumping to results gives very limited symbol sense.

  There is a definite corrolation between symbol sense in programming and in math. It also goes the other way: those coming into programming with good symbol sense from math are most likely to have an easy time in programming.

Bridging Ideas

Tie the importance of algorithms in math with some of the approach to algorithms in Computer Science.

Algorithms in Math

• A large part of math is algorithms: numerical algorithms in primary school, and more symbolic algorithms in secondary school.
• Both are traditionally practiced by rote repetition
• Where technology is used (graphing...), student as consumer of computer packages/devices

Bring in Computer Science

Algorithm:

Clear, step by step instructions to complete
a task in a finite amount of time

• Computer Science is all about algorithms.
• An important part is have programming languages to do the lengthy repeated steps.
• Historically there has been a steep learning curve for languages

Present Situation

• Good special-purpose beginner languages for fun, graphics, and an introduction to algorithms: Squeak [1], Scratch [2], Alice [3], TurtleArt [4]

• Accessible, free, general purpose, powerful language: Python [6].

  Python is sometimes referred to as algorithms or pseudocode with colons. it has excellent features for beginners, for high-level math constrcts, and also for professional programmers (1/3 of code at Google in Python).

All Together with Python

• The issue with CAS was the black box - gave little symbol sense

• Stating the algorithms you are using in Python is quite easy, and also is highly symbolic - to succeed you need to develop good symbol sense!

• Essentially discuss and write your own mini CAS - and then you intellectually own it.

  Expressing in Python is also a modeling/translation exercise - an important skill.

• You are totally free to take advantage of the fruits of your labor and understanding to solve large sized problems instantly.

  You can also elaborate the code to solve a wider set of problem types.

Math, Computer Science, Other Sciences

• This approach is centered on math, not making students into software engineers or computer scientists.

• Obviously the exposure to easy algorithms in a neat language helps give exposure to some computer science ideas.

  With computing so central to our lives this is reasonable, and not otherwise done in a No Child Left Behind, 3 R's only world. I will resist the temptation to get on my soapbox about all the ways average people see computers as infallible black boxes - really bad if you want to get into a discussion of electronic voting, security, phishing, medical robots, robot armaments...

• In particular, greater ease with calculation algorithms is going to be helpful in science classes.

• Further integration, math + computer algorithms + science would be a next step.

Get to Python!

• Idle, shell
• Shell: arithmetic, variables
• Shell Sample [9]: algorithm/function definition
• Shell: math features: big numbers, modulus, gcd, gcd2
• ? Canned Shell: intro to object-oriented syntax and programming a simple first object class

Real Math

• Then the first real mathematical example: Create a new kind of object for rational numbers:
• Rational: attributes numerator, denominator (abbreviate num, denom)
• If we call a particular fraction frac2, we can refer to frac2's numerator. In Python: frac2.num
• operations with fractions defined in Rational class
• When writing code for fractions, generally there is the current fraction, always explict, called self.
• idea for math development is very incremental! compressed here!

Rational

• Start with constructor, one thing to do: print
• Test
• How define infix operation * ? Use special method name __mul__
• Operate on current fraction (self) and another parameter (say frac2). Operation will return the new resulting fraction.
• Add pieces to later versions, decisions, type conversion, place in lowest terms....

Questions?

aharrin@luc.edu

This slideshow at http://cs.luc.edu/~anh/HS/pyalgebra.html

All resources in http://cs.luc.edu/~anh/HS/pyalgebra.zip