TMC 12 - Some other "AnyQs" I've always had about "real-world" problems but been too ashamed to admit in public that I have

I am so appreciative of Dan Meyer's digital media problems and set-ups as well as his wholehearted spirit of collegiality. I have made what I'm sure must have been perceived as strange or totally off-the-wall comments or observations, and he has never been anything but gracious, kind, and supportive, both online and in person. Sometimes this has involved beer, but I like to think it has mostly to do with his innately generous and collaborative spirit.

So at my session at Twitter Math Camp 12, I felt brave enough to admit to some of the questions I've found myself having as a non-native speaker of math teaching who walks among you. I confessed that they do not sound like the typical questions I feel are expected to be generated by students, although there are plenty of students in math classrooms who, like me, are non-native speakers.

The perplexing thing is, they generated a lot of interest and conversation about on-ramps for students into a state of flow while doing mathematical activity, so I thought I would make a list of them here. So without editing, here is a list of the questions I prepared as part of my thinking as I was working through the issues of flow for students to whom the physics-oriented world-around-us questions are not the most natural ones to raise.

I often look at Dan's digital media problems and set-ups and find myself wondering...

• Does it always work that way?
• Does it ever deviate?
• Are there any rules of thumb we can abstract from observing this process?
• Are there any exceptions? If so, what? If not, why not?
• How long have people known about this?
• Who first discovered this phenomenon?
• How was it useful to them in their context?
• How did they convince others it was an important aspect of the problem?
• Did the knowledge it represents ever get lost?
• If so, how/when was it rediscovered?
• How did this discovery cross culture? How did it cross between different fields of knowledge?
• What were the cultural barriers/obstacles to wider acceptance of these findings as knowledge?
• What were the implications of a culture accepting this knowledge?
• Why do I feel like the only person in the room who ever cares about these questions?

It made me realize I object to the characterization of mathematics as the exclusive slave to physics. It also makes me want to introduce students to other fields (such as economics, financial modeling, forecasting and projections, free cash flow analysis, business planning and marketing planning).

It also made me realize that I am not, in fact, alone.

SOLVE - CRUMPLE - TOSS in Algebra 1: hommage à Kate Nowak

Kate Nowak creates some of the most innovative and engaging practice activities anywhere -- especially for those skill/concept areas that are more like scales and arpeggios than like discovery/inquiry lessons. Some skills, like basic math facts, simply need to be practiced. This is true not because students need to be worn down but rather because it takes the mind and body time and first-hand experience to process these as matters of technique. It takes time to get used to the new realities they represent.

Nowhere is this more true than in tinkering with the multiple different forms and components of linear equations in Algebra 1. No sooner have students gotten the hang of finding the intercepts of a line than they're asked to find the slope. They figure out how to find the slope and the y-intercept, and they're given the slope and a non-intercept point. They figure out how to crawl toward slope-intercept form, but fall on their faces when asked to convert to standard form. Standard form, point-slope form, slope-intercept form, two points and no slope, it's a lot of abstraction to juggle. Mastery is part vocabulary work, part detective work, part scales and arpeggios, and part alchemy of different forms. It's a lot to take in.

Enter Kate's Solve - Crumple - Toss activity. I have loved this practice structure since the day I first read about it, but I have struggled with the fact that the most engaging part of the activity destroys the paper trail/evidence. This was less important with high school students, but it is really important with middle schoolers, I find, because they are so much more literal.

For today's linear equation-palooza in class, I created a basic "score sheet" for each student and I numbered each of the quarter sheets on which I glued blocks of problems (4-6 problems per mini-sheet). I also differentiated them from "Basic" level (Basic-1 through -4, Level 2-1 through -4, etc), so that students could choose their own levels. Students were also invited to work in pairs or groups of three because I find it encourages mathematical language use and increases risk-taking. It also seems to be more fun.

After the "Solve" part of the activity, students brought their solved mini-sheets to me to be checked. If they completed the problems correctly, they got a stamp on their score sheet and proceeded to the back of the classroom where I'd set up the Tiny Tykes basketball hoop over the recycling bin. There they completed the "Crumple" and "Toss" stages, awarding themselves a bonus point on the honor system if they made the shot. Then they returned to the buffet table of problems and chose a new mini-sheet.

Because my middle school students like to bank extra credit points toward a test wherever they can, I like to attach these to practice activities such as this one or Dan Meyer's math basketball. Being more literal and concrete than high school students, middle schoolers seem to find great comfort in the idea that they can earn extra credit points ahead of time in case they implode on a quiz or test. What they don't seem to realize -- or maybe they do realize and they just aren't bothered by it -- is that if they participate in the process, they win no matter what. Either they strengthen their skill/concept muscles and perform better and more confidently on the test; or they feel more confident and less pressured because they have banked a few extra-credit points for a rainy day; or both.

It was fun to hear my previously less-engaged students infused with a rush of sudden, unanticipated motivation to tease apart a tangled ball of yarn they have previously been unmotivated -- or uncurious -- to unravel. And something about the arbitrary time pressure of trying to complete as many problem sheets as possible in a short period was also fun for them. I'm feeling a little ambivalent about not having found the secret ingredient of intrinsic motivation in this required blob of material. But I am grateful that, once again, an unexpected game structure generated what the late Gillian Hatch called "an unreasonable amount of practice."

The last word goes to the one student who put it best: "The crumpling is definitely the most satisfying part."

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 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.