Two years ago, I began using *unit* as an organizing theme in the math content course for future elementary teachers. That led to many adventures, including a TED-Ed video and new ways of talking to my colleagues about fractions, decimals and place value.

That work continues, but it has become part of my instructional practice; one of my habits of mind.

This year, I am thinking about *sameness, *and about helping my students to notice and pay attention to sameness. The formal name is *equivalence*, but I am not so worried about the vocabulary and formal definitions here.

I am concerned with helping students understand something about how mathematics views and uses sameness.

It is awkward at first, as any new teaching moves are. But it got us some good stuff recently.

We are studying functions. Our grounding metaphor for functions is vending machines. We discussed the following collection of vending machines the other day.

- This is my favorite vending machine of all time. The banana vending machine. It dispenses only bananas. It is like the constant function. More on this below.
- There are two ways to get the Pocari Sweat in a can. Two inputs, same output. That’s OK. It’s not one-to-one, but it’s a function.
- You put in a quarter, you turn the knob. Sometimes you get a die. Sometimes you get a top. Sometimes you get a ball. This is not a vending machine, really. Same input gets you different outputs. That’s a problem in the vending machine world, and in the world of functions.
- The battery vending machine is one-to-one. Each battery type has its own button to push.
- Put a dollar into this one, get a dollar out. Put in five dollars, get out five dollars. The output has the same value as the input. This is the identity function.

We discussed these in class one day. Then we opened the next class session by having students brainstorm with their partners specific functions with the traits exemplified by the vending machines. We divided up responsibilities for recording these functions on the classroom whiteboards.

Here is what our boards looked like after the large group (45 students) discussion. (Click to make legible.)

In order:

- Lots of good stuff here.
*x*=2 is not a function because, as a vending machine, it would take your money and not put anything out. All input, no output. The idea that we can write*y*=5 as*y*=5+0*x*was important. More importantly, this led a student to ask* about y=5, “Can it be a variable if it’s always the same value?” - Our example the previous day had been
*absolute value*. They weren’t ready to venture much beyond this. As a class, they struggled to identify two*x*-values that would generate the same*y*-value. We need to work on that. But I have mentioned that this is College Algebra, right? Students have placed here, or worked their way here through developmental math. Either way, the idea of producing example points to demonstrate properties of a function has not been schooled into them yet. I’m on it. - Again, +/– square root was the prior day’s example. I love +/–
*x*as an extension of the technique. Love that. And square root of*x*is not right. We’ll come back to that. Having a permanent record of the difference will be helpful. - Wow. Just wow. That was our example from the previous day. Not even a
*y*=*x*+3 in the bunch! Work to do here. - Now we’re having fun. I love the . Same function, different notation. I finished off our work by asking whether is the same as .

Which (finally!) brings us back to sameness.

My students are highly accustomed to writing . But they are not accustomed to thinking about what this *means*. Because when , that equation is not true. The question then becomes, *In what sense are these the same?*

And that points us to the very heart of the discipline.

In mathematics, we decompose things according to their attributes, and we focus on one (or two, or…) of these attributes at a time, *disregarding all of the others*. Formally, when we write , we mean “These two expressions are the same for all but a finite number of values of *x*.” We don’t say that, of course, but that is the essence of the equal sign here.

We returned to the sameness question with this video.

Are the two outputs the same? How? Are they different in any way? How? Again, mathematical sameness requires us to specify the precise ways in which two objects are alike.

We will return to machine number 3 above in class shortly. If you just want to get “a cheap plastic toy” out of the machine, then you get that every time. It’s a function. If you want to get “a top” out of the machine, then you get something different every time. Is it a function? Depends on what you mean by “same”.

Much more work to do. I’ll keep you posted.

*I recently argued that learning is *having new questions to ask*. This student was learning about what *variable* means, and had a question to ask that she maybe could not have articulated before this.