Math 109 – Jeffrey Rabin
Helpful Advice: Math 109 is intended to help you make the transition from the lower-division
calculus sequence to upper-division mathematics courses. The emphasis in calculus is on learning to
do concrete calculations and compute numerical answers correctly. In upper division mathematics
courses there is a dramatic shift of emphasis to constructing and understanding the proofs of
mathematical facts (theorems). The focus is on the precise meanings of definitions, how the truth of
a theorem is established, how an entire structure of theorems is built up by logical reasoning from a
set of axioms, and how the language and notation of mathematics are used to express such reasoning.
Problems are solved by logical reasoning rather than rote calculation, and their solutions (proofs)
take the form of a short essay explaining the reasoning rather than a string of equations. How much
detail to include in a proof depends on who the intended reader is; mathematically sophisticated
readers can fill in missing details themselves. You should write your proofs as if the reader were one
of your classmates, or yourself months from now when you’ve forgotten how this particular proof
went: the proof should be clear and convincing to such a reader.
Learning to read proofs is as much a part of this course as learning to write them. Read both
textbooks! Expect this to take time: the meaning of each sentence must be digested as well as the
logical steps that lead from each sentence to the next. Stop frequently to try to summarize in your
own words the main points of what you have read. Be sure you understand the reasoning used in the
examples and theorems in the book. Try to make up additional examples yourself. Try alternative
ways of working the examples and see which ones work and which do not.
A useful technique when reading or writing a proof is to organize it in a “modified two-column
format”, a distant relative of the two-column proofs you may have seen in high-school geometry.
Column one contains the individual steps of the proof itself, and column two contains questions you
may have about them: how is this step justified, why does it occur now and not later, what does this
symbol mean, how did they come up with this idea, wouldn’t it be clearer another way? On your next
pass through the proof you should try to answer some of these questions.
Your participation in class is vital. A proof is not a proof until it is convincing to the reader, so
question anything that is not clear. I need your feedback to tell me what concepts I need to spend
more time on. Other students need your feedback to sharpen their skills at writing and speaking
mathematics. Math is supposed to make sense, so don’t be satisfied until it does!