TEN TRUE STATEMENTS - Using Cognitive Load Theory to Build Toward Mastery of Proofs - Geometry

 I've been reading and thinking a lot about cognitive load theory in Geometry class, thanks to Michael Pershan, Greg Ashman, Dylan Wiliam, and Ilana Horn. 

I've pared back what I ask students to do using a new structure I've been calling "Ten True Statements." It could be twelve or eight or nine, but ten is a nice number. Here's the basic idea.

Students are given a problem that includes a diagram and a statement, but my instructions to them are extremely non-pathway-specific. I ask them to generate at least ten true statements about the situation. I given them a specific amount of time and then I yell, "GO!"

I circulate, but only provide just-enough of a hint to table groups to help them get themselves unstuck. The purpose here is to learn how to ask for help and not just stay stuck.

Here's one of the problems they did on Thursday:

I consider this activity purely generative. Students need practice in brainstorming.

I want them to lose themselves in flow so they can practice using their reference materials to develop as many ideas (aka "true statements") about the figure as they can, together with justification. 

I don't care about the order of statements. I don't care if statements are relevant to a proof pathway. 

The habits of mind I am trying to cultivate are to learn how to brainstorm more gently with their minds without judgment; to use their tools as a memory aid; and to document their thinking process.

My theory of action is this: the more practice they have in generating true statements and in deriving new true statements from previous true statements they have generated, the easier it will be for them to learn how to put their true statements and justifications into order.

I am trying to focus their working memory just on the generation of true statements. 

All four classes are really loving this activity, so I have to go find more suitable problems for the week.

Angle Measuring Practice & Fine Motor Skills

My 10th grade Geometry classes missed two critical years of in-person schooling in middle school.

One thing I've noticed is that these students seem to have more trouble than I had anticipated, and one of the things they seemed to struggle with most is working with a physical protractor in 3-dimensional space. The idea of using a physical tool to measure a spatial object seemed very foreign to almost everybody.

Every time I encounter something like this in our post-pandemic world, I've learned to ask myself what impact distance learning may have had on the students who were stuck at home. My training, my experience, and my own research have taught me that our physical organism moves towards health, so long as we assist it. That makes me want to treat this problem not as a deficit of mind but rather as a gap in experience.

I realized I needed to create an activity that would backfill this gap in experience and empower students to move forward from where they are.

So here is my Angle Measuring Practice activity from today. There may be typos or my own silly measuring errors because I'm tired. 

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Start by printing and hanging angles #1 - 12 around the room. Kids at each table number (#1 - 9) start their measuring journey at their corresponding angle number. Everybody measures every angle. Table members compare measurements and call me over for a read through. We check for understanding -- did you accidentally start your measuring from 180 rather than 0? Clarify that. Support kids at measuring stations by asking/showing where the vertex goes. How do you align one side of the angle against the protractor?

Kids will start clarifying for each other. This is good.

When they complete the circuit, whole tables called me over for a check. We talked about estimation and levels of precision. 

Then I gave them level #2 with instructions. Now they have to check their own work, using what they know about linear pairs and the sum of their measures. 

When they finished, they did level 3.

I don't know what it is about hanging stuff around the room and getting kids standing up, but it works. By the time they finished the circuit of the room, they were deep into the work.

Physical collaboration is powerful. 

This reminded me to use it.

Proof Portfolios

Over the last five years of teaching proofs in Geometry, I have learned two things: (1) the most effective student understanding comes from writing about their proof process, not from the proving itself, and (2) the most effective feedback process for students is a peer-to-peer reciprocal feedback process.

So this year, when I had to be out of school for a few days, I designed a Proof Portfolio project for them to do in my absence.

Each day had four small, reasonable proofs students had to do — and they could collaborate on these. But then... they had to write a number of short-answer reflections to analysis questions based on their own proofs in the day's set.

In addition, they had to find a peer to trade with and to give a rubric-based peer review and reflection.

In my class, they did this for several consecutive days. I made it worth a quiz/project grade.

When I returned, there was a great deal of wailing and moaning and gnashing of teeth about How Hard This Project Was and How Hard They All Worked.

It was clear that this project was a rite of passage for my classes.

But as I'm reading their work, I am blown away by how much they seem to have learned!

Their mastery of proof is not perfect. But it is authentic and it is growing. And to me, that is the most important point at this stage.

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I made up four days' worth of activities. Each day is two double-sided pages (proofs & reflections).

Here is a link to the G-drive folder with the four PDFs:

https://drive.google.com/open?id=1Mcb-AueXujpiWI2wD1FkuXAGTtWFjKAY

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More photos:

Things That Work #1: Regular Vocab Quizzes in Geometry

One of the things that worked incredibly well last year—and which I want to extend this year—is regular vocab quizzes in Geometry.

Vocabulary is the gating factor for success in a problem-based, student-centered Geometry class. If you can't talk about geometry, you can't collaborate about geometry.

I learned the value of extremely routine-looking vocabulary quizzes when I taught 8th grade English with Alec MacKenzie, Linda Grady, and Kelly Starnes. At the beginning of the school year, the copy room delivered us each a giant stack of very basic matching quizzes: numbered terms in the left-hand column, lettered definitions on the right. Each student got a vocabulary workbook at the beginning of the school year. Every week we assigned a new chapter/list. Every week we gave a matching quiz. And then we would trade and grade them.

At some level, I recognize that this sounds stultifying. But at another level, it was incredibly empowering for the students. Everybody understood exactly what was being asked and expected. And everybody saw it as an opportunity to earn free points. Students gave each other encouraging written comments and cheered each other on. They saw their scores as information—not as judgment. They used what they knew to make flash cards or Quizlet stacks. They quizzed each other. They helped each other.

And nobody ever complained about the regularly scheduled vocab quiz. It was a ritual of our course.

Vocab quiz for initial unit on circles

In my first few years of teaching Geometry, I have noticed that the kids who make the effort to integrate and use the vocabulary and specialized terms tend to succeed. And the kids who don't use the language of geometry suffer. So I decided to use what I know to raise the number of kids who know and use the vocabulary by instituting regular vocabulary quizzes for the relevant lessons or chapters as we go.

Many of my discouraged math learners sprang to life when I assigned this task. They pulled out flash cards, folded sheets of binder paper in half lengthwise, and started organizing the information they wanted to integrate. In most of my classes, I noticed that the highest-status math students often seemed to get stuck while the weaker students knew EXACTLY where to start and what to do.

It was a revelation.

It also ensured that everybody spent a little quality time on the focus task of preparing for the vocab quiz on Thursday or Friday. And this, in turn, meant that everybody was a little more ready to use the correct and appropriate mathematical vocabulary in our work. They noticed more because the owned more.

Because these were "for a grade," kids put their shoulder into it. My colleagues in other departments commented about my students taking two or three available minutes during passing period to quiz each other.  It gave them hope.

Now I want to create a full set of vocab quizzes for my whole year. 

A few implementation notes:

• I collect and shred/recycle all of the quizzes after I enter their scores so I can reuse the same quizzes from year to year. If I don't have your quiz, you can't get a score. I am strict about this.
• Every new vocabulary term does not have to get quizzed, but lessons or units where there is a huge vocabulary burden that gets front-loaded deserves its own vocab quiz. I have been surprised to discover how many lessons are more vocabulary-intensive/language-intensive than I had realized.
• Correct use of technical language is self-reinforcing. Once I introduce a new term, I mercilessly ask kids to remind each other of the definitions for 15 seconds in their table groups. Getting one kid to call out the correct definition to the whole class is not the point here. Getting 36 kids to all speak the definitions or the terms in their table groups is.

UPDATE: D'OH! I can't believe I forgot the most important implementation note I wanted to remind myself about!!!

• There should be many more definitions in your right-hand list than there are terms in your left-hand list. Also definitions can be re-used. This way there isn't a zero-sum outcome if someone misses an answer.

Scaffolding Proof to Cultivate Intellectual Need in Geometry

This year I'm teaching proof much more the way I have taught writing in previous years in English programs, and I have to say that the scaffolding and assessment techniques I learned as a part of a very high-performing ELA/Writing program are helping me (and benefiting my Geo students) a lot this year.

I should qualify that my school places an extremely high value on proof skills in our math sequence. Geometry is only the first place where our students are required to use the techniques of formal proof in our math courses. So I feel a strong duty to help my regular mortal Geometry students to leverage their strengths wherever possible in my classes. Since a huge number of my students are outstanding writers, it has made a world of difference to use techniques that they "get" about learning and growing as writers and apply them to learning and growing as mathematicians.

We are still in the very early stages of doing proofs, but the very first thing I have upped is the frequency of proof.  We now do at least one proof a day in my Geometry classes; however, because of the increased frequency, we are doing them in ways that are scaffolded to promote fluency.

Here's the thing: the hardest thing about teaching proof in Geometry, in my opinion, is to constantly make sure that it is the STUDENTS who are doing the proving of essential theorems.

Most of the textbooks I have seen tend to scaffold proof by giving students the sequence of "Statements" and asking them to provide the "Reasons."

While this seems necessary to me at times and for many students (especially during the early stages), it also seems dramatically insufficient because it removes the burden of sequencing and identifying logical dependencies and interdependencies between and among "Statements."

So my new daily scaffolding technique for October takes a page from Malcolm Swan and Guershon Harel (by way of Dan Meyer).

I give them the diagram, the Givens, the Prove statement, and a batch of unsorted, tiny Statement cards to cut out.

Every day they have to discuss and sequence the statements, and then justify each statement as a step in their proof.

This has led to some amazing discussions of argumentation and logical dependencies.

An example of what I give them (copied 2-UP to be chopped into two handouts, one per student) can be found here:

-Sample proof to be sequenced & justified

Now students are starting to understand why congruent triangles are so useful and how they enable us to make use of their corresponding parts! The conversations about intellectual need have been spectacular.

I am grateful to Dan Meyer for being so darned persistent and for pounding away on the notion of  developing intellectual need in his work!

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.

DIY Geometry Vocabulary Game, courtesy of the MTBoS (a collaborative effort)

Through an amazing collaborative effort on Twitter that took, like, all of 15 minutes, the collective hivemind of the #MTBoS came up with a great way to teach/reinforce vocabulary using Maria Anderson's tic-tac-toe style of Block games.

Most of the needed resources are referenced here on my Words into Math blog post.

I just added a bunch of new files to the Math Teacher wiki, including blank vocab cards so the kids can make up their own practice cards.  I make cards in Pages, so I'm also including the PDF and an exported Word doc version.

BLANK TEMPLATE Words into Math Block game cards.pages

BLANK TEMPLATE Words into Math Block game cards.pdf

1-3 Words into Math Block game cards LEVEL 1 SIDE A.doc

1-3 Words into Math Block game cards LEVEL 1 SIDE B.doc

1-3 Words into Math Block game cards LEVEL 1 SIDE B.pages

1-3 Words into Math Block game cards LEVEL 1 SIDE A.pages

Here's a link to the gameboard.

And voilà! A vocab activity for Geometry is born.

When we receive the Oscar for this production, credit should go to Teresa Ryan (@geometrywiz), Julie Reulbach (@jreulbach), Kate Nowak (@k8nowak , aka The High Priestess), Sam Shah (@samjshah), Tina Cardone (@crstn85), Michael Pershan (@mpershan), and if there's room left in the credits, me too.


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UPDATE (08-Aug-14):

Two things:

Thing 1 - I made an actual set of Words into Geometry cards (double-sided) because I want to be ready to have students actively practice. You can find the file here on the Math Teacher's Wiki: Words into Geometry game cards.

Thing 2 - I'm decided to use lima beans and pinto beans as counters for block games in Geometry because I am going to have a lot more students than I've had in the past! These will never go out of style and will always be replaceable as needed.