Zodiac review: In which there was mad flow

Peter Liljedahl speaks and writes often about the concept of flow in the context of mathematics education, building on the earlier work of Mihály Csíkszentmihályi. According to Liljedahl, flow is “that moment where we are so focused and so absorbed in an activity that we lose all track of time, we are un-distractible, and we are consumed by the enjoyment of the activity. As educators we have glimpses of this in our teaching and value it when we see it” (Liljedahl, 2016b). The concept of flow thus offers a productive way to talk about student engagement, and offers a way to think about environments that may occasion engagement in our classrooms. In particular, flow arises in an environment “wherein there are clear goals, immediate feedback, and there is a balance between the challenge of the activity and the abilities of the doer” (Liljedahl, 2016b).

In order to occasion and nurture flow, Liljedahl argues, teachers should be intentional in providing hints and extensions that keep students in balance between their abilities and the challenge of the current task (2016a). Although I typically opt for this approach—that is, offer the same task to all student groups, and provide hints and extensions as necessary—, today, I found success in offering students choice of tasks (coupled with hints and extensions from me, as necessary).

Reframing a classroom culture problem

tl;dr: Sometimes, a “problem” in the classroom is really a symptom of a bigger issue.

Here’s the backstory. Lately, I’ve been struggling with a lack of engagement in whole-group discussions after small-group work. I was growing increasingly frustrated that the majority of the students – especially the students who had a good grasp on the concepts – didn’t volunteer to share their thinking, and that they had no qualms about chatting with their friends when a student (or me) did choose to put forth a question or share an idea. Increasingly, I was feeling like I was losing them, and searched for explanations ranging from summer being around the corner, to deficiencies on the part of my students (kids these days!), to the possibility that I am a terrible teacher for whom there is, frankly, no hope. (During a recent conversation, Jimmy Pai described teaching as a balancing game between overconfidence and despair, and these days I’ve been leaning heavily to the right.)

“MY numbers!” (On the ownership of knowledge)

The other day, I presented some students with the following game:

Last cookie is a game for two players. In this game, a number of cookies are divided between two cookie jars, and each jar has at least one cookie. Each player can take cookie in one of two ways:

1. By taking any number they like from just one jar or
2. By taking the same amount from both jar.

The winner is the person who takes the last cookie(s).

Some of you may recognize this as Wythoff’s game; on the NRICH website, which has a great online version of the game, it’s called Last Biscuit (I modified the name for a Canadian audience). If you aren’t familiar with the problem, I highly encourage you to give it a go – very easy to understand and play, but the optimal strategy is not particularly obvious.

I assigned this game as homeplay (get it?? because it’s a game… terrible, I know). Then, the other day, we picked it up again after about a week of rumination.

On McNugget math (and being less helpful)

Another great Pi Club (still a working title) meeting.

Today, we worked on the McNugget problem (a case of the Frobenius/coin problem), which goes something like this:

Chicken McNuggets come in packages of 6, 9, and 20. Assuming money is not a factor and that you can only buy full packages, what is the largest number of Chicken McNuggets that you cannot buy?

(Of course, I brought McNuggets for the occasion. You know, as manipulatives…)

Pi Club: Round 2

First things first, Pi Club has a new member – and SHE is a wonderful addition to the group! WOOP WOOP, girl power!

Today, we worked on two problems. First, the lightbulb problem (found here):

There are 100 light bulbs lined up in a row in a long room. Each bulb has its own switch and is currently switched off. The room has an entry door and an exit door. There are 100 people lined up outside the entry door. Each bulb is numbered consecutively from 1 to 100. So is each person.

Person No. 1 enters the room, switches on every bulb, and exits. Person No. 2 enters and flips the switch on every second bulb (turning off bulbs 2, 4, 6…). Person No. 3 enters and flips the switch on every third bulb (changing the state on bulbs 3, 6, 9…). This continues until all 100 people have passed through the room.

What is the final state of bulb No. 64? And how many of the light bulbs are illuminated after the 100th person has passed through the room?

They didn’t take too long to work this one out. Great conversations were had.

I think I started a math club

It began with one student. I noticed pretty early in the semester that grade 9 math was old hat for him and that he needed a challenge, so we started to meet once a week or so to talk math and work on some interesting problems. (During our first meeting, we proved that the square root of 2 is irrational.)

Then, he brought a friend. Who eventually brought two more friends. (If you’ve been doing the math, you might expect 8 students at our next meeting; alas, there were only 5. But wait, that’s five students who want to do math outside of math class!) We did a few problems, but mostly spent time discussing the idea of doing independent projects that the students would present to the class on the last day before Christmas break.

Today was the first session that really felt math club-y: I ordered some pizza, gave the students the Crossing the Bridge problem (thanks, Sadie!) and some white boards, then set them loose.