Capstone course in Algorithmic Geometry

[pbierre]

I’m posting this to bring attention to my team’s development of a capstone course where advanced geometric problem-solving and 21st century spatial applications are explored.    The opportunity is for upper-level math students (beyond Pre-Calc) to learn geometry problem-solving the way it has evolved in our tech industries, as a seamless integration of vector math and computer science.

The core of Algorithmic Geometry is to explore the choices for numerical representations that are well suited to developing spatial software apps.   The entranceway is implementing Cartesian points [ x y ] as 2D point vectors p =  [ x  y ], and developing a 2D distance method (currently using Java) which processes points p1 , p2 as the input.   There is no prior programming experience required….we teach the Java programming fundamentals as part of this Math course.

By the end of a 1-year course, students have learned how to solve “wicked difficult” math problems, not by being told the answer, but by exploratory invention of their own solutions (usually in creative sketching mode on paper).   Note that we study problems that generally require software computation to tame the complexity.  Examples:

• intersection of 3D line and plane –> point
• intersection of 2 spheres –> 3D circle
• intersection of sphere and 3D circle –> 2 points
• closest approach 3D LineSeg of two skew lines
• distance trilateration (GPS positioning – where am I based on sensing distances to 3 satellites?)
• directional positioning (where am I in 3D based on sensing direction to 3 known points?)

We use a predominantly Project-Based-Learning (PBL) pedagogy.

For a more thorough treatment of the core representations that signify the shift to doing geometry in partnership with software writing, see my NCSSSMST Nashville talk:

http://www.algogeom.org/files/AlgorithmicGeometry.mov

For 2-Year Progress Report (including inventive-problem-solving assessment results), see:

Click to access TwoYearReport.pdf

Dr. McCallum, I’m concerned that the CCSSM does not seem to embrace software programming as a toolset students should be comfortable using if they aspire to becoming powerful applied math practitioners.   The Computer Science classes offered in high school barely scratch the surface of real-number computing, and it benefits greatly to be taught algorithmic geometry by Math teachers who understand vector math.   We are still early in the process, and have too few Math teachers trained (2) to say we have a PD model.

What we have is a solid 1-year course curriculum, and achievement results with the first 28 students.   The payoff is the complexity of problems able to be solved, and the enhanced relevance of being able to undertake projects in computer vision, 3D wireframe graphics synthesis, molecular modeling, GPS navigation, mobile robot platform, camera-based navigation, and robot arm motor coordination.

The heightened relevance is an outgrowth of the fact that the spatial applications that are reshaping our world are the result of interdisciplinary Math-CS, and more specifically, implementation of vector math in software, with immersion in computer graphics.

We see the adjustments (shifts) in core representations, and the informatic thinking behind them, as highly desirable to be able to claim that the CCSSM is preparing young people to inherit the software world they inhabit.   Let us know how we can help.

[Bill McCallum]

Hi, thanks for all this information. CCSS does not have any programming standards per se, because our job was to describe the mathematics students should learn. Programming in a mathematics curriculum is a tool, not a topic … it looks like you have thought a lot about how to use it as a tool to learn mathematics.

[Pierre Bierre]

Thanks, Bill.    The question of bringing high-level programming languages into Advanced Math as a toolset poses logistical challenges, granted.   There is a substantial payoff if we can solve this, in that students are able to craft their own numerical representations, and thus operate at a creative level traditionally monopolized by professional mathematicians.  (E.g., on the AlgoGeom Final Exam, students are asked to design a parametric representation for a generalized Torus, the topic never having been broached in class — 79% were successful).

The long-term implication for CCSSM is the potential for greater cohesion in introducing students to vector math.   Geometry is a natural, fertile platform for initial work with vectors.   As soon as students are comfortable with Cartesian coordinates [ x y ], they are ready to extrapolate to 3D points [ x y z ].   The Greek distance formulas scale up naturally and intuitively.    The problems discussed elsewhere in this blog (midpoint of Line Segment, and proportional splitting of a Line Segment) introduce the repeating pattern (across x  y and z ) of arithmetic operations that begs for a streamlined formalism and notation.

The next question then becomes, “how can we represent spatial directions in a way that scales naturally from 2D –> 3D?   This is where direction vectors enter the scene.    Once this representational innovation is absorbed, students have a potent foundation that is straightforward in 3D.   This foundation is the launching pad for advanced problem-solving.

The major adjustment to CCSSM content is to soften up allegiance to slope and angle enough to allow a more up-to-date alternative (coming from the scientific software world) for spatial direction, and strongly directional features (such as the tilt of a 3D circle).

This modernization will happen naturally so long as educators are encouraged to pose deep mathematical questions, such as, what is a good way to represent direction in 2D space that scales up to 3D as intuitively as do Cartesian points and Euclidean distances?

Software programming is not necessary to these concepts, although sketching and computer graphics are the keys to creative immersion.

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