This week I’ve had some great lessons, and some awful ones. Looking back at what I had done differently in the good versus not-so-good lessons, I realized that one of the biggest differences was the amount of “telling” I was doing in one class versus another. It didn’t matter that I had amazing visuals and was super enthusiastic about the content I was teaching; if I talked too much, students would start to zone out. Compound this with the fact that we are distance learning and all of my students are English language learners, we now have wi-fi/connectivity, audio, and language learning issues all thrown into the mix.

The one who does most of the talking, is doing most of the learning.

(Something I’ve heard from multiple sources throughout my teaching career)

At this point, I have to slap myself on the wrist because I *know* better, so I need to* do* better.

In yesterday’s class, I consciously made an effort to talk less and ask more questions. I also explicitly told my students that my goal as a teacher is to never tell them an answer, but to just show them the way. *Classic* case of easier-said-than-done.

I realize, with a sudden mixture of nervousness, trepidation, and excitement as I’m writing this, that this might be the *first time* in five years of teaching that I have *really* made a conscious effort to take Cathy Fosnot’s advice to heart. She writes,

Don’t try to fix the mathematics; work with the mathematician. The point is not to fix the mistakes in the children’s work or to get everyone to agree with your answer, but to support your students’ development as mathematicians.

Cathy Fosnot

On the surface, I’d like to think I was doing my best to project a calm, neutral tone as I jotted down notes while students shared their thinking. I wrote everything down, regardless of whether their strategy was “right” or “wrong”. Meanwhile, it felt like Hermione Granger was living in the back of my mind jumping up and down going “Pick me, pick me!! I know the answer!!” Talk about my “rescue the student” instincts being on overdrive!

In the past, I would have eventually given in to those instincts and immediately correct any mistakes that came to my attention. I tell myself that this is okay, because if I don’t, my students will continue to make those fundamental math errors, divide by zero, and initiate the end of the universe as we know it. I also think that deep down, that hidden behind this instinct is fear, fear that I can’t help them get where they need to go without just giving them the answer. Although, impatience is an equally guilty accomplice here in my crime of robbing students of a perfectly good learning moment.

This time, however, I tell myself a different story. I learn to trust myself and my students a little bit more by just letting them get where they need to go, in their own time and in their own way. This too, is a little scary.

## The Lesson

To add some context, here’s a bit about how my lesson went.

The goal of today’s lesson was to introduce the idea of trigonometric identities, collect some strategies that may be helpful in identifying whether a given statement is true or false, and then work on moving towards what it means to rigorously prove the truth or falsity of a mathematical statement.

I began the class by doing a modified version of an “Always, Sometimes, or Never True” activity with radicals (trying to introduce some interleaving here) from the Mathematics Assessment Project and called it “Truths and Lies”. I asked students to tell me which statements they believed to be truths and which they thought were lies, and to share their thinking on Padlet.

After about ten minutes of individual think time, I selected a few student strategies and had students explain them to the class. Here’s what we came up with:

Strategies Used:

- Plug in a number for
*x*and check to see if both sides are equal - Start with LS or RS, use algebra to show it is equal to the other side
- Assume the statement is true. Square both sides of the equation. (If both sides are equal after squaring, then the statement is true).

Next, I showed them this image about the different Levels of Convincing from Robert Kaplinsky’s site.

We then revisited each strategy and I asked students to mentally place each of these strategies fell on the spectrum of least to most convincing. Ideally, I would have given more time for students to really think this part through, but since we are doing distance learning, I was eager to get to the real meat of today’s activity, which was to prove trigonometric identities. From there, I took on the role of prosecutor and started to stir up some trouble.

For instance, in statement 1), we can demonstrate the statement is false by finding a value of *x* that shows LS does not equal RS, however, I argued that x = 5 worked, so wouldn’t that make the statement true? What I’m getting at here is that I want students to be able to articulate what exactly are we asking when we ask whether or not a statement is true? That it must be true all the time? Or only some of the time?

Then, I attempted to tackle the “squaring both sides” strategy… Couldn’t we also use same reasoning to show that 1 = -1?? (Can you see why?)

At this point something really amazing happened, and that was when a student interrupted me and said, “Ms. Soo, I just thought of another way to explain this!” The student was able to connect what we were doing to our study of transformations of functions from a previous unit.

I couldn’t —

keep my poker face, that is. This was me:

For the remainder of our lesson I had students work independently on the following:

My goal for them was to use different strategies and methods to try and “prove” or “justify” which were truths and which were lies. Students always have the option of messaging me privately for hints or advice if they were stuck, very few did.

After about 15 minutes, I asked students to send me pictures of their work and we could start talking about some strategies they used. The one mistake I see students make when “proving” trigonometric identity is to start by assuming the statement is true and start manipulating both sides of the equation.

Instead of telling students WHY they can’t do that, I referenced my earlier example of how, by the same logic, we can prove that that 1 = -1 and asked them, WHERE did the mistake occur?

*Let’s assume. the statement 1 = -1 to be a true statement.*

*Next, let’s square both sides of the equation.* *Doing so, I get*

*1=1*

*Therefore, it must be the case that 1 = -1. *(End of proof).

Getting students to where I wanted them to be was really challenging because many were focused on the math, and not the logic of the argument itself. They focused more on the operation of ‘squaring’ and how we need to keep in mind both positive and negative square roots, which is certainly a valid piece of mathematical insight, but again not where I needed them to be.

Since we only had about 5 minutes of class left, I decided to pause the discussion there and ask students to write me a 3 – 5 sentence of the strategies we used to justify whether a statement is true or false.

## What Went Well

I stuck with my goal.

## Where I Need Help

Right now, students still don’t understand what a proof is. I want students to be able to articulate that while plugging in values, and graphing both sides of an equation are helpful strategies to show why a statement *might* be true, they don’t constitute enough rigour to show that a statement is *always* true.

How do I get students to this point without just handing them the answer? How can I do this effectively in an online setting? They also have a common assessment (assignment) coming up in which they will be asked to prove trigonometric identities, and the quickly approaching deadline makes me feel anxious to default to just tell students the answer.

Any tips, suggestions, or feedback would be greatly appreciated! Please leave your comments below.