| Category | Difficulty |
|---|---|
| HW | 6 |
| Exams | 5 |
Required introduction math course for ECE/CS and math majors. This course is intended to teach students how to write proofs, which will be useful in theory classes, and helps build a more rigorous math foundation in general. Topics that are covered include but are not limited to number theory, infinite and finite sets, induction, relations/functions, and combinatorics.
21-128/15-151 are technically the honors version of 21-127, but besides extra credit questions and giving the option to use LaTeX, all the classes are essentially the same. You can argue that offering extra credit makes the honors classes easier to get a higher grade.
Even though concepts is supposed to be an introductory math class, the class is known to be notoriously difficult for people who have never seen proofs before. While it is true that proofs are hard to pick up from the start, I think another reason why this course is hard for students is that they don't have certain fundamental math skills to pick up lectures quickly in concepts.
For example, every student who enters concepts should be able to do all of the worksheets in the following packets easily, otherwise the student is missing concepts in Number Theory, Counting and Probability:
- http://agmath.com/media//DIR_57727/1$20Counting.pdf
- http://agmath.com/media//DIR_57727/Probability.pdf
- http://agmath.com/media//DIR_57727/NumberTheory.pdf
In general, students who have participated in competition math like AMC/AIME/USAMO find this class much easier than students who don't. If you want to be more prepared for concepts, it helps to do proof math and competition math beforehand.
Concepts is one of the first classes at CMU where you realize getting the right answer doesn't mean you get 100%...with proofs, even if you feel like you "get it," properly communicating that you understand how to prove the problem can be much more difficult. Proofs will feel unnecessary harshly graded at first, but the purpose of the course is to teach you how to be rigorous in your proofs. Every time you lose points on a proof, try to understand why your explanation is flawed. A good way to start to write proofs is to mimic what you do in class. Lectures provide all the knowledge necessary to do well on homework assignments so consistent attendance is important.
Concepts exams contain questions that are easier versions of the HW, but might be content that you haven't necessarily seen before, so you can't just regurgitate proofs from the HW. The difficulty for exams usually comes from that they have to be completed in short time periods, usually the 50-minute class block you have, which means you have around 10-minutes per problem -- which can be quite little to do a proof. This means if you get stuck on an exam, you're pretty much toast. To best prepare for exams, you should know how to identify what to do for problems very quickly, meaning you need a plan of attack for every problem. If you can figure out how to do a problem, then you'll be fine, but if you are stuck, you should certainly have a backup plan to try to earn some partial credit. If you are stuck on a proof, demonstrating that you understand the course content will often give significant partial credit. For example, providing a base case on an inductive proof will certaintly lead to partial credit even if the problem is not completed. Many professors also offer extra credit problems on exams.
- If you notice you're missing a lot of points on your homeworks for poor proof techniques, you should definitely consult a TA and ask for advice on how to make your proofs better. Going to office hours for homework and exams is always helpful but they tend to get very crowded before exam and homework due dates.
- Learning LaTeX and writing your homeworks in it can benefit you substantially later on in your CMU career. While it certainly has a high learning curve to start off, it will pay dividends. It also helps you write better proofs since it's a lot easier to revise your work, and you'll eventually learn to format your work to communicate your ideas better. Some professors will give extra credit on homework if your answers are formatted in LaTeX.
- Practicing competition math problems can help you with your speed and problem solving skills. There are also many resources online from other universities with similar proof-based courses that can provide practice problems for the topics covered in class.
The course notes for the class are written by a past TA. You can find them at https://infinitedescent.xyz/dl/infdesc.pdf. The notes are quite dense, so I think they are good for skimming before you take the class and also to help you organize your thoughts after you learn the topics.