Calculus for Infants

My wife and I are expecting our first child in February, so these are definitely items that should be on our registry, wouldn’t you agree?

I saw these first on thinkgeek.com. Introductory Calculus for Infants is currently out of stock, but Amazon.com has it! I haven’t seen any legitimate reviews of these books yet (no reviews on amazon.com), so I’ll have to provide a review once I get them. Introductory Calculus is a Calculus book, Andre Curse is about infinite recursion, as the cover subtly suggests.

This is the most important thing to be thinking of as I prepare for fatherhood, right?

Why are infinite series so hard to grasp?

I’ve posted on infinite series a few times before. But I was inspired to touch on the topic again because I saw this post, yesterday, over at the Math Less Traveled. Actually, the post isn’t really about infinite series as much as it is about p-adic numbers and zero divisors. I’m excited to read more from Brent on this subject. But I digress.

The point I want to make with this post is that students struggle with wrapping their minds around convergent infinite series, and yet they live with them all the time. Students have inconsistently held beliefs about infinite sums.

The simplest convergent series is a geometric series \sum_{n=1}^\infty a_n=a_1r^{n-1} which converges to \frac{a_1}{1-r}. The easy proof of this fact goes like this: we look at the sum formula for a finite geometric series, s_n=\frac{a_1(1-r^n)}{1-r} and we notice that

\lim_{n\to\infty}\frac{a_1(1-r^n)}{1-r}=\frac{a_1}{1-r}

for |r|<1.

But this proof isn’t very satisfying for the student encountering infinite series for the first time ever. Evaluating the limit feels like ‘magic.’ The idea of adding up an infinite amount of things and getting a finite value is unsettling. I admit, it sounds like quite a lot to swallow. That being said, however, students have no problem declaring the infinite series

0.3 + 0.03 + 0.003 + 0.0003 + \cdots

to be 1/3. It’s not “close to” 1/3, it’s not “approaching” 1/3, it IS EQUAL TO 1/3. And my Precalculus students already accept this as fact. So without even thinking about it, they’ve been living with convergent infinite series all along. Hah!

Once they finally shake their denial, they can more easily accept the convergence of other infinite series like \sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}. At first when students encounter a series like this, they think, “surely we can’t say the sum is EQUAL to \frac{\pi^2}{6}. It must be close to \frac{\pi^2}{6} or approach it, but equal to?” But the same students make no such distinction with 0.3+0.03+0.003+\cdots = \frac{1}{3}.

So there it is. An inconsistently held belief about infinite sums. To the student: You cannot have it both ways. Either you must agree with, or deny, both of the following equations:

0.3+0.03+0.003+\cdots = \sum_{n=1}^\infty 0.3(0.1)^{n-1}=\frac{1}{3}

1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\cdots = \sum_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6}

But to believe one equation is true and the other is only ‘kind of’ true is inconsistent. I rest my case. 🙂

The Manga Guide to Calculus

This summer I finally finished reading the Manga Guide to Calculus by Hiroyuki Kojima and Shin Togami. Here are my two cents:

The Manga Guide to Calculus is chocked full of great mathematics and lots of quality comic art (the author went to great lengths to ensure it was authentic manga, with illustrations by popular artist Shin Togami).

That being said, I don’t think anyone could ever learn Calculus using this book. In fact, I think Kojima must know that. He never claims this can be used as a textbook replacement. The math isn’t presented in a very systematic way, and there are very few real exercises for the reader. Right from the beginning he puts heavy emphasis on linear approximation. He takes a very different approach to presenting Calculus than a math book would. It is a story most of all. Kojima, in his preface, says its a great book for those who already have Calculus knowledge–both for those who love Calculus and for those who have been “hurt by it.” I tend to agree.

As for the story, well, it’s a bit contrived. But what story that tries to smuggle in some math doesn’t seem a little contrived? Sometimes it’s a bit of a stretch and the story suffers. You should still give it a chance, though.

So to those looking for a Calculus textbook, you need to look elsewhere. For instance, I was looking for things I might be able to use in the Calculus class I teach, but couldn’t find much usable content. But for those interested in math and are looking for a fun read, I would recommend picking it up.

Teaching domain and range incorrectly

What’s wrong with these high-school math questions?

  1. State the domain of the function f(x)=\frac{1}{x}.
  2. Where is the function f(x)=\ln(x) undefined?
  3. State the range of the function f(x)=x^2.

As a math teacher, I’ve asked these questions before too. But I always ask them with a bit of a cringe. Do you see what’s wrong with them?

Domain

A function is only well-defined when it is defined with its domain. A function f is an association between two sets A and B that assigns only one element of B to each element of A. The set A is called the domain and the set B is sometimes called the codomain. If  f is a function mapping elements from A into B, then we often write f:A\to B. For instance, consider the function f:\mathbb{R}\to\mathbb{R} defined by

f(x)=e^x

The domain of this function is \mathbb{R}, since that’s how the function is defined. Notice I explicitly gave the domain right before defining the function rule. Technically, this must always be done when defining any function, ever.

We might ask a student, “What is the domain of f(x)=e^x?” But this is a poor question. The function rule isn’t well-defined by itself. There are many possible domains for this function, like the set of integers \mathbb{Z}, the set of positive reals \mathbb{R}^+, or rational numbers between 20 and 30. What a teacher probably means is, “What is the largest possible subset of \mathbb{R} that could be used as the domain of f(x)=e^x?” In this case, the answer is \mathbb{R}.

So I hope you see why question (1) at the beginning of this post is not a very precise question. Likewise, question (2) is not very precise either. “Where is \ln{x} undefined?” has multiple answers. The answer the teacher is looking for is (-\infty,0]. The question would be better worded, “What real values cannot be in the domain of \ln{x}?”

 

Range

What about the range? The codomain in the example f(x)=e^x is also \mathbb{R}. But the range of f is (0,\infty). The range of a function is defined to be the set of all y\in B such that there exists an x\in A with f(x)=y.

But the range depends on the choice of domain. So asking questions like (3), “State the range of the function f(x)=x^2” aren’t well defined for all the same reasons as above. The desired answer is probably [0,\infty). But the domain of f(x)=x^2 could be the integers, in which case the range is the non-negative integers. We’re not told. So in the case of (3), the more precise question would read, “State the range of the function f(x)=x^2 with x\in\mathbb{R}.”

 

Should we change our teaching?

Maybe. But maybe not. I think I’ll still ask the questions in the imprecise way I started this post. Using the more precise questions would be unnecessarily confusing for most students. But we as teachers should be aware of our slightly incorrect usage, and be ready to give a more precise and thoughtful answer to students who ask.

That being said, I think there’s room for more set theory and basic topology at the high school level. I’m a bit sad I didn’t learn the words onto, surjective, one-to-one, injective, bijective, image, and preimage until very late in my post-high-school studies. I’m not sure all students are ready for such language, but we shouldn’t ever shy away from using precise language. That’s part of what makes us mathematicians.

I ❤ precise language!

Physics of Angry Birds

A nice projectile motion application, suitable for almost any level of high school math.

Here’s an excerpt of the Wired.com article by Rhett Allain:

You know the game, I know you know. Angry Birds. I have an attraction to games like this. You can play for just a little bit at a time (like that) and each time you shoot, you could get a slightly different result. Oh, you don’t know Angry Birds? Well, the basic idea is that you launch these birds (which are apparently angry) with a sling shot. The goal is to knock over some pigs. Seriously, that is the game.

But what about the physics? Do the birds have a constant vertical acceleration? Do they have constant horizontal velocity? Let’s find out, shall we? Oh, why would I do this? Why can’t I just play the dumb game and move on. That is not how I roll. I will analyze this, and you can’t stop me.

 

[Hat tip: Tony Sanders]


 

PEMDAS Problems

Inspired by this post at Education Week, by David Ginsburg, and a fruitful discussion in the comments, I thought I’d make my own post about the order of operations. I feel the need to do this because of my own experiences teaching Algebra 1 and correcting the habits of Algebra 2, Precalculus, and Calculus students. The problem is often with their misunderstanding of the ubiquitous “PEMDAS” approach to order of operations (parentheses, exponentiation, multiplication, division, addition, subtraction).

The classic example given by those who have discussed this before is an expression like the following. This example shows the problem with instructing students to simply “add then subtract”:

8-4+1

You can easily see that blindly following PEMDAS would give you an answer of 3, not 5. In reality, addition and subtraction can be done in any order, provided that you think of the expression this way:

8+(-4)+1

I have to constantly reinforce in my students the notion that the “-” goes with the 4. It owns it. That’s the right way to think of it, since subtraction (and division) are just shorthand for addition by the additive inverse (and multiplication by the multiplicative inverse). That’s right kids, subtraction is an illusion. The typical Real Analysis approach is to use the field axioms to define the properties of addition. Here they are:

  1. If x and y are in the field, so is x+y (closure).
  2. x+y=y+x (commutativity)
  3. (x+y)+z=x+(y+z)=x+y+z (associativity)
  4. There exists an element 0 such that 0+x=x for all x (identity)
  5. For each x there is an element called -x such that x+(-x)=0 (inverse)

There are almost identical ones for multiplication, and then one more axiom for distribution. Just listing the field axioms for addition  helps us in the discussion of order of operations. As I said above, subtraction is never really mentioned. We only later define x-y to mean x+(-y).

Similarly, division is an illusion. Or more precisely, it is defined in terms of multiplication. We define x/y to mean x\cdot(1/y).

The other important thing to notice is that addition and multiplication are binary operations, meant to be functions of two variables. So the expression x+y+z is defined to mean (x+y)+z=x+(y+z).

But the problems with PEMDAS don’t stop with multiplication and addition. There are many other issues with the order of operations as well. We lie to students when we just tell them they’ll encounter those limited operations in tidy situations. Here’s a perfectly legitimate expression with which PEMDAS would flounder:

-2^{3^2}+\left|\frac{2\sin{0}+3!}{2-\sqrt{9}}\right|\pmod{3}

The reason the expression is so difficult for PEMDAS is because PEMDAS doesn’t acknowledge the existence of other functions or grouping symbols, and ignores the possibility of nested exponentiation.

As another somewhat related example, I often have Calculus students who evaluate the following expression incorrectly (because they misuse their calculator):

e^{(0.05)(10)}

And some high school students at every level still struggle with evaluating these expressions:

  • -2^2
  • Given f(x)=-x^2+x, evaluate f(-2).

The wikipedia article on Order of Operations is very clear on all this, but I wanted to just highlight some of the common misconceptions. For more, feel free to check out that article. I’m in no way recommending that PEMDAS be removed from math curriculum, but that it be used with great care. And by the way, the expression above evaluates to 1.

 

Mathematical modeler fails to learn from history

Posted by guest blogger Dr. Gene Chase.

For all of you who love mathematical modeling and love (unintentional) humor, here’s a link for you.

Apparently researcher M. M. Tai invented a method for finding the area under “glucose tolerance and other metabolic curves,” a method which has now come to be called — in American Diabetes Association circles at least — “Tai’s model.”

We of course call it the Trapezoidal Rule. ::sigh:: As historian George Santayana once said, “Those who cannot remember the past are condemned to repeat it.”

[Hat off to Slashdot for breaking the news.]