# Outcomes, Objectives, and Skills

## Outcomes (the big picture)

1. Grapple with the complications and limitations of various approaches to 'truth.'
2. Recognize and understand deductive reasoning and its role in science and mathematics.
3. Use deductive reasoning to solve problems and answer questions related to geometry and the measurement of space and time.
4. Understand the development and limitations of geometry, deductive reasoning, and our conceptions about measuring space and time.
5. Gain confidence in your own abilities to grapple with and grasp concepts in math and science.

## Objectives (the pieces)

1. Translate problems stated in technical language into natural language. Formulate problems that arise in natural language in appropriate, technical terms.
2. Recognize and analyze geometrical proofs. Invent and properly formulate proofs of basic theorems
3. Recognize and analyze logical arguments. State and use the basic laws of logic.
4. Describe the features and historical significance of the axiomatic method. Describe Gödel's work in establishing limits on the method itself.
5. Describe the features and importance of non-Euclidean geometries.
6. Explain various implications of the concept that the laws of physics are the same for all observers. Outline how the laws of physics show the speed of light to be a universal constant.
7. Distinguish between 'arbitrary' and 'relative' in reference to measurement, and suggest ways to deal with both.
8. Describe the effect of high relative velocity on the measurement of space and time.
9. Provide convincing explanations of the experimental evidence for time dilation.

## Skills (the practice)

1. Apply the Pythagorean theorem and distance formula to solve geometrical and physical problems. Analyze proofs of the Pythagorean theorem.
2. Use congruence theorems to prove theorems about parallel lines and triangles.
3. Derive or justify, as appropriate, and use formulas for the areas of triangles, rectangles, and circles.
4. Distinguish between a line and a geodesic. Identify geodesics in various geometries.
5. Recognize and analyze logical paradoxes and their relationship to general deductive systems.
6. Describe and understand the basic features of Gödel's incompleteness theorem.
7. Verify the invariance of distance when various transformations are applied in planar geometry. Discuss the analogy of the spacetime interval as a distance between two events.
8. Describe effective means for synchronizing clocks for use in measuring time and space.
9. Comfortably measure time in meters and distance in seconds.
10. Calculate the time rate of change of various quantities. Comfortably manipulate formulas which deal with time rates of change.
11. Use equations to represent waves and distinguish what part of the wave equation represents the velocity of the wave.
12. Distinguish between quantities which are additive and quantities which are not. Explain the conditions under which velocities are additive and the conditions under which they are not.
13. Use the spacetime interval to solve problems related to special relativity.