On the importance of scaffolding within the zone of proximal development

I don't know about you, but my experiences with even the best Shell Centre tasks (their Formative Assessment Lessons, or "FALs") have been hit or miss. These are among the best anywhere, and while I generally love the ideas of their tasks, in practice, I almost always need to tweak their implementations to make them work in my classroom and with my students.

I wonder if this is not inevitable, given the highly customized nature of scaffolding.

I experienced this phenomenon yet again today when I used their Ferris Wheel task, which I brought out for my extremely able but easily discouraged Precalculus students:

    http://map.mathshell.org/materials/download.php?fileid=1252

The purpose of this task is to jump-start students' understanding of modeling and graphing trigonometric functions, using the movement of a ferris wheel cart together with some scaffolded information. The pre-assessment task scaffolded the process well, except for the crazy way they laid out the equations. But then when I ran the card sort task in my 1st period classroom, it was a complete trainwreck. The changes in set-up seemed to yank the problem out of the zone of proximal understanding and launched it into the stratosphere. When something is new, my students are easily thrown (and discouraged), so when they shifted from providing the period to providing the number of revolutions in some arbitrary number of minutes, my students' heads exploded. There was a domino effect of cascading fear and consequences — the equations are set up strangely and are given in degrees rather than radians (after all our hectoring!), the vocabulary changed from revolutions to rotations, etc etc etc.

So during my prep, I made new versions of the cards — ones that would allow my 5th and 7th period classes to recognize the new material and to connect it to things that were still strange, but at least strange in ways they could notice and recognize.

The differences were dramatic. 1st period groups were upset because they were unable to make what they considered meaningful progress on the task (in spite of excellent mathematical conversations) and left feeling unsteady and discouraged. Here is the work from an especially confident and capable 1st period group:

My two later classes, on the other hand, were able to attack the task, making meaningful connections, spot patterns, and really solidify their understanding. Here is the work from an often-cautious and un-confident (but still very capable) end-of-day group:

All in all, it was a good reminder to me to avoid changing horses in mid-stream when I am trying to help learners build conceptual understanding through good scaffolding!

Slogging through the middle – in praise of wallowing

in praise of wallowing in the middle of stuff -- marinating, versus just discovering

As much as I love discovery learning, there is a dirty little secret I really hate to have to admit: discovery is, in some ways, a cheap thrill.

It's easy to get excited about something new. It's harder to get enthusiastic for the middle. And this creates problems when it comes to developing fluency and deep conceptual understanding. I think that's why my students (who are all superb students, probably in the top top tier of all public high school students) run into troubles in math.

Sustaining enthusiasm through the MIDDLE of mastering something that is conceptually very difficult is what's really hard. But it's also where there is the greatest payoff.  Getting students over their resistance and into flow is where situational motivation and projects can work wonders.

Also wallowing.

Sometimes this means helping students learn how to practice — how to develop what I tell them is called "mental motor skills." With my Geometry students this week, that amounted to  giving everybody an index card first thing every day and playing "Let's Pretend" (i.e., let's pretend this is a quiz) and write down the five ways we know of proving lines parallel. I set the timer for 90 seconds and called out, "GO!." We all wrote down the five techniques we now know — corresponding angles, alternate interior angles, same-side interior supplementary angles, both lines parallel to a third line, and both lines perpendicular to a third line.

We traded and "graded" each other's index cards, remembering to find something positive that others did and circling their errors.

Then I yelled, "Flip over your card and do it again."

The first day we actually did this four times. I handed out new index cards and offered to destroy the evidence of any previous mistakes. And then we did it again — two more times.

With my Precalculus students, I had them flesh out Quadrant 1 of the unit circle on their index card.

And again 3 more times.

From years and years of piano training, I'm used to having to practice a new figure multiple times. I know how to repeat it multiple time across multiple different days so I can burn it into my mind and body.  But many students seem to have missed that learning episode.

So I am giving it to them now.

With my Precalculus students, we then practiced mental event rehearsal through guided imagery afterwards. "Close your eyes and visualize Quadrant I. Find pi over 4 and visualize a point there on the circle. Picture its cosine and sine. Now go back to (1, 0) and travel to pi over 4 in the NEGATIVE direction. Where is the traveling point now? What are its cosine and sine? What is sine of pi over 4?"

"New point: now visualize 2 pi over 3. What is the cosine of 2 pi over 3?" I pause. "Now go back to (1, 0) and travel to NEGATIVE 2 pi over 3."

Over and over, to help them learn how to engage their will in the service of their own best interests.

The Secret of Flowchart Proofs #officesupplies

Two words for you:

Tiny. Post-Its:

UPDATE —

OK, so maybe there is still a tiny bit more to say about this.

One of my Geometry colleagues and I have students who are still struggling with two-column proofs (which are required for us), so we decided to add in Flowchart Proofs. He went first, introducing them in his 1st Block class, and I introduced them in my 2nd Block class, except... being obsessed with office supplies, I decided to chop up some tiny Post-It notes into REALLY tiny Post-It notes (i.e., I cut them into halves). This way they fit better into students' notes and really gave the idea of representing a single idea.

They also dramatically simplified the process of rearranging steps if you realized you had not ordered them in the most effective way.

At first I had students write their "Reason" underneath the tiny Post-It with their "Statement." But then I realized it would be better if the "Reason" were written at the bottom of the tiny Post-It. That way, if you have to move anything around, there is much less to erase.

Once students are satisfied with their flowchart, they can draw in arrows and add any Latinate flourishes from which they believe their proof will benefit.

Surfacing and studying studying misconceptions via Talking Points

In a class of 36 students, where it can be, shall we say, difficult for me to do formative assessment on every student every day, the Talking Points structure gives me a great way to surface and deal with student misconceptions by getting students to surface, discuss, and correct them.

Par example...

Our sequencing for Geometry has gotten totally screwy this year because of some new district requirements around the Common Core. Having learned how to do all of the basic constructions, we are now finally approaching the unit test on parallel lines and their angles. There are so many possible errors in understanding that can happen around these, I wanted to create a group work activity to address them. I also want to change groups up this week, so I am using this as community-building as well.

I've discovered it is a good idea to weave together community-building statements, growth mindset statements, metacognitive self-monitoring statements, and Always / Sometimes / Never statements. The Talking Points structure lets me accomplish multiple goals simultaneously, which is something I need to do with such big classes.

An editable version of this set of Talking Points is on the Group Work Working Group (#GWWG14) wiki page on the TMC14 wiki.

Constructions Castles

Einstein was right — imagination IS more important than knowledge.

I used to think of that quote a lot as I walked the Princeton campus, through quadrangles and past trees that — legend had it — he had crashed into on his bicycle while he was lost in his thoughts.

I passed his house on Mercer Street every week on my way to music lessons, and I wondered if his sister had not had the door painted it bright red so that he might notice it and not bump into it.

When I realized I would have to teach both constructions AND proof this year in Geometry, I first thought of them as a Big Obstacle. But seven weeks in and I have fallen in love with constructions. I created this Constructions Castle project to give students plenty of practice doing constructions while also giving them a chance to develop their understanding of how shapes and angles fit together.

I think their work speaks for itself.

Files are on here (zip file with all tools).

UPDATE (10/22/14):

The more students do this project, the more awesomeness is revealed:

Formative Assessment PLUS Practice Activity: "Lightning Rounds" Review With Whiteboards

I've been looking for a way to do formative assessment that also doubles as a practice activity, and I seem to have stumbled on something good with my new "Lightning Rounds Review With Whiteboards activity. It uses some of what I appreciate most from both Steve Leinwand and Dylan Wiliam.

We have large-ish whiteboards — enough for one per table group — in our classrooms. Since I've also got around 36 students in each of my classes, I've been drowning with my older methods of checking for understanding. With 175 total students, even a 1-minute-per-exit-ticket assessment can take ages to go through — plus all the time it takes me to make it up.

This activity has been working much better. It also takes less prep up front and allows me to engage more deeply with the students and/or groups who need more attention. Assessing 9 tables' work at a glance is much more manageable and less exhausting than trying to flip through 36 pieces of student work. And of course, that is better for everybody.

Here is how it worked today.

In Precalculus, we're working on evaluating trig functions of real numbers on the unit circle. Students and groups are strongly encouraged to use their unit circles and notes to help them gain confidence and fluency.

Each table group gets a board, three markers, and an eraser. On the document camera, covering the non-current rows with a folder, I reveal one row of problems, which contains three problems: (a), (b), and (c).

Students discuss and work the problems, and when they are done, they hold up their table's whiteboard for a "check-in."

I glance around the room and check three answers at a time, calling out, "Table 2, you are checked in and correct!" "Table 6, you are checked in and correct!" "Table 1, you need to reexamine part (b)."

Because there are three problems in each "round," there's plenty of time for groups to make an error, assess their work and reconsider, and re-present their findings. Because there are only nine tables reporting in, it is manageable from both a teacher and a student perspective.

Plus, of course, since there is a group whiteboard and markers to play with, those groups who finished quickly and received their checkpoint are happy to doodle or play tic-tac-toe or offer funny editorial comments or cartoons while other groups receive some attention or extra time.

Last night was our Back To School Night, and student clubs were selling food as fundraisers for their clubs. Two of my students discovered that my classes' parents and I were helpless in the face of their delicious goodies. At one point, I had told them, Look, I don't have cash; my purse is locked upstairs in my desk.

Around the 6th or 7th round, Table 7 included an editorial comment on the side of their board: "We heart Dr. S! Also, you owe us $2. :)" While everybody started working the next round of problems, I unlocked my purse out of my closet and paid my debts. Some good laughs were had by all, and it was a nice piece of mathematical community-building.

The files I used are on the Math Teacher Wiki.

All in all, an easy, effective practice activity, with embedded formative assessment and community-building built right in. Not bad for a too-hot Friday!