Category Archives: Math Education

Teaching and Grace

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Thanks to my dad for the hat tip. If you haven’t yet seen it, this talk (available in text & audio) is a MUST READ for all teachers. Thank you, Francis Su, for sharing your thoughts on teaching and grace!

Your accomplishments are NOT what make you a worthy human being.

I highly recommend, over and above any other teaching book I’ve read, Teaching with Love and Logic. The message of that book is similar to the message Professor Su shares. Let kids know you like them for who they are, not for how they perform in your class!

Cubic polynomials and tangent lines

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Just read an article in the most recent NCTM Mathematics Teacher magazine called “Students’ Exploratory Thinking about a Nonroutine Calculus Task” by Keith Nabb. I really, really enjoyed this article. Maybe for some this isn’t new, but I didn’t know this fact:

Average two of the roots of a cubic polynomial. Draw a tangent line to the cubic at this point. Did you know it will always pass through the third zero?? Incredible!

Here’s a nice site that I just googled that goes through one proof. However, the charm of the article mentioned above is that there are many interesting proofs that students came up with, some of which are more or less elegant (brute force algebra with CAS, Newton’s Method, just to name two of the four strategies mentioned in the article).

I wish I could give you the whole article, but you have to have an NCTM membership to see it. Here’s the link, but you’ll have to log in to actually see it.

“Japanese” Multiplication

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My brother sent me a link to this video that teaches “Japanese” multiplication (thanks Tim!):

I learned about this technique in my History of Math class, and Vi Hart talked about it in a video back in 2011:

She does a nice job showing why there’s nothing particularly special about this Japanese “visual” multiplication. Here are a few reasons why it’s not better, as far as I’m concerned:

  1. It’s not faster (sometimes it is, but most of the time not). As Vi points out, counting the number of dots in a rectangle by hand is ridiculous.
  2. It’s painful when the numbers are bigger than 1, 2, or 3 and when there are more than 2 digits in the numbers (just try multiplying 976 x 8937 for example).
  3. Zeros make things difficult (use dashed lines?)
  4. Carrying is still required.
  5. It’s perhaps more error prone, since it relies on your counting all the intersections.

In the end, to multiply two numbers you still have to multiply all their digits by each other and deal with carries, no matter which method you choose. I think it’s still worth teaching various methods of multiplication to students in an effort to make the abstract more concrete.

Syllabus Design

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Has anyone ever considered designing a syllabus that reads like a glossy brochure? Check out this history professor’s blog post about his syllabus “extreme makeover.” Incredible!

From Hangen’s blog

Could it work with a math syllabus? I think so! Is it worth doing? Maybe. Not sure. But it does look like fun :-).

 

[HT: John Fea’s blog]

 

Fearless Symmetry

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I come to you today with a recommendation for the book Fearless Symmetry by Avner Ash and Robert Gross. I started it this summer and finally had a chance to finish it over the Christmas break. I didn’t understand the last half-dozen chapters, but my dad did warn me that would happen. I wouldn’t even attempt reading it unless you’ve already been exposed to some undergraduate mathematics. But if you have, or if it’s been a while and you need a refresher, I highly recommend the book.

In the book, Ash and Gross attempt to explain some of the math underlying Wiles’ proof of Fermat’s Last Theorem. So you can understand why the math gets a bit hard at the end.

Along the way, you’ll get a very conversational, well-written, fun-loving introduction to the Absolute Galois Group of the Algebraic numbers. This is a group that is so complicated and messy and theoretical that we can only explicitly write down two elements of the group. In order to talk about it, we need representations, which the authors also introduce in a gentle way. In particular, we need linear representations.

Elliptic curves become very important too. I have studied elliptic curves in two of my classes before, but I really liked the way they introduced them here: We know everything about linear equations (highest exponent 1), and everything about conics (highest exponent 2 on x and y), but suddenly things become very interesting when we allow just ONE of the exponents (on x) to jump to 3. These are elliptic curves. Amazingly, you can define an arithmetic on the points of an elliptic curve that yield both a GROUP and an algebraic VARIETY. Incredible. Of course, the authors introduce what a variety is too.

After reading this, I also gained a much bigger view of abstract algebra–a course I’ve taken, but I found myself guilty of seeing the trees but not the forest. I loved the way Ash and Gross introduce the group SO3 and relate it to A4 with the rotations of a sphere inside a shell. Very nice visualization!

I could go on, but just know that there are lots of little mathematical gems scattered throughout this book. It’s a refreshing jaunt through higher-level mathematics that will demystify some of the smart-sounding words you’ve been afraid to ask about :-).

Go check it out!

TI Calculator Emulators

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Online Emulator

Check out this online TI-83 Plus emulator! This just came across my radar from Hackaday.

It requires that you upload a (legally acquired) rom, but once you do, this seems like it would be a very good ‘on-the-go’ resource for presentations, teaching, or just any other time and place when you might need at graphing calculator.

I don’t have a TI-83 plus rom lying around, and I tried a regular TI-83 rom (which I did happen to have) but it didn’t seem to work for me. Hmm.

Mobile Devices

[updated] I now recommend Wabbitemu as the best emulator on the computer and for Android devices. It accepts a very wide range of rom files and has a nice feature set. The whole process is pretty user friendly. Here’s a link to the app in the Google Play store and here’s their website where you can download the desktop app.

Another great emulator for Android is Andie Graph which can be obtained in the following way (instructions come from our student, Jim Best):

  1. Download the app “Andie Graph” on your market for the phone
  2. Go to this link on the phone to download the ROM.
  3. There will be ads promoting other products so click on the bottom link that says DOWNLOAD.
  4. Once this has finished downloading, Run “Andie Graph”
  5. Go to settings by pushing the little icon on the phone itself that looks like a garage door or a tool box.
  6. Go down to ROM and select the ROM you downloaded. If the app doesn’t find the ROM, then you can search for it from the app in the phone.

Jim also suggests this calculator if you have an Apple product:

There is an app that is a type of TI-83. It is called RK-83 on the app store for apple products such as the iPhone and iPod touch. This is a $0.99 app that has the same functionality as a TI-83. It does not have the best of reviews but for $0.99, its worth a shot. There is also an app by the same creator that has better reviews but it is an 89.

Of course, there are scads of other great calculators out there if you’re willing to give up the look and feel of the TI experience. Desmos is a popular choice and works nicely on all platforms but isn’t a powerhouse of a calculator.

Plain-old Software

And as far as plain-old desktop software goes, here are some great emulators:

I actually prefer the Wabbitemu and Rusty Wagner emulators to the TI-SmartView emulator, even though our school has purchased copies for all of the math teachers.

Rom Files

In almost all the above cases, you’ll need to obtain a rom file for the calculator you’re interested in emulating. This is like the brains of the calculator. The emulator is just the pretty buttons and interface that run on top of the rom.

To obtain the rom file for your calculator, follow the instructions at ticalc.org here. I actually preferred using the instructions here for my TI-84 Plus SE. You can create your own rom file, or you can try to hunt for a download somewhere. This page has a bunch of downloads, but it’s not comprehensive. Remember that, legally, you must own the physical calculator if you download and use one of these rom files.

Why Calculus still belongs at the top

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AP Calculus is often seen as the pinnacle of the high school mathematics curriculum*–or the “summit” of the mountain as Professor Arthur Benjamin calls it. Benjamin gave a compelling TED talk in 2009 making the case that this is the wrong summit and the correct summit should be AP Statistics. The talk is less than 3 minutes, so if you haven’t yet seen it, I encourage you to check it out here and my first blog post about it here.

I love Arthur Benjamin and he makes a lot of good points, but I’d like to supply some counter-points in this post, which I’ve titled “Why Calculus still belongs at the top.”

Full disclosure: I teach AP Calculus and I’ve never taught AP Statistics. However I DO know and love statistics–I just took a grad class in Stat and thoroughly enjoyed it. But I wouldn’t want to teach it to high school students. Here’s why: For high school students, non-Calculus based Statistics seems more like magic than mathematics.

When I teach math I try, to the extent that it’s possible, to never provide unjustified statements or unproven claims. (Of course this is not always possible, but I try.) For example, in my Algebra 2 class I derive the quadratic formula. In my Precalculus class, I derive all the trig identities we ask the students to know. And in my Calculus class, I “derive” the various rules for differentiation or integration. I often tell the students that copying down the proof is completely optional and the proof will not be tested–“just sit back and relax and enjoy the show!”

But such an approach to mathematical thinking can rarely be applied in a high school Statistics course because statistics rests SO heavily on calculus and so the ‘proofs’ are inaccessible. I’d like to make a startling claim: I claim that 99.99% of AP Statistics students and 99% of AP Statistics teachers cannot even give the function-rule for the normal distribution.

Image used by permission from Interactive Mathematics. Click the image to go there and learn all about the normal distribution!

Image used by permission from Interactive Mathematics. Click the image to go there and learn all about the normal distribution!

In what other math class would you talk about a function ALL YEAR and never give its rule? The normal distribution is the centerpiece (literally!) of the Statistics curriculum. And yet we never even tell them its equation nor where it comes from. That should be some kind of mathematical crime. We might as well call the normal distribution the “magic curve.”

Furthermore, a kid can go through all of AP Statistics and never think about integration, even though that’s what their doing every single time they look up values in those stat tables in the back of the book.

I agree that statistics is more applicable to the ‘real world’ of most of these kids’ lives, and on that point, I agree with Arthur Benjamin. But I would argue that application is not the most important reason we teach mathematics. The most important thing we teach kids is mathematical thinking.

The same thing is true of every other high school subject area. Will most students ever need to know particular historical facts? No. We aim to train them in historical thinking. What about balancing an equation in Chemistry? Or dissecting a frog? They’ll likely never do that again, but they’re getting a taste of what scientists do and how they think. In general, two of our aims as secondary educators are to (1) provide a liberal education for students so they can engage in intelligent conversations with all people in all subject areas in the adult world and (2) to open doors for a future career in a more narrow field of study.

So where does statistics fit into all of this? I think it’s still worth teaching, of course. It’s very important and has real world meaning. But the value I find in teaching statistics feels VERY different than the value I find in teaching every other math class. Like I said before, it feels a bit more like magic than mathematics.**

I argue that Calculus does a better job of training students to think mathematically.

But maybe that’s just how I feel. Maybe we can get Art Benjamin to stop by and weigh in!

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*In our school, and in many other schools, we actually have many more class options beyond Calculus for those students who take Calculus in their Sophomore or Junior year and want to be exposed to even more math.

** Many parts of basic Probability and Statistics can be taught with explanations and proof, namely the discrete portions–and this should be done. But working with continuous distributions can only be justified using Calculus.