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This just in from dy/dan: Jo Boaler (Stanford prof, author of What’s Math Got to Do With It and inspiration for Dan Meyer’s “pseudocontext” series) is offering a free online course for “teachers and other helpers of math learners.” The course is called “How To Learn Math.”
“The course is a short intervention designed to change students’ relationships with math. I have taught this intervention successfully in the past (in classrooms); it caused students to re-engage successfully with math, taking a new approach to the subject and their learning. In the 2013-2014 school year the course will be offered to learners of math but in July of 2013 I will release a version of the course designed for teachers and other helpers of math learners, such as parents…” [emphasis is original]
I’ve been disheartened this year to realize how limited my toolset is for convincing students to broaden their thinking about the meaning of math. Every year, I tangle with students’ ingrained humiliation in the face of their mistakes and sense of worthlessness with respect to mathematical reasoning. I model, give carefully crafted feedback, and try to create low-stakes ways for them to practice analyzing mistakes, understanding why math in physics gives us only “evidence in support of a model” — not “the right answer”, and noticing the necessity for switching representations. This is not working nearly as well as it needs to for students to make the progress they need and that I believe they are capable of.
I hope this course will give me some new ideas to think about and try, so I’ve signed up. I’m especially interested in the ways Boaler is linking these ideas to Carol Dweck’s ideas about “mindset,” and proposing concrete ideas for helping students develop a growth mindset.
Anyone else interested?
Michael Pershan kicked my butt recently with a post about why teachers tend to plateau in skill after their third year, connecting it to Cal Newport’s ideas such as “hard practice” (and, I would argue, “deep work“).
Michael distinguishes between practice and hard practice, and wonders whether blogging belongs on his priority list:
“Hard practice makes you better quickly. Practice lets you, essentially, plateau. …Put it like this: do you feel like you’re a 1st year teacher when you blog? Does your brain hurt? Do you feel as if you’re lost, unsure how to proceed, confused?If not, you’re not engaged in hard practice.”
Ooof. On one hand, it made me face my desire to avoid hard practice; I feel like I’ve spent the last 8 months trying to decrease how much I feel like that. I’ve tried to create classroom procedures that are more reuseable and systematic, especially for labs, whiteboarding sessions, class discussions, and model presentations.
It’s a good idea to periodically take a hard look at that avoidance, and decide whether I’m happy with where I stand. In this case, I am. I don’t think the goal is to “feel like a first year teacher” 100% of the time; it’s not sustainable and not generative. But it reminds me that I want to know which activities make me feel like that, and consciously choose some to seek out.
Michael makes this promise to himself:
It’s time to redouble my efforts. I’m half way through my third year, and this would be a great time for me to ease into a comfortable routine of expanding my repertoire without improving my skills.
I’m going to commit to finding things that are intellectually taxing that are central to my teaching.
It made me think about what my promises are to myself.
Be a Beginner
Do something every summer that I don’t know anything about and document the process. Pay special attention to how I treat others when I am insecure, what I say to myself about my skills and abilities, and what exactly I do to fight back against the fixed-mindset that threatens to overwhelm me. Use this to develop some insight into what exactly I am asking from my students, and to expand the techniques I can share with them for dealing with it.
Last summer I floored my downstairs. The summer before that I learned to swim — you know, with an actual recognizable stroke. In both cases, I am proud of what I accomplished. In the process, I was amazed to notice how much concentration it took not to be a jerk to myself and others.
Learn More About Causal Thinking
I find myself being really sad about the ways my students think about causality. On one hand, I think my recent dissections of the topic are a prime example of “misconceptions listening” — looking for the deficit. I’m pretty sure my students have knowledge and intuition about cause that I can’t see, because I’m so focused on noticing what’s going wrong. In other words, my way of noticing students’ misconceptions is itself a misconception. I’d rather be listening to their ideas fully, doing a better job of figuring out what’s generative in their thinking.
What to do about this? If I believe that my students need to engage with their misconceptions and work through them, then that’s probably what I need too. There’s no point in my students squashing their misconceptions in favour of “right answers”; similarly, there’s no point in me squashing my sadness and replacing it with some half-hearted “correct pedagogy.”
Maybe I’m supposed to be whole-heartedly happy to “meet my students where they are,” but if I said I was, I’d be lying. (That phrase has been used so often to dismiss my anger at the educational malpractice my students have endured that I can’t even hear it without bristling). I need to midwife myself through this narrow way of thinking by engaging with it. Like my students, I expect to hold myself accountable to my observations, to good-quality reasoning, to the ontology of learning and thinking, and to whatever data and peer feedback I can get my hands on.
My students’ struggle with causality is the puzzle from which my desire for explanation emerged; it is the source of the perplexity that makes me unwilling to give up. I hope that pursuing it honestly will help me think better about what it’s like when I ask my students to do the same.
Interact with New Teachers
Talking with beginning teachers is better than almost anything else I’ve tried for forcing me to get honest about what I think and what I do. There’s a new teacher in our program, and talking things through with him has been a big help in crystallizing my thoughts (mutually useful, I think). I will continue doing this and documenting it. I also put on a seminar on peer assessment for first-year teachers last summer; it was one of the more challenging lesson plans I’ve ever written. If I have another chance to do this, I will.
Work for Systemic Change
I’m not interested in strictly personal solutions to systemic problems. I won’t have fun, or meet my potential as a teacher, if I limit myself to improving me. I want to help my institution and my community improve, and that means creating conditions and communities that foster change in collective ways. For two years, I tried to do a bit of this via my campus PD committee; for various reasons, that avenue turned out not to lead in the directions I’m interested in going. I’ve had more success pressing for awareness and implementation of the Workplace Violence Prevention regulations that are part of my local jurisdiction’s Occupational Health and Safety Act.
I’m not sure what the next project will be, but I attended an interesting seminar a few months ago about our organization’s plans for change. I was intrigued by the conversations happening about improving our internal communication. I’ve also had some interesting conversations recently with others who want to push past the “corporate diversity” model toward a less ahistorical model of social justice or cultural competence. I’ll continue to explore those to find out which ones have some potential for constructive change.
Design for Breaks
I can’t do this all the time or I won’t stay in the classroom. I know that now. As of the beginning of January, I’ve reclaimed my Saturdays. No work on Saturdays. It makes the rest of my week slightly more stressful, but it’s worth it. For the first few weeks, I spent the entire day alternately reading and napping. Knowing that I have that to look forward to reminds me that the stakes aren’t as high as they sometimes seem.
I’m also planning to go on deferred leave for four months starting next January. After that, I’ve made it a priority to find a way to work half-time. The kind of “intellectually taxing” enrichment that I need, in order for teaching to be satisfying, takes more time than is reasonable on top of a full-time job. I’m not willing to permanently sacrifice my ability to do community volunteer work, spend time with my loved ones, and get regular exercise. That’s more of a medium-term goal, but I’m working a few leads already.
Anyone have any suggestions about what I should do with 4 months of unscheduled time starting January 2014?
I wrote recently about creating a rubric to help students analyze their mistakes. Here are some examples of what students wrote — a big improvement over “I get it now” and “It was just a stupid mistake.”
The challenge now will be helping them get in the habit of doing this consistently. I’m thinking of requiring this on reassessment applications. The downside would be a lot more applications being returned for a second draft, since most students don’t seem able to do this kind of analysis in a single draft.
Understand What’s Strong
“I thought it was a parallel circuit, and my answer would have been right if that was true.”
“I got this question wrong but I used the idea from the model that more resistance causes less current and less current causes less power to be dissipated by the light bulbs.”
“The process of elimination was a good choice to eliminate circuits that didn’t work.”
“A good thing about my answer is that I was thinking if the circuit was in series, the current would be the same throughout the circuit.”
Diagnose What’s Wrong
“The line between two components makes this circuit look like a parallel circuit.”
“What I don’t know is, why don’t electrons take the shorter way to the most positive side of the circuit?”
“I made the mistake that removing parallel branches would increase the remaining branches’ voltage.”
“What I didn’t realize was that in circuit 2, C is the only element in the circuit so the voltage across the light bulb will be the battery voltage, just like light bulb A.”
“I looked at the current in the circuit as if the resistor would decrease the current from that point on.”
“I think I was thinking of the A bulb as being able to move along the wire and then it would be in parallel too.”
“What I missed was that this circuit is a series-parallel with the B bulb in parallel with a wire, effectively shorting it out.”
“What I did not realize at first about Circuit C was that it was a complete circuit because the base of the light bulb is in fact metal.”
“I thought there would need to be a wire from the centre of the bulb to be a complete circuit.”
“I wasn’t recognizing that in Branch 2, each electron only goes through one resistor or the other. In Branch 1, electrons must flow through each resistor.”
“I was comparing the resistance of the wire and not realizing the amount of distance electrons flowed doesn’t matter because wire has such low resistance either way.”
“My problem was I wasn’t seeing myself as the electrons passing through the circuit from negative to positive.”
“In this circuit, lightbulb B is shorted so now all the voltage is across light bulb A.”
“When there is an increase in resistance, and as long as the voltage stays constant, the current flowing through the entire circuit decreases.”
“After looking into the answer, I can see that the electrons can make their way from the bottom of the battery to the middle of the bulb, then through the filament, and back to the battery, because of metal conducting electrons.”
“To improve my answer, I could explain why they are in parallel, and also why the other circuits are not parallel.”
“I can generalize this by saying in series circuits, the current will stay the same, but in parallel circuits, the current may differ.”
“From our model, less resistance causes more current to flow. This is a general idea that will work for all circuits.”
I went looking for a resource about “growth mindset” that I could use in class, because I am trying to convince my students that asking questions helps you get smarter (i.e. understand things better). I appreciate Carol Dweck‘s work on her website and her book, but I don’t find them
- concise enough,
- clear enough, or
- at an appropriate reading level for my students.
What I found was Diana Hestwood and Linda Russel’s presentation about “How Your Brain Learns and Remembers.” The authors give permission for non-profit use by individual teachers. It’s not perfect (I edited out the heading that says “You are naturally smart” … apologies to the authors) and it’s not completely in tune with some of the neuroscience research I am hearing about lately, but it meets my criteria (above) and got the students thinking and talking.
Despite her warning that it’s not intended to stand on its own and that the teacher should lead a discussion, I’d rather poke my eyes out than stand in front of the group while reading full paragraphs off of slides. I found the full-sentence, full-paragraph “presentation” to work on its own just fine (CLARIFIED: I removed all the slides with yellow backgrounds, and ended at slide 48). I printed it, gave it to the students, and asked them to turn in their responses to the questions embedded in it. I’ll report back to them with some conversational feedback on their individual papers and some class time for people to raise their issues and questions — as usual, discussion after the students have tangled with the ideas a bit.
The students really went for it. They turned in answers that were in their own words (a tough ask for this group) and full of inferences, as well as some personal revelations about their own (good and bad) learning experiences. There were few questions (the presentation isn’t exactly intended to elicit them) but lots of positive buzz. About half the class stayed late, into coffee break, so they could keep writing about their opinions of this way of thinking. Several told me that “this was actually interesting!” (*laugh*) I also got one “I’m going to show this to my girlfriend” and one, not-quite-accusatory but clearly upset “I wish someone had told me this a long time ago.” (*gulp*)
I found a lot to like in this presentation. It’s a non-threatening presentation of some material that could easily become heavily technical and intimidating. It’s short, and it’s got some humour. It’s got TONS of points of comparison for circuits, electronic signal theory, even semiconductors (not a co-incidence, obviously). Most importantly, it allows students to quickly develop causal thinking (e.g. practice causes synapses to widen).
Last year I found out in February that my students couldn’t consistently distinguish between a cause and a definition, and trying to promote that distinction while they were overloaded with circuit theory was just too much. So this year I created a unit called “Thinking Like a Technician,” in which I introduced the thinking skills we would use in the context of everyday examples. Here’s the skill sheet — use the “full screen” button for a bigger and/or downloadable version.
It helped a bit, but meant that we spend a couple of weeks talking about roller coasters, cars, and musical instruments. Next year, this is what we’ll use instead. It’ll give us some shared vocabulary for talking about learning and improving — including why things that feel “easy” don’t always help, why things that feel “confusing” don’t mean you’re stupid, why “feeling” like you know it isn’t a good test of whether you can do it, and why I don’t accept “reviewing your notes” as one of the things you did to improve when you applied for reassessment.
But this will also give us a rich example of what a “model” is, why they are necessarily incomplete and at least a bit abstracted, and how they can help us make judgement calls. Last year, I started talking about the “human brain model” around this time of the year (during a discussion of why “I’ll just remember the due date for that assignment” is not a strong inference). That was the earliest I felt I could use the word “model” and have them know what I meant — they were familiar enough with the “circuits and electrons model” to understand what a model was and what it was for. Next year I hope to use this tool to do it the other way around.
This morning, my students are reading about negative feedback and assessing the information provided using our standard rubric, which asks them to summarize and write their questions. They’re finding it difficult to understand, almost too confusing to summarize. I remind them that that’s ok — to summarize what they can, if they can. I also tell them to write questions as they read, not to wait until the end of the passage to write them down.
Especially, I remind them that common cause of “getting stuck” is waiting until they understand the paragraph before writing down a question. The problem, of course, is that you might not be able to understand the passage until after the question is answered. Waiting for understanding before asking questions is like waiting to be fit before going to the gym.
I have this conversation with one student:
Student: “What I’m afraid of is, if I get partway through the paragraph and write a question, then I get later in the paragraph and write down another question, I’ll get to the end and realize, Oh, that’s what it meant, and I won’t need to ask that question any more.”
Me, joking: “So what happens then? What horrible consequence ensues?”
Student: “I have to kill an eraser!”
Me: “No need to erase it. Just write a note that says, ‘oh, now I get that… [whatever you just understood]. Have you ever noticed how often I do that on your quizzes and papers? I write questions as I’m reading, then I cross them out when I get to the end and write a note that says “never mind, I see that you’ve answered the questions down here.”
Student: [noncommittal shrug, smiling, seems willing to try this]
I think that’s an ok way to get the point across. I sit back down. Then I need to be a smart ass. I go back to chat with the same student. “You know, from our conversation earlier, it sounded like you were saying, ‘I’m afraid that if I ask questions, I’ll get it.’ “
My point, of course, is that asking questions, thinking through our questions, and clarifying to ourselves what question we mean to ask can be an important part of sense-making, and can even help us answer our own questions. But that’s not how it comes across to the student. Now he’s been backed into a corner, shown the absurdity of something he just said. He scrambles to defend his statement. “No, what I meant was that if I ask questions while I’m reading, I might get to the end and not understand my… [pause] I can’t put it into words.”
Notes to self
- Students sometimes think they should delay asking questions until after they have understood something. This causes deadlock and frustration. Strategize about this with students.
- Pointing out someone’s misconception, especially in the middle of class, does not usually result in a graceful acknowledge of “oh, yeah, that doesn’t really make sense, does it?” It usually results in backpedaling and attempts to salvage the idea by re-interpreting, suggesting that I didn’t understand them, or saying “I understand it, I just can’t put it into words.”
- The phrase “I understand it but I just can’t put it into words” is highly correlated with “You just pointed out a misconception to me and now I must save face by avoiding your point at all costs.” Use this clue to improve.
- Dear Mylène, you think you’re too highly evolved to use “elicit-confront-resolve” to address student misconceptions, but you’re mistaken. It’s causing students to avoid their misconceptions instead of facing them. Find a way to do something else.
Is school like a grocery store of ideas? Learning should result in understanding and action, but we’re not always clear about what kind. There’s a big difference between understanding the organization of the grocery store layout, and understanding how to grow food yourself.
I live in farm country. Changes to zoning bylaws can draw protests of hundreds of people. People know where and how and by whom their food is produced. We also know how we affect the system — even if it’s only through our consumer choices and by-election votes. We’re engaged with the production narrative of our food.
I recently finished reading Shop Class as Soul Craft (thanks, John). I hope you’ll overlook the silly title because, though the book has its flaws, it’s also full of useful and refreshingly unusual ideas. One of the less surprising ones is that we have a responsibility to know the production narrative of our stuff, as well as our food. Knowing who makes what, and why, can take us past catchphrases (buy local) and teach us about class, agency, and democracy, if we let it. (Update: my review is on Goodreads)
I’ve been thinking about these because of a recent post on Educating Grace about what “sense-making” is, and why it sometimes diverges from understanding. I don’t know the answer, but the question is becoming urgent in my classroom. Brian Frank weighs in with a comment, and Grace responds with an even more perplexing post.
Here’s an excerpt from Brian’s comment:
Without knowing how to participate in the creation, telling, and changing of stories, learning science stories is no different than learning myths… The more we make our disciplines exclusionary, the more myth-making we do.
I haven’t fully wrapped my head around this. It’s starting to sound as though what my classroom needs more of is the production narrative of our ideas.
Frank Noschese just posed some questions about “just trying something” in problem-solving, and why students seem to do it intuitively with video games but experience “problem-solving paralysis” in physics. When I started writing my second long-ish comment I realized I’m preoccupied with this, and decided to post it here.
What if part of the difference is students’ reliance on brute force approaches?
In a game, which is a human-designed environment, there are a finite number of possible moves. And if you think of typical gameplay mechanics, that number is often 3-4. Run left, run right, jump. Run right, jump, shoot. Even if there are 10, they’re finite and predictable: if you run from here and jump from exactly this point, you will always end up at exactly that point. They’re also largely repetitive from game to game. No matter how weird the situation in which you find yourself, you know the solution is some permutation of run, jump, shoot. If you keep trying you will eventually exhaust all the approaches. It is possible to explore every point on the game field and try every move at every point — the brute force approach (whether this is necessary or even desirable is immaterial to my point).
In nature, being as it is a non-human-designed environment, there is an arbitrarily large number of possible moves. If students surmise that “just trying things until something works” could take years and still might not exhaust all the approaches, well, they’re right. In fact, this is an insight into science that we probably don’t give them enough credit for.
Now, realistically, they also know that their teacher is not demanding something impossible. But being asked to choose from among infinite options, and not knowing how long you’re going to be expected to keep doing that, must make you feel pretty powerless. I suspect that some students experience a physics experiment as an infinite playing field with infinite moves, of which every point must be explored. Concluding that that’s pointless or impossible is, frankly, valid. The problem here isn’t that they’re not applying their game-playing strategies to science; the problem is that they are. Other conclusions that would follow:
- If there are infinite equally likely options, then whether you “win” depends on luck. There is no point trying to get better at this since it is uncontrollable.
- People who regularly win at an uncontrollable game must have some kind of magic power (“smartness”) that is not available to others.
And yet, those of us on the other side of the lesson plan do walk into those kinds of situations. We find them fun and challenging. When I think about why I do, it’s because I’m sure of two things:
- any failure at all will generate more information than I have
- any new information will allow me to make better quality inferences about what to do next
I don’t experience the game space as an infinite playing field of which each point must be explored. I experience it as an infinite playing field where it’s (almost) always possible to play “warmer-colder.” I mine my failures for information about whether I’m getting closer to or farther away from the solution. I’m comfortable with the idea that I will spend my time getting less wrong. Since all failures contain this information, the process of attempting an experiment generally allows me to constrain it down to a manageable level.
My willingness to engage with these types of problems depends on a skill (extracting constraint info from failures), a belief (it is almost always possible to do this), and an attitude (“less wrong” is an honourable process that is worth being proud of, not an indictment of my intelligence) that I think my students don’t have.
Richard Louv makes a related point in Last Child in the Woods: Saving Our Children From Nature-Deficit Disorder (my review and some quotes here). He suggests that there are specific advantages to unstructured outdoor play that are not available otherwise — distinct from the advantages that are available from design-y play structures or in highly-interpreted walks on groomed trails. Unstructured play brings us face to face with infinite possibility. Maybe it builds some comfort and helps us develop mental and emotional strategies for not being immobilized by it?
I’m not sure how to check, and if I could, I’m not sure I’d know what to do about it. I guess I’ll just try something, figure out a way to tell if it made things better or worse, then use that information to improve…
This will be my first September using the “skill-based” assessment and grading system. So although I’ve used it for two semesters, I’ve never done the “sales pitch” to an incoming class. For the last two weeks I’ve been in a fog, trying to figure out how to introduce so many (probably foreign) ideas at once, to a bunch of people I didn’t know and who have no particular reason to trust me (yet). It seemed like every idea depended on every other idea, so nothing could go first.
Dan Goldner proposed an elegant solution: have them assess me. It gives me a chance to show (not tell) what assessment and grading will look like. At the same time, it exposes my philosophy about teaching and learning, introduces some of the concepts that will run through our semester (i.e. “make mistakes understandable”), and opens a conversation about what good teaching is. I plan to scaffold it with a survey about their learning experiences and goals.
But for now, I’ve got a very rough draft of what a skill sheet would look like if it was for assessing me, not a student. Think I’m way off base? Wrong emphasis, missed something important, need to find more student-friendly language? I hope you’ll let me know.
I’m on a jag about what confusion is and whether it’s necessary for learning. My latest Gordian knot is about how confusion relates to pseudoteaching.
It seems that some condition of readiness has to happen before students can internalize an idea. Obviously they will need some background knowledge, and basics like enough sleep, etc. But even when my students have the material and social and intellectual conditions for learning, it often seems like there’s something missing. To improve my ability to promote that readiness, I have to figure out what the heck it is. I’m wondering if confusion is part of the answer.
Dan Goldner writes that students must have “prepared a space in their brain for new knowledge to fit into” — that they must have found some questions that they care about.
Grace points to the need for conflict in a good story. She advocates creating a non-threatening “knowledge gap” using either cognitive dissonance or curiosity.
Dan Meyer, obviously, has made it an art form. He calls it “perplexity” and distinguishes it from confusion (or sometimes describes it as a highly fruitful kind of confusion). If I’m reading it right, perplexity = conflict + caring about the outcome.
Rhett Allain has a great post about the “swamp of confusion” (go look — the map illustration is worth it). He points out that a lifetime of pseudoteaching can convince students that working through confusion is impossible, or that teachers design courses to go around confusion, so that if you feel confused, either the teacher is incompetent, or you did something wrong. He also pulls out some of the assumptions about “smartness” that people often hold about confusion: “If this IS indeed the way to go, I must be dumb or I wouldn’t be confused.”
Finally, the word “confusion” comes up in Derek Muller’s points about using videos to present misconceptions about science (videos that explained the “right answers” were clear but ineffective; videos that included common misconceptions were confusing but effective).
What I get in my classroom, which often gets called confusion, is conflict + anger. Or possibly conflict + fear, or conflict + not caring (it’s possible that “not caring” is made out of anger, fear, and/or fatigue). Just a guess: students get angry when they think I’ve created conflict that is unnecessary, or when they think I’ve created it carelessly. These are worth thinking about. Conflict can be threatening or exhausting. Have I created the right conflict? Is my specific method of creating conflict going to improve our learning, or did I use videos/whiteboards/particle accelerators because I think they’re fun and cool and make me look like a “with it” teacher?
Given that my students use the word “confusion” for a lot of situations where the next move is not immediately clear, I bet they would call all of these things confusion.
Which ones encourage learning? Are any of them necessary? Next year, I think I will ask students to make a note in their skills folder (portfolio-like thing with loose-leaf in it) to record confusions, so we can get a better grip on it.
In the meantime, I’m not having trouble with intellectual conflict. By all the accounts above, the conflict is not just an inevitable side-effect but one of the main components of learning. We’ve got lots of it to go around, and I hope that opening a conversation about it earlier in the semester will help students understand it as part of learning. Bringing to light our conflicts is part of what allows us to transform them into new understanding.
That leaves me with the “non-threatening” part, the “caring enough about the outcome to want to resolve it” part, and the “skills for dealing with it” part.
I attended a webinar today about the pros and cons of flipped classrooms (i.e. information gathering such as video lectures or textbook-reading happen at home; experimenting, exploring, and inquiring happen in class). There was lots of great discussion and food for thought. Several presenters brought up this important point: A video lecture is still a lecture. Sure, it has some advantages. But why are we (video) lecturing at all? Lectures were born in the days when only one person owned a copy of the book. If you wanted to know what was in it, they would read it to you. In medieval Latin (the language of European scholars pre-Gutenburg), lecture means “to read.”
This alone is not sufficient evidence to either keep or get rid of lectures. Nowadays, the word “lecture” doesn’t always mean “reading the book at you.” Sometimes it means “storytelling,” sometimes it means “asking short questions of one student at a time,” sometimes it means “direct instruction,” sometimes it means “modelling my work or my thinking,” sometimes it means “teacher talking, broken occasionally by outbursts of student discussion.” I’m not interested in “are these useful tools.” Of course they are. My question is, “are these the best tools for my purpose.” The answer to that is more difficult, also more dependent on my purposes and my students.
There are a few topics where I don’t think lecturing is the best tool for my purpose, but I do it anyway (the inner workings of a P-N junction, for example). If I’m going to lecture, a 5-min video buys me at least an hour, considering that it would take me 15-20 min in class, plus repetition for students who were absent or needed to go over it again. The reason I do it, just as Jerrid Kruse mentions, isn’t that I think it’s ideal; it’s that I haven’t found a suitable collection of examples or a good way to guide a discovery process. So ultimately, the PD I need isn’t a lecture about why I should move away from lectures; it’s a guided exploration where I can explore my intractable problems with some guidance (inquiry-discover-model-constructivist-project-engaging-self-directedness: not just for students anymore). Somehow, we need to create that course.