Learning Activities and Submitted Work

Summary

 
Most of your learning in the course will come about through the following learning activities and assessments of learning. Some additional detail is provided further down this page.

Narrative note-taking

 
The famous mathematician George Polya once said "Mathematics is not a spectator sport". By that he meant, you learn math by doing rather than just by reading or listening. One type of doing is solving problems, which is often given the highest priority in math education. However, an equally important companion in doing math is high-quality note-taking.

Even within note-taking, the most common kind is what occurs in the classroom: short notes to copy down what occurs in class, with the aim of substituting for perfect memory. But the better kind is narrative note-taking in which you build an entire narrative, a story, about the material you are studying. By telling yourself the full story, in a kind of dear-diary format, you have the best shot at "bringing it all together" and retaining for the long-term.

So ... what you will do is build a narrative of the key concepts in linear algebra that you will submit pieces of electronically. Think of this as on-going review notes that teach yourself now (a review) and later in life (recalling the most important concepts).

What to write:

  • The main goal in narrative note-taking is to explain key points in the material to yourself, in your own words, and neatly, with pride.
  • Include a few original examples that you have constructed, or diagrams that explain something to yourself. When you construct something original, put that in your notebook with a box (borders) around it, bringing that to our attention.
  • For any theoretical result, explain what the theorem or proposition is saying, with an example to clarify if appropriate. Do not include the proof, but you can create something original with an annotated proof, which fills in details or intuition.
  • You can create original content in other ways too. For example you can intuitively explain a definition or fact with your own example.
  • Important: You will submit your narrative notes by taking a scan and converting to PDF. Doing a good job with these notes will earn you get-back points (see the About section).

Notebook organization:
Organize your notebook into three sections:

  • Part I for modules 2-6.
  • Part II for modules 7-10.
  • Part III for modules 11-13.
You will scan each of these and upload as PDFs.

Which modules and sections to especially focus on:

  • Module 2: 2.4, 2.5, 2.7
  • Module 3: 3.2, 3.4-3.8
  • Module 4: 4.1-4.4, 4.6, 4.8, 4.10-4.12
  • Module 5: 5.1-5.6
  • Module 6: 6.1-6.6
  • Module 7: 7.0-7.3, 7.6, 7.7.
  • Module 8: The definitions in Section 8.1, the meaning of the statements in each of Theorems 8.1, 8.2, 8.3.
  • Module 9: 9.1, 9.3, 9.5, 9.6
  • Module 10 (optional for undergrads): 10.1-10.3, 10.5
  • Module 11: 11.0-11.2
  • Module 12: 12.0-12.3
  • Module 13: 13.0, 13.2, 13.4, 13.7

In-class exercises

 
We will touch upon many in-class (module) exercises in class itself, as one of the main learning activities of the course. However, because we have limited time in class, we will assign some to complete ahead of time, possibly rush through some of these in class, and skip yet others. You will need to complete these on your own after class.

The best time to complete in-class exercises is before the next class. It will increase your learning efficiency by helping you keep up with the next class. If, on the other hand, you postpone completing these, you will merely accumulate more and more "unfinished business" that will slow down your rate of learning.

Get organized by creating a main directory for the course, then one sub-directory for the in-class exercises of each module. For example, in the module3 subdirectory, you will have all your code and answers for in-class exercises in Module 3.

For in-class exercises that have plain-text answers or pictures, put these down in a single PDF for each module.

Submission:. See the submission instructions.

Proofs

 
One goal in this course is to strengthen your proof skills, no matter how modest the skill level is. You do NOT have to have this skill coming into the course. All you need is the right attitude: the goal of improving them.

Why work on proof skills? Why not just learn about results and leave the proving to others? There are several reasons. First, proofs are possibly the most rigorous form of reasoning and persuasion, the gold standard for scientific discourse. Second, many proofs (but not all) convey intuition: they explain why a result is what it is. Third, proofs also make assumptions and its important to understand them. Too often, a weak theoretical result is used to justify some ill-advised action simply because, "there's math behind it". Lastly, there is deep satisfaction to be gained in proving something yourself: it is proof, if you'll excuse the double-entendre, that you've understood something at its deepest level.

We're going to work on proof skills in two ways:

  1. Proofs submitted in module exercises/assignments. The first, conventional way, is when you write up your proofs for the module exercises that ask for proofs, and for assignments that feature proofs.
  2. In-class proofs. The coursework page lists specific proof-requiring questions. What this means:
    • You will need to have worked on these proofs BEFORE coming to class.
    • We will call on students randomly to write their proofs on the board and explain them.
    • The main idea is not to have something perfect, but instead to have an idea as a seed for discussion.

Let's talk about "proof anxiety," at the moment in class you are presenting a proof. Remember, the goal is NOT to succeed flawlessly (although that will be appreciated). It is to try. In fact, there's more learning that occurs in class when proofs go awry.

Submitting proofs. Those proofs that are part of your module exercises or assignments should be submitted as a PDF (see below) and included in the appropriate zip file uploaded to Blackboard.

Submitting module/Assignment code and PDFs

Notice that some module exercises call for proofs, others call for code, and yet others ask for demonstrating something or answering a question. Let's clarify how the submission works:

  • Make one folder for each module. So, you will have, for example, a module3 folder.
  • All the code you write for module3 should be in this folder.
    Important: Include all the source files needed for compilation, otherwise we will not be able to assign points.
  • Some module exercises ask for proofs, and yet other exercises ask for plots or answering a question. Use a single PDF for each module that will have your response to all of these. So, for example, your module3.pdf should your responses to exercises 1-3, 7, and so on, include Ex-23 (which is a proof).
    Important: Your text answers will need to be supported by reasoning. Simply answering "Yes" is not sufficient.
  • Then, make a zip of the folder and submit to Blackboard.

Writing math with Latex

Latex or its precursor, Tex, is the preferred formatting language used to write mathematics. It's a language in the sense that HTML is a language: a variety of commands that describe content, layout, and presentation. Unlike HTML, however, it's harder to learn at first and can sometimes be annoying if you want to do complicated things.

A brief history. In the old days, math formatting was either non-existent for ordinary users or available only to sophisticated publishers with expensive machines. Besides, there were no standards and it was impossibly to share anything electronically. Computer science pioneer Donald Knuth (you should read about him) developed Tex and Metafont to address both font generation (Metafont) and mathematical typesetting (Tex). It took the world by storm. Latex is a small improvement over Tex devised by another computer science pioneer, Leslie Lamport, to simplify Tex by building on top of Tex. It is now the standard for math typesetting.

Note: Metafont uses the Bezier curves you saw in the first class.

It works roughly like this. You first write down your math on paper. Then, you create a plain text file with latex commands and the math in it. This file will be named something like myexample.tex. Then you run "latex" like you would compile a Java program, except that you'll be compiling the file myexample.tex. The result will be a PDF file that you can print.

We will not be teaching you to use Latex. There are a zillion tutorials and examples. It's a good exercise to go through the steps: download and install, then try someone else's examples, then write a few of your own.

Grade breakdown

About grades and points: