Friday, 21 June 2013

MAT: Memorize This

Ah yes, the debate about memorizing concepts versus enduring understanding. I've had opinions about that for a while. There even seems to be immediate relevance on Twitter (hi @PaiMath!), so I'll spill this out into a post for you, the reader, to think about. Even though I should be marking papers.


1) IT'S UNNECESSARY. This is valid. For many reasons. The primary one being Google. But yes, why should I have to memorize the times tables when I have a calculator sitting right in front of me??

Here's the thing. It takes time to punch something into a search engine, or a calculator. "Sure!", you counter, "all of three seconds!" Consider that a student will need to multiply numbers many times over the course of a week. Let's conservatively say 10 times. 10x40 weeks means 400 times in a year. Which is 1200 seconds. Which is 20 minutes of your life WASTED to verify "4 times 3 is 12".

Why does the student not just know that after the 20th time? Alternatively, maybe they DO, but it's become so rote, so ingrained to just "punch it in", that they'll do it even though they're pretty sure they know the answer! By not memorizing, we're making people second guess themselves - if they were even bothering to think in the first place.

There's also this problem.

2) DIFFERENTIATION. Also valid. There isn't only one way to solve a problem, and we should be able to learn how to attack something from different angles. As long as we're right in the end, who cares what algorithm we use to get there, right?

Counterpoint. To find the vertex of a parabola, there are a bunch of methods, including graphing, completing the square, and partial factoring. So who cares, right? Jump to the following year - you need to change to standard form for a conic. So just complete the square... what do you MEAN you don't know how to complete the square?!

In Grade 10, we built this wonderful second floor, and all the students designed their own floor plans. And it was awesome. But now we're making the third floor, and some of those plans are MISSING a supporting wall. That's a building code violation! "But," comes the protest, "I don't NEED that wall!" Not if you were done building, no, but you're still going.

For those who don't know conics, consider simply comparing two fractions. Standard practice is to use a common denominator. Conventional wisdom is that a common numerator works just as well. Let's move along to adding fractions. How exactly does a common numerator help here?

SIDEBAR: I think this 'differentiation' is what's causing education "gaps" from year to year. It's not that the material WASN'T taught, it's that something that turned out to be rather key wasn't emphasized. Why? Well, teachers aren't psychic, students are forgetful... and no one had to memorize it.

(Disclaimer: Ontario high schools don't teach conics any more. So the above metaphor may be equal parts invalid and ironic.)

3) IT'S TEDIOUS AND BORING. Not to mention probably a bit painful too. Nothing sucks the joy out of anything faster than having to simply memorize a bunch of mathematical facts - for apparently no reason.

Don't you know your history??
Yet there's that old saying, "Those who cannot remember the past are condemned to repeat it". (George Santayana, often paraphrased as 'those who do not read history are doomed to repeat it'.) It applies here. Because if you don't remember "how to multiply by 4", that's 20 minutes of your life gone, per year, which probably amounts to at least a couple hours through twelve years of school - and we're not even getting to using math when shopping at the grocery store.

Sometimes, a little tedium is necessary. Again with proverbs, "Short term pain for long term gain". If you memorize the algorithm for completing the square NOW, you will find things a lot easier down the road! Trouble is, humans are notoriously BAD at helping out their future selves; case in point, a teacher blogging instead of doing their marking.

4) NO LEARNING IS OCCURRING. So true. Machines memorize stuff! Shouldn't we be better than that?

Yes, yes, we should.


Honestly, I don't have a good counterpoint this time. Aside from the fact that you are learning something - you're learning how to memorize. There's something to that, given all the damn passwords our machines require of us these days.

Which is a weak protest. Moreover, since the point to education IS learning, we'd better kill all the memorization, never mind any of those other counter-arguments! Or so the story goes.

But not so quick, memorization lovers. You're on shaky ground too.


1) EFFICIENCY. "If you wish to make an apple pie from scratch, you must first invent the universe." (Carl Sagan) Screw that, I've got all the ingredients memorized already, lets bake the sucker.

By doing that, you're missing out on so much. The universe, in fact. You're also stuck eating apple pie forever, unless you take a step back to see there might be other fruit options available. Or what if an ingredient is missing, temporarily forgotten? Do you know enough to substitute?

Turning it back to maths, there was a student some years ago who was really good at remembering things. This may have been his go-to method... until Grade 12, when there was finally too much to commit to memory. And without seeing how all the pieces interlocked along the way, what was there to fall back on? For that matter, was there even an alternative method available to the student?

2) USEFULNESS. You will need this information later in life! Remember, if you don't know completing the square, you'll be in trouble when we hit conics!

Because yeah, every student in the world is going to take the Grade 12 course that has conics in it. (I grant that many parents FORCE that decision, but let's try to be realistic.) If you can predict exactly what a Grade 5 student is going to need five years down the road, either you've got their life mapped out for them (you're doing it wrong), or you somehow know them better than they know themselves (people change).

Pictured: Not a fertility goddess
Back to the metaphors, if a building is only going to be one story high, you don't need to worry about installing a staircase, no matter how "useful" it would be. "But," comes the protest, "What if you later realize you want more floors??" Well, then it might not be a bad idea to consider a redesign of the ground floor too. Because displaying that life-size statue of a fertility goddess by the entrance may no longer be as high of a priority either.

3) FLUENCY. If you don't know the basics, how can you understand anything beyond that? If you want to build high, you need a solid foundation!

Okay. So at what point is it no longer a foundation? More to the point, when do we actually build the damn house? Because, oh my God, if counting is the foundation for addition is the foundation for multiplication is the foundation for area is the foundation for geometric properties is the foundation for algebraic proofs, kill me now.

That's not fun. Maths IS fun, and playing around with things. Or it's supposed to be.

I wonder if part of the social stigma on maths is that, after 50 years of memorization, people have forgotten it even can be fun. You wouldn't hear people say, "Yeah, I never got the hang of having fun" - hahaha, big joke.

Frankly, if the only real justification we can provide for memorizing is "You need to know the fundamentals!": 1) no you don't, see above; 2) apparently you're psychic, see above again; 3) true motivation has to be internal, not external.


At the risk of overgeneralizing:

Memorization comes from a time when we couldn't look things up very easily, when complex calculations necessitated a slide rule, and when we were assured that knowing math facts would get you a good job. People bought into that because, well, I suspect it was mostly true. But that time has passed.

Standards based mathematics now exists because calculators and computers have made things easy if all you want is an answer, because society has made "math sucks" a thing, and because there is no guarantee that knowing math will help you become employed. People (more importantly, students) won't blindly buy into memorizing any more.

And really, in the end, IS it necessary? No. But is it HELPFUL? Oh yes. Thus the best plan would seem to be, have the students learn through standards, and then once they've got the understanding, commit that stuff to memory.

Good thing there's absolutely no chance of people getting mixed messages.

Something I tweeted earlier today.

Thursday, 20 June 2013

MIX: Yi Am On Twitter

Welcome to my Yi series, installment 2. Circumstances prod me to continue the series before the summer. This time it's because I want to explain who I am, at a time when it feels relevant among people I know on Twitter. Also, I've been tweeting for 51 weeks now - so the upcoming anniversary is as good a time as any to look back.


Publicity stunt.

Yeah, I didn't join twitter for the math pedagogy ("Math-Twitter-Blog-O'Sphere", aka MTBoS), though it's mostly where I've ended up. That said:
1) I was aware of the math community prior to tweeting, through Professional Development sessions. It seemed a bit overwhelming.
2) I've since concluded that people don't follow you for what you do, they follow you for who you are. More on that below.

As implied, I joined in to promote my math fiction blog "Taylor's Polynomials", which is about the personification of equations. (What? Not a real-life blog with lessons? Yeah, I really don't fit into your traditional boxes.) I'd been running the stories for precisely a year at that time.

Some of the accounts I followed first (aside from friends) were Dan Meyer (duh), Rick Mercer (I'm Canadian) and Mr Burke Math - because he writes an awesome and clever webcomic about math, which is sort of what I was doing too. You should check him out, by the way. He's currently blogging about the Regents exams.

Through July and August 2012 I sent out 134 tweets (70 were retweets). In that time, I accumulated less than 10 followers. Thanks to information on this website, I was able to learn how to access my twitter archive, so here's a few of the things I sent out:

First two tweets ever; reverse order
Sometimes I would tweet out links to news articles
I started this second (non-fiction) blog on August 3rd 2012
Why yes! I did make a video for Mystery Teacher Theatre 2000.
I even stumbled into a GlobalMath discussion

And through it all, I would be tweeting updates to my personification of math web serial (which at the time, I didn't even know to classify as a "serial"). After 60 days of this? Not even 10 followers. Comparatively, I was following at least 40 or 50 others by then. THAT SAID, I was rarely tweeting AT anybody, I was merely broadcasting. You get lost in the shuffle doing that.

Then there's what I said earlier. People don't follow you for what you do, they follow you for who you are.
Enough. No one cares.


From September to June, teaching is my life. I'd be deluding myself to think I was anything but a teacher, because it's so all consuming, even when you don't factor in the chances of running into students on the weekend. It also didn't help that the Ontario Liberals legislated all educators back to work last Fall, even though we weren't even in a legal strike position. Go solidarity.

I'm either referring to me or the government. Or both.

So given my absolute failure to garner a fiction readership, I started to follow more teachers. (Of the maybe 50 people I was following then, less than 20% were teachers.) Then on October 21st I had to stop my web serial for personal reasons, so let's just say that freed up some extra time too.

By November I was actively tweeting back at folks, and commenting on blogs. (Tina Cardone with her "Day in the Life" puzzle was, I think, my first.) I eventually resumed my Web Serial to start 2013... and yeah, to this day, it receives about half the hits that this blog does (maybe 20 there, as opposed to 40 here). But honestly, that serial IS what brought me in! It's also the reason I now juggle math followers with writers and serial accounts. Though is it still the only reason I'm tweeting today?

Not so much.


Yeah, it didn't always do that. I used to be "A HS Math Teacher with SciFi leanings who tweets - because equations are people too". But in March 2013, when I realized what the heck I'd actually been DOING so passionately for the last two years of my life (ie- writing a serial), I brought it to the forefront.

Before you attack the gazebo, find out if it's bigger on the inside.

In her "Infinite Tangents" podcast, Ashli has a set of questions that she often asks of her guests, one of them being "If you weren't teaching, what would you be doing?". I am fairly certain I'd be an editor; perhaps even a writer. In fact, the story of how I got into teaching will be the topic for a future Yi, because some might find it interesting.

Of course, you can only be successful as a writer if people actually READ what you write. Would anyone read my stories? Well, the fiction isn't working out (yet), but... these days, at least some of the time, people are reading up to 140 characters of mine. And there are times when I know that those messages are making others think - or even better, smile.

That's why I am on Twitter.

Becoming a better teacher myself is honestly more of a fringe benefit. I suspect I'm actually at my BEST when I'm on the sidelines of the "MTBoS", pointing people towards the mainstream, or highlighting stuff cooler than me, or even just making observations about what it is I'm noticing. Being the weird outlier.

The only downside is the occasions when I see no comments and feel rather lonely.

Sunday, 9 June 2013

TCH: Teachers Work 8 hr Days

Where "teachers" is defined as me, "8 hr days" is defined as the mode, and the month in question was May 2013. Let's break it down further, shall we?

The mean workday is 6.77 hrs, the median is 8 hrs, the mode is 8 hrs.  But the first thing to notice is that this is for ALL 31 days of the month, including weekends. In that graph, every day under 6 hrs was work done on a weekend - with the exception of 4 hrs of work on Holiday Monday.

Since no one works EVERY day (even I have two days at zero), we equalize. Wipe out the eight Sunday/Saturday data.  The month is left with 22 "working" days (23 if you include the holiday).  The mean is now 9 hrs, the median is 8.5 hrs, and the mode, well, still 8 hrs.  Except there's now 12 hours of work unaccounted for, so we put that back in... and the mean "work day" rolls up to 9.55 hrs.  (If you're wondering what counts as "work", I'll refer you back to the rules in the post that started this venture.)

That actually feels slightly LOW as an average, when you take into account:
1) I have a tendency to short change myself when rounding off to the half hour.
2) The days were all pretty typical, with a 4 hr day balancing a 12 hr day. The true "skewness" of a teaching distribution is better seen around report card time, with lots of high impact days.

Also, again, this is me.  Newer teachers may have to work more, people who get to teach the same course twice in a day may be able to work less.  I have NO idea what's typical.  Feel free to speak up and tell me I'm a slacker, or a workaholic.


Now, it was suggested previously that I break things down by day, to seek patterns. Good idea. As May had more "Wed/Thurs/Fri" days, I'll take the individual daily totals and divide by either 4 or 5.

Monday is a bit lower, but again remember there was a holiday in May, when I worked but did no teaching.  Observations:
1) Tuesday seems to be my powerhouse day; I start to trail off towards the end of the week.
2) I *SEVERELY* overestimate the amount of work that I can accomplish on weekends. Again, possible personal bias in that weekend work tends to involve multitasking, so I invariably chop my times in half... but seeing this makes me feel like, heck with it, I should just stay later during the week.

Given how the month ran to a total of 210 hours worked, I next became curious how much time I actually slept in May. If we assume 7 hrs a night for 31 days, that's 217 hours. Which is probably low - there are some weekends I sleep in - but if we add in my commuting time (20+ mins twice a weekday), one could probably say I'm involved with work/teaching just as much as I am sleeping.

The "Social Media" wedge there is for 1.5 hrs daily, on average, which might also be low.  I also publish a mathematical web serial, which sucks a couple hours out of me per week, but I suspect is even less than the commute, so would be in "Other".  As is stuff like "meals", "lawn care" or "doing the dishes".


Okay, just for kicks, here's the numbers for the last 2.5 weeks of April: 165.5 hrs worked.  Mean 8.7 hrs, Median 9.5 hrs, many modes (4, 7, 11.5, 14.5).  But that's across 19 days.  So, equalizing again to only "working" days gives (ready for this?) a mean of 12.75 hrs per day.  Because of report cards and the school play alongside everything else.  But I wasn't recording yet during the first part of that month, so it's kind of artificial.

Now, June only has the report cards, no play, so these last three weeks should be a piece of cake or something, yeah?  ....  Oh wait.  Exams.


Saturday, 8 June 2013

ETC: Being The Outlier

Here's the thing about social media - it's wildly inaccurate.
1) The friendship paradox ensures your friends are more popular than you. They experience the same effect. It's maths.
2) The ingroup bias ensures we overestimate the value of our immediate group at the expense of unknown outsiders. 
3) There is also a natural tendency to FB share and TWTR retweet items only at the extreme good or extreme bad ends of the spectrum.

This all leads to what I'll dub the "sidelines" effect. You're in a group, and the awesomeness of it is happening all around you, and you're contributing... yet somehow you're not quite a part of it.

I posted this up the other day:

At the time, I quantified myself back in a wishy-washy 140 character kind of way, but I think more needs to be said. Because I'm often an outlier not just in the "Math Twitter Blogosphere", but in a more general sense. More to the point, I suspect I'm not alone.

For instance, when someone in the last days of teaching says to me "Solidarity!" my knee jerk response trends to "Liquidarity!" or even "Sewing Machine!". Which means you, as the other party, are perfectly within your rights to nod slowly and go talk to someone else.

After all, it's the sort of thing that probably works better in person, when you can hear the tone of my voice and see the expression on my face. (Assuming it works at all.) That's hard to convey in text, and is impossible for you to picture if you only KNOW me through Twitter or Facebook. So, more often than not, I'll hold my tongue.


I won't.


Not seriously.


See, I like provoking that sort of reaction. When I was young, it kept the bullies off balance. Now that I'm older, I use it more for entertainment purposes (and because I can't always turn it off). It also cushions me from the occasional harsh realities of life. For instance, the bleakness of how alone I so often feel because, oh God, I'm so ignored and people don't GET me!!

Kleenex! Sewing machine.

But yes, in all seriousness, I am changing. So are you. We all do, we all MUST change and adapt, otherwise we're not learning from our mistakes. But it's not always on a conscious level, and it tends to happen because of one of two things:
1) You're changing for yourself, not for other people.
2) You're changing in small bite-size pieces. Otherwise you implode.

My blog is one example of this. Back at the end of March I posted up "Why Do You Blog?", essentially a rebuttal to Michael Pershan's "Do it for others, not yourself". And I got comments, both on that post, and through Twitter, which forced me to re-evaluate, and re-design how I post. I DID want to help others, so change happened.

Naturally, within a week, the world went silent again.


As per my point 3 above, when something is really BAD, you tend to hear about it. (For instance, whatever common core is, it needs to be stopped.) Or when something is really GOOD, you tend to hear about it. (Nguyening.) Everything else is just... out there. Maybe garnering a 'like' or 'favourite'. Until it either finds a megaphone, or gets a following, or receives enough feedback to morph into one of those extremes. Before that, you're participating, but off on the sidelines.

You're me.

And that's another reason why I get random sometimes. I know I'm more than mediocre. I'm more than just one guy in the group!

Yet I AM just one person, and all the good social media things REQUIRE the group. There's feedback, there's collegiality, and you become more than the sum of your parts. But I just said that's NOT me. I am NOT good in groups, and I'm certainly not a leader. Hence, the "sidelines" effect. Hence why I say I'm an outlier.

I also grant that I don't have the feedback knack... randomness aside, I generally only speak up when I have something to say that's particularly (a) profound, or (b) inane. (See, even I only do extremes.)

In conclusion then, this post must fit into one of those two categories.

I'll leave it up to you, the reader, to decide. Have I been able to convey how someone can feel perfectly alone in a crowded room, merely standing by on the sidelines as the group dynamic awesomeness continually happens around them? Or does this post come across more as a pretentious hack whining about how nobody truly "understands" them?

For something more profound: My Choose Your Own Exam
For something more inane: Seeking New Personifications
For something with more violin: Lindsey Stirling Music Video

Because Transcendence. The violin there also works as a pretty good instrumental background while you're reading this post. Damn, that probably should have been my thesis...

Wednesday, 5 June 2013

MIX: Yi Write Series 5

Welcome to my Yi series, installment 1. I had planned to start the "Why I" series in the summer, but circumstances would seem to warrant an earlier entry.

Because at a rather difficult time in the school year, I'm writing about suicidal tendencies.


Let me start by laying out the following:
1) I don't currently have suicidal tendencies.
2) I used to. (Write what you know, eh?)
3) I am going through a rather difficult time right now.

Part of me was wondering if I should even bother to lay this out, given the limited number of people who keep up with my web serial. But seeing as it's probably my friends who do... yeah, you're the ones who are important to me, and I don't want you getting the wrong idea.

First, the timing is really rather coincidental, even apropos. I wrote about the parabola and her knife almost six weeks ago (I keep a buffer for the story), and her hints stretch back even further than that. I didn't feel particularly bad at the time I wrote it. Circumstances changed.

Current context: The reason I'm feeling like life is difficult at the moment comes from a few prongs. We're getting closer to exams, so lots of wrap-up tests, meaning more marking and more students stressing out over their current mark, all of which downloads into more work and stress on me. I also have to finish setting my exams, and summatives, for three separate courses AGAIN (there were some timetabling changes back in September), and thank the gods for my colleagues or I'd be beating my head into a wall, no question.

It doesn't help that after a year of trying to manage things locally for Cappies and the theatre, I have absolutely nothing tangible to show for it. Oh, tons of intangibles, lots of good feelings and the like, but nothing physical. I wasn't even in the show programme this year (of all years, when extra curriculars were going to hell) so yeah, when it was the cause of my one major spaz for the year, little depressed about that.

All of which is to say, I'm very much down on myself for my inability to accomplish what feels like the simplest things. Like motivating students (which actually isn't so simple), marking papers (I'm blogging now rather than doing it), and running clubs (having technical issues among other things). I can't even walk across a room without running into a chair.

But I'm not about to take real life as far as I do in Taylor's Polynomials: Series 5. Where math is taking depression to the next level.


That should be obvious. Because it's hated. And when you're hated enough, you start to hate yourself, and to think that if you weren't there, everybody else would be happy. Which is a lot of nonsense, but not when you're at the centre of it.

In retrospect, a better question might be, how can one personify mathematics and NOT have it become suicidal? At least, given the culture we're living in today.

Parabola has become the actual voice for it. Partly because she's one of my original three, but also because I think she's my favourite, and I have a thing for torturing my favourite characters. Also because students seem to hate her more than any other function. Oh, and because of ViHart. In particular, the way one of her videos built up cardioids by metaphorically slamming the parabola to the ground and stomping on her. (Bullying much?) I've actually referenced that video in my serial too, way back in September. 

Things haven't been any easier on Para since then. I foreshadowed the knife. It's subtle, look for it. And things will get worse for her.

One of my math characters is going to die.


Sorry. To make it this far in the post, I figured you were reading already.

See, after two years at this, I've kind of accepted that personified math is not as much of a turn on as personified history.

Moreover, I still haven't found my audience. I think I've determined that it's NOT students, or math teachers, or writers... so for now, it's still me. Me and the few people who enjoy the bizarre way I think.

In the back of my mind, I'm also telling myself that I'm ahead of my time. That in about two years, math will be more interesting, and people will have the desire to scroll through years of backlog. (At least one mentioned they've tried - hi Audrey M, if you happen to be reading!) In the meantime, the writing is vaguely cathartic, and also, I hope, socially relevant.

That's why I write Series 5.

By the way, anyone notice that, along with the depression angle, there's a subplot dealing with same sex couples? Yeah, that's not only fan service, it's going somewhere. Where? Well, that's a good question. My buffer runs out in two weeks, I need to refill it.

In the meantime, I teach. And occasionally suffer from low self esteem.

Such is life.

"Back up. One of your math characters is going to DIE?!"