Tuesday, September 4, 2012


This summer, I was a TA for a (awesome and super-fun) pre-algebra class.  One of the questions that my students asked me frequently is what exactly IS pre-algebra?  This was a question that I never had a good answer to.  I could tell them what's in the curriculum, but that's what the textbook's table-of-contents is for.  There's no mathematical discipline of "pre-algebra."  You can't be a pre-algebraist (though I suppose you might be a post-algebraist some day.)*

*The "pre-____" class titles in general rankle me.  They devalue whatever you're actually trying to teach.  

This brings me to the Pre-Calculus Dilemma, or what do we put in-between geometry and calculus?  I certainly don't remember what I took in those two years.  It clearly wasn't anything that fueled a passion for mathematics (how that started is another story.)

Certainly some parts of the alg2/precalc curriculum are valuable and necessary as prerequisites for further studies.  But a lot of it is just silly.  Particularly if you're a student who doesn't plan to go on to take calculus.    One way to fix this might be to shift the balance towards modeling real life situations - NOT fakey from-the-textbook "application" problems, but real situations.  In this way, mathematical understanding can be rooted in the physical world.  Using math to model real-life situations gives it meaning.  A learning cycle in such a class might look like:
  • Look for questions around you
  • Develop the math necessary to form a model
  • Look for other situations in which this new math can apply
  • In those situations, find a need for more new mathematics.  GOTO 1.

A natural source for situations to model is in science, but there's also opportunities for analysis in the social sciences, or literature, or problems in the school's community at large.  

I think that such a program of modeling and investigation, combined with some of the traditional elements of the classes could be much more successful at giving students an understanding of what mathematics is and how it is used.  And isn't that the whole point?  

Friday, August 24, 2012

My Favorite Lesson - "How Big is Chicago?"

This is one of the first activities I developed entirely on my own, and the first one I ever taught to an actual class of students.  I've now done it four times, and it's been met with mostly success.  It's an activity that I really like and plan to continue improving.  It's worked very well with classes of 5th-7th graders, but can be adapted for older students.  I've fit it into a 50-minute block, and while time is tight it's definitely doable in that timeframe.

How Big is [Your City]? 

What you'll need:
  • Rulers
  • Markers/drawing supplies
  • A class set of maps (at least one per student or group, but make extras in case they get really into it!).  I use maps of the City of Chicago because that's where I've taught the lesson.  You can [should] use your own city/county/state/ward.  Interesting shapes are better than squarey ones.  More on the maps:
    • The maps need to have a scale.
    • You might be tempted to use large paper for your maps.  It's more trouble than its worth!  Maps on 8.5"x11" paper are just the right size, and are perfectly legible.  
    • Try to get as "clean" a map as possible.  Students will get distracted by whatever miscellaneous printing is on the map (roads, El lines, district numbers, anything!)
    • This is the map I use  [PDF alert!]. I have a paper copy that has the city border outlined that I use for making student copies.  It's not perfect, but it's the cleanest one I could find of the city that also had a scale.  
Start by asking students some various ways to measure the "size" of a city.  Answers here can vary, it's more to get them thinking.  Answers have ranged from the standards of population and land area to thoughts about economic power, tourism, or a city's fame. (this part can be cut for time.)

Now, introduce the task - use the map to estimate the land area of your city.  The problem solving skills of the students will probably influence the amount of guidance that you have to give from this point.  If they're pros at compound shapes and calculating areas, then just hand out the maps and let them roll.  Otherwise, this might be the time to introduce some vocabulary like partition, dissection, approximation, or maybe work a quick example with a simpler map (this part is where I need to do some work before teaching it again.)

Hand out the maps and supplies.  The goal here is that they'll start drawing rectangles / triangles / parallelograms that approximate the edges of the mapped area.  This is where the spare maps come in handy, because they will want to re-start their drawing, either because of a mess-up or because they thought of a better way halfway through.

Once they've got a partition that resembles the map outline (you can have them check with you before proceeding,) they should start measuring the shapes and calculating areas.  I find it's easiest to get a total area in square inches or centimeters, then do one unit conversion at the very end.  Encourage students to write their measured dimensions directly on the map - it'll be easier for them and you to find errors if all the dimensions are in one spot rather than scattered around (calculations should be on separate paper.)

The scale conversion has presented a problem for some of my students.  Most of them get that on the map (my map at least) 1 inch = 3 miles, but some then try to use this same factor to convert square inches to square miles.  One way that helps explain the conversion without algebra is to draw a picture like this:
I think a lot of students struggle with the idea of what exactly square/cubic/etc units are, and how they compare to linear units.  This activity might help them figure out the difference between them.

If you've done it right, the kids will be burning to know what is the actual area of the city?  I like to show them this list of large US cities.  It's particularly nice because you can sort by population, land area, and pop density.  Beware that this list doesn't include small-but-large-area cities (the top 4 US cities by land area alone are actually all in Alaska [as per this list].)

I generally conclude at this point, but here are some extension questions I've pondered:
  • Using your method, how close do you think you could get to the actual area?  How many shapes would this take?  
  • How would you design a computer program/algorithm to calculate areas using this method?
If you're tech savvy and have the resources in your class you might try to import the map file into Sketchpad or Geogebra so that students can dynamically adjust and recalculate the areas of their shapes.  In this case, the activity becomes less about measuring and precision and more about getting the partition just right.  You lose the kinesthetic aspect of physically drawing and measuring, but gain the ability to experiment with many more types of shapes.

Let me conclude with two examples of student work.  This is from the first time I taught the lesson, and most of the class didn't actually make it to the measuring phase.  One pair made it all the way through and got an extremely close estimate for the land area.  This is their map:
These students even went back and refined their original number by subtracting the area of the small triangle in the bottom left.  The map tells you that they had a problem-solving plan when they started drawing.  

This second map is from a group that didn't make it through to the end calculation, but it illustrates some of the pitfalls of doing this activity:
Particularly in the top left you can see how they started to make shapes fit willy-nilly rather than focusing on the end calculation.  These students could have used more support and guidance as they worked on the project.  

Thanks for reading if you've made it this far.  I hope you liked this activity as much as I do.  Please let me know your thoughts/suggestions in the comments.  

Thursday, August 23, 2012

What I Learned From the Guy Playing Guitar at 3AM in the Amtrak Dining Car

So I was on the train from Boston to Chicago, and (it being a train rather than a bed) I couldn't sleep.  I headed down to the dining car, and found a few restless folks reading, and one guy loudly playing guitar.  In between his second and third renditions of Folsom Prison Blues, he was talking about writing songs.  One time, he said, he'd got access to a bunch of nice guitar gear, but try as he might he still couldn't play the song that was in his head.  He said he'd been so frustrated that called up his dad (who was also a guitarist) and yelled at him "you did this to me!".  He was stuck, but that didn't stop him from playing on, trying to get that song out.

I'm not a musician, but I know that feeling too.  It's the same feeling I get when I'm trying to learn a new math concept.  It's horrible, knowing that the key that will unlock new understanding is just right there, but there's some kind of block - maybe I'm looking at it in the wrong way, or missing some previous point, or maybe just not in the right frame of mind.  But I endure the frustration because I know that it all become worth it when I finally do get it.

That's also the feeling I want my students to have.  Doing math in school isn't about the answers, it's about the process.  About knowing how to deal with a problem that seems impossible.  About experiencing defeat followed by the thrill of success.  About not giving up.  Then it is our job as teachers to deliver the kind of problems where students can have these experiences.

In this video, Paul Lockhart describes mathematical arguments as "reason poems."  I think he's right - math is more than just solutions.  It's an art.  It's no coincidence that I identified on some level with the frustrated guitarist - after all, we're both in the business of creating beauty.

Edit/PS: I experienced one of those moments of mathematical revelation just this morning while reading the Wikipedia article on Hypercubes.  I'd seen the cool/trippy animations, and heard explanations before, but never really got it.  It was this picture that did it:

It's amazing how just one change in perspective can bring on whole new worlds of meaning.  Yay math!

Sunday, August 19, 2012


Hi!  My name is Kyle, and this is my first. blog. post. ever.  I'm really thrilled at this nice welcome mat that the math ed blog community has laid out for folks like myself.  Thanks to everyone (even though I don't yet know who most of you are!)

In terms of teaching experience, I'm a young-un:  I start my student teaching at a high school in Chicago in a few weeks.  I'm really excited to finally get to use all of my ideas with actual students.

As for this blog, I plan to use it as a bit of a reflection space, and as a place to show and refine ideas about how and what to teach. 

My first post for the Math Blogger Initiative is below, and explains the name of the blog.  Hopefully it's interesting :)

What's a k-gram?

Have you ever had a lesson just rolling around in your head?  Just lurking, waiting, for the one day that you finally get a chance to teach it?  For nearly two years now, ever since I started my teaching program, I've had a half-formed idea for a project/lesson/unit on stars.  Not the ones in the sky, but the ones you can draw on paper.  You're probably familiar with five and six-pointed stars, but stars can also be drawn with any (natural) number of points.  Now 2, 3, and 4 pointed stars aren't very exciting [try to draw them!], but when you get up to seven points a neat thing happens - you can actually form multiple types of stars!  (Try it if you don't believe me - draw two sets of seven points, then draw a star skipping one point each time.  Draw another star skipping two points each time.)  In general, shapes like this are called n-grams (or pentagrams, heptagrams, dodecagrams, etc.)  There's lots of neat math that you can do with these shapes: angle sums, prime numbers, modular inverses.  There's a lot of material that can make a good lesson here too - it's numerical, visual, and tactile.  You can draw a star, or make a star with sticks and string, or even make a human star. 

Of course for now these are all just ideas, that I've not had the chance to put into practice.  But they represent something that I can aspire to in my teaching - to make my lessons investigative, artistic, interactive, and mathematically rich.  Maybe one day I'll even have the joy of teaching real actual students about n-grams!  When naming the blog, I couldn't resist substituting k for n, since it's my initial and all.  I also like the association with a tele-gram - a form of communication - which is what I hope for this blog to be: a way to send ideas and feedback back and forth between others who think math and teaching are awesome and who want to deliver the best instruction possible.