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.

“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.”

Unless you are getting the same dollar back that you put in, I wouldn’t call this the identity function.

As an example, during the financial crisis, many companies found that the assets they were carrying on their books with a value of x dollars, could not actually be turned into x actual dollars. Many of the loan programs that the government arranged for the banks were in place specifically to solve this “identity” problem (which the economists seem to like to call a liquidity problem, rather than a “your assets aren’t worth anything close to what you say they are worth” problem).

If I graph x = 2, I do have “output.” I just have no way of predicting what it will be, based on my input of x.

Thank you for this. I skip the whole functions chapter in our textbook because I think what they ask students to do is lame. But I think I can include some of what you’ve done here.

I like the vending machine metaphor. Metaphors are awesome.

When I thought of the change machine, I was like, “Jack puts a $1 bill in the machine. Gets 4 quarters. Now he puts a $5 bill in, and gets 20 quarters. Later: a guy puts “some bills” in the machine and gets 20 quarters. What just happened?”

Answer? He either put in 5 $1 bills or 1 $5 bill, but we don’t know which. i.e. not a 1-to-1 function.

JamesandMikeseem to be expressing the same objection about thechange machine as identity functionexample. These objections come up in class. They are important because they illustrate my point about sameness. What the dollar I put in and the four quarters I get back have in common—the way they are the same—is their value.In mathematics, we have to identify the attribute we are going to pay attention to. I claim that we

couldpay attention to the value of the money going in and coming out, while ignoring all other attributes (its form, to whom it belongs, year of issue, et cetera). If we do, then we can let x=value of money inserted, y=value of money dispensed, and we have y=x. By definition, this is the identity function.But viewing money that way requires an abstraction—the very abstraction that is at the heart of doing real mathematics.

Now, the metaphor breaks down (as

allmetaphors do). I cannot put in 100 pennies, even though they have the same value as a one-dollar bill. So stating the domain in an abstract way doesn’t really work.In James’s example, we know that Jack put in something with the value of 5 dollars. It’s a 1-to-1 function when we pay attention only to values.

I made the M&M video to initiate this conversation. Most of us will accept that those two bags are the same. And we can also accept that if I give James one bag, and Mike one bag, they did not each get “the same bag”. Instead, they got bags that are the same in some respects. But maybe one bag has more broken M&Ms. Maybe one has more brown M&Ms than the other. Are they still the same? Not if I care about the color distribution. All

samenessrequires us to pay attention to some attributes and to ignore others. Mathematical sameness (i.e.equivalence) makes these ideas rigorous.As for the x=5 example from

xiousgeonz…I retract my previous claim. I now see that x=5 is like vending machine 3 above. Same input gives multiple outputs (and, importantly, there is only one input). We talked about this in class today.Thanks for playing along, everyone!

The value question is interesting. Say that you put in a $1 bill and get back a different $1 bill. Both have the same value as currency, but could have different values in other situations.

Perhaps the 2nd dollar bill is rare and worth $10 to a collector, or perhaps it has an unusual serial number and some Wall St. trader would like very much to use it in games of Liar’s Poker and would pay $50 for it. It isn’t just the “legal tender” piece of the $1 bill that can bring value.

Just to be a curmudgeon (because I really do like the change machine metaphor), I find your argument that the

valueis the only quantity we care about to be problematic, in thatallvending machines then represent identity functions. I put in $1 worth of currency and get back $1 worth of Twinkie, or whatever. Presumably I personally find the value of the input to be equal to the value of the output in every single vending machine situation, otherwise I wouldn’t use vending machines. Or, rather, I find the output value to beat leastas great as the value I put in (vending machine arbitrage, anyone?), in which case the metaphor becomes less clear.Yours curmudgeonly,

Lusto

The “fungible” comment above provides a great example for a college algebra class about mathematical definitions and precision.

Among the more famous legal cases around the idea that “money is fungible” is described here:

http://www2.gtlaw.com/pub/alerts/2005/1201.asp

It shows that “money is fungible” is quite a powerful argument – indeed powerful enough to help carry the day in a multi-million dollar lawsuit.

But if we’ve learned anything from this blog, it is that just because an argument is good enough for an American billionaire, doesn’t mean it will carry the day around here. And so we need to ask the question: does the idea of “money is fungible” rise to the level of a true mathematical statement?

The answer seems to be no. See here, for some easy counterexamples:

http://en.wikipedia.org/wiki/List_of_most_expensive_coins

The list on the Wikipedia page gives 10 examples where someone was willing to trade far more than the face value of the currency for a piece of money. In these cases we see that money was indeed not fungible – dictionary definitions, and case law be damned.

Of course mathematical history is filled with many other interesting examples. A single counterexample could have shown Fermat’s Last Theorem to be false. A single counterexample could show the Goldbach Conjecture to be false. A single counterexample from Euler did show that not all Fermat Numbers are prime.

Precision is one of the beautiful things about mathematics. Statements such as “money is fungible” can be regarded as “true” in many, many situations – linguistic, legal, blog comments, and etc – while nonetheless not rising to the level of a true mathematical statement.

For the student in a college algebra class, how “money is fungible” can be true enough to carry the day in a multimillion dollar legal case but still not true enough to be considered a mathematically precise statement would, I think, be quite interesting. In the specific case about the vending machine representing the identity function, I’m sure the discussion would also be quite interesting.

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