Back on the MTBoS Bandwagon!

The first quarter of the school year is over and grades are done, which means (among other things) that I finally have time to blog again! Time to catch up with some of the great MTBoS missions of the past few weeks. I'm responding in this post to MTBoS Mission #3. It's been a while.

I've played with the Desmos online graphing calculator before and liked it, so I used this mission to explore the Daily Desmos blog. Part of the reason I didn't explore Desmos much in the past was that it didn't seem to do anything GeoGebra couldn't. (Full disclosure: I am a huge GeoGebra fan.) But Daily Desmos gave me some pretty good evidence that the simple interface and streamlined visuals of the Desmos graphing calculator make it a potentially better independent exploration tool than GeoGebra.

Every few days the folks at Daily Desmos post a pair of related graphs, one Basic and one Advanced, with the deceptively simple instructions to use Desmos to recreate the graphs. The easy problems can often be solved with Algebra 1 level knowledge of linear and quadratic functions, while the difficult problems can be time-consuming even for undergraduate math majors. This, of course, means serious bragging rights if a student in your high school classroom manages to solve one of the Advanced problems.

The Daily Desmos problem that hooked me and subsequently ate up an hour of my Sunday afternoon is here, and it's based on a (much, much) simpler problem here. The Basic version of the problem requires graphing ten different linear equations which can be found by simply finding the slopes and y-intercepts of the lines in the graph.

Way more fun than a set of ten practice problems, but essentially the same thing.

The Advanced version requires finding the envelope curve for the family of lines in the Basic version. The given graph is below.

Looks like a fun Sunday afternoon, right?

When I saw this graph, I thought to myself, "Piece of cake. Just generalize the equations in the Basic version!" I did that, and then I had to parametrize the equation, then combine the resulting parametric equations back into a function, then square the function and make it implicit to allow for both positive and negative square roots. [SPOILER ALERT] After some serious algebraic gymnastics, I ended up with the equation and graph shown here. It turns out it's just a parabola with an oblique axis of symmetry!

So what did I take away from this that could help my teaching? Well, for students naturally inclined toward math, the Advanced problems on the Daily Desmos site are engrossing. And for those not naturally inclined, the simplicity of the Daily Desmos approach is brilliant. It doesn't give these students anything they don't need. It just gives them a picture and a simple question: "Can you make yours look like this?" For some of these problems there may be more than one solution, leading to interesting and heated debates among students. Really, I've seen this happen. Even if there's only one final solution, there may be many routes to get there. Daily Desmos problems require ingenuity and develop problem-solving abilities. But unlike some problems I toss at students, they know for certain that they've reached the end. If their graph looks exactly like the one they were aiming at, they're done! That's very appealing for them.

For now, I'd like to use Daily Desmos problems for extra credit or for independent work on those weird half-day schedules, but the approach could lead to some seriously cool unit design.

All a-Twitter

This post is in response to Exploring the MTBoS Mission #2.

I've had a Twitter account for about a year and never tweeted once until last week, so I am definitely a newcomer to the Twitterverse. When I started my Twitter account my intent was to use it as a professional development tool. I never followed through on that because of the overwhelming sense of chaos I felt as I looked at my feed. There's just so much STUFF piling in all the time. It seemed impossible to keep up with it all. And to be honest, I was (and probably still am) put off by some of the linguistic doomsday prophets who warn of the dangers of reducing discourse to 140-character spurts.

After using Twitter a little more concertedly this week, I'm still overwhelmed by the inexorable march of tweets streaming across my feed. However, I've seen some of the advantages of the 140-character limit--for one thing, it's a lot harder for people to dominate conversations the way they can on Facebook. You'll never see anyone write TLDR in response to a tweet. I've also seen what Justin Lanier (writing on the MTBoS blog linked above) calls the "funny combo of synchronous and asynchronous communication" that happens on Twitter.

Probably the best part of Twitter (and weirdest, for me) is the ability to follow and even tweet at people whose work you admire. There are blogs I've read for a while written by gurus in my field, and I can send a tweet their way just like that! Nothing stops me! I don't have to track down an email address and carefully craft an introductory letter hoping that I can disguise my inner fanboy enough to sound like a competent professional, all so I can maybe hear back from someone I look up to. I can just tweet them! And on their side of things, they don't have to worry about 1,000-word emails from fans. One hundred and forty characters, max.

In summary, I'll continue using Twitter, but I don't know if it will be a go-to tool for me.

The Meno Problem

If you've ever read Plato's Meno, you probably recall the conversation in which Socrates tries to prove to Meno that all knowledge is recollection by helping Meno's slave boy recall some geometry. Socrates does this in his typical style, through a series of questions:

Soc. Mark now the farther development. I shall only ask him, and not teach him, and he shall share the enquiry with me: and do you watch and see if you find me telling or explaining anything to him, instead of eliciting his opinion. Tell me, boy, is not this a square of four feet which I have drawn?
Boy. Yes.
Soc. And now I add another square equal to the former one?
Boy. Yes.
Soc. And a third, which is equal to either of them?
Boy. Yes.
Soc. Suppose that we fill up the vacant corner?
Boy. Very good.
Soc. Here, then, there are four equal spaces?
Boy. Yes.
Soc. And how many times larger is this space than this other?
Boy. Four times.
Soc. But it ought to have been twice only, as you will remember. 
Boy. True. 
Soc. And does not this line, reaching from corner to corner, bisect each of these spaces?
Boy. Yes.
Soc. And are there not here four equal lines which contain this space?
Boy. There are.
Soc. Look and see how much this space is.
Boy. I do not understand.
Soc. Has not each interior line cut off half of the four spaces?
Boy. Yes.
Soc. And how many spaces are there in this section?
Boy. Four.
Soc. And how many in this?
Boy. Two.
Soc. And four is how many times two?
Boy. Twice.
Soc. And this space is of how many feet?
Boy. Of eight feet.
Soc. And from what line do you get this figure?
Boy. From this.
Soc. That is, from the line which extends from corner to corner of the figure of four feet?
Boy. Yes.
Soc. And that is the line which the learned call the diagonal. And if this is the proper name, then you, Meno's slave, are prepared to affirm that the double space is the square of the diagonal?
Boy. Certainly, Socrates.
Soc. What do you say of him, Meno? Were not all these answers given out of his own head?
Men. Yes, they were all his own. 

I often try to teach Socratically, using questions to guide students into answering their own questions. Recently, though, a comment from a student got me thinking. She complained that when I'm helping her, she feels like she understands the concept perfectly, but when she goes home and starts here homework, she has no clue what to do. I know she can't be the only one who feels this way. So does this question and response style of teaching actually work?

I have to admit I doubt Socrates' claim that all of his pupil's answers were "given out of his own head"--some of these answers are so obviously fed to the boy that it makes me wonder if Plato was being serious when he wrote this. I doubt even more that the boy took away a lasting understanding of the relationship between linear dimensions and area. My question, then, is: how do good teachers use questions to guide student thinking in a way that both gives students structure and enables them to become independent of a teacher's guidance? Any ideas out there?

The math clock

The clock below was given to me by my wife's parents for my birthday this year. It makes a great addition to my classroom, and my students get a kick out of it. There's just one small mathematical inaccuracy...

I pointed out the expression for 9:00 to my department chair, and she posed a question: what if I could fix it? Or even better, what if I could get my students to fix it? We could add some more complex expressions for the 12:00, 2:00, and 8:00 slots, and maybe get rid of that pesky long division symbol at 11:00, 3:00 and 5:00. Since one of my classes is currently working on solving multi-step linear equations, an opportunity presents itself...

Fix the Math Clock!

First, I'll provide each of my students with a list of the expressions and equations already on the clock, and tell them that there's one that's not quite accurate. I'll ask them to simplify or solve each one (whichever word applies--a great time to review the difference between the two terms) to figure out which expression/equation doesn't correspond exactly to the number it's supposed to represent. Once they figure it out, I'll also lament loudly that the equations and expressions are way too easy for my students, and they deserve more of a challenge when they want to know what time it is.

Then I'll give each student circular piece of paper cut to fit my lovely clock, and a fistful of markers. Their job will be to create twelve equations, one for each clock number. These equations must take more than one step to solve, and they must require the use of the distributive property, combining like terms, or adding variable terms to both sides of the equation. Then they'll swap equations with a neighbor and check each other to make sure the solutions of these equations come out correctly. Once they're good to go, they'll create a new clock face featuring their equations along with any other creative touches they care to add.

The best one will replace the clock face that's already there!

Any thoughts to make this task richer?

The questions that keep me up at night

Ok, yes, I'm a math teacher, and yes, I started this blog to join the online math education community. I want to share and steal good lesson ideas, good ways to frame mathematical topics, good explorations, good applications, good assessment tools, good classroom technology. Yes, absolutely. But I should be honest: the questions I obsess about, the ones I debate with friends and brood over while starting out windows, are not mathematical.

Don't misread me. I love math. I love its beauty and rigor. As far as I know, mathematics--and here I lump formal logic in with math--is the only subject in which a result can be proved deductively and then held as absolute truth, at least in the context of certain axioms. Still, my first love is philosophy. So the questions that keep me up at night (until 9:30pm) are philosophical. I can't stop questioning what Nel Noddings calls "the aims of education."

Here's a sampling of the questions that puzzle and motivate me:

• To what extent ought we to coerce students? Every day I essentially coerce students to learn math. They may not want to learn it, but the school says they must take four years of math, and I say they must perform certain tasks satisfactorily to pass my class. Yet theorists like John Dewey and even A. S. Neill appeal to me with their emphasis on student empowerment and free choice. Is there a way to empower students in a public school setting and still meet the standards imposed by the powers that be?
• What does equity really mean in education? I get this question from Noddings and from Matthew Crawford (Shop Class as Soulcraft). We tend to assume that equity means providing everyone with a college preparatory education, and so we push all students to attend college. It does seem that providing only a selection of students with an education aimed at college would create a deplorably aristocratic situation. Or is there a way to build a system that honors all learning styles and interests, and doesn't assume that the goal of earning a college degree is innately better than the goal of becoming a really good auto mechanic?
• Do grades destroy intrinsic motivation? Thinkers like Alfie Kohn abhor the use of external motivators to get students to learn, and this includes everything from candy to grades. I sympathize with this to some extent. The use of incentive-based learning models from kindergarten on tends to obscure the fact that many things in life are worth doing just because they're worth doing. But don't we need some way to measure student growth and learning? Is there potentially a hybrid model that starts kids with incentives and then weans them from such motivators?

I consider myself lucky to be able to engage in both theory and praxis when it comes to these questions. To me, the appeal of being a teacher comes from my dual role as philosopher and practitioner. In the future this blog will probably feature more practical reflections on pedagogy, but I think it's crucial never to divorce practical concerns from their theoretical underpinnings. So I begin here.