I wanted to see what students were capable of on their own. No peer discussion. No Desmos. I was particularly interested to see whether they could write an equation to describe the relationship they see unfolding visually, numerically, and graphically. Some could. Some could not.

As I walked throughout the room, I took note of the incorrect equations students were writing down. There were four in particular that caught my eye. Students were clearly doing some relevant work with calculating slope and (trying, anyway) to identify the y-intercept… However, the way they mashed it all together left something to be desired.

On the Fly…

So as my last few students were finishing their work, I threw together this slide, featuring the pattern they had been working with, four “out-in-the-wild” incorrect equations, and a not-so-accidental suggestion that one of the equations is correct (just to keep ’em on their toes):

We then took a blank sheet of printer paper, folded it in half, and unfolded it. Two workspaces on the front, and two on the back. We then filled each workspace with an equation as well as a table of the actual values…

…plus a show-your-thinking-on-the-page run through of each x-value, evaluated in the given equation.

In each case, the expression values didn’t match the actual values, and the equation proved to be an imposter. While this “let’s evaluate” approach was rather typical, and the discussion was somewhat predictable, the results were nevertheless quite powerful. The major issues we identified and addressed were:

1. Students lacked a go-to strategy for verifying whether their equation was valid. It’s almost as if they viewed the progression (from numerical to algebraic) as a one-way street. Our conversation helped them see how to go from numerical to algebraic, and then back again to confirm. I probably should have equipped them with this skill some time ago. Better late than never.
2. Students were working with some faulty y-intercept assumptions. The evidence suggests (and their comments confirm) that of those who got this wrong, most went hunting for the y-intercept where it’s often found: at the top of the table. “The y-intercept occurs when the x-value is zero…” is not something that has stuck with them, despite visual, verbal, and numerical attempts to drive that home. I’m cautiously optimistic that today’s conversation smashed that misconception for most students.

Side Benefit

While most students wrote an equation with a rate of change of 4 new circles per stage, I did have a few “slope stragglers” suggest equations like y = x + 4 and y = 10x + 4. As we evaluated multiple x-values in each of four equations, and left a record of our evaluating on the board, the similarities and differences rose to the surface. The connection between the coefficients of our faulty equations (1, 4, 4, and 10) and the common differences between expression results caused a few more light bulbs to turn on.

Future Use

We’ll continue to explore Visual Patterns in small groups most the time, but I think I’ll include a dose of individual formative assessment, followed by small-group or whole-class error analysis now and again. We unpacked a lot of misconceptions today, and made a number of valuable connections as well, all in a rather short period of time. Anything that draws out misconceptions so we can smash ’em to bits through class discussion is worth bringing back for an encore.

Within each year, I’ve never taught more than two sections of the same course. In fact, one year I taught seven different courses. (Assuming, of course, that you’ll allow me to count Algebra 2 Honors as distinct from Algebra 2.)

With all of this variety from year to year (and class period to class period), I’ve come to rely on classroom routines not only to preserve my sanity, but also to challenge myself to pursue growth.

The latest initiative I’m considering? It’s right in line with my recent bent on extracting every ounce of growth-mindset goodness from the abundance of mistakes we (teachers and students) all make in the classroom.

With that background in mind, here’s what I intend to put in front of my students (and myself) during the last few minutes of class each Friday for the rest of the semester.

There’s no guarantee I’ll stick with this routine, or that we’ll benefit from it. But I am holding out hope—on both counts, in fact.

As for my actual goals with this prompt, I’m looking to further develop my students’ growth mindset. In particular, I want to foster a sense—across the entire classroom—that mistakes are actually opportunities, and that they can help launch us into the next level of awesome-ness.

As for sharing my thoughts here… There’s a sense of commitment that comes with writing about something.

I’m counting on you to hold me to it in the comments.

One practice I’ve been using for years is Assessment Corrections, where students identify, correct, and reflect on recent assessment mistakes. (Here’s the form I use to guide the process.)

One practice I’ve wanted to use but never followed through on is starting math department meetings with a five-minute discussion of a mistake from one of our students.

Whatever the format, I’m convinced that my own teaching practice would benefit from an increase in thoughtful error analysis. I suspect the same is true for many other teachers. And since developing new habits is difficult, I’ve been considering how to incorporate math-mistakes-style reflection into my classroom through simple, repeatable routines.

Two Truths and a Lie, Math Style

Here’s what I’ve come up with:

1. Display three solutions to a problem. Ideally, include photos of original student work. If that’s not possible for some reason, photos of teacher-written solutions on post-it notes or index cards will do the trick. So will typed-out solutions.
2. Facilitate student reflection and discussion. There is much to be gained from each of the following: individual student reflection, discussion in pairs or small groups, and whole-group discussion. Whatever format (or combination) you choose, lead students to think about the assumptions behind each mistake, whether there are fragments of correctness in a given mistake, and how they might help a classmate who was prone to such a mistake.

Looking Ahead

I’m excited to use this approach more often in my classes throughout the rest of the semester. Will it be perfect? Probably not. I can imagine some errors that simply won’t fit in this side-by-side-by-side format. And I’m uncertain whether a projected slide is as useful as a paper handout, or if I should lean toward the latter. Even with these lingering questions, I do know this: This routine has the potential to add considerable value to my classroom, as it provides students with repeated opportunities to develop their sense-making and argument constructing/critiquing abilities. And if I had to rank the CCSS Standards for Mathematical Practice in order of importance, it’s likely I would place those two (SMP 1 and SMP 3) at the top.

Questions

How do you incorporate thoughtful discussion of math mistakes into your classroom or department? Do you have ideas for how to make my “two wrongs and a right” approach better? Drop a line in the comments below.

And if you make a mistake, it can be an absolute bear to track it down. Is your issue differentiation? Evaluating a function? Or is your weak spot related to what’s happening visually in linear approximation?

Here’s a problem from last year’s AP Calculus review workbook:

There’s a lot of great work on the page.

• Take the derivative? Check.
• Find the slope of the tangent line? Check?
• Find the equation of the tangent line? Check.

But that’s where things fall apart.

Now, imagine you’re a calculus student. You’ve been hammering away at this thing for several minutes. Maybe you don’t even remember what the problem’s asking for in the first place. You get to this point, you scan the original problem, see an input value of 4.2, and presto-whammo, plug it in and out comes 0.4. “Great news, everyone! That’s on the list! Well done, folks. On to the next problem!”

Only, that answer is entirely wrong. So how do you debrief this problem with a student, small group, or class, so they can see the source of the error? For problems with even a sliver of something graphical, Desmos has become my go-to tool for helping students find and fix their errors.

Here’s what I built with a pair of students last year (with a link to the live graph here):

Students can’t use Desmos on the AP exam (for now, anyway), so I’m not trying to permanently sidestep what they ultimately must be able to do sans technology (or with a device from that “other” graphing calculator company). But what we can do in class with Desmos is build a better visual/conceptual sense of what’s happening in this problem so they’ll be more prepared for something similar in the future.

Here’s a short list of what this Desmos graph did for us in this scenario:

1. We offloaded the algebraic and numerical work of finding the derivative (and evaluating it for a particular x-value) and built the tangent line in a matter of seconds (rather than minutes). Mentally, we’re still fresh, and ready to focus on what the problem is really asking us to do: compare the function and the tangent line.
2. We gave things specific names so we could call on them in our time of need. Okay, that may sound a little dramatic. But think about why we even bother with function notation. Why give a function a name? Well, why did your parents give you a name? So they could call on you! (“Alfred! Get down here and pick up your comic books!”) So then, why do we give functions names? Because function notation is on the Chapter 8 test in Algebra 2? No! We give functions names so we can call on them. So they can do our bidding. If you don’t name it, it’s difficult to put a function to work for you. Give it a name? Now our wish is its command. (It sounds a little bit like we’re going to take over the world, with math as our trusty sidekick.)
3. We gave things specific names so we could keep clear in our own minds the various moving pieces in the problem. There’s a function. We called that f. There’s a tangent line. We called that l (or t, or whatever). As we approach the end of the problem, and we start looking for the error, it’s easier to avoid simply evaluating f(4.2) or l(4.2), because we know we’re dealing with both f and and l. (How could we forget?! We named them! We practically gave birth to them.)
4. We visualized the error with a beautiful little orange bar, and in doing so imprinted on our minds (for future problems) what linear approximation error looks like.
5. We dropped a slider in so we could answer a hundred related problems in a matter of seconds, further clarifying for the confused student (or teacher; I was terrified of linear approximation my first two years of teaching Calculus) what the problem is really about, and all with a nifty, dynamic burst of compare-and-contrast.

Do you need Desmos to teach this stuff? Maybe not. But given the option, I’ll use it every time. Last year in my classroom, we got a lot more out of a five minute Desmos-supported conversation than we did in a whole day of business-as-usual notes and examples.

So this year? We went straight to Desmos and put the emphasis right where it belongs.

P.S. I love GIFs.

I’m a big fan of Michael Pershan’s project Math Mistakes. If you’ve never checked it out, it’s worth exploring. And while I’m meddling with your life, here’s a tip for your entire department: Start each meeting by spending five minutes exploring one of the mistakes posted on Michael’s site. On a rotating basis, have one member of the department share a “provocative” math mistake from the blog (or maybe even one from his or her own classroom). And once duly provoked… Cue the discussion!

Amble

I included the following uninspiring question on a recent assessment:

The first two assessments I graded included the following responses:

Student 1

Student 2 ((My apologies for the retype. I added some feedback before snapping a photo.))

Comment Fodder

So here’s my question (er, set of questions) for you:

1. Are these mistakes equally egregious?
2. What misconceptions are contained in the first mistake? How would you address them?
3. What misconception are contained in the second mistake? How would you address them?
4. Would you grant any credit for either response, and why?

I asked him to complete as many of the problems as he could during the next 10 minutes or so. Then we’d chat about what he got right, what he got wrong, and what he skipped. A little while later he had this:

Trouble with Radicals, But Not with Rational Exponents?

When we first sat down, I figured he would have trouble with rational exponents. Nearly all of my students (this year and in the past) who struggle with this topic have almost no trouble with square roots in radical form, some trouble with cube roots in radical form, and (if they have any issues at all) massive problems with rational exponents.

This student’s struggle was more or less the exact opposite of what I typically see. He had trouble with radicals (square roots, cube roots, anything in radical form), and almost no trouble with rational exponents. (My conjecture on #21 is that the times table in his brain has a blank spot at 8 times 8.)

Concept vs Notation

The evidence clearly shows that this student doesn’t have a conceptual deficiency. Instead, his struggle is with…

@mjfenton @mpershan Notation, notation, notation.

— Justin Lanier (@j_lanier) October 2, 2013

Remediation for notation is usually fairly simple. We talked for a minute or two about how radicals (unknown and unfamiliar to him) relate back to rational exponents (known and familiar). Several minutes later, he came back with this:

Things are looking up, even if they’re still not perfect.

Shifting My Approach

This exchange has me rethinking the direction of the conceptual/notational connection I’ve been trying to draw out for years. In working with an expression raised to the 1/2, I’ve always angled our conversations toward (and silently rejoiced inside when a student shouts):

I treated square roots like the native language, the most helpful representation, and rational exponents as this foreign thing that needs to be converted back to familiar territory.

It’s true that students are more familiar with radicals (at least in my experience with middle schoolers), but I’m quickly starting to believe that rational exponents are dramatically more informative when it comes to thinking conceptually (and when it comes to working procedurally).

August, Every Year

When students enter my classroom, our first discussion about exponents (which invariable happens within the first couple weeks of school) goes more or less like this:

Me: What does mean?

Students: Eight!

Me: No, not “What is its value?” What does it mean, what does it represent?

Students: Oh (why didn’t you say so). 2 times itself three times.

Me: What?! You mean…

(two)

(times itself once)

(times itself a second time)

(times itself a third time)

Me: There. 2 times itself three times. Wait… That’s…

Students: No, you got it wrong. That’s 2 to the fourth!

Me: But you said…

Students: Yeah, but we didn’t mean…

Me: Grrr…

TWO MINUTE TIME WARP…

Me: And that’s why it’s more useful to say it like that. So, how would you state the meaning of this:

Students (in unison, with a three-part harmony): Four factors of 10!

Me: Perfect!

I want them to express powers this way for a number of reasons. At the very least, saying it the other way is flat out wrong. But describing as “b factors of a” has proven immensely useful in developing properties of exponents (which, for what it’s worth, I don’t hate as much as many in the MTBoS, probably because I’m easily entertained, and maybe also because my simple brain enjoys finding and justifying simple patterns).

Okay, So… What Exactly Are We Talking About?

By now, of the 12 people who started reading this post, and the three who are still reading, at least two of you are wondering: What does this have to do with rational exponents and your struggling student?

Well, several weeks ago in Algebra 1 (the above student’s class) we had our first discussion of rational exponents. As usual, I was trying to elicit from them the idea that “to the 1/2” can be thought of as “square root,” and so on.

But a few students—bless their little hearts—wondered: Why?

And another student—bless his heart—applied our beloved “b factors of a” phrasing to come up with this:

What would that even mean? I knew, and you know, too, because we’ve seen this movie before (or at least accidentally read a spoiler in some blog comment or Facebook news feed overpopulated by comments you were never interested in reading in the first place; I digress).

But my students didn’t have half a clue what “half a factor of…” would mean, and I was on the edge of my seat to see where they would take this. (Correction: I was standing. But I fully expect I was standing on the edge of wherever it was that I was standing.)

What They Saw

After a few more minutes of discussion, here’s what they saw and (more or less) how they described it.

• If you need to find “some number” to the 1/2, write the number as two identical factors. Then “take” one of them. That’s your answer.
• If you need to find the value of “some number” to the 1/3, write the number as three factors of the same number. Then “take” one of them. That’s your answer.

What I Wondered

I’d never have a conversation on rational exponents take that turn, so now I was curious… What would my students do with other rational exponents? The next day, on their Topic 2 assessment, I invited students to attempt two challenge problems on the back:

1. Find the value of
2. Find the value of

The results were mixed, but a majority of those who attempted the problems were spot on. Here’s a sample:

I guess in some ways this doesn’t differ much from the classic treatment:

because

But it somehow strikes me as different, as offering more potential for extension, at least in the form my student wrote it on the review sheet that inspired this post. And now I’m wondering another thing. Would these same students—without any additional instruction from me—be able to evaluate ?

My guess is some of them could, and I expect they’d treat it like this:

Write 8 as three identical factors, take 2

In fact, I’d wager that with a brief class discussion, most of them would be equipped to handle any of these:

Write 16 as four identical factors, take 3

Write 100 as two identical factors, take 3

Write 4 as two identical factors, take 5

Write 100,000 as five identical factors, take 3

For My Next Trick, I’ll Be Misusing the Word Partition

This idea of partitioning a number into identical factors and selecting a portion of those factors feels an awful lot like multiplying whole numbers by rational numbers:

Write 20 as four identical terms, take one

Write 21 as three identical terms, take two

Write 8 as two identical terms, take five

I don’t know if you can use partition in that sense (the factors sense), but I couldn’t shake the notion that these two problem types have a lot more in common than I ever thought before. (Maybe now that I’ve rambled all over the page I’ll be able to get some sleep at night. Or was it the kids waking up in the middle of the night that was disturbing my sleep… Too tired to remember.)

Your Thoughts?

I’m a little bit nervous about hitting “publish” on this one. I feel like there are four likely responses to the post—for anyone persistent enough to ramble (as in walk) through to the end of these ramblings (as in babble):

1. Thanks for blathering, but I have no idea what you just said.
2. Thanks for nothing, everyone already knew everything you said.
3. Thanks for trying, but I think you need a mathematical intervention to work out some of your own misconceptions.
4. Thanks for sharing, that’s a nifty connection. I might use it one day with my own students.

If you made it this far, let me know which of those reactions best describes your own. Or go off script and drop a more thoughtful comment.

Either way, thanks for playing!