Tag Archives: equivalence

Wiggins questions #2

Question 2

“Solving problems typically requires finding equivalent statements that simplify the problem” Explain – and in so doing, define the meaning of the = sign.

This question is a strange one. It really isn’t how I would define problem solving, and I certainly wouldn’t include equality as a major component underlying problem solving.


I suppose he is getting at the idea that expressing equations in equivalent forms sometimes reveals different details of a problem.

For instance, I have created a new measure for cylinders: the circumradial measure. You add the radius and height. Then multiply this sum by the circumference.

C_M = (r+h) \cdot (2\pi r)

In exploring this measure, one might end up restating this formula in equivalent terms, as:

C_M = 2\pi r^2+2\pi rh

This is more recognizable as a formula for surface area of a cylinder. The form of the equation affects how we think about the relationship it expresses.

What does the equal sign mean?

This is an important question. There is lots of research about it (CGI folks have worked on it, for instance). Three quick points:

  1. The equal sign means that the two things on either side have the same value as each other.
  2. We often teach in ways that lead students to think that the equal sign means and now write the answer.
  3. You can’t really understand much about algebra with the conception that (2) fosters. You need (1).

Finally, there are deep ideas underlying the equal sign. Equivalence is the mathematical way of talking about sameness. Stating the meaning of sameness precisely in mathematics turns out to be tricky and interesting work, and is a foundation of modern algebra.


Sameness in College Algebra

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. vending.machine.collage

  1.  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.
  2. 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.
  3. 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.
  4. The battery vending machine is one-to-one. Each battery type has its own button to push.
  5. 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:

  1. 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+0x 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?”
  2. 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.
  3. 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.
  4. Wow. Just wow. That was our example from the previous day. Not even a y=x+3 in the bunch! Work to do here.
  5. Now we’re having fun. I love the y=\frac{x}{1}. Same function, different notation. I finished off our work by asking whether y=\frac{x^{2}}{x} is the same as y=x.

Which (finally!) brings us back to sameness.

My students are highly accustomed to writing \frac{x^{2}}{x}=x. But they are not accustomed to thinking about what this means. Because when x=0, 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 \frac{x^{2}}{x}=x, 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.

What did you learn?

One thing Malke Rosenfeld and I agreed on over breakfast the other day is that the question, What did you learn? makes us uncomfortable. Weird, right? We are teachers and find both answering and asking this question makes us uncomfortable.

I have many reasons for not liking the question: that it implies the process has ended; that when I ask it of my students, they may be inclined to say what they think I want to hear; that it doesn’t invite further questions; on and on.

Being asked this question in Malke’s (fabulous) workshop* led me to something new, though.

New to me, anyway.

This coming school year, I will characterize learning—for myself and for students—in the following way.

Learning is having new questions to ask.

If I have learned something, it is because I can ask questions that I previously could not. Some examples…

example 1: Algebra II

Reading Nicholson Baker’s article on Algebra II in Harper’s [behind pay wall; also available at your local library. And seriously, a Harper’s subscription is like $15 a year.] recently, I didn’t learn anything. Much of what he had to say about the course and the way students experience it is pretty familiar and the tone resonated with many of my feelings. But when I read Jose Vilson’s response to it, I had questions. Jose writes,

If someone said, “Let’s end compulsory higher-order math tomorrow,” and the fallout happens across racial, gender, class lines, then I could be convinced that this was a step towards reform.

I wondered whether I would view Algebra II differently if I were a man (or woman) of color. I wondered yet again about the place and effect of developmental math and College Algebra on the economically and culturally diverse population of my community college. I have new questions to ask, so I learned something from my colleague Mr. Vilson that I didn’t learn from Mr. Baker.

And you are reading Jose Vilson’s blog on a regular basis, right? If not, now would be a good time to start.

Example 2: Percussive Dance

At Malke’s workshop this week, I began asking about:

  • the relationship between variable and attribute,
  • the importance of decomposing things by their attributes and paying attention to one of these attributes at a time, and whether that is a fundamental characteristic of mathematical activity,
  • whether a characteristic of a novice is an inability to distinguish noise from pattern,
  • how children’s experiences with sameness in their non-mathematical lives informs and constrains their ability to work with sameness in mathematics,
  • whether I was taking seriously my responsibility and opportunity to use physical classroom space for student learning, and
  • what kinds of equivalence relations we could use in Malke’s percussive dance work, and whether we can form a group from the resulting elements, together with composition (my hunch is yes and that the resulting group is non-Abelian, but I haven’t worked out the details).

Now you should watch Malke in action. I’ll be surprised if this 3-minute video doesn’t give you some new questions to ask.


See, in math classes asking questions is usually a sign that you have not learned.

“Any questions?” is a signal to students to speak up if they don’t get what has just been explained.

We have it all backwards.

It shouldn’t be, “What questions do you have?” [I hope you have none so that I can tell myself you learned something.] 

It should be, “What new questions can you ask?” [I hope you have some because otherwise our work is having no effect on your mind.] 

*Asked by someone who is not Malke, for the record.