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Sean Gonzales |
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Math 1 Foundations of Lower Division Mathematics
(Fall 2024)
Welcome to Math 1: Foundations of Lower Division Mathematics!
The transition from high school to university-level mathematics is challenging. Of course the material becomes more complex, but there is more it it than that. The whole style of learning mathematics is different at university, and takes time to adjust to. So what are the major differences?
- More Conceptual. The material will emphasize conceptual understanding rather than rote computation. While each course will build a toolkit of essential computational techniques, why and how they work is what really matters. You'll need to understand why specific methods are effective under certain conditions and how to adapt them when assumptions change. Some concepts may require weeks of concentrated thought to fully internalize.
- Much Faster Pace. Nearly every lecture introduces a new concept or technique, assuming a solid grasp of previously covered foundational material. If you fall seriously behind it can be really difficult catching up.
- More In-Depth Computations. Examples and problems often involve multiple parts that combine various techniques. Developing a strong intuition is crucial for progressing from point A to point B through a sequence of logical steps.
- Serious Applications. Real-world problems frequently present complexities that necessitate sophisticated mathematical techniques for accurate modeling. Lower-division math courses are structured to swiftly bring students up to speed, balancing thoroughness with practicality.
The transition from high school to this next level clearly involves significant adjustments. Drawing from my experience teaching lower-division mathematics, Prof. Alex Paulin and I have designed Math 1 to help you navigate this challenging shift. Even very foundational topics, which you may already know well, will be approached from a university-level perspective with a strong emphasis on conceptual understanding and systematic problem-solving. The course follows the same style as other lower-division courses and comprehensively covers all essential topics necessary for success in future classes. Click on the tabs below to see the plan.
I hope you enjoy the class!
Sean Gonzales
If you need to contact me or your UGSI about an urgent matter, please use the following guidelines. Our contact information can be found at this link.
- Lecture absence: you do not need to email any of us. Lecture attendance is not recorded.
- Discussion absence: contact your UGSI.
- Homework extension: contact your UGSI.
- Personal emergency (*): contact me and your UGSI.
(*) A personal emergency typically means a family or medical emergency that will impact your performance in the course.
Going into lower division mathematics classes, a solid grasp of all key precalculus topics will be assumed. In this course we will cover all of that and more. In particular we will also explore extra topics like limits, approximations, and error bounds to name a few. These are classic points of confusion and we will look at them in great detail, fully preparing you for how they will appear in future courses. Every topic in the course will be covered in comprehensive prerecorded video lectures each between 1 and 1.5 hours. Here are some tips about how to get the most out of them.
- Pay attention to the details. These lectures will likely be in more depth than what you are used to. Simply grasping the basic idea of a concept won't suffice. Your goal should be to comprehend each concept in granular detail. Every point covered in the videos is crucial to the topic, so if something isn't clear, don't ignore it. Pause the video, reflect on it, and if needed, jot down your questions to discuss them during office hours or in the discussion section later on. Understanding these nuances is key to mastering the material.
- Be very cautious of other math resources online. This course is structured to be as self-contained as possible. Each exercise has been written to utilize only the concepts taught in the video lectures. If you find yourself struggling with a problem, revisit the corresponding video lecture rather than turning to random online sources. Such sources almost always lack sufficient detail and can lead to a false sense of understanding. Additionally, avoid relying on tools like ChatGPT. They won't contribute to your learning, especially since AI is still terrible at mathematics.
- Make sure you understand the examples. The exercises in the worksheets and homework are directly related to examples in the lecture. Realy make sure you really understand the approach that is taken to solve problems in the videos.
- Don't skip any videos. Even if the topic is familiar to you, there will be subtleties that you have likely not seen before. I recommend watching them the evening before the corresponding in-class lecture.
- Treat it like a proper in-person lecture. Have my notes close at hand. Even better, take your own notes in real time. It'll be excellent preparation for future courses. Don't watch it at twice speed only half paying attention.
- Come back to them later. Video lectures offer great flexibility for learning. Being able to revisit them allows you to reinforce your understanding of complex topics or clarify any confusion you may have. This ability to review at your own pace can significantly enhance comprehension and retention of the material. Make use of it.
In addition to the comprehensive video lectures, you will attend in-person class twice a week for 1.5 hours each. I will give an overview of the main points of the current topic, and go over more examples that weren't covered in the video lecture. These "lectures" will serve as a supplement to the video lecture, while also providing you an opportunity to ask questions and clear up any confusion that you may have encountered while watching the video lecture.
- Watch the video lecture beforehand. The in-person lectures will be less comprehensive than the corresponding video lectures. I strongly recommend watching the video lecture the evening before class.
- Come prepared with questions. While you take notes during the video lecture, make sure to write down any questions you have about the material. During the in-person class, I will be happy to take questions and go over examples that will help to answer your questions.
- Don't skip the in-person class. The major downside to video lectures is that you cannot ask clarifying questions to the lecturer. By interacting with me directly in class, you will be able to gain a full understanding of the material.
To get hands on experience with the material, you'll also attend in-person discussion sections lead by experienced undergradatuate student instructors (UGSIs). These will last an hour and you'll be working through carefully designed worksheets. Your first discussion section will be on Wednesday 10/23.
These sessions provide an excellent opportunity to explore the material as a smaller group, with each section consisting of around 20-30 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. Here are some tips to get the most out of these sessions.
- Mastering Mathematics is a challenge and most challenges are best approached as a group. Regard your discussion section not only as an opportunity to understand the course material better but also as a place to connect with classmates for collaborative work outside of class. Having other people to bounce ideas off is incredibly valuable. The more perspectives on a difficult problem the better.
- Don't feel apprehensive about participating. Many of of us get nervous speaking in public, doubly so when it comes to mathematics. You may think that only you are struggling with a concept, but that is never ever true. If something's confusing, ask questions. If the instructor asks a question, try answering it. It can be scary at first but over time it won't be and the payoff will be enormous in the long run
- Be active in group work. 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 following link. Your UGSI will contact you with details before the start of class.
You can only attend the section you are officially enrolled in. If you have enrollment questions contact: enrollment@math.berkeley.edu
Here is information about all UGSIs.
You'll be following fixed worksheets in your discussions. After you've had section, they'll be posted here along with solutions. Come to office hours if you want to talk about the worksheet problems after discussion section.
In addition to lectures and discussion section, you will have access to office hours. Office hours are your 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 Fridays 12:30pm-1:30pm.
Here is office hours information for all UGSIs. You can attend any UGSI's office hours.
There will be one homework assignment per week, each covering topics from that week. These will be due on the following Monday at 11:59pm on Gradescope (an online platform to submit assignments). You can access Gradescope from this link. Here are instructions for how to upload your work. If you have issues submitting your work, contact your UGSI.
You may discuss the homework problems with your classmates, but you must write your solutions on your own. Make use of discussion sections and office hours 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 mathematics classes 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 may struggle to complete the homework assignment if you start on the night before it is due. Work consistently in small installments.
- 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. Make sure you attempt them. Almost every problem will be very similar to one done in the video lectures or the workseets. Consult those if you're unsure of where to start.
- Seek help, but only after seriously thinking about a problem on your own. If you're struggling with a challenging problem you should spend at least 30 minutes on your own thinking about it carefully. Even if you fail to make a breakthrough, this is still more worthwhile than giving up after a couple of minutes and talking to peer, posting on the online discussion forum, or looking up a random source on the internet. Knowing what doesn't work is just as important knowing what does. The questions will be very closely related to the video lectures and worksheets. Those should be where you look for help first. In future courses you'll be solving problems that require days (or weeks) of dedicated thought. Now is the time to hone this skill and get comfortable with not being able to immediately solve a problem.
Here is a link to the homework.
Homework Solutions will be made available shortly after submission at the above link.
At the end of the course you'll be given a take-home final assessment. This will be due on Friday, December 13th at 11:59pm on Gradescope (an online platform to submit assignments). You can access Gradescope from this link. It'll really be like another homework assigment, covering the whole course, but with more of a focus on the last third.
The final assessment will be posted in the same folder as the homework. Here's the link.
Below is the weekly schedule for the course. If you click on any of the topic tabs you'll find videos and lecture notes, which will be added as the course progresses. We are covering fourteen different topics and you should always look at the appropriate video lecture before the class on it the next day. Here's my basic advice for how to plan your weeks.
- Discussion Section. Attendance is part of your grade so you need be there. You can miss two without penalty. Each discussion worksheet is divided into three parts. Some are longer than others, so don't worry if you only managed to cover part A in discussion. That's totally fine. You can look at the more challenging questions later and discuss them in office hours. The homework will be very like the worksheet problems.
- Go to office hours. This will be a good time to talk about worksheet problems you couldn't cover in the morning, points of confusion in the video lectures, or homework.
- Watch the video lecture in the afternoon or evening. The video lectures are very detailed. Set aside a couple of hours to watch each one. They will require your full attention. Pause if you need time to absorb something. Follow along with the notes provided (or write your own!) and mark points of confusion. Raise them in class or office hours.
| When | What |
|---|---|
| Before we start! |
This is a warm up video to make sure you are comfortable with three important pieces of notation: the equals symbol, the implies symbol, and the if and only if symbol. Using these correctly is critical for this and all future mathematics classes. If you have time, this is something you should talk about in your first discussion session. |
| Week 9 (10/21-10/25) | |
| Week 10 (10/28-11/1) | |
| Week 11 (11/4-11/8) | |
| Week 12 (11/11-11/15) | |
| Week 13 (11/18-11/22) | |
| Week 14 (11/25-11/29) | |
| Week 15 (12/2-12/6) | |
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, weekly homework assignments (graded on correctness, not completion), 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. More precisely your grade will be calculated as follows:
| Homework | 40% |
| Participation in Discussion Section | 40% |
| Final (Take-Home) Assessment | 20% |
Participation is based on attendance in discussion sections. You may miss three discussion sections without penalty. A final assessment must be submitted.
An overall percentage score of 70 or above will result in a pass.