Try bringing the problem to virtual reality

One fundamental problem: in Algebra 2 there are few good examples and fewer (zero?) relevant to my students’ daily lives.  The solution, so far, has been elusive;  wrapping a context around the equation being studied or finding a situation that you can help your students explore in a mathematical way are my best attempts so far.

Let me extract what I mean from that jungle of punctuation and parenthesis. My goal is to help my students see how they can superimpose mathematical models upon their reality to help them do stuff.  Unfortunately, at the level of math that I am supposed to be teaching, their skill level with the stuff is actually still too low to be useful in any kind of exploratory way – they know just enough to do what I’m doing and see that it works.  That is, they aren’t going to stumble upon the formulation of a logistic population model by themselves, but if I show them such a model they can understand what it means and how to manipulate it.

If we take this too far, we run in to the problem of being overly specific in our instruction and robbing our students of the chance to be creative or even inquisitive.  When we are talking about volumes of circular cylinders we have to narrow it down for them, and often our only way of doing so is to narrow it down all the way.  “You see, kids, since V=\pi r^2 h, h = 6.0 cm (or even just 6) and r=2.0 cm, the volume is… I mean, what is the volume, kids?  And why do you all look so bored?”

But what choice do we have?  Real cylinders don’t actually follow this law, when measured by students – they fall somewhere within 5 and 500%, depending on the quality of measuring going on.  Also, measuring real cylinders takes like five hundred hours.  Are you going to let your kids use an entire hour to get the five data points they need to see a quadratic pattern?  Even if you know some of those points will be completely wrong?

Computer models offer us the opportunity to narrow the problem space without narrowing it right down to the nub of the solution.  Without going so far as to say that every math teacher should be able to program a computer (I’ll do that later), I’d like to extol some of the benefits of modeling problems with something like geogebra.

Consider the following geogebra applet, which took me about 10 minutes to cook up (I’m a geogebra whiz, though).  Drag the bottom right corner of the cylinder to change the radius of the cylinder.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now). If you're reading this in an RSS aggregator, try following the link to the actual page.

Here, I can let my kids explore!  There is a confined space that I have carefully constructed for them and within that space there are no rules.  They can do whatever they want.  In this universe I can ask questions like “Can you find a cylinder with a volume of 2000?” or “what happens when you double the radius of a cylinder?”  I can ask them to make a graph of volumes against radii, and they will enjoy doing it (you may have to trust me on this if you haven’t used an interactive program like geogebra with your class yet).

And from this problem space they can actually figure out the formula for a cylinder with a height of 10.  They can try many experiments very quickly, and with satisfying proficiency.  When they make the graph of volume vs. height radius, they will do so with a speed that lets them really enjoy that the shape it makes is a parabola.  And the discussion taking you from cylinders with heights of 10 to cylinders of height 20, to cylinders of height 5, to cylinders of height h will be delighting because they not only know what you are talking about but freaking invented what you are talking about.

I haven’t even talked about the best part yet, you guys.  The purest virtue remains unsung, if you can believe it.  So, here I come with the biggest freaking bell-ringing mallet I can find: the feedback kids get from their computers is absolutely unjudging, unbiased, unhelpful, instant, and 100% correct.  Kids don’t have relationships with their computers.  Kids don’t ask computers for help, and computers mercilessly avoid volunteering any.  There is no “Clever Hans” effect happening with computers.  Kids can try an idea and if they’re wrong their computer shows them so without the slightest hint about why.  So, guess where that why has to come from?

Whew.  More on this later (with more examples, hopefully).


PS: As I was writing this post, one of my students submitted an honors project from last month (due any time in the next month).  Drag the points around.  Do you have any doubt that this student thoroughly understands how to graph an exponential function through two points?  She created this file from scratch.  Double-click a and b to see how she did it.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

PPS: For teachers with students more or less skilled than mine: you can adjust any problem space tighter or looser, right?  If your kids don’t understand graphing or squaring yet, let them vary the height of the cylinder instead of the radius.  If your kids were talking about volumes of cylinders with their parents when they were 8, let them vary the height and the radius at the same time.  The basic benefits remain.

PPPS: Bonus points for anyone who can concisely explain how to make a cylinder like that in geogebra.

Curse your sudden but inevitable betrayal!

So, on Friday I passed out the Algebra 2 CPM books I convinced my school to buy.  And today, the very first lesson out of the book… I’m disappointed.

Ok, not REALLY disappointed.  The book has us doing some basic practice problems for a day, finding parabolic sections that go up 4 feet and are 8 feet wide, or that go down 15 feet and are 40 feet wide, etc.  The problems are structured in such a way that students will discover a method for algebraically determining a in y=a(x-h)^2+k, which is great.  It also brings back the idea of domain and range, and asks the kids to define their own, which I think is fantastic (another post: domain and range should be used more often in a more sensible way).  So, as boring practice problems go, these one at least have some inherent self-discovery lurking.

But every single problem is boring, and worse, every single problem suffers from the classic burden of pre-printed, static materials: they must explicitly identify all of the information needed to solve the problem, and they don’t give any extraneous information.  The students need not think about anything outside of the printed page.

So, I’m replacing this, the very first lesson after my free-time-enabling, curriculum-guiding, all-but-cocktail-mixing miracle books arrived, with one of my own.  I based it on Dan Meyer’s WCYDWT: Projectile Motion photo, but made a couple of improvements:

  1. I reworked the geogebra file so that the photograph could be moved by the students.  More importantly, the photograph zooms correctly when students zoom in and out.  I did this by:
    1. Creating a point, called A.
    2. Creating a line with the equation y = y(A), giving us a horizontal line that will follow A around.
    3. Creating a point called B on the line from the last step.
    4. Inserting the photo, then using its properties to specify that A be one of its corners, and B another.  Now, students can resize the photo and move it anywhere they like, but cannot rotate it.  I made the line invisible, and made it an auxiliary object so that it would not appear in the object list.
  2. After looking at the photo (briefly) as a whole class, I split the students into groups of four, and gave them this worksheet (docx).Small groups are the freaking way to go, man.  Even in my small Algebra class (only 12 students!) if I have a full-class discussion, 5 or 6 kids aren’t going to say anything.  In groups of 4, maybe I have one kid per day not speaking and sharing.  I imagine the difference in participation percentages would be exaggerated in classes of 30.

So, CPM is totally great, but some of my commenteers were right: it is not mixing my drinks for me.  I think my lesson was an improvement over CPM’s – but it’s important to remember that, though I changed details of this lesson, I am still operating in a research-proven structure and curriculum.  I don’t need to worry that this lesson would have been better delivered in a month, or three months ago.

Mathtype Challenge

To keep our linked list in order, here’s a reference to the end of the mathtype challenge chain: http://function-of-time.blogspot.com/2010/02/mathtype-challenge.html

Interestingly, I use the built-in equation editor in Word 2007 instead of MathType.  The integration (with Word, not with limits) is tighter, but some features are missing.

Summary and pro-tips from my video below:

  • I got through Equation 13, with no number line for #5, poorly formatted equations on #6, and a few typos.  I was going to get the number line from geogebra by turning off the y-axis, selecting the proper part of the x-axis, and hitting ctrl+shift+c to copy the selection as an image to paste back into word.
  • Alt, N, E, I gets you a new equation object.  Money.
  • Word 2007’s equation editor will automatically do a bunch of stuff you feel like it should.  Just write a few expressions to get a sense of how it auto-corrects.  It’s nice.  It’s not exactly latex, but you get the sense that maybe the programmers know latex themselves.
  • \int_a^b space gets you a nice integral with limits from a to b.  Niiice.   My biggest time-saver of the year.
  • In Word 2007 you can add your own auto-correct entries, and there are a ton built in.  \div becomes \div, \pi becomes \pi, etc.

The video: http://screencast.com/t/MTY5ZTllZDUt

One way to convince kids of finite infinites:

The shape in question was \int_1^\infty \! \frac{1}{x^2} \, dx, and when I claimed its area was finite, my students didn’t believe me.  Actually, my students didn’t even know what I meant.

Back when my improv was as good as my planning, I used to love digressing into topics that I found especially interesting, and I indulged myself again this week by spending 10 minutes talking about the idea that a number that never stops getting bigger might have a limit.  The idea I focused on this week to exemplify my point was the difference between 1 and 0.99999… “repeating.”  I showed, through long subtraction, that there was no difference between the two numbers (the two representations of the same number?).  I went to the “ends” of 1.0000… and 0.9999… and did the 1st-grade trick of borrowing from a more significant digit.  When my students said, “but you’re still left with a 0.00..01!” I said, “well, we gave up too early!” and added a few digits.  It was this last bit that really helped them understand, and I saw that wave of comprehension go across their faces (all of them, I think!).

Since my calculus class is not an AP class, I struggle with the decision to include limits at all.  You don’t need them to explain the other concepts of calculus, and since I’m definitely not going to get all the way down to basic limit proofs it makes sense to me to skip them altogether.  This breakthrough is a big one, though.  If I can create a lesson that leads students to this thought process… I’m in!

Teaching note-taking with notes templates

One of the questions I struggle with is “what is my class for?”  This has been a question forever, of course: is my most important lesson about Algebra, about being ready for college, about independent thinking, about forming a community, about the connection between me and my students?  It’s easy to get bogged down in this but it’s also easy to see that the answer is not Algebra alone.

One of the other things I try to teach my students is the skill of taking notes, and my approach revolves around templates for notes for each class.  At the beginning of the year, their templates will be almost completely filled out, and colorfully organized.  I use the trick of leaving everything but the f____ l____ of some words blank to give kids some ownership of the notes but to simultaneously make sure that kids have really important points written down.  There are spaces at the top and bottom of each template for a title (a super-short summary) and a summary of the class (a couple of sentences max).

A complex notes template

A complex notes template (click to download pdf)

As the semester progresses, we talk occasionally about what from the class would be important to remember, and how the students could have guessed at what material to write on their own notes template.  On a couple of days in the second or third month I give them the notes template at the end of class, and we take a little time for them to realize what they missed, and what they included on their own notes that I didn’t.  And the templates get simpler.

A simpler template (click to download PDF)

Creating these templates for class every day is a fair amount of work, but I find that focusing around the template can actually help me focus my lesson.  And, in those halcyon days of my imagined future self, in which I am following the same sequence in two consecutive years and sipping white russians and reading Robertson Davies instead of working frantically until 10 PM and still going to bed unsatisfied, these notes templates can be reused.

I don’t have data for you about how well these work.  I don’t grade the notes and I don’t control any part of the experiment.  Some students say they like them, and some don’t.  I can tell you this:

  • When a student comes to my office hours and asks a question, I can ask him to pull out his notes on the subject.  If he can’t do that, I’ve immediately identified a problem in his work habits.  Better still, it’s a problem that we can manage and check in on the next week.  If he can do that, he is empowered to help himself, and good study habits are fortified.
  • I am showing kids what it means to take good notes.  They can compare their own notes to mine.  I am no longer just expecting them to figure this skill out and getting frustrated when they don’t.
  • The kids who keep all of the templates in order have a binder they can really be proud of.  It looks nice.
  • I don’t have to give kids a ton of time to accurately copy complex diagrams or equations.  I just put those things on the template.
  • You can really supplement a powerpoint presentation with the right stuff on a complementary handout.  And there are some fun classroom activities to be had by subtly changing a few of the handouts in ways that students won’t notice right away :D

I’d love to hear of any other ideas for teaching note taking, or any responses to mine.  My method is based on the Cornell notes system.

Percentages don’t have the power to express a grade.

The traditional model for grades in a class lacks the flexibility required to reflect what I really think of a student.  When I used weighted categories (e.g. 50% exams, 30% homework, 20% class participation), I found that some of the students passing my class didn’t really seem to deserve it, and some of the students failing my class really should have been passing.  “Well, adjust your weights,” you say, and that’s a good idea: I made several improvements and was progressively more satisfied with my results.

But.

One test of the 29 I’ll give this semester deals with simplifying exponential expressions.  If a student gets 100% on each test except the exponential simplification test, on which he gets a 0%, his average will be 96.5%, A+, Honors.  He doesn’t have to worry about exponential simplification at all, and he can just move on and never learn it.  I’m not suggesting that this hypothetical kid be made to retake Algebra 2, of course.  I’m suggesting that he be required to learn exponential simplification.

So, in my class, I require that every student earn at least a 3/5 on every single skill that we study.  Then, I require an overall average of 75% on top of that minimum requirement.  Students get some leeway, and they do not need to master every single subject (I understand that there are time constraints involved in my students lives, and that they may not really care about my class).  However, they can’t just do well on a lot of skills and decide they’re not even going to bother with one.  I am not willing to send a kid who can’t simplify exponential expressions at all to the precalc teacher.

The same philosophy can extend to homework.  If you think homework is vital, make it a requirement of passing.  If you don’t think it’s vital, don’t.  Averaging test scores with homework scores is harmful because it dilutes the meaning of your tests and the meaning of your homework.  Averaging mathematically destroys information!

At my school we only have three grades, Pass, No Pass, and Honors.  Each grade has certain clearly stated requirements that I give the students at the beginning of the semester.  I think that a teacher using letter grades could more clearly define what a C was and what an A was by stating the objectives vital for that award than he or she can by trying to come up with a formula to fit every student.  We shouldn’t be afraid to use some criteria that cannot be expressed with percents.

Why CPM is killing my blog

First of all, I should say that I am not getting paid by CPM in any way (except for in the free sample books that they’re happy to send out to anyone who asks).  I’m about to give it a pretty freaking positive review, tainted only by the melancholy of knowing a superior.  CPM, if you want to start paying me after reading this post, that would be fine.  riley@larkolicio.us.

College Preparatory Mathematics (CPM.org) has a boring name but the most exciting lesson materials and curriculum I’ve ever seen.  Now, I’m only 26, everyone, so you might skip the shakers and go straight for the salt mine, but I have been using sample lessons from CPM’s algebra 2 and calculus books for the better part of two months, and I am literally moved to extol the benefits I’ve seen to each of my 118 readers.  Bulleted, for your convenience:

  • The sequencing is beautiful.  It makes so much sense.  In big ways, like “teach derivatives and antiderivatives together,” and also in small ways, like “finding the derivative of a logarithmic function will be a great way to practice implicit differentiation,” and also in tiny ways, like “think about this, and then this,” and then as often as not I see my kids beaming with pride and excitement at their discoveries.  CPM is very well thought out.
  • The teacher’s manual tells teachers what the authors were thinking, which allows you to be flexible.  It tells you which problems are most important, and which ones are important for topics coming up soon, so if you want to skip a topic or lesson, you know what you can drop and what you can’t.  If you have an awesome lesson idea, the teacher’s manual helps you figure out how to swap the CPM lesson out and swap yours in without missing something that will be essential later.
  • The books are really lesson plans.  The courses include homework, assessments, and classwork.  They include materials for each lesson.  There is literally a full, well-explained lesson plan for each of like 160 days of class.  Purchasing the courses gives you three free days of professional development (re. how to work with CPM materials) and it gives your kids free access to solutions and additional practice online.

It’s this third bullet that has me in an existential crisis.  This is why I haven’t been writing much lately: I’m not doing much lesson planning!  CPM has done all of the work I used to do, and they’ve done it better.  They’ve tested their materials on thousands of students and hundreds of teachers over the course of the last twenty years (or so?  I don’t actually know, sorry).  How can I presume to add anything to this after 5 years of teaching by myself?  I haven’t even been taking good notes, for crying out loud!  These books have been a huge hit to my (admittedly distended) ego for the last couple of months.

One of my coworkers really put this crisis into perspective for me.  She said, as if this were obvious, that I’m supposed to be a teacher, not a curriculum developer.  My immediate reaction, which scares me retroactively, was that there is no distinction – this is what I’ve been doing for all five years!  Thinking about what’s important to teach my students and then figuring out the best way to teach it to them!  That’s what’s so fun!  Right?

But imagine this drastic role change: it is my job to deeply understand math and to guide my students through lessons created by people who have spent WAY MORE TIME ON IT THAN I CAN EVER SPEND IN MY WHOLE LIFE.  It’s my job to support my students, encourage them, lead them to additional resources or work to fit their interests into the class.  It’s my job to assess their progress and support them in failure and success.  It’s my job to show them the habits of successful and happy intellectual adults.  It’s my job to deal with schedule hiccups and individual circumstance to give the students the smoothest experience possible.

It might not be my job to figure out what is important for my students to learn, or how they will best learn it. Holy fuck.

One of my first reactions to this realization was, “well, I’m in the wrong job.  I want to work on curriculum.”  But I’ve been ceding entirely to CPM in my Algebra 2 class for over a month now and the increased depth I’ve attained in my relationships with students during class is satisfying in a new way.  I’m hoping my school will officially adopt CPM for Algebra 2 and Calculus next year, so I can start from the beginning of the year next year, and I can really give it a shot.  But you might see significantly fewer posts from me about the totally rad new lesson idea I’m developing.  Because maybe teachers don’t develop lesson ideas!

Oof.  This idea still shakes me.  It’s so radical.

PS: Teaching is taking less time than it ever has before, btw.  Pretty nice side effect.