## Pseudoteaching: What Was Complex Becomes Routine

John Burk and Frank Noschese are running a pseudoteaching meme I encourage you to explore for yourself. Their definition:

Pseudoteaching is something you realize you’re doing after you’ve attempted a lesson which from the outset looks like it should result in student learning, but upon further reflection, you realize that the very lesson itself was flawed and involved minimal learning.

Recall also Deborah Loewenberg Ball:

I think we’ve all had the experience of giving so much structure and help that the problem becomes a simple routine problem when it wasn’t originally.

Scene

I don’t like to think about my student teaching year but for the sake of the exercise let me describe my PACT lesson, which, if you’re in California education, you know is the summative assessment of your worth as a preservice teacher. It was a precalculus class and the objective was an understanding of rose petal polar functions like r = 3cos(2theta). Students will be able graph the family. Students will be able to describe how each coefficient affects the graph. Etc.

As I recall it, I was up at the front of the classroom with a TI ViewScreen showing them graph after graph, asking them to determine what the a is doing to the graph of r = asin(3theta).

Then how does b affect the graph of r = 2sin(btheta)?

Then how do sine and cosine differ?

Then I assessed their knowledge of rose petal functions with a worksheet of graphing problems.

Breakdown

I thought my students understood the behavior of r = acos(btheta) on a deep level but they were only responding to superficial patterns in notation. (My supervisor positively thumped me for that one.) Many of you readers â€” even the straggling humanities instructors we haven’t yet scared off â€” can see from the graphs above that the petals double when b is even, that the petal length is equal to a but do any of you understand why? Do you know why the graph of r = cos(2theta) has four petals? Could you tell me why the tip of the first sine petal is always on the polar axis? (Not for nothing, the scaffolding in the worksheet also ensures I’m only getting pattern mimicry out of my students.)

Without that understanding, if I attach so much as a negative sign to that function, my students are toast. If I change a coefficient to a fraction, they’re toast. If I change sine or cosine to tangent, they’re toast. That inflexibility is an outcome of pseudoteaching.

Resolution

Unless you understand what an algorithm is going to do, it isn’t going to make sense to you.

The students have to develop the algorithm themselves. Given a second chance at that mess, I’d get students in groups of three or four and let each student pick a member of the family of the functions â€” “Okay, you do r = 1cos(2theta). I’ll do r = 2cos(2theta). You do r = 3cos(2theta).” Rather than watch me mashing buttons at the front of the room, students would graph their functions by hand and then summarize their findings to each other and then the class. Maybe with a poster â€” your call. Questions from the teacher would then include everything in the breakdown above, everything I missed six years ago.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Tom Hoffman

February 21, 2011 - 1:09 pm -

You know, it is kind of shocking that high school students can learn that kind of stuff at all, or that they’d be expected to, or anyone cares.

2. #### Pwolf

February 21, 2011 - 1:51 pm -

I’ve had similar thoughts about the same lesson, but having the kids graph those things by hand from scratch is pulling teeth–then again if I didn’t teach this right, maybe I didn’t teach plotting in polar well either.

Next time I have to teach this, I’ll probably have them match tables of values, functions, graphs, and descriptions of the graphs, kind of like that UK maths resource suggests.

3. #### Jason Dyer

February 21, 2011 - 2:08 pm -

Rather than watch me mashing buttons at the front of the room, students would graph their functions by hand and then summarize their findings to each other and then the class.

Devil’s advocate looming: while the students would get the benefit of the process of exploration, would they understand the “why” part any better? In this specific case at the very least I would include translation between rectangle and polar coordinates and some arrows showing the points of matching.

I know from working with preservice K-6 teachers this semester that pulling out the “why” (even on adding two numbers) is extremely hard and requires nurturing and practice far beyond what explorations provide. I’ve had decent luck getting them to make algorithm explanations with two columns (a How and a Why) but that’s entering into Way Too Much Time Spent To Be Practical in High School Class territory.

4. #### Dan Meyer

February 21, 2011 - 2:22 pm -

Tom: You know, it is kind of shocking that high school students can learn that kind of stuff at all, or that theyâ€™d be expected to, or anyone cares.

Get him, boys!

Pwolf: Iâ€™ve had similar thoughts about the same lesson, but having the kids graph those things by hand from scratch is pulling teeth

and

Jason: Iâ€™ve had decent luck getting them to make algorithm explanations with two columns (a How and a Why) but thatâ€™s entering into Way Too Much Time Spent To Be Practical in High School Class territory.

The constructivism multiplier, and all that. But I think the students have to see the petal return to the origin a few times. I mean, even if they’re looking at a calc-generated table but graphing the points by hand, that’s better conceptually than just teaching them “equation turns into flowery-thingy.”

5. #### Jen

February 21, 2011 - 3:26 pm -

I dunno, as a student, I’m pretty sure that I’d have found â€œequation turns into flowery-thingyâ€ interesting enough to look at and play with (and maybe get to make my own flowery-thingy, rather than just look at other ones.

Then, about a week later, maybe even on the test, although often later, I’d get the concepts. They’d just sort of dawn on me and all those things the teacher had said that seemed rather pointless (though were clearly important since s/he said them many times) would suddenly come into focus.

So, while I agree that just grinding through isn’t likely to stick, then again some grinding and some drawing or at least plotting and lots of repetition may oftentimes lead to delayed onset comprehension.

6. #### Sam Critchlow

February 21, 2011 - 7:21 pm -

Some thoughts on the specifics of the graphs etc.: the Grapher built in to most macs has the capability to parametrize a variable (a) via a slider. Parametrizing a and b (perhaps with a step of .5 or .1?) might convey the examples more efficiently than a series of graphs. Better still if there were a way for the students to control the parameters without flipping between the r= and graph screen.

This, of course, does little to nothing to explain the “why”. I struggle with the same problem when teaching transformations in rectangular coordinates. Easy to show the various rules of shifts, reflections, etc. and to come up with mnemonics to remember the rules, but difficult for me to explain in a comprehensible manner. Spent hours creating powerpoint slides trying, but given a week most students remembered almost nothing of the “how” let alone the “why”.

7. #### Alex Eckert

February 21, 2011 - 8:32 pm -

I’ve always thought you were a little too harsh on your 1st-year-teacher-self, second-year-teacher-self, and as not so subtly stated here, your preservice-teacher-self. Maybe it’s all by way of letting others know that even the great Dan Meyer was not so great once (even though I’m sure most of us on here would dispute that).

Personally I don’t think you’d be where you are now if you weren’t where you were then.

My goal, with teaching in general, isn’t so much for them to grasp the why now. It’s to grasp an appreciation for searching for the why now, so that when they’re my age they can’t stand but to go a day without trying to figure out why to something. I can’t, and it makes things awfully fun.

8. #### Alex Eckert

February 21, 2011 - 8:36 pm -

Ha…just re-read my post and realized how Cypress Hill – esque, circa 1992, my middle paragraph (sentence) sounded….

“How do you know where I’m at if you haven’t been where I’ve been……understand where I’m comin’ from???”

9. #### Laura

February 22, 2011 - 3:56 am -

@ Jason:

I don’t know that Dan is implying that he would toss out these ideas, let the kids “explore,” and expect them to arrive at the “why” as a result. I believe in guided inquiry, and carry this out as often as I can/plan with my second graders. Giving the kids time to explore something gives them a frame of reference during the discussion of “why.” And I think it’s important to remember that understanding is a precursor to articulating that understanding. So…students will be able to understand “why” before they’re able to clearly explain the “why.”

10. #### Maria

February 22, 2011 - 5:56 am -

I love this self reflection. It so mirrors my own struggling reality in the here-and-now. Thanks for the boost in confidence for a newby (whose not so ‘new’ in the chronological sense).

11. #### ClimeGuy

February 22, 2011 - 6:24 am -

John and Frank’s definition:

Pseudoteaching is something you realize you’re doing after you’ve attempted a lesson which from the outset looks like it should result in student learning, but upon further reflection, you realize that the very lesson itself was flawed and involved minimal learning.

reminded me of Alan Schoenfeld’s paper When Good Teaching Leads to Bad Results: The Disasters of “Well Taught” Mathematics Courses which might be of interest here.

12. #### Frank Noschese

February 22, 2011 - 9:53 am -

Hi Dan,

Awesome post. We want students to find patterns, but we also want them to be able to transfer the inner workings of those the patterns to new problems. Your polar functions lesson solidly shows how pseudoteaching fails to do this.

Thank you so much for taking time out of your busy #gradskool and speaking schedule to put together this post. The differences between good lessons and better lessons are nuanced, and you’ve helped bring those subtleties to light.

Many thanks again,
Frank

13. #### Brian Wyzlic

February 22, 2011 - 10:36 am -

One thing that was repeated a lot during my undergrad courses was “play with it.” While this is not something I could say without context to a group that includes high school males, that is what I asked my pre-calc students to do. All my students either had a graphing calculator or could borrow one overnight. Towards the end of class, we went over how to put the calculator in polar form (they already know what polar graphing is, we just hadn’t done much with it yet), and we look at a few graphs involving sin and cos. Then they take some time in class to play with it, and their assignment for the night is to play around with polar graphing and find some patterns.

The next day, we have a whole mess of patterns that have been found, and as a class, we can explore why those patterns exist. It works pretty well, though it does not rule out pseudoteaching without the next day exploration.

14. #### Amy

February 22, 2011 - 11:12 am -

I explored this on my own as a high school student. As soon as I started drawing out the waves, I noticed that whether or not the wave had point symmetry affected the number of petals, and it was easy to see and explain how the “b” could manipulate that point symmetry.

…. at least that’s what I remember, it’s been YEARS since I’ve explored polar coordinates!

Signed,
Humanities-Type

15. #### Jason Dyer

February 22, 2011 - 12:45 pm -

@Dan: Not disagreeing — I’ve run polar coordination exploration in class before — just being cautious. It didn’t read like a complete lesson to me yet.

Soâ€¦students will be able to understand â€œwhyâ€ before theyâ€™re able to clearly explain the â€œwhy.â€

@Laura: This is the knotty bit I’d love to unravel: how do you tell when this happens? I’ve certainly done explorations where students had the patterns down as brainlessly as if I’d just told them a recipe. I’d love to avoid that, but sometimes it’s a hard thing to check. The standard approach is to add a wrinkle and recheck for understanding (like mentioned in the example, adding a negative) but that’s not foolproof.

16. #### Luke Hodge

February 22, 2011 - 2:20 pm -

It seems to me that asking what the parameters a & b do for the functions r=asin(bt) provides a nice visual surprise, still has a little bit of meat to it, and avoids the tedium of graphing by hand.

Plotting points in hopes of a graph taking some sort of form is not an approach (at least not a first approach) that I would want to encourage. I would much rather see if I can figure anything out by looking at the equation first – then pick some nice points to plot.

I think the “why” part of this is accessible if you focus on values of t that minimize/maximize r – which is also a relatively appetizing question. Finding values of t, by hand, which minimize/maximize r is not tedious & very clearly shows why you get the number of loops that you do – it is all about the period & whether some loops will overlap previous loops. I think most people would likely need a hint or instructions to focus on max/mins for r, though.

17. #### Fachler

February 22, 2011 - 2:26 pm -

I think they key here was the end realization: by facilitating performances during which students created (and then probably evaluated), a teacher would be able to accurately gauge understanding, which is the goal–not just performance or knowledge. I think this is where the task analysis folks miss the boat.

I believe students should be in a constant state of production. As one of the “straggling humanities teachers” still following along in the discussion, it’s easy for me: students are always writing, revising, conferring, and creating class-wide and personal language arts-based projects. For the example above, I might ask, “What were the students creating with these rose petal formation graphs?”

It’s tough to get kids to understand something if the final output is just graphs. Graphs for what purpose? The Teaching for Understanding (TfU) framework calls the steps along the way “understanding performances” because the students understands more deeply with each scaffolded output (from initial inquiry, to guided, to independent).

As you have already pointed out the aesthetic qualities of those rose petals, perhaps an art-based summative output wherein students had to manipulate the equations for intended results (i.e. students sketch first, plan out the art, then have to troubleshoot the math to make it work) might have engendered transfer.

18. #### Dave Smith

February 26, 2011 - 7:13 pm -

Thanks all, for a very thought provoking discussion.

I discovered my own psuedoteaching on a big scale one semester when I read the NSTA’s wonderful little pamphlet “Ask the Right Questions.” In it they suggested using additional wait time after answers, not just after questions, and perhaps even adding in a little gesture to ask for more. I went in full of confidence to the next day’s class (upper level college geology class that I was very pleased with – lots of engagement, lots of good answers to my questions, etc). Well, first there was the panic stricken look in the eyes of my students. More?! What do you mean more?! Then there was the uncomfortable pause. Then there was the complete nonsense that came out of their mouths. Then it was my turn to panic (Oh no! They haven’t actually learned anything! Now what?!). It really was remarkable – The first couple of words in response to each of my questions were generally a correct short answer, and the next ten or twenty or more words were confused, rambling, or just plain wrong.

What I eventually learned from that experience was that I had to let/make the students do a LOT more talking. Only once students were talking did I have any way to know whether they had any idea what was going on. I began to focus my teaching on data sets. “Here are first person accounts of different kinds of volcanoes, what patterns do you notice?” Yes it took time and so I cut down my curriculum to 6 big ideas in a semester (15 class/lab hours per idea). I did much less preparation for specific class activities and much more lying awake at night thinking “Is this really the most important idea for this unit, or can I get away with leaving them confused for now?”

@Frank Noschese this is why I have been a huge fan of Modeling Instruction (modeling.asu.edu) ever since I first read about it.

Thanks for the journey back into polar coordinates, too. I had to use Excel to generate a table and scratch out a sketch graph before I began to understand what was going on, even though I clearly recall learning how to predict the number of petals once upon a time.

Dave

19. #### Dave Smith

February 26, 2011 - 7:24 pm -

Just a quick note on graphing by hand. Ever since I took a short course from Win Means at SUNY Albany on Mohr’s Circle (a graphical method for working with tensors), I have been much more likely to use sketch graphing and have a much greater appreciation for the potential of a simple sketch to convey useful and even reasonably precise quantitative information. When I realized I did not understand the functions above, I started in with SpaceTime on my iPad, but the finished graph lacked the information to show me how the petals connected to the trig function, especially since it converted all the coordinates to x, y pairs. I grabbed a handy envelope and went to work on a sketched polar plot, going from cos(theta) to cos (2theta) and within 5 or 10 minutes I had the basics figured out.

Dave

20. #### Jovan Miles

February 28, 2011 - 6:41 am -

Hey Dan, I always read but rarely comment!

Your point about algorithms not making sense to people unless they know what the algorithm is going to do just hit home with me.

I’m currently working as an instructional math coach with 2 alternative schools that serve special education students.

I’m looking over diagnostic tests and noticing a pattern of students simply not knowing what the algorithm is supposed to do…

I’m going to share this post with my teachers and hopefully they’ll get something out of it.

21. #### Launa

October 25, 2011 - 2:29 am -

I love that you hold yourself to this amazing high standard: that every kid learns at a high level, every class.

I’m an old (humanities) teacher working with a new age group at a new school, and realize just how rarely I am able to get up to that level in my new setting. Whereas before I could convince myself that I was there a lot — now, hardly at all.

I’m looking, instead, at progress-over-time, when I feel like a failure. Here at six weeks into the semester they have certainly learned, even though I could hardly see it unfolding on a daily basis, and had so many pseudoteaching fails along the way.

Thanks for the reminder of exactly what I am striving for.

22. #### Sue VanHattum

March 25, 2012 - 7:22 am -

>Could you tell me why the tip of the first sine petal is always on the polar axis?

First cosine petal, right?