Random groupings

popsicle sticks

I recently had an epiphany. It came from Ilana Seidal Horn.

I was reading her book, Strength in Numbers, and she was addressing status in the classroom. Her definition is status is the perception of students’ academic ability and social desirability. Here’s an excerpt that blew my mind.

Unless we address underlying conceptions of smartness, we risk reverting to the commonly help belief that group work benefits struggling students because smart students help them. As long as we have a simplistic view of some students as smart and others as struggling, we will have status problems in our classroom. Students quickly pick up on assessments of their ability. For example, when teachers arrange collaborative groups to evenly distribute strong, weak, and average students, children will figure out that scheme and rapidly learn which slot they fill….If mathematics is rich enough, the strengths of the different students come into play, rendering the common mixed-ability grouping strategy useless. (p.29)

Truth. Talk about unraveling so many years of my teaching career in one paragraph.

A day later I noticed this tweet from Frank Noschese:

Bam. Just like that I was finished with strategic grouping.

Each seat in the room is assigned a number and every Monday students select a numbered popsicle stick upon entering the room. I’m coining them destiny sticks.

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This week, after the first go around with the new approach, I immediately got lots of “this is a great idea” and “I love this!” from the students. Full steam ahead.



I’ve contracted a SEVERE throat infection. I cannot speak at all.


I recently wore this sign on my neck during class. I placed it around my neck and went about the business of being the teacher in the classroom.

At first there was high levels of concern. There were comments like “Are you ok?” or  “Stay away from me, I don’t want it.”

And then one student remembered to having seen my speaking just fine earlier in the day. There were snickers, laughs, and smiles all around. They were picking up what I was putting down.

I silently gestured them through a slide that explained that although I couldn’t verbally communicate, this didn’t mean that they couldn’t learn. There was a handout. They have brains. They had each other. No excuses. The focus was recursive sequences.

I don’t think this idea would work well with any class, but it was an awesome experience with those I tried it with.

There were natural leaders who volunteered to demo examples on the board for the class. Some kickstarted conversations about what they noticed and wondered. There were others who made their way across the room to seek help or give it to other groups. And yet others did their thing on their own or with a partner.

Throughout all of this, I said nothing and did very little. For the entire period I peered over shoulders, pointed out (literally) interesting steps in student work, and wrote on the board the time remaining before the exit slip was to be administered. I reminded them (with a slide) that they needed to ensure everyone in the class could be successful on the exit slip, as it was the measure of their success as a team. (They ended up being 90% proficient.)

I closed by reiterating with them the fact they need me far less than they think they do. I’m no gatekeeper of knowledge.



Developing rational exponents through geometric sequences

Soon I’ll be teaching rational exponents in algebra 2. I’ve never found an intuitive way of teaching it…until now. Thanks Bowen.

The approach leverages geometric sequences. I’d love to regurgitate it, but this tweet from Bowen is my source and sums it up:

The unit prior focused on sequences with a heavy emphasis on geometric sequences, so this is the perfect bridge to developing this idea that most students find confusing. It all comes back to repeated multiplication, as it should.

In the past, I’ve usually had students enter various expressions (e.g. 100^(1/2)) into their calculators to stumble upon the relationship between rational exponents and radicals:


But this painfully ignores the mathematics behind exponentiation and instead lures them into believing that these two concepts are magically connected through a few keystrokes on their calculator. It treats rational exponents as an isolated concept and unrelated to repeated multiplication.

After discovering my new strategy for teaching rational exponents, I found this video from Vi Hart on logarithms. The similarities run deep.


Now I can’t wait to teach logarithms.



Thank you hint tokens.


A few weeks ago I stumbled across the idea of a hint token. Think of it as a get out jail free card, but for the classroom. While working on a task, groups can trade one for a hint from me.

Loving this idea, I immediately went to implement it. This, I thought, would be a great way to give students more ownership over their learning and hopefully learn to rely more on one another. The first time around we were studying sequences and I gave each group two hint tokens in the form of Jolly Ranchers (thanks Sam).

What happened was something unexpected: no tokens were used.

They may have simply wanted to eat the candy afterwards, I’m not sure. I wouldn’t doubt it. That said, what was most impressive was how they worked interdependently to solve the problems. I was essentially ignored.

Afterwards, I realized how empowered I was. The kids need me far, far less than I think. Understanding something and feeling something are very different phenomena. I’ve always known that my students should need me less, but I now know how that feels. It’s incredible. I even communicated this to the kids and saw the realization in their faces. They felt the same way.

This experience has had a dramatic affect on my teaching. What’s ironic about this is that you’d think I would move to incorporate hint tokens every day. The thing is, I’m not. Instead, I ensure that students have the opportunity to own their learning and sit longer in each other’s thoughts. It usually consists of 10 minutes of focused, small-group discussion and productive struggle during every lesson (I have 42 minutes class periods). During this time, I provide no any assistance of any kind.

As a result, it’s common for me to pull up next to a group, watch and listen. Before, they would be inclined to ask me something simply because they could. Now, they forget I’m even there. I silently assess their thinking the entire time – which reminds me of a live version of video-based PD.

It’s a win for everyone. They purposefully and interdependently think through a problem, which spurs engagement and ownership, and I get valuable insight into their thinking that serves as a driving force for the rest of the lesson…and beyond.

Thank you hint tokens. Thank you for facilitating this change in the culture of my classroom.

I’m left thinking that this shift may be directly related the class chemistry I’ve developed this year – which has cultivated a willingness to learn and explore amongst the kids. In other words, the tokens could have simply been what I needed use in order to realize the new path that learning is taking in my class. I don’t know. Maybe next year I will need the tokens. We’ll see.