Chords and Secants and Tangents, oh my!

Once you get to circles, chords, tangents, secants, arcs, and angles in Geometry, it's all just a nightmare of things to memorize — which means things to forget or confuse.

Rather than have students memorize this mishmash of formulas, I decided to have them focus on recognizing situations and relationships. And a good way to do that is with a four-door foldable.

Make it and use it.

Humans are tool-using animals. So let's do this thing!

Doors #1, 2, 3, and 4 each have only a diagram on their front door with color-coding because, as my brilliant colleague Alex Wilson always says, "Color tells the story."

Inside the door, we wrote as little as we possibly could. At the top we wrote our own description of the situation depicted on the door of the foldable. What's the situation? Intersecting chords. What is significant? Angle measures and arc lengths. What's the relationship? Color-code the formula.

Move on to Door #2.

We did the same thing with the chord, secant, and tangent segment theorems.

These two foldables are the only tools students can use (plus a calculator) for all of the investigations we have done with this section. They are learning to use it like a field guide — a field guide to angles and arcs, chords, secants, tangents. I hear conversation snippets like, "No, that can't be right because this is an intersecting secant and tangent situation."

My favorite is the "two tangents from an external point" set-up, which we have dubbed the "party hat situation" in which n = m (all tangents from an external point being congruent).

A big part of the success of this activity seems to come from teaching students how to make tools and to advocate for their own learning. Help them make their tools, set them loose, and get out of their way.

That's a pretty good strategy for teaching anything, I think.

TMC #14 Group Work Working Group Morning Session – Annotated References & Framework

I'm having a lot of fun planning the Group Work Working Group morning session for Twitter Math Camp 2014, and it's time to start sharing.

Here is the background material I'm using for developing the group work morning sessions. Please note that this is NOT required reading!  Recreational reading only! So please don't freak out!  :)

I wanted to give people a sense of the framework and background I'd like us to start from so attendees can decide whether this morning session will be right for them. I also wanted to provide links and titles to valuable materials.

These are listed in order of relevance to the Group Work Working Group morning session — they are not in formal bibliographical form.

National Academies Press, How People Learn (downloadable PDF here)
This amazing free book provides the framework within which we'll consider the use of group work. I am especially keen for us to explore how we can develop and implement tasks that fit within their (approximately) four-stage cycle for optimizing learning with understanding while also fitting with our own individual school and district requirements. In a nutshell, the four stages are as follows:

STAGE 1 - a hands-on introductory task designed to uncover & organize prior knowledge (in which collaboration cultivates exploratory talk to uncover and organize existing knowledge)

STAGE 2 - initial provision of a new expert model (with scaffolding & metacognitive practices) to help students organize, scaffold, & develop new knowledge (in which collaboration provides a setting to externalize mental processes and to negotiate understanding)

STAGE 3 - what HPL refers to as "'deliberate practice' with metacognitive self-monitoring" (in which collaboration provides a context for advancing through the 3 stages of fluency with metacognitive practices)

STAGE 4 - transfer tasks to extend and apply this new knowledge & understanding in new and unfamiliar non-routine contexts

Malcolm Swan, "Collaborative Learning in Mathematics" (downloadable PDF here)
A short and highly readable summary of Swan's instructional design strategy for collaborative tasks, including notes on his five types of mathematical activities that constitute the bulk of the Shell Centre's formative assessment MAP tasks and lessons.

Malcolm Swan, Improving learning in mathematics: challenges and strategies (downloadable PDF here)
An in-depth introduction to Swan's approach to designing and using the kind of rich tasks offered by the Shell Centre and the MARS and MAP tasks.

Chris Bills, Liz Bills, Anne Watson, & John Mason, Thinkers (can be purchased from ATM here)
The richest source book imaginable for ideas for activities to stimulate mathematical thinking. Often credited by Malcolm Swan and Dylan Wiliam.

Anne Watson & John Mason, Questions and Prompts for Mathematical Thinking (can be purchased from ATM here)
The richest source book imaginable for variations on questioning and prompting strategies.

Dylan Wiliam, Embedded Formative Assessment
This book is a gold mine. Don't leave home without it.

Thoughts On Making Math Tasks "Stickier"

Last year, the book that changed my teaching practice the most was definitely Dan Pink's Drive: The Surprising Truth About What Motivates Us. It helped me to think through how I wanted to structure classroom tasks in order to maximize intrinsic motivation and engagement.

This year, the book that is influencing my teaching practice the most would have to be Made To Stick: Why Some Ideas Survive and Others Die by Chip and Dan Heath. I bought it to read on my Kindle, and I kind of regret that now because it is one of those books (like Drive) that really needs to be waved around at meaningful PD events.

The Heath brothers' thesis is basically that any idea, task, or activity can be made "stickier" by applying six basic principles of stickiness. Their big six are:

1. Simple
2. Unexpected
3. Concrete
4. Credible
5. Emotional
6. Story

The writer in me is bothered by the failure of parallel structure in the last item on this list (Seriously? SERIOUSLY? Would it have killed you to have used a sixth adjective rather than five adjectives and one noun? OTOH, that does make the list a little stickier for me, because my visceral quality of my reaction only adds to the concreteness of my experience, so there is that). But that is a small price to pay for a very useful and compact rubric. It also fits in with nicely with a lot of the brain-based learning ideas that @mgolding and @jreulbach first turned me on to.

This framework can also help us to understand — and hopefully to improve —a lot of so-so ideas that start with a seed of stickiness but haven't yet achieved their optimal sticky potential.

I wanted to write out some of what I mean here.

For example, I have often waxed poetic about Dan Meyer's Graphing Stories, which are a little jewel of stickiness when introducing the practice of graphing situations, yet I find a lot of the other Three-Act Tasks to be curiously flat for me and non-engaging. Some of this has to do with the fact that I am not a particularly visual learner, but I also think there is some value in analyzing my own experience as a formerly discouraged math learner. I have learned that if I can't get myself to be curious and engaged about something, I can't really manage to engage anybody else either.

Made To Stick has given me a vocabulary for analyzing some of what goes wrong for me and what goes right with certain math tasks. The six principles framework are very valuable for me in this regard, both descriptively and prescriptively. For example, Dan's original Graphing Stories lesson meets all of the Heath brothers' criteria. It is simple, unexpected, concrete, credible, emotional, and narrative. The lesson anchors the learning in students' own experience, then opens an unexpected "curiosity gap" in students' knowledge by pointing out some specific bits of knowledge they do not have but could actually reach for if they were simply to reach for it a little bit.

But I would argue that the place where this lesson succeeds most strongly is in its concreteness, which is implemented through Dan's cleverly designed and integrated handout. At first glance, this looks like just another boring student worksheet. But actually, through its clever design and tie-in to the videos, it becomes a concrete, tangible tool that students use to expose and investigate their own curiosity gaps for themselves.

Students discover their own knowledge gap through two distinct, but related physical, sensory moments: the first, when they anchor their own experiences of walking in the forest, crossing over a bridge, and peering out over the railing as they pass over (sorry, bad Passover pun), and the second, when they glance down at the physical worksheet and pencil in their own hands and are asked to connect what they saw with what they must now do.

This connection in the present moment to the students' own physical, tangible experience must not be underestimated.

Watching the video — even watching a worldclass piece of cinematography — is a relatively passive sensory experience for most of us.

But opening a gap between what I see as a viewer and what I hold in my hands — or what I taste (Double-Stuf Oreos!), smell, feel, or hear — and I'm yours forever.

"My work here is done."

This way of thinking has given me a much deeper understanding of why my lessons that integrate two or three sensory modalities always seem to be stickier than my lessons that rely on just one modality. Even when the manipulatives I introduce might seem contrived or artificial, there is value in introducing a second or third sensory dimension to my tasks. In so doing, they both (a) add another access point for students I have not yet reached and (b) expose the gap in students' knowledge by bringing in their present-moment sensory experiences. And these two dimensions can make an enormous different in students' emotional engagement in a math task.

Life on the Number Line - board game for real numbers #made4math

UPDATE: Here is a working link to the zip file: https://drive.google.com/open?id=0B8XS5HkHe5eNNy10MWZVSDNKNnc

Last year I blogged about my work on a Number Sense Boot Camp, so I won't rehash all of that here. This year I want to give the follow-up on how I used it last year, what I learned, and how I'm going to use it this year in Algebra 1.

This was my breakthrough unit last year with my students. It anchored our entire Chapter 2 - Real Numbers unit and really solidified both conceptual understanding and procedural fluency in working with real numbers, the real number line, operations on real numbers, and both talking and writing about working with real numbers. We named it Life on the Number Line.

Here's how the actual gameboards, cards, and blank worksheets looks in action (sans students):

I sure hope I didn't make a bonehead mistake in my example problem!

The most effective thing about this activity was that it compressed a great deal of different dimensions of learning into the same activity, requiring learners to work simultaneously with the same material in multiple dimensions. So for example, they had to think about positive and negative numbers directionally in addition to using them computationally. They had to translate from words into math and then calculate (and sometimes reason) their way to a conclusion. They had to represent ideas in visual, verbal, and oral ways. And they had to check their own work to confirm whether or not they could move on, as no external answer key was provided.

Since they played Life on the Number Line for multiple days in groups of three or four players comprising a team who were "competing" in our class standings, learners felt that the game gave them an enormous amount of practice in a very short amount of time. Students also said afterwards that they had liked this activity because it helped them feel very confident about working with the number line and with negative numbers in different contexts.

I also introduced the idea of working toward extra credit as a form of "self-investment" with this game. For each team that completed and checked some large number of problems, I allowed them to earn five extra-credit points that they could "bank" toward the upcoming chapter test. Everyone had to work every problem, and I collected worksheets each day to confirm the work done and the class standings.

What I loved about this idea was that students won either way — either they had the security blanket of knowing they could screw up a test question without it signifying the end of the world, or they got so much practice during class activities that they didn't end up actually needing the five extra credit points!

Students reported that they felt this system gave them an added incentive to find their own intrinsic motivation in playing the game at each new level because it gave them feelings of autonomy, mastery, and purpose in their practice work.

The game boards were beautifully laminated by our fabulous office aide but do not have to be mounted or laminated. The generic/blank worksheets gave students (and me) a clear way of tracking and analyzing their work. And the game cards progressed each day to present a new set of tasks and challenges.

All of these materials are now also posted on the Math Teacher Wiki.

Let me know how these work for you!

UPDATE 10/27/2016: Here is a working link to a zip file of all the components for this: https://drive.google.com/open?id=0B8XS5HkHe5eNNy10MWZVSDNKNnc

Rational Expressions Treasure Hunt (or, Intrinsic Motivation for the Intrinsically Unmotivat-able Bits)

We are coming up on our state testing window (STAR testing in California, ugh), which means it's time to finish "covering" all the so-called key standards material in preparation for the last-minute test prep and hand-wringing.

In our Algebra 1 curriculum, the last unit before state testing is the Rational Expressions unit, which I find to be one of those inherently frustrating and procedural sections.

The good news of the rational expressions unit is that it gives Algebra 1 students an opportunity to see how some of the foundational skills and concepts they've been learning can come together to provide some very sophisticated conceptual and computational tools.

The bad news is, students are tired -- tired of the routine of observation - guided practice - independent practice - followed by a chapter test. And they're just plain tired in the sense that they've absorbed a lot in every class they're in and basically their brains are kind of full.

So I reviewed my notes on Dan Pink's book Drive to see how I could weave together his ideas of autonomy, mastery, and purpose. And that gave me the following ideas:

• AUTONOMY - students needed a unit which relies more on self-checking than on getting a teacher seal of approval (or a rubber stamp);
• MASTERY - students seemed to need a unit which would emphasize tracking their mastery of skills and concepts, which meant giving them lots of opportunities both to do and to spot patterns in their doing;
• PURPOSE - students needed a sense of purpose to the overall activity, rather than the "bigger picture" sense of Dan Meyer's WCYDWT (What Can You Do With This?) -  to provide some light-hearted gratification or, as some might put it, a cheap thrill. :-)

As these ideas whirled around, I felt envious of Kate Nowak's wonderful circumcenters lesson (from Geometry) that harnessed fun clues with the need to draw circumcenters on a map of the school as a motivation for hunting down numbered markers which she had tucked around the school.

It occurred to me that I could use a treasure hunt idea (organized more as a geocaching or letterboxing hunt) as an intrinsically self-motivating reward system -- coupled with a series of self-check-able problem sets/worksheets -- to give students a little break from routine and a reason to work together purposefully to practice simplifying rational expressions and finding the excluded values.

So here are the components I created:

1. a set of graduated worksheets (one for each day) for them to use as practice problems
2. a handwritten Answer Key that I slipped into sheet protectors and tacked up on the front bulletin board for students to check their work against
3. a "rubber-stamping station" where I or they would stamp their completed AND checked worksheets to place in their...
4. group folder
5. a Ziploc bag or sheet protector stapled to the inside of the group folder where they could store their treasure tokens
6. a set of "clue" cards, one for each worksheet's treasure token, that students would take together with a...
7. Hall Passes (which clearly states that I have given them permission to snuffle around in the designated area to find their treasure tokens and come back to my classroom)
8. one weatherproof plastic Gladware container PER treasure box/worksheet to contain the treasure tokens for each worksheet/level (I used Woodsies stars that I bought at Michael's and labeled them 11-2a or 11-2b to correspond to their worksheets, but you could also use punched-out paper or cardboard shapes of any type); treasure boxes were labeled with masking tape and also contained a marker and an index card on which students would write their group's initials when they removed a token

A couple of observations. First of all, your "clue cards" will be different from mine based on your school and where you wish to send your students on their searching. Being a California indoor-outdoor type of school that is built into a hillside, we have a lot of beautifully xeriscaped/landscaped pocket gardens that made for good treasure box hiding places. So my clues said stuff like, "Facing room ELEVEN, TWO TIMES TEN paces or so to your right, you'll find a bathtub fit for a bird. Look below the bushes behind..." Or "Head toward LUCKY 13 and look for the LAVENDER HEDGE. The treasure box is tucked behind..."

Students responded positively to this shift toward autonomy. They delight in completing a practice task that grants them permission to snuffle around a little outside in the fresh air. And they seem to love unraveling the clues, maybe because they love reading and thinking about themselves and their own environment. The Hall Pass requirement gave them a clear set of expectations and accountability, and even my least accountable students have been taking it seriously, which is a refreshing change of pace for all of us.

Their assessment will come from my review of their group project folder -- its tokens, its completed and corrected worksheets, and the presence or absence of worksheets for each student in the group.

The worksheets and sample hall passes and clues are up on Box.net

The Big Ideas inside the 'big ideas'

As I've been organizing my job search materials, I've been reflecting on some of the Big Ideas I have learned are the most important among the too-many "big ideas" our textbook and state standards emphasize for Algebra 1.

One of the problems with the state standards is that they can't let go of anything as being less important than anything else. Which is why there are 26 overall standards, plus embedded sub-standards inside the standards, and the whole thing is a nasty ball of yarn to try and untangle.

One thing I've tried this year, which seems to be working well, is to choose my emphases based on what developmental psychologists have discovered about children's mathematical development. I am particularly grateful for the work of the British psychologists Terezinha Nunes and Peter Bryant, who also do a lot of work with Anne Watson of Oxford's math education program. Nunes and Bryant's book, Children Doing Mathematics, has really blown my mind open to what they call the "generative" quality of children's mathematical development and number sense — that is to say, kids develop their sense of number and of mathematics in layers, the way some inkjet printers work, with each pass of the printhead setting down another layer that completely transforms the image that is emerging on the paper.

In Nunes and Bryant's synthesis, as well as in their summary of others' research, kids' understanding of quantity is revealed to be an extremely fluid, dynamic, and multi-faceted set of tools. As they put it, "a successfully developed understanding of number comes from four distinct developmental threads" which they summarize as:

1. the ability to COUNT discrete OBJECTS
2. a deep familiarity with a wide range of QUANTITIES OF QUALITATIVELY DIFFERENT KINDS (such as both countable and uncountable quantities)
3. the ability to COMPARE QUANTITIES OR COLLECTIONS of objects, assessing both similarities and differences regardless of their qualitative kind(s)
4. the ability to use established notation for all of these (Nunes and Bryant, pp. 1-20)

Yet, as they have observed, our current curriculum tends to address only #1 and #4.

This helped me understand something I had been struggling with for a long time — namely, the fact that there are many incomplete understandings children develop that are sufficient for them in their context, BUT that are insufficient over the long term as a foundation for mature mathematical understanding. These are the mathematical versions of ideas like "the tooth fairy" or "Santa Claus." They are enabling fictions that are developmentally appropriate in their time and place, though they are not at all what we want our young adults to rely on by the time we release them out into the grown-up world of mathematics.

One of these incomplete understandings — one that drives university-level mathematicians like Keith Devlin head-banging mad — is MIRA, or the idea that Multiplication Is Repeated Addition. Their argument is that this is such a stunted understanding of multiplicative reasoning that it threatens to undermine the very foundations of civilization, dammit.

But in truth, like the idea of the tooth fairy or Santa Claus, MIRA does have a legitimate place in a child's generative mathematical development — as long as his or her teachers understand that, like the idea of the tooth fairy, it is an incomplete understanding that is meant to be expanded upon into a much richer and more scalar concept of multiplication.

The place where I am finding MIRA to be an extremely useful tool is with Algebra 1 students making their first forays into the abstractions of algebraic reasoning, which is to say, in dealing with polynomial arithmetic. I say this because young adolescents are such intensely concrete thinkers. When I ask them to consider combining like terms such as 5 elephants and 3 elephants, they can easily understand what I am asking for. But the moment we start investigating the idea of combining 5x^2 and 3x^2, their heads explode. Things only get worse when they are asked to combine 5x^2, 3x^2, and  6x^3. It seems like they forget everything they have ever known about the combining of like terms, and they start adding or multiplying exponents and or worse things than most of you can imagine.

This is the place where some teachers find algebra tiles to be helpful. But I find that algebra tiles have a grammar and a rhetoric of their own that is not easily extensible into polynomial arithmetic beyond quadratic thinking. Also with their color-coding, they also add in moving parts I find my students are not yet ready to think about.

But guide their attention away from abstraction for a moment, and ask them what happens if they combine 5 dogs and 6 apples. They understand the logic of ConcreteLand completely. In this case, I have found, getting them to think about 5x^2 the way they think about 5 dogs and about 6x^3 the way they think about 6 apples, and their conceptual understanding shoots through the roof. I can even use Brahmagupta's idea of "fortunes, debts, and ciphers" (positive numbers, negative numbers, and zero) to help students think about what happens in a trading economy where I might "owe them" 3 dogs (or 3 x^3) as we barter our algebraic quantities away for each other's lunch components.

Students still need a lot of practice and experience with this whole crazy abstract insanity to cement their understanding in place, and they can be expected to relapse several times into believing that they need to add exponents instead of thinking about coefficients as quantifiers. But eventually the idea of quantifying (and if need be, combining) x^2s the way they count and quantify dogs or apples gives them a surer footing as they begin to construct a new and deeper understanding of multiplying variables. And that is something I can eventually build a rich and scalar concept of multiplication on top of — on that would be appropriate to eventually deliver to Professor Devlin's lecture hall.

I do this knowing that even within a few months, this conceptual framework will be revised and replaced with other incomplete understandings many times over. But I do so knowing that I am teaching my students how to learn by giving them tools for understanding how to build tools that help them understand what the heck they are doing.