Alexander Paulin

Alexander Paulin
apaulin@berkeley.edu

Department of Mathematics
796 Evans Hall
University of California, Berkeley

me | research | teaching | CV

Math 1 Foundations of Lower Division Mathematics (Fall 2023)

Mathematics is an essential skill set that plays a critical role in achieving success across a wide range of majors. If you are starting your lower-division classes at Berkeley and find yourself facing a significant increase in rigor and abstraction, it can be quite intimidating, especially if you feel there are gaps in your foundational knowledge. Maybe you had a negative experience with math in high school but don't want to spend an entire semester on a pre-calculus class before diving into Math 1A. Or perhaps you've realized you need extra support a few weeks into the Fall semester and are unsure where to turn.

If you can relate to these concerns, then this course is specifically designed for you. Math 1, titled "Foundations of Lower-Division Mathematics," is a condensed seven-week, in-person course offered twice consecutively during the Fall semester. Focusing on fundamental concepts, this two-unit course will provide you with a solid base to excel in UC Berkeley's lower-division Math curriculum. It is intended to be taken alongside your introductory calculus classes, serving as a building block for your mathematical studies. The course features weekly in-person lectures delivered by Professor Paulin (that's me!).

However, Math 1 offers more than just covering core mathematical topics. Building a strong network of friends and peers is crucial for a successful undergraduate journey, and Math 1 recognizes this. Establishing this support system early on is vital, and a significant portion of your time will be dedicated to smaller discussion sections with ten to fifteen fellow students. These sessions are led by trained undergraduates known as Undergraduate Student Instructors (UGSIs), who were in a similar position not long ago. Attendance in these small-group discussions is mandatory, as they provide valuable learning opportunities.

In addition to lectures and discussion sections, you'll have access to various resources facilitated by expert peer tutors through the Student Learning Center (SLC). These resources include weekly topic reviews, small-group learning pods, and drop-in office hours. While attendance for these resources is not mandatory, they offer flexibility in mastering the material based on your preferred learning style.

It's important to note that Math 1 has a unique structure compared to other classes at Berkeley due to its seven-week duration. The course is divided into two consecutive sessions: the first session begins in week 2, and the second session starts in week 9. Both sessions cover the same content. This arrangement allows students who realize they need additional support later in the semester, and have missed the enrollment deadline for the first session, to still have the opportunity to benefit from Math 1.

Math 1 follows a pass/not-pass grading system, alleviating additional stress as you embark on your academic journey at Berkeley. We understand that you already have plenty to handle, and Math 1 aims to minimize any unnecessary pressure. Your final grade (P/NP) will be determined based on your active participation in discussion sections, completion of weekly homework assignments, and a final take-home assessment at the end of each session. Rest assured that the focus is on your engagement and progress rather than assigning letter grades.

Lectures are where I will introduce and explore the material. Throughout the semester, I'll do my best to make you part of the process, giving you the chance to try problems yourself and ask questions. If there's something you feel unsure of, don't feel nervous about putting your hand up. If you are thinking it, then so are lots of other people in the room. I'm here to guide you towards proper understanding, not deliver dry uninterupted monologues. The more like a back and forth conversation, the better!

The course is seven weeks, and will run twice consecutively. The LEC 001 session will run from from week 2 to week 8; the LEC 002 session will run from week 9 to week 15. For a detailed weekly schedule see below.

LEC 001 (Week 2 - Week 8) 12.30pm-2.00pm TT, 2060 VLSB

LEC 002 (Week 9 - Week 15) 12.30pm-2.00pm TT, 2060 VLSB

Attendance is not mandatory (although, I'd strongly advise that you go).

Here is a work-in-progress textbook for the course.

Here are the in-person lecture notes and handouts. This link will be updated as the course progresses

Discussion sections, led by Undergraduate Student Instructors (UGSIs), will take place every Monday, Wednesday, and Friday, lasting for approximately an hour. Your first discussion section will be on Wednesday August 30th. These sessions provide an excellent opportunity to delve into the material as a group, with each section consisting of around 10-15 students. In these interactive discussions, you will collaborate with your peers to tackle challenging problems, enabling you to truly grasp and excel in the subject matter. To maximize your learning experience during these sessions, here are some helpful tips to keep in mind:

Mastering Mathematics can be a formidable challenge. However, like many endeavors in life, transforming it into a social experience can greatly enhance its rewards and productivity. Regard your discussion section not only as an opportunity to understand the course material better but also as a potential avenue to connect with classmates for collaborative work outside of class. By fostering a social learning environment, you can make the process of learning Mathematics more enjoyable and find like-minded individuals to collaborate with, further enhancing your understanding and overall success in the subject.

There's no need to feel apprehensive about participating in group discussions by asking questions or providing answers. Remember, even if you happen to be incorrect, it's highly likely that others in the group are thinking the same thing. It's important to realize that there are often multiple approaches to problem-solving, and hearing different perspectives from your peers can significantly enrich your understanding of the subject matter. So embrace the opportunity to share your thoughts and learn from others; it will definitely deepen your mathematical comprehension.

Your UGSI will frequently split the main group into smaller groups where you'll work on problems more collaboratively. Be an active participant in these. Even if you're not completely confident in your approach, still share it.

For a discussion section schedule go to the folllowing links: LEC 001 and LEC 002. Attendance at discussion section is mandatory.

You can only attend the section you are officially enrolled in. If you have enrollment questions contact: enrollment@math.berkeley.edu

Here is contact information for all UGSIs.

You'll be following fixed worksheets in your discussions. After you've had section, they'll be posted here along with solutions.

In addition to lectures and discussion section, you will have access to daily office hours. Office hours are you're chance to talk to me or a UGSI. They are a really good way to get to know your instructors better and get help with any aspect of the course. These are also a space for you to work in small study groups at your own pace if you'd like.

In office hours you can talk about any aspect of the course (and beyond). If you've spent time on a homework problem and have stalled, come to office hours; If you're unsure about your academic trajectory, come to office hours; If you want to learn what mathematics research is about, come to office hours; If you're struggling with the course and don't know what to do, come to office hours; If you think you might need a reference from a professor in the future, come to office hours; If you just want a chat, come to office hours.

I'll be having office hours every 3pm-4pm Tuesday and Thursday in 796 Evans Hall. As a default GSI office hours are in-person, although some may be conducted online via Zoom.

Here is a link with the full office hour schedule.

Topic Reviews:
Focus on the learning process and strategies to master each concept.

Learning Pods
A guided learning expericence in small groups of two or three.

Drop-In Sessions
An open space for students to review material with tutor support.

While attendance for these resources is not mandatory, they offer flexibility in mastering the material based on your preferred learning style.

The course will cover the essential topics required to excell in UC Berkeley lower division mathematics classes. Below you'll find a basic roadmap for the course. Clicking on any tab will open a link to video lectures covering the relevant topic in depth. Because Math 1 is 7 weeks we won't be able to cover all of these topics. However, the diagram below is something you'll have access to for your whole time at Berkeley, and is there to provide targetted support when you need it.

Here is the homework.

Homework assignments are due each Friday at 11.59pm on Gradescope. Here are instructions for how to upload your work. If you have issues submitting your work, contact your UGSI. Your lowest homework score will be dropped.

You may discuss the homework problems with your classmates, but you must write your solutions on your own. Make use of discussion sections, office hours, etc. if you need assistance, but in the end, you should still write up your own solutions.

Lectures are where new concepts are introduced, but homework is where much of the real learning happens. It's where you internalize the abstract ideas and discover for yourself how they can be used to solve problems. From my experience the main distinction between those who succeed in this course and those who don't, is how they treat the homework. Here's some advice about how to approach it:

Be organized. Don't leave things for the last moment. (You'll struggle to complete the homework assignment if you start on the night before it is due.) Work consistently in small installments. Some homework can take 6 hours to complete properly.

As you progress through the homework you'll notice it increasing in difficulty. The more difficult problems are often the most important, giving you the opportunity to really master the material. If you're struggling with a challenging problem you should spend at least 30 minutes on your own, thinking about it in depth. Even if you fail to make a breakthrough, this is still more worthwhile than giving up after a couple of minutes and talking to a peer or consulting an answer scheme. In future courses you'll be solving problems that require days (or weeks) of dedicated thought. Now is the time to hone this skill.

If you've spent serious time exploring a problem and are still struggling, then is the time to seek help: Speak to your peers, come to my or a GSIs office hours, go to the SLC and talk to a tutor. The last thing you should do is look at a solution manual. Actively speaking to someone about a problem is much better than passively reading a solution.

Homework Solutions will be made available shortly after submission.

Below is the week by week schedule for the course.

When	What	Where
Week 1 (8/21 - 8/25)
Week 2 (8/28 - 9/1, LEC 001 starts)
Week 2 (8/28 - 9/1, LEC 001 starts)
Week 3 (9/4 - 9/8)
Week 3 (9/4 - 9/8)
Week 4 (9/11 - 9/15)
Week 4 (9/11 - 9/15)
Week 5 (9/18- 9/22)
Week 5 (9/18- 9/22)
Week 6 (9/25 - 9/29)
Week 6 (9/25 - 9/29)
Week 7 (10/2 - 10/6)
Week 7 (10/2 - 10/6)
Week 8 (10/9 - 10/13, LEC 001 ends)
Week 9 (10/16 - 10/20, LEC 002 starts)
Week 9 (10/16 - 10/20, LEC 002 starts)
Week 10 (10/23 - 10/27)
Week 10 (10/23 - 10/27)
Week 11 (10/30 - 11/3)
Week 11 (10/30 - 11/3)
Week 12 (11/6- 11/10)
Week 12 (11/6- 11/10)
Week 13 (11/13 - 11/17)
Week 13 (11/13 - 11/17)
Week 14 (11/20 - 11/24)
Week 15 (11/27 - 12/1, LEC 002 ends)
Week 15 (11/27 - 12/1, LEC 002 ends)

Homework 40%

Participation in Discussion Section 40%

Final (Take-Home) Assessment 20%

The lowest homework score will be dropped. Participation is based on attendance in discussion sections. You may miss two discussion sections without penalty. A final assessment must be submitted.

An overall percentage score of 70 or above will result in a pass.