“Student Learning Can Only Be Described, Not Measured,” by Rog Lucido

Rog Lucido has written an interesting article against standardized testing, and suggesting better alternatives. (Hat-tip to Susan Ohanian.) He argues that the numerical aggregation of final test scores is not valid and therefore not meaningful, and that subjective assessments together with verbal descriptions are meaningful and valid. (This post first appeared at my other (now … Read more

A Neat Trick For Determining The Integrals Of exp(x) cos x and exp(x) sin x

The standard method (typically found in first-year calculus textbooks) for determining the integrals $\int e^x \cos x \, {\rm d}x$ and $\int e^x \sin x \, {\rm d}x$ is to integrate by parts twice. If you haven’t seen the standard method, I’ll show you how to do the first one; the second one is similar. … Read more

“No Student Left Untested,” by Diane Ravitch

Measuring teacher effectiveness by the performance of students on standardized tests is insane. New York State has just signed on to a particularly dangerous form of this insanity. Diane Ravitch has clearly explained the insanity and its destructive consequences in No Student Left Untested, in the New York Review of Books (hat-tip to Observational Epidemiology). … Read more

Both Students And Professors Need Certification, and the Elsevier Boycott

I’ve written before about the evils of grading (for example, see here and here), the main purpose of which is to make certifying students easy. Our current grading system in mathematics is counterproductive to learning (students are inhibited from engaging in essential learning activities out of the fear that is naturally induced by typical high-stakes … Read more

A Number Riddle, Updated With Solution, And Some Comments On Iterative Playgrounds

Two weeks ago I posted a puzzle sent to me by my nephew Matthew: 1 is 3, 3 is 5, 5 is 4, and 4 is cosmic. Why is 4 cosmic? What happens to the other numbers? As I mentioned in the earlier post, I slept on this before solving it. I initially thought about … Read more

How Much Mathematics Should A Student Memorize? Part 6, Derivatives Of Exponential And Logarithmic Functions

In teaching calculus many, many times over the years, I strove to present to my students my approach to learning and mastering the subject. Part of this approach can be summarized by the slogan memorize the minimum As a teacher, I took it as part of my responsibility to help students identify the essential core … Read more

Failing … To Learn

Dr. Brian Goldman is an emergency-room medical doctor and host of the excellent CBC radio program White Coat, Black Art. As I was tidying up some computer files I came across some notes from one of his CBC radio appearances from 18 May, 2011. Goldman was discussing mistakes in the context of medical practice, and … Read more

Daniel Coyle On The Yin And Yang Of Learning

Unencumbered time and enchantment … why can’t we have more of that in our schools? Oh right, it’s because we’re too busy cramming vital content into our poor students’ heads … and then crushing their spirits as we drag them through high-stakes tests. The worst of it is that after 13 years of this, we … Read more

How Much Mathematics Should A Student Memorize? Part 5, The Multiplication Table

There has been a war in the mathematics education world for the past few decades about whether students should master basic skills, or whether they should use calculators or software for basic skills to save time and energy for higher-level thinking. More and more people nowadays are seeing this for what it is: a false … Read more