S1: Welcome to the Hillsdale College K-12 Classical Education podcast, bringing
S1: you insight into classical education and its unique emphasis on
S1: human virtue and moral character, responsible citizenship, content, rich curricula,
S1: and teacher led classrooms. Now, your host, Scott Bertram.
S2: Thanks for listening. The Hillsdale College K-12 Classical Education podcast
S2: is part of the Hillsdale College Podcast Network. More episodes@podcast.hillsdale.edu
S2: or wherever you get your audio. You also can find
S2: more information on topics and ideas discussed on this show
S2: at our website, k12.hillsdale.edu. Great teachers don't merely know and
S2: love their content. They captivate students with it. The best
S2: lessons are full of wonder, engagement, and curiosity. But it
S2: can be daunting to try to replicate these moments over
S2: and over again with some initial work and lots of practice.
S2: Teachers can infuse their class time with these elements without
S2: running in circles. Doctor Jonathan Gregg, assistant professor of education
S2: at Hillsdale College, spoke on how to infuse your pedagogy
S2: with wonder at a September 2025 center for Teacher Excellence
S2: event on the Art of teaching mathematics. Though his ideas
S2: are presented in the context of mathematics, countless applications can
S2: be found for every classroom subject. Here's Doctor Gregg.
S3: I'm going to talk about pedagogy. Pedagogy is not easy.
S3: The Greek of pedagogy just means a child leader. And
S3: children don't like to be led. I have some of them.
S3: They're bad at at following. And so pedagogy is necessarily hard.
S3: It's complex. It's different every day. And it's it's hard
S3: to kind of what you. The difficulty, I think, in
S3: pedagogy is that as teachers, we search for something stable.
S3: We want something. We want to eliminate the chaos in
S3: the classroom, right? There is you deal with enough chaos
S3: every day. And so you want something to rely on.
S3: But pedagogy has to be new. It has to be exciting.
S3: It has to be something that that you can't just
S3: repeat day after day after day after day, because then
S3: it becomes boring and we lose. We lose the sense
S3: of wonder here. And so I want to try to
S3: inhabit that space a little bit today to think about
S3: how do you keep things exciting? How do you keep
S3: things wonderful without making making yourself run in circles wildly
S3: or making yourself plan for hours and hours every night?
S3: How do you how do you do something that's habitual
S3: and yet still wonderful? This is kind of the thing
S3: that I'm interested in here. I think that's the core
S3: of core of pedagogy because. So I'll start with a
S3: quote as well. This is, you know, one of those
S3: quotes that you read and you just you can't get
S3: it off of your mind. It kind of haunts you.
S3: I read this, I read this a few years ago,
S3: and I think I think it's it haunts me. Right.
S3: The capacity to wonder is among man's greatest gifts. It's
S3: so true. And if you think back, even. Just think
S3: back to your good lesson. Think back to a good
S3: lesson that you did. I guarantee you wonder showed up somewhere.
S3: And actually, this is the one place where I do
S3: think students will and children will follow. They'll follow what
S3: they wonder about. They'll follow what, uh, what's wonderful. Um,
S3: and so I think this quote is if we're looking
S3: for a, a beginning of pedagogy, a first philosophy of pedagogy,
S3: I think this has got to be it. People are
S3: people actually goes on. And he says this, he says,
S3: really the situation is this, the deeper aspects of reality.
S3: That's what we want students to get are apprehended in
S3: the ordinary things of everyday life and not in the
S3: sphere cut off and segregated from it. The sphere of
S3: the essential or whatever it may be called. It's in
S3: the things we come across in the experience of everyday
S3: life that the unusual emerges and we no longer take
S3: them for granted. In that situation corresponds with the inner experience,
S3: which has been regarded as the beginning of philosophy, the
S3: act of marveling, kind of see how this is going
S3: to work? You present something. It's unusual. Students wonder at it,
S3: and then they start to think. And that's philosophy. And
S3: this is where where you can kind of see how
S3: wonder is going to be the thing that's possible as
S3: a proof here. And honestly, we can't we can't go
S3: too far in math without talking about about wonder as
S3: a maybe not a proof, but at least some evidence
S3: for it. I have to be careful using the word
S3: proof around around this, this crew, you guys have some
S3: high standards for this, but at least as some evidence. Right?
S3: Here we go. Here are some quotes on wonder. We
S3: want students. We want to get students to wonder about hypotheses.
S3: This was Doctor Dave Gabler four hours ago. Mathematics should
S3: instill in students an ever increasing sense of wonder. This
S3: was doctor Tom Treloar three hours ago and we mathematicians
S3: like to wonder if there's anything else we can do
S3: with this. This was Doctor David Murphy. Not even one
S3: hour ago. All right, so these guys are in here
S3: talking about wonder, right. I wasn't in Miss Jackson's talk
S3: but I guarantee you wonder showed up. I'm gonna go
S3: back and look at the transcript. But it's there, right?
S3: It's there. But what I can't do is stand up
S3: here and say, go get your students to wonder. That's
S3: not a thing we can do. That's that's not helpful
S3: as teachers. So the real question is how do we
S3: get students to wonder and how do we habitually do that?
S3: This is the question for today. It's not an easy question.
S3: How do we habitually get students to wonder? Because I
S3: think if we can, we're already doing the kind of
S3: things that we should be doing in our math classroom, right?
S3: You've already kind of won as a teacher. Wonder leads
S3: to philosophy. So what is wonder and how do we
S3: do it? Okay, let's go to Peter one more time. Socrates.
S3: is talking about Socrates here. He's actually about the Titus.
S3: Titus is a dialogue between Socrates and a mathematician. All right.
S3: We saw Theaetetus actually show up on Doctor Murphy's slides here.
S3: And he says Socrates brings Theaetetus to the point of
S3: admitting his ignorance with his shrewd and kindly but staggering
S3: and astonishing questions, questions that stagger and astonish with wonder.
S3: And there, for the first time in the theater, without
S3: solemnity or ceremony, almost. By the way, though fresh as
S3: dawn appears, the thought that has become commonplace in the
S3: history of pedagogy. The beginning of philosophy is wonder. You
S3: see it again here. But notice what he's doing with wonder.
S3: Saying look wonder happens in two ways. Yes, it's kind
S3: of staggering and astonishing thing. So it's actually a really
S3: lovely word, right? One of the best words in the
S3: English language are the words that we equivocate on, that go,
S3: that go that have two different definitions, right? Think of
S3: the word witness is one of my favorite examples here. Witness.
S3: Witness is a beautiful word, right? It means both to
S3: observe something taking place and to share about the thing
S3: that you saw, right? And the fact that we use
S3: the same word for it means when you see something,
S3: you are obligated. You have a responsibility to share it. Actually,
S3: what you guys are doing, right, you're witnessing to your students. Okay. Anyways,
S3: I'm off track. I'm off track. But wonder is the
S3: same thing, right? Where questions that stagger and astonish with wonder. Right.
S3: First of all, wonder means that I'm staggered. I don't
S3: know something. Right. I recently wondered how to put Freon
S3: in my car because I didn't know how to do it,
S3: and it was 120 degrees in my car and I
S3: was upset. I don't know how to do this. And
S3: I was staggered and I wondered, and then what did
S3: I do? I did what any self-respecting person would do.
S3: Got on YouTube right. And figured it out. But there's wonder.
S3: We are staggered. We then go and figure the thing out.
S3: And that's where learning takes place. But first you have
S3: to be dislocated, right? I wonder about this. I wonder
S3: what's going on. I'm confused. It's almost like you're you're
S3: you're dislocated. You're shocked. Socrates actually calls himself a torpedo fish,
S3: which is an electric eel. Apparently that the idea is
S3: he's a he's just shocking people. He's like, what about this?
S3: What about this? And you're shocked. I think students you
S3: can shock students into wondering. But also this astonishment. Say, oh,
S3: I wonder at that thing that's marveling. It's this, it's
S3: this or that students can have as well. So we
S3: see this dual understanding of wonder, right. And an invocation
S3: of the Socratic method. So let's piece some of this
S3: together and think about what pedagogy looks like. So I
S3: present to you pedagogy. Thank you. Pedagogy. I'll take your question. No,
S3: I need some labels. Yes, we need some labels. Okay,
S3: so don't take the pictures yet. Wait for the labels.
S3: Here we go. Actually, don't take any pictures. Just email
S3: me for the slides. But let's get some labels here
S3: because this shape is meaningless. This is not a meaningful shape.
S3: This is not a polyhedron. Right? So what's going on here?
S3: Number one. Right. I have a lightning bolt. I think
S3: this is what we think of as an opening question. Right?
S3: I'm envisioning the teacher kind of as as Zeus here, right?
S3: Chucking a lightning bolt in the middle of students. And
S3: what's going to happen is they're going to this is
S3: the shock part of wonder, right? You're going to be dislocated.
S3: You're not going to understand the thing. Right? And you
S3: can see these blue lines. This is ideally what student
S3: responses are going to be that students can go have
S3: a bunch of different potential ideas out of this wonder.
S3: Wonder leads to this kind of exploration, right? And then
S3: these kind of curlicue lines, this is what I'm calling
S3: follow up questions that select different student responses and then
S3: follow up on them until we get to the green thing.
S3: This is the truth. Okay. The thing we want to
S3: teach the learning goal, if you will. What you want
S3: to teach the student. Okay. And then what we have
S3: are these maybe some examples, maybe some practices, some applications.
S3: Once we learn this thing, then we practice it, then
S3: we do it. Then we. But I think the idea
S3: here is that you, uh, this is kind of how
S3: one might learn in the, in the field of wonder
S3: or using wonder where you're invoking the Socratic method, where
S3: there's both shock, right? Staggering. And then there's potentially astonishment
S3: down here. Okay. I think every good lesson, or at
S3: least a lot of good lessons are going to follow
S3: something like this in a math classroom. Okay. And maybe
S3: here's one way of, of proving, again, this is not
S3: a proof, but some evidence for this. If we look
S3: at the dominant pedagogical approaches that we are surrounded with,
S3: I think there's two dominant pedagogical approaches. I think what
S3: you might call the progressive and what you might call
S3: the dogmatic, the dogmatic, I think, is this here's the truth.
S3: Let me hand you the formula. Let me hand you
S3: how to do it. Practice it over and over and
S3: over and over again until it gets drilled into your face. Okay,
S3: that's the dogmatic pedagogy. Or you have progressive pedagogy, which
S3: kind of throws out an idea and then says, go students, explore,
S3: be empowered, go to your heart's content. Right? But it
S3: doesn't really believe that there's any truth to be found, right?
S3: It's about whatever students want to do, right? And it
S3: just cuts off the top half of this versus cutting
S3: off the bottom half of this. Right? But somehow in
S3: between these two extremes, there's something holistic where we have
S3: wonder at the beginning and wonder at the end from
S3: wonder to wonder, right? That's, I think, what a pedagogical
S3: approach looks like. But this shape is, is just a shape.
S3: Let me illustrate. All right. So I'm going to teach
S3: you guys. You guys are students. I will be teacher.
S3: And and I'm going to teach you guys pretend you're
S3: first graders. This is a first grade class. Okay, good.
S3: Good morning, boys and girls. I'm really excited to be
S3: with you guys today. Guess what. I want to teach
S3: you guys a new word. Can you guys say this
S3: with me? Say this after me. The word is multiplication.
S3: You guys say that with me. Very good. Multiplication. Now
S3: multiplication is repeated. Addition multiplication is very good. So if
S3: I have a problem like three plus three plus three,
S3: plus three plus three, that's a lot of threes, don't
S3: you think? My hand is almost tired writing up all
S3: of these threes. Right. So would you guys like to
S3: learn a better way for writing that I think we would.
S3: So I can write this as five and then I
S3: need a new symbol. I'm not going to use addition anymore.
S3: Can everyone make me the symbol of addition with your hands? Right. Okay.
S3: I need a new symbol. So what we're going to do?
S3: Hold your addition. Oh. New symbol. Right? We have an X5X35X3.
S3: And just like I have a word for this, what
S3: word do I use for this plus right. I'm going
S3: to use a new word for my multiplication. I'm going
S3: to use times. Say it with me. So five times
S3: three because it's three five times. Some of you guys
S3: are legitimately like, oh wow. Okay, so can we figure
S3: out what Three five times three is. Can you guys
S3: count with me? Three. Six. Nine. 1250. Very good. Five
S3: times three is equal to 15. And you guys just
S3: did your first multiplication problem. Okay? And cut and cut. Okay,
S3: so we can keep going, right. But what kind of
S3: pedagogy was that? Right. That's dogmatic pedagogy. Okay. It's entertaining,
S3: but it's handing you the four way to do this.
S3: Here's how to do this. Here's what the words mean.
S3: I'm handing you this thing. Is it good? Do I
S3: need to teach kids this? Yes. Is this where I
S3: want to start? No. Right. Because if you're first graders,
S3: I can walk in and say, I'm going to teach
S3: you guys a new word, multiplication. But if I walk
S3: into my ninth grade class, I'm going to teach you
S3: guys a new word logarithm. Mr. Greg, why we don't care, right? Okay,
S3: so and slowly. This is why students start to hate
S3: math is because we teach it pretty. One of the
S3: reasons we teach it pretty dogmatically. And there. There's no
S3: buy in, right? There's no there's nothing that they're doing mathematically.
S3: They're not actually engaging in mathematics. They're just receiving the
S3: results of mathematics. Right. Doctor Murphy said mathematics is a
S3: human activity. We need to get them doing the mathematics. Okay. Good.
S3: Let me try again. Good morning boys and girls. I'm
S3: really excited to be with you guys today. Go ahead
S3: and take out your devices. I've preloaded the module to
S3: explore today's lesson. It's called multiplication, but you guys go
S3: ahead and explore on your own. Figure it out, see
S3: what you can make of what's going on. And I'll
S3: just be over here in case you need to answer
S3: any questions. And you guys can go ahead and explore
S3: and we'll talk at the end just to see. We'll
S3: share how we'll do a little share out to see
S3: what people thought of your chance to explore And cut. Okay,
S3: so what kind of progressive what kind of pedagogy here, right?
S3: This is what progressive pedagogy looks like or feels like.
S3: It's not. There are other ways to do this, but
S3: really what it does is it the goal is for
S3: the teacher to fade into the background, to have students
S3: do the exploration on their own, right? The teacher doesn't
S3: get to have a role to play that this is
S3: that it's it's actually oppressive for me to impose my
S3: truth upon you. Right. And I think this, as you
S3: guys know, it's it just doesn't work, right. Students don't
S3: aren't able, by and large to do this. And what
S3: it does, I think, is it bifurcates students into those
S3: who can figure it out on their own and those
S3: who can't. Right. I think one of the one of
S3: the really unspoken things of progressive pedagogy is that it's
S3: attempt to kind of empower all students equally. The pedagogy
S3: is the thing that ends up bifurcating the students into
S3: the haves and the have nots. It's really problematic, actually.
S3: The methods of progressive pedagogy undermine their stated goals, in
S3: my opinion. I can talk more about this if you want,
S3: but okay, let me try one more time. Morning boys
S3: and girls. I've got a question for you guys today.
S3: Which of these two problems is easier to figure out
S3: how many fingers we have as a class, or to
S3: figure out how many pets we have as a class.
S3: You guys are bad first graders. Fingers. Pets. Fingers. Pets.
S3: My brother has a cat. Okay. Right. But this is
S3: what you want. This is what you want. Okay, so, fingers,
S3: let's try the pets one. What do I have to
S3: do to figure out the pets? What do I have
S3: to do? I've got to ask people. Right. Okay, good.
S3: Let's ask. How many pets do you have? Five. How
S3: many pets do you have? Ten. How many do you have?
S3: Does my little sister count as a pet? No. Fast forward. Pets, pets.
S3: Now I've got all everyone's pets on the board. Now,
S3: how do I figure out how much we have all together?
S3: I have to use addition. Why do I have to
S3: use addition? Ah, because we're sitting all together. Good. So
S3: let's add fast forward. We have 312 pets as a class. Whew!
S3: That is a lot of pets. I hope we don't
S3: bring them in here. So let's try the fingers one.
S3: How do I figure out how many fingers we have
S3: as a class? Can I do the same way? Can
S3: I do it the same way? Yes. No. Who thinks yes?
S3: Who thinks no? Why do you think? Yes. Sure. Why
S3: do you think no? Okay, okay. So let's try. Let's
S3: try it the same way here. Right. How many fingers
S3: do you have? Ten. How many things do you have?
S3: I have eight. Two of them are thumbs, right? Yeah. Okay, good. Yeah.
S3: This is what you. But again, this is what you
S3: want in the classroom. You want a little drama, you
S3: want a little energy, right? Yeah, right. And I get
S3: a couple people in here. Do I need to keep
S3: asking people? I don't why not? Everyone has the same
S3: number of fingers. That's interesting. interesting. What do I need
S3: to know to solve this problem? How many people there are?
S3: I need to know that. And how many fingers. If
S3: someone's missing a finger. Yeah. How many fingers each person has. Right.
S3: This is a different kind of problem, right? So. Okay.
S3: And I'll cut there because now what am I going
S3: to do? I'm going to do I'm here now, right.
S3: And now I'm going to say like, let's write up
S3: the multiplication problem. What you guys just did with the pets.
S3: That's multiplication. That's not addition. Something new right? Okay. So
S3: go ahead and cut. And you could see how a
S3: lesson looks like when you walk in and say which
S3: is easier. Fingers or pets? Lightning bolt question right. No
S3: one knows what's going on here. They're like, what? This
S3: is math class. You're talking about fingers and pets. I
S3: got reactions. I at least got two reactions, right? I
S3: got fingers and I got pets. And both are defensible pets.
S3: Probably there's fewer pets than fingers. It would be easier, right?
S3: Then what did I do? Notice the kind of pedagogy
S3: that I employed. What did I do? Did I say
S3: a single statement? I didn't say a statement. One time.
S3: I say one statement. Right. This is what the Socratic
S3: method looks like. By asking, right. Picking the two responses
S3: and asking a bunch of follow up questions, I was
S3: able to get them to tell me the things that
S3: we need for a multiplication problem, namely, namely number of
S3: groups and number in each group. That's what multiplication is.
S3: We need two things and we can figure out the
S3: total that way. But they're the ones that did that thing.
S3: They engaged in the activity of multiplication before. And we
S3: didn't just kind of let it be there. And then
S3: we're going to formalize it into what what we actually
S3: want and practice it like we would like we would normally,
S3: but we can see what we you know, we by
S3: taking this Socratic method. And is there wonder at the end.
S3: I don't know, but at the very least they should
S3: feel that, oh, in this world there are multiplicative situations
S3: and additive situations. Is a fourth grader being able to
S3: say that? No, but I can say what other problems
S3: are like the fingers problem toes. Yeah, right. You guys
S3: are bad first graders. How about some non body part
S3: ones and ask him to think right wheels on a car,
S3: maybe something like this where they're noticing that the world
S3: actually that multiplication is a response to the world and
S3: math is woven into the cosmos and all. And what
S3: we're doing is we're noticing patterns in the cosmos, right?
S3: Our first grader is going to be able to say
S3: that to, you know, but are they going to slowly
S3: feel that math starts to be this thing that I
S3: walk in? I put a question on the board or
S3: ask a question and we mess around with it and
S3: play with it until some truth emerges. Until we discover
S3: some truth. That's, I think, where we can get something
S3: like wonder in the math class. This type of a shape. Okay.
S3: So again, I still don't think this shape is all
S3: that helpful because how do you come up with these
S3: opening questions? What follow up questions do you ask? How
S3: do you do these things? Okay. So that's what I
S3: want to do. Don't let me off the hook for
S3: being impractical still, right? I have to give you guys
S3: some tangible advice for these things. This is what we need.
S3: So let's do it. Okay. Here's how you do opening questions. Go.
S3: New new. New talk. Opening questions. Socrates says, come then
S3: to me. He's talking to a mathematician here, actually. Right.
S3: And do your best to answer the questions I'll ask you.
S3: And if I abstract and expose your ideas? Because I
S3: discover upon inspection that the conception which you have formed
S3: is a vain shadow. Do not quarrel with me, for
S3: it would be wrong for me to admit falsehood, or
S3: to stifle the truth. Right. Come to me and expose
S3: your ideas. That's what an opening question does. Exposes student thinking,
S3: exposes ideas. Okay, so here we go. What makes for
S3: a good opening question? Number one has to be accessible
S3: to all students. Okay. The very first thing out of
S3: your mouth is something that some students can't grapple with
S3: or understand. You're going to lose half your class. Okay.
S3: There's got to be something that every student can think about, right?
S3: Which is easier? Fingers or pets? Everyone can have an opinion.
S3: That's no access barrier there. Okay. Number two, it has
S3: to be challenging to all students because there's no easier
S3: way to lose half your class than for the first
S3: thing out of your mouth to be something they already
S3: know the answer to. Okay, so it has to be
S3: both accessible, right? Which is easier, fingers or pets? No
S3: one really has written the has written the book on this.
S3: No one. This isn't a thing that one can really
S3: even know, right? This is challenging in some ways to everyone,
S3: at least to think about. Right? Number three, it has
S3: to provoke multiple thoughtful student responses. I need this is
S3: not just a yes or no right or wrong answer
S3: that I'm starting with. It has to be multiple thoughtful responses.
S3: I want to scatter my students. I want them to
S3: be dislocated. I want them to wonder. But number four,
S3: it has to be directly oriented to the truth. This
S3: isn't just some progressive. Think about this guys. Do what
S3: you want, right? Follow your dreams. No. Right. And in particular,
S3: I think these two and these two are really tough
S3: needle to thread. How do you get something that's accessible
S3: and challenging to everyone. And how do you get something
S3: that's Towards the truth. Enough that it's going to get
S3: you there. But open ended enough that it's going to
S3: get students thinking in multiple ways. It's a high bar
S3: for opening questions. Okay. I don't think there's as high
S3: a bar for follow up questions, which is good. But again,
S3: don't let me off the hook. What kinds of things
S3: make for good opening questions? What are some good examples? Okay,
S3: so what I've done, I took all my lesson plans
S3: that I ever wrote and put them in one massive pile,
S3: and I started just sorting them into different kinds of
S3: opening questions. Okay. And I came up with five piles.
S3: So here are my five piles. There's probably more opening questions,
S3: but here are the ones that I know that work. Okay.
S3: Number one, we get our students. Each one of these
S3: has a kind of posture that I want students to adopt, right?
S3: So number one, I think these are noticing questions. I
S3: think this is the very basic of opening questions. We
S3: ask students to notice, say, what's going on, right? Usually
S3: this is a why question. Noticing is observe. See something
S3: and try to explain, observe and explain. Observe and explain.
S3: See something and explain to me what's going on, what's
S3: going on here, what's going on here? Why is this
S3: taking place? The very, at the very least, walk in
S3: with a why question at the very beginning. Why is
S3: this right? I think it's especially true for generating formulas.
S3: Here are some examples. Okay. Why can we not use
S3: our method for adding 23 plus 45 for the question
S3: 23 plus 48. Notice that this doesn't work. And explain.
S3: Notice and explain. Notice and explain. Why is subtracting a
S3: negative and positive? How big a box do I need
S3: to carry stuff? Notice stuff comes in different sizes and
S3: how to explain if you're trying to teach volume, how
S3: big a box do I need to carry that thing?
S3: What thing are you carrying? I don't know a thing,
S3: a turtle. How big a box where what you're doing
S3: is you're asking students to think and then forcing them
S3: to articulate things. They say, well, right. Instead of me
S3: being like, volume has three dimensions length, width, and height, right?
S3: Ask them to put stuff in boxes, see what happens.
S3: They're going to tell you that things have three dimensions
S3: you got to worry about. They're going to tell you
S3: how big space is. Students can do this. They can
S3: come up with this stuff. It takes the right opening question.
S3: So actually fun fact, I would give this same talk
S3: to any of the disciplines. I think this is how
S3: we teach literature history science best anyways, right. Here are
S3: some other questions. Why is being a hero so difficult? Literature.
S3: Why do the Articles of Confederation fail? History. Why do
S3: feathers fall slower than bowling balls? Science. Everything. We are
S3: driven by these noticing type of questions. Okay, but that's
S3: very basic level. Okay, there are better ones. Number two,
S3: I asked my students to notice. Ask my students to compare. Okay.
S3: It's like noticing on steroids. Noticing. Noticing. Times two. Notice. Notice. Compare. See?
S3: What's the difference between really good ones, right? This is
S3: different than this notice. Notice. Explain the differences. Right. Juxtaposing
S3: two related ideas. It's often phrased as an er or
S3: est question. If you teach like third, fourth, fifth grade,
S3: those humans are just little extremists, right? This is what
S3: they want to do with their life anyway, right? Just
S3: lean into it. Ask them best. Worst math is a
S3: lot of about what's the best way to do something.
S3: Which is easier. Figuring out figures of pets. Which is easier,
S3: this problem or this problem? Which is faster? Start in
S3: the ones place the hundreds. Place the difference between this
S3: and this. This one's really great. This. This factoring one.
S3: The second one on there. This is how I teach
S3: my eighth graders how to factor. I divide them class
S3: in half and I say, give them some reward. Like,
S3: you know. All right, a little test for you guys
S3: right off the bat. Uh, first, uh, fastest way to
S3: get a solution gets half the homework off or whatever, right?
S3: And they're all like, they're all in now. Like, yeah, okay, good, good.
S3: Let's do it right. Going down right. And I give
S3: one half. I pass out this problem. They flip it
S3: over and I give the other half this problem. And
S3: you know, the the half that I get this problem
S3: is done in 10s. The other half is still going.
S3: I'm like. And so they're raising up their whiteboards. I'm like, good,
S3: good good good good good good. And this guy, this,
S3: this whole half is just getting angrier and angrier and
S3: angrier and angrier. Like what? You gave them a different problem.
S3: I was like, I don't think I did. I think
S3: it's the same problem, right? It's the same problem. Same problem. Right.
S3: It's unfair. Right. And, and I'm like, oh, you, you
S3: think this is a hard problem? What's hard about this?
S3: They're like, there's two X's. There's a squared, right? And
S3: I'm like, oh, you want some ways to take this
S3: hard problem and turn it into an easier problem? Yeah.
S3: Oh let me. And they beg me to teach him
S3: how to factor. Right. And I am benevolent and so
S3: I do. Right. This is. But the idea is just look,
S3: this is hard. Oops. This is hard. This is easy.
S3: Let's learn how to do it right. Easy stuff. Right. Okay.
S3: It's like shooting little angsty eighth graders in a barrel.
S3: It's great. It's great pedagogy, right? Same in other disciplines
S3: as well, right? What's the difference between the way that
S3: Elizabeth envisions marriage, the way that Darcy envisions marriage? Great
S3: opening question. Oops. Uh, was there ever a moment when
S3: you'd be decisive and moment need to be cautious? If
S3: I'm teaching the Battle of Gettysburg. Oh, man. I'm opening
S3: with that question. Day one. They had to be real
S3: decisive to get that high ground right. Day three, just
S3: the boldness run amok, right? Number four. Right. Or the
S3: last one, which is harder to roll a ball up
S3: a hill to keep it stable at the top of
S3: the hill. Good comparison. Okay. I asked them to notice,
S3: to compare, to create questions of how would you or
S3: design Your kindergarten first, second grade teacher. They should be drawing.
S3: They should be designing all of these things they can do.
S3: Draw me this. Draw me something. Design me something. Create
S3: me something. The idea here is to use get students
S3: using their imaginations. We want our math classes to be imaginative.
S3: Notice how this is great to get all students doing
S3: different things right. This is going to provoke multiple responses
S3: to create. This is one of the beautiful things about
S3: human beings is we all have different creativity. If I
S3: asked you guys to write a poem, you guys would
S3: all hopefully write something different. Otherwise it'd be a little creepy, right?
S3: Design a game that seems fair but isn't. If I'm
S3: teaching some expected value. That's a great question, right? Design
S3: me a problem that can be solved for this bar model, right?
S3: Flip it on its head. Literature. Design me a defensive
S3: strategy for Tom Robinson. Design me a scenario of the
S3: Roman system of government particularly struggle with. Design me something
S3: that seems alive but isn't. If. I'm sorry, I'm doing
S3: all these other disciplines here, but if I'm trying to
S3: teach the seven qualities of life right? Movement, science, knowledge, movement, respiration, sensation, growth, reproduction, excretion, nutrition. Okay, great.
S3: These are my seven classes. I can walk in and say,
S3: all right, kids, today we're going to learn the seven
S3: qualities of life. Write them down. Write them down. Write
S3: them down. Right. It's not really pedagogy, right? That's just
S3: dogmatic teaching. It's easy. Right? Design me something that seems
S3: alive but isn't. Go work with the person next to you.
S3: Explore for a little bit. This isn't just like explore
S3: to your heart's content, but try some stuff and then
S3: let's come back and talk about it. Right. And my
S3: follow up questions write themselves. Maybe they come up with,
S3: I don't know, a robot, a virus Wind. Uh, stories,
S3: I don't know. Plenty of things seem alive, but aren't.
S3: And I can ask them, why do you think that
S3: seems alive? Why does it seem alive? How do you
S3: know that it's not alive? I bet you they can
S3: get all seven of those things. Or at least six, right? Okay, good. Notice. Compare.
S3: Create number for struggle. Okay. Um. The question in the
S3: back of your head should be, is this accessible to everyone?
S3: If I make them struggle with stuff. No. Right. In
S3: that they're going to struggle. But since I'm asking everyone
S3: to struggle, it's weirdly accessible, right? As soon as if
S3: you're a kid that's going to that I put a
S3: problem on the board and they're like. But then they
S3: look around their classmates and everyone's going, now I've leveled
S3: the playing field again, right? This is accessible to everyone.
S3: I'm not asking for insights here as much as I'm
S3: asking them to grapple with the thing, right? To struggle
S3: to try some stuff, right. I think we struggle in
S3: two ways. Something initially that seems impossible becomes clearer upon investigation,
S3: or something that initially seems simple, that becomes more impossible
S3: under investigation. So a couple examples here. Right? 706 divided
S3: by three trying to teach remainders for the first time.
S3: Throw it up there. 76 divided by three go! Mr. Greg.
S3: It doesn't come out right. Oh, okay. What are we
S3: going to do? Here's $706. You guys just made it
S3: divided amongst yourselves, right? Two to the X equals 13.
S3: Oh it's great. I'm trying to teach. I'm trying to
S3: teach logarithms. I walk in, right. You guys are the
S3: ninth graders. Two to the X equals 13. All right.
S3: Go for it, guys. Now your ninth graders, right? Oh, Mr. Greg,
S3: why do you do this to us? Oh, what's hard
S3: about this one? There's no twos that get a 13, Mr. Greg. Oh, okay.
S3: How about this one? Two plus x equals 13. Do
S3: you know that one? 11. How'd you do that one? Ah,
S3: we just subtracted two. Okay. Right. All right. Two times
S3: x equals 13. Six and a half. How'd you do that?
S3: One divided by two. We know these. All right. Well,
S3: two x equals 13. Is the answer bigger or less
S3: than one bigger. How do you know? Two to the
S3: one is two. Is it bigger? Less than two bigger.
S3: It's bigger. Less than three. It's still bigger. It's between
S3: 3 and 4. Mr. Greg. Oh, okay. Well, what's hard
S3: about this? How'd you know to divide by two when
S3: we had a multiplication problem? Because it's the opposite of it.
S3: We're not going to use inverse operations. They're just not
S3: going to say it, but they will. You know, they're
S3: never going to use the right words that you want,
S3: but they'll say, say, say. Okay. Well, what's hard about
S3: this one? And some kid will say, oh, well, we
S3: just need is a magic thing that undoes the exponentiation.
S3: I'm like, oh, is that what you want? Right. And
S3: they've begged me for logarithms. Right. I mean, like, again,
S3: this take take this one. Right. I'm trying to teach
S3: the fundamental theorem of calculus. I put this on the board.
S3: They're like, how much area are. It doesn't even have
S3: any lines. Mr.. Greg. Okay, I'll draw some lines. Fine. Great. Now,
S3: how much area? What's hard about this? It's not a square, Mr. Greg.
S3: It's got curves. Oh, okay. Is the answer bigger? Less
S3: than one. I can see the the follow up questions
S3: right themselves. Right. Like bigger. Bigger. How do you know? Well,
S3: there's one square here, and it's bigger. Right. How do
S3: you get a better. How do you get a better approximation?
S4: I could use some smaller squares.
S3: Sometimes you have to pull this out and then some
S3: kid will raise their hand. Usually the smart aleck of
S3: the smart alecks would be like, why don't we just
S3: use infinite rectangles, Mr. Greg? I'm like, yes, let's do that.
S3: Let's do that. Right? But again, like, instead of walking
S3: in and being like, so now I have a curve
S3: and then I have area and I'm going to divide
S3: this up into infinite rectangles. Like the kids are just like, what?
S3: They can do this stuff, right? You just have to
S3: ask the right opening question to get them there. Okay.
S3: Struggle questions are all over the place. Lastly, prediction. So
S3: the last one I like to do notice, compare, create, struggle, predict.
S3: These are what if questions, right. Where they're either gathering
S3: evidence or reasoning forward or they're imagining what's going to
S3: happen if something were different. What if questions. Okay. What
S3: if there were no tends to borrow from. Oh, no.
S3: What if we could take the square root of a
S3: negative number? Imagine what if there's an A term that's
S3: not actually not that great of an opening question, but
S3: what if there were an a term for this quadratic?
S3: What if frog toed shoes and toad were in frog shoes?
S3: What if Hitler hadn't tried to invade Russia? What if
S3: we heat this up? We do these types of questions
S3: all the time, getting them to predict each one of
S3: these postures. Right? Notice, compare, create, struggle, and predict is
S3: going to get kids wondering about the thing. Okay? And
S3: so when you walk into your classroom, I challenge you.
S3: See if you can put a question on the board
S3: or a ask them, say to say, all right, boys
S3: and girls, I've got a question for you guys. If
S3: that's the first words out of your mouth every single time,
S3: it's not going to work right away, right? They're not
S3: going to be used to this and be like, what?
S3: This is not math. But if every day you walk
S3: in and say, good morning, I've got a question for
S3: you guys. Good morning. I've got a question for you guys.
S3: Good morning. I've got a question for you guys. Slowly
S3: they're going to start to realize that math is not
S3: just repeating and regurgitating the thing, but it's engaging in
S3: the act of mathematics. It's doing math. It's figuring stuff out.
S3: It's imagining, it's hypothesizing, it's playing around with stuff. You're
S3: involving them in what mathematicians actually do. Okay. I do
S3: think this is then a thing that can become somewhat habitual,
S3: because you can't be inventing opening questions every single night,
S3: but you can just say, all right, say, I'm going
S3: to commit to doing this one thing every single day. Right?
S3: And I have five where you sit down to lesson plan.
S3: Maybe you have ten minutes, maybe you have five minutes
S3: to plan for this thing. Is there a question I
S3: can ask that gets my kids either noticing, comparing, creating, struggling,
S3: or predicting? One thing that I like to do a
S3: little bit is to challenge myself not to write the
S3: key formula, definition, or process on the board until the
S3: students say it. I can't always do this. I fail
S3: a lot of times, but I like to challenge myself.
S3: Can I get them to tell me groups and in
S3: each group and total, can I get them to say
S3: those words? Sometimes I fail, but at least I'm trying
S3: to do that type of thing. Can I get them
S3: to say we need a process that undoes the exponentiation?
S3: Can I get a student to say that? And then
S3: I say, yes, that's what we need. That's what we do. Okay, good.
S3: New talk, follow up questions. Where am I at? What
S3: time is it? We don't have time, do we? What time?
S3: We got time. I'm ignoring. Ignoring the clock. The good
S3: news is, follow up questions I think are easier than
S3: opening questions. I think there's a lot of bad opening questions.
S3: I think there's very few bad follow up questions. Okay.
S3: And I mean that sincerely. Right. Where again, what we
S3: have here is this example from the Theaetetus. If you
S3: want to know what it means to just teach Socratically,
S3: if you've ever wondered what that word means, this is
S3: what it means to do this, because this is what
S3: Socrates does. Once more, Theaetetus, I repeat my old question
S3: what is knowledge? Opening question. This is a struggle. Struggle
S3: question what is knowledge? There's no good answers to that, right?
S3: The knowledge is perception. That's a student response. And then
S3: what does Socrates do? He says, bravely said, that's the
S3: way you should express your opinion. Now let's examine together
S3: this conception of yours and see whether it's a true
S3: birth or a windegg. Windegg. This is just an empty
S3: shell with no. I don't know if you've ever gotten
S3: one of those from a grocery store. Okay, good. Okay,
S3: so what we have here is this shape where I
S3: prompt student thinking. Usually with an opening question, I get
S3: a response and then I ask a follow up question
S3: based on that response. Right? So there's other options. I
S3: could say what's five plus eight? Someone could say 13
S3: and I could say good and ask a different question.
S3: That's not a follow up question. It's five plus 813.
S3: How'd you figure that out? It's using the response as
S3: the thing that you go back and forth. This is
S3: where you get a little bit of dialogue is the
S3: idea here. Okay, so. I'm going to skip through this
S3: a lot. Let's see why our follow up question important.
S3: They help students make sense of their experiences, right? They
S3: help you to think deeper about the thing. They ask
S3: you to dwell on meaningful ideas and problems, probe into
S3: the reasons for truth and not just truth themselves. We
S3: care about truth, especially in math, but we care about
S3: the discovery of truth. Challenge students to provide explanations that
S3: go beyond one word answers. One of the best definitions
S3: of math. We've seen a couple of them. One of
S3: them we haven't seen. Paul Lockhart says mathematics is the
S3: art of explanation. Okay. Right. Get them explaining. You've already won. Right.
S3: If you're getting students explaining their thinking, you've already been victorious.
S3: I was I was doing a couple of years ago,
S3: I was doing a model lesson in kindergarten. So I
S3: was I was doing a kindergarten and we were doing
S3: calendar time and I was having a blast. Calendar time
S3: is amazing, right? And we it was Tuesday or something,
S3: and we put the 14 in the slot of the calendar.
S3: It was the 14th. And so I asked, what number
S3: are we going to put in the Friday slot? Right. Okay,
S3: maybe not the best opening question, but a question. Right.
S3: And a girl raises her hand. She goes 17. I
S3: was like, yes. How'd you figure that one out? And
S3: the look on her face, I'm never going to forget it.
S3: And she was like. How did I figure that out? Like,
S3: she had never realized that, like, you could have a
S3: thought and then attach another thought to that thought, right?
S3: It's this like it's this. Oh, I how did I
S3: figure that out? And she was like, she didn't say
S3: she didn't give a super long explanation. She didn't say, well,
S3: it's three days away. And so I added 3 to
S3: 14 and she was like, well, we're going to put
S3: a 15 and then a 16 and then a 17.
S3: And I was like, what I've done is I've, as
S3: a kindergartener, I've got her to explain her thinking, I've
S3: already won. It doesn't matter what she says at that point,
S3: the fact that kindergartners can say, I have ideas and
S3: then I can explain how I get those ideas, that's
S3: a win. And it happens because we follow up on
S3: student thinking, okay, good, we're open, struggling students. Best way
S3: to do this? I talk about this in just a sec, right?
S3: You don't know what's going on. 71 times 14, I
S3: don't know. Well, is it bigger? Less than one. Bigger.
S3: How do you know? Well, 14 is bigger than one.
S3: And I did a multiplication. Multiplication makes things bigger. Why? Right.
S3: Because multiplication, I mean. Right. You're getting them to explain
S3: to you math because you've asked them a question and
S3: then asked them some follow up questions instead of just
S3: like 71 times 14. I don't know who can, who can,
S3: who can help, right? Turning away from that student rather
S3: than going deeper with that student. Okay, good. So situations
S3: for follow up questions. Number one. Silence. It's an important one.
S3: Notice I said student responses. Silence is a response. If
S3: I ask a question and I get silence or I get,
S3: I don't know. That's a situation. For a follow up question.
S3: It's tempting to either go in a different direction or to.
S3: Answer the question yourself, but I don't know. So is
S3: a place where I would then ask a student a
S3: follow up question. I would dig in. What students need
S3: to know is, I don't know is a perfectly acceptable
S3: response in your classroom. I'm not willing to think about
S3: it is not an acceptable response. Okay. There's a difference.
S3: If I say something and you say, I don't know,
S3: and then I ask you a follow up question about this.
S3: If I say something like, what's hard about this? That's
S3: a follow up question about what you don't know. You
S3: have to be willing to think about this with me, right?
S3: Put that rule on your on your wall somewhere. I
S3: don't know is an acceptable response. I'm not willing to
S3: think about it is not an acceptable response. Okay. Number
S3: two wrong answers site for a follow up question. What's
S3: the temptation as a teacher? Fix it. Correct it. Right.
S3: Wrong answers. How do I dig in to that wrong answer?
S3: I will even solicit wrong answers in the classroom. If
S3: you've never done this in your classroom, put a problem
S3: on the board and say wrong answers only. It's kind
S3: of great. Put a clock on the board for your
S3: second graders. What time is it? Wrong answers. Only 781, right? Great.
S3: They say, what's the. And you say, what's the most
S3: tempting wrong answer? And help them to articulate the errors
S3: that we might make as students. Right. They start to
S3: filter themselves if they're imagining the wrong answers. You want
S3: the wrong answers in the classroom. Wrong answers are gold.
S3: Wrong answers are the only times we are going to
S3: teach someone something. To go from wrong answer to right answer.
S3: Number three correct answers. What's the temptation here to move on? Great.
S3: Move on. I did it, yes. Done. Instead of saying,
S3: how'd you figure that out? What was the key? How
S3: can we remember this? Notice the follow up questions that
S3: are there with a right answer. Lastly, a question from
S3: a student. What's the temptation? Answer it. Right. Especially if
S3: you get a good question. How do I reframe that
S3: question and turn it back on the on the student
S3: who asked it? And not in some cheesy way like, well,
S3: what do you guys think? Not a good follow up question.
S3: How do you turn that back on the student? Right.
S3: Notice I think I've covered almost everything here. Right. Every
S3: situation can be done with a follow up question. Their
S3: only limits are time and exhaustion, both of which are
S3: legitimate concerns. Right? You only have so much time. You
S3: are going to get exhausted, right? But look, if you're
S3: seeing everything that students do as an opportunity, that's the key.
S3: Every single thing a student responds. That's an opportunity for
S3: me to ask about what's going on, right? Okay. So
S3: types of follow up questions. Here's how you do them.
S3: Number one. Most basic, you press for explanation. Pressing for explanation. Basically,
S3: it's saying, why did you respond the way that you responded? Right.
S3: That's all that question is. You got to be a
S3: little careful. You don't seem like a jerk. Or you're
S3: just like, ask a question, I get silenced. I'm not
S3: going to be like, well, why are you guys silent
S3: on this? That's not what you want. But I can
S3: ask a question and I get silenced and I can say, oh,
S3: that's good. What's what's hard about this question? It's basically saying,
S3: explain to me why you guys are silent here. What's
S3: genuinely hard about this? Right? I think math is hard.
S3: I think we should lean into that. Okay. Number two comparison.
S3: We saw this as an opening question. It's actually a
S3: really good follow up question as well, juxtaposing it with
S3: another thing. For example, let's try a wrong answer. If
S3: I say five plus three and someone says seven, I
S3: can say, well, five plus two is seven. What's five
S3: plus three? That's comparison follow up question. I'm saying compare
S3: it with this and rethink, compare and rethink. It's not
S3: the best way to respond to wrong answers, but it's
S3: a possibility. Okay. Number three, hook, hook question goes back
S3: to something that's basic and then scaffolds it back up.
S3: Use a basic response as a launching pad for a
S3: deeper question. Often it's a thing that you asked, you
S3: got silence or you got 1 or 2 hands. 1
S3: or 2 hands in a math class is silence. Okay,
S3: you get 1 or 2 hands. Don't call on 1
S3: or 2 of those guys. Don't do it. It tells
S3: every other kid they can turn off their brains. Okay,
S3: if I get 1 or 2 hands, I'm going to say,
S3: oh yeah, this is a hard one. Is it bigger
S3: or less than one? I like the idea is go
S3: to a basic one and build it back up. We
S3: saw this today. Doctor Treloar did this beautifully. Right. How
S3: do you divide by a fraction? Silence. Okay. What's addition? What?
S3: Go to a basic one and build it up. What
S3: are the different ways we think of addition? What are
S3: the different ways you think of subtraction? Right. Build it
S3: back up. Opening question. Follow up question. Okay. Number four. Ramification.
S3: Probing the implications of this response. How do we apply it?
S3: What does it matter? Why does it make sense? What's
S3: the key? Are all problems like this? You could see
S3: the the potential follow up questions here about what this
S3: means for us. So what I've done is this. Okay, great.
S3: So what we have here are the situations on this
S3: axis and the kinds on this axis. And then just
S3: some examples. Not all of all these work, but just
S3: to show you there's so many different follow up questions
S3: that I think we can do for every single situation. Now,
S3: don't print this off and put it on a little
S3: clipboard and be like, well, this is a wrong answer.
S3: So now I'm going to use a hook, right? Don't
S3: do that. That's not the point of this exercise. But
S3: the point is just to say every single situation can
S3: be responded to with every single kind of question. That's
S3: not like all it is, is it's asking students to
S3: think more. You've got to develop your habit of this,
S3: which is easier. Fingers are pets. Fingers. Why do you
S3: think fingers? What's hard about that? What's easy about that? Pets.
S3: Why do you think pets. It's just getting in the
S3: habit of seeing student responses as opportunities. Right. Okay, so
S3: some of my favorite ones, I'll put these on the
S3: board for a sec. Uh, I think why is this
S3: hard is a really good one. What's the key? What's
S3: your strategy? All of these things students love talking about strategies.
S3: My six year old, I play a lot of war
S3: with him, and he's always wanting to tell me this
S3: strategy for the game. I'm like, it's a pure game
S3: of luck, you moron, right? I don't call my son
S3: a moron. I'm always eager to hear his strategy. He's like,
S3: whenever you play a nine, I'm gonna try to play
S3: a ten. I'm like. Great. How's that going for you? Yeah. No,
S3: he's a good kid. He's going to need to get
S3: some athletic scholarships, but he's going to be a good kid. Okay.
S3: Some that I don't really like up here. Right. Anyone
S3: have any questions? Not great. Just doesn't get much traction, right?
S3: How do you feel? I don't really care. Okay, good.
S3: So how do you habituate this one? I think if
S3: you're thinking about ways to prep for this. I don't
S3: think you need to prep for it as much. Right.
S3: The follow up questions write themselves a lot of times
S3: as long as you start getting the habit of it,
S3: but including an anticipated student responses section, what my students
S3: say and how might I respond? Right. Number two, think
S3: about the different situations. Test yourself as you're thinking about
S3: your own classroom. How do I respond to silence? Do
S3: I do it well? Do I do it poorly? How
S3: do I respond to wrong answers? Try to make some
S3: renewed habits to respond to some of these in different ways.
S3: Specifically with this kind of Socratic pedagogy, right? One thing
S3: that I find really helpful is to count the number
S3: of questions that I ask that can be answered with
S3: a single word, and then count the number of questions
S3: that beg a sentence. At least one of my absolute
S3: proudest moments as a teacher. I was teaching sixth grade,
S3: and I don't know, I forget what question I asked,
S3: but a kid raised his hand and he gave an answer.
S3: He's like, I think it's seven. And because you always
S3: make us explain how we got it, here's how I
S3: got it. And I was like, yes, I have broken them. Right?
S3: But that's the idea. You want to do this so
S3: habitually that they get in the habit of explaining their thinking, right?
S3: That's the thing that you're interested in, right? They one
S3: word answers are good, but we have to get students
S3: talking about the math. Notice, notice this is hard, right?
S3: This is hard as a teacher, right? It requires actually
S3: more teacher involvement to teach Socratically a few takeaways here.
S3: Number one, a deeply rooted understanding of and love of
S3: mathematical truth requires an exploration of the mystery that surrounds it. Okay.
S3: Number two Questions with opening and follow up are the
S3: single most important thing that distinguishes a wonder based pedagogy.
S3: If you're interested in wonder, you want to try to
S3: teach socratically. Okay. Number three. I think this wonderful teaching
S3: is going to require some habituation. The habit of teaching
S3: with wonder is a shape. It's not a formula, but
S3: it is a repeatable shape. That thing is repeatable. It's
S3: new every day, but it's a repeatable shape. It's a
S3: thing that students can become habituated to. Number two, I
S3: think you can build a culture where students are going
S3: to respond to questions. This requires perseverance, but students can
S3: get in the habit of doing it. They know that
S3: you're going to be asked to explain their thinking, to
S3: do mathematics. Teaching is always inventive, but I think it
S3: adheres to a set of questions that can become ingrained
S3: in teaching practice. Let me finish with the end of
S3: this quote. Right. The capacity to wonder is among man's
S3: greatest gifts. It signifies that the world is profounder, more
S3: all embracing and mysterious. And the logic of everyday reason
S3: had taught us to believe.
S2: That was Doctor Jonathan Gregg, assistant professor of education here
S2: at Hillsdale College. Thank you for listening to the Hillsdale
S2: College K-12 Classical Education podcast. I'm Scott Bertram. We invite
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