Functions that commute

At the bottom of this post is a one-liner that I use in my classes the first time I present a theorem where two functions are permitted to commute. At many layers of the mathematics curriculum, students learn about that various functions can essentially commute with each other. In other words, the order in which the operations is performed doesn’t affect the final answer. Here’s a partial list off the top of my head:

Arithmetic/Algebra: $a \cdot (b + c) = a \cdot b + a \cdot c$ . This of course is commonly called the distributive property (and not the commutative property), but the essential idea is that the same answer is obtained whether the multiplications are performed first or if the addition is performed first.
Algebra: If $a,b > 0$ , then $\sqrt{ab} = \sqrt{a} \sqrt{b}$ .
Algebra: If $a,b > 0$ and $x$ is any real number, then $(ab)^x = a^x b^x$ .
Precalculus: $\displaystyle \sum_{i=1}^n (a_i+b_i) = \displaystyle \sum_{i=1}^n a_i + \sum_{i=1}^n b_i$ .
Precalculus: $\displaystyle \sum_{i=1}^n c a_i = c \displaystyle \sum_{i=1}^n a_i$ .
Calculus: If $f$ is continuous at an interior point $c$ , then $\displaystyle \lim_{x \to c} f(x) = f(c)$ .
Calculus: If $f$ and $g$ are differentiable, then $(f+g)' = f' + g'$ .
Calculus: If $f$ is differentiable and $c$ is a constant, then $(cf)' = cf'$ .
Calculus: If $f$ and $g$ are integrable, then $\int (f+g) = \int f + \int g$ .
Calculus: If $f$ is integrable and $c$ is a constant, then $\int cf = c \int f$ .
Calculus: If $f: \mathbb{R}^2 \to \mathbb{R}$ is integrable, $\iint f(x,y) dx dy = \iint f(x,y) dy dx$ .
Calculus: For most differentiable function $f: \mathbb{R}^2 \to \mathbb{R}$ that arise in practice, $\displaystyle \frac{\partial^2 f}{\partial x \partial y} = \displaystyle \frac{\partial^2 f}{\partial y \partial x}$ .
Probability: If $X$ and $Y$ are random variables, then $E(X+Y) = E(X) + E(Y)$ .
Probability: If $X$ is a random variable and $c$ is a constant, then $E(cX) = c E(X)$ .
Probability: If $X$ and $Y$ are independent random variables, then $E(XY) = E(X) E(Y)$ .
Probability: If $X$ and $Y$ are independent random variables, then $\hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y)$ .
Set theory: If $A$ , $B$ , and $C$ are sets, then $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ .
Set theory: If $A$ , $B$ , and $C$ are sets, then $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ .

However, there are plenty of instances when two functions do not commute. Most of these, of course, are common mistakes that students make when they first encounter these concepts. Here’s a partial list off the top of my head. (For all of these, the inequality sign means that the two sides do not have to be equal… though there may be special cases when equality happens to happen.)

Algebra: $(a+b)^x \ne a^x + b^x$ if $x \ne 1$ . Important special cases are $x = 2$ , $x = 1/2$ , and $x = -1$ .
Algebra/Precalculus: $\log_b(x+y) = \log_b x + \log_b y$ . I call this the third classic blunder.
Precalculus: $(f \circ g)(x) \ne (g \circ f)(x)$ .
Precalculus: $\sin(x+y) \ne \sin x + \sin y$ , $\cos(x+y) \ne \cos x + \cos y$ , etc.
Precalculus: $\displaystyle \sum_{i=1}^n (a_i b_i) \ne \displaystyle \left(\sum_{i=1}^n a_i \right) \left( \sum_{i=1}^n b_i \right)$ .
Calculus: $(fg)' \ne f' \cdot g'$ .
Calculus $\left( \displaystyle \frac{f}{g} \right)' \ne \displaystyle \frac{f'}{g'}$
Calculus: $\int fg \ne \left( \int f \right) \left( \int g \right)$ .
Probability: If $X$ and $Y$ are dependent random variables, then $E(XY) \ne E(X) E(Y)$ .
Probability: If $X$ and $Y$ are dependent random variables, then $\hbox{Var}(X+Y) \ne \hbox{Var}(X) + \hbox{Var}(Y)$ .

All this to say, it’s a big deal when two functions commute, because this doesn’t happen all the time.

I wish I could remember the speaker’s name, but I heard the following one-liner at a state mathematics conference many years ago, and I’ve used it to great effect in my classes ever since. Whenever I present a property where two functions commute, I’ll say, “In other words, the order of operations does not matter. This is a big deal, because, in real life, the order of operations usually is important. For example, this morning, you probably got dressed and then went outside. The order was important.”

You just can’t, Nemo!

During in-class discussions, students often but inadvertently make the same mistakes over and over again… say, thinking that $\sqrt{a^2+b^2} = a + b$ or $\displaystyle \frac{d}{dx} (uv) = \displaystyle \frac{du}{dx} \cdot \frac{dv}{dx}$ . Naturally, such mistakes need to be corrected, but hopefully politely and in a memorable way.

After the third or fourth such repetition of the same mistake during a semester, I’ll try to lighten the mood by saying, “You think that you can do these things, but you just can’t, Nemo!”

Pop culture reference:

See also my one-liner about the Third Classic Blunder: https://meangreenmath.com/2013/06/07/laws-of-logarithms/

A bit of a fixer upper

I always encourage students to answer occasional questions in class; naturally, this opens the possibility that a student may suggest an answer that is completely wrong or is only partially correct. Naturally, I don’t want to discourage students from participating in class by blunting saying “You’re wrong!” So I need to have a gentle way of pointing out that the proposed answer isn’t quite right.

Thanks to a recent movie, I finally have hit on a one-liner to do this with good humor and cheer: “To quote the trolls in Frozen, I’m afraid your answer is a bit of a fixer-upper. (Laughter) So it’s a bit of a fixer-upper, but this I’m certain of… you can fix this fixer-upper up with a little bit of love.”

If you have no idea about what I’m talking about, here’s the song from the movie (you can hate me for the rest of the day while you sing this song to yourself):

The Lighter Side of Teacher Evaluation

More about W. James Popham can be found here: http://www.corwin.com/authors/508599

Every math major should take a public-speaking course

Harvey Mudd College requires their math majors to take a public-speaking course specifically intended for math majors. From http://horizonsaftermath.blogspot.com/2014/04/every-math-major-should-take-public.html:

No matter what we all do after college . . . [we] will have to speak to people. Every one of us will have a limited amount of time that we can convince someone else to see our point of view.

I recommend reading the whole article.

Accommodating Blind Students Taking Mathematics

This was another interesting pedagogical article from the February/March 2014 issue of FOCUS, published by the Mathematical Association of America. I had never heard of the Nemeth code, the braille for mathematical notation.

http://digitaleditions.walsworthprintgroup.com/display_article.php?id=1639571

Flipping to Offer Low-Enrollment Courses

I just read a very interesting article about how an instructor at Ohio Dominican University is simultaneously teaching several upper-level mathematics courses with low enrollment by flipping the classroom (using the catchy title One-Room Schoolhouse). Here’s the article: http://digitaleditions.walsworthprintgroup.com/display_article.php?id=1639570

The Failure of Test-Based Accountability

From Marc Tucker’s blog on Education Week:

In my last blog, I pointed to the data that shows that, after 10 years of federal education policies based on test-based accountability, there has been no perceptible improvement in student performance among high school students (which, when you get right down to it, is what really matters) as a whole, or when the data are broken down by different groupings of disadvantaged students. There is little doubt—whether test-based accountability is being used to hold schools accountable or individual teachers—that it has failed to improve student performance.

That should be reason enough to abandon it. But it is not. The damage that test-based accountability has done goes far deeper than a missed opportunity to improve student achievement. It is doing untold damage to the profession of teaching…

Test-based accountability and teacher evaluation systems are not neutral in their effect. It is not simply that they fail to improve student performance. Their pernicious effect is to create an environment that could not be better calculated to drive the best practitioners out of teaching and to prevent the most promising young people from entering it. If we want broad improvement in student performance and we want to close the gap between disadvantaged students and the majority of our students, then we will abandon test-based accountability and teacher evaluation as key drivers of our education reform program.

But no one, certainly not me, would argue that we should not hold our professional educators accountable for their performance. The question is, what would accountability look like if we actually regarded our teachers as professionals doing professional work, instead of interchangeable blue-collar workers doing blue-collar work? That is the question I will deal with in my next blog.

I encourage you to read the whole thing: http://blogs.edweek.org/edweek/top_performers/2014/02/the_failure_of_test-based_accountability.html

The following video made the rounds a few months ago and ties in with the above point. It is less about the shortcomings of the Common Core than our leaders’ fixation with quantifying educational output. As the speaker says well, “If everything I learned in high school is a measurable objective, then I have not learned anything.”

Getting the right answer the wrong way

I just read “But My Physics Teacher Said… A Mathematical Approach to a Physical Problem,” which was a very interesting pedagogical article concerning the teaching of calculus. Here’s the central problem:

I included on their exam a question involving average velocity. I gave the students a quadratic function and asked them to calculate the average velocity over a given interval… One of my students… got the final numerical answer correct, but he hadn’t used the average velocity formula he had learned in our course. Instead… he had calculated the average of the velocities at the end points of the given interval. When I explained this to him, he stated that he didn’t understand the difference because he had learned the latter formula to calculate average velocity in his physics class.

It turns out that this alternative approach always work under the condition of constant acceleration (i.e., a quadratic function), and since constant acceleration is such an important special case in freshman physics, the formula was presented and the student remembered the formula. Of course, the student probably was not aware of the formula was only generally true under this specific circumstance.

After some pedagogical reflection, the author concluded

My student and I both learned from this experience. He gave me the opportunity to look at a familiar topic with the eye of a physicist, and I taught him the importance of context when using a formula. Specific adventures such as the one my student and I encountered will undoubtedly strengthen my approach to teaching this course and my students’ ability to think like mathematicians.

The full article can be found at http://digitaleditions.walsworthprintgroup.com/publication/?i=187509&p=19.

Rejection regions

Sage words of wisdom that I gave one day in my statistics class:

If the alternative hypothesis has the form $p > p_0$ , then the rejection region lies to the right of $p_0$ . On the other hand, if the alternative hypothesis has the form $p < p_0$ , then the rejection region lies to the left of $p_0$ .

On the other hand, if the alternative hypothesis has the form $p \ne p_0$ , then the rejection region has two parts: one part to the left of $p_0$ , and another part to the right. So it’s kind of like my single days. Back then, my rejection region had two parts: Friday night and Saturday night.