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!

Arithmetic of Complex Numbers Placemat Activity - Algebra 2 + Complex Instruction (CI)

Just because you have an all-groupwork and all-Complex Instruction (CI) format doesn't mean you don't need practice activities too.

Our Algebra 2 kids were getting the concepts of complex numbers and complex conjugates, but were still kinda shaky in terms of fluency in working with them.

Based on ideas I stole borrowed a long time ago from the fabulous Kate Nowak (@k8nowak, http://function-of-time.blogspot.com) and the equally fabulous Rachel Kernodle (@rdkpickle, http://sonatamathematique.wordpress.com ), I proposed a placemat activity to my ever-game Algebra 2 teaching team and they dove right in.

Set-Up
We have typical CI four-person table teams set up in each of our rooms, with each person assigned a specific role based on where they're seated at the table. Our roles are Facilitator, Resource Manager, Recorder/Reporter, and Team Captain, although of course, your mileage may vary. Each role has specific tasks they are expected to perform; for example, only the Resource Manager may call the teacher over for a group check-in or a group question (in our program, teachers only accept and answer group questions).

Each table was given:

• two, double-sided "placemat" sheets for doing work in the center of the table
• a set of problem cards (there are four sets, one for each round of play; to simplify clean-up and organization, I printed each round of cards single-sided on a different color of paper, one set per table group. I've got 7 tables in my room, so I made seven sets of cards. I laminated them and clipped them together, but hey, that's just me)
• the sum to which all four answers for any given round should add up

The sum for each round was written on the whiteboard, though it could have been projected via document camera or Keynote/Powerpoint slide.

Objectives
We had mathematical objectives for the activity as well as CI or norms-based, group work objectives. My students in particular needed reinforcement in group work norms and collaboration. Our objectives were:

     Math Objectives

• achieve greater fluency in the arithmetic of complex numbers (including the distributive property)
• deepen understanding of and fluency with the powers of i
• deepen understanding of and fluency with complex conjugates

     Group Work Objectives

• work in the middle of the table
•  same problem, same time (no one moves on until everyone moves on)
• using table group members as resources

The next time I run this activity, I will definitely give a Participation Quiz because the group work norms are so beautifully reinforced in this activity.

How We Ran It
Recorder/Reporter writes the sum in the central oval of the first side of the placemat. Each group member gets a problem card for round 1 (problem a) and works his or her problem on his or her quadrant of the placemat.

When everybody is finished with their problem, the Facilitator facilitates the addition of all four answers. If they add up to the given sum for that round, the Resource Manager calls the teacher over for a "checkpoint" and the next set of cards for the subsequent round of work.

If their answers don't add up to the given sum, they need to work together through everybody's work on the placemat to diagnose what went wrong and where, as well as how to fix it. Then when they've fixed it, they call the teacher over for a checkpoint and the next set of cards for the subsequent round.

Group Work Benefits — Reinforcing Norms
For my classes, the greatest benefits of this activity came from the fact that it forced students to work in the middle of the table, to use each other as resources, and to talk mathematics. Getting kids to work in the middle of the table is the hardest part of CI, in my view, because it goes against the grain of most of their in-school conditioning. The placemat format makes it nearly impossible NOT to work in the middle of the table. And once they're doing that, it seemed like everything else ran pretty smoothly.

I especially liked the fact that this activity created a context in which students experienced an intellectual need for the using the rules of arithmetic for complex numbers and for the powers of i. It was situationally motivated, but extremely targeted.

Sums for Each Round 
The sums for each round are as follows (if you find an error, please speak up):

• Round 1 (problem a):   26-73i
• Round 2 (problem b):  0
• Round 3 (problem c):  165
• Round 4 (problem d):  2 – 48i

PDF Files for the activity
These are available also on the Math Teacher Wiki on the Algebra 2 page. If you haven't visited the Math Teacher Wiki, you don't know what you're missing.

• Complex Numbers Arithmetic – Problem Cards
• Complex Numbers Arithmetic – Placemat Master