# Outcomes, Objectives, and Skills

## Outcomes (the big picture)

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

## Objectives (the pieces)

- Translate problems stated in technical language into natural language. Formulate problems that arise in natural language in appropriate, technical terms.
- Recognize and analyze geometrical proofs. Invent and properly formulate proofs of basic theorems
- Recognize and analyze logical arguments. State and use the basic laws of logic.
- Describe the features and historical significance of the axiomatic method. Describe Gödel's work in establishing limits on the method itself.
- Describe the features and importance of non-Euclidean geometries.
- 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.
- Distinguish between 'arbitrary' and 'relative' in reference to measurement, and suggest ways to deal with both.
- Describe the effect of high relative velocity on the measurement of space and time.
- Provide convincing explanations of the experimental evidence for time dilation.

## Skills (the practice)

- Apply the Pythagorean theorem and distance formula to solve geometrical and physical problems. Analyze proofs of the Pythagorean theorem.
- Use congruence theorems to prove theorems about parallel lines and triangles.
- Derive or justify, as appropriate, and use formulas for the areas of triangles, rectangles, and circles.
- Distinguish between a line and a geodesic. Identify geodesics in various geometries.
- Recognize and analyze logical paradoxes and their relationship to general deductive systems.
- Describe and understand the basic features of Gödel's incompleteness theorem.
- 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.
- Describe effective means for synchronizing clocks for use in measuring time and space.
- Comfortably measure time in meters and distance in seconds.
- Calculate the time rate of change of various quantities. Comfortably manipulate formulas which deal with time rates of change.
- Use equations to represent waves and distinguish what part of the wave equation represents the velocity of the wave.
- 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.
- Use the spacetime interval to solve problems related to special relativity.