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.

Compound Inequalities Treasure Map

Never underestimate the power of novelty to help you engage certain students.

I just spent the last hour and a half-long block period with my jaw on the floor, watching in amazement as my most discouraged, 12th grade College Prep Math students worked productively and peacefully on, of all things, the analysis and solving of compound inequalities.

During my prep, I turned a boring worksheet into a treasure map. And that turned a boring requirement into a very peaceful and enjoyable period.

As she was leaving, one girl asked, Could we please do more work like this?

I'll take that as a compliment!