Math 1 Foundations of Lower Division Mathematics (Fall 2026)
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, 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!
Professor Paulin
No one can properly learn mathematics by passively watching lectures or videos. To get hands on experience with the material, you'll attend three hours per week of active learning lab. In these session we will bounce back and forth between watching short expository videos and working on carefully selected problems (in groups or solo). Attendance at the lab is mandatory (and will count towards your final grade). The lab will take place at the following times.
Math 1 Lab: 3.30pm-5pm TuTh, Wheeler 212
The first lab in on Thursday of Week 1
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 lab 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 nervoius 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 asnswering 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. You'll often be working in a small group where you'll focus on problems more collaboratively. Be an active participant in these. Even if you're not completely confident in your approach, still share it.
In addition to lab you will have access to weekly office hours held be 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.
Every week you'll submit a homework assignments, each covering topics from the previous week. These will be due on Monday 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.
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.
There will be two (1-hour) midterm exams and one (2-hour) final exam.
As per university policy, if you do not sit the final exam, you automatically fail the course.
Here is the (vague) schedule:
Midterm: Week 9
Final Exam: TBD
Calculators are NOT be allowed for the exams. That's the policy for all university math classes. That may seem scary, but it means the exams are designed not to require challenging mental arithmetic. That's a good thing!
Cheatsheets are not be allowed for the exams. Making one yourself is a very good way of preparing, but it can't be taken into the test.
To obtain full credit for an exam question, you must obtain the
correct answer and give a correct and readable
derivation or justification of the answer. Unjustified correct
answers will be regarded very suspiciously and will receive little or
no credit. The graders are looking for demonstration that you
understand the material. To maximize credit, cross out incorrect
work. We will be scanning all exams so you will get them back electronically.
After the midterm, there will be a brief window when you can request a regrade.
If you are unsure about making a regrade request consult myself or a UGSI beforehand. Regrade requests may result in a lowering of your grade.
As per university policy, final exams cannot be regraded.
All special exam accomodations for DSP students will be arranged by us and not the DSP proctoring service. If you are a DSP student with exam accomodations we will contact you directly the week before an exam.
It goes without saying that cheating is unacceptable. Any student caught cheating will be
reported to higher authorities for disciplinary action.
There will be no make-up exams, unless there are truly exceptional circumstances.
Here is my basic advice on how to do well in exams.
Take a moment to look through the whole exam at the start. We can all agree that exams are no fun. I have always been very anxious taking exams and something that always helped was looking through the exam for a few minutes at the start. Find a question you know how to approach and tackle that first. It'll help settle your nerves.
Success is in the details. It's not enough that you have a broad superficial understanding of a topic. You must appreciate the subtleties and how they turn up in problems.
Diagnose and solve systematically. Once you've identified what type of problem you are dealing with, think about what's in your toolkit. How does one approach such a problem in a methodical way? You want to be familiar and comfortable with the standard procedures to deal with different problems. This is a skill for life and is the essence of analytic problem solving.
At the end of each week each of the tabs below will go live, with links to the notes, videos, and homework.
This is a warm up 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.
Week 2 (8/31–9/4)
Week 3 (9/7–9/11) Labor Day 9/7
Week 4 (9/14–9/18)
Week 5 (9/21–9/25)
Week 6 (9/28–10/2)
Week 7 (10/5–10/9)
Week 8 (10/12–10/16)
Week 9 (10/19–10/23)
Week 10 (10/26–10/30)
Week 11 (11/2–11/6)
Week 12 (11/9–11/13) Veterans Day 11/11
Week 13 (11/16–11/20)
Week 14 (11/23–11/27) Thanksgiving 11/25–11/27
Week 15 (11/30–12/4)
Grades are calculated as follows.
Category
Percentage of Grade
Notes
Lab Attendance and Participation
30%
Three labs can be missed without penalty.
Homework Assignments
10%
The lowest 2 scores will be dropped.
Midterm
20%
Covers material on first half of the class
Final Exam
40%
Covers the whole course. If the final score is better than the midterm it will repalce it.
Your final letter grade will ultimately be decided by your ability to demonstrate a crisp understanding of the material and the ability to apply it to a diverse set of problems.
Broadly speaking I will be looking for the following criteria for each letter grade:
A-/A/A+: A clear demonstration that the central concepts have been fully understood;
Computational techniques (and their many subtleties) have been mastered and can be applied accurately to a diverse problem set;
A strong understanding of how the abstract concepts can be applied to many real world applications.
B-/B/B+: Demonstration that the central concepts have been reasonably understood, but perhaps with minor misunderstandings;
Core computational techniques have been reaonably understood (but generally not key subtleties) and can be applied fairly accurately to a fairly large problem set;
Reasonable understanding of how the abstract concepts can be applied to some real world applications.
C-/C/C+: Demonstration that the central concepts have been vaguely understood, but with major misunderstandings; Core computational techniques have been poorly understood and can be applied accurately only in the most standard examples; Weak understanding of how the abstract concepts can be applied to even basic real world applications.
This is not high school. In general, letter grades will no longer directly correspond to the familiar percentages (70-80 for a C, 80-90 for a B, 90-100 for an A). University exams are more challenging than highschool exams and it would not be fair to impose a system this rigid. The eventual grade bins will be decided once the course is complete, although I'll let you know roughly how you are doing after each midterm. To be as fair as possible, I will also take into account the historic average of the class. This means that if I set an exam which is very difficult it will be taken into account in the final letter grades.
I know that grades are important, and how tempting it is to fixate on getting specific percentage scores. However, do remember that every moment spent trying to micromanage your grade is time spent away from actually mastering the material. The most effective way to do well is to focus all your attention on the mathematics.
Please note that incomplete grades, according to university policy, can be given
only if unanticipated events beyond your control (e.g. a
medical emergency) make it impossible for you to complete the course,
and if you are otherwise passing (with a C- or above).
I have no control over enrollment issues. For question about enrollment contact Marsha Snow, or email enrollment@math.berkeley.edu.
I know almost all of you have used AI to help with academics at some point. I'm not going to tell you that you shouldn't use AI, indeed, it can really enhance the learning process. However, there are good and bad ways to use it. You don't need to take my word for it. Here's what Gemini correctly has to say about the matter:
The High Cost of the Shortcut
Using AI to bypass the "struggle" of math homework might feel like a life hack, but from a cognitive and academic perspective, it is a recipe for long-term failure.
Math is less about "finding the answer" and more about building the mental machinery to solve problems you haven't seen before.
Why relying on AI can be disastrous...
1. The "Cognitive Atrophy" Effect
Learning math requires a specific type of mental effort called germane cognitive load. This is the energy your brain uses to build "schemas", permanent mental maps of how numbers and logic work.
Without AI: You struggle, your brain creates new neural pathways, and the concept moves into long-term memory.
With AI: You skip the struggle. Research shows this leads to "cognitive atrophy," where your critical thinking skills shrink because those "muscles" aren't being used.
2. The Illusion of Competence
When you see an AI generate a solution, you experience passive comprehension. However, there is a massive gap between recognizing a solution and generating one. (The same goes for looking at practice exam solutions before spending serious time working on the questions.)
The Data: A 2024 study found that students using AI to complete math practice scored 48% higher on homework but 17% lower on actual exams where the AI was unavailable (and remember, in this course homework is work only 10% of your grade, while exams are 80%)
Immedaitely feeding your homework and project problems directly into AI is a guaranteed recipe for disaster in the long term.
3. Missing the "Hidden" Skills
Math homework is a training ground for:
Pattern Recognition: Relating new problems to old ones.
Resilience: Sitting with a problem without giving up.
Precision: Noticing small details like minus signs.
By offloading this to an AI, you do not develop these critical skills.
4. The Risk of "Hallucinations"
AI models are "stochastic parrots", they predict the next likely word but don't "calculate" like a calculator. They can confidently provide incorrect proofs, and without your own knowledge, you won't realize the error until it's graded.
How to Use AI productively...
Shift from using AI as a ghostwriter to using it as a tutor.
Feature
Disastrous Use (The Crutch)
Productive Use (The Coach)
Goal
Get the answer to turn in.
Understand the logic to learn.
Prompt
"Solve this problem for me."
"Explain the first step of this problem."
Verification
Copy the output exactly.
Try to solve it first, then compare.
Result
High homework grade, failed exam.
Solid understanding, long-term skill.
To turn AI from a cheating tool into a high-level tutor, you must use Scaffolded Prompting. This forces the AI to guide your thinking rather than doing the heavy lifting for you.
1. The Master Template
Copy and paste this prompt to set the "rules of engagement" with the AI:
"I am working on [Topic]. I want you to act as a supportive math tutor. Do not give me the final answer. Instead, look at the problem I provide and ask me what I think the first step should be. If I am stuck, give me a small hint or a similar example to help me figure it out myself. Here is the problem: [Insert Problem]"
2. Calculus Case Study: The Power Rule
In a First-Year Calculus course, the Power Rule is the foundation for derivatives. Here is how a productive interaction looks:
Problem: Find the derivative of f(x) = 3x4 - 5x + 2
User: How do I find the derivative of this function?
AI Tutor: I'd be happy to help. The Power Rule states that for any term xn, the derivative is nxn-1. Looking at the first term, 3x4, what do you think happens to that exponent of 4 when we apply the rule?
Note: The AI identifies the rule but leaves the calculation to the student.
User: Does the 4 move to the front and multiply by the 3 to get 12?
AI Tutor: Exactly! Now that we have 12 as the coefficient, what happens to the power of x? Does it stay as a 4, or does it change?
Note: The AI confirms the correct logic and prompts the student for the next specific step.
User: It becomes a 3. So the first part is 12x3.
AI Tutor: Perfect. Now let's look at the second term: -5x. Keep in mind that x is the same as x1. What is the derivative of a linear term like that?
3. Why This Works
Active Recall: You are forced to remember rules (like x having an invisible exponent of 1).
Procedural Memory: By typing the steps, you "code" the logic into your brain.
Confidence Building: You finish the problem knowing why the answer is correct, making you ready for the exam.
Long Story Short: Do not use AI to write your homework and project solutions. You'll get crushed when it comes to exams, which is where your grade is really determined. Not to mention the fact that you'll be underwater in later classes. If you do use it, let it act as a tutor, as outlined above.
