On remembering to allow time and space to refill the well: lessons from Ursula Le Guin

A long time ago, in a galaxy far, far away, I studied at a women's writing workshop called "Flight of the Mind" in the forests of central Oregon.

It was an amazing experience, organized and run in ways that taught me about a lot of things in addition to honing my writing craft — insights that come from wisdom rather than knowledge, the kind of deep-rooted knowledge that comes from the soul rather than from the intellect.

One of the most important lessons I learned there was the need to stay with my own efforts and to protect my creative life — even to the point of becoming fierce in defense of my own needs. Regeneration of energies in preparation for a new beginning is what I am thinking of now, and I was blessed to have this lesson hammered home one day at lunch with the legendary writer Ursula Le Guin on the deck overlooking a bend the wild McKenzie River, in the dappled shade of ash trees.

We were talking about how many new threads and ideas we wanted to follow after the workshop, but how tired we were and afraid of running out of steam.

She told us something that has stayed with me and that seems applicable to life as a teacher as well.

She said, "You need to build in time to 'refill the well.' You've been working hard, and experiences like this one draw out even more energy and insight from you. The temptation is to go home and dive into the work with renewed intensity, but the reality is that you need to allow time for your spirit to refill the well. Be sure to give yourselves time and space to allow it to soak in. This is time that is not wasted."

Later that week, I watched her climb out onto a boulder a good way from the shore. She climbed gingerly, but with quiet confidence. When she reached the spot where she wanted to sit, she took her time settling in and hugged her knees to her chest, watching the water rushing over the clear pebbles on the river bottom.

I always have to remind myself of this when I hit this point in the process. One time is ending and the next time has not yet begun. I am tempted to allow this between-time to be gobbled up or frittered away, either in cleaning up the old year or in setting up the new year. I forget the need to just sit still and let myself rest and digest. I forget that this time is not wasted.

I am refilling the well.

Thank you, Ursula, for the gift of this memory.

Mathematical Language Manifesto — including mathematical language skills in SBG

Yeah, I'm talking to you. But mostly I'm really writing this for myself.

One of the biggest holes I see in our SBG skills lists is the skill of using correct mathematical language. Not just the names of things, but the verbs.

Oh, the verbs, they are killing me.

Here's an example of the toxicity of mathematical babytalk.

In the math lab at lunch time, I listen in on the students trying to answer tutors' or teachers' questions. "Um, you times it by two...?" or "You minus one." or "You put three."

Which makes me say to myself, You WHAT?

What bothers me even more is hearing how other teachers respond to this. Most of them don't. They simply cringe and try to ignore it, as if overlooking the use of babytalk will improve the doing of the mathematics on paper and in the mind.

And I have not noticed magical thinking to be an especially effective intervention in the classroom.

I started listening in with regularity and I noticed two related correlations:

• When students sound ignorant, they tend to be treated with subtle (or not-so-subtle) contempt.
• When students sound knowledgeable, they tend to be treated treated with respect.

This gave me pause. It also made me wonder if I am doing this too. Which naturally gives rise to worry.

So since the only way I know to address this kind of blind spot is head-on, I've decided to address it head-on. No shame, no blame. Just another couple skills for students to master on the skills checklist.

I want to set my students up to sound knowledgeable and be taken seriously as math learners. And that means I have to encourage their courage by means equipping them with the language skills (both oral and written) to get themselves taken seriously in the math classroom as well as in the world outside.

So I'm adding two mathematical language skills to the skills checklist this year -- both for Algebra 1 and for Algebra 2.

1. Use the appropriate mathematical names for arithmetic operations (noun forms) when answering short-response questions
2. Use the appropriate verb forms of basic arithmetic operations when answering short-response questions

Same SBG rules as ever: Just show me you can do it twice perfectly and you'll be off the hook for those questions in the future — forever.

But secretly, in my heart of hearts, I'll be hoping that by that point, they will already have internalized a better set of language habits.

Number Sense Boot Camp - Request for Feedback and Input

Maybe all of your Algebra 1 students showed up on Day 1 every year with a solid and fluent grasp of basic number sense, but mine sure didn't... and it scared the crap out of me. And then afterwards it haunted me, ALL   YEAR    LONG . . .

The stuff they didn't get was just mind-boggling to me:

• subtracting
• adding a negative number
• the basic concepts of the real number line
• fractions
• measuring
• counting
• basic ops with fractions
• absolute value (any related topic)

I mean, this is basic citizenship numeracy stuff, on the same order as basic literacy.

So since this does seem to be a general condition I am likely to encounter anywhere I am likely to teach, I decided to develop a "Number Sense Boot Camp" unit I could use to start the year off with, diagnose critical number sense deficits, use as an occasion for teaching basic classroom routines, give students a chance to dust off (or remediate) their basic arithmetic skills, and basically give us all a fighting chance of getting to some introductory algebra work.

Another thing that worked this year was stealing adopting game-like practice structures, such as those advocated by Kate Nowak in New York state and by the late Gillian Hatch in the U.K. As Gillian Hatch said, a game can provide "an intriguing context" as well as "an unreasonable amount of practice" in vocabulary, reasoning, procedural skills, generalizing, justifying, and representation than they might otherwise be inclined to do. As Hatch said, it also seems able to lead students "to work above their normal levels." As anyone who has tried any of Kate's practice structures can attest, there is something about introducing this playful element that really gets students to dive in.

IDEA #1
One thing I did this past year that worked for many individual students was to do some specific work with the real number line. I made a printable number line and gave each person their own number line (downloadable from Box.net folder) and a plastic game piece to use with it as a calculating device.

Since the rudiments and rules of board games have such wide currency in our culture, most students found this a helpful physical metaphor that gave them both conceptual understanding and procedural access to basic counting, addition, and subtraction experience that had eluded them in their previous nine to eleven years of schooling.

These had the added benefit of conferring prestige upon those who had shown up for extra help and received their very own set (though I gladly handed them out to anybody who requested one).


IDEA #2

Emboldened by my initial success, I realized could expand the idea of a number line "board game" to use as a basic structure for practice – both in using the number line and in many other basic number sense activities.

It even dawned on me that this could be made extensible by having different kinds of "task cards," depending on whether a player has landed on an even number, on an odd number, or on the origin (a decent justification for considering even- and odd-ness of negative numbers here ; go argue over there if you have a problem with this).


Players move by rolling one regular die and one six-sided pluses-and-minuses die (+ and –) (kids seem to need grounding in the positive and negative as moving forward and backward idea). Kids earn "points" in the form of game money, which could carry over and be used to purchase certain kinds of privileges (such as a "free parking" pass for a day when they don't have their homework to turn in).
Your thoughts?

UPDATE: 
Here are links to the different game boards, along with descriptions of each.
Basic Printable Number Line For Use With a Game Piece:
http://www.box.net/shared/eyy4nvhbtn5xx1qdc9j2

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 1-up version (for use with your basic at-home printer):
http://www.box.net/shared/nv0sdz65hy5p3hv8ix1x

Printable Number Line Game Board With Spots For 3 Sets of Question Cards - 3-up version (prints a 24" x 24" poster at FedEx Kinko's--costs about 2 dollars):
http://www.box.net/shared/d4uly1arl88lm3qu9vsm

I made Game Card files using Apple's Pages software (for Mac OS X) and MathType equation editor. You can use these as templates or make your own:
http://www.box.net/shared/s6ha4ol1o6tk0xltp1y1
http://www.box.net/shared/7or8g5klub7jiymshq0f
http://www.box.net/shared/5vq6cmpcd9f9lq1qhud1

Here is a link to the folder itself if you'd like to share and upload your own documents or samples:
http://www.box.net/shared/ftzkun7cvi5vxgvanvh5

Please share any experience or insights you have with them. Enjoy!


AND FUTHERMORE:
Julia (@jreulbach on Twitter who blogs at ispeakmath.wordpress.com) has started a Number Sense Boot Camp page on the Math Teacher Wiki where you can share and find other Number Sense Boot Camp ideas and activities. Available at http://msmathwiki.pbworks.com/w/page/42105826/Number-Sense-Boot-Camp .

UPDATE - 14-Sep-11:

It's only been one day since I introduced the tournament of "Life on the Number Line" but I am already excited about how well this is working out. It is exposing ALL kinds of misconceptions and misunderstandings about adding a negative and about interpreting negative and positive as movement along the number line. Students are playing individually as a "team," and the team with the highest number of correctly worked problems will win 10 free points (2 problems using the 5-point rubric for each person) on next Friday's unit test.

     Since they are surfacing all kinds of misunderstandings about + and - movement on the number line, this is leading to vast amounts of mathematical conversation to get it figured out. So basically, they are teaching each other about adding negatives and subtracting negatives and interpreting that as movement along the number line. 

     I can see that each day it will make sense to give some daily "notes" at the start of class on clearing up common misconceptions I've seen the previous day in students' work so they can solidify their conceptual understanding as well as their procedural fluency a little more each day.

     Best moment yesterday: a girl looked up at me beaming and said, "This is way more fun than doing math!"

     I said, "Good!" but I was thinking, "You have no idea how much math you are actually doing!" :-)


ANOTHER UPDATE:
Here are the game cards to use on the first day: http://msmathwiki.pbworks.com/w/file/45547360/1st%20batch%20of%20game%20cards.pdf


And here is a generic worksheet (front and back) you can print out and give to the kids to use as their template:
http://msmathwiki.pbworks.com/w/file/45547628/generic%20worksheet%20for%20Life%20on%20the%20Number%20Line.pdf

If you have only a ton of basic 1-6 6-sided dice, use Post-Its to make two (2) plus-and-minus dice for students to use with one (1) regular numbered die. This is a good task to give to a student helper. ;-)

FINAL UPDATE:

Four final things:

Thing #1
This unit confirmed me for that kids really do need active, multi-day practice in "living life on the number line" to gain a sense of positives and negatives as directions WHILE AT THE SAME TIME they are developing a sense of positives and negatives as additive quantities. It's not enough for us to just wave the idea of life on the number line at students. It doesn't make sense to them. They really needed experience alternating between (a) positives and negatives as indications of directional movement and (b) positives and negatives as additive or subtractive quantities in the process of deepening their additive reasoning skills.

Thing #2
Right before we started, I had the bright idea to give every group TWO +/- dice and ONE six-sided number die. If you don't mind my saying so, this ended up being a master stroke because it forced students to think about rolling (–)(–)(3) and rolling (–)(+)(3) and every possible combination thereof. This one thing alone might have done the most to deepen their sense of additive reasoning and of +/– as directions of movement.

Thing #3

Here's a link to a zip file that contains ALL of the game cards I created for this unit (on the math teacher's wiki): Game Cards- ALL

For all those who have asked and those who are thinking of asking, I'll say that my school uses the California edition of the McDougal Littell Algebra 1 textbook (by Larson, Boswell, Kanold, and Stiff). For this reason, the game cards are targeted at each of the lessons in Chapter 2. However they are not tied to that textbook and could easily be used with any curriculum or textbook (just sayin').

Thing #4

I'll have to take a photo of the final game boards our instructional aide mounted and laminated for us. They are a true work of art!

Unmediated Experience, part 2 — another use of primary texts in the math classroom

While gathering problems for my logarithms and exponential functions unit from some of my favorite textbook sources, I noticed that John Napier's birth and death dates (1550-1617) looked kind of familiar — really familiar, in fact.

In fact, they're almost the same as Shakespeare's birth and death dates (1564-1616).

And they're awfully close to Galileo's birth and death dates too (1564-1642).

So why not at least mention the historical context of logarithms?

Getting students writing and reflecting in math class

I used this idea in Algebra 2 during our polynomial graph sketching unit to support their conceptual understanding of end behavior, and it worked surprisingly well.

Most of my students had not had much truck with the idea of infinity, much less with the idea of approaching it, so we started by talking about how the graph of a function is a representation of a specific story — like a novel — or like the life story of a human being. Every person's life story is completely unique, yet everyone has an experience of both birth and death (the beginning and end being like end behavior). Everyone's birth story may be different -- some people are born in a taxi cab, some are born at home, some are born in a hospital, some are born under water, and yet every single one of these births has the very fact of having gone through birth in common. Similarly, death is the ultimate "end behavior" of life. We cannot know exactly how or when a specific person's death may come, but we do know that it is going to happen some time after they have been born and have lived for some duration of time.

We also talked about the movement of a function graph across the axis at its x-intercepts — how at certain key transformative moments, a person's life crosses from negative into positive territory -- or it may move in the opposite direction.

Unmediated Experience — part 1the case for using primary texts in the math classroom

Because I was a literary historian before I became a math teacher, it bothers me to see how little direct contact our math students have with primary source documents.

It bothers me a lot.

In the humanities, primary sources are the lifeblood of the curriculum. We don't limit students to reading about  historical or cultural artifacts, events, or texts. We bring students directly into relationship with those texts themselves.

What kinds of distorted ideas might a person develop if she or he had never wrestled with the original text of the preamble to the Declaration of Independence —

When in the course of human events, it becomes necessary for one people to dissolve the political bands which have connected them with another, and to assume among the powers of the earth, the separate and equal station to which the Laws of Nature and of Nature's God entitle them, a decent respect to the opinions of mankind requires that they should declare the causes which impel them to the separation.

Or the original text of the preamble to the U.S. Constitution —

We the people of the United States, in order to form a more perfect union, establish justice, insure domestic tranquility, provide for the common defense, promote the general welfare, and secure the blessings of liberty to ourselves and our posterity, do ordain and establish this Constitution for the United States of America.

Pedagogically, in the humanities, we find great value in connecting directly with the words and thoughts, hopes and dreams, and even biases and delusions of those who came before us. When we do so, we connect with what is most powerful — and most human — in the enterprises and events we choose to investigate.

So why, I wonder, do we not do the same thing in the math classroom — at least from time to time?

Mathematics is a cultural and historical phenomenon. Acts of mathematics are performed by human beings who were born and who lived in times that were both similar to and different from our own. Giving students some experience of direct access to primary texts is an easy and cost-effective way to give them a basis — and context — for their own relationships with mathematics.

At least, that's one of the things that helped me the most when I first decided to cultivate my own relationship with math teaching and learning.

I think this is one of the most compelling — and least well-articulated — benefits of Dan Meyer's WCYDWT pedagogy.

The Rough Guide to the SBG Rubric

I've been wrestling with the question of whether or not to include a 3.5 in my 4-point rubric, but the problem is, I keep coming up with one big reason against it:

Nothing seems to motivate a student on the brink of insight like a score of "3 out of 4."

Nothing breathes life into a student's determination like that *forehead smack* they get when they realize what a careless, boneheaded, nearly unforgiveable error has kept them from achieving a perfect score on a skills quiz.

Their mistake is usually something pretty witless -- a negative sign, say, that got dropped during a hurried calculation, or an inability to decipher their own messy chicken-scratch work. Wait, that's not really fair to chickens. Their scratching is much more deliberate.

I've heard all the begging for an extra half-point, all the groveling, cajoling, sullen threats of self-punishment. And I'm not buying it. Nope, sorry. Not convinced.*

Sorry.

Seeing how they suffer reminds me of something I've learned through years of meditation practice -- namely, that as a teacher, it would be uncompassionate of me to take their struggle away from them.

That struggle is the sign of their genuine engagement with the mathematics. When they are suffering the loudest is the exact moment when they are most involved in the process of doing mathematics.

That heart-piercing moment of angst at their own careless error -- and the lengthy, painful, and often howling soliloquy that follows shortly thereafter -- is a critical part of their learning process. And -- speaking only for myself -- I've come to realize that I do them a disservice when I lessen the sting of that 3 out of 4 score.

How the heck can half a point make such a difference?

The best answer I can come up with is, I don't know. I only know that it does.

All I know is that for some reason, that 3 out of 4 pricks at their ego, it pokes at them, it makes them a little crazy. It jump-starts a hunger for the kind of focus and determination they will need to get themselves to the next level.

And I should probably be honest with you about something.

I like that kind of clarity in a student.

This is the point that most of my students usually never get pushed past. It's the same place where I myself -- as a once-discouraged math learner -- almost allowed myself to get left behind.

The teacher tends to move on, driven by an overburdened curriculum, while the bulk of the class trudges along behind him or her, dragging their sorry butts and their overstuffed backpacks that are filled with the kind of crumpled and torn papers that could only result from a person rolling with them in the dirt or using them as padding on a bed of nails.

For whatever reason, a 3.5 out of 4 just doesn't seem to have the same sting. In fact, it seems to take the sting out of marginal failure, reducing the urgency to give that last little oomph of a push that gets a person to the next level.

I think that, in a weird way, that little affront is kind of good for them. It has nutrition in it -- nutrition that they need.

-----------

* Now, if a student were to threaten to smash her iPhone to smithereens, that is a mental health threat that I would take very seriously. But this is basically the adolescent equivalent of threatening to hold their breath until they pass out.