PEMDAS Problems

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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.

 

Predicting Partitions

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This recent news from the American Institute of Mathematics:

January 20, 2011.   Researchers from Emory, the University of Wisconsin at Madison, Yale, and Germany’s Technical University of Darmstadt discovered that partition numbers behave like fractals, possessing an infinitely-repeating structure.

In a collaborative effort sponsored by the American Institute of Mathematics and the National Science Foundation, a team of mathematicians led by Ken Ono developed new techniques to explore the nature of the partition numbers. “We prove that partition numbers are ‘fractal’ for every prime. Our ‘zooming’ procedure resolves several open conjectures,” says Ono.

Accompanying this result was another achievement developing an explicit finite formula for the partition function. Previous expressions involved an infinite sum, where each term could only be expressed as an infinite non-repeating decimal number.

Counting the number of ways that a number can be ‘partitioned’ has captured the imagination of mathematicians for centuries. Euler, in the 1700s, was the first to make tangible progress in understanding the partition function by writing down the generating series for the function. These new results involve techniques which could have applications to other problems in number theory.

[original article]

Wired.com also just reported on it, and you can find their coverage here.

MathOverflow

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I’m sure most of the mathematical community already knows all about  mathoverflow.net, but just in case, thought I’d post a link here. It was news to me…perhaps just because I’m not really in the collegiate academic math community. (Though the wikipedia article says it was created in 2009, so it’s still in its infancy.) But it’s definitely a resource I’ll be using as I do my own mathematical investigations and my grad coursework. I’ve enjoyed the original stackoverflow.com for looking up answers to questions, and in two instances so far, asking my own. StackOverflow and sites like them fill a vital role in the online community–quick, thorough answers to pointed questions.

Geogebra has new skills

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A new version of Geogebra has been released, in beta. It’s called Geogebra 5.0, and you can see the news about it here. Or, here’s a direct link to launch it right away. Thanks to The Cheap Researcher for the lead on this. As readers of this blog may already know, I love Geogebra!

One of the main highlights is that Geogebra now supports 3D manipulations. Awesome! However, don’t get too excited–it doesn’t let you graph anything except planes. No surfaces. It will do geometric constructions, like spheres and prisms. Using parametric equations and the locus feature, you can coax it into rendering spirals or other space curves. [edit: I figured this was possible, but it actually wasn’t. Not sure why.]

Another highlight, which I find even more exciting, is that Geogebra now has a built in CAS. Here’s a screen shot of me playing around with a few of its features. It also has a ways to go, especially for those who are used to more robust systems like Mathematica/Maple/Derive/TI-89. But this is a great step in the right direction, and 10 points for the open-source camp!

Notice that it can work with polynomials in ways you would expect, it can symbolically integrate and derive (simple things), perform partial fraction decomposition, evaluate limits, and find roots. Here are a few more things it can do. Strangely, it had problems finding the complex roots of a quadratic (easy), but not a cubic (hard). Just take a look at my screen shot. Seeing that it did okay finding the complex roots, I wondered if it could also plot them for me. I started by entering (copying and pasting) the complex zeros as points in Geogebra, which worked. But then I discovered the new ComplexRoot[] function which approximates the roots and plots them on the coordinate plane all at once. Cool! Here’s the screenshot:

The seven complex roots of f(z)=z^7+5z^4-z^2+z-15

As you can see, I asked for the roots of a 7th degree polynomial. Since the polynomial had real coefficients, notice that every zero’s conjugate is also a zero, as we’d expect. And we also expect that at least one solution of an odd-degreed polynomial will be real (notice this one has only one real root, approximately 1.22).

That’s all I’ve discovered so far. I’ll let you know if I come across anything else exciting. Keep in mind that this is beta, so the final release will likely have all the bugs worked out and more features.

Vi Hart’s Blog

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It’s high time I gave a bit of press to Vi Hart’s Blog. If you haven’t checked it out, do so right away. It’s brilliant.  A number of people have pointed me to her blog, including one of my Calc students. Her little math videos are fresh, funny, and insightful. Denise, at Let’s Play Math, gave her some press too, which is what reminded me to finally make this post. Here’s the video Denise highlighted (the most recent of Vi’s creations):

This is particularly appropriate because there were a couple of us in our math department discussing this very question: In total, how many gifts are given during the 12 Days of Christmas song? It’s a nice problem, perfect for a Precalculus student. Or any student. Here’s a super nice explanation of how to calculate this total, posted at squareCircleZ. But before you go clicking that link, take out a piece of scrap paper and a pencil and figure it out yourself!

Here’s another nice video from Vi Hart:

You could spend a lot of time on her site. Here’s another awesome video. I’ll have to have my Precalculus class watch this one when we do our unit on sequences and series.

And you’ve got to love the regular polyhedra made with Smarties ,  right?

Plus, Vi Hart plays StarCraft, which is awesome too.  Back in the day, I really loved playing. I haven’t played in a while, and I certainly haven’t tried SC 2 yet, because then I’d never grade my students’ papers.

Bottom line is, you need to check out all the playful stuff Vi Hart is doing at her blog. Happy Wednesday everyone!

 

Learning from Mistakes

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Watch this inspiring TED talk:

As a teacher, I know I’m succeeding when students feel free to take risks in the classroom. But letting students take risks is risky for a teacher. I’m a bit OC, and that’s the thing that keeps me from letting students take on more creative tasks. There’s also a good argument for teaching with a more systematic approach in such a way that all necessary topics get covered with a reasonable amount of depth in a reasonable amount of time. This reminds me of the discussion surrounding Paul Lockhart’s essay (my response here).

What if we Graded Toddlers?

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From techdirt.com

What If We Gave Toddlers An ‘F’ In Walking?

from the rethinking-education dept

theodp writes “To improve math and science education, Physics prof Dr. Yung Tae Kim thinks professors and teachers should take a page from skateboarding. ‘The persistence and the dedication needed in skateboarding — that’s what we need to be teaching,’ explains Kim. ‘No one says to a toddler, ‘You have ten weeks to walk, and if you can’t, you get an F and you’re not allowed to try to walk anymore.’ It’s absurd, right? But the same thing is true with math and science education. If you want to learn trig or calculus, it’s set at such a pace in schools that it guarantees that only the absolutely best students will learn it.’ Kim says it’s possible to ‘polish the turd’ of high school and college education, and lays out his plan for doing so in Building A New Culture Of Teaching And Learning (YouTube: parts 1–2–3), a video drawn from a farewell talk he gave to his Northwestern students. There’s more on The Way of Dr. Tae at DrTae.org and PhysicsOfSkateboarding.com.”

I was just discussing the same point with my father in law this past week. Our education system needs to change in fundamental ways if we want students to truly learn at their own pace. We do a bit of a disservice to students who need to take the material a little slower. There’s nothing wrong with taking things slowly. Likewise, we do disservice to students who could complete the coursework in half the time.

[Hat tip: Tim Chase, as usual :-)]