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Part of my mandate as a teacher of Principles of Mathematics 12 (ie. roughly Precalc, for you Americans) includes teaching mathematical modeling.  My textbook is filled with little subsections that boldly proclaim, “MODELING!”.  It’s one of the mathematical processes that are supposed to be woven together across all of the curriculum content I’m covering.

I am wrapping up a unit on trig functions and it’s time to hook into the “MODELING REAL-WORLD SITUATIONS” content.  Now, I personally have a serious love-hate relationship with trig functions.  Being a graduate of a computer engineering program means I’ve seen them a LOT in my formal education.  Trig integrals have nearly killed me on multiple occasions.  On the other hand, playing with trig functions in an electronics lab is awesome, and being able to visually comprehend trig graphs is probably the only reason I managed to pass a course on Communication Systems.  So part of me really, really wants them to get this.

But there’s one big problem: when I look through the textbook for real-world, all I see is textbook perfection.  The opener they use is tide-level data from Nova Scotia – except that they’ve stripped the real data down to this:

This is a complete and utter fabrication.  That sine wave is friggin’ perfect.

Here’s the reality.

Messy peaks that don’t always line up.  Some kind of weird alternating pattern hiding in them as well that totally makes sense if you stop for a second and think about how far the earth has turned in 12h.

You know what?  It’s not perfect, it’s reality.  And our model, based on a single sinusoid, is never ever going to match that reality perfectly.  And that’s just fine, but for some reason the textbook seems deathly afraid of letting students realize this.  The really ironic bit is that the real data is already incredibly close to the model, and yet they still couldn’t bring themselves to let students deal with even a tiny bit of messy reality.

So that’s what led me to this.  I’ll just start off by saying this image probably only scores a C on the WCYDWT rubric but somehow this kind of worked anyway.

(source: Steam Game and Player statistics)

Opening question was an obvious one: “Which part of the graph do you notice first?”

After students pointed out the weird downward spike in the middle, I moused over that part and talked a bit about the numbers and what this was graphing.  (The site this image comes from has that graph in a Flash applet that gives exact values when you mouseover the graph.)  The story went something like this:

This graph shows the number of users connected to the online PC gaming service Steam over the past 48 hours.  That spike in users online probably represents about 500,000 really ticked-off customers who can’t get at their online game.

We can see they got people back online pretty quickly.  Which is good, because you don’t want to give 500,000 ticked-off time to start posting on forums on a Sunday afternoon.  YOU DON’T WANT TO ANGER THE INTERWEBS.

So imagine you’re working at Steam.  It’d be really nice to have some kind of system that alerts you automatically when something like this happens, because you don’t want to be at the office all weekend watching this.

So, ignoring the programming for now and just thinking about the math … how can we come up with a system that catches this?

Brainstorming session ensued!  Ideas – great ideas! – came from the room and hit the whiteboard.  We started off with four big ideas that were just point-form statements:

• goes down too quickly
• drops below a threshold
• below the avg for that time of day
• doesn’t fit the pattern

This made me so happy.  From there we turned some of these into something we could calculate; we talked about turning these ideas into “math”.  One thing I loved about this brainstorm is that the third item on the list was outside the scope of the class, but at least as good of a solution as what I was guiding them to.

Then we unpacked the big one: “doesn’t fit the pattern”.  What pattern? Can we model it?  Cue discussion / lecture on trig graphing where I showed them how to construct a sinusoidal model, and afterward we checked how accurate it was.  (Not very, but probably enough to fit our task of catching a large drop in user connectivity.)

My own evaluation?  This lesson isn’t a great WCYDWT – it required a lot of me talking, and I had to do some storytelling.  The question wasn’t short.  It got students participating who weren’t normally confident of their skills, but it didn’t get everyone involved.

But I had students asking, nearly begging me at the start of class to wrap up before 7pm to catch the Canucks game.  This lesson went straight through to 7:15 and they didn’t even notice until they were a couple minutes into individual work afterward.  Something must have gone right.

As per Dan Meyer’s curriculum approach: What Can You Do With This?

Live data is at http://store.steampowered.com/stats/ but this image matters.

Edit: The full story is here.

Normally when my brain cross-links game design and education, I try to temper my enthusiasm by remembering that I like to relate nearly everything to games at some point, somehow, and not everyone else has this disease.  But I can’t let this one go.

I’m going to attempt to write about difficulty in game design then talk a bit about the Super Meat Boy design process, namely when it comes to how we approached dealing with difficulty. …

Dealing with difficulty is one of the key challenges I face every time I bring a math lesson into my classroom.  This kind of design analysis gets my attention.

I’ll skip over drawing comparisons to the history of platformer design and the history of mathematics education, but the parallels are there, at least in the caricatured form you hear when teachers gripe.  (“…back before the make-it-fun-and-easy crowd got a hold of the curriculum”, etc)

How could we make a seemingly aggravatingly difficult game into something fun that the player could get lost in?

This is what I can’t let go. This is the question I stare down when I start to question how I’m presenting that next lesson.  This is the question that makes me rethink what I’m doing when I’m writing the next big unit test.

Go, read the article if you haven’t yet.  Then come back.

It’s when I start to look at the solution to the design problem that I suspect Edmund McMillen has it easier than we do in the classroom.

Here are the key points to summarize:

1. Keep it small.
2. Keep the action constant.
3. Reward success.
4. Extend the challenge as people master the basics.

How many of these could be applied to the classroom to improve things?  Where does it break down?  (I’ve got some ideas but let’s get some discussion going in the comments first.)

From William Gibson‘s blog:

A You write from 10AM til whenever. Is research a separate activity?
Q I don’t regard research as a separate activity. From anything. Everything is research. Relatively little great stuff turns up for me as a result of deliberately looking. Life is crowd-sourcing. In a good way.
A The reason I ask is that research tends to wander off into the weeds so easily, especially on the internets.
Q But they hide the good stuff *in the weeds*!

This is so many kinds of good.  Professional development shouldn’t be limited to seminars and workshops.

I can’t completely explain it – either it was channeled procrastination, or my top-down brain demanding more structure. Either way, I found I couldn’t get this first unit test written until I committed myself to a standards-based assessment strategy and wrote up a full rubric for that unit’s learning standards.

I’m glad I went through with it. I was kind of wussing out and just going to put together “the usual” – write tests, add up scores, call that a grade. Now I feel much more in control of how this is going to come together. I’m still unsure of whether I want to expose the system to the students; we’ll see. But at least when students come to me after bombing a test, I’ll have a framework for getting them to demonstrate future mastery of those skills. Even better, it means I can split apart the re-demonstrations into subparts, rather than having them rewrite an entire test at once (which, being a night school course, we don’t really have time for).

What was interesting about the write-up experience is that our provincial standards are *so close* to being directly usable for a solid standards-based (concept-based? whatever) system. They’re already written up in nice bite-sized chunks of understanding. The problem is, they’re totally imbalanced. One unit that takes about 1/7th of your class time over the year contains 1/4 of the total number of standards. Worse, in other grades there are individual “standards” that have subpoints that encompass an entire unit on their own.

Now that I type this up, I suppose arguably I’m making the same classic mistake – assuming that an accurate summative grade should come from simply adding up all of the individual standard marks. But with a nice manageable number of concepts listed, keeping them weighted equally makes the entire system more accessible for students as well as for yourself. Students can glance at their scorecard and know immediately how they’re doing and what to focus on mastering. (In theory.)

After the break I’ll c&p the provincial standards, and my own reworking I came up with including marking rubrics. I’d love to hear critique in the comments! (I pulled a sneaky trick with my third ‘concept’; I can’t decide yet if that was evil or not.)

Provincial “Prescribed Learning Outcomes”:
Transformations
B1 describe how vertical and horizontal translations of functions affect graphs and their related
equations:
− y = f(x − h)
− y − k = f(x)
B2 describe how compressions and expansions of functions affect graphs and their related equations:
− y = af(x)
− y = f(kx)
B3 describe how reflections of functions in both axes and in the line y = x affect graphs and their related
equations:
− y = f(−x)
− y = −f(x)
− y = f −1(x) 1
B4 using the graph and/or the equation of f(x), describe and sketch
f (x)
B5 using the graph and/or the equation of f(x), describe and sketch |f(x)|
B6 describe and perform single transformations and combinations of transformations on functions and
relations

My version:
1. Understand the relationship between horizontal and vertical
translations of a graph of a function and the corresponding
changes to the equation of a function.
_Rubric:_
1: No consistent understanding shown.
2: Shows general understanding but makes frequent errors in
conversion from graph to algebra and back.
3: Good understanding of translations when isolated (not
combined).
4: Consistent understanding of translations in isolation and
when combined with other transformations.

2. Understand the relationship between horiz. and vert. scalings
(compressions / expansions) of the graph of a function and the
corresponding changes to the equation. (Includes horizontal /
vertical reflections.)
_Rubric:_
1: No consistent understanding shown.
2: Shows general understanding but makes frequent errors in
conversion from graph to algebra and back. (eg. changes
compressions to expansions or vice versa)
3: Good understanding of compressions/expansions but has
trouble with reflections, or with combining in other
translations.
4: Consistent understanding of compressions / expansions /
reflections in all forms and when mixed with other
transformations.

3. Understand relationship between graphical and algebraic
representations of the following transformations: absolute
value, reciprocal, and reflection on the line y=x.
_Rubric:_
1: No consistent understanding shown for any of the above.
2: Shows good understanding of one of the three
transformations.
3: Demonstrates good understanding of two of the above.
4: Understands all three of the above, and can sketch the
reciprocal of a function.

Here’s the let’s-get-thinking warm up opener I used for my first Math 12 class last night.  I shamelessly stole Jason Dyer’s idea and turned it into a three-page set of puzzles.

I handed out the first double-sided page, got them going on that, and when people finished that then I handed them the extra hard follow-up.  I had them sitting in groups, gave everyone their own copy but encouraged them to discuss how to solve them.  By about 40 min, almost everyone had solved the first two sides and some were as far as the last (incredibly evil) puzzle.

Afterwards I showed them a quadratic equation and asked how many people felt comfortable factoring it to solve.  About four hands went up.  The rest of them were surprised when I told them they’d already done it.  I unpacked some of the good stuff going on in there a bit, probably got too wordy and I think I could’ve made the transition from puzzle to algebra better – maybe with a “reveal” puzzle that had more of the usual algebraic notation / structure embedded in it. Anyway, whatever, they were thinking and doing math for over half an hour on the first day of a night class – I call that a win.

Here are the files. The PDF files are ready to print; the .svg files are the source files made in Inkscape.  If you download Inkscape (a free, open-source vector graphics program) you can modify the puzzles and make your own fairly easily.  (Cut-and-paste the circles, and there’s an arrow tool to connect them.)

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Canada License.

I just lost 18min of my prep time for Monday watching a Wolfram Alpha researcher give a talk on whether or not using W|A for homework is “cheating”.  The loss was that by the end, the talk had devolved into a false dichotomy of hand calculations vs. computer-based calculations.

I agree that we need to integrate tech like W|A into our classrooms, and more importantly into our assessments.  I also agree that to reach that point, we need to re-evaluate what the goals of hand-calculations are in math curriculum, and probably need to make serious cuts.

The problem of the all-or-nothing is that that kind of thinking has already been abused for years.  Elementary educators who struggle with math anxiety have used the arguments against “rote learning” as an excuse for purely calculator-based arithmetic training.  These students then get passed along and struggle with later work where it’s assumed that you can simply spot common factors because you’re familiar with your multiplication tables.

Does this matter?  Here’s the real problem: any career / lifestyle will carry with it some level of implicitly required mathematical ability in which you don’t want to pull out a computer or even your freaking iPod calculator.  This varies wildly depending on your career, from trades to warehouse work to core math skills as an engineer, but in every lifestyle some amount of rote learning and mental algorithmic skill is irreplaceable.  Math education needs to elevate people’s numeracy to an appropriate level for their life.

This isn’t just an argument for paper-based work; I want to see estimation actually taught well for once.  (Textbooks are inherently horrible at teaching estimation.)

So, don’t pretend this is all-or-nothing.  Admit it’s messy.  Then let’s dive into the real work of figuring out just how much is trash that we need to throw away.

* I’m throwing around big words because it’s quicker, easier, and I have a 1.5-yr-old next to me waiting for me to get off the computer and take him outside.  Sorry.

The update!  I am not only employed as a teacher-on-call, but I am now hired to teach an evening class of Math 12 through the district’s continuing-ed program.  This means a mostly adult group of students, widely varying levels of ability and recent math experience, and a fantastic opportunity for me to teach an upper-level math course.

It looks like I won’t likely have access to a digital projector or any of that other fancy-shmancy edumacational technology.  So I’m focusing on good ideas for how to manage notes with this group so that we can make the best of the whiteboards.  My wife has taught using a modified Cornell notes technique with some good stuff in there; I think I’d mod it further but there’s something worth stealing from there.  These other bloggers’ ideas are also theft-worthy: samjshah’s binder checks, or Kate Nowak’s homework quizzes.

Mixed in there somewhere is my desire to have downloadable notes in some format.  Bringing camera to snap pictures of whiteboards and uploading, maybe?

Also, follow-up to my official unofficial pro-d, Letters to a Young Mathematician was completely fantastic and I recommend it to any and all math educators.  I was going to blog some specific bits of awesome, but that didn’t happen and now I’ve returned the book and am far too busy prepping for Math 12 next week.  (Did I mention it starts this Monday?)

Yes, my head is spinning around full of a dozen innovative ideas I need to experiment with.  No, I am not going to try them all at once and explode.

Now to dive into the review material and see if I can pull a good collaborative challenge out of there somehow…  This is going to be interesting.

First, the good news: I’m officially employed as a teacher-on-call.  This is fantastic news but still pretty surreal; the month and a half between practicum and now feels like it’s been an eternity.  But I’m sure that the first day or two of work as a sub will have an extreme “jump into the deep end” effect and I’ll remember how to swim in no time.

Today is a professional development (pro-d) day; for any non-teachers out there, that means a day allocated for teachers to get further training.  A paid day, if you’re a full-time teacher; just a day without work for me.  It has me thinking about the pro-d wishlist I already have stacked up, in the form of books I’ve started reading, books I want to start reading, video tutorials I haven’t finished working through, etc.  So even if I don’t get through any of these today, I thought I’d get my entire pro-d backlog list down and make myself feel like, hey, I’m actually kind of disturbingly ambitious and I should be happy if I even get through a couple of these in the near future!

Books:

• Elementary Number Theory, Underwood Dudley
  • Started reading / working through; learned about diophantine equations, congruences; lots more good stuff waiting. ps. it’s awesome having a book on your shelf by an author named “Underwood Dudley”. It’s also awesome having a number theory book you got for free, written in the 70′s back when number theory was still an area that was proud for being math-for-math’s-sake with no immediate practical application. (In other words, written before public-key cryptography.)
• The Colossal Book of Mathematics, Martin Gardner
  • Just grabbed this from the library. It’s a great collection of Gardner’s recreational mathematics topics; I expect I’ll read through some select chunks and then return it. Definitely want to finish reading the bits on topology.
• Letters to a Young Mathematician, Ian Stewart
  • getting this from the library today

Online pro-d

• Unity tutorials by Alec Holowka of Infinite Ammo
  • Unity has huge potential for high school Info Tech / Game Design courses as it’s now 100% free!
• Google Sketchup video tutorials
• Getting to Grips with Latex – Mathematics (less on my radar as I’m no longer prepping Latex-based math worksheets in emacs’ org-mode, as I’m no longer prepping math anything at the moment)
• And of course, keeping up on a small pile of math/teaching/software-art/digital-media/etc blogs via Google Reader

Long term:

• Grab my wife’s Abstract Algebra text and learn myself some more maths.
• Topology: anyone recommend a great textbook or other resource to teach myself this?  I keep loving the recreational bits I’ve seen here and there, but wonder if I’m only seeing an incredibly thin slice of the topic and/or if it’s still as interesting as it sounds if I tackle it more comprehensively.
• Eventually figure out the category theory -> monads -> functional programming connection that I caught a glimpse of last summer.

(I have this thing where I feel like I need to fill the gaps in my math training, if I’m going to turn myself into an excellent math teacher.  I have a huge applied-math chunk of training via engineering, but I’m pretty weak on proofs and abstract algebras and all of the other upper-level things that aren’t calculus.  I don’t know how far this will last, but I figure it’s a healthy motivation to nurture.  Even if it only gets me a little ways into a number of advanced topics, I’m sure that’ll help.)

I guest blogged on translating Vector TD (a Flash-based tower defense game) into a math lesson over at Teaching College Math.  Go give it a read!

The other day I found the weirdest source of teacher inspiration: reading MLIA (MyLifeIsAverage) over my little sister’s shoulder at my in-laws’ house.

Today, my schedule got switched around so instead of having history 6th hour I have it 2nd. As soon as I walked in I noticed that my teacher didn’t have a brittish accent like he normally does. After having him for nearly half a school year I learned that he went through the day using different accents for each of his classes. Guess who is now officially my new favorite teacher.MLIA

(source)

There was another good teacher one I read that day, but I couldn’t remember how it went so I tried searching “favorite teacher”.  Guess what?  There are so many “new favorite teacher” posts that it’s already a cliché.  People snark about it in the comments regularly.  I find that strangely encouraging – kind of a weird sign that yes, kids really do want to like their teachers.

Anyway, I couldn’t find the other one I had seen, but here are a few more gems.

During our first lab in my Honors Chem class, my teacher asked me to flip a switch to turn on the fans. There was only one there, but as I flipped it he looked terrified and shouted, “No!! Not that one!!” I turned it off and jumped backwards, scared I just blew up the school, and the entire class turned to stare at me. After a second he added, “No I’m totally kidding. That’s right,” and continued doing his work normally. Hello, favorite teacher. MLIA.

Today, I was looking for my math teacher on the second floor of my school. I only found my English teacher walking into the staff lounge. So, I walked downstairs only to find my English teacher walking out of his room. I sarcastically asked him if he had a Teleporting machine. He suspiciously glanced back and forth down the hall ,stared at me seriously and whispered “You found out, this is our secret now”. then, winked at me and walked away as if nothing had happened. Best teacher ever. MLIA

Students and teachers alike are back in classes here today – and I’m at home sorting out what paperwork to do next so that someone hires me.

There aren’t any externally-posted positions for me to apply to right now other than “Teacher-on-Call” (ie. substitute teacher).  I’m feeling strangely ambivalent towards the idea.  The obvious advantages are that you don’t have to take your work home with you as a TOC; no planning, no prep, no marking, no report cards.  This was sounding really, really good by the end of my practicum, but now that I’ve already had a good chunk of time away from lesson / unit planning I’m less certain.

At the hiring panel during my final week on campus, one district HR rep tried to convince us that TOCing is where we should want to be right now – seeing how other teachers do things and learning from their their tricks.  She had a point, but I don’t think snagging people’s “tricks” is going to cut it in the long run.  I’ve barely scratched the surface at planning and implementing the kind of classroom I want to be a part of.  As a TOC, I’m going to be walking into someone else’s room every day, teaching someone else’s lesson.  Until I spend more time wading into the deep stuff, trying to structure challenges for students that keep them hooked in without boring or breaking them, I’m not really getting any closer to mastering this thing.  I’m also not likely to see any lesson plans that push my own boundaries in terms of cooperative learning, student inquiry, WCYDWT / media-driven stuff, etc.

The obvious disadvantage to TOCing is that kids try to get away with murder when there’s a sub.  (At least I know my class did when I was in high school. But they were exceptional; every now and then when students were nuts in my practicum, I’d stop and remember my own grade 9 class and realize that things could be a LOT worse.)  Again, this is good and bad.  The flip side is that this’ll give me experience in an area I’d like to get a better grip on.  I’ve already had a trial-by-fire which has given me a good head start so I don’t feel helpless or hopeless.  The real disadvantage here, hidden beneath the obvious one, is that I’ll get no experience in setting down long-term classroom expectations and building a good learning environment.

So, meh. First things first, though: time to get hired, pay the bills and get access to internal district job postings.  And if I start to feel really stagnant in terms of planning, I can always get my Moodle server running, pick a course and plan something for the heck of it.