Monday, July 30, 2012

#Made4Math: Poster Channels

Don't expect a lot of these: it's not my thing.  BUT I LOVE this killer idea and I'd be a cruel troll not to share it with the world!


Well, if your school is anything like mine, you have a stack of construction paper laying around.  Probably 12" x 18" and looks something like this:

And if your school is anything like mine, nothing ever sticks to the wall.  If it's not extremely caustic and permanent, it will fall down.  Even duct tape won't work.  Hot glue might be tolerated, but we all know the custodians will give you an evil eye for using it.

Years ago I bought a giant melamine board and cut it into 1' x 1' squares so I'd have a classroom set of white boards (a la THIS).  When I bought that I walked past a bunch of J-Channel and I'm pretty sure my mind asked itself #wcydwt?

Of course, the J-Channel is designed for paneling, but I got to thinking that it might be a great way to hold vocab on a word wall.  So I bought some of the 8 foot J-Channel and a few double-sided pieces.  Sadly, at the time, all they had was brown in stock, so I got brown:

I put them on the wall with caulk and/or hot glue a tiny bit more than 12 inches apart - so the construction paper will fit into it easily.  The result:

Over time I have laminated my vocab word posters and any time I need them I just slip them into the poster channel.  I have them in different parts of the room for different classes, if need be.  It's pretty great.

I'll give this warning: some of the channel may be warped and you need to watch out for that. Make sure it's straight.

So there you have it.  What might be my one and only Made 4 Math Monday.

Friday, July 27, 2012

When is Imperfection OK?

This is not a Khan reaction blog, so I doubt I'll say much on the subject for awhile.  After this post anyway.

I am not a Khan hater or a Khan gusher.  I'm just a guy excited about possibilities. Christopher Danielson and Michael Paul Goldenberg recently wrote a critique of the Khan Academy for Valerie Stauss titled How well does Khan Academy teach?  In that piece, they assert that Khan lacks pedagogical content knowledge (PCK).  They also claim:
(1) the examples Khan chooses appear selected at random and thus are, perhaps unsurprisingly, often quite poor. They are prone either to create further confusion, or to fail to address fundamental questions students are likely to have; and 
(2) Khan’s explanations are frequently off target in addressing likely student questions that experienced teachers would anticipate and elicit.
I agree with Danielson & Goldenberg.

The two go on to illustrate Khan's shortcomings within the subjects of decimals and using an equals sign, and describe how his lessons perpetuate misconceptions.

I do feel like it's important to stave off these misconceptions.  As a high school teacher it drives me crazy when kid thinks a rectangle can look like this: , but not this:  ("that's not a rectangle it's a square, dummy!").  I hate it when kids use equals signs improperly.  And I will probably lose it the next time a kid tells me that 42=8.

With that in mind, I ask this question:

Is it better to educate more people with an inferior product, or fewer people with a better product that limits misconception?

And after much debate inside of my brain, I think I am leaning toward the former.

There is no question that Sal Khan is a novice teacher that teaches imperfect mathematics.  But he is making that math accessible to more people than ever before.  And I think I believe that if his students learn enough imperfect mathematics to open doors for other learning, it's a worthy trade-off.  So for the student who is not  going to need a deep understanding of math in her career, KA is doing just fine.  There is little or no harm done here.

A student who will need mathematics for their career, however, will want a more perfect teacher who squashes misconceptions and allows for deeper understanding.  And that student will most likely get that through their rigorous high school and college coursework.  So Khan is irrelevant to that student's learning.

And of course, as KA gets better and better (and there's no reason to think that it won't), and as the technology proliferates to higher-quality instructors, the end-game here is a high-quality education accessible to all at their individual pace.  It's coming gradually, and that's exciting.  And when it does, the debate inside my brain will be moot.

Thursday, July 26, 2012

Processing Khan: Taking Lessons from Others

Turns out nothing drives traffic to your blog these days like posting a pseudo-defense of Khan Academy.  Wow, this subject really gets people fired up!

Me, I just get annoyed or intrigued.  I totally get that Sal Khan is a novice teacher, but that doesn't really fire me up.  I get it that he and others over-hype his site from time to time, but that doesn't fire me up either.  I get annoyed with the ferocity with which some in the teaching establishment criticize KA, and I am intrigued by the possibilities that the platform offers as as we skid into a future less defined by seat time. 

How we move forward as a community is a centerpiece of a discussion Robert Talbert and I have been having.  I suggested that the educational establishment should find ways to guide Sal Khan and KA and he suggests another route:
I think what KA really needs is competition. ... If KA had even one or two comparable alternatives that did take a more intentional approach and did use sound, research-tested design principles in its design, then we could stop focusing on Khan Academy so much and get on to the work of improving student learning. It might even force KA to step up its game.

So we come to this:  We all have a goal of the best possible math education accessible to all.  What's the best way to achieve this goal?

To inspect this, it's useful to look at Khan through the lenses of some other innovations that have changed established products or practices.

LENS: Khan Academy is Napster and the Educational Establishment is the Record Industry

Background: In 1999, Napster was released and it seemed that overnight everyone was using it.  Like Khan, it offered a product that was inferior to CD's (overcompressed sound files) and like Khan, it distributed this inferior product to many more people than thought possible.

The record industry sued Napster and forced it to shut down.  Though it tried to convert its free file sharing service to a subscription service (in many ways a very early version of itunes), it could not adapt fast enough and died.

Lesson: The educational establishment doesn't have a monopoly on content or pedagogy so a lawsuit or cease and desist order is unlikely to say the least.  We can learn from the technology, however.  Although Napster died, file sharing has flourished since its demise.  This method of distribution has gotten better and better.  The record companies now embrace the technology and have made their product available through iTunes and others.

Perhaps the same will happen with Khan.  The educational establishment will adopt the method of mass distribution, but offer a better, more pedagogically sound product.  I really do believe that if KA doesn't adapt quickly enough, it will die.  Because, frankly, we're right on their heels.

LENS: Khan Academy is Walmart and the Educational Establishment are Mom & Pop Stores

Background: Walmart comes in and squishes independent stores out of the market.

Lesson: Oddly enough, the educational establishment is disconnected enough to be sort of like a loose network of independent mom & pop stores.  In a lot of ways, we all kind of do our own thing and get squeezed all the time by the latest fad.  Only relatively recently have we begun to collaborate (both locally and online).

I think there is pretty good evidence that brick and mortar schools are not going anywhere any time soon.  Like Walmart, Khan offers a cheaper product more cheaply, so we as educators should take a good look at what our value proposition really is.  Right now it's quality.  Khan's got us on service.  We need to improve our ability to individualize and distribute but maintain our quality.

And oddly enough, Khan's priorities are inverted from ours.  He needs to maintain his ability to individualize and distribute while increasing quality.

LENS: Khan Academy is TiVo and the Educational Establishment are Company-Owned DVR's

Background: TiVo first emerged in the early 2000's as the leading DVR on the market.  As cable companies and satellite providers have offered their own DVR's, TiVo's market share has waned.  They recently partnered with DIRECTV to offer a TiVo DVR through the satellite provider.

Lesson: Much like TiVo was leading the pack with the new DVR technology in the early 2000's, KA is leading the pack with their technology in 2012.  Khan will likely not go away, but as the establishment offers similar services, KA's relevance will wane.  Eventually we may see full partnerships develop between Khan and established school districts.

So again the question:  We all have a goal of the best possible math education accessible to all.  What's the best way to achieve this goal?

I think Robert is right.  A competitive environment will make both established educators and KA better.  We are truly in a battle for market share and whoever offers the best product - one that is individualized, accessible, and pedagogically sound - will gain market share while the other will lose market share.  Khan's ability to adapt quickly should give KA an advantage, and unless there is a legitimately viable alternative I would expect Khan to thrive as it improves.

Those who are worried about KA & online learning displacing them have a legitimate concern, I believe.  If it realizes its potential it should gain market share.  Add to that nearly constant reductions in ed budgets, and I don't think it's a stretch to say many of us may lose our jobs.

And although others assert it, I don't see any reason why someone can't learn mathematics just as well from an online course as a traditional face-to-face course.

Finally, it seems like the trend is toward a "blended" approach that merges classroom and online experience.  This to me is really exciting.  If technology can free some time up for me to focus my time and energy on what is most important, I am all for it.  And I think what Khan's doing is a piece to that puzzle.

Sunday, July 22, 2012

5 Math Topics I'm in Love With

This is the flipside of 5 Math Topics I'm Bored With.  These are topics I love to teach for various reasons, and believe these are generally underused topics.  Many are casualties of our standardized curricula.

5) Cake Cutting
Fair division problems are the coolest!  If you've never studied cake cutting beyond "I cut, you choose," you really need to because it's an absolute treat (pun intended).  Life can be fair, you just need to learn the mathematics behind it.  And it gets pretty intense once you move beyond 2 or 3 participants.  People from Edina love this one!

4) Partial Fractions
OK, this is probably the most boring on this list.  But partial fractions is an activity that gets pretty intense algebraically, and the level of that intensity is not overwhelming.  Partial fractions force you to factor polynomials and set up & solve systems of equations.  On one hand it's just another process, but on the other, it's an algebraically rich topic.  (By the way, I almost put conic sections here...also another nice topic that reinforces completing the square.)

3) Truth Tables
My 8th grade math teacher's name was Gerry Warkel.  Best math teacher I've ever had in my life.  Mr. Warkel started off our 8th Algebra class with truth tables and other logical ideas like the square of opposition.  Although this didn't have direct application to the other things we did in Algebra 1, it got the juices in our brains flowing and I really do believe the foundation of logical reasoning made a big difference throughout the course.  Plus, this is another topic that is fits well into almost any high school math class.

2) Combinatorics
In high school we don't go much beyond a question like, "how many ways can we form an 8-person committee" types of problems.  But you don't have to dig all that much deeper to get into some really cool conceptual mathematics.

My personal favorite goes something like, "Ten people are ordering ice cream cones and they each have a choice between vanilla, chocolate, or strawberry. How many different possible ice cream orders are there?"

Spoiler alert: the answer involves filling 12 spaces with 10 stars and 2 bars:

 *  *  |  *  *  *  *  *  *  *  |  * 
-  -  -  -  -  -  -  -  -  -  -  -

So you  partition the ten orders into the three flavors (in this example, 2 vanillas, 7 chocolates, and 1 strawberry), and your answer turns out to be

where n is the number of choices that need to be made and k is the number of possible choices.

For me, it doesn't get any more fun than that!!!  And there's plenty more where that comes from.  Combinatorics are a super-rich, under-utilized branch of mathematics.

1) Financial Math
I am trying to restrain myself here but I'll warn you I get pretty fired up about this one.  Why don't we teach the mathematics of investments, loans, mortgages and credit cards?  Is there anything more directly applicable to LIFE than this?  So why don't we teach it?  Because it's not on the effing test, that's why.  I don't know who this idiot is who decided this wasn't important, but someone did, probably because we are trying to get every kid "college-ready," forgetting that "life-readiness" is what should be most important.

Now that I'm firmly on my soapbox, let me be a little more specific.  Our kids do, for the most part, learn about compound interest through Pert.  But these problems have a very superficial context, and if we dug just a little deeper, we could teach kids how to make life choices based on a deeper understanding of the mathematics.  For crying out loud, these kind of choices are only a click away

Or take mortgages.  How many of us really knew how the hell to buy a house before we did it the first time?  Did you know what was involved in the closing costs and if it would be better to roll them into the loan or pay for them outright?  Did you know what points were?  When we buy a house, we are told a monthly payment, and we might even break it down by principle vs. interest.  But do we create the amortization schedule?  How much more interest would we pay over time if we did a 20-year loan vs. 30?  And you might pay more interest, but how will inflation affect the actual buying power that money will have in 20 years?

How about cars?  Will it be better to lease or own, and if we buy should we take the low APR or the cash back?

Shoot, we should be all strung up for educational malpractice for not helping kids understand student loans at even the most basic level!

But the big one is credit cards.  Most credit cards calculate a monthly interest based on something they call average daily balance.  The mathematics behind ADB are pretty straightforward.  Yet, I'll venture to guess that most MATH TEACHERS don't even know what this is or how cards calculate it.  There's a shitload of math on the back of every credit card solicitation, and if we take a day, yeah ONE DAY, to simply explain to the kids what all of those rates mean, I'm telling you we could make a difference in their lives far beyond any impact we could make teaching some more abstract branch within our field.

But sadly, I don't have time for this, and neither do you.  Someone decided it was more important to make sure students can prove that (x2 + y2)2 = (x2y2)2 + (2xy)2 can be used to generate Pythagorean triples (common core, page 64).

Saturday, July 21, 2012

5 Math Topics I'm Bored With

I better preface this with this disclaimer: some of what I say here may be sacrilegious.  Please try not to be offended.  These are the equivalent to dinosaurs and the Iditarod in elementary school.  Topics that in my opinion are overdone and cause me to roll my eyes a bit.

5) Roman Numerals
This is number 5 because I actually do see value in introducing kids to an alternate mathematical language.  But at the end of the day this is overdone (or at least poorly done) and can be a distraction from legitimate mathematical learning.

4) Tessellations
Someone please tell me what value tessellation projects have aside from the fact that they are an easy break from what can be a pressurized curriculum.  Is there anything challenging about chopping off a piece of a square and pasting it on the other side?  There are few topics easier than transformations of figures.  Sure, tessellations reinforce the concepts of translations and rotations, but how much does the topic really contribute to sound mathematical thinking within the minds of our students?  Bottom line: it's a fun project but has very little substance.

3) Scientific Notation
Forget that "science" is part of the name and therefore should be taught in science (just joking, sort of).  Interpreting scientific notation is little more than memorizing another notation.  It has value when interpreting that pesky "E" in our calculators, but that's about it.  And the contrived problems that go along with it are even more nauseous.

2) Recitation/celebration of the digits of Pi
Admittedly, I can get this far: 3.141592634.  Yay for me I guess.  But honestly, who really cares.  I know we're supposed to embrace it, but if I have to hear one more nerdy kid recite Pi to the 40th digit I am going to lose it.  I'd love to go with 22/7 just to tick these decimal-loving fractionphobes off!

1) Fractals
One word describes fractals: "neato."  The fact that we can perform iterations and the recursion forms a cool picture is certainly "neato."  But in their core, recursions are flat-out boring, and if they didn't come with these pretty pictures we'd move onto bigger and better things.  For example, check this picture out:

Neato, right?  Go ahead and plot this by hand and then tell me how much fun you had doing it.

Thursday, July 19, 2012

Two Views of Probability

Here's takeaway #2 from our IB Higher Level Mathematics training in New Mexico:

Source: Wanda Bussey, IB Trainer, Montezuma NM 2012
Again, I have in the past ethereally made mention that "different ways of thinking work in different situations," but I will absolutely call out the two views explicitly in the future.

I personally gravitate toward the second type, but, frankly, my comfort with probability problems has hampered my ability to teach it in a coherent way.

Wednesday, July 18, 2012

Tools Have Jobs

Unfortunately, I won't be able to make Twitter Math Camp (#TMC12), because I am in Montezuma, New Mexico being trained for our IB Higher Level Mathematics course.

So far there's been a lot of talk about the structure of our classes and changes in curriculum & internal assessments.  But my biggest takeaway so far is this: tools have jobs.

I have been working to shift my thinking, likening the content students learn to tools in a toolbox.  So the challenge comes from (1) making sure that the kids have these tools and (2) making sure they know what tools to use in which situations.  So why haven't I been explicit about what jobs each of these tools are useful for?  Here's a slide from Wanda Bussey, our instructor this week:

Admittedly, I haven't thought this through for all content throughout the year, but at face value it seems like something so obvious that I am embarrassed that I hadn't thought to be this explicit with the jobs of all of the tools!  Tools have jobs!

Thursday, July 12, 2012

Bored with Khan Critiques

I am more excited about Khan's Mongolian Barbecue than these critiques of Khan Academy.

If you're wondering what I'm talking about, click here for some background info.  Or here or here or here or here.  Geez, people really love ripping on Khan.

I know where this is coming from.  Khan has been anointed messiah of education doing little more than creating how-to videos and distributing them for free.  The United States, looking for any answer we can find to the problem of poor academic performance when compared with other countries, grabs onto Khan's metaphoric leg wide-eyed and hopeful.  The educational establishment recoils, becomes defensive, and tries to balance the praise with objective criticism.

In the process we look petty and entrenched.

Correct me if I'm wrong on any of these, but I think these are generally accepted truths about KA:
  • Khan does a generally good job of teaching procedures and processes.
  • Simply learning procedures and processes without depth of concept can hinder mathematical understanding.
  • Khan and backer Bill Gates often overextend or poorly articulate the effectiveness of KA and its ability to enhance student learning.
  • Although the instruction is not innovative, the concept of free distribution on such a large scale is.  Other innovations like virtual classrooms, when used correctly and with sound pedagogy have the potential to radically change education for the better.
Wait, hold on, seems like I need to insert a picture of a kid at a computer.

OK good, that feels better.

So with those truths in mind, why do we need to get snarky?  Sure, we are causing positive change, but at what cost?  Frankly, I think we are making ourselves look like jealous buffoons.  I get the impression that KA has a goal of pedagogical soundness.  Is this the best way to help them achieve that goal?

Sal Khan is not a dummy.  He is clearly working through some of the same pedagogical misconceptions we all worked through (and continue to work through).  How can we best help him through his personal journey without alienating him or causing him to be defensive?  His goal is the same as ours.  Does it advance the cause of mathematical education to put him down and be hyper-critical?

I say no.

Wednesday, July 11, 2012

Super Exponent Spectacular

I rolled out of bed Monday to teach our summer Algebra 2 course. The main teacher of the course is in Thailand on his honeymoon. (Married rich, I guess.)

When I got there I realized it was time to teach exponent rules. Which meant that it was time for one of my favorite lessons of the year, the

click for file

This lesson is nothing groundbreaking.  I show kids expressions to simplify, they work the problems on white boards at their desks, show them to me, and I give them immediate feedback.  What makes this lesson so fun for me is the music I have playing in the background and the energy around it.  I dim the lights and make it a party.

I have a suggested playlist here: SES youtube playlist.  I keep these as audio files with playlist embedded into the slide show, but this was the easiest way for me to assemble them for you.  With my radio background, I took great care to assemble these songs.  There are a few rules that go along with them:

  • You MUST always start with Machine Gun by the Commodores.  It's even OK to play it twice.
  • Only after the funk vibe has been established can you deviate a little bit.  You should always return to it though at some point.
  • Ball of Confusion by the Temptations should optimally be played when a more difficult slide comes up.
  • Journey - Feeling that Way/Anytime is required.  I put a live version on the playlist, but the album version is preferred.  Youtube would let me add it to the playlist, but not PLAY it in the playlist.  (Why, youtube, why?)
  • Current hits should be used here and there, but not overdone.  Instrumental versions are encouraged for lyric-heavy songs.

Finally, the slide show is designed to highlight similar, but different scenarios within expressions.  This is by no means the holy grail of problems, and the curve is probably steep toward the end.  There is not enough rational exponent problems.  When I put this together I just threw in a couple of really difficult ones at the end in case we get there and I needed to keep them busy until the bell.  Usually we don't get there.  I have adjusted this activity for different classes/levels and it works like a charm every time!!