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 approachthat 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).

The unit is trigonometry (Foundations and Pre-calculus Mathematics 10), and we are a few days away from the exam. I wanted students to engage with a variety of problems involving triangles where they could apply their understanding of trigonometric ratios and the Pythagorean theorem, and initially considered a “speed dating” review (see this post). However, I decided to change the format, wanting students to spend the lesson immersed in doing mathematics, rather than listening and sharing, even if it meant they engaged with a smaller number of problems. (I still love the speed dating idea, and would like to give it a go sometime in the future.)

Luckily, I had already made up a series of 12 problems for the speed dating activity, which I labeled with the astrological signs (also Nat’s idea), which is why I’m referring to it as the Zodiac review.  Some of the problems were taken straight from the textbook, or adapted slightly. My tweak was to add an approximate difficulty level to each problem (1-4, with 1 being the easiest). This morning, I made a few copies of each problem and stuck them to the board with magnets. I also put up the answers (not solutions) for verification. See photo below.

I grouped students randomly (as I do every day), provided each group with a marker and a whiteboard, and gave the following instructions:

  • Decide, as a group, where you would like to start; if it’s too easy, go up a level, if it’s too hard, go down. Once you’re done with a problem, put it back on the board.
  • Try to work through as many of the astrological signs as you can, but it’s not a race. Keep track of which ones you have done, and take photos of your work as “notes.”

Any questions? Nope. The students got right to work; some grabbed more than one problem right off the bat. Now, when offering this kind of choice to students, there is always the fear that they will choose wrong – especially that they will take the easy way out, and stick with the simpler problems. This is not what happened. Check out which problems flew off the board first:

A few comments, noticings, and takeaways:

  • Since students were working on different tasks and not encountering new concepts, there was no need for a whole-group discussion to close the lesson. I focused on facilitating communication within groups, as I discuss here.
  • A few of the groups spent the majority of the class working on one of the level four problems. I did not hear one “this is too hard” or “let’s try an easier problem” – they wanted to, needed to solve it, and by golly they would, as if their life depended on it. When I sensed that they were stuck, I directed groups working on the same or a similar problem to talk to each other (side note: I did not plan for this, but I now realize this is a good reason to make several copies of the same problem for this activity). All of these groups eventually found the solution, and were able to experience a nice connection to the previous unit (systems of equations).
  • For the most part, students were engaged until the bell rang, which caught us all off guard. In several cases, I was surprised to see students who tend to shy away from participating taking the lead on a problem. Cell phones were being used as calculators or to take photos of the work. I hesitate to confirm (despite the title of this post) that the students were in flow because the experience is very personal, but the outward signs were there.
  • One group ran up to me at the end of class to excitedly share how they had solved one of the problems using Desmos. The problem is as follows:

You and your friend Michael are 38 m apart, both west of Big Ben. From your vantage point, the angle of elevation to the top of Big Ben is 65°. From Michael’s vantage point, the angle of elevation is 49,5°. What is the height of Big Ben?

They drew a diagram, and figured that the tangent ratio would be involved. They reasoned that since the tangent ratio is essentially the slope of the hypotenuse of a right triangle, they plotted the line y = (tan65)x, to represent your vantage point. Then, they plotted the line y = (tan49.5)x + b to represent Michael’s vantage point, adjusting b until the distance on the x-axis was 38. Zooming out, they found the point of intersection of the lines (84, 97), from which they concluded that the height of Big Ben is about 98 m. My recreation of the graph appears below.

You guys. I couldn’t even. What an awesome connection between trigonometry, linear relations, and systems of equations – three of the major topics of the course. They couldn’t have demonstrated any better their deep understanding of the concepts at hand.


Many hours later, I’m still buzzing from this class.

Because today was a shortened day, I will give students some time to continue their work tomorrow. Since none of the groups worked through even half of the problems, I will put up the same questions I offered today, but have added a few more level 3 and 4 problems into the mix. (Since I ran out of Zodiac signs, I used the names of some Harry Potter creatures… close enough.)

If you’re interested in the problems, I’ve uploaded a PDF here; it’s heavier on higher-level problems (of course, depending on your students, you may assign different levels to the problems). Adjust as you please.

Let’s see if we can keep up the mad flow for two days in a row…


Liljedahl, P. (2016a). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick , & E. Pekhonen (Eds.), Posing and solving mathematical problems: Advances and new perspectives. New York, NY: Springer.

Liljedahl, P. (2016b). Flow: A framework for discussing teaching. In Proceedings of the 40th Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 203-210).

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