One way to find out how well you know math fundamentals is to take a timed math test. In the U.S., over two million students per year take the SAT (which includes a math portion) as part of the college admissions process. A multitude of online resources, including Khan Academy, are available to help with […]

Continue## Should You Solve Every Problem In a Math Textbook?

When you’re studying a topic, a course acts as a filter for the material: Rather than having to consider every reference on a subject, you get a nicely curated subset that the instructor believes fulfills the goals of the course. Rather than having to read an entire textbook, you get a list of sections, or […]

Continue## How Long Should You Work on a Math Problem Before Looking at the Answer?

When you’re studying math on your own, a question you must continually ask yourself is how much time to spend on the problem you’re working on. That question also comes up when you’re taking a traditional class, though your options are more limited in that case because of the fixed class schedule. I wrote about […]

Continue## Why are Textbook Proofs so Short?

A mathematical proof can be the size of a novella. For example, Andrew Wiles’s published his famous proof of Fermat’s Last Theorem in two journal articles covering 129 pages. But proofs in introductory textbooks like Rosen often contain just a few sentences. It might seem obvious that these proofs are short because they’re easy. But […]

Continue## Example Proof: Using the Div Operator with a Negative Argument

Last week, I suggested a process for getting better at writing proofs. To illustrate that process, here’s an example of how to use it to prove a theorem from Rosen.

Continue## How to Practice Writing Proofs

Proof-writing skills are important for all college-level math. But there’s a special relationship between proofs and discrete math. In the “Goals of a Discrete Mathematics Course” section in the preface to his textbook, Rosen puts Mathematical Reasoning first in the list. He writes: Students must understand mathematical reasoning in order to read, comprehend, and construct […]

Continue## How to Read a Math Textbook

If you search for advice on how to read a math textbook, you’ll find plenty to choose from. I’ll link to some of my favorites at the end of this article. But first, here’s an overview of my experience reading math books, and what techniques I found to be useful.

Continue## What is Discrete Mathematics?

In their first 13 or so years of school, students cover a standardized math curriculum. Last week, I covered how Khan Academy approaches that curriculum. Notably absent from that list are many topics in discrete mathematics. But what is discrete mathematics, anyway? I’ll answer that in two ways: with a definition, and with a curriculum.

Continue## The Khan Academy Math Course System

Khan Academy’s math program is designed to help children and young adults learn and practice a particular set of math skills. These skills map to school curricula like the US Common Core, so that when students learn something on the site, it translates to success in the classroom. But Khan Academy can also help adults […]

Continue## Post-Mastery Practice Examples from Khan Academy

The process of learning math involves mastering thousands of small skills. Khan Academy has exercises that help you practice the first 1500 or so of these skills. But as I discussed last week, the Khan Academy mastery system only gets you to an initial level of mastery. The topic for this week: how to continue […]

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