Principles of Pedagogy in Mathematics

lessons are full of wonder, engagement, and curiosity. But it can be daunting to try to replicate these moments over and over again with some initial work and lots of practice. Teachers can infuse their class time with these elements without running in circles. Doctor Jonathan Gregg, assistant professor of education at Hillsdale College, spoke on how to infuse your pedagogy with wonder at a September 2025 center for Teacher Excellence

event on the Art of teaching mathematics. Though his ideas are presented in the context of mathematics, countless applications can be found for every classroom subject. Here's Doctor Gregg.

the classroom, right? There is you deal with enough chaos every day. And so you want something to rely on. But pedagogy has to be new. It has to be exciting. It has to be something that that you can't just repeat day after day after day after day, because then it becomes boring and we lose. We lose the sense of wonder here. And so I want to try to inhabit that space a little bit today to think about

how do you keep things exciting? How do you keep things wonderful without making making yourself run in circles wildly or making yourself plan for hours and hours every night? How do you how do you do something that's habitual and yet still wonderful? This is kind of the thing that I'm interested in here. I think that's the core of core of pedagogy because. So I'll start with a quote as well. This is, you know, one of those quotes that you read and you just you can't get

it off of your mind. It kind of haunts you. I read this, I read this a few years ago, and I think I think it's it haunts me. Right. The capacity to wonder is among man's greatest gifts. It's so true. And if you think back, even. Just think back to your good lesson. Think back to a good lesson that you did. I guarantee you wonder showed up somewhere. And actually, this is the one place where I do think students will and children will follow. They'll follow what

they wonder about. They'll follow what, uh, what's wonderful. Um, and so I think this quote is if we're looking for a, a beginning of pedagogy, a first philosophy of pedagogy, I think this has got to be it. People are people actually goes on. And he says this, he says, really the situation is this, the deeper aspects of reality. That's what we want students to get are apprehended in the ordinary things of everyday life and not in the sphere cut off and segregated from it. The sphere of

the essential or whatever it may be called. It's in the things we come across in the experience of everyday life that the unusual emerges and we no longer take them for granted. In that situation corresponds with the inner experience, which has been regarded as the beginning of philosophy, the act of marveling, kind of see how this is going to work? You present something. It's unusual. Students wonder at it,

and then they start to think. And that's philosophy. And this is where where you can kind of see how wonder is going to be the thing that's possible as a proof here. And honestly, we can't we can't go too far in math without talking about about wonder as a maybe not a proof, but at least some evidence for it. I have to be careful using the word proof around around this, this crew, you guys have some high standards for this, but at least as some evidence. Right?

Here we go. Here are some quotes on wonder. We want students. We want to get students to wonder about hypotheses. This was Doctor Dave Gabler four hours ago. Mathematics should instill in students an ever increasing sense of wonder. This was doctor Tom Treloar three hours ago and we mathematicians like to wonder if there's anything else we can do with this. This was Doctor David Murphy. Not even one hour ago. All right, so these guys are in here

talking about wonder, right. I wasn't in Miss Jackson's talk but I guarantee you wonder showed up. I'm gonna go back and look at the transcript. But it's there, right? It's there. But what I can't do is stand up here and say, go get your students to wonder. That's not a thing we can do. That's that's not helpful as teachers. So the real question is how do we get students to wonder and how do we habitually do that? This is the question for today. It's not an easy question.

How do we habitually get students to wonder? Because I think if we can, we're already doing the kind of things that we should be doing in our math classroom, right? You've already kind of won as a teacher. Wonder leads to philosophy. So what is wonder and how do we do it? Okay, let's go to Peter one more time. Socrates. is talking about Socrates here. He's actually about the Titus. Titus is a dialogue between Socrates and a mathematician. All right.

We saw Theaetetus actually show up on Doctor Murphy's slides here. And he says Socrates brings Theaetetus to the point of admitting his ignorance with his shrewd and kindly but staggering and astonishing questions, questions that stagger and astonish with wonder. And there, for the first time in the theater, without solemnity or ceremony, almost. By the way, though fresh as dawn appears, the thought that has become commonplace in the

history of pedagogy. The beginning of philosophy is wonder. You see it again here. But notice what he's doing with wonder. Saying look wonder happens in two ways. Yes, it's kind of staggering and astonishing thing. So it's actually a really lovely word, right? One of the best words in the English language are the words that we equivocate on, that go, that go that have two different definitions, right? Think of the word witness is one of my favorite examples here. Witness.

Witness is a beautiful word, right? It means both to observe something taking place and to share about the thing that you saw, right? And the fact that we use the same word for it means when you see something, you are obligated. You have a responsibility to share it. Actually, what you guys are doing, right, you're witnessing to your students. Okay. Anyways, I'm off track. I'm off track. But wonder is the same thing, right? Where questions that stagger and astonish with wonder. Right.

First of all, wonder means that I'm staggered. I don't know something. Right. I recently wondered how to put Freon in my car because I didn't know how to do it, and it was 120 degrees in my car and I was upset. I don't know how to do this. And I was staggered and I wondered, and then what did I do? I did what any self-respecting person would do. Got on YouTube right. And figured it out. But there's wonder. We are staggered. We then go and figure the thing out.

And that's where learning takes place. But first you have to be dislocated, right? I wonder about this. I wonder what's going on. I'm confused. It's almost like you're you're you're dislocated. You're shocked. Socrates actually calls himself a torpedo fish, which is an electric eel. Apparently that the idea is he's a he's just shocking people. He's like, what about this? What about this? And you're shocked. I think students you

can shock students into wondering. But also this astonishment. Say, oh, I wonder at that thing that's marveling. It's this, it's this or that students can have as well. So we see this dual understanding of wonder, right. And an invocation of the Socratic method. So let's piece some of this together and think about what pedagogy looks like. So I present to you pedagogy. Thank you. Pedagogy. I'll take your question. No, I need some labels. Yes, we need some labels. Okay,

so don't take the pictures yet. Wait for the labels. Here we go. Actually, don't take any pictures. Just email me for the slides. But let's get some labels here because this shape is meaningless. This is not a meaningful shape. This is not a polyhedron. Right? So what's going on here? Number one. Right. I have a lightning bolt. I think this is what we think of as an opening question. Right? I'm envisioning the teacher kind of as as Zeus here, right?

Chucking a lightning bolt in the middle of students. And what's going to happen is they're going to this is the shock part of wonder, right? You're going to be dislocated. You're not going to understand the thing. Right? And you can see these blue lines. This is ideally what student responses are going to be that students can go have a bunch of different potential ideas out of this wonder.

Wonder leads to this kind of exploration, right? And then these kind of curlicue lines, this is what I'm calling follow up questions that select different student responses and then follow up on them until we get to the green thing. This is the truth. Okay. The thing we want to teach the learning goal, if you will. What you want to teach the student. Okay. And then what we have are these maybe some examples, maybe some practices, some applications.

Once we learn this thing, then we practice it, then we do it. Then we. But I think the idea here is that you, uh, this is kind of how one might learn in the, in the field of wonder or using wonder where you're invoking the Socratic method, where there's both shock, right? Staggering. And then there's potentially astonishment down here. Okay. I think every good lesson, or at least a lot of good lessons are going to follow

something like this in a math classroom. Okay. And maybe here's one way of, of proving, again, this is not a proof, but some evidence for this. If we look at the dominant pedagogical approaches that we are surrounded with, I think there's two dominant pedagogical approaches. I think what you might call the progressive and what you might call the dogmatic, the dogmatic, I think, is this here's the truth. Let me hand you the formula. Let me hand you how to do it. Practice it over and over and

over and over again until it gets drilled into your face. Okay, that's the dogmatic pedagogy. Or you have progressive pedagogy, which kind of throws out an idea and then says, go students, explore, be empowered, go to your heart's content. Right? But it doesn't really believe that there's any truth to be found, right? It's about whatever students want to do, right? And it just cuts off the top half of this versus cutting

off the bottom half of this. Right? But somehow in between these two extremes, there's something holistic where we have wonder at the beginning and wonder at the end from wonder to wonder, right? That's, I think, what a pedagogical approach looks like. But this shape is, is just a shape. Let me illustrate. All right. So I'm going to teach you guys. You guys are students. I will be teacher. And and I'm going to teach you guys pretend you're first graders. This is a first grade class. Okay, good.

Good morning, boys and girls. I'm really excited to be with you guys today. Guess what. I want to teach you guys a new word. Can you guys say this with me? Say this after me. The word is multiplication. You guys say that with me. Very good. Multiplication. Now multiplication is repeated. Addition multiplication is very good. So if I have a problem like three plus three plus three, plus three plus three, that's a lot of threes, don't you think? My hand is almost tired writing up all

of these threes. Right. So would you guys like to learn a better way for writing that I think we would. So I can write this as five and then I need a new symbol. I'm not going to use addition anymore. Can everyone make me the symbol of addition with your hands? Right. Okay. I need a new symbol. So what we're going to do? Hold your addition. Oh. New symbol. Right? We have an X5X35X3. And just like I have a word for this, what word do I use for this plus right. I'm going

to use a new word for my multiplication. I'm going to use times. Say it with me. So five times three because it's three five times. Some of you guys are legitimately like, oh wow. Okay, so can we figure out what Three five times three is. Can you guys count with me? Three. Six. Nine. 1250. Very good. Five times three is equal to 15. And you guys just did your first multiplication problem. Okay? And cut and cut. Okay, so we can keep going, right. But what kind of

pedagogy was that? Right. That's dogmatic pedagogy. Okay. It's entertaining, but it's handing you the four way to do this. Here's how to do this. Here's what the words mean. I'm handing you this thing. Is it good? Do I need to teach kids this? Yes. Is this where I want to start? No. Right. Because if you're first graders, I can walk in and say, I'm going to teach you guys a new word, multiplication. But if I walk into my ninth grade class, I'm going to teach you

guys a new word logarithm. Mr. Greg, why we don't care, right? Okay, so and slowly. This is why students start to hate math is because we teach it pretty. One of the reasons we teach it pretty dogmatically. And there. There's no buy in, right? There's no there's nothing that they're doing mathematically. They're not actually engaging in mathematics. They're just receiving the results of mathematics. Right. Doctor Murphy said mathematics is a

human activity. We need to get them doing the mathematics. Okay. Good. Let me try again. Good morning boys and girls. I'm really excited to be with you guys today. Go ahead and take out your devices. I've preloaded the module to explore today's lesson. It's called multiplication, but you guys go ahead and explore on your own. Figure it out, see what you can make of what's going on. And I'll just be over here in case you need to answer any questions. And you guys can go ahead and explore

and we'll talk at the end just to see. We'll share how we'll do a little share out to see what people thought of your chance to explore And cut. Okay, so what kind of progressive what kind of pedagogy here, right? This is what progressive pedagogy looks like or feels like. It's not. There are other ways to do this, but really what it does is it the goal is for the teacher to fade into the background, to have students

do the exploration on their own, right? The teacher doesn't get to have a role to play that this is that it's it's actually oppressive for me to impose my truth upon you. Right. And I think this, as you guys know, it's it just doesn't work, right. Students don't aren't able, by and large to do this. And what it does, I think, is it bifurcates students into those who can figure it out on their own and those

who can't. Right. I think one of the one of the really unspoken things of progressive pedagogy is that it's attempt to kind of empower all students equally. The pedagogy is the thing that ends up bifurcating the students into the haves and the have nots. It's really problematic, actually. The methods of progressive pedagogy undermine their stated goals, in my opinion. I can talk more about this if you want, but okay, let me try one more time. Morning boys

and girls. I've got a question for you guys today. Which of these two problems is easier to figure out how many fingers we have as a class, or to figure out how many pets we have as a class. You guys are bad first graders. Fingers. Pets. Fingers. Pets. My brother has a cat. Okay. Right. But this is what you want. This is what you want. Okay, so, fingers, let's try the pets one. What do I have to do to figure out the pets? What do I have to do? I've got to ask people. Right. Okay, good.

Let's ask. How many pets do you have? Five. How many pets do you have? Ten. How many do you have? Does my little sister count as a pet? No. Fast forward. Pets, pets. Now I've got all everyone's pets on the board. Now, how do I figure out how much we have all together? I have to use addition. Why do I have to use addition? Ah, because we're sitting all together. Good. So let's add fast forward. We have 312 pets as a class. Whew! That is a lot of pets. I hope we don't

bring them in here. So let's try the fingers one. How do I figure out how many fingers we have as a class? Can I do the same way? Can I do it the same way? Yes. No. Who thinks yes? Who thinks no? Why do you think? Yes. Sure. Why do you think no? Okay, okay. So let's try. Let's try it the same way here. Right. How many fingers do you have? Ten. How many things do you have? I have eight. Two of them are thumbs, right? Yeah. Okay, good. Yeah. This is what you. But again, this is what you

want in the classroom. You want a little drama, you want a little energy, right? Yeah, right. And I get a couple people in here. Do I need to keep asking people? I don't why not? Everyone has the same number of fingers. That's interesting. interesting. What do I need to know to solve this problem? How many people there are? I need to know that. And how many fingers. If someone's missing a finger. Yeah. How many fingers each person has. Right. This is a different kind of problem, right? So. Okay.

And I'll cut there because now what am I going to do? I'm going to do I'm here now, right. And now I'm going to say like, let's write up the multiplication problem. What you guys just did with the pets. That's multiplication. That's not addition. Something new right? Okay. So go ahead and cut. And you could see how a lesson looks like when you walk in and say which is easier. Fingers or pets? Lightning bolt question right. No one knows what's going on here. They're like, what? This

is math class. You're talking about fingers and pets. I got reactions. I at least got two reactions, right? I got fingers and I got pets. And both are defensible pets. Probably there's fewer pets than fingers. It would be easier, right? Then what did I do? Notice the kind of pedagogy that I employed. What did I do? Did I say a single statement? I didn't say a statement. One time. I say one statement. Right. This is what the Socratic

method looks like. By asking, right. Picking the two responses and asking a bunch of follow up questions, I was able to get them to tell me the things that we need for a multiplication problem, namely, namely number of groups and number in each group. That's what multiplication is. We need two things and we can figure out the total that way. But they're the ones that did that thing. They engaged in the activity of multiplication before. And we

didn't just kind of let it be there. And then we're going to formalize it into what what we actually want and practice it like we would like we would normally, but we can see what we you know, we by taking this Socratic method. And is there wonder at the end. I don't know, but at the very least they should feel that, oh, in this world there are multiplicative situations and additive situations. Is a fourth grader being able to say that? No, but I can say what other problems

are like the fingers problem toes. Yeah, right. You guys are bad first graders. How about some non body part ones and ask him to think right wheels on a car, maybe something like this where they're noticing that the world actually that multiplication is a response to the world and math is woven into the cosmos and all. And what we're doing is we're noticing patterns in the cosmos, right?

Our first grader is going to be able to say that to, you know, but are they going to slowly feel that math starts to be this thing that I walk in? I put a question on the board or ask a question and we mess around with it and play with it until some truth emerges. Until we discover some truth. That's, I think, where we can get something like wonder in the math class. This type of a shape. Okay. So again, I still don't think this shape is all that helpful because how do you come up with these

opening questions? What follow up questions do you ask? How do you do these things? Okay. So that's what I want to do. Don't let me off the hook for being impractical still, right? I have to give you guys some tangible advice for these things. This is what we need. So let's do it. Okay. Here's how you do opening questions. Go. New new. New talk. Opening questions. Socrates says, come then to me. He's talking to a mathematician here, actually. Right.

And do your best to answer the questions I'll ask you. And if I abstract and expose your ideas? Because I discover upon inspection that the conception which you have formed is a vain shadow. Do not quarrel with me, for it would be wrong for me to admit falsehood, or to stifle the truth. Right. Come to me and expose your ideas. That's what an opening question does. Exposes student thinking, exposes ideas. Okay, so here we go. What makes for a good opening question? Number one has to be accessible

to all students. Okay. The very first thing out of your mouth is something that some students can't grapple with or understand. You're going to lose half your class. Okay. There's got to be something that every student can think about, right? Which is easier? Fingers or pets? Everyone can have an opinion.

That's no access barrier there. Okay. Number two, it has to be challenging to all students because there's no easier way to lose half your class than for the first thing out of your mouth to be something they already know the answer to. Okay, so it has to be both accessible, right? Which is easier, fingers or pets? No one really has written the has written the book on this. No one. This isn't a thing that one can really even know, right? This is challenging in some ways to everyone,

at least to think about. Right? Number three, it has to provoke multiple thoughtful student responses. I need this is not just a yes or no right or wrong answer that I'm starting with. It has to be multiple thoughtful responses. I want to scatter my students. I want them to be dislocated. I want them to wonder. But number four, it has to be directly oriented to the truth. This isn't just some progressive. Think about this guys. Do what

you want, right? Follow your dreams. No. Right. And in particular, I think these two and these two are really tough needle to thread. How do you get something that's accessible and challenging to everyone. And how do you get something that's Towards the truth. Enough that it's going to get you there. But open ended enough that it's going to get students thinking in multiple ways. It's a high bar for opening questions. Okay. I don't think there's as high

a bar for follow up questions, which is good. But again, don't let me off the hook. What kinds of things make for good opening questions? What are some good examples? Okay, so what I've done, I took all my lesson plans that I ever wrote and put them in one massive pile, and I started just sorting them into different kinds of opening questions. Okay. And I came up with five piles. So here are my five piles. There's probably more opening questions,

but here are the ones that I know that work. Okay. Number one, we get our students. Each one of these has a kind of posture that I want students to adopt, right? So number one, I think these are noticing questions. I think this is the very basic of opening questions. We ask students to notice, say, what's going on, right? Usually this is a why question. Noticing is observe. See something and try to explain, observe and explain. Observe and explain.

See something and explain to me what's going on, what's going on here, what's going on here? Why is this taking place? The very, at the very least, walk in with a why question at the very beginning. Why is this right? I think it's especially true for generating formulas. Here are some examples. Okay. Why can we not use our method for adding 23 plus 45 for the question 23 plus 48. Notice that this doesn't work. And explain. Notice and explain. Notice and explain. Why is subtracting a

negative and positive? How big a box do I need to carry stuff? Notice stuff comes in different sizes and how to explain if you're trying to teach volume, how big a box do I need to carry that thing? What thing are you carrying? I don't know a thing, a turtle. How big a box where what you're doing is you're asking students to think and then forcing them to articulate things. They say, well, right. Instead of me being like, volume has three dimensions length, width, and height, right?

Ask them to put stuff in boxes, see what happens. They're going to tell you that things have three dimensions you got to worry about. They're going to tell you how big space is. Students can do this. They can come up with this stuff. It takes the right opening question. So actually fun fact, I would give this same talk to any of the disciplines. I think this is how we teach literature history science best anyways, right. Here are

some other questions. Why is being a hero so difficult? Literature. Why do the Articles of Confederation fail? History. Why do feathers fall slower than bowling balls? Science. Everything. We are driven by these noticing type of questions. Okay, but that's very basic level. Okay, there are better ones. Number two, I asked my students to notice. Ask my students to compare. Okay. It's like noticing on steroids. Noticing. Noticing. Times two. Notice. Notice. Compare. See?

What's the difference between really good ones, right? This is different than this notice. Notice. Explain the differences. Right. Juxtaposing two related ideas. It's often phrased as an er or est question. If you teach like third, fourth, fifth grade, those humans are just little extremists, right? This is what they want to do with their life anyway, right? Just lean into it. Ask them best. Worst math is a lot of about what's the best way to do something.

Which is easier. Figuring out figures of pets. Which is easier, this problem or this problem? Which is faster? Start in the ones place the hundreds. Place the difference between this and this. This one's really great. This. This factoring one. The second one on there. This is how I teach my eighth graders how to factor. I divide them class in half and I say, give them some reward. Like, you know. All right, a little test for you guys

right off the bat. Uh, first, uh, fastest way to get a solution gets half the homework off or whatever, right? And they're all like, they're all in now. Like, yeah, okay, good, good. Let's do it right. Going down right. And I give one half. I pass out this problem. They flip it over and I give the other half this problem. And you know, the the half that I get this problem is done in 10s. The other half is still going. I'm like. And so they're raising up their whiteboards. I'm like, good,

good good good good good good. And this guy, this, this whole half is just getting angrier and angrier and angrier and angrier. Like what? You gave them a different problem. I was like, I don't think I did. I think it's the same problem, right? It's the same problem. Same problem. Right. It's unfair. Right. And, and I'm like, oh, you, you think this is a hard problem? What's hard about this?

They're like, there's two X's. There's a squared, right? And I'm like, oh, you want some ways to take this hard problem and turn it into an easier problem? Yeah. Oh let me. And they beg me to teach him how to factor. Right. And I am benevolent and so I do. Right. This is. But the idea is just look, this is hard. Oops. This is hard. This is easy. Let's learn how to do it right. Easy stuff. Right. Okay. It's like shooting little angsty eighth graders in a barrel.

It's great. It's great pedagogy, right? Same in other disciplines as well, right? What's the difference between the way that Elizabeth envisions marriage, the way that Darcy envisions marriage? Great opening question. Oops. Uh, was there ever a moment when you'd be decisive and moment need to be cautious? If I'm teaching the Battle of Gettysburg. Oh, man. I'm opening with that question. Day one. They had to be real decisive to get that high ground right. Day three, just

the boldness run amok, right? Number four. Right. Or the last one, which is harder to roll a ball up a hill to keep it stable at the top of the hill. Good comparison. Okay. I asked them to notice, to compare, to create questions of how would you or design Your kindergarten first, second grade teacher. They should be drawing. They should be designing all of these things they can do. Draw me this. Draw me something. Design me something. Create me something. The idea here is to use get students

using their imaginations. We want our math classes to be imaginative. Notice how this is great to get all students doing different things right. This is going to provoke multiple responses to create. This is one of the beautiful things about human beings is we all have different creativity. If I asked you guys to write a poem, you guys would all hopefully write something different. Otherwise it'd be a little creepy, right? Design a game that seems fair but isn't. If I'm

teaching some expected value. That's a great question, right? Design me a problem that can be solved for this bar model, right? Flip it on its head. Literature. Design me a defensive strategy for Tom Robinson. Design me a scenario of the Roman system of government particularly struggle with. Design me something that seems alive but isn't. If. I'm sorry, I'm doing all these other disciplines here, but if I'm trying to

teach the seven qualities of life right? Movement, science, knowledge, movement, respiration, sensation, growth, reproduction, excretion, nutrition. Okay, great. These are my seven classes. I can walk in and say, all right, kids, today we're going to learn the seven qualities of life. Write them down. Write them down. Write them down. Right. It's not really pedagogy, right? That's just dogmatic teaching. It's easy. Right? Design me something that seems

alive but isn't. Go work with the person next to you. Explore for a little bit. This isn't just like explore to your heart's content, but try some stuff and then let's come back and talk about it. Right. And my follow up questions write themselves. Maybe they come up with, I don't know, a robot, a virus Wind. Uh, stories, I don't know. Plenty of things seem alive, but aren't. And I can ask them, why do you think that seems alive? Why does it seem alive? How do you

know that it's not alive? I bet you they can get all seven of those things. Or at least six, right? Okay, good. Notice. Compare. Create number for struggle. Okay. Um. The question in the back of your head should be, is this accessible to everyone? If I make them struggle with stuff. No. Right. In that they're going to struggle. But since I'm asking everyone to struggle, it's weirdly accessible, right? As soon as if you're a kid that's going to that I put a

problem on the board and they're like. But then they look around their classmates and everyone's going, now I've leveled the playing field again, right? This is accessible to everyone. I'm not asking for insights here as much as I'm asking them to grapple with the thing, right? To struggle to try some stuff, right. I think we struggle in two ways. Something initially that seems impossible becomes clearer upon investigation, or something that initially seems simple, that becomes more impossible

under investigation. So a couple examples here. Right? 706 divided by three trying to teach remainders for the first time. Throw it up there. 76 divided by three go! Mr. Greg. It doesn't come out right. Oh, okay. What are we going to do? Here's $706. You guys just made it divided amongst yourselves, right? Two to the X equals 13. Oh it's great. I'm trying to teach. I'm trying to teach logarithms. I walk in, right. You guys are the ninth graders. Two to the X equals 13. All right.

Go for it, guys. Now your ninth graders, right? Oh, Mr. Greg, why do you do this to us? Oh, what's hard about this one? There's no twos that get a 13, Mr. Greg. Oh, okay. How about this one? Two plus x equals 13. Do you know that one? 11. How'd you do that one? Ah, we just subtracted two. Okay. Right. All right. Two times x equals 13. Six and a half. How'd you do that? One divided by two. We know these. All right. Well, two x equals 13. Is the answer bigger or less

than one bigger. How do you know? Two to the one is two. Is it bigger? Less than two bigger. It's bigger. Less than three. It's still bigger. It's between 3 and 4. Mr. Greg. Oh, okay. Well, what's hard about this? How'd you know to divide by two when we had a multiplication problem? Because it's the opposite of it. We're not going to use inverse operations. They're just not going to say it, but they will. You know, they're never going to use the right words that you want,

but they'll say, say, say. Okay. Well, what's hard about this one? And some kid will say, oh, well, we just need is a magic thing that undoes the exponentiation. I'm like, oh, is that what you want? Right. And they've begged me for logarithms. Right. I mean, like, again, this take take this one. Right. I'm trying to teach the fundamental theorem of calculus. I put this on the board. They're like, how much area are. It doesn't even have

any lines. Mr.. Greg. Okay, I'll draw some lines. Fine. Great. Now, how much area? What's hard about this? It's not a square, Mr. Greg. It's got curves. Oh, okay. Is the answer bigger? Less than one. I can see the the follow up questions right themselves. Right. Like bigger. Bigger. How do you know? Well, there's one square here, and it's bigger. Right. How do you get a better. How do you get a better approximation?

ask the right opening question to get them there. Okay. Struggle questions are all over the place. Lastly, prediction. So the last one I like to do notice, compare, create, struggle, predict. These are what if questions, right. Where they're either gathering evidence or reasoning forward or they're imagining what's going to happen if something were different. What if questions. Okay. What if there were no tends to borrow from. Oh, no. What if we could take the square root of a

negative number? Imagine what if there's an A term that's not actually not that great of an opening question, but what if there were an a term for this quadratic? What if frog toed shoes and toad were in frog shoes? What if Hitler hadn't tried to invade Russia? What if we heat this up? We do these types of questions all the time, getting them to predict each one of these postures. Right? Notice, compare, create, struggle, and predict is

going to get kids wondering about the thing. Okay? And so when you walk into your classroom, I challenge you. See if you can put a question on the board or a ask them, say to say, all right, boys and girls, I've got a question for you guys. If that's the first words out of your mouth every single time, it's not going to work right away, right? They're not going to be used to this and be like, what? This is not math. But if every day you walk in and say, good morning, I've got a question for

you guys. Good morning. I've got a question for you guys. Good morning. I've got a question for you guys. Slowly they're going to start to realize that math is not just repeating and regurgitating the thing, but it's engaging in the act of mathematics. It's doing math. It's figuring stuff out. It's imagining, it's hypothesizing, it's playing around with stuff. You're

involving them in what mathematicians actually do. Okay. I do think this is then a thing that can become somewhat habitual, because you can't be inventing opening questions every single night, but you can just say, all right, say, I'm going to commit to doing this one thing every single day. Right? And I have five where you sit down to lesson plan. Maybe you have ten minutes, maybe you have five minutes

to plan for this thing. Is there a question I can ask that gets my kids either noticing, comparing, creating, struggling, or predicting? One thing that I like to do a little bit is to challenge myself not to write the key formula, definition, or process on the board until the students say it. I can't always do this. I fail a lot of times, but I like to challenge myself. Can I get them to tell me groups and in each group and total, can I get them to say

those words? Sometimes I fail, but at least I'm trying to do that type of thing. Can I get them to say we need a process that undoes the exponentiation? Can I get a student to say that? And then I say, yes, that's what we need. That's what we do. Okay, good. New talk, follow up questions. Where am I at? What time is it? We don't have time, do we? What time? We got time. I'm ignoring. Ignoring the clock. The good news is, follow up questions I think are easier than

opening questions. I think there's a lot of bad opening questions. I think there's very few bad follow up questions. Okay. And I mean that sincerely. Right. Where again, what we have here is this example from the Theaetetus. If you want to know what it means to just teach Socratically, if you've ever wondered what that word means, this is what it means to do this, because this is what Socrates does. Once more, Theaetetus, I repeat my old question

what is knowledge? Opening question. This is a struggle. Struggle question what is knowledge? There's no good answers to that, right? The knowledge is perception. That's a student response. And then what does Socrates do? He says, bravely said, that's the way you should express your opinion. Now let's examine together this conception of yours and see whether it's a true birth or a windegg. Windegg. This is just an empty shell with no. I don't know if you've ever gotten

one of those from a grocery store. Okay, good. Okay, so what we have here is this shape where I prompt student thinking. Usually with an opening question, I get a response and then I ask a follow up question based on that response. Right? So there's other options. I could say what's five plus eight? Someone could say 13 and I could say good and ask a different question. That's not a follow up question. It's five plus 813. How'd you figure that out? It's using the response as

the thing that you go back and forth. This is where you get a little bit of dialogue is the idea here. Okay, so. I'm going to skip through this a lot. Let's see why our follow up question important. They help students make sense of their experiences, right? They help you to think deeper about the thing. They ask you to dwell on meaningful ideas and problems, probe into the reasons for truth and not just truth themselves. We care about truth, especially in math, but we care about

the discovery of truth. Challenge students to provide explanations that go beyond one word answers. One of the best definitions of math. We've seen a couple of them. One of them we haven't seen. Paul Lockhart says mathematics is the art of explanation. Okay. Right. Get them explaining. You've already won. Right. If you're getting students explaining their thinking, you've already been victorious. I was I was doing a couple of years ago,

I was doing a model lesson in kindergarten. So I was I was doing a kindergarten and we were doing calendar time and I was having a blast. Calendar time is amazing, right? And we it was Tuesday or something, and we put the 14 in the slot of the calendar. It was the 14th. And so I asked, what number are we going to put in the Friday slot? Right. Okay, maybe not the best opening question, but a question. Right. And a girl raises her hand. She goes 17. I

was like, yes. How'd you figure that one out? And the look on her face, I'm never going to forget it. And she was like. How did I figure that out? Like, she had never realized that, like, you could have a thought and then attach another thought to that thought, right? It's this like it's this. Oh, I how did I figure that out? And she was like, she didn't say she didn't give a super long explanation. She didn't say, well,

it's three days away. And so I added 3 to 14 and she was like, well, we're going to put a 15 and then a 16 and then a 17. And I was like, what I've done is I've, as a kindergartener, I've got her to explain her thinking, I've already won. It doesn't matter what she says at that point, the fact that kindergartners can say, I have ideas and then I can explain how I get those ideas, that's a win. And it happens because we follow up on

student thinking, okay, good, we're open, struggling students. Best way to do this? I talk about this in just a sec, right? You don't know what's going on. 71 times 14, I don't know. Well, is it bigger? Less than one. Bigger. How do you know? Well, 14 is bigger than one. And I did a multiplication. Multiplication makes things bigger. Why? Right.

Because multiplication, I mean. Right. You're getting them to explain to you math because you've asked them a question and then asked them some follow up questions instead of just like 71 times 14. I don't know who can, who can, who can help, right? Turning away from that student rather than going deeper with that student. Okay, good. So situations for follow up questions. Number one. Silence. It's an important one.

Notice I said student responses. Silence is a response. If I ask a question and I get silence or I get, I don't know. That's a situation. For a follow up question. It's tempting to either go in a different direction or to. Answer the question yourself, but I don't know. So is a place where I would then ask a student a follow up question. I would dig in. What students need to know is, I don't know is a perfectly acceptable response in your classroom. I'm not willing to think about

it is not an acceptable response. Okay. There's a difference. If I say something and you say, I don't know, and then I ask you a follow up question about this. If I say something like, what's hard about this? That's a follow up question about what you don't know. You have to be willing to think about this with me, right? Put that rule on your on your wall somewhere. I don't know is an acceptable response. I'm not willing to think about it is not an acceptable response. Okay. Number

two wrong answers site for a follow up question. What's the temptation as a teacher? Fix it. Correct it. Right. Wrong answers. How do I dig in to that wrong answer? I will even solicit wrong answers in the classroom. If you've never done this in your classroom, put a problem on the board and say wrong answers only. It's kind of great. Put a clock on the board for your second graders. What time is it? Wrong answers. Only 781, right? Great. They say, what's the. And you say, what's the most

tempting wrong answer? And help them to articulate the errors that we might make as students. Right. They start to filter themselves if they're imagining the wrong answers. You want the wrong answers in the classroom. Wrong answers are gold. Wrong answers are the only times we are going to teach someone something. To go from wrong answer to right answer. Number three correct answers. What's the temptation here to move on? Great. Move on. I did it, yes. Done. Instead of saying,

how'd you figure that out? What was the key? How can we remember this? Notice the follow up questions that are there with a right answer. Lastly, a question from a student. What's the temptation? Answer it. Right. Especially if you get a good question. How do I reframe that question and turn it back on the on the student who asked it? And not in some cheesy way like, well, what do you guys think? Not a good follow up question. How do you turn that back on the student? Right.

Notice I think I've covered almost everything here. Right. Every situation can be done with a follow up question. Their only limits are time and exhaustion, both of which are legitimate concerns. Right? You only have so much time. You are going to get exhausted, right? But look, if you're seeing everything that students do as an opportunity, that's the key. Every single thing a student responds. That's an opportunity for me to ask about what's going on, right? Okay. So

types of follow up questions. Here's how you do them. Number one. Most basic, you press for explanation. Pressing for explanation. Basically, it's saying, why did you respond the way that you responded? Right. That's all that question is. You got to be a little careful. You don't seem like a jerk. Or you're just like, ask a question, I get silenced. I'm not going to be like, well, why are you guys silent

on this? That's not what you want. But I can ask a question and I get silenced and I can say, oh, that's good. What's what's hard about this question? It's basically saying, explain to me why you guys are silent here. What's genuinely hard about this? Right? I think math is hard. I think we should lean into that. Okay. Number two comparison. We saw this as an opening question. It's actually a really good follow up question as well, juxtaposing it with

another thing. For example, let's try a wrong answer. If I say five plus three and someone says seven, I can say, well, five plus two is seven. What's five plus three? That's comparison follow up question. I'm saying compare it with this and rethink, compare and rethink. It's not the best way to respond to wrong answers, but it's a possibility. Okay. Number three, hook, hook question goes back to something that's basic and then scaffolds it back up. Use a basic response as a launching pad for a

deeper question. Often it's a thing that you asked, you got silence or you got 1 or 2 hands. 1 or 2 hands in a math class is silence. Okay, you get 1 or 2 hands. Don't call on 1 or 2 of those guys. Don't do it. It tells every other kid they can turn off their brains. Okay, if I get 1 or 2 hands, I'm going to say, oh yeah, this is a hard one. Is it bigger or less than one? I like the idea is go to a basic one and build it back up. We saw this today. Doctor Treloar did this beautifully. Right. How

do you divide by a fraction? Silence. Okay. What's addition? What? Go to a basic one and build it up. What are the different ways we think of addition? What are the different ways you think of subtraction? Right. Build it back up. Opening question. Follow up question. Okay. Number four. Ramification. Probing the implications of this response. How do we apply it? What does it matter? Why does it make sense? What's

the key? Are all problems like this? You could see the the potential follow up questions here about what this means for us. So what I've done is this. Okay, great. So what we have here are the situations on this axis and the kinds on this axis. And then just some examples. Not all of all these work, but just to show you there's so many different follow up questions

that I think we can do for every single situation. Now, don't print this off and put it on a little clipboard and be like, well, this is a wrong answer. So now I'm going to use a hook, right? Don't do that. That's not the point of this exercise. But the point is just to say every single situation can be responded to with every single kind of question. That's not like all it is, is it's asking students to think more. You've got to develop your habit of this,

which is easier. Fingers are pets. Fingers. Why do you think fingers? What's hard about that? What's easy about that? Pets. Why do you think pets. It's just getting in the habit of seeing student responses as opportunities. Right. Okay, so some of my favorite ones, I'll put these on the board for a sec. Uh, I think why is this hard is a really good one. What's the key? What's your strategy? All of these things students love talking about strategies.

My six year old, I play a lot of war with him, and he's always wanting to tell me this strategy for the game. I'm like, it's a pure game of luck, you moron, right? I don't call my son a moron. I'm always eager to hear his strategy. He's like, whenever you play a nine, I'm gonna try to play a ten. I'm like. Great. How's that going for you? Yeah. No, he's a good kid. He's going to need to get some athletic scholarships, but he's going to be a good kid. Okay.

Some that I don't really like up here. Right. Anyone have any questions? Not great. Just doesn't get much traction, right? How do you feel? I don't really care. Okay, good. So how do you habituate this one? I think if you're thinking about ways to prep for this. I don't think you need to prep for it as much. Right. The follow up questions write themselves a lot of times as long as you start getting the habit of it, but including an anticipated student responses section, what my students

say and how might I respond? Right. Number two, think about the different situations. Test yourself as you're thinking about your own classroom. How do I respond to silence? Do I do it well? Do I do it poorly? How do I respond to wrong answers? Try to make some renewed habits to respond to some of these in different ways.

Specifically with this kind of Socratic pedagogy, right? One thing that I find really helpful is to count the number of questions that I ask that can be answered with a single word, and then count the number of questions that beg a sentence. At least one of my absolute proudest moments as a teacher. I was teaching sixth grade, and I don't know, I forget what question I asked, but a kid raised his hand and he gave an answer.

He's like, I think it's seven. And because you always make us explain how we got it, here's how I got it. And I was like, yes, I have broken them. Right? But that's the idea. You want to do this so habitually that they get in the habit of explaining their thinking, right? That's the thing that you're interested in, right? They one word answers are good, but we have to get students talking about the math. Notice, notice this is hard, right?

This is hard as a teacher, right? It requires actually more teacher involvement to teach Socratically a few takeaways here. Number one, a deeply rooted understanding of and love of mathematical truth requires an exploration of the mystery that surrounds it. Okay. Number two Questions with opening and follow up are the single most important thing that distinguishes a wonder based pedagogy. If you're interested in wonder, you want to try to

teach socratically. Okay. Number three. I think this wonderful teaching is going to require some habituation. The habit of teaching with wonder is a shape. It's not a formula, but it is a repeatable shape. That thing is repeatable. It's new every day, but it's a repeatable shape. It's a thing that students can become habituated to. Number two, I think you can build a culture where students are going to respond to questions. This requires perseverance, but students can

get in the habit of doing it. They know that you're going to be asked to explain their thinking, to do mathematics. Teaching is always inventive, but I think it adheres to a set of questions that can become ingrained in teaching practice. Let me finish with the end of this quote. Right. The capacity to wonder is among man's greatest gifts. It signifies that the world is profounder, more all embracing and mysterious. And the logic of everyday reason had taught us to believe.

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