Monthly Archives: October 2011

The College Board and Calculators: Some Thoughts

What follows are some of my more personal thoughts about a project that I’ve worked on with several other teachers.  We wrote a letter to the College Board to express our hope that they will update their calculator policies to better reflect current technologies and the classroom practices that they make possible.  You can read and also sign the letter we’ve composed here:

http://calculatorpetition.wordpress.com

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In 1997, the College Board drew a line in the sand that was a good but temporary solution for a novel situation: they created a policy that embraced the advent of the graphing calculator.  At the time, it was a new tool that was rapidly changing the way that calculus and all of high-school mathematics was being taught.  At the same time, the College Board tried to distinguish between a graphing calculator and a full-fledged computer, believing one to be an appropriate and fair tool to be used on their tests and the other not.  The way they drew that defining line was the commonsense idea of banning devices with QWERTY keyboards.  At the time, this token gave them the distinction they were after.

Over the past fifteen years, mathematical technology has grown both more powerful and more ubiquitous.  One way that this has played out is that the divide between graphing calculator and computer has greatly narrowed.  Anyone who has seen the latest Texas Instruments device—the TI-Nspire CX CAS—would recognize it to be a souped-down computer with a ban-evading ABCDEF keyboard.  With the radical changes to technology in the past fifteen years—not to mention how these have begun to shape classroom practice—make it time for the College Board to revisit their calculator policies.

It’s most compelling feature? Permitted on the SAT and APs.

Bottom line: we should not be sharing inferior tools with our students because of an outdated policy.

My own connection to this story began about the time of the policy was put in place, when I was in high school.  It’s pretty wild to think that in that bygone era, I had a box that I could carry around with me that contained such powerful mathematical tools.  It was truly a marvel.  Not that I thought so much of it, of course.  After all, I never had the experience of learning much of high school mathematics without a graphing calculator.  I loved my      TI-89.  It could multiply out (x+1)^{8}  .  It could give exact answers like \frac{\sqrt{3}}{2}  .  I could explore all kinds of crazy functions by graphing them.  And I used it for years to great effect.

Still, when I lost my graphing calculator a few years ago—saddened though I was—I did not buy another one.  By that point, I had begun to carry it around more out of nostalgia than for use’s sake.

These days, when I need to do some arithmetic quickly, I’ll often type the problem into a Google search bar or into the calculator on my phone.  When I want to graph something, I use Grapher or Winplot or GeoGebra—all of which are on my laptop.  When I need to crunch some big computation, I use Wolfram Alpha or, more recently, Mathematica.

I never yearn for a graphing calculator.  I’m convinced that the tools that I use are more powerful, more accessible, and more useful than stand-alone graphing calculators.

Because of this, I want to share these awesome tools with my students.

When I give my students the opportunity to create awesome GeoGebra sketches like this one, there is a trade-off.  They aren’t getting as much practice using a TI-84 or what-have-you.  This year it’s not so much an issue for me, since just one of my students will take the AP Calculus exam.  But I know that it will be more of a consideration in future years, and I know that it already is and has been an issue for teachers around the country.

We live and teach in a world where there are more and more one-to-one laptop programs and where a significant minority of students walk around with cell phones that can run cheap apps that outstrip the capabilities of top-of-the-line graphing calculators.  In such situations, graphing calculators are at best redundant and at worst an impediment.

Texas Instruments—to pick an example at random—is in a hard place.  The new bells and whistles that they continue to add to their single-purpose, stand-alone devices are weak attempts at staying relevant.  At a math teaching conference over the summer, a TI representative was practically apologetic in pitching us the new version of the TI-Nspire.  And yet there is a real sense in which they are in an enviable position.  Even if their sales numbers are low for their newer models–because folks are slow to move to technologies that either seem like not much of an improvement or already surpassed by other products–they have a captive audience.  As long as they are suggested and required for the SATs and APs, TI can sell us 84s and 89s ad infinitum–that is, as long as the College Board calculator policy stands.

It seems to me that two forces are keeping the graphing calculator industry in business.  One is understandable inertia.  Many teachers and schools and districts have invested a lot of time, effort, and money into incorporating graphing calculators into their classrooms.  Teachers understand how to use TI-84s and 89s in their classrooms and do a great job of it.  That’s awesome.  I have no frustration about that.  New tools and new technologies take time to become the mainstream.  I’m not looking for any kind of bloody revolution.  May these calculators be used in great ways for years to come!

Inertia is natural and fine, but I cannot abide impediment.  That is what the College Board calculator policy is—an impediment.  It stands in the way of the education community even imagining real technological progress.  It’s keeping graphing calculators on indefinite life support–pull the standardized-test plug, and they would die a slow but natural death.  That is why I’m so passionate about raising this issue and working for change—not because I think that old technology needs to be replaced by new technology, but because the College Board policy is preventing us from collectively considering the full space of the mathematical experiences we can share with our students.

As educators, I feel like we too often feel beholden to the College Board and standardized tests and take them as monolithic givens.  We betray this in the way that we talk: “But they have to take the SATs,” and “What about the AP scores?”  But the College Board is just a human institution that can change over time, and we can help that change to happen.  In fact, we are best positioned to do so.

Issues about standardized tests range far beyond these questions about calculator policy.  One thing I like about the issue of calculator policy is that it’s circumscribed and a manageable size.  If some teachers can pull together in a unified voice to say that we believe that the College Board’s policy about technology needs updating–for the good of our students–then we may both achieve a small goal and feel the power of our collective say.

The College Board provides us and our students with a service.  They certify that our students can perform certain tasks and think in certain ways.  As such, they should cater to our needs, and not the other way around.  We should call the tune.

Please read the open letter.  If you find yourself nodding in agreement, then please consider signing it and sharing it with others:

http://calculatorpetition.wordpress.com

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Math Munch!

If you’ve come here looking for Math Munch, click here!

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If you’re not here looking for Math Munch, let me explain.  This past weekend, Paul and Anna and I created a new blog called Math Munch.  It’s an idea that I’ve been kicking around for a while now–a venue for curating some of the fabulous mathematical content that exists on the internet for my students.

I’ve learned so much math and found so much math to enjoy through the internet that it’s just crazy.  I mean, I love good math books as much as anyone, and person-to-person mathematical discourse is irreplaceable.  But the math content that exists on the internet is astoundingly deep and varied and gorgeous. I want my students to have these gems as part of their relationship with mathematics, and I want them to have an empowered and seeking relationship with learning where they know that they can reach out for knowledge that will delight and inspire them.

This kind of independent and personal-growth mindset has to come from somewhere.  The usual classroom techniques of chalk and talk don’t go far enough, at least in my experience.  It has taken me years to build for myself the habits and touchstones that make for a fruitful experience of learning with the internet as a tool.  Connecting students with the wealth of resources that the world has to offer is a first step in realizing that this is even possible.

Math isn’t just in me the teacher.  It’s not just in my classroom, the worksheets that I share and the problems we put up on the board–as good as I try to make these.  The truth is that there is an enormous, multifarious, unbelievably rich ecosystem of mathy people and stuff that is out there.  I have some significant experience with it and want my students to have an experience of it, too.  Math Munch is an attempt to make that happen.  Like it says in the inaugural post, “Here you will find links to lots of cool mathy things on the internet.  We’ll post some new items each week for you to enjoy.  We hope you are as inspired and excited by these creations as we are!”

Each week I’ll ask my fifth graders to do a journal entry about something they’re jazzed about from the mathematical internet.  I’m hoping these researches will inform and guide the way they choose to spend their free-choice time in class and the way they spend their time outside of school.  We’ll see where things lead!

So finally, the proximate cause of this post is that I got several search hits on I Choose Math that were clearly trying to find Math Munch.  Hence the redirect. But I’m also so glad to share this new venture with you!  You’re of course welcome to read Math Munch yourself, as well as to share it with your students as you find useful.  And if you have some cool mathy links that more people should know about, please send them my way!

“Bon appetit!”

“Modeling” – Part 1

In what I’m pretty sure was the first-ever instance of a live-tweeting of our weekly math department meeting, Paul sent out a tweet about a presentation I was giving in which he called me out to blog about its contents.  So here it goes.

First, a backdrop: this year for the first time, our department is reserving a chunk of our weekly meeting to share curriculum.   Nice.   I signed up to be the sharer for the second week.  I wasn’t sure what I was going to present.  Many of the things that I do in my classes I’ve shared with my colleagues over the course of time in less formal ways, and I didn’t want to be redundant.  In the end, I decided to share two different takes on mathematical “modeling” that I’ve done with my middle schoolers.  This post is about the first one.

Here is a first worksheet of an investigation of something that’s called a “measure task”:

The idea is to take some mathematical property that has some intuitive meaning and to attempt to formalize it—in particular, to try to come up with a way of measuring it numerically.  I think this “squareness” measure task is an awesome one because it has a very low threshold and there are many ways that students can approach it, regardless of their prior knowledge.

For example, last year I did this “squareness” worksheet with a faster-paced seventh grade class and a slower-paced sixth grade class.  The seventh graders had a fair amount of experience with expressing quantities algebraically, both from prior years and from our previous units.  They went straight for comparisons of length and width, coming up with formulas like \frac{l}{w}   and \frac{l+w}{w}  .  This brought up interesting questions—like whether these different expressions would give the same ordering of the rectangles, and whether they were really capturing our intuitions.  On the latter point, notice how the formula \frac{l}{w}   results in widely different measures for the squareness of 1×5 and 5×1 rectangles.

My sixth graders approached this problem through the lens of fractional parts.  They decided to measure squareness by the (smallest) fraction of the rectangle that needs to be chopped off in order to make a square.  Lacking algebraic sophistication, they landed on a simpler, “uncoordinatized” method for measuring squareness that was visceral and re-upped a core concept that we had been discussing.

I’m pretty sure that I first heard about this kind of task from Avery.  He’s even blogged about it here.  Some fabulous resources for finding out more—both in terms of theory and practice—are these two articles:

‘Creating Measures’ Tasks by Swan and Ridgway

Formulating Measures by Schwartz

In my presentation, I said that the measure task paradigm is both flexible and mathematically potent–in terms of content, lesson structure, and aims.  Measure tasks can be used to tackle mathematical concepts from the most elementary to the most advanced.  Like I said, I’ve used the “squareness” task with 6th and 7th graders, and I’m sure that this and similar tasks could be done with elementary schoolers.  On the other hand, some of the tasks that my colleagues brainstormed—like measuring how curvy a curve is—would be great tasks for a calculus class.  Once you get rolling, there are so many mathematical concepts that can be approached in this manner.

The lesson structure of measure tasks is also flexible.  They can serve as brief intros to a new concept at the beginning of a period that then swiftly moves on to other pedagogical modes.  They can be one-off lessons.  They can be used as review or extension activities, or even as luxurious several-day explorations.  As with content, the time-frame and commitment needed to bring measure tasks into a classroom are both as flexible as could be.

Finally, the goals that one might have for introducing measure tasks to students are varied.  The focus could be on particular content (rectangles or squares, in my example), applying concepts in new contexts (fractions or algebraic translation), or just on the mathematical process of creating, testing, modifying, and comparing.

I found talking (and now writing) about measure tasks to be exciting–I can’t wait to use them more often in my classes!  My colleagues really got into the idea as well, making great connections and brainstorming task ideas of their own.

Brainstorm in the comments?

Next up: Part 2.

A First Plank Across The Feedback Swamp

Much of my geometry class is built around a series of what I call Investigations. My students just wrapped up their work on the second one of the year.  This Investigation explores different kinds of geometric properties through a set of problems–position, size, shape, connection, and dimension.  For the Investigation, students can try their hand at several of the problems, but after initial forays they choose one problem to dig into and then do a write-up about their results.  You can view the collection of problems here: Investigation #2.

On Wednesday, my students turned in their write-ups and we had time for most of them to do a short presentation about their work.  On their warm-up for the day were a couple of  housekeeping questions, as well as the following:

What kind of feedback do you want on your first two Investigation write-ups? Are there parts of your work for which you are especially interested in my feedback?

Over the summer, I wrote about a minor epiphany that hit me about my struggles with giving useful and timely feedback to my students about their work.  In short, I always end up feeling swamped and overwhelmed by wanting to “do right” by my students–to give them the individualized attention that I know they deserve. To help to get me around this sinkhole, I realized that I should be asking my students about the kind of feedback they want.  I figured that this would make the task of giving feedback feel less like an infinite task where I needed to be all-seeing and say the “right” things and more like a conversation where the goal is to be relevant and helpful.

In teaching, of course, nice theories need to be borne out in practice.  What would my students say when I asked them what kind of feedback they wanted?

Here are a few:

“I would like some pointers on how to write a clearer math paper.”

“I would actually like very harsh feedback.  No sparing of feelings please.”

“Things I could have done more precisely.”

“I would like feedback about how clear I am in explaining and if my calculations are correct.”

“I don’t know.”

These are all great first stabs, including the last one.  These responses will each help to focus my reader’s eye and will shape the comments I give to individual students.

By asking and continuing to ask my students about what feedback they want on their assignments, I hope–and dare even expect–that they will become more reflective about their work, both upon its completion and during its progress.  I can already see it making me feel more comfortable and confident in giving feedback.  And I know that it will help me to better serve them and to let them know that I care about them and that I want to help them to meet their goals and to flourish.