Zandra de Araujo: All right, welcome! Glad to see so many of you today at 3:30 on a Tuesday. I know we’ve changed the time on this one, so hopefully you are able to join live if you haven’t previously.

​We’ll get started in just about a minute. Just about a minute. 

If you want to introduce yourself in the chat, feel free to do so. Maybe your name and where you’re joining us from, and you can share your role if you’d like as well. I’d love to see where people are from. If you have a minute, you should be able to access the chat.

​Thanks for getting us started, Olga. I appreciate that.

​Mia! Nice to see you!

​Awesome! Thanks, y’all, for putting your names in. It’s nice to see a lot of familiar names, if not faces.

​We’re talking about a topic today that I’m very passionate about, so I’m very excited for you all to join today.

​Awesome! So, I know your time is short and valuable, so, you can continue introducing yourselves in the chat if you’d like. It’s nice to see connections, and you can, you know, reach out to people that you are familiar with. I’m Zandra De Araujo, I’m the Director of Mathematics at the Lassinger Center at the University of Florida. (Sorry if the university busted your brackets like it did mine this week for March Madness.)

​I’m joined by Hope from our communications team, and she can help me navigate the chat and admit people into the room. Also online are Jose de Leon and the math team are having a group watching party today, because we had a team retreat. So, you have lots of good company with you today. The people that brought you the math matrix are all kind of in the room, so to speak.

​So, this is the third of the five webinars this spring around math proficiency, and today we’re focusing specifically on designing rich mathematics tests, which is a passionate topic for me. It was related to my dissertation research and other things that I’ve done since that time.

​So, I’m going to briefly recap the first and second webinars. This is going to be a very small snippet. They are all available on demand afterward online. You will get an email with the link to this one, and they’re all on the website, so if you want to watch them again, or if you want to watch them for the first time, you can go to that website and see what I had to say, and always feel free to follow up with me and email me.

​The first webinar was on the left-hand side of the screen here, which is mathematical proficiency. We talked about what mathematical proficiency is, and we talked about it being a rope from the National Research Council’s lovely report from 2001, adding it up. And so, math is more than just procedures, it’s also conceptual understanding, strategic competence, productive disposition, liking mathematics, adaptive reasoning, being able to see math as useful and applicable to the real world and things like that. 

So we talked a lot about that in that first webinar. And so here’s a cheat sheet in case you need it, but again, they’re on demand, so you can go back and watch that.

​Then we also talked in that webinar about how I asked ChatGPT, like, what are the characteristics of a brilliant mathematician? What distinguishes a brilliant mathematician? 

And so, these are the five things that came up. A deep curiosity and passion for understanding, persistence, creativity, active engagement, deep thinking, patience and sustained focus.

 And we talked about, what does that mean for our classrooms? Are we doing things in our classrooms every day that kind of align with this vision of mathematics? And if not, what might we want to start doing to better align our classrooms with this view of mathematics from time to time so students start to love mathematics as much as we do?

​In the second webinar, we focused on math discussions. So math discussions are hugely, hugely important if you’ve taken the Math Matrix, which is our statewide micro-credential. If you have questions about that, you can reach out to me.

​We talk about math discourse a lot, because it’s how we learn and communicate our learnings in mathematics.

​We talked about the teacher’s role in this, in questioning and explaining mathematical thinking with students or fostering students to do the explaining, as it were. We talked about some common math talk moves that you might use, such as repeating students’ reasoning so the rest of the class can hear it, having another student repeat it, adding on to their reasoning, applying their reasoning to other situations, providing wait time or restating something in other words to see if that captures the sentiment the students were trying to voice.

​These were good moves that teachers should keep in their back pocket to kind of get students to explain their reasoning in the mathematics classroom and get them talking about mathematics.

​We also talked about closed questions and open questions, and how in most mathematics classrooms, we use a lot of closed questions, like, “how did you solve the problem? What’s the answer? Etc.” And we want to move toward more open questions, like, “tell me about how you solved that problem, or tell me about another way you could have done that. What have you tried so far?” Those kinds of things.

​And finally, we talked about, you can ask questions about the process students do to solve a problem. So, like, “what kind of thinking did you use? What kind of tools did you use?” That kind of thing. Or you could just ask about the product that they had. “What answer did you get?” “What did you draw?” Etc. 

We know that we typically use too many closed product questions, and we want to start to pepper in, if we’re not already, more open process questions into our classrooms. And so that was kind of a big point of the last webinar session and we also talked about some strategies to do so. So again, you can go back and watch that on demand if you want to find out more.

​Today, we get to talk about tasks, and tasks are some of my favorite topics, as I’ve said before.

​So that’s today, math tasks. Just so you know, next time is going to be about connections, and then we’ll wrap it up with math habits at the end. So I’ll have more information on the next webinar at the end of this one.

​So I say tasks, and people might think of tasks in different ways. So this is an example of a task.

​What I mean by task, it’s just a segment of a classroom activity that students are engaging with to learn something. So here’s a classic task. It’s called the staircase task. You’ve probably seen it before.

​This is an example of something that could be called a task.

​This is also an example of something that can be called a task. So, choose two equations to create a system of linear equations that you would solve by substitution. (I give a shout out to my colleague, Sam Otten, from the University of Missouri. He has a really nice article where he and his brother wrote about this task, and I love it. I use it all the time. Thanks, Sam.)

​But this is also a task. Now, these are pretty rich tasks, and a lot of people might be like, “oh, but what about, you know, tasks that look like this?” I would say this is also a type of task. It serves a different purpose, so if you think about the strands of math proficiency.

This is definitely more aligned with a computational or procedural task, but we can do a lot to enhance it, and that’s some of what we’re going to talk about today.

​So a task is just a segment of classroom activity, it’s like a problem, or a set of problems that kind of cohere around some topic.

​What we’re really going to talk about today, and what I talked about last time a little bit, is that we want to focus on rich tasks. Now, at the Center, we’ve kind of defined rich tasks in a particular type of way. We’ve talked about it being at the intersection of having high cognitive demand, like the amount of thinking required, allowing for multiple solution strategies, like different approaches to solving it. So it doesn’t say, use the quadratic formula to solve this problem. It would have more open pathways for it.

​It fosters reasoning and creativity, and it allows for multiple entry points, what a lot of people refer to as, like, low floor tasks, high ceiling. So you can have different ways to get at the solution, because there’s not just one pathway that’s always going to work.

​So we’re going to talk about rich tasks a bit, and then I’m going to give you some specific strategies for enhancing existing tasks to make them more rich.

​First, I’m going to dig into each of these components a little bit more deeply. They’re ways to talk about mathematics tasks, and so I want you to be well-versed in them. 

Not every task should or will be rich. That is very important. Students are not expected to do every task as a rich mathematics task.

​However, what we know from research is that students do not have a lot of opportunities to engage with rich tasks. They get a lot of procedures, a lot of low cognitive man tasks and a lot of procedure tasks.

​If we really want to embed mathematical thinking and reasoning standards and the mathematical practices into our classrooms and get students to engage in deep conversations and rich mathematical discourse, you gotta use some rich tasks, because if you don’t, they’re not going to have a lot to talk about. 

So you really want to use rich tasks strategically and consistently, I would say. Not every task, but consistently throughout a lesson, you do want to engage students in some rich tasks.

​So, cognitive demand is probably the most well-known framework for talking about how much thinking a task requires students to engage with.

​I will preface this by saying what’s high cognitive demand for one student at one level of understanding may not be the same for another student. So it’s kind of dependent on the learner and where they’re at and how much background knowledge they have. So this is a broad framework, but you can apply it as it makes sense.

​So, cognitive demand, the amount of thinking it takes to solve a task. I’m going to start with non-mathematical tasks. There are a lot of tasks that kind of start out and look mathematical, but actually, they’re not really mathematical. They don’t really have a clear math focus, or that focus gets lost as students implement it. So I like to give this example. This is from my dissertation study.

​And there was one teacher, and this task started out as mathematical. They were making mobiles, like the things you hang over a crib, and they were making them in the shapes of quadrilaterals, and they were putting the properties of quadrilaterals on them, and then pictures of them, etc.

​That all is very mathematical. However, they spent a very long time decorating them and things like that, so long that it started to lose the mathematical focus. So common in the literature is that things that start out as mathematical may transform into non-mathematical tasks. Or, sometimes we give students things that seem cool and fun, and they’re actually not really super mathematical in orientation. 

One example of this might be a word find. So, if teachers give a word find, and it has mathematical terms, but the students don’t really need to know what the terms mean or anything mathematical about it, they just have to look for the letters that match and circle them in a word find. It’s not really a mathematical task, it’s a non-mathematical task. It kind of has the appearance of math at first glance, but when you think about the type of thinking students have to engage with. It’s not mathematical, it’s just matching letters, and it could be any word. So that’s what we mean by non-mathematical tasks.

​Memorization tasks, so the next two categories are both what we call low cognitive demand. So they don’t require as much or as complex mathematical thinking.

​Memorization tasks are, like, just regurgitating facts. You’re telling facts that you don’t have to figure out, you just memorize them. That’s why it’s called memorization. They can be solved without a procedure, and they have no or limited connection to concepts or meanings underlying them. It’s really just, like, responding to something. So if you know your times table is really fast, and you’re just repeating them, because, like, they’re memorized. That’s a memorization-level task.

​Something like this for a student that’s already learned their properties of arithmetic. In this case, if you’re just matching these things, it’s something that you know, you’re just recalling this. You’re not, like, kind of having to figure it out or do deep thinking. It’s kind of a recall at that point, so that’s a low cognitive demand task.

​We are not saying they’re not helpful and we’re not saying they don’t have a place in the curriculum. What we’re saying is that that is low cognitive demand. There doesn’t require a high level of sophisticated thinking to engage with this, if it’s dependent on the learner, of course, like I said earlier.

​The most common category of cognitive demand I’ve seen in research studies and in my own observations of classrooms is procedures without connection. So, procedures are very common.

​They’re algorithmic, they require limited cognitive demand and there’s little connection to the concepts undermining them. It’s really not recalling facts, you’re recreating procedures. So, you’re calculating something, but there’s one pathway there. You’re not doing a lot of deep thinking about the why behind it.

​So, solve all of these, stacked addition problems.

​That’s a low cognitive demand task. It does take thinking, but it’s pretty procedural in orientation, and you don’t have to make a lot of connections. You’re usually just cranking out an algorithm.

​Again, necessary, but shouldn’t be every single task that students have access to, or even 85% of the tasks that students have access to.

​So, we started with no mathematical ideas, or mathematical focus, then we went to the low categories, memorization and procedures without connections, and then there’s two categories of high cognitive demand. One is procedures with connections.

​So that’s when you’re doing a procedure, but there’s some kind of connection to the underlying why or how. So they connect to ideas. You might have to draw a picture or explain your thinking in some kind of way.

​So something like this. Using base 10 blocks, show that .292 is less than 0.3, or three tenths. So you’re showing something, it could be a procedural-type problem, but you’re using base 10 blocks to make a connection there. So there’s a procedure, but with a connection to a representation in this instance.

​And then finally, doing math, which are very rare and probably should be less common, because they take a lot of cognitive effort, and they’re very deep, and they’re hard to find good ones. So they require complex non-algorithmic thinking, they require students to explore and understand the math, and you have to do a lot of self-monitoring and thinking about your own processes to engage with them successfully.

​So these types of problems, where we have some kind of graph, and you’re either interpreting or making a story about it, you’re describing what you see in the graph and doing some complex thinking, maybe even producing your own graph, that’s kind of what we think about when we think about a doing math task.

​Most of the time, you don’t need to think about all five of these levels. Really, you can just think about, is it lower, is it high, do I have a mix of them throughout my lessons, etc.

​Multiple entry points, again, lots of ways for kids to enter the task, no matter where they’re in terms of background knowledge, those low floor tasks.

​So, here’s one example. So, you have these different subtraction problems at the top here.

​And it says, solve the problems. You must use each of the strategies one time. So there’s lots of different ways to approach it, so if you’re really good at counting up, you can think about, “Which one do I want to start with counting up?” And I would hope that maybe they did the one on the top left, because that’s a nice counting up one, they’re only, you know, four apart.

​But that might not be what another student thinks of first, so it’s not forcing them into one entry point, it’s kind of allowing them to make some choices about how they enter this problem.

​I like this one because I kind of created the subtraction problems to align better with one of these strategies, and so it’s nice for students to kind of think about, “hmm, which one do I want to use the standard algorithm for?” And if they use it for the first one on the left, you’re kind of like, “huh, that’s an interesting choice, I wonder why they did that?” It gives you a lot of information as the teacher as well.

​Multiple solution strategies are just, can they potentially solve it in multiple different ways?

​So, a common problem, like, what is the sum of 123 and 275? If you allow students to use and implement different solution strategies.

​This allows for multiple solution strategies. Maybe you encourage them to use an open number line. Maybe you encourage them to use base 10 blocks. Maybe they just decide to use the algorithm. Any of these things are possible.

​And that’s the point, is, like, they can use multiple solution strategies to solve it.

​Another example, Xander wanted to add 123 and 275. She started by adding 20. What should she do next and why? Again, same kind of mathematical goal, a different way of getting at it, but it does honor different solution strategies, especially if you foster them or encourage them amongst your students.

​And finally, I think this is the one that I’m maybe most passionate about, is reasoning and creativity.

​If you go back to the very first webinar where we talked about the characteristics of brilliant mathematicians, creativity is so important.

​And when we asked on the listening tour that we did a few years ago, like, “do you think math is creative?” A lot of people said no. They think it’s very black and white, very bland, not creative at all, but actually, creativity is what is the most beautiful and fun part of mathematics, and so we really want to foster that.

​If we don’t think math is creative, maybe reignite that view of mathematics for ourselves.

​So here’s one of my favorite kinds of strategies from our practice-driven PD, a project that we have that we did for a task that would be fostering creativity.

​So, this is called Super Wrong Kind of Right, and some of you may have seen this before. So, you’re asking students to create a wrong answer, and it’s not accidentally wrong. It’s on purpose. “Tell me what could be a wrong answer for this problem? Like, this task, what is a wrong answer?”

So tell me a number that is less than 3 fourths but more than two-thirds. And I don’t want you to tell me the right answer, I want you to tell me a wrong answer. And this allows students to get creative, and it also fosters reasoning, because you’re going to ask them also, “How do you know it’s wrong?” So they could tell you one, but they have to say, like, how do you know one is wrong?

​And that takes a lot of reasoning. And also, it takes a little creativity. They can come up with an answer that’s, like, just a little bit wrong, or super wrong, and that’s kind of the point, and that’s where some of the creativity comes in.

​Some people are like, well, wrong is wrong, there’s no degree of wrongness. I would argue that most of us think some things are more wrong than others in math class, and so this is a fun way to kind of encourage students to think about, like, what is a wrong answer, and why is it wrong?

​It also has, like, a low entry point, because there’s only one truly wrong answer, and that is the correct answer in this instance, because you don’t want the correct answer. You want a wrong answer. So there’s infinitely many wrong answers to this question. It’s just reasoning and explaining how you know it’s wrong. So you can do this with pretty much any computational answer problem.

​Okay, so that’s what makes a task rich when we talk about rich tasks at the Center. It’s higher in cognitive demand, so it’s not at those lower levels. It’s like procedures with connections or doing mathematics.

​It has the opportunity for students to solve it using multiple solution strategies, different ways of solving it. It fosters reasoning and creativity, which we love.

​And it also has multiple ways to enter it, so you don’t have to know something for sure to get in. There’s probably lots of different ways that you could get at this problem, which aligns nicely with the multiple solution strategies.

​If you think about the ropes of math proficiency, having rich tasks really helps you get at all of the strands instead of just one. So if you do low-level tests, low cognitive demand tasks, they’re not rich, you’re probably getting at procedural fluency fairly well.

​But are you getting at adaptive reasoning? Like, the ability to justify your reasoning? Probably not. Are you getting at the why behind the math? Also, probably not.

​Are you helping students see math as useful? Maybe. Maybe they think procedures are useful, but probably it’s better to see it’s useful if you can create it or apply it to different situations.

​And then, can you ask mathematical questions, and answer them mathematically? Probably, if you’re not getting the opportunity to engage with rich tasks, you’re probably not doing so. So that’s kind of how we’re thinking about it.

​Okay, so the second half of this is just some strategies for enhancing existing tasks. So a lot of us are very busy. We don’t have a lot of time to go sifting through a thousand websites to say, I really need a rich task today. Like, let me spend hours looking for just the right task.

​I would argue a better use of your time is saying, this is what I have in front of me, I know it’s a task that meets my math goal, but how do I just zhuzh it up a little bit to make it a little better? So that’s kind of where I’m coming from for today. Certainly, there’s great sources for mathematics tasks.

I was a board member, and I’m a longtime member of the National Council of Teachers of Math. They have lots of collections of great tasks, you can go there. I have a lot of other websites I tend to go to if I’m trying to look for ideas, but this is like, hey, any task you have, how do you just make it a little bit better?

​So, we’re gonna get the most out of every task. Again, we don’t have to do every single task, but occasionally, we just want to get more math out of a task.

​And I think about it, like, if you have a lemon, and you’re just, like, a typical task is just like, alright, you got lemon juice. Like, that’s great. But we’re gonna get so many different things out of a task. We’re gonna use all the parts of the task to really exploit the mathematical potential of the task, so that they’re rich and exciting. So instead of just, like, ‘hey, it’s a lemon, I can make lemon juice,’ all these different ways to get math out of the task is kind of where we’re going today.

​Why do we care about this? I would say, if you did simple tasks and you just connected them like this, like, hey, it’s a procedural task. It doesn’t have a lot of thinking connections, and I’m gonna do another task, and it doesn’t have a lot of thinking connections. If you think about the knowledge that students are building from this.

​It’s pretty weak.

​If you miss any one of those tasks, there’s not a lot of connections, not a lot of map they got out of it, so they might have gaps in their understanding. When we get more out of each task, they’re making more connections to more ideas, and they’re building this really nice web of knowledge and understanding and connections to other things.

​It’s kind of like that rope again. So that, like, even if they don’t remember one procedure or something, it’s okay, because they can make sense of things, because they’re connected in a more brilliant and beautiful way. And so that’s what we’re really trying to do with, expanding what we can get out of a math task.

​As a teacher, I think about, like, what do I do before the kids start working on the task? What do I do as they’re working on the task? And what do they do after they’re working on the task?

​So we’re gonna kind of talk about tasks, we’re gonna look at before, before the kids start on it, like, when I’m setting it up. We’re gonna look at during, what should I be doing during to get more math out of the task?

​And then, what should I do after, at the end? If you have questions, you can put them in the chat. We’ll do Q&A at the end of this, for sure.

​Okay, before, so you have a task. It might be okay, it might be meh, it might be great, but anyways, you have this task, you’re gonna use it, and what do you do with the students to get more math out of it before they actually start solving it? That’s where we’re at.

​First thing you can do. You probably have an idea of what this image is, especially if you grew up in the United States, and you’ve had children, or read books to children. Goldilocks is how I think about this strategy.

​So, before students start working on something, all you do is you have them estimate, right? And this is particularly good for a procedural task.

​Like, can you tell me an answer that’s too large? How do you know?

​Can you tell me an answer that’s definitely too small? How do you know? This connects back to Goldilocks, of course, because, like, she sat in the chairs, it’s too big, it’s too small, this one’s just about right. And then, like, what’s an answer that’s probably just about right? How do you know?

​Now, as a teacher, I am sneakily, or maybe not as sneakily, trying to get my students to have some number sense, to estimate, to grow their understanding. I’m also getting some reasoning out of them, because what might happen is my students might just keep changing, flip, or multiply, invert, whatever you want to say, and they don’t have to do a lot of thinking. It’s a low-level task.

​So if we want to enrich it, why not just ask them, “do you know something that is going to be definitely too big of an answer? How do you know?”

​So, for example, if I said 10, I know 10 is definitely too big.

​Okay, well, how do I know 10 is definitely too big?

​Well, because, like, if it was divided by halves, I know two halves go in each one, and there’s 5, so a half would have gone into 5, 10 times, but 2 thirds is bigger, so it should go in fewer times. That kind of reasoning, right? So we want students to develop this number sense, and Marin asked what grade level this task, whatever grade level you’re working with, fraction division, right? So this example is just fraction division.

​So say you were using a fraction division task and you wanted them to practice, before you let them go and start solving, you would just ask them this. You could do this in very little grades, too. Like, you could do it in a first grade classroom where you’re adding whole numbers. So say you’re adding two two-digit numbers, like 12 plus 25.

​You can start and say, before we start adding, maybe they’re gonna work with blocks, maybe they’re gonna, practice the algorithm, whatever they’re doing. Before you start solving it, what’s something that we know is going to be just too big for this? And they might say 100, and you’ll be like, well, how do you know 100’s way too big? And then they have to think, like, well, 100’s gonna be too big, because it’s only one ton and two tons. It’s not going to magically go up to 100, or something like that.

​What’s something that’s way too small? Zero, because I have something, I’m adding some more to it, it’s not going to be zero in this instance.

​And then what’s something just about right? You don’t have to use all three, you could use just the bigger and smaller ones, but it’s a good strategy to get students to anticipate and think and reason before they just start calculating without thinking, which is what happens a lot of the time.

​The next one is a Notice in Wonder, and again, I told you I love NCTM, I do.

​Notice and wonder, and Annie Fetter also is kind of credited with this idea, but, like, a lot of us are using this, because this is really spread, but what do you notice about the problem, and what do you wonder? It’s getting kids to think about the situation and the task at hand before they start solving.

​A lot of the problems in math class are kids just start pulling numbers and operating, and they’re not really thinking about what’s in front of them. So a lot of the strategies we have are to slow them down and increase their reasoning and sense-making.

​So, back to the staircase task before they solve it, which is a hard task, right? Like, this is a high-conteman task already, but it doesn’t mean that we can’t get some more math or thinking out of it. So before they solve it, you could have a classroom discussion, or a partner discussion or an independent reflection.

​Like, what do you notice about this task? It could be the images, it could be part of the task. What do you wonder?

​Like, maybe they say, “I notice it’s getting taller.”

Great, what do you mean, it’s getting taller? “The blocks are getting stacked higher.”

Great, that’s a great noticing. Did anybody else notice something different?

​They might wonder, “Am I gonna have to figure out what happens to the hundredth one, or something like that?” That’s a great wondering, right? So you’re asking them to notice and wonder as a starting point for reasoning that they’re going to make sense of this problem.

​And then this one, these are visual cues to myself to remember, this one’s just like, have the students huddle up and talk about the problem before they get started. Often, the teacher is always kicking off the problem, and then the students just go.

​If you encourage the students to make sense of the problem before they start, they just pick up on things that they’re not sure of, they talk about the context, and they get some more math thinking out of it. So it’s a good idea to just have them huddle up and say, “Okay, look at this problem and talk about it. What’s a game plan that you have?”

​You see this task, you’re choosing two equations to make a system, like, what’s your strategy going to be? In this instance, like, what are you going to look for? If you’re solving by substitution, maybe they would say, I’m gonna look for one that has an X or a Y by itself, because that’s going to make substitution easier in this instance, or something like that, right? 

​Asking them to talk about a strategy before they start solving it is a great way to get more math out of the problem, also to get fewer problems or questions because you end up answering the same question to every kid when they start, because they haven’t really thought about it. This way, they can talk to their peers, they can get some ideas on the table, and then they can go work on it. So that’s before.

​Okay, so those were three strategies before, now we’re going to go to during. So you’ve done some of these things, and now kids are working on the problem. What’s your job as a teacher? Your job is to get the kids thinking about their own thinking, so metacognition. You’re trying to, like, walk around, or look at what they’re doing, and you’re like, “hey, tell me about what you’re doing there.” You’re trying to get them to think about their own thinking, because again, kids don’t do enough thinking about their own thinking. We really want to foster that.

​So your job is to ask good questions, notice some interesting things as you walk around. I’m a researcher by training. I think about it as, like, a small research experiment. I’m looking around, trying to understand, “Oh, what are you doing? That’s interesting. Oh, what are you doing?” It’s giving you informative assessment, ideas and, like, information.

​But it’s also helping me help the students reason and sense-make from the math in front of them. So we want to get them to think about what they’re doing, not just to process, like, you know, procedures, but also, like, “why are you doing that? Can you explain? Can you tell me what you’ve tried, what you haven’t?”

​Ask a lot of questions. That’s your job, asking questions. And, you probably know the answer to a lot of these questions. Sometimes you don’t, but your job is just to surface them and get the kids thinking. You probably have an inner dialogue when you solve math problems. Students may not yet have that.

​So what questions would you ask in your own mind, and can you ask those of students, so they can start having their own internal dialogue with themselves?

​“How confident are you in your approach? That’s a great question.” It’s not scary, because you’re just asking, are you confident or not? And if they’re not, you can ask more questions to see what they need to be confident. “What’s gone well so far? Something’s probably gone well so far.”

​Maybe it’s just I got my pencil out, and I found my pencil. Some days are like that. Maybe it’s like, “hey, I figured out that this is the wrong answer,” that’s a great success. Like, one more pathway that you’re closer to.

​“Tell me about what you’ve tried so far. Are there any tools that could help you?” Like, if they’re struggling sometimes, you might just ask, like, “is there a tool that might be helpful?” Maybe it’s a calculator. Maybe it’s a hundreds chart, maybe it’s a manipulative, and then you can give that to them and see if that helps them.

​“What questions do you have for me?” So, like, do you have any questions that’s different than, what questions do you have for me? Because you’re just assuming you have questions, and you’re encouraging them to ask.

​So these are kind of my go-to as I work with students around the room, just to check in with them. Again, trying to get them to think about their own thinking, and also understanding where they’re at as a teacher, so I can drive my instruction afterward.

​And finally, after. So they’re done! The thinking shouldn’t stop when they’re done with a solution.

​It’s an opportunity to go deeper, because they’ve already solved it once, or they’re at some kind of natural stopping point. Now they have some prior knowledge you know about, you can build on and go deeper. So afterward, the math doesn’t stop.

​So, the first thing you could do, they just solved a problem forwards, you can reverse it and ask them, could you go backwards? And what I mean by that is this.

​Typical problems have a start and a finish. In procedural problems, it’s usually in algebra class, for example. You start with an equation, a linear equation, and you say, solve for X.

​So if they just did that, one way to reverse it would be, “hey, is there another equation that would have also given you the solution for? Could you reverse the problem?” 

It allows for a lot more flexibility in what they produce. It’s creativity, it’s reason-making, there’s an infinite number of answers in this instance, like, what equations have solutions to four? There’s so many. There’s infinitely many. So, have students reverse their thinking. It’s a great idea to do. If they just solved a problem and they got a difference of five, say.

​Is there another problem that would have also given you a difference of five? What is it? Could you come up with another one? You could put constraints on it and get a lot more thinking out of it. So it’s kind of like a doing and undoing move, which is really important for algebra. 

A lot of algebra is doing something and undoing it, doing it and undoing, and you’re kind of fostering that reasoning in an earlier grade level, potentially, or in the later grade levels as well.

​Say they just solved it forward. Five divided by two-thirds is whatever it is. Now you take the solution, and you say, “Hey, are there other fractions I could have divided to get that same solution?”

​It’s a good question and it follows directly on what they just did. So it gets more thinking, it gets more possibilities, and there’s a lot to talk about. It’s a much richer problem as well, so reversals are great for this.

​I call it oranges and apples, but it’s comparing problems. So, say you just did a problem. “Do you know another problem that was similar? Have we done another problem that was similar? Are there other problems on the sheet that look similar? What’s a problem that might be different than this one that we’ve done recently? Why is it different?”

​If you did a whole sheet of problems, “what was the big idea on this sheet of problems? Why did we do this sheet of problems?”

​Asking students to kind of compare and look across problems is super beneficial to them.

​It gets more reasoning, sense-making, it helps them look for structure in mathematics, which is very important and it also just is fun. Like, okay, I did all these problems, what was the same? It’s helping them think about their thinking. So, comparing problems is a really important skill, and it’s an easy one to do after they’ve already finished a sheet of problems or a particular problem.

​For this one, again, it’s a patterning task. Have we done other patterning tests? What approaches did we take when we did those patterning tests? Is this different from another pattern name task? How so? Like, you know, maybe that one didn’t grow in the same way.

There’s a lot of connections that students can make, but they have to start to think about these kinds of things, and we need to encourage them by asking these types of questions.

​And finally, they solved it one way. Could they have solved it a different way? That’s one of the things that you can ask. Another thing is, they’ve already solved it. Can you ask a good question, or ask them to share something besides the answer?

​So, give them a choice of what to share. So, you just solved this problem. Could you share a different correct solution or solution strategy? Could you share a picture that you could have drawn to help you, or that you did draw? Could you share your strategy with the class?

​Could you share a question that this problem raised for you? Can you share a wrong solution that you knew was wrong because you tried it on the way and you figured out it was the wrong answer? Can you share another question you can now answer? So you’ve solved this, does that help you answer something else that you might ask?

So it’s kind of like, encourage them to potentially solve it a different way. But then also, at the end, you could just ask them, “can you share something that’s not just the answer?” And that’s, like, a really good way to get more thinking out of kids. Also, students who don’t feel confident sharing a right answer because they’re not sure if they’re right.

​Might share one of these, which is great, because we want more students sharing and more voices heard in the classroom, and validating that it’s important to share your thinking, even if it’s not correct or you’re not sure if it’s correct.

​So for this one, say a student already solved it.

​What strategy led you to picking those two equations? Or did you graph? Did you use a tool? Could you have solved it a different way? Could you have picked two different equations? You know, there’s lots of questions you can ask afterwards, rather than just, “What did you get for the point of intersection,” or “What solution did you get?”

​Okay, so if you want to try this later, you can take a task like this. This is the cow and chicken problem that I do love.

​Where there’s a field of animals, and for some reason, you only know how many animals, and you only know how many legs there are, but some are cows, some are chickens.

​How many cows are there? So you can try this with yourself later. If you’re in a PLC or something, you can talk about this later. What could you do before the students work on this to get more math out of it? During it, what questions might you ask after? What might you follow up on or do related to this?

​So, afterward, you could say, do you also know how many chickens there are, if you know how many cows, etc, that kind of thing.

​So, to summarize, we’re almost done.

​If you want to get more math and make tasks richer, before, you might do a Goldilocks situation, where you ask for something that’s too big, too small, or just about right. You could do a Notice and Wonder to just see what they’re understanding, what they don’t, what strategies they might anticipate, that kind of thing. You could have them huddle up and talk about a plan for the particular task that they have in front of them.

​As they’re working on it, you’re gonna ask really good questions to get them thinking about their own thinking. If you work with very small children, I found some questions that work well, because metacognition’s really hard when you’re very small. So you might ask the kids, “what did your brain do? Like, what were you thinking in your brain, or what did your brain say? Or, huh, how did you figure that out? Did you do something in your brain? Did you have a picture?” You know, things like that. There’s ways to elicit things from kids that will get them in the habit of sharing their thinking. It just might differ from a five-year-old to a 15-year-old, so you have to be cognizant of that.

​And then afterwards, what you can do is you can ask them to share something that’s not just the answer, you could share a question they still have, or a wondering they still have, or, a wrong turn they made during the solution strategy. You could ask them to solve it a different way. You could ask them to reverse the problem. So, like, what’s another question that would have the same solution kind of situation?

​You could ask them to compare it to a different problem they’ve already done, or one that they’ve seen before, find the differences, the key ideas, etc.

​Again, if you want to practice, here’s one for later that you can practice with. Three, actually. Here’s three different tasks. None of them are particularly rich, so you can think about, like, could you use any of these strategies to make them richer and more engaging, and bring out more math proficiency and more talk?

​And with that, I’m gonna come to a close. I would just plug that our next webinar is on April 14th, which is a Tuesday; we’re gonna do it at noon. Again, I will send the recordings out, you can register, and if you can’t make it, they will be posted later, but I love to have people live, so that’s really great.

​I hope to see many of you on April 14th, which is just about 3 weeks away. If you have any questions, please put them in the chat. Otherwise, thank you for coming, and it’s nice to see everyone.

​Oh, are there webinars geared towards Algebra 2 students? These are all gonna be broad K-12, in this series.

​But eventually, down the line, if there’s a desire to see more, higher-level content, drop me an email or let me know, because we can host other webinars, too. So, if there’s topics or grade levels that you’d really like to see expanded toward, let me know, and we can put that on our plan for this summer or this fall. So, thank you for the question.