Tag Archives: reflection

In what ways can math be a part of your life?

A goal I have for this coming year—or rather a commitment I’ve made, as John Spencer reminds me—is to more consciously cultivate values in my classroom.  Specifically, I want to help my students to think about learning and about mathematics.  Thoughts have been kicking around in my head for a while, and so I’d like to make a first stab at sharing some of them.  Hopefully you’ll be able to give me some feedback and/or find something here useful to your own thinking.

I don’t often get asked “why are we learning this?” or “when are we going to use this?” or other “what’s the point?” questions.  Maybe it’s the kids, or me, or the stuff we do, or who knows.  I do know that this is a live question, whether it comes out or not, and that it’s a question where math teachers usually find themselves on the defensive.

So regardless of whether the questions comes up, I want to address them and teach about them.  Especially:

  • In what ways can math be a part of your life?
  • In what ways will math be a part of your life?

Getting kids asking these question in personal ways is most important—of themselves, their peers, their parents, their parents’ friends, other teachers, and so on.  I’m planning to put together some assignments, journal reflections, and/or free-choice projects to help them to seek out answers.

But I also have my own answers that I want to share with them, and with you.  It’s in four parts.  It’s not meant to be canonical—just the way I’m thinking right now.

  • Math can be a useful tool in everyday activities. When shopping, managing money, estimating, or reading about and discussing issues, some mathematical ideas and skills come up time and again.  These might be lumped under the generic heading “numeracy”.
  • Math can be a social token.  The classes you take, what you score on the SAT, whether you consider yourself a “math person” or not.  Your relationship and experiences with mathematics can be and often are a status symbol, a tattoo, for your whole life.  It can be a burden, a weapon, a bridge, a currency, or a welcome mat.
  • Math can be a tool to investigate the world.  There are parts of mathematics that are essential tools in any specific human pursuit—especially the physical and social sciences.  If you want to do physics or economics, play poker or create mathematically-inspired art, there’s some specific parts of math that you’re going to need to know for each one.
  • Math can be a constant companion and source of enjoyment.  You can do math for fun.  You can explore parts of math that excite you, work on problems and puzzles that are tough but gratifying, let math concepts become your playthings and color your life.

A chart came to mind as I pondered these different roles that math can play.

I especially want to point to the distinction between required and opt-in categories.  There are mathematical situations we’ll all encounter, regardless of the lifepaths we take.  At some point we’ll all do, for instance, some money-related calculations, and at some point we’ll find ourselves in a social-token situation.

Other math experiences are opt-in.  There is math that will be very important to understand if you want to be a physicist or a juggler, but that you can easily enough ignore if you don’t.  There are of course questions about what kinds of foundations students need in various mathematical disciplines so that they aren’t later shut out of opportunities.  Drawing the line between a math education for a generalist and a specialist is a live question for me (and maybe you), but I feel like the governing math ed consensus wrongly forces kids to specialize in ways they’d rather not.  I think this has to do with how unclear the meaning of “math is useful in life” usually is.

It also has to do with the fact that math as a social token is such an enormous but often understated or unseen force in our world.

I’m particularly excited about sharing this idea of math as social token with my students, especially in the context of the other three roles.  I feel like it’ll help them take greater command of their own mathematical identities.

Students have a fair amount of control over these four aspects of their mathematical lives, and they certainly have control over their attitudes towards these aspects. I think that pointing them out could be wonderfully empowering for students.

Classroom experiences and institutional structures that support these different values of course look different from each other.  I feel like just articulating them to myself helps me to think about how I’d like to talk about math with my students.

What do you think of my answers?  What would you add, subtract, merge, or reframe?  Or do you have a different way of thinking about this altogether?

And if you can point to resources for having kids think about these issues, I’m all…clicks?

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.

Free-Choice Time: Gearing Up

Finding the balance of freedom and structure that is good for kids—who are of course all different from each other—is hard.  Still, I know that giving my students a fair shake at some real autonomy is a principle that I’m willing to work for and stand by.  My dear friend and colleague Paul Salomon and I collaborated last year on making free-choice time happen for our middle schoolers.  For me, this meant that every Friday, my sixth and seventh graders would take their SBG skills quizzes; after finishing these, they were free to pursue whatever they pleased.

Inspired by my mom, I distributed a bingo grid of different activities at the beginning of the year to encourage my kids to dig into a variety of tasks.  It looked like this:

last year's bingo board

(BTdubs, Bloxorz was a huge hit.  You should definitely give it a whirl.)

I thought I would roll out other boards as the year progressed.  I didn’t.  The school year, as school years are wont to do, took on a crazy, unkempt life of its own.  While I introduced a few new activities to my classes over the course of the year, the activities that I put on offer at the beginning were basically the ones that students worked on.  In addition, I never got around to providing the resources for students to actually do some of the activities, so this cut down on their options even more.

To better serve our soon-to-arrive new batch of students, Paul and I are aiming to improve upon our last year’s attempts to provide students with structure and rich resources during their free-choice time.  This year I’m giving out a board that has categories of activities, rather than individual things to do.  Like so:

free-choice bingo board

Really broad stuff, plus a center square that rivals Zombo.com in making the sky the limit.  To accompany the board will be a list of some possible activities for them to pursue:

Those are the ones that I’ve penciled in for the start of the year.  The full brainstormed list currently stands as follows:

…although that will probably be out of date as soon as I publish this post.  Over the course of the year, the list of pursuits that my students will have in hand will expand to include more items from this larger list.  I’m planning to limit the options at the start for a few reasons.  First, I didn’t want all of the possibilities to overwhelm kids.  Second, for some of the activities on the full list, I want to have the chance to introduce the activity in a whole-class setting—for instance, to have a day or two where we talk about cellular automata, run some by hand, and take a look at the Game of Life.  Finally, I figure for the sake of diversity, it won’t hurt to put off the unveiling of Bloxorz for a little while.

An important part of my plan to make free-choice time an opportunity for my students to grow is to have them record and reflect upon the progress they make on their free-choice projects, as well as to help them to set goals for themselves.  Students will be able to write their goals into their bingo grid, and they’ll also have opportunities to goal-set and reflect in their journals.  By design, the items in the list above aren’t very specific.  Yes, here is a Rubik’s Cube.  But what’s your goal?  Try to figure out how to solve it on your own?  Watch some videos online and try to internalize some algorithms?  Solve a cube in under five minutes?  Take one apart?  Only you can decide.   My thought is that putting this decision down in writing–even if it will change a few days from now–can give kids more empowered visions of themselves and of mathematics.  That’s what I’m betting on, anyway.

I’ll have a poster-sized copy of the bingo board on the wall of my classroom.  When a kid completes a goal in one of the categories, she’ll get to write her name into that square on the poster.  She’ll also get to share her accomplishment with her classmates, either by announcing it, making a presentation, or displaying it in the room.

It may feel to you—I know it does to me—that in my description of this supposedly free-choice time, I’m emphasizing structure over freedom.  My excuse is that I don’t have live, kicking, and wooly fifth graders to share this stuff with yet, to run rampant in my classroom and test and break and remold my structures—until they aren’t really even structures any more, but rather the culture of our classroom.  Right now, the build-up of thoughts and hopes just lies in potentia, waiting to go live and spring forth.  I can’t wait to share with you the awesome things my students do!

There’s a lot of good stuff on that list of activities, and I’m excited to uncover, stumble across, and get suggestions for even more.  And I feel that the list is not only good, but that it includes things that often have no place in our mathematics classrooms.  And so if I may…

<ascends soapbox>

We as teachers, by the tasks and opportunities we provide in our classrooms, define what mathematics is to our students.  If we don’t model activities like reading books about math, creating mathematical games, or communicating with “outside” people who are interested in math, then our students may never encounter these activities.  And to me, that’s a lot scarier than a kid missing out on any particular fact, theorem, or skill.

<descends soapbox>

To conclude, let me say that I’m working to find ways of incorporating more free-choice time into my high school courses, but this is a challenge.  While I’ve made room for choice in my high school courses, free-choice feels more difficult.  In the usual mold, middle school is more student-centered, while high school is more content-centered.  Still, just typing this out has got me thinking about how I might construct a similar board for my high schoolers to encourage them to try out different kinds of tasks over the course of the year in a non-deadline way.  Stay tuned for that, and for lots of sharing of how free-choice goes down with my fifthies!

At the moment, though, I would love to hear your suggestions for pursuits for my middle schoolers to take on.  Help that list grow!

SBG: Affective Quizzing

When has a student mastered a skill?  Part of the purpose of Standards-Based Grading is to help us answer this question.  (I’d even say it helps us to articulate the question in the first place.)  Different teachers establish skills mastery in their versions of SBG in different ways–for instance, getting everything right on a skill quiz, or getting everything right on two skill quizzes in a row.  In these versions, once a student achieves a certain outcome, she is exempt from further quizzes on that skill.  For other teachers, a kid is never “done” with a skill until the year is over; skills continue to come up on new assessments, and the student’s grade for that skill is his most recent one.  On top of these, I’m sure there are many other awesome variations and hybrids that people have devised that work well for their classrooms.  I’d love to hear about your own twist.

This past year I was in the “one-and-done” camp, and I’m planning on staying there.  That said, I think I will be keeping a better eye out for occasions when a student is struggling with the mechanics of a skill that he has previously quizzed well on.  That may be a good time for a conversation and perhaps reviewing and requizzing.  (Without grades in the picture, there won’t be anything punitive about this–just an opportunity to learn.)  But I’m looking forward to including a new facet to my skills quizzes themselves this year.  Like so:

"How did that go for you?"

It’s that last bit.  I’m curious to see how asking students about their quizzing experiences affects the whole assessment process.  I can’t say exactly where the idea to do this came from, but I think it was at least partially inspired by a student of mine from last year.  She would often write notes to me on her quizzes explaining what she still felt fuzzy on.  This was on quizzes she was choosing to take, buffet-style.   I found that her notes both gave me a better sense for her understanding and provided a really natural way for us to start conversations about remediation.  While there were times that she struggled with our course material, her ability to self-evaluate helped to make her year a successful one.

I’m sure you’ve taught both over-confident and under-confident students.  I feel that by asking “How did that go for you?” on quizzes I may sometimes hear anxiety and doubts from under-confident students even when they get everything right.  In the past, I’ve just checked off their correct responses, said “good job” and moved on.  They were “done” with that skill, despite not feeling at home with it.  No doubt they were in the short-term relieved, but perhaps they were left feeling uneasy about this skill in the long run.  That isn’t what mastery should look like.  By putting that quick gut check at the bottom of each quiz, I’m trying to give my students a safe and immediate place to tell me about their relationship with the skill, right after they’ve shown me their attempts to apply it.  And if they’re telling me they don’t feel like they’ve mastered it yet–even with a perfect paper–then they haven’t.  More steps need to be taken.

I’m sure you can fill in the corresponding things that could transpire with an over-confident student.

Of course, it’s entirely possible that a kid will pose as more confident than he really is, or decide not to respond to the question at all.  To me, it’s not important that all of my students share their feelings toward their skills with me in this way.  If providing the occasion helps even a few of them to do so, the fruits of that will make me feel like it’s well worth asking.

To sum up, I’ve come to think that any answer to the question “When has a student mastered a skill?” needs to include, “when she feels like a master of it.”
I hope that finding room in my classroom for the affective aspect of learning skills will make for more effective assessment and feedback.

(Ah, there’s the punchline.  It turns out I’m not a spelling dummy after all.)