Advice for Calculus Students

Discipline

• Learn how to use your time well.
  • Be diligent. It's amazing how much you can accomplish if you just keep plugging.
  • Be strategic. "Is this an effective use of my time?"
  • Be focused. Keep asking yourself, "What is going on in my mind? Am I learning?"
  • Don't get diffused with too many activities.
• Have vision.
  • Feed your motivation. Talk with your instructors. Talk with people who are where you would like to be. "What is this used for?" "What kinds of problems will it equip me to solve?" "What are the main goals of the course?"
  • Keep the purpose of the course in view.
    • Focus on learning rather than grades.
    • Don't operate on fear of small failures.
    • Lift your head long enough to see where you are going.
• Focus on the essentials.
  • Look for the main idea. Identify the common threads.
  • Focus on basic understanding and long-term retention.
  • Be willing to dig; get to the bottom of it. When you don't understand something, take the time to dig down until you get to the root of the concept you are missing, even if (especially if) it means reviewing material from a prerequisite course.

Independent Study

• New material: Learn how to read a math book.
  • Reading a math book properly requires pencil and paper. Follow along with the book. Treat example problems, theorems, and formulas as exercises with solutions. Try to work each exercise ahead of the book.
  • Know the definitions and boldface terms. Use the index to find things.
  • Study the table of contents to see the main topics.
  • Be able to work each example problem. The example problems are designed to cover all the basic ideas. They are a great tool for review.
• Exam preparation: Learn how to commit mathematics to memory.
  • Recognition knowledge isn't enough. You need to be able to do the problems without looking at a book.
  • Don't blindly memorize. This results in superficial knowledge and you won't retain it. You won't see how the material fits together, you won't know when you can apply it, and it will balloon the quantity of material you have to commit to memory.
  • Study the patterns in the formulas. For example, if the result of antidifferentiating has a trig function that begins with a "c", the answer has a minus sign.
  • Organize the material. Systematizing clarifies points of confusion and gives you a clean framework of knowledge on which to build. Use each exam as an opportunity to write summaries. The essence of a math courses can usually be summarized on a single sheet.
  • Understand how to derive each formula.
    One of the best ways to test your knowledge of algebra, differentiation, and integration techniques is to work your way through a table of identities and prove each one.
  • Understand why the theorems are true.
    If you don't know why they are true, you won't know when you can apply them.

Learning Community

• Seek an academically supportive social environment.
  • Befriend people you can learn from. Those who walk with the wise will be wise.
  • Have worthwhile conversations with people about things that matter.
  • Don't waste time in mindless talk and activity.
• Help one another out.
  • Build community: try to give a little more than you take. When enough people do this, it transforms the social dynamic.
  • Don't be a slave to note-taking. Have a couple note-buddies. Form a note-taking pool. One person can take notes while the others listen.
• Help other people.
  • Answering questions is one of best ways to solidify your own understanding.
  • Don't do others' work for them. Encourage people to do their own thinking.
• Ask others for help.
  • The purpose of getting help is to become able to do the problems without help.
  • Take a few minutes to try to answer your own question before asking someone.
  • Don't be afraid to ask for help when you are stuck.
  • Don't hesitate to speak up when you are lost. Humility is the path to knowledge.

Resources

• Help your teaching assistant help you.
  • Highlight what you don't understand; don't hide it. If your teaching assistant checks work, write questions on your work.
  • Come to office hour with pointed questions.
• Take advantage of the math lab.
  • Work in an environment where you can ask questions when you get stuck.
  • Strive for independent understanding.
• Don't become dependent on the solutions manual.

Lecture

• Know what the lecturer is going to talk about. Pre-read the sections. Identify what questions each section is trying to answer. If you learn the definitions of the bold-face terms before coming to class, you will be in a much better position to understand the lecture.
• Be able to do what the instructor did without looking at your notes.
• If you anticipate where your instructor is going, you will be able to follow.
• Understand what you write. There is no point in taking notes that you never read.