Solving systems of equations Solving a single linear equation with one unknown quantity is a task that is studied in elementary school. One proceeds in high school to study more complex equations involving more involved functions. Many problems in mathematics and its scientific applications amount to solving equations, and especially to solving systems of equations. … Read more
Concentration is a key skill for becoming a good learner. This is especially true in studying mathematics and physics, where you are continually engaging with new concepts. You also frequently have to mobilize all of your thinking power to grapple with complex problem-solving tasks. This often takes time, so you need to sustain your focused … Read more
Daily practice is essential for learning. Right now I am beginning to learn how to play the piano, which is an interesting experience for me as a teacher. It’s quite fun to be an absolute beginner at something, and experience all the joys and frustrations that come with learning something new. As an absolute piano … Read more
I recall reading about a calculus teacher who taught in a gorilla suit one day to make his class more entertaining. This was lauded by some as a good thing. It is absolutely not a good thing! The unintended message is that calculus is so intrinsically boring that clowning around is needed to make it … Read more
We must concentrate to learn effectively, a point that I have experienced numerous times in my life. The myth of multi-tasking has been thoroughly debunked by now (for example, see here), but nevertheless many of us try to learn things while trying to do something else. This is ineffective and just ends up causing stress. … Read more
The main purpose of this web site is to help students learn about mathematics and physics, and in particular to provide guidance in how to approach your studies in a professional and effective way. I’ll be posting every week on these topics, and will also be regularly uploading various learning materials. But if there is … Read more
How does the value of $\cos^{-1} \left ( 0.1 \, \textrm{cm} \right )$ differ from $\cos^{-1} \left ( 0.1 \, \textrm{m} \right )$? This is an interesting question, especially because the question of units in trigonometric and inverse trigonometric functions is rarely discussed in mathematics textbooks. It is discussed in physics textbooks, so you may … Read more
In many physics textbooks, the explanation for lift on a flying airplane is that the top of a plane’s wing is longer than the bottom, and so air must travel faster across the top than the bottom, and therefore the pressure is lower above the wing than below. (This difference in pressure associated with different … Read more
If you’ve had the pleasure of standing right next to the roaring water falling over the precipice at Niagara Falls (and for me, it’s the sound that’s most impressive), then it’s hard to believe that the flow once naturally stopped for more than 24 hours! How could this have happened? Very strong winds jammed ice … Read more
Anne Stone and Jeff Nichols beautifully make the case against standardized testing at the New York Times SchoolBook blog (hat-tip to Susan Ohanian). Here is an excerpt: In light of current controversies around testing and teacher evaluation, let’s do a little thought experiment. How would Miss Snug have handled this lesson if it were occurring … Read more