First cosine petal, right?

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.

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.

Dave

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

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.

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.

So…students will be able to understand “why” before they’re able to clearly explain the “why.”

@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.

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

Signed,
Humanities-Type

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.