“A Day In…” posts are averaging 1997 words per post. Holy wow! Time for a shorter one.

## The Setting

3rd Period, Wednesday, April 3, 2013

Honors Algebra 1

## How Things Went Down

Bells rang. Sets were found. Homework was checked. Estimations were made, reasons were given, the answer was shown.

And then, the lesson began. (Full disclosure: I wrote the lesson a year after reading this, and was even more influence by a jigsaw-puzzle-building activity—solo, solo, then tag-team—I heard about from some friends who work here.)

**Me:** “Does anyone have a magic phone with a stopwatch?”

**Student R:** “I do.”

**Me:** “Awesome. Get ready. (Pause.) Ready?”

**Student R:** “Yep.”

**Me:** (Walking to the front of the room with my bucket of binder clips…) “The rules are as follows: I am allowed to use my left hand only, one clip at a time. Got it?”

**Everyone:** “Uh… What are you talking about?”

**Me:** “Ready?”

**Everyone:** “Okay, we still have no idea what you’re talking about. But sure, whatever.” (This is a paraphrase.)

**Me:** (While dumping the binder clips on the floor…) “Student R, give me a countdown.”

**Student R:** “3… 2… 1… Go!”

My task then becomes clear to the students, as I proceed to pick up and toss the binder clips into the bucket as fast as my left hand will let me (one clip at a time, mind you). I’m right-handed, so this takes a while. 100 seconds to be exact. (Two years in a row, 100 seconds exactly.)

It gets a little awkward after about 30 seconds (70 seconds to go!!!) so I banter with the students for about 20 seconds, invite them to hum the Jeopardy theme music for another 30 seconds, and ask them to cheer me on for the last 20 seconds. Some oblige, some do not. (Hey, that’s not unlike the rest of my experience in teaching!)

At that point we record my time. I then dump the clips on the group a second time. I ask for a volunteer. (“Thanks, Student J!”) This brave volunteer then picks up the clips as fast as he can using two hands, one clip at a time (per hand). His time is 59 seconds. (60 seconds last year.)

Then the fun part, essentially stolen from the world of Three Act Math Tasks: Students make an estimate for how long they think it will take the Fenton-and-Student-J-Tag-Team to pick up the clips (same individual rules apply).

Guesses are made, clips are dumped, the stopwatch is readied, and the clip cleanup commences.

We’re an amazing team, so we finish the task in 40 seconds.

From that point the lesson is rather predictable, so I won’t bore you with the details (though we did have some great conversations in this “predictable” portion because of the seeds planted in the introduction).

## What I Liked

The lesson was fun to teach, and the kids were definitely engaged.

I love the extra buy in from students that I get simply by asking them to guess before we measure, calculate, etc..

All the guesses were reasonable! No one offered the absurd (yet tempting, for the totally lost) answer of 100 + 59 = 159 seconds. Why? Because the setting/context/problem type was set before the students in such a tangible way. “Of course the tag team will finish faster!”

## What I Didn’t Like

The lesson doesn’t do a good job of building on the reasoning students were engaged in during the introduction once we transition to a search for more efficient solutions. By no means do I dive headfirst into a “watch and mimic” approach. But the students who had no idea how to approach the problem in the first place (i.e., the students who could do no more than make an educated guess) are still unable to do more than make an educated guess.

There is a decent amount of semi-downtime for students in the first 10 minutes of class. The advantage here is that we create the data as a class. The disadvantage is that only a few of us are actually involved in generating the data. I don’t have a fix for this yet, but I would like to involve more students or decrease the downtime (or both).

## How I’ll Get Better

Immediately after teaching the lesson I began brainstorming improvements for next year. This is my attempt. My goal was to create something that would help students develop two efficient approaches that emphasize/promote *understanding* in the midst of finding the solution, but that didn’t require me to be a central part of the conversation while it unfolded.

I was happy with the handout and excited to use it sooner rather than later, so instead of waiting until next year I presented it to my students the day after the first lesson. I was pleased with the results, as students learned efficient methods *without* abandoning their reasoning. (Sadly, this abandonment-of-reason-for-the-sake-of-efficiency happens too often for many of my students, especially when we transition from estimates and arithmetic approaches to algebraic ones.) And while they didn’t develop the methods entirely on their own (to expect that of them at this point in the year would require that I’ve expected similar things all year long, which sadly I have not), there was a lot of great conversation followed by some favorable assessment results a few days later.

## Questions

Need some inspiration before you head to the comments? Consider responding to one or more of these:

- What do you think of the first handout (Day 85 Notes)? What do you like, what would you change, and why?
- What do you think of the second handout (Day 85 Practice)? What do you like, what would you change, and why?
- Do you have any ideas for helping me solve the “downtime” issue described above? Or is it a non-issue, and I should just relax?
- I want to help my students grow in their ability to develop efficient problem solving strategies on their own. What sorts of things can I do throughout the year to help them improve in this regard?

## Comments 3

Michael

A lovely lesson here. I am pleased to hear that no one simply added your times. How many simply took the average of your times? This is obviously a bad estimate and I think we can talk kids out of that with examples like yours. However, I wonder about the persistence of an idea like this one (similar to the exponent conversation that Christopher Danielson has picked up on) I can imagine that kids would focus on some aspect of the average difference in your times and try to shoehorn that into this situation.

I have passed this along to my Algebra I colleagues. Thanks!

mrdardy,

Actually, 100% of the guesses were less than 59 seconds (the time of the student) and greater than ~30 seconds (half the time of the student). With the visual/tangible introduction, students seemed to grasp the idea that the tag team should finish faster than either person working alone, but that my contribution wouldn’t be quite as helpful as having two students team up (that’s where the 30 seconds comes into play).

After we measure the tag team time, students did wonder if finding the difference of the two times (100 – 59 = 39) would always provide a good estimate. “Suppose two students, each with a time of 59 seconds, worked together. Would their tag team time be 59 – 59 = 0?” That question allowed students to reason that the difference is not the way to go, and was just coincidence here.

One semi-disappointing and unintended consequence that showed up about a week after the lesson: One student applied our “efficient” strategies (those developed on the second handout) to a different type of problem, demonstrating that she doesn’t really understand what’s going on in either scenario, and she’s just relying on mimicry. So the work never ends. But that’s what I signed up for, and a significant part of why I like my job.

Thanks for sharing your comment and question! I hope your colleagues enjoy using some aspect of the lesson. If they have feedback (whether positive, or things they would improve), send it my way.

Pingback: Mark Dubowitz