WEBVTT - Principles of Pedagogy in Mathematics 0:00:04.960 --> 0:00:10.120 Welcome to the Hillsdale College K-12 Classical Education podcast, bringing 0:00:10.119 --> 0:00:13.920 you insight into classical education and its unique emphasis on 0:00:13.920 --> 0:00:19.200 human virtue and moral character, responsible citizenship, content, rich curricula, 0:00:19.200 --> 0:00:23.400 and teacher led classrooms. Now, your host, Scott Bertram. 0:00:23.920 --> 0:00:28.520 Thanks for listening. The Hillsdale College K-12 Classical Education podcast 0:00:28.520 --> 0:00:35.720 is part of the Hillsdale College Podcast Network. More episodes@podcast.hillsdale.edu 0:00:35.840 --> 0:00:38.760 or wherever you get your audio. You also can find 0:00:38.760 --> 0:00:41.919 more information on topics and ideas discussed on this show 0:00:41.920 --> 0:00:48.680 at our website, k12.hillsdale.edu. Great teachers don't merely know and 0:00:48.680 --> 0:00:52.960 love their content. They captivate students with it. The best 0:00:52.960 --> 0:00:57.000 lessons are full of wonder, engagement, and curiosity. But it 0:00:57.000 --> 0:00:59.860 can be daunting to try to replicate these moments over 0:00:59.860 --> 0:01:03.580 and over again with some initial work and lots of practice. 0:01:03.700 --> 0:01:07.100 Teachers can infuse their class time with these elements without 0:01:07.140 --> 0:01:11.740 running in circles. Doctor Jonathan Gregg, assistant professor of education 0:01:11.740 --> 0:01:15.940 at Hillsdale College, spoke on how to infuse your pedagogy 0:01:15.940 --> 0:01:19.700 with wonder at a September 2025 center for Teacher Excellence 0:01:19.700 --> 0:01:23.300 event on the Art of teaching mathematics. Though his ideas 0:01:23.300 --> 0:01:27.020 are presented in the context of mathematics, countless applications can 0:01:27.020 --> 0:01:31.220 be found for every classroom subject. Here's Doctor Gregg. 0:01:31.340 --> 0:01:34.979 I'm going to talk about pedagogy. Pedagogy is not easy. 0:01:35.380 --> 0:01:40.979 The Greek of pedagogy just means a child leader. And 0:01:41.100 --> 0:01:44.500 children don't like to be led. I have some of them. 0:01:44.500 --> 0:01:49.500 They're bad at at following. And so pedagogy is necessarily hard. 0:01:49.660 --> 0:01:55.100 It's complex. It's different every day. And it's it's hard 0:01:55.100 --> 0:01:58.310 to kind of what you. The difficulty, I think, in 0:01:58.310 --> 0:02:01.710 pedagogy is that as teachers, we search for something stable. 0:02:02.150 --> 0:02:05.390 We want something. We want to eliminate the chaos in 0:02:05.390 --> 0:02:09.790 the classroom, right? There is you deal with enough chaos 0:02:09.790 --> 0:02:12.710 every day. And so you want something to rely on. 0:02:12.710 --> 0:02:14.950 But pedagogy has to be new. It has to be exciting. 0:02:14.950 --> 0:02:17.070 It has to be something that that you can't just 0:02:17.070 --> 0:02:19.109 repeat day after day after day after day, because then 0:02:19.110 --> 0:02:22.470 it becomes boring and we lose. We lose the sense 0:02:22.470 --> 0:02:25.030 of wonder here. And so I want to try to 0:02:25.190 --> 0:02:26.990 inhabit that space a little bit today to think about 0:02:26.990 --> 0:02:29.910 how do you keep things exciting? How do you keep 0:02:29.910 --> 0:02:35.550 things wonderful without making making yourself run in circles wildly 0:02:35.950 --> 0:02:38.829 or making yourself plan for hours and hours every night? 0:02:38.870 --> 0:02:42.070 How do you how do you do something that's habitual 0:02:42.430 --> 0:02:44.350 and yet still wonderful? This is kind of the thing 0:02:44.350 --> 0:02:46.070 that I'm interested in here. I think that's the core 0:02:46.070 --> 0:02:49.150 of core of pedagogy because. So I'll start with a 0:02:49.150 --> 0:02:51.070 quote as well. This is, you know, one of those 0:02:51.070 --> 0:02:53.230 quotes that you read and you just you can't get 0:02:53.230 --> 0:02:55.670 it off of your mind. It kind of haunts you. 0:02:55.889 --> 0:02:57.250 I read this, I read this a few years ago, 0:02:57.290 --> 0:03:00.050 and I think I think it's it haunts me. Right. 0:03:00.090 --> 0:03:05.730 The capacity to wonder is among man's greatest gifts. It's 0:03:05.730 --> 0:03:08.010 so true. And if you think back, even. Just think 0:03:08.010 --> 0:03:10.490 back to your good lesson. Think back to a good 0:03:10.490 --> 0:03:14.290 lesson that you did. I guarantee you wonder showed up somewhere. 0:03:14.450 --> 0:03:16.690 And actually, this is the one place where I do 0:03:16.690 --> 0:03:20.410 think students will and children will follow. They'll follow what 0:03:20.410 --> 0:03:25.490 they wonder about. They'll follow what, uh, what's wonderful. Um, 0:03:25.490 --> 0:03:29.050 and so I think this quote is if we're looking 0:03:29.050 --> 0:03:32.970 for a, a beginning of pedagogy, a first philosophy of pedagogy, 0:03:32.970 --> 0:03:34.609 I think this has got to be it. People are 0:03:34.650 --> 0:03:36.370 people actually goes on. And he says this, he says, 0:03:36.370 --> 0:03:40.290 really the situation is this, the deeper aspects of reality. 0:03:40.570 --> 0:03:43.450 That's what we want students to get are apprehended in 0:03:43.450 --> 0:03:45.930 the ordinary things of everyday life and not in the 0:03:45.930 --> 0:03:47.890 sphere cut off and segregated from it. The sphere of 0:03:47.890 --> 0:03:49.810 the essential or whatever it may be called. It's in 0:03:49.810 --> 0:03:52.170 the things we come across in the experience of everyday 0:03:52.170 --> 0:03:55.700 life that the unusual emerges and we no longer take 0:03:55.700 --> 0:03:58.900 them for granted. In that situation corresponds with the inner experience, 0:03:58.900 --> 0:04:01.619 which has been regarded as the beginning of philosophy, the 0:04:01.620 --> 0:04:04.580 act of marveling, kind of see how this is going 0:04:04.620 --> 0:04:11.060 to work? You present something. It's unusual. Students wonder at it, 0:04:11.260 --> 0:04:14.220 and then they start to think. And that's philosophy. And 0:04:14.220 --> 0:04:16.140 this is where where you can kind of see how 0:04:16.140 --> 0:04:19.099 wonder is going to be the thing that's possible as 0:04:19.100 --> 0:04:21.659 a proof here. And honestly, we can't we can't go 0:04:21.660 --> 0:04:24.940 too far in math without talking about about wonder as 0:04:25.220 --> 0:04:28.100 a maybe not a proof, but at least some evidence 0:04:28.100 --> 0:04:29.420 for it. I have to be careful using the word 0:04:29.420 --> 0:04:32.260 proof around around this, this crew, you guys have some 0:04:32.260 --> 0:04:34.860 high standards for this, but at least as some evidence. Right? 0:04:34.860 --> 0:04:37.060 Here we go. Here are some quotes on wonder. We 0:04:37.060 --> 0:04:39.980 want students. We want to get students to wonder about hypotheses. 0:04:40.940 --> 0:04:46.180 This was Doctor Dave Gabler four hours ago. Mathematics should 0:04:46.180 --> 0:04:49.540 instill in students an ever increasing sense of wonder. This 0:04:49.540 --> 0:04:54.640 was doctor Tom Treloar three hours ago and we mathematicians 0:04:54.640 --> 0:04:56.240 like to wonder if there's anything else we can do 0:04:56.240 --> 0:04:59.200 with this. This was Doctor David Murphy. Not even one 0:04:59.200 --> 0:05:02.280 hour ago. All right, so these guys are in here 0:05:02.279 --> 0:05:06.640 talking about wonder, right. I wasn't in Miss Jackson's talk 0:05:06.640 --> 0:05:09.320 but I guarantee you wonder showed up. I'm gonna go 0:05:09.320 --> 0:05:11.400 back and look at the transcript. But it's there, right? 0:05:11.440 --> 0:05:14.600 It's there. But what I can't do is stand up 0:05:14.600 --> 0:05:19.320 here and say, go get your students to wonder. That's 0:05:19.320 --> 0:05:21.840 not a thing we can do. That's that's not helpful 0:05:21.839 --> 0:05:24.880 as teachers. So the real question is how do we 0:05:24.880 --> 0:05:28.239 get students to wonder and how do we habitually do that? 0:05:28.680 --> 0:05:30.760 This is the question for today. It's not an easy question. 0:05:30.760 --> 0:05:33.359 How do we habitually get students to wonder? Because I 0:05:33.360 --> 0:05:36.000 think if we can, we're already doing the kind of 0:05:36.040 --> 0:05:38.719 things that we should be doing in our math classroom, right? 0:05:38.760 --> 0:05:42.320 You've already kind of won as a teacher. Wonder leads 0:05:42.320 --> 0:05:45.720 to philosophy. So what is wonder and how do we 0:05:45.720 --> 0:05:50.640 do it? Okay, let's go to Peter one more time. Socrates. 0:05:50.700 --> 0:05:54.460 is talking about Socrates here. He's actually about the Titus. 0:05:55.020 --> 0:05:58.140 Titus is a dialogue between Socrates and a mathematician. All right. 0:05:58.180 --> 0:06:01.420 We saw Theaetetus actually show up on Doctor Murphy's slides here. 0:06:01.660 --> 0:06:04.780 And he says Socrates brings Theaetetus to the point of 0:06:04.779 --> 0:06:08.900 admitting his ignorance with his shrewd and kindly but staggering 0:06:08.900 --> 0:06:13.620 and astonishing questions, questions that stagger and astonish with wonder. 0:06:13.980 --> 0:06:15.780 And there, for the first time in the theater, without 0:06:15.779 --> 0:06:18.260 solemnity or ceremony, almost. By the way, though fresh as 0:06:18.260 --> 0:06:20.300 dawn appears, the thought that has become commonplace in the 0:06:20.300 --> 0:06:23.180 history of pedagogy. The beginning of philosophy is wonder. You 0:06:23.180 --> 0:06:27.700 see it again here. But notice what he's doing with wonder. 0:06:28.100 --> 0:06:33.219 Saying look wonder happens in two ways. Yes, it's kind 0:06:33.220 --> 0:06:36.740 of staggering and astonishing thing. So it's actually a really 0:06:36.740 --> 0:06:38.380 lovely word, right? One of the best words in the 0:06:38.380 --> 0:06:41.380 English language are the words that we equivocate on, that go, 0:06:41.620 --> 0:06:45.419 that go that have two different definitions, right? Think of 0:06:45.420 --> 0:06:49.630 the word witness is one of my favorite examples here. Witness. 0:06:49.670 --> 0:06:52.390 Witness is a beautiful word, right? It means both to 0:06:52.589 --> 0:06:56.150 observe something taking place and to share about the thing 0:06:56.150 --> 0:06:58.150 that you saw, right? And the fact that we use 0:06:58.150 --> 0:07:00.950 the same word for it means when you see something, 0:07:00.950 --> 0:07:04.550 you are obligated. You have a responsibility to share it. Actually, 0:07:04.550 --> 0:07:07.870 what you guys are doing, right, you're witnessing to your students. Okay. Anyways, 0:07:07.910 --> 0:07:10.110 I'm off track. I'm off track. But wonder is the 0:07:10.110 --> 0:07:15.150 same thing, right? Where questions that stagger and astonish with wonder. Right. 0:07:15.310 --> 0:07:19.630 First of all, wonder means that I'm staggered. I don't 0:07:19.630 --> 0:07:24.670 know something. Right. I recently wondered how to put Freon 0:07:24.670 --> 0:07:28.310 in my car because I didn't know how to do it, 0:07:28.310 --> 0:07:30.390 and it was 120 degrees in my car and I 0:07:30.390 --> 0:07:32.790 was upset. I don't know how to do this. And 0:07:32.790 --> 0:07:35.270 I was staggered and I wondered, and then what did 0:07:35.270 --> 0:07:38.630 I do? I did what any self-respecting person would do. 0:07:38.670 --> 0:07:43.230 Got on YouTube right. And figured it out. But there's wonder. 0:07:43.310 --> 0:07:46.950 We are staggered. We then go and figure the thing out. 0:07:46.950 --> 0:07:49.090 And that's where learning takes place. But first you have 0:07:49.090 --> 0:07:52.490 to be dislocated, right? I wonder about this. I wonder 0:07:52.490 --> 0:07:55.290 what's going on. I'm confused. It's almost like you're you're 0:07:55.290 --> 0:08:02.130 you're dislocated. You're shocked. Socrates actually calls himself a torpedo fish, 0:08:03.090 --> 0:08:06.730 which is an electric eel. Apparently that the idea is 0:08:06.730 --> 0:08:09.770 he's a he's just shocking people. He's like, what about this? 0:08:09.810 --> 0:08:12.890 What about this? And you're shocked. I think students you 0:08:12.890 --> 0:08:17.610 can shock students into wondering. But also this astonishment. Say, oh, 0:08:17.650 --> 0:08:21.530 I wonder at that thing that's marveling. It's this, it's 0:08:21.530 --> 0:08:24.450 this or that students can have as well. So we 0:08:24.450 --> 0:08:29.010 see this dual understanding of wonder, right. And an invocation 0:08:29.010 --> 0:08:31.050 of the Socratic method. So let's piece some of this 0:08:31.050 --> 0:08:33.610 together and think about what pedagogy looks like. So I 0:08:33.610 --> 0:08:42.770 present to you pedagogy. Thank you. Pedagogy. I'll take your question. No, 0:08:44.730 --> 0:08:46.860 I need some labels. Yes, we need some labels. Okay, 0:08:46.900 --> 0:08:48.780 so don't take the pictures yet. Wait for the labels. 0:08:48.780 --> 0:08:50.540 Here we go. Actually, don't take any pictures. Just email 0:08:50.580 --> 0:08:52.260 me for the slides. But let's get some labels here 0:08:52.260 --> 0:08:54.540 because this shape is meaningless. This is not a meaningful shape. 0:08:54.540 --> 0:08:58.740 This is not a polyhedron. Right? So what's going on here? 0:08:58.740 --> 0:09:01.500 Number one. Right. I have a lightning bolt. I think 0:09:01.500 --> 0:09:04.620 this is what we think of as an opening question. Right? 0:09:04.740 --> 0:09:08.380 I'm envisioning the teacher kind of as as Zeus here, right? 0:09:08.820 --> 0:09:13.020 Chucking a lightning bolt in the middle of students. And 0:09:13.020 --> 0:09:14.780 what's going to happen is they're going to this is 0:09:14.780 --> 0:09:16.979 the shock part of wonder, right? You're going to be dislocated. 0:09:16.980 --> 0:09:19.459 You're not going to understand the thing. Right? And you 0:09:19.460 --> 0:09:22.860 can see these blue lines. This is ideally what student 0:09:22.860 --> 0:09:26.100 responses are going to be that students can go have 0:09:26.140 --> 0:09:30.420 a bunch of different potential ideas out of this wonder. 0:09:30.700 --> 0:09:34.140 Wonder leads to this kind of exploration, right? And then 0:09:34.140 --> 0:09:36.660 these kind of curlicue lines, this is what I'm calling 0:09:36.700 --> 0:09:43.300 follow up questions that select different student responses and then 0:09:43.300 --> 0:09:46.640 follow up on them until we get to the green thing. 0:09:46.679 --> 0:09:49.600 This is the truth. Okay. The thing we want to 0:09:49.600 --> 0:09:51.800 teach the learning goal, if you will. What you want 0:09:51.840 --> 0:09:54.680 to teach the student. Okay. And then what we have 0:09:54.679 --> 0:09:57.440 are these maybe some examples, maybe some practices, some applications. 0:09:57.440 --> 0:09:59.600 Once we learn this thing, then we practice it, then 0:09:59.600 --> 0:10:02.160 we do it. Then we. But I think the idea 0:10:02.200 --> 0:10:06.800 here is that you, uh, this is kind of how 0:10:07.040 --> 0:10:11.880 one might learn in the, in the field of wonder 0:10:12.200 --> 0:10:15.520 or using wonder where you're invoking the Socratic method, where 0:10:15.520 --> 0:10:21.200 there's both shock, right? Staggering. And then there's potentially astonishment 0:10:21.200 --> 0:10:24.960 down here. Okay. I think every good lesson, or at 0:10:24.960 --> 0:10:26.600 least a lot of good lessons are going to follow 0:10:26.600 --> 0:10:29.520 something like this in a math classroom. Okay. And maybe 0:10:29.559 --> 0:10:32.640 here's one way of, of proving, again, this is not 0:10:32.640 --> 0:10:34.840 a proof, but some evidence for this. If we look 0:10:34.840 --> 0:10:39.640 at the dominant pedagogical approaches that we are surrounded with, 0:10:39.640 --> 0:10:43.540 I think there's two dominant pedagogical approaches. I think what 0:10:43.540 --> 0:10:46.460 you might call the progressive and what you might call 0:10:46.460 --> 0:10:51.380 the dogmatic, the dogmatic, I think, is this here's the truth. 0:10:51.380 --> 0:10:53.420 Let me hand you the formula. Let me hand you 0:10:53.420 --> 0:10:55.620 how to do it. Practice it over and over and 0:10:55.620 --> 0:10:58.860 over and over again until it gets drilled into your face. Okay, 0:10:59.179 --> 0:11:02.860 that's the dogmatic pedagogy. Or you have progressive pedagogy, which 0:11:02.900 --> 0:11:06.980 kind of throws out an idea and then says, go students, explore, 0:11:07.340 --> 0:11:10.620 be empowered, go to your heart's content. Right? But it 0:11:10.620 --> 0:11:14.020 doesn't really believe that there's any truth to be found, right? 0:11:14.100 --> 0:11:17.140 It's about whatever students want to do, right? And it 0:11:17.140 --> 0:11:20.540 just cuts off the top half of this versus cutting 0:11:20.540 --> 0:11:23.980 off the bottom half of this. Right? But somehow in 0:11:24.020 --> 0:11:27.819 between these two extremes, there's something holistic where we have 0:11:27.820 --> 0:11:30.620 wonder at the beginning and wonder at the end from 0:11:30.620 --> 0:11:33.900 wonder to wonder, right? That's, I think, what a pedagogical 0:11:33.900 --> 0:11:39.140 approach looks like. But this shape is, is just a shape. 0:11:39.140 --> 0:11:42.910 Let me illustrate. All right. So I'm going to teach 0:11:42.910 --> 0:11:45.870 you guys. You guys are students. I will be teacher. 0:11:47.230 --> 0:11:50.670 And and I'm going to teach you guys pretend you're 0:11:50.670 --> 0:11:53.030 first graders. This is a first grade class. Okay, good. 0:11:53.590 --> 0:11:55.750 Good morning, boys and girls. I'm really excited to be 0:11:55.750 --> 0:11:58.190 with you guys today. Guess what. I want to teach 0:11:58.190 --> 0:12:01.750 you guys a new word. Can you guys say this 0:12:01.750 --> 0:12:05.910 with me? Say this after me. The word is multiplication. 0:12:05.950 --> 0:12:10.750 You guys say that with me. Very good. Multiplication. Now 0:12:10.790 --> 0:12:20.270 multiplication is repeated. Addition multiplication is very good. So if 0:12:20.270 --> 0:12:25.470 I have a problem like three plus three plus three, 0:12:25.630 --> 0:12:30.150 plus three plus three, that's a lot of threes, don't 0:12:30.150 --> 0:12:33.830 you think? My hand is almost tired writing up all 0:12:33.830 --> 0:12:36.950 of these threes. Right. So would you guys like to 0:12:36.990 --> 0:12:40.650 learn a better way for writing that I think we would. 0:12:40.690 --> 0:12:47.010 So I can write this as five and then I 0:12:47.010 --> 0:12:51.050 need a new symbol. I'm not going to use addition anymore. 0:12:51.090 --> 0:12:55.210 Can everyone make me the symbol of addition with your hands? Right. Okay. 0:12:55.929 --> 0:12:57.569 I need a new symbol. So what we're going to do? 0:12:57.610 --> 0:13:08.330 Hold your addition. Oh. New symbol. Right? We have an X5X35X3. 0:13:08.450 --> 0:13:11.010 And just like I have a word for this, what 0:13:11.010 --> 0:13:14.209 word do I use for this plus right. I'm going 0:13:14.250 --> 0:13:17.730 to use a new word for my multiplication. I'm going 0:13:17.770 --> 0:13:24.170 to use times. Say it with me. So five times 0:13:25.050 --> 0:13:30.770 three because it's three five times. Some of you guys 0:13:30.770 --> 0:13:35.370 are legitimately like, oh wow. Okay, so can we figure 0:13:35.370 --> 0:13:38.590 out what Three five times three is. Can you guys 0:13:38.590 --> 0:13:46.230 count with me? Three. Six. Nine. 1250. Very good. Five 0:13:46.270 --> 0:13:49.270 times three is equal to 15. And you guys just 0:13:49.270 --> 0:13:55.589 did your first multiplication problem. Okay? And cut and cut. Okay, 0:13:55.630 --> 0:13:59.830 so we can keep going, right. But what kind of 0:13:59.870 --> 0:14:05.590 pedagogy was that? Right. That's dogmatic pedagogy. Okay. It's entertaining, 0:14:05.830 --> 0:14:09.750 but it's handing you the four way to do this. 0:14:09.750 --> 0:14:11.750 Here's how to do this. Here's what the words mean. 0:14:11.790 --> 0:14:14.510 I'm handing you this thing. Is it good? Do I 0:14:14.510 --> 0:14:16.829 need to teach kids this? Yes. Is this where I 0:14:16.830 --> 0:14:20.830 want to start? No. Right. Because if you're first graders, 0:14:20.950 --> 0:14:22.270 I can walk in and say, I'm going to teach 0:14:22.270 --> 0:14:24.750 you guys a new word, multiplication. But if I walk 0:14:24.750 --> 0:14:26.470 into my ninth grade class, I'm going to teach you 0:14:26.470 --> 0:14:36.800 guys a new word logarithm. Mr. Greg, why we don't care, right? Okay, 0:14:36.840 --> 0:14:40.120 so and slowly. This is why students start to hate 0:14:40.120 --> 0:14:42.200 math is because we teach it pretty. One of the 0:14:42.200 --> 0:14:46.440 reasons we teach it pretty dogmatically. And there. There's no 0:14:46.440 --> 0:14:50.280 buy in, right? There's no there's nothing that they're doing mathematically. 0:14:50.280 --> 0:14:54.440 They're not actually engaging in mathematics. They're just receiving the 0:14:54.440 --> 0:14:58.120 results of mathematics. Right. Doctor Murphy said mathematics is a 0:14:58.120 --> 0:15:02.360 human activity. We need to get them doing the mathematics. Okay. Good. 0:15:02.400 --> 0:15:05.080 Let me try again. Good morning boys and girls. I'm 0:15:05.080 --> 0:15:07.880 really excited to be with you guys today. Go ahead 0:15:07.880 --> 0:15:11.640 and take out your devices. I've preloaded the module to 0:15:11.680 --> 0:15:15.280 explore today's lesson. It's called multiplication, but you guys go 0:15:15.320 --> 0:15:18.280 ahead and explore on your own. Figure it out, see 0:15:18.280 --> 0:15:20.560 what you can make of what's going on. And I'll 0:15:20.560 --> 0:15:22.840 just be over here in case you need to answer 0:15:22.840 --> 0:15:25.600 any questions. And you guys can go ahead and explore 0:15:25.600 --> 0:15:27.680 and we'll talk at the end just to see. We'll 0:15:27.680 --> 0:15:29.280 share how we'll do a little share out to see 0:15:29.280 --> 0:15:35.100 what people thought of your chance to explore And cut. Okay, 0:15:35.140 --> 0:15:38.340 so what kind of progressive what kind of pedagogy here, right? 0:15:38.380 --> 0:15:41.300 This is what progressive pedagogy looks like or feels like. 0:15:41.340 --> 0:15:43.420 It's not. There are other ways to do this, but 0:15:43.420 --> 0:15:46.580 really what it does is it the goal is for 0:15:46.580 --> 0:15:49.820 the teacher to fade into the background, to have students 0:15:49.820 --> 0:15:53.260 do the exploration on their own, right? The teacher doesn't 0:15:53.260 --> 0:15:56.060 get to have a role to play that this is 0:15:56.980 --> 0:16:00.340 that it's it's actually oppressive for me to impose my 0:16:00.340 --> 0:16:04.740 truth upon you. Right. And I think this, as you 0:16:04.740 --> 0:16:08.020 guys know, it's it just doesn't work, right. Students don't 0:16:08.540 --> 0:16:10.740 aren't able, by and large to do this. And what 0:16:10.740 --> 0:16:13.340 it does, I think, is it bifurcates students into those 0:16:13.340 --> 0:16:15.020 who can figure it out on their own and those 0:16:15.020 --> 0:16:17.580 who can't. Right. I think one of the one of 0:16:17.580 --> 0:16:23.140 the really unspoken things of progressive pedagogy is that it's 0:16:23.140 --> 0:16:28.060 attempt to kind of empower all students equally. The pedagogy 0:16:28.220 --> 0:16:31.180 is the thing that ends up bifurcating the students into 0:16:31.180 --> 0:16:35.070 the haves and the have nots. It's really problematic, actually. 0:16:35.110 --> 0:16:38.390 The methods of progressive pedagogy undermine their stated goals, in 0:16:38.390 --> 0:16:40.110 my opinion. I can talk more about this if you want, 0:16:40.150 --> 0:16:43.470 but okay, let me try one more time. Morning boys 0:16:43.470 --> 0:16:45.670 and girls. I've got a question for you guys today. 0:16:45.990 --> 0:16:48.750 Which of these two problems is easier to figure out 0:16:48.750 --> 0:16:51.870 how many fingers we have as a class, or to 0:16:51.870 --> 0:16:53.790 figure out how many pets we have as a class. 0:16:54.670 --> 0:16:57.870 You guys are bad first graders. Fingers. Pets. Fingers. Pets. 0:16:58.270 --> 0:17:02.030 My brother has a cat. Okay. Right. But this is 0:17:02.030 --> 0:17:04.590 what you want. This is what you want. Okay, so, fingers, 0:17:04.910 --> 0:17:07.030 let's try the pets one. What do I have to 0:17:07.030 --> 0:17:09.790 do to figure out the pets? What do I have 0:17:09.790 --> 0:17:12.070 to do? I've got to ask people. Right. Okay, good. 0:17:12.070 --> 0:17:14.949 Let's ask. How many pets do you have? Five. How 0:17:14.950 --> 0:17:17.350 many pets do you have? Ten. How many do you have? 0:17:17.390 --> 0:17:21.750 Does my little sister count as a pet? No. Fast forward. Pets, pets. 0:17:23.510 --> 0:17:26.270 Now I've got all everyone's pets on the board. Now, 0:17:26.270 --> 0:17:28.270 how do I figure out how much we have all together? 0:17:29.390 --> 0:17:30.970 I have to use addition. Why do I have to 0:17:30.970 --> 0:17:35.169 use addition? Ah, because we're sitting all together. Good. So 0:17:35.170 --> 0:17:40.650 let's add fast forward. We have 312 pets as a class. Whew! 0:17:40.970 --> 0:17:42.370 That is a lot of pets. I hope we don't 0:17:42.369 --> 0:17:45.130 bring them in here. So let's try the fingers one. 0:17:45.290 --> 0:17:46.930 How do I figure out how many fingers we have 0:17:46.970 --> 0:17:49.609 as a class? Can I do the same way? Can 0:17:49.609 --> 0:17:52.169 I do it the same way? Yes. No. Who thinks yes? 0:17:52.530 --> 0:17:55.650 Who thinks no? Why do you think? Yes. Sure. Why 0:17:55.650 --> 0:17:59.170 do you think no? Okay, okay. So let's try. Let's 0:17:59.170 --> 0:18:01.369 try it the same way here. Right. How many fingers 0:18:01.369 --> 0:18:03.810 do you have? Ten. How many things do you have? 0:18:04.130 --> 0:18:07.810 I have eight. Two of them are thumbs, right? Yeah. Okay, good. Yeah. 0:18:07.850 --> 0:18:09.889 This is what you. But again, this is what you 0:18:09.890 --> 0:18:11.530 want in the classroom. You want a little drama, you 0:18:11.530 --> 0:18:16.330 want a little energy, right? Yeah, right. And I get 0:18:16.330 --> 0:18:18.890 a couple people in here. Do I need to keep 0:18:18.930 --> 0:18:25.330 asking people? I don't why not? Everyone has the same 0:18:25.330 --> 0:18:30.550 number of fingers. That's interesting. interesting. What do I need 0:18:30.550 --> 0:18:35.030 to know to solve this problem? How many people there are? 0:18:35.070 --> 0:18:39.990 I need to know that. And how many fingers. If 0:18:39.990 --> 0:18:46.310 someone's missing a finger. Yeah. How many fingers each person has. Right. 0:18:46.470 --> 0:18:50.030 This is a different kind of problem, right? So. Okay. 0:18:50.070 --> 0:18:52.510 And I'll cut there because now what am I going 0:18:52.550 --> 0:18:55.990 to do? I'm going to do I'm here now, right. 0:18:56.430 --> 0:18:59.149 And now I'm going to say like, let's write up 0:18:59.150 --> 0:19:02.910 the multiplication problem. What you guys just did with the pets. 0:19:02.950 --> 0:19:06.710 That's multiplication. That's not addition. Something new right? Okay. So 0:19:07.270 --> 0:19:09.550 go ahead and cut. And you could see how a 0:19:09.550 --> 0:19:12.590 lesson looks like when you walk in and say which 0:19:12.590 --> 0:19:18.270 is easier. Fingers or pets? Lightning bolt question right. No 0:19:18.270 --> 0:19:21.190 one knows what's going on here. They're like, what? This 0:19:21.190 --> 0:19:23.550 is math class. You're talking about fingers and pets. I 0:19:23.590 --> 0:19:27.320 got reactions. I at least got two reactions, right? I 0:19:27.359 --> 0:19:32.080 got fingers and I got pets. And both are defensible pets. 0:19:32.119 --> 0:19:35.480 Probably there's fewer pets than fingers. It would be easier, right? 0:19:35.760 --> 0:19:37.960 Then what did I do? Notice the kind of pedagogy 0:19:37.960 --> 0:19:42.240 that I employed. What did I do? Did I say 0:19:42.240 --> 0:19:46.359 a single statement? I didn't say a statement. One time. 0:19:46.359 --> 0:19:49.080 I say one statement. Right. This is what the Socratic 0:19:49.080 --> 0:19:53.760 method looks like. By asking, right. Picking the two responses 0:19:53.760 --> 0:19:56.400 and asking a bunch of follow up questions, I was 0:19:56.400 --> 0:19:59.840 able to get them to tell me the things that 0:19:59.840 --> 0:20:03.240 we need for a multiplication problem, namely, namely number of 0:20:03.240 --> 0:20:06.880 groups and number in each group. That's what multiplication is. 0:20:07.200 --> 0:20:09.080 We need two things and we can figure out the 0:20:09.080 --> 0:20:12.679 total that way. But they're the ones that did that thing. 0:20:12.720 --> 0:20:16.960 They engaged in the activity of multiplication before. And we 0:20:16.960 --> 0:20:19.280 didn't just kind of let it be there. And then 0:20:19.280 --> 0:20:21.840 we're going to formalize it into what what we actually 0:20:21.840 --> 0:20:24.640 want and practice it like we would like we would normally, 0:20:24.640 --> 0:20:27.940 but we can see what we you know, we by 0:20:28.140 --> 0:20:31.260 taking this Socratic method. And is there wonder at the end. 0:20:31.300 --> 0:20:34.419 I don't know, but at the very least they should 0:20:34.420 --> 0:20:42.340 feel that, oh, in this world there are multiplicative situations 0:20:42.660 --> 0:20:46.060 and additive situations. Is a fourth grader being able to 0:20:46.060 --> 0:20:48.540 say that? No, but I can say what other problems 0:20:48.540 --> 0:20:54.180 are like the fingers problem toes. Yeah, right. You guys 0:20:54.180 --> 0:20:56.940 are bad first graders. How about some non body part 0:20:56.940 --> 0:21:00.740 ones and ask him to think right wheels on a car, 0:21:00.980 --> 0:21:03.820 maybe something like this where they're noticing that the world 0:21:03.820 --> 0:21:07.940 actually that multiplication is a response to the world and 0:21:07.940 --> 0:21:10.220 math is woven into the cosmos and all. And what 0:21:10.220 --> 0:21:12.419 we're doing is we're noticing patterns in the cosmos, right? 0:21:12.460 --> 0:21:14.060 Our first grader is going to be able to say 0:21:14.060 --> 0:21:15.740 that to, you know, but are they going to slowly 0:21:15.740 --> 0:21:19.380 feel that math starts to be this thing that I 0:21:19.420 --> 0:21:21.060 walk in? I put a question on the board or 0:21:21.060 --> 0:21:24.950 ask a question and we mess around with it and 0:21:24.950 --> 0:21:28.869 play with it until some truth emerges. Until we discover 0:21:28.869 --> 0:21:32.550 some truth. That's, I think, where we can get something 0:21:32.550 --> 0:21:36.670 like wonder in the math class. This type of a shape. Okay. 0:21:36.710 --> 0:21:41.350 So again, I still don't think this shape is all 0:21:41.350 --> 0:21:46.630 that helpful because how do you come up with these 0:21:46.630 --> 0:21:50.669 opening questions? What follow up questions do you ask? How 0:21:50.670 --> 0:21:52.869 do you do these things? Okay. So that's what I 0:21:52.869 --> 0:21:55.070 want to do. Don't let me off the hook for 0:21:55.070 --> 0:21:57.189 being impractical still, right? I have to give you guys 0:21:57.190 --> 0:22:00.230 some tangible advice for these things. This is what we need. 0:22:00.470 --> 0:22:05.430 So let's do it. Okay. Here's how you do opening questions. Go. 0:22:05.869 --> 0:22:12.670 New new. New talk. Opening questions. Socrates says, come then 0:22:12.670 --> 0:22:15.350 to me. He's talking to a mathematician here, actually. Right. 0:22:15.470 --> 0:22:17.990 And do your best to answer the questions I'll ask you. 0:22:18.910 --> 0:22:22.490 And if I abstract and expose your ideas? Because I 0:22:22.490 --> 0:22:24.690 discover upon inspection that the conception which you have formed 0:22:24.690 --> 0:22:26.730 is a vain shadow. Do not quarrel with me, for 0:22:26.730 --> 0:22:29.169 it would be wrong for me to admit falsehood, or 0:22:29.170 --> 0:22:31.610 to stifle the truth. Right. Come to me and expose 0:22:31.609 --> 0:22:36.370 your ideas. That's what an opening question does. Exposes student thinking, 0:22:36.410 --> 0:22:40.290 exposes ideas. Okay, so here we go. What makes for 0:22:40.290 --> 0:22:43.970 a good opening question? Number one has to be accessible 0:22:43.970 --> 0:22:49.010 to all students. Okay. The very first thing out of 0:22:49.010 --> 0:22:51.690 your mouth is something that some students can't grapple with 0:22:51.690 --> 0:22:55.690 or understand. You're going to lose half your class. Okay. 0:22:55.770 --> 0:22:58.290 There's got to be something that every student can think about, right? 0:22:58.330 --> 0:23:01.850 Which is easier? Fingers or pets? Everyone can have an opinion. 0:23:02.690 --> 0:23:06.490 That's no access barrier there. Okay. Number two, it has 0:23:06.490 --> 0:23:12.209 to be challenging to all students because there's no easier 0:23:12.210 --> 0:23:14.970 way to lose half your class than for the first 0:23:14.970 --> 0:23:16.530 thing out of your mouth to be something they already 0:23:16.530 --> 0:23:20.050 know the answer to. Okay, so it has to be 0:23:20.109 --> 0:23:23.190 both accessible, right? Which is easier, fingers or pets? No 0:23:23.190 --> 0:23:27.550 one really has written the has written the book on this. 0:23:28.390 --> 0:23:31.030 No one. This isn't a thing that one can really 0:23:31.030 --> 0:23:33.830 even know, right? This is challenging in some ways to everyone, 0:23:33.830 --> 0:23:35.910 at least to think about. Right? Number three, it has 0:23:35.910 --> 0:23:40.949 to provoke multiple thoughtful student responses. I need this is 0:23:40.950 --> 0:23:43.190 not just a yes or no right or wrong answer 0:23:43.190 --> 0:23:47.070 that I'm starting with. It has to be multiple thoughtful responses. 0:23:47.190 --> 0:23:49.109 I want to scatter my students. I want them to 0:23:49.109 --> 0:23:53.150 be dislocated. I want them to wonder. But number four, 0:23:53.350 --> 0:23:56.830 it has to be directly oriented to the truth. This 0:23:56.830 --> 0:24:01.830 isn't just some progressive. Think about this guys. Do what 0:24:01.830 --> 0:24:07.830 you want, right? Follow your dreams. No. Right. And in particular, 0:24:07.830 --> 0:24:11.350 I think these two and these two are really tough 0:24:11.350 --> 0:24:14.310 needle to thread. How do you get something that's accessible 0:24:14.310 --> 0:24:16.710 and challenging to everyone. And how do you get something 0:24:16.710 --> 0:24:21.119 that's Towards the truth. Enough that it's going to get 0:24:21.119 --> 0:24:24.000 you there. But open ended enough that it's going to 0:24:24.000 --> 0:24:27.440 get students thinking in multiple ways. It's a high bar 0:24:27.440 --> 0:24:32.159 for opening questions. Okay. I don't think there's as high 0:24:32.200 --> 0:24:36.480 a bar for follow up questions, which is good. But again, 0:24:36.920 --> 0:24:40.280 don't let me off the hook. What kinds of things 0:24:40.280 --> 0:24:42.800 make for good opening questions? What are some good examples? Okay, 0:24:42.840 --> 0:24:46.760 so what I've done, I took all my lesson plans 0:24:47.600 --> 0:24:50.400 that I ever wrote and put them in one massive pile, 0:24:50.560 --> 0:24:52.959 and I started just sorting them into different kinds of 0:24:52.960 --> 0:24:56.920 opening questions. Okay. And I came up with five piles. 0:24:57.200 --> 0:24:59.600 So here are my five piles. There's probably more opening questions, 0:24:59.600 --> 0:25:01.760 but here are the ones that I know that work. Okay. 0:25:01.800 --> 0:25:05.480 Number one, we get our students. Each one of these 0:25:05.480 --> 0:25:08.560 has a kind of posture that I want students to adopt, right? 0:25:08.800 --> 0:25:11.719 So number one, I think these are noticing questions. I 0:25:11.720 --> 0:25:16.940 think this is the very basic of opening questions. We 0:25:16.940 --> 0:25:20.939 ask students to notice, say, what's going on, right? Usually 0:25:20.940 --> 0:25:26.900 this is a why question. Noticing is observe. See something 0:25:27.340 --> 0:25:31.220 and try to explain, observe and explain. Observe and explain. 0:25:31.460 --> 0:25:35.020 See something and explain to me what's going on, what's 0:25:35.020 --> 0:25:38.419 going on here, what's going on here? Why is this 0:25:38.420 --> 0:25:43.379 taking place? The very, at the very least, walk in 0:25:43.380 --> 0:25:45.940 with a why question at the very beginning. Why is 0:25:45.940 --> 0:25:48.820 this right? I think it's especially true for generating formulas. 0:25:49.580 --> 0:25:54.860 Here are some examples. Okay. Why can we not use 0:25:54.859 --> 0:25:57.139 our method for adding 23 plus 45 for the question 0:25:57.140 --> 0:26:03.659 23 plus 48. Notice that this doesn't work. And explain. 0:26:04.820 --> 0:26:08.260 Notice and explain. Notice and explain. Why is subtracting a 0:26:08.260 --> 0:26:11.180 negative and positive? How big a box do I need 0:26:11.180 --> 0:26:15.190 to carry stuff? Notice stuff comes in different sizes and 0:26:15.310 --> 0:26:18.550 how to explain if you're trying to teach volume, how 0:26:18.550 --> 0:26:20.350 big a box do I need to carry that thing? 0:26:22.630 --> 0:26:25.750 What thing are you carrying? I don't know a thing, 0:26:27.390 --> 0:26:31.070 a turtle. How big a box where what you're doing 0:26:31.070 --> 0:26:34.790 is you're asking students to think and then forcing them 0:26:34.790 --> 0:26:38.150 to articulate things. They say, well, right. Instead of me 0:26:38.150 --> 0:26:42.990 being like, volume has three dimensions length, width, and height, right? 0:26:43.030 --> 0:26:46.070 Ask them to put stuff in boxes, see what happens. 0:26:46.430 --> 0:26:48.190 They're going to tell you that things have three dimensions 0:26:48.190 --> 0:26:50.189 you got to worry about. They're going to tell you 0:26:50.190 --> 0:26:54.510 how big space is. Students can do this. They can 0:26:54.510 --> 0:26:57.550 come up with this stuff. It takes the right opening question. 0:26:57.550 --> 0:27:02.990 So actually fun fact, I would give this same talk 0:27:03.470 --> 0:27:08.030 to any of the disciplines. I think this is how 0:27:08.030 --> 0:27:12.510 we teach literature history science best anyways, right. Here are 0:27:12.510 --> 0:27:16.930 some other questions. Why is being a hero so difficult? Literature. 0:27:17.170 --> 0:27:21.170 Why do the Articles of Confederation fail? History. Why do 0:27:21.210 --> 0:27:26.530 feathers fall slower than bowling balls? Science. Everything. We are 0:27:26.570 --> 0:27:29.650 driven by these noticing type of questions. Okay, but that's 0:27:29.650 --> 0:27:32.169 very basic level. Okay, there are better ones. Number two, 0:27:32.170 --> 0:27:36.210 I asked my students to notice. Ask my students to compare. Okay. 0:27:37.450 --> 0:27:47.330 It's like noticing on steroids. Noticing. Noticing. Times two. Notice. Notice. Compare. See? 0:27:47.369 --> 0:27:51.369 What's the difference between really good ones, right? This is 0:27:51.369 --> 0:27:56.930 different than this notice. Notice. Explain the differences. Right. Juxtaposing 0:27:56.970 --> 0:28:01.370 two related ideas. It's often phrased as an er or 0:28:01.410 --> 0:28:06.130 est question. If you teach like third, fourth, fifth grade, 0:28:07.050 --> 0:28:10.290 those humans are just little extremists, right? This is what 0:28:10.290 --> 0:28:14.510 they want to do with their life anyway, right? Just 0:28:14.510 --> 0:28:18.109 lean into it. Ask them best. Worst math is a 0:28:18.150 --> 0:28:20.310 lot of about what's the best way to do something. 0:28:22.470 --> 0:28:25.070 Which is easier. Figuring out figures of pets. Which is easier, 0:28:25.070 --> 0:28:30.109 this problem or this problem? Which is faster? Start in 0:28:30.109 --> 0:28:32.630 the ones place the hundreds. Place the difference between this 0:28:32.630 --> 0:28:39.030 and this. This one's really great. This. This factoring one. 0:28:39.030 --> 0:28:41.470 The second one on there. This is how I teach 0:28:41.790 --> 0:28:45.790 my eighth graders how to factor. I divide them class 0:28:45.790 --> 0:28:49.230 in half and I say, give them some reward. Like, 0:28:49.270 --> 0:28:51.550 you know. All right, a little test for you guys 0:28:51.550 --> 0:28:56.190 right off the bat. Uh, first, uh, fastest way to 0:28:56.230 --> 0:28:59.830 get a solution gets half the homework off or whatever, right? 0:29:00.990 --> 0:29:04.510 And they're all like, they're all in now. Like, yeah, okay, good, good. 0:29:04.510 --> 0:29:07.550 Let's do it right. Going down right. And I give 0:29:07.550 --> 0:29:10.320 one half. I pass out this problem. They flip it 0:29:10.320 --> 0:29:14.320 over and I give the other half this problem. And 0:29:14.360 --> 0:29:16.520 you know, the the half that I get this problem 0:29:17.480 --> 0:29:19.720 is done in 10s. The other half is still going. 0:29:19.760 --> 0:29:22.400 I'm like. And so they're raising up their whiteboards. I'm like, good, 0:29:22.440 --> 0:29:24.840 good good good good good good. And this guy, this, 0:29:24.840 --> 0:29:27.000 this whole half is just getting angrier and angrier and 0:29:27.000 --> 0:29:30.280 angrier and angrier. Like what? You gave them a different problem. 0:29:30.280 --> 0:29:32.120 I was like, I don't think I did. I think 0:29:32.160 --> 0:29:35.840 it's the same problem, right? It's the same problem. Same problem. Right. 0:29:37.000 --> 0:29:42.680 It's unfair. Right. And, and I'm like, oh, you, you 0:29:42.720 --> 0:29:44.520 think this is a hard problem? What's hard about this? 0:29:44.520 --> 0:29:48.080 They're like, there's two X's. There's a squared, right? And 0:29:48.080 --> 0:29:50.920 I'm like, oh, you want some ways to take this 0:29:50.920 --> 0:29:54.520 hard problem and turn it into an easier problem? Yeah. 0:29:55.080 --> 0:29:57.719 Oh let me. And they beg me to teach him 0:29:57.720 --> 0:30:02.800 how to factor. Right. And I am benevolent and so 0:30:02.800 --> 0:30:06.400 I do. Right. This is. But the idea is just look, 0:30:06.440 --> 0:30:10.900 this is hard. Oops. This is hard. This is easy. 0:30:11.060 --> 0:30:14.500 Let's learn how to do it right. Easy stuff. Right. Okay. 0:30:14.540 --> 0:30:17.340 It's like shooting little angsty eighth graders in a barrel. 0:30:17.380 --> 0:30:22.860 It's great. It's great pedagogy, right? Same in other disciplines 0:30:22.860 --> 0:30:24.580 as well, right? What's the difference between the way that 0:30:24.580 --> 0:30:28.540 Elizabeth envisions marriage, the way that Darcy envisions marriage? Great 0:30:28.540 --> 0:30:33.100 opening question. Oops. Uh, was there ever a moment when 0:30:33.100 --> 0:30:35.860 you'd be decisive and moment need to be cautious? If 0:30:35.860 --> 0:30:38.860 I'm teaching the Battle of Gettysburg. Oh, man. I'm opening 0:30:38.860 --> 0:30:42.220 with that question. Day one. They had to be real 0:30:42.220 --> 0:30:45.020 decisive to get that high ground right. Day three, just 0:30:45.020 --> 0:30:48.700 the boldness run amok, right? Number four. Right. Or the 0:30:48.700 --> 0:30:49.980 last one, which is harder to roll a ball up 0:30:49.980 --> 0:30:51.780 a hill to keep it stable at the top of 0:30:51.780 --> 0:30:55.780 the hill. Good comparison. Okay. I asked them to notice, 0:30:55.780 --> 0:31:03.220 to compare, to create questions of how would you or 0:31:04.100 --> 0:31:09.840 design Your kindergarten first, second grade teacher. They should be drawing. 0:31:09.880 --> 0:31:12.440 They should be designing all of these things they can do. 0:31:13.080 --> 0:31:15.760 Draw me this. Draw me something. Design me something. Create 0:31:15.760 --> 0:31:19.040 me something. The idea here is to use get students 0:31:19.040 --> 0:31:23.720 using their imaginations. We want our math classes to be imaginative. 0:31:26.080 --> 0:31:29.400 Notice how this is great to get all students doing 0:31:29.400 --> 0:31:32.280 different things right. This is going to provoke multiple responses 0:31:32.280 --> 0:31:34.280 to create. This is one of the beautiful things about 0:31:34.280 --> 0:31:37.400 human beings is we all have different creativity. If I 0:31:37.400 --> 0:31:39.200 asked you guys to write a poem, you guys would 0:31:39.200 --> 0:31:43.480 all hopefully write something different. Otherwise it'd be a little creepy, right? 0:31:45.760 --> 0:31:50.160 Design a game that seems fair but isn't. If I'm 0:31:50.160 --> 0:31:54.959 teaching some expected value. That's a great question, right? Design 0:31:55.000 --> 0:31:56.960 me a problem that can be solved for this bar model, right? 0:31:57.000 --> 0:32:02.120 Flip it on its head. Literature. Design me a defensive 0:32:02.120 --> 0:32:05.570 strategy for Tom Robinson. Design me a scenario of the 0:32:05.570 --> 0:32:08.570 Roman system of government particularly struggle with. Design me something 0:32:08.570 --> 0:32:13.970 that seems alive but isn't. If. I'm sorry, I'm doing 0:32:13.970 --> 0:32:15.690 all these other disciplines here, but if I'm trying to 0:32:15.690 --> 0:32:29.530 teach the seven qualities of life right? Movement, science, knowledge, movement, respiration, sensation, growth, reproduction, excretion, nutrition. Okay, great. 0:32:29.570 --> 0:32:31.530 These are my seven classes. I can walk in and say, 0:32:31.530 --> 0:32:33.210 all right, kids, today we're going to learn the seven 0:32:33.210 --> 0:32:35.970 qualities of life. Write them down. Write them down. Write 0:32:35.970 --> 0:32:39.530 them down. Right. It's not really pedagogy, right? That's just 0:32:39.570 --> 0:32:44.490 dogmatic teaching. It's easy. Right? Design me something that seems 0:32:44.490 --> 0:32:49.410 alive but isn't. Go work with the person next to you. 0:32:49.850 --> 0:32:52.730 Explore for a little bit. This isn't just like explore 0:32:52.730 --> 0:32:54.730 to your heart's content, but try some stuff and then 0:32:54.730 --> 0:32:56.970 let's come back and talk about it. Right. And my 0:32:56.970 --> 0:33:00.570 follow up questions write themselves. Maybe they come up with, 0:33:00.610 --> 0:33:05.150 I don't know, a robot, a virus Wind. Uh, stories, 0:33:05.470 --> 0:33:08.470 I don't know. Plenty of things seem alive, but aren't. 0:33:09.390 --> 0:33:10.830 And I can ask them, why do you think that 0:33:10.830 --> 0:33:13.510 seems alive? Why does it seem alive? How do you 0:33:13.510 --> 0:33:15.990 know that it's not alive? I bet you they can 0:33:15.990 --> 0:33:22.390 get all seven of those things. Or at least six, right? Okay, good. Notice. Compare. 0:33:22.390 --> 0:33:30.710 Create number for struggle. Okay. Um. The question in the 0:33:30.710 --> 0:33:34.670 back of your head should be, is this accessible to everyone? 0:33:34.670 --> 0:33:40.190 If I make them struggle with stuff. No. Right. In 0:33:40.230 --> 0:33:43.790 that they're going to struggle. But since I'm asking everyone 0:33:43.790 --> 0:33:47.310 to struggle, it's weirdly accessible, right? As soon as if 0:33:47.310 --> 0:33:48.430 you're a kid that's going to that I put a 0:33:48.430 --> 0:33:51.990 problem on the board and they're like. But then they 0:33:51.990 --> 0:33:55.950 look around their classmates and everyone's going, now I've leveled 0:33:55.950 --> 0:33:57.910 the playing field again, right? This is accessible to everyone. 0:33:57.910 --> 0:33:59.950 I'm not asking for insights here as much as I'm 0:33:59.950 --> 0:34:03.320 asking them to grapple with the thing, right? To struggle 0:34:03.320 --> 0:34:05.160 to try some stuff, right. I think we struggle in 0:34:05.200 --> 0:34:10.040 two ways. Something initially that seems impossible becomes clearer upon investigation, 0:34:10.400 --> 0:34:13.560 or something that initially seems simple, that becomes more impossible 0:34:13.560 --> 0:34:18.800 under investigation. So a couple examples here. Right? 706 divided 0:34:18.800 --> 0:34:22.840 by three trying to teach remainders for the first time. 0:34:23.760 --> 0:34:29.759 Throw it up there. 76 divided by three go! Mr. Greg. 0:34:29.800 --> 0:34:34.360 It doesn't come out right. Oh, okay. What are we 0:34:34.400 --> 0:34:37.760 going to do? Here's $706. You guys just made it 0:34:37.800 --> 0:34:41.120 divided amongst yourselves, right? Two to the X equals 13. 0:34:41.560 --> 0:34:44.200 Oh it's great. I'm trying to teach. I'm trying to 0:34:44.200 --> 0:34:46.560 teach logarithms. I walk in, right. You guys are the 0:34:46.560 --> 0:34:49.279 ninth graders. Two to the X equals 13. All right. 0:34:49.280 --> 0:34:56.360 Go for it, guys. Now your ninth graders, right? Oh, Mr. Greg, 0:34:56.360 --> 0:35:00.100 why do you do this to us? Oh, what's hard 0:35:00.100 --> 0:35:07.820 about this one? There's no twos that get a 13, Mr. Greg. Oh, okay. 0:35:08.460 --> 0:35:11.219 How about this one? Two plus x equals 13. Do 0:35:11.219 --> 0:35:16.540 you know that one? 11. How'd you do that one? Ah, 0:35:16.540 --> 0:35:19.740 we just subtracted two. Okay. Right. All right. Two times 0:35:19.739 --> 0:35:25.180 x equals 13. Six and a half. How'd you do that? 0:35:25.180 --> 0:35:29.620 One divided by two. We know these. All right. Well, 0:35:29.620 --> 0:35:31.460 two x equals 13. Is the answer bigger or less 0:35:31.460 --> 0:35:36.819 than one bigger. How do you know? Two to the 0:35:36.820 --> 0:35:39.020 one is two. Is it bigger? Less than two bigger. 0:35:39.060 --> 0:35:41.130 It's bigger. Less than three. It's still bigger. It's between 0:35:41.130 --> 0:35:46.779 3 and 4. Mr. Greg. Oh, okay. Well, what's hard 0:35:46.780 --> 0:35:49.540 about this? How'd you know to divide by two when 0:35:49.540 --> 0:35:53.460 we had a multiplication problem? Because it's the opposite of it. 0:35:53.900 --> 0:35:55.739 We're not going to use inverse operations. They're just not 0:35:55.739 --> 0:35:58.200 going to say it, but they will. You know, they're 0:35:58.200 --> 0:35:59.680 never going to use the right words that you want, 0:35:59.719 --> 0:36:02.359 but they'll say, say, say. Okay. Well, what's hard about 0:36:02.360 --> 0:36:05.480 this one? And some kid will say, oh, well, we 0:36:05.480 --> 0:36:08.440 just need is a magic thing that undoes the exponentiation. 0:36:08.840 --> 0:36:11.160 I'm like, oh, is that what you want? Right. And 0:36:11.160 --> 0:36:16.040 they've begged me for logarithms. Right. I mean, like, again, 0:36:16.080 --> 0:36:18.839 this take take this one. Right. I'm trying to teach 0:36:18.840 --> 0:36:21.360 the fundamental theorem of calculus. I put this on the board. 0:36:21.719 --> 0:36:27.680 They're like, how much area are. It doesn't even have 0:36:27.680 --> 0:36:32.040 any lines. Mr.. Greg. Okay, I'll draw some lines. Fine. Great. Now, 0:36:32.040 --> 0:36:37.680 how much area? What's hard about this? It's not a square, Mr. Greg. 0:36:38.200 --> 0:36:40.759 It's got curves. Oh, okay. Is the answer bigger? Less 0:36:40.760 --> 0:36:44.680 than one. I can see the the follow up questions 0:36:44.680 --> 0:36:49.959 right themselves. Right. Like bigger. Bigger. How do you know? Well, 0:36:49.960 --> 0:36:52.520 there's one square here, and it's bigger. Right. How do 0:36:52.520 --> 0:36:54.800 you get a better. How do you get a better approximation? 0:36:55.160 --> 0:36:56.690 I could use some smaller squares. 0:36:56.969 --> 0:36:58.529 Sometimes you have to pull this out and then some 0:36:58.530 --> 0:37:00.730 kid will raise their hand. Usually the smart aleck of 0:37:00.730 --> 0:37:03.049 the smart alecks would be like, why don't we just 0:37:03.050 --> 0:37:07.489 use infinite rectangles, Mr. Greg? I'm like, yes, let's do that. 0:37:08.330 --> 0:37:11.729 Let's do that. Right? But again, like, instead of walking 0:37:11.730 --> 0:37:13.210 in and being like, so now I have a curve 0:37:13.210 --> 0:37:14.649 and then I have area and I'm going to divide 0:37:14.650 --> 0:37:18.009 this up into infinite rectangles. Like the kids are just like, what? 0:37:18.610 --> 0:37:20.450 They can do this stuff, right? You just have to 0:37:20.489 --> 0:37:22.410 ask the right opening question to get them there. Okay. 0:37:22.930 --> 0:37:27.609 Struggle questions are all over the place. Lastly, prediction. So 0:37:27.610 --> 0:37:30.890 the last one I like to do notice, compare, create, struggle, predict. 0:37:30.930 --> 0:37:36.530 These are what if questions, right. Where they're either gathering 0:37:36.530 --> 0:37:39.890 evidence or reasoning forward or they're imagining what's going to 0:37:39.930 --> 0:37:47.810 happen if something were different. What if questions. Okay. What 0:37:47.810 --> 0:37:51.290 if there were no tends to borrow from. Oh, no. 0:37:54.670 --> 0:37:56.469 What if we could take the square root of a 0:37:56.469 --> 0:38:00.230 negative number? Imagine what if there's an A term that's 0:38:00.230 --> 0:38:01.870 not actually not that great of an opening question, but 0:38:02.270 --> 0:38:04.029 what if there were an a term for this quadratic? 0:38:04.270 --> 0:38:06.870 What if frog toed shoes and toad were in frog shoes? 0:38:07.350 --> 0:38:09.710 What if Hitler hadn't tried to invade Russia? What if 0:38:09.710 --> 0:38:14.310 we heat this up? We do these types of questions 0:38:14.310 --> 0:38:16.950 all the time, getting them to predict each one of 0:38:16.950 --> 0:38:20.750 these postures. Right? Notice, compare, create, struggle, and predict is 0:38:20.750 --> 0:38:24.549 going to get kids wondering about the thing. Okay? And 0:38:24.550 --> 0:38:26.870 so when you walk into your classroom, I challenge you. 0:38:27.150 --> 0:38:29.590 See if you can put a question on the board 0:38:29.590 --> 0:38:32.790 or a ask them, say to say, all right, boys 0:38:32.790 --> 0:38:35.750 and girls, I've got a question for you guys. If 0:38:35.750 --> 0:38:38.790 that's the first words out of your mouth every single time, 0:38:38.989 --> 0:38:41.710 it's not going to work right away, right? They're not 0:38:41.710 --> 0:38:43.109 going to be used to this and be like, what? 0:38:43.110 --> 0:38:47.469 This is not math. But if every day you walk 0:38:47.469 --> 0:38:49.230 in and say, good morning, I've got a question for 0:38:49.230 --> 0:38:50.989 you guys. Good morning. I've got a question for you guys. 0:38:50.989 --> 0:38:52.590 Good morning. I've got a question for you guys. Slowly 0:38:52.590 --> 0:38:54.759 they're going to start to realize that math is not 0:38:54.760 --> 0:38:58.040 just repeating and regurgitating the thing, but it's engaging in 0:38:58.040 --> 0:39:04.040 the act of mathematics. It's doing math. It's figuring stuff out. 0:39:04.040 --> 0:39:09.480 It's imagining, it's hypothesizing, it's playing around with stuff. You're 0:39:09.480 --> 0:39:16.120 involving them in what mathematicians actually do. Okay. I do 0:39:16.120 --> 0:39:18.959 think this is then a thing that can become somewhat habitual, 0:39:19.960 --> 0:39:24.160 because you can't be inventing opening questions every single night, 0:39:24.160 --> 0:39:28.319 but you can just say, all right, say, I'm going 0:39:28.360 --> 0:39:31.560 to commit to doing this one thing every single day. Right? 0:39:31.840 --> 0:39:35.080 And I have five where you sit down to lesson plan. 0:39:35.120 --> 0:39:38.960 Maybe you have ten minutes, maybe you have five minutes 0:39:40.320 --> 0:39:43.200 to plan for this thing. Is there a question I 0:39:43.200 --> 0:39:46.360 can ask that gets my kids either noticing, comparing, creating, struggling, 0:39:46.360 --> 0:39:52.020 or predicting? One thing that I like to do a 0:39:52.020 --> 0:39:56.020 little bit is to challenge myself not to write the 0:39:56.020 --> 0:39:59.700 key formula, definition, or process on the board until the 0:39:59.700 --> 0:40:02.980 students say it. I can't always do this. I fail 0:40:02.980 --> 0:40:05.340 a lot of times, but I like to challenge myself. 0:40:05.340 --> 0:40:08.299 Can I get them to tell me groups and in 0:40:08.300 --> 0:40:10.900 each group and total, can I get them to say 0:40:10.900 --> 0:40:14.700 those words? Sometimes I fail, but at least I'm trying 0:40:14.739 --> 0:40:16.259 to do that type of thing. Can I get them 0:40:16.260 --> 0:40:20.020 to say we need a process that undoes the exponentiation? 0:40:20.780 --> 0:40:23.700 Can I get a student to say that? And then 0:40:23.700 --> 0:40:28.180 I say, yes, that's what we need. That's what we do. Okay, good. 0:40:29.980 --> 0:40:33.060 New talk, follow up questions. Where am I at? What 0:40:33.060 --> 0:40:36.940 time is it? We don't have time, do we? What time? 0:40:38.260 --> 0:40:43.339 We got time. I'm ignoring. Ignoring the clock. The good 0:40:43.340 --> 0:40:45.860 news is, follow up questions I think are easier than 0:40:45.900 --> 0:40:49.140 opening questions. I think there's a lot of bad opening questions. 0:40:49.140 --> 0:40:52.680 I think there's very few bad follow up questions. Okay. 0:40:53.200 --> 0:40:56.759 And I mean that sincerely. Right. Where again, what we 0:40:56.760 --> 0:40:58.719 have here is this example from the Theaetetus. If you 0:40:58.719 --> 0:41:00.759 want to know what it means to just teach Socratically, 0:41:00.760 --> 0:41:03.319 if you've ever wondered what that word means, this is 0:41:03.320 --> 0:41:06.759 what it means to do this, because this is what 0:41:06.760 --> 0:41:10.120 Socrates does. Once more, Theaetetus, I repeat my old question 0:41:10.120 --> 0:41:13.800 what is knowledge? Opening question. This is a struggle. Struggle 0:41:13.800 --> 0:41:17.600 question what is knowledge? There's no good answers to that, right? 0:41:18.120 --> 0:41:23.560 The knowledge is perception. That's a student response. And then 0:41:23.560 --> 0:41:26.279 what does Socrates do? He says, bravely said, that's the 0:41:26.280 --> 0:41:29.040 way you should express your opinion. Now let's examine together 0:41:29.040 --> 0:41:32.000 this conception of yours and see whether it's a true 0:41:32.000 --> 0:41:37.560 birth or a windegg. Windegg. This is just an empty 0:41:37.560 --> 0:41:39.879 shell with no. I don't know if you've ever gotten 0:41:39.880 --> 0:41:44.520 one of those from a grocery store. Okay, good. Okay, 0:41:44.560 --> 0:41:48.370 so what we have here is this shape where I 0:41:48.410 --> 0:41:52.009 prompt student thinking. Usually with an opening question, I get 0:41:52.010 --> 0:41:55.850 a response and then I ask a follow up question 0:41:55.850 --> 0:41:59.810 based on that response. Right? So there's other options. I 0:41:59.810 --> 0:42:02.770 could say what's five plus eight? Someone could say 13 0:42:02.810 --> 0:42:04.890 and I could say good and ask a different question. 0:42:04.890 --> 0:42:08.410 That's not a follow up question. It's five plus 813. 0:42:08.450 --> 0:42:11.410 How'd you figure that out? It's using the response as 0:42:11.410 --> 0:42:13.130 the thing that you go back and forth. This is 0:42:13.130 --> 0:42:14.969 where you get a little bit of dialogue is the 0:42:14.969 --> 0:42:19.770 idea here. Okay, so. I'm going to skip through this 0:42:19.770 --> 0:42:22.690 a lot. Let's see why our follow up question important. 0:42:22.930 --> 0:42:26.489 They help students make sense of their experiences, right? They 0:42:26.530 --> 0:42:27.969 help you to think deeper about the thing. They ask 0:42:28.050 --> 0:42:32.170 you to dwell on meaningful ideas and problems, probe into 0:42:32.170 --> 0:42:36.609 the reasons for truth and not just truth themselves. We 0:42:36.650 --> 0:42:39.450 care about truth, especially in math, but we care about 0:42:39.450 --> 0:42:42.890 the discovery of truth. Challenge students to provide explanations that 0:42:42.890 --> 0:42:45.290 go beyond one word answers. One of the best definitions 0:42:45.290 --> 0:42:47.190 of math. We've seen a couple of them. One of 0:42:47.190 --> 0:42:49.549 them we haven't seen. Paul Lockhart says mathematics is the 0:42:49.550 --> 0:42:55.070 art of explanation. Okay. Right. Get them explaining. You've already won. Right. 0:42:55.350 --> 0:42:58.870 If you're getting students explaining their thinking, you've already been victorious. 0:42:58.910 --> 0:43:01.310 I was I was doing a couple of years ago, 0:43:01.310 --> 0:43:05.469 I was doing a model lesson in kindergarten. So I 0:43:05.469 --> 0:43:06.910 was I was doing a kindergarten and we were doing 0:43:06.910 --> 0:43:09.910 calendar time and I was having a blast. Calendar time 0:43:09.910 --> 0:43:12.150 is amazing, right? And we it was Tuesday or something, 0:43:12.150 --> 0:43:15.270 and we put the 14 in the slot of the calendar. 0:43:15.310 --> 0:43:18.910 It was the 14th. And so I asked, what number 0:43:18.950 --> 0:43:22.670 are we going to put in the Friday slot? Right. Okay, 0:43:22.710 --> 0:43:25.350 maybe not the best opening question, but a question. Right. 0:43:25.550 --> 0:43:27.469 And a girl raises her hand. She goes 17. I 0:43:27.469 --> 0:43:30.669 was like, yes. How'd you figure that one out? And 0:43:30.670 --> 0:43:32.430 the look on her face, I'm never going to forget it. 0:43:32.430 --> 0:43:37.750 And she was like. How did I figure that out? Like, 0:43:37.830 --> 0:43:40.710 she had never realized that, like, you could have a 0:43:40.710 --> 0:43:43.950 thought and then attach another thought to that thought, right? 0:43:43.989 --> 0:43:48.120 It's this like it's this. Oh, I how did I 0:43:48.120 --> 0:43:50.320 figure that out? And she was like, she didn't say 0:43:50.400 --> 0:43:54.319 she didn't give a super long explanation. She didn't say, well, 0:43:54.320 --> 0:43:55.870 it's three days away. And so I added 3 to 0:43:55.870 --> 0:43:57.719 14 and she was like, well, we're going to put 0:43:57.719 --> 0:43:59.400 a 15 and then a 16 and then a 17. 0:43:59.400 --> 0:44:01.680 And I was like, what I've done is I've, as 0:44:01.680 --> 0:44:05.680 a kindergartener, I've got her to explain her thinking, I've 0:44:05.680 --> 0:44:08.360 already won. It doesn't matter what she says at that point, 0:44:08.400 --> 0:44:11.000 the fact that kindergartners can say, I have ideas and 0:44:11.000 --> 0:44:14.880 then I can explain how I get those ideas, that's 0:44:14.880 --> 0:44:18.200 a win. And it happens because we follow up on 0:44:18.200 --> 0:44:23.040 student thinking, okay, good, we're open, struggling students. Best way 0:44:23.040 --> 0:44:25.240 to do this? I talk about this in just a sec, right? 0:44:26.120 --> 0:44:32.759 You don't know what's going on. 71 times 14, I 0:44:32.760 --> 0:44:38.120 don't know. Well, is it bigger? Less than one. Bigger. 0:44:38.360 --> 0:44:41.440 How do you know? Well, 14 is bigger than one. 0:44:41.440 --> 0:44:45.340 And I did a multiplication. Multiplication makes things bigger. Why? Right. 0:44:45.380 --> 0:44:48.580 Because multiplication, I mean. Right. You're getting them to explain 0:44:48.580 --> 0:44:51.739 to you math because you've asked them a question and 0:44:51.739 --> 0:44:54.620 then asked them some follow up questions instead of just 0:44:54.660 --> 0:44:57.779 like 71 times 14. I don't know who can, who can, 0:44:57.900 --> 0:45:01.620 who can help, right? Turning away from that student rather 0:45:01.620 --> 0:45:06.220 than going deeper with that student. Okay, good. So situations 0:45:06.219 --> 0:45:11.620 for follow up questions. Number one. Silence. It's an important one. 0:45:12.620 --> 0:45:16.500 Notice I said student responses. Silence is a response. If 0:45:16.500 --> 0:45:19.339 I ask a question and I get silence or I get, 0:45:19.380 --> 0:45:24.540 I don't know. That's a situation. For a follow up question. 0:45:24.540 --> 0:45:28.980 It's tempting to either go in a different direction or to. 0:45:31.180 --> 0:45:36.940 Answer the question yourself, but I don't know. So is 0:45:36.940 --> 0:45:38.819 a place where I would then ask a student a 0:45:38.820 --> 0:45:42.560 follow up question. I would dig in. What students need 0:45:42.560 --> 0:45:47.040 to know is, I don't know is a perfectly acceptable 0:45:47.040 --> 0:45:51.440 response in your classroom. I'm not willing to think about 0:45:51.440 --> 0:45:55.239 it is not an acceptable response. Okay. There's a difference. 0:45:55.719 --> 0:45:57.720 If I say something and you say, I don't know, 0:45:57.719 --> 0:46:00.040 and then I ask you a follow up question about this. 0:46:00.080 --> 0:46:03.760 If I say something like, what's hard about this? That's 0:46:03.760 --> 0:46:07.360 a follow up question about what you don't know. You 0:46:07.719 --> 0:46:10.160 have to be willing to think about this with me, right? 0:46:10.400 --> 0:46:13.719 Put that rule on your on your wall somewhere. I 0:46:13.760 --> 0:46:16.360 don't know is an acceptable response. I'm not willing to 0:46:16.360 --> 0:46:21.720 think about it is not an acceptable response. Okay. Number 0:46:21.719 --> 0:46:27.120 two wrong answers site for a follow up question. What's 0:46:27.120 --> 0:46:31.840 the temptation as a teacher? Fix it. Correct it. Right. 0:46:31.880 --> 0:46:35.840 Wrong answers. How do I dig in to that wrong answer? 0:46:36.760 --> 0:46:41.730 I will even solicit wrong answers in the classroom. If 0:46:41.770 --> 0:46:43.730 you've never done this in your classroom, put a problem 0:46:43.730 --> 0:46:46.649 on the board and say wrong answers only. It's kind 0:46:46.650 --> 0:46:50.609 of great. Put a clock on the board for your 0:46:50.610 --> 0:46:57.690 second graders. What time is it? Wrong answers. Only 781, right? Great. 0:46:57.770 --> 0:46:59.690 They say, what's the. And you say, what's the most 0:46:59.690 --> 0:47:04.810 tempting wrong answer? And help them to articulate the errors 0:47:04.810 --> 0:47:07.569 that we might make as students. Right. They start to 0:47:07.610 --> 0:47:10.330 filter themselves if they're imagining the wrong answers. You want 0:47:10.330 --> 0:47:13.410 the wrong answers in the classroom. Wrong answers are gold. 0:47:14.210 --> 0:47:16.170 Wrong answers are the only times we are going to 0:47:16.170 --> 0:47:18.850 teach someone something. To go from wrong answer to right answer. 0:47:20.450 --> 0:47:26.570 Number three correct answers. What's the temptation here to move on? Great. 0:47:26.570 --> 0:47:29.969 Move on. I did it, yes. Done. Instead of saying, 0:47:30.170 --> 0:47:32.210 how'd you figure that out? What was the key? How 0:47:32.210 --> 0:47:34.770 can we remember this? Notice the follow up questions that 0:47:34.770 --> 0:47:39.589 are there with a right answer. Lastly, a question from 0:47:39.590 --> 0:47:46.230 a student. What's the temptation? Answer it. Right. Especially if 0:47:46.230 --> 0:47:48.190 you get a good question. How do I reframe that 0:47:48.190 --> 0:47:50.030 question and turn it back on the on the student 0:47:50.070 --> 0:47:52.950 who asked it? And not in some cheesy way like, well, 0:47:52.950 --> 0:47:57.270 what do you guys think? Not a good follow up question. 0:47:58.030 --> 0:48:00.790 How do you turn that back on the student? Right. 0:48:01.230 --> 0:48:05.110 Notice I think I've covered almost everything here. Right. Every 0:48:05.110 --> 0:48:08.910 situation can be done with a follow up question. Their 0:48:08.910 --> 0:48:13.989 only limits are time and exhaustion, both of which are 0:48:13.989 --> 0:48:18.670 legitimate concerns. Right? You only have so much time. You 0:48:18.670 --> 0:48:22.190 are going to get exhausted, right? But look, if you're 0:48:22.190 --> 0:48:27.310 seeing everything that students do as an opportunity, that's the key. 0:48:28.230 --> 0:48:31.589 Every single thing a student responds. That's an opportunity for 0:48:31.590 --> 0:48:34.470 me to ask about what's going on, right? Okay. So 0:48:34.750 --> 0:48:36.810 types of follow up questions. Here's how you do them. 0:48:36.810 --> 0:48:45.170 Number one. Most basic, you press for explanation. Pressing for explanation. Basically, 0:48:45.170 --> 0:48:51.010 it's saying, why did you respond the way that you responded? Right. 0:48:51.050 --> 0:48:55.170 That's all that question is. You got to be a 0:48:55.170 --> 0:48:58.969 little careful. You don't seem like a jerk. Or you're 0:48:58.969 --> 0:49:00.890 just like, ask a question, I get silenced. I'm not 0:49:00.890 --> 0:49:02.690 going to be like, well, why are you guys silent 0:49:02.730 --> 0:49:05.210 on this? That's not what you want. But I can 0:49:05.210 --> 0:49:07.770 ask a question and I get silenced and I can say, oh, 0:49:07.810 --> 0:49:11.770 that's good. What's what's hard about this question? It's basically saying, 0:49:12.250 --> 0:49:15.170 explain to me why you guys are silent here. What's 0:49:15.210 --> 0:49:19.530 genuinely hard about this? Right? I think math is hard. 0:49:19.570 --> 0:49:25.410 I think we should lean into that. Okay. Number two comparison. 0:49:25.450 --> 0:49:27.170 We saw this as an opening question. It's actually a 0:49:27.170 --> 0:49:31.169 really good follow up question as well, juxtaposing it with 0:49:31.170 --> 0:49:35.700 another thing. For example, let's try a wrong answer. If 0:49:35.700 --> 0:49:38.299 I say five plus three and someone says seven, I 0:49:38.340 --> 0:49:40.700 can say, well, five plus two is seven. What's five 0:49:40.739 --> 0:49:44.540 plus three? That's comparison follow up question. I'm saying compare 0:49:44.580 --> 0:49:47.779 it with this and rethink, compare and rethink. It's not 0:49:47.780 --> 0:49:49.660 the best way to respond to wrong answers, but it's 0:49:49.660 --> 0:49:57.300 a possibility. Okay. Number three, hook, hook question goes back 0:49:57.300 --> 0:50:00.940 to something that's basic and then scaffolds it back up. 0:50:02.340 --> 0:50:05.020 Use a basic response as a launching pad for a 0:50:05.020 --> 0:50:07.660 deeper question. Often it's a thing that you asked, you 0:50:07.700 --> 0:50:11.530 got silence or you got 1 or 2 hands. 1 0:50:11.530 --> 0:50:15.820 or 2 hands in a math class is silence. Okay, 0:50:16.060 --> 0:50:18.070 you get 1 or 2 hands. Don't call on 1 0:50:18.070 --> 0:50:20.980 or 2 of those guys. Don't do it. It tells 0:50:20.980 --> 0:50:24.020 every other kid they can turn off their brains. Okay, 0:50:24.060 --> 0:50:25.460 if I get 1 or 2 hands, I'm going to say, 0:50:25.500 --> 0:50:27.980 oh yeah, this is a hard one. Is it bigger 0:50:27.980 --> 0:50:30.899 or less than one? I like the idea is go 0:50:30.900 --> 0:50:33.560 to a basic one and build it back up. We 0:50:33.560 --> 0:50:37.680 saw this today. Doctor Treloar did this beautifully. Right. How 0:50:37.680 --> 0:50:46.280 do you divide by a fraction? Silence. Okay. What's addition? What? 0:50:47.400 --> 0:50:49.960 Go to a basic one and build it up. What 0:50:49.960 --> 0:50:51.279 are the different ways we think of addition? What are 0:50:51.280 --> 0:50:53.440 the different ways you think of subtraction? Right. Build it 0:50:53.440 --> 0:50:59.120 back up. Opening question. Follow up question. Okay. Number four. Ramification. 0:51:01.239 --> 0:51:05.360 Probing the implications of this response. How do we apply it? 0:51:05.400 --> 0:51:07.280 What does it matter? Why does it make sense? What's 0:51:07.280 --> 0:51:10.240 the key? Are all problems like this? You could see 0:51:10.239 --> 0:51:13.959 the the potential follow up questions here about what this 0:51:13.960 --> 0:51:19.760 means for us. So what I've done is this. Okay, great. 0:51:20.960 --> 0:51:24.319 So what we have here are the situations on this 0:51:24.600 --> 0:51:27.120 axis and the kinds on this axis. And then just 0:51:27.120 --> 0:51:32.850 some examples. Not all of all these work, but just 0:51:32.850 --> 0:51:35.810 to show you there's so many different follow up questions 0:51:35.810 --> 0:51:41.090 that I think we can do for every single situation. Now, 0:51:41.250 --> 0:51:43.250 don't print this off and put it on a little 0:51:43.250 --> 0:51:45.730 clipboard and be like, well, this is a wrong answer. 0:51:45.730 --> 0:51:49.050 So now I'm going to use a hook, right? Don't 0:51:49.050 --> 0:51:51.450 do that. That's not the point of this exercise. But 0:51:51.450 --> 0:51:54.810 the point is just to say every single situation can 0:51:54.810 --> 0:51:57.129 be responded to with every single kind of question. That's 0:51:57.130 --> 0:52:00.650 not like all it is, is it's asking students to 0:52:00.690 --> 0:52:05.170 think more. You've got to develop your habit of this, 0:52:06.530 --> 0:52:09.089 which is easier. Fingers are pets. Fingers. Why do you 0:52:09.090 --> 0:52:15.330 think fingers? What's hard about that? What's easy about that? Pets. 0:52:15.370 --> 0:52:17.810 Why do you think pets. It's just getting in the 0:52:17.850 --> 0:52:21.170 habit of seeing student responses as opportunities. Right. Okay, so 0:52:21.890 --> 0:52:23.690 some of my favorite ones, I'll put these on the 0:52:23.690 --> 0:52:28.370 board for a sec. Uh, I think why is this 0:52:28.790 --> 0:52:33.150 hard is a really good one. What's the key? What's 0:52:33.150 --> 0:52:38.870 your strategy? All of these things students love talking about strategies. 0:52:40.230 --> 0:52:43.150 My six year old, I play a lot of war 0:52:43.150 --> 0:52:46.790 with him, and he's always wanting to tell me this 0:52:46.790 --> 0:52:51.549 strategy for the game. I'm like, it's a pure game 0:52:51.550 --> 0:52:56.310 of luck, you moron, right? I don't call my son 0:52:56.350 --> 0:53:01.910 a moron. I'm always eager to hear his strategy. He's like, 0:53:01.950 --> 0:53:03.750 whenever you play a nine, I'm gonna try to play 0:53:03.750 --> 0:53:11.989 a ten. I'm like. Great. How's that going for you? Yeah. No, 0:53:11.989 --> 0:53:15.830 he's a good kid. He's going to need to get 0:53:15.830 --> 0:53:21.110 some athletic scholarships, but he's going to be a good kid. Okay. 0:53:23.590 --> 0:53:26.790 Some that I don't really like up here. Right. Anyone 0:53:26.790 --> 0:53:31.130 have any questions? Not great. Just doesn't get much traction, right? 0:53:31.170 --> 0:53:38.290 How do you feel? I don't really care. Okay, good. 0:53:38.330 --> 0:53:43.170 So how do you habituate this one? I think if 0:53:43.170 --> 0:53:45.330 you're thinking about ways to prep for this. I don't 0:53:45.330 --> 0:53:47.009 think you need to prep for it as much. Right. 0:53:47.050 --> 0:53:49.169 The follow up questions write themselves a lot of times 0:53:49.170 --> 0:53:50.770 as long as you start getting the habit of it, 0:53:50.770 --> 0:53:54.330 but including an anticipated student responses section, what my students 0:53:54.330 --> 0:53:58.330 say and how might I respond? Right. Number two, think 0:53:58.330 --> 0:54:01.529 about the different situations. Test yourself as you're thinking about 0:54:01.530 --> 0:54:05.330 your own classroom. How do I respond to silence? Do 0:54:05.330 --> 0:54:06.969 I do it well? Do I do it poorly? How 0:54:06.969 --> 0:54:11.489 do I respond to wrong answers? Try to make some 0:54:12.650 --> 0:54:15.170 renewed habits to respond to some of these in different ways. 0:54:15.210 --> 0:54:18.210 Specifically with this kind of Socratic pedagogy, right? One thing 0:54:18.210 --> 0:54:21.169 that I find really helpful is to count the number 0:54:21.170 --> 0:54:23.209 of questions that I ask that can be answered with 0:54:23.210 --> 0:54:25.570 a single word, and then count the number of questions 0:54:25.660 --> 0:54:29.140 that beg a sentence. At least one of my absolute 0:54:29.140 --> 0:54:31.940 proudest moments as a teacher. I was teaching sixth grade, 0:54:32.300 --> 0:54:35.979 and I don't know, I forget what question I asked, 0:54:35.980 --> 0:54:38.700 but a kid raised his hand and he gave an answer. 0:54:38.739 --> 0:54:41.140 He's like, I think it's seven. And because you always 0:54:41.140 --> 0:54:43.020 make us explain how we got it, here's how I 0:54:43.020 --> 0:54:48.460 got it. And I was like, yes, I have broken them. Right? 0:54:49.060 --> 0:54:51.379 But that's the idea. You want to do this so 0:54:51.420 --> 0:54:55.100 habitually that they get in the habit of explaining their thinking, right? 0:54:55.180 --> 0:54:59.060 That's the thing that you're interested in, right? They one 0:54:59.060 --> 0:55:01.380 word answers are good, but we have to get students 0:55:01.380 --> 0:55:05.299 talking about the math. Notice, notice this is hard, right? 0:55:05.340 --> 0:55:08.100 This is hard as a teacher, right? It requires actually 0:55:08.100 --> 0:55:12.419 more teacher involvement to teach Socratically a few takeaways here. 0:55:12.420 --> 0:55:15.299 Number one, a deeply rooted understanding of and love of 0:55:15.300 --> 0:55:20.980 mathematical truth requires an exploration of the mystery that surrounds it. Okay. 0:55:22.340 --> 0:55:25.200 Number two Questions with opening and follow up are the 0:55:25.200 --> 0:55:28.719 single most important thing that distinguishes a wonder based pedagogy. 0:55:29.040 --> 0:55:31.239 If you're interested in wonder, you want to try to 0:55:31.239 --> 0:55:35.799 teach socratically. Okay. Number three. I think this wonderful teaching 0:55:35.800 --> 0:55:38.400 is going to require some habituation. The habit of teaching 0:55:38.400 --> 0:55:41.480 with wonder is a shape. It's not a formula, but 0:55:41.480 --> 0:55:45.000 it is a repeatable shape. That thing is repeatable. It's 0:55:45.000 --> 0:55:47.920 new every day, but it's a repeatable shape. It's a 0:55:47.920 --> 0:55:50.719 thing that students can become habituated to. Number two, I 0:55:50.719 --> 0:55:54.040 think you can build a culture where students are going 0:55:54.080 --> 0:55:59.000 to respond to questions. This requires perseverance, but students can 0:55:59.000 --> 0:56:02.600 get in the habit of doing it. They know that 0:56:02.640 --> 0:56:05.040 you're going to be asked to explain their thinking, to 0:56:05.080 --> 0:56:09.960 do mathematics. Teaching is always inventive, but I think it 0:56:09.960 --> 0:56:13.720 adheres to a set of questions that can become ingrained 0:56:13.719 --> 0:56:19.400 in teaching practice. Let me finish with the end of 0:56:19.400 --> 0:56:23.370 this quote. Right. The capacity to wonder is among man's 0:56:23.370 --> 0:56:28.130 greatest gifts. It signifies that the world is profounder, more 0:56:28.130 --> 0:56:31.770 all embracing and mysterious. And the logic of everyday reason 0:56:31.770 --> 0:56:32.930 had taught us to believe. 0:56:33.290 --> 0:56:36.969 That was Doctor Jonathan Gregg, assistant professor of education here 0:56:36.969 --> 0:56:40.690 at Hillsdale College. Thank you for listening to the Hillsdale 0:56:40.690 --> 0:56:46.330 College K-12 Classical Education podcast. I'm Scott Bertram. We invite 0:56:46.330 --> 0:56:50.210 you to like us on Facebook. Search for Hillsdale College, 0:56:50.250 --> 0:56:54.210 K-12 Classical education. You also can follow us on Instagram 0:56:54.210 --> 0:57:00.730 at Hillsdale underscore k12. That's Hillsdale underscore k12 on Instagram. 0:57:01.090 --> 0:57:05.049 Thank you for listening to the Hillsdale College K12 Classical 0:57:05.050 --> 0:57:13.050 Education Podcast, part of the Hillsdale College Podcast Network. more@podcast.hillsdale.edu 0:57:13.210 --> 0:57:15.169 or wherever you get your audio.