 This is not so much to give you things you can do, activities and such, but more addressing the issue that we all have to face, which is that kids learn math at different rates. And what does one do about it? So I'm going to share what I've learned in my decades. In the high school classroom, I'm retired now for a few years from the classroom. I do other stuff in math education. So the standard responses, I know of three fairly standard ways that people deal with the fact that kids learn at different rates. One is tracking, where the concept is that kids are placed, quote, where they belong, which there's many problems. First of all, the tracks are not parallel, but they diverge. So as soon as you track, the kids in the upper track move faster, the kids in lower track move slower, and pretty soon, they're not even doing the same thing within just a couple of months. Anyway, and as you'll see, if you haven't yet, in NCTM's new high school catalyzing change in high school document that's coming up, I've been a member for a very long time. And for the first time, they're coming out questioning tracking, which I think is a healthy development. But anyway, I talked in a school where we did not track and learned a lot in the process. Another standard solution is acceleration, where some kids are put a year, two years ahead of their classmates. And I think to a limited extent, maybe having kids one year ahead from others, that's, in my view, pretty much OK. But the problem is that it's gotten out of control. And in many schools and districts, they have maniacal parents who really want to push the kids way beyond what's reasonable. And so you see these two dogs here, racing is not a good way to learn math. Digging is more appropriate. So anyway, and then the third thing is so-called differentiation. When I was young, it was called individualization. And basically, when you're trying to give different work to different kids, and it's problematic in a number of ways. First of all, it's too much work, because one class becomes many classes. It's just too much prep. Another problem is stronger students tend to resent it. Like, why do I have to do more? And then you can't use the classroom as a learning community. So it's problematic. So what I'm going to present is a non-traditional response to the fact that kids learn at different weights. And I'll just start by saying, it's true that kids learn at different weights. That's just the fact we all have to deal with. So the philosophy underlying the approaches I'm going to present, this kind of a strategy aligns with your stronger students' support for the weaker students. Both thoughts are super important. You can't omit one or the other. There are many kind-hearted teachers whose heart goes out to the weaker students and figure, well, the strong students will just be themselves. And that's not true. The strong students are still kids. They still have responsibility. And they're crucial in many ways. Politically, they're crucial, because their parents are the ones who are likely to put pressure on you if you're not challenging their kids and complain to administrators and so on. They're important philosophically, because they're the next generation of math teachers, among other things. They're essential from that point of view. It's part of our job. And they're essential pedagogically, because they're the ones who can help you teach the class and have the class move forward. So it's crucial to have them on your side. So one way is what I call the Goldilocks strategy, that every day you try and do something that's too difficult, something that's too easy. What's hurt if you do something that's too easy? Nothing. What's hurt if you do something too difficult? Well, nothing. Not everybody has to get everything. But if you do something that's too difficult, it keeps your stronger students engaged and interested. And of course, something that's just right. But the worst advice I ever got as a beginning teacher was, aim for the middle, I was told. That's just wrong. Aim for all the levels. Though you think of it also as an elevator, you have to stop them on the floors. So some ways to do this, one is that I'm not going to talk about. One is the importance of classroom discourse. So number of talks, good discussions that you have with the whole class about challenging problems or difficult concepts, whatever. I mean, that's essential. Another key ingredient is good material, rich problems and activities, so-called low threshold, high ceiling, or group-worthy activities. So no time for those things. But not because I don't value them, just because that's not what the talk is about. Group work, I think, is a kind of a crucial ingredient. So what I've done is randomly selected groups every other week. But I'm told that changing groups every single day works as well or better, so I haven't tried that. But the key is that the groups should be random. The engineering your groups, too many downsides. First of all, the kids start thinking, well, why am I in this group? Am I supposed to be the smart kid? Or am I supposed to be the stupid kid? Or not worth it. Keep it random, and then you have different outcomes that are worthwhile. Sometimes all the fast kids are together, or they're really excited. And that gives you more time for the rest of the class. Sometimes you have some groups that are just really weak. You can help them. You can focus some attention on them. And much of the time, the groups are mixed. So randomness is good. In my approach, students mostly work independently, but are expected to help each other. So you always have access to your neighbors when solving problems. And of course, sometimes they have work where they really need to collaborate. But my idea isn't collaboration all the time. It's just collaboration as needed. And if a group doesn't function well, I just talk to the kid who needs an intervention rather than make speeches. In other words, instead of saying, it's nice to help your neighbors, I say, Lucy, can you help Tom with number three? So you just keep asking for exactly the behavior you want, exactly when it would be helpful. Push your chair in. Anything that will make the groups work better, but you ask it directly from specific kids to do on the spot rather than through generalized philosophizing from the front of the room. And you keep doing this, and they develop good habits, and less intervention is required. And this is not at the exclusion of whole class discussion. When you notice that groups are getting stuck, stop everything and talk as a whole class. Even if when you're talking as a whole class, not everybody's going to get it. But then you go back to the groups, and those who got it can spread the word. So anyway, so group work, I think, is a key ingredient in the heterogeneous class. Another key ingredient is extending exposure to topics. In other words, not limiting something you could just teach and then move on, but you've only taught it to a small number of kids. If you extend exposure, you're reaching more kids without harming anyone. So how does one do that? The philosophical underpinning is constant forward motion. That's crucial to keep the strong students on your side. If your course grinds to a halt because a bunch of kids don't get it, it becomes really boring for the kids who do get it. So you have to have forward motion. You also have to have eternal review because when you move forward, somebody is not quite there yet. So it's not one or the other. You, as a teacher, are drawn to one or the other. I'm certainly a forward motion guy, and I had colleagues who are much nicer than I am who are eternal review people. And both are good. It's not like one, and if you specialize in one or the other, you're not doing your job. You have to do both. So how is one ingredient lagging homework? So you'll teach it. Can you see my pointer here? I'm gonna assume, oh no. Okay, can you see my pointer? Now we can see it, Henry. Okay, so let's say you're teaching a particular topic. Work on it in class during the first week of that topic. Do not assign homework on it at all. That means that the first exposure is gonna be in class so that if it's challenging, kids have each other and have you to help. The next week, you're starting something new, but the homework is on the previous week's topic. So now you've doubled the exposure because week one, we did it in class. Week two, we're doing it in homework. And when I say week, this is approximate, but basically something like a week. And then don't quiz right away, quiz in week three. And then ask for test corrections in week four. At my school, we called these Recycle. And so now you've done four weeks on the topic which if you believed your own fantasies, you could have done it in one week. You could have done like taught something, given it in homework, quizzed and thought that everything was fine. But in reality, there's somebody who needs the four weeks, somebody who just needs two weeks, somebody who just needs three weeks, but you stretch it out like this and everybody catches up. So maybe not everybody, but way more people than in the one week approach. But the beauty of this is it doesn't cost you anything because it doesn't take extra time. You're just rearranging the time. So it's not gonna slow you down at all. In fact, it might speed you up because you're moving on to the next topic. You're constantly moving up forward. So anyway, so lagging homework and assessments, that's one ingredient to increase exposure. Another is use cumulative tests, maybe not every quiz, but once in a while, you have a test or other assessment, something they do at home or whatever that is about everything or it's about many things. So that topics don't go away. And if topics don't go away, eventually kids resign themselves to learn them. That's the reality. While if you hear the question, how long are we gonna be doing logarithms? That's a kid who's already given up. I'm just hoping you're gonna move on. But we don't give up. We want them to learn it. So extend exposure. Another way to expose exposure that's really powerful but mysteriously not well known is separating related topics. The normal thing which is done in textbooks and seems normal just because we're used to it is I better teach scientific notation because we just did exponents. Or I should teach cosine and tangent all at the same time because it's all related. Or logarithms are just the inverse of exponentials. We just did exponential. Let's do logarithms now. Bad idea. I know everybody does it. It's still a terrible idea. The reason it's a bad idea is that as soon as you separate them, you're really helping everyone. You separate them. It makes it more interesting for your stronger students because you don't seem to be staying on this one thing forever. And it's great for your weaker students because when you come back, like at my school we did the tangent ratio in one semester and then the sine and cosine in another semester. Well, when we did the sine and cosine, we had to review the tangent. So that made sense. You'd say, remember, you might do a review it in homework. So then you've extended exposure to the tangent and doing anything more than once is more effective than just doing it once. So anyway, separate related topics. Again, this takes no extra time. And making support available outside of class because no matter, even if you do all these things, some kids still are gonna need more help. But much less so than if you were trying to march through the usual way. Okay, so these are structural strategies to extend exposure. So it's rearranging things. And they don't take more time and they carry the message of gross mindset which has all of a sudden become very popular. I visit schools a lot now in my retirement and I see these bulletin balls. My brain can grow and so on. So gross mindset, sort of pep talks on bulletin balls. And, okay, but more effective is to actually make it possible for the growth to occur. And when you have the kind of setup I'm suggesting where you're taking, giving kids a lot of time to learn stuff, they get the idea, I can do this. I don't have to do it by Friday. I can do it, I'll get it eventually. So it sort of moves away from the maniacal time pressure which is both in the calendar. You have to know this by Friday and in the clock. You have to finish the quiz by 10 o'clock. You still have the corrections to do next week. So anyway, that stuff you can start doing tomorrow if you want, I mean, it's just rearranging things. Unfortunately, it's not quite enough. All classes are heterogeneous, even if you track, you're still gonna have a range of kids. And the suggestions I've made up to this point have the advantage that they're easy to do if you have the, feel yourself have the growth mindset to do things differently. But they're not sufficient in and of themselves if they're not combined with some changes in pedagogy. And so I'm gonna focus on one particular approach which has been a big part in my teaching, using tools, learning tools. So I'll just give some examples. So manipulatives, I taught elementary school for about 10 years before coming to high school. So I had some exposure to manipulatives, for example, the pattern blocks. So day one of my geometry class, we did an activity which you can download from my website called angles around the point where you surround the point with these blocks. And that forces you to pay attention to the concept of angle. And I don't know if you've had the experience where kids can't seem to use a protractor. And the reason they can't is they don't exactly know what they're measuring. This an activity of putting these blocks around the point makes a lot of points, so to speak. And you sort of get a feel for what the angles are. Okay, so the manipulatives are helpful kind of the ground floor to learn what the angles are. But then you can do some pretty fancy work with the same manipulatives. If you look at this arrangement of pattern blocks, it can be used to figure out what if there was pi for regular polygons. So the pi for the area of a regular polygon, for a regular dodecagon, which is what you see in this image, that happens to be exactly three. That's a really fun exploration to do, supported by these same manipulatives. And so high school kids, I mean, I guess you guys sort of range from middle school to high school, but high school kids, you might think that, oh no manipulatives, that's baby stuff. In reality, you bring out the pattern blocks and the same kid who's like, wearing heels and makeup and looking as adult as she can, she sees these and she goes, yay, Kindergarten blocks! And so there actually is a chance for them to not be so grown up. Here's another one. These are pentaminos, traditional thing in recreational math and really good to teach scaling. So here's one of them. They all have area five. They're made of five individual squares. So if, but if I double the dimensions, instead of one inch, I have two inches. What happens to the area? Well, now I'm, you know, this was five square inches, but here it's four times five, even though I only doubled the dimensions. That's because I doubled width and height. And so it's the ratio, it's square. Then you can triple them and then you need nine pieces to cover it. So I spent a period just solving these puzzles and then we put, you know, kids solve them, color them, put them on the bulletin board. And then we have that as a reference as we work on scaling and what happens to area. For some reason a hard idea because it's not straight proportion, but having that reference on the wall is super helpful. And it does take some class time, but worth it. But also it can be taken further. So these are, they're called the super 10 grams and they all have area two, two square inches. They're made up of four half squares, you know, four white isosceles triangles. And so here they are scaled. So the first one is just one piece. The second one is two pieces. And the third one is four and then it's eight and nine. So remember what happens when I double the dimensions from here to here, I need four times the area. So what did I do to the dimensions to go from here to here? It's only twice the area. And if this is one inch, what's my perimeter for this shape and what's my career for this shape? So all of a sudden you have, you know, you're starting with these puzzles of manipulatives and you have to deal with a scaling factor of square root of two. And that gives you, you know, and then, you know, if you know that this is square root of two inches, then this is two inches and so on. So, you know, there's a lot of, I mean, I'm sorry, this is not two. Yeah, it's two. So anyway, so you manipulatives can start at a basic level and get pretty fancy. Here's another example, algebra manipulatives. This is the lab gear, which I created. So you ask the students to make a rectangle. Everybody knows what you're saying. Do that before you say, today we're gonna learn the distributive law. You say, let's make a rectangle. And so this is one of the possible rectangles that we could make. And then we measure the sides in the area and then we find that two x square plus four x is two x times x plus two. You do this a bunch of times. You make a bunch of rectangles and pretty soon the kids have a reference on how the distributive law works. And it's no longer a thing to memorize. It's just a thing you get. And, but then at a more advanced level, here's an image of rectangles. So these are all made with x squared, eight x's, and then some ones, different numbers of ones. So I have kids make, the instructions are, make a rectangle with x squared, eight x's, and as many ones as you want, including zero. And pretty soon we have all these rectangles, including a square. Okay, but now let's look at the graphs that correspond to these. And at that point I showed this image. Let's look at the graphs and let's see if there's a relationship between the graph and the blocks. And, you know, if you look at it, you'll see that there are plenty of relationships. You know, you can find the intercepts, the x-intercept, the y-intercept, the vertex, all that information is lurking within these blocks. These are the blocks down here rearranged. I attempted to arrange them into a square, couldn't do it, except in this case. And look what happened to the graph. This case I was missing one and look where the vertex is. It's just one short. Here I was missing four and that's where the vertex is and so on. So a lot of math in this one figure, this one image, and a really good way to review many ideas. You know, what's an intercept? You know, how is the vertex related to the x-intercepts, et cetera? How does this all connect with completing the square? So again, using manipulatives is helpful to your weaker students. It's also helpful to your stronger students because you can get a lot of debt. Oh, another of this example, which I came up with decades ago, has now become a standard thing that many people do, which I'm very excited about. The idea is make a square, make as many squares as you can on a geobold. And then find their areas. So it's a little tilted because I took a photo at an angle, but this here is a square. And that's already interesting. How do I know these are right angles? Anyway, and then I embedded it into a bigger square to find its area by subtracting these triangles. So I have students find as many squares as they can and their areas, as many sized squares as they can and the area. And they do this a bunch of times. And pretty soon you have the proof of the Pythagorean theorem which is the same thing with letters. Now I used to do it, the proof with letters and the kids politely nodded or looked out the window. But now they all get it because they've done it like 10 times with numbers. So this becomes a completely obvious proof that they totally get. So from reaching a small number to reaching a large number thanks to manipulatives. So, okay, technology, I mean we're all on technology as we speak. I see it as a way to make things more visual like manipulatives can. Different from manipulatives because electronic technology is hypnotizing. Physical manipulatives get kids talking. Electronic tools make kids focus on the screen, less so with smaller devices. But anyway, you can make things visual, you can make things interactive and you can give kids an opportunity to be creative. So here's for example, completing the square made visual. So we do this after doing a lot of work with the physical blocks about completing the square. This is an algebra two thing. I've long given up on teaching, completing the square with meaning to younger kids. But algebra two, you can do it. And we started with x squared plus bx and we rearranged it. So we needed to divide the b in half in order to be able to make a square here. So you do this a bunch of times with blocks and then you have kids look at this and write up their explanation of it. So all 3D stuff, so this is about the volume of a pyramid. You can see that the volume doesn't change when I keep the base and the height constant but it does change if I change one of those. And then you can sort of work towards a proof by seeing how a prism can be divided into three pyramids. So there's still work to do but at least you can see how the proof works. This is an example of an activity which is probably the most popular activities I've designed where you tell kids you can make these with y equals mx plus b, just you have to do it. You have to use many, many functions and it's better than the, there's a style of using the computer which I hate which is graph y equals x. Now graph y equals 2x. Now graph y equals 3x. Now graph y equals 4x. What do you notice? The kids might say, well, the line is getting more jagged. I mean, that's a legitimate thing to notice. Not what I'm after but they're not doing the work. You're doing the work by telling them what to graph. In this design, in this activity, they have to make the design, they have to come up with the formulas and it's a more interesting, by far, because they're not being guided by the nose and it's fun, you get these interesting images and it's more effective because you're doing the thinking, the student is doing the thinking. And notice, I don't say A, B, C, D because I like them to choose, so they find their level and they don't have to do them in the same order so it reduces, which one are you on? The unhealthy competitiveness. And there's like hard ones working within. Like this one, you have to know exactly what you're doing to get the intended result, this bottom one. And this one is easy unless you wanna do it with Y equals Mx plus B, which in which case it's hard. So anyway, I tell them, oh yeah, you can skip this one. And then I say, yeah, but last year somebody got it. Anyway, and it's fun. So, and then it's a place for creativity where you can start with an empty screen. And this was done back in the day, we're using the Capri software, but you could do it in GeoGebra where construct a circle through a given point, you know, tension to a given line. And that's hard to do. This is like the end of our construction unit and the kid figures it out. Like it's like so proud and walks around the home telling you already the strategy. And then the amazing thing is that you have actually the locus of these centers is a parabola. And you know, we don't do parabolas in my geometry class but they've heard of them. And it's kind of an amazing thing that we made a parabola from just points and lines and circles. So anyway, the sort of thing that works best with technology. And then, you know, there are some schools and districts where kids are not allowed to use technology because it's a crutch. And, you know, like you can all the schools where you can't use a calculator until algebra too. You know, for example, this happens. And, you know, you shouldn't use a calculator to do 10 times 32. I mean, I agree with that, you shouldn't. But if somebody has a broken leg, we let them use a crutch and you have more chance to get across to the kid who does 10 times 32 using a calculator if you're not torturing them about it. You know, the calculator is allowed and then you have a conversation about powers of 10. You know, they're more able to listen to you. So my approach is calculators are legit anytime unless I say no calculator because there's some lessons that I want to do with no calculator and kids totally get that, it's not a problem. But the default is use whatever technology is appropriate. So anyway, manipulatives, technology and other tools which are neither. For example, this is called a function diagram. And, you know, these two forms are familiar. The table and the graph. So like on the table I have point, this is my formula. And then two, for plug-in two, I get one. That gives me this point. I can connect the points on the line. This is the same thing as the table just this is my Xs and this is my Ys. My axes are parallel instead of being perpendicular. And there's all kinds of interesting learning to do using this tool to my website. But many, many really interesting lessons all the way from basic arithmetic to the chain rule in calculus. So anyway, so this is a part of my tool-based strategies. But the tool is not a physical, it's just pencil and paper. All this, this is the 10 centimeter circle where you can actually see, you know, with your eyes the sign, the sign, the cosine and the tangent. They're right there, you just, you can count them. You can measure them with your centimeters. So you can do, you can give them, before they've even learned any twig, as an introduction to twig, you can give them one of these, how high is the flagpole problems and then show how this device can answer the problem. And you can find that ratio by just reading it off on here. So, anyway, so the idea is that you have to, you know, if you, if you, anything important, you should teach more than one way. If it's important and you're just doing it one way, it's not, you're not gonna reach as many kids. So multiple representation will do all these things if you use the, you know, if you use many representation. The reason I highlighted geometric is because this one is underused. These are reasonably fashionable, you know, numeric, symbolic and graphical and applied, you know. But, you know, not for geometric representations when you count. Anyway, they do all these things, you know, multiple representations of different ways to get into something, a way to review something without having it be boring. You have to do a lot of review, but it doesn't have to be, you don't have to review the same way as you did it the first time. That's just, you know, it's boring to the kid who got it the first time and you lose them and you lose, you lose that kid, you lose all class criticism. So review is essential, but do it in multiple, in other representation. It's a way to extend exposure. It's a way to deepen understanding that the beginner knows a way to do something. An expert knows many ways. So it's a deeper understanding. And it increases variety, you know. So the whole tool thing and multiple representation, one of the benefits is math class isn't every day the same. Work, the moment I remember from my beginning teaching, I was in my second year of teaching high school and I was teaching algebra one and this girl comes up to me at the end of class and says, you know, maybe second or third day, is every day gonna be like this? And I write hard things because yeah, everything was gonna be like this and that got me going. I'm looking for more variety in the math class. So anyway, tool rich pedagogy, you know, gets more motivation because it's more fun and interesting. It lowers the threshold, gives more access to more kids, raises the ceiling, more challenge and deeper understanding. So anyway, I think number one, extending exposure, number two, use tools. Those are my messages. Bad news, tools are not magic. It's not, like if the kids do the activities with the pattern blocks or with the geovol, doesn't mean they learn anything. It just means they're engaged. It's all in the discussion that you have to lead. It's all in the applications. It's all in the, you know, it's not using the tool that's not confer knowledge by magic. It's just an arena for discussion and reflection and collaboration which is where the math happens. So I'm not trying to claim any magic here. Periodically, you have people, you know, sort of consultants who will say, you know, do this and all your problems will be solved. That's not how it works. You have to know a lot of stuff and you have to use a lot of stuff. And, you know, that's what makes being a teacher fun is that you, you know, there's all this stuff you can learn on how to do the job well. So, so add tools to your toolbox that don't expect miracles. And then just to do, to use tools and multiple presentations seriously, you pretty much have to teach fewer topics, which that's hard because you may not have a choice. And even if you have a choice, how do you decide? You know, but fewer topics, if you pick the most important topic, use your chance for more depth on what you teach and reaching more kids and kids are remembering stuff. So how, how, how to proceed? Well, okay, my, my, my choices, less paper, pencil, multi-digital arithmetic, less complicated factory. So I can't believe how many hours people sink into factoring increasingly difficult expressions. Less complicated equation solving, less complicated manipulation of radicals. The concepts are important and ever more important. The manipulations less so because they can be done by machine. So, so, so that's one way to, to narrow your, you know, to gain some time to, to teach better. Another way is let go of obsolete topics. You know, there are topics that just are no longer relevant, you know, because of technology, you know, like, I don't know, the rational theory, you know, you can look for the roots, whether the rational amount by looking at graphs, you know, that it's no longer a crucial topic. So if anything, for example, isn't in common core, please don't teach it. And even if it is, in some cases, you shouldn't teach it. Topics are not connected to anything. Now the most important topics should be interconnected with each other. Topic that's just sitting out there and not related to anything. Maybe you can let that go. Topics that you do because you did it last year. I mean, many of you are young, so that's not an issue, but believe me, many teachers do things because they did them last year. Super hard topics, save them like topics where you know only two kids are gonna get it. Don't waste everybody's time on it, you know, but, you know, get these kids to come to math club and meet with them during lunch on those topics. And then you may have some personal favorite favorites which you should drop. I have a question mark here because if you love something, it's probably worth doing, but maybe not every single one of your personal favorites. So anyway, that's all I have to say. So how am I doing on time? Oh, not bad. Did great on time. Thank you so much, Henry. This was fascinating. I'm so excited. I really personally like your make the design lesson and I'm gonna already be using that. That was phenomenal. So thanks so much for sharing. So I'm happy to answer questions if... Oh, wait, wait, before I answer questions, if there are any, I'm doing two summer workshops this summer. One on visual algebra, grades seven to 11, and they're both in Argus. And the other, algebra two, Treg, Precaf, probably called No Limits. And so there's information about that on my website, matheducationpage.summer. And links about everything I said, you can find at matheducationpage.org slash talks.html. And if you care about what I did, at PCMI in 2006, you can look that up on my website as well. So anyway, but okay, so any questions? I have a question, Henry. In terms of use, everything you said is like spot on, I love it. How do you work with a district or a school that's very rigid or old school and get them to come around to this? Do you have any tips of how to approach because I think that's going to be an issue with many schools? Yeah, so here's what I would say. If you can only make small changes, make small changes. If you can find one colleague and engineer it so that next year you're teaching the two sections, different sections of the same thing and you collaborate, you have no idea how that can have an impact because what happens is it's sure to go better for you and the colleague and people are gonna stop noticing. At least that close my experience. If you can change a district, change a school, if you can change the whole department, change one person and you just have to stop. And then if you can't do any of that, do the best you can within your class. But part of it is, I don't think it's, there's no quick way to say, let's all do this tomorrow. Nobody's gonna do it. People are busy, people have believed their own beliefs. They have their own experiences, they believe what they do work, otherwise they wouldn't be doing it. So you kind of have to lead by example, but it's a lot more powerful. As soon as you have two people, it then becomes, oh, she's interesting. As soon as you have two people, it becomes the thing that people can do. And so anyway, those are all my thoughts. Thank you. Henry, this is Kathy. I'm interested to know about the curriculum that you perhaps used when you were in the school district and not tracking math classes. Yeah, so, I guess the short answer is we developed our own stuff. Much of it you can find on my website, but it's not a, you won't see it as a one whole coherent thing. We draw a lot from my algebra book, which is free download on my website. We drew a lot from my geometry labs, free download on my website. And we have other things, other materials that I'm also sharing on my website. Our experience was that with the algebra tool was our toughest challenge. We tried one textbook after another and it was really hard. It was like too hard for some kids, too easy for other kids. That's really what pushed us to develop our own stuff. Now, I mean, I don't know what's available, unfortunately, so I mean, my experience was largely with materials I developed. And I'm hoping that with NCTM questioning tracking that more materials will become available for that. But, I don't know, I mean, I think that the, I mean, I would say that I don't, I'm not that familiar with other curriculars, but for example, the CPM books are geared, more discovery oriented, more group oriented, so likely work better with the heterogeneous class. But frankly, don't ask me any more than that because I don't know. I suspected that would be what your answer would be. And I'm so excited to reconnect with you after all these years. And I think I just need to take advantage of your website more. Yeah, there's a ton of stuff there and now that I'm tired, I'm happy to answer emails. So, yeah, you know, this is part of my motivation for leaving the classroom. I was a teacher for 42 years and then I thought, well, now I'd like to work with adults. So, if that, you know, I'm happy to be helpful. I appreciate that. You have me very excited to work with young teachers. Right. So how does this work? I'm not seeing everybody who talks. Oh, I don't have my video on Henry because, you know, when it's still 10 below and I haven't taken a shower, it's not a problem. Okay. Sean, you had a question, right? Yeah, can you guys hear me? Yeah. Okay, so with lagging homework, do students get confused by learning new stuff but doing the homework from the previous week or how do you talk about that, like keep track of that? Yeah, I love that question. So, I mean, there's two levels to the question. One is, do the students get confused? Do they get confused? In other words, in the first week or two, my whole school did the lagging thing after once it became established that how well that worked. But the first couple of weeks for the ninth graders and new kids in the school, they were really baffled, you know, like, what, this isn't a, what, you know, this was just, and that's really revealing about how deep that culture goes. It's, that's math class in the US, you know, for some reason. So, however, they quickly get used to it after a few weeks, that's just how this school works. And then they became more convinced than you are, you know, like, if I say, okay, and you're gonna have to do this on homework, today, you know, you want us to do this, today we just learned this, you know, and same with quizzes, you know, like, you know, we are struggling with something difficult and they say, it's not gonna be on the quiz tomorrow. I said, no, no, of course not. It's not even gonna be on the homework, you know. So, over time, this reduces a lot of anxiety for the kids because they trust that they have a chance, you know. And it allows you to make things challenging because, you know, they're less freaked out, you know. The other thing is, the idea, like, you know, there's a certain style of teaching which is, I'm gonna, we're gonna do this intensely, class, homework, quiz, while you still remember it so you can do well on the quiz. That is insane. That is completely insane. If our goal is for them to do well on the quiz, let's just give them the answers, you know. They need to remember this in the future. That we're teaching this, not for the quiz, we're teaching this for next year and the year after. So, the fear that they're gonna forget it from today and between now and next week, something is wrong with that picture, you know. So, this part of it is kind of a culture shift that we're teaching this so you'll understand it and you'll always know it. And that's different from we're teaching this so you'll know it's, you know, in the immediate future. Cool. And then, I guess a follow-up would be, I would like worry about some students just totally falling behind if I didn't have good ways of checking in and seeing that they're staying with it. Do you have any suggestions there or ideas that you use to check in on their understanding? Well, so when you're in class, I mean, that's the thing. If the work is happening, I mean, this is like part of the people who do the flipped classroom, which I had long periods that didn't need to flip the classroom. But one of the advantages which you can do, even if you don't flip the classroom, if they work right there in front of you, you can see, you know, you walk around, you see who gets it, who doesn't get it. Number one, number two, you're still checking homework, however you do it. My approach was I walk around and kids get zero, one or two on their homework. And you can see, you know, the zeros, kids who didn't do it, is it, you know, then you've narrowed, you've narrowed the number of kids you need to look into. Did they not do it because they couldn't? Did they not do it because they were flakes, you know? And, you know, you still have quizzes. I mean, you're still keeping track. It's really not that different from what you're doing. It's just rearranging the time. So, you have more information if the initial work is in the classroom. If the initial work is at home, and let's say they copy it from somebody, you have no information. If the initial work is in class, you see it. So, I'd say it doesn't take anything away to do it this way. It gives you more information. That's my experience. And yeah, so any others? Okay, sorry, one more. No, go for it. So, I started using equation maths and algebra problems this year. And I noticed that a lot of students were like, I hate this, even though they were totally struggling on combining like terms and things, they were just really against using their hands and drawing pictures. So, I was thinking that it'd be great if next year I could introduce it as a game without even talking about solving equations, kind of like how they do at PCMI, how we play with things, then we realized, oh, we've been doing this all along. So, I was trying to come up with the rules for equation maths. I don't know, so far I came up with, you can flip everything if you flip it all at the same time. And then you have to find out what the rectangle equals, the green rectangle equals. Yeah, so, what level do you teach? Algebra one. Middle school, high school? Middle school, algebra one. So, they've already been kind of ingrained with how do you solve an equation? See, okay, I'll just say something, you feel free to ignore it. In my view, starting with equation solving is a mistake. Equation solving should be late. Do stuff, for example, I mean, if you start with the manipulative, you say make a rectangle, that's my interesting. It doesn't even look like math. And it's gonna be different kids from the kids who are good with Xs and Ys who are good at making a rectangle. And you're building from that, you make a lot of rectangles and you discuss them about length and width and length and width equals area and so on and so forth. It's only after you've done that that you reverse it and you say, what's X plus two times X plus three? Because that is already a slightly higher level question. And if you've made a lot of rectangles, that's not that hard, but you have to have made a lot of rectangles upfront. So, I would say doing work with manipulative number one. With manipulative number one. Number two, doing, apparently now, I think the common core lingo is modeling where you start with numbers and problems. And with numbers, and then you get to the Xs. You don't start with the Xs. You start with numbers, you get, you know, you study a particular situation, you get a table of numbers from the facts of the situation. Like I get 10 cents an hour for washing dishes. But they pay me a dollar upfront. Okay, so, you know, let's make a table. How much have I made, you know, after one hour or two hours or three hours? Let's graph that. It's only at the end of that that we talk about, you know, what's the formula? Because we have the numbers to work with, it becomes more accessible. It also seems like there's a reason for it because we're figuring out the dishwashing situation and so on. So equations, you should do a bunch of that before you solve equations. Because by then, if they've worked with the manipulatives and they've worked with the, you know, so-called real world problems, solving equations has, you can put some meaning on it. Otherwise, if you start with equation solving, it's like pleasing the teacher. They want me to put, as they put it, am I supposed to put, to get x by itself, you know? And what are the rules for this? And there's like so many rules. And is this a one-step equation or is this a two-step equation? You know, I never even teach any of that stuff. But by doing it later, you teach, you can have general rules, like you can change all the signs or, you know, and often they can find rules themselves and then, you know, with the blocks and then you translate it into the symbols. So, I don't know, I mean, go to my website, take a look at some of the algebra stuff I have on there. It might be useful. So. Hi, Bob. Can I say something? Can you hear me? Yeah. All right. First, I am so happy I did not miss this session. I really, I really learned a lot from your presentation. And when you were explaining about, a start with an equalities or real life problem where everybody can do the math, simple math, it just gave me a clear idea of what is the low going to high ceiling tasks when they get to the formula in the end, talk about the formula in the end. The way you explained it, that gave me a clear idea of what is low, high ceiling tasks. Thank you so much. Great. Okay, well, maybe we have a lot of questions. Am I considered going to your workshop? You're very welcome, but we have a few, it's costly, but we have a huge discount for the first 10 or so public school teachers. So there's still some scholarships for public school teachers. So can you move quickly? Okay. Even though that was, that would be my birthday. Uh-oh. That day, it's going to be my birthday, I will still consider going there. I could get this scholarship. Thank you. Okay, so I guess this wraps it up. All right, thank you so much, Henry. This was absolutely phenomenal. All of this will be posted, the recording will get posted up to our YouTube channel. Henry, if you wouldn't mind sending me your slides so we could also post those on our PCMI website. If that's okay with you, you can send those to me and then I'll get those posted. Yeah, they may already be on my website and if not, I'll send you the link. If they are, I'll just send you the link. Is that okay? Okay, then we'll have them, we then put the link, we'll put the link up on the PCMI pages as well so everything's right there with the recording. Okay, great. Thank you so much. Okay, thank you, bye-bye. Thanks everybody, this was great. Thank you, Henry. Bye-bye.