Zandra de Araujo: Great! So, continue to introduce yourselves in the chat, if you’d like, and I’m going to get started. This should be a fun, quick one today, on student habits, engagement and math proficiency. This is the last one of our series of five webinars.

So, if anybody’s made it to all five, you can put it in the chat, because I’m just curious if anybody has watched all five, either synchronous or asynchronously.

I’m Zandra De Araujo, I’m the Director of Mathematics at the Lastinger Center. Hope Miller is on to help with any tech challenges that I might encounter, and you can use the chat if you have any questions.

Hey, Pam and Valentin, yay! Thank you for all five, you’ve seen a lot of me. Nice to see everyone today. 

Okay, so as per usual, I’m gonna start with a recap. These are not dependent on one another, but if you want to check out any of the prior webinars, they are going to be posted on our website. We will share that at the end, or Hope may put the link in there.

But we’ve had five, or well, we will have had five by the end of today. Today is the fifth one. I’m just going to recap the focus of the prior four.

So, Pam and Valentino, since they’ve been to all five, the first one was on mathematical proficiency, and so we talked about the strands of mathematical proficiency, how being good at math, or proficient at math, does not mean that you can just do things quickly and accurately. That’s certainly part of it, but it is one string of five to make a strong rope of understanding.

There’s also the ability to pose questions, to enjoy mathematics, to understand the why behind it etc. And we talked about the actions of teaching, which include setting goals, math tasks, math connections, math discussions and habits. And that is roughly the order through which we’ve done these webinars.

So, here’s the strands again.

We talked about what makes a brilliant mathematician, too, and we’re gonna go back to that today, so I’m not gonna spend time on that.

Webinar two, after we talked about what mathematical proficiency is, then we talked about math discussions, because they’re hard to do, but they’re also super important. So if you want to build the why of mathematics, you want to be able to pose questions and justify your reasoning, you have to have math discussions.

And actually, they’re pretty fun, but again, they’re hard. The classroom management is harder, getting students to understand what it means to explain your thinking is harder, and managing what comes up, because things will come up that you do not expect in those conversations. Managing all that is a challenge, but it’s definitely a challenge worth working toward.

And so we talked about some little steps along the way that you could do to get more engaging math discussions with your students.

So we talked about explaining mathematical thinking and questioning, so how the teacher plays into both those aspects.

And we also talked about some math talk moves that are really common. There’s a great book by Chapin et al. that talks about math talk moves. So, like, repeating or restating students’ thinking, adding on, having them add on to each other’s thinking, giving wait time, which is really hard sometimes.

So, like, counting mentally to 10 before you respond, to give people space to think before they respond.

And that kind of thing. So, that was webinar two, if you’re interested.

We also talked about closed and open questions, and how there’s been a lot of research on this. And that, by and large, we as humans and as teachers tend to ask closed questions. Like, what’s the answer? What’d you do? Very closed questions, like, what’s the name of this thing? 

But we want to open it up from time to time to say, tell me about your process. Tell me about a different way you could have solved that. Can you tell me what else you tried that maybe didn’t come to fruition in this problem? That kind of thing. So thinking about the types of questions you ask, and not always asking close questions.

And then we also talked about how you can ask questions about what they got at the end, which is great, and we need to do that, but we should also ask questions about the process that got them there. And so, oftentimes, an open question will lend itself better to kind of, like, the process. Product questions often are closed questions, they’re just like, hey, what’s the answer that you got?

And so we want to open up our questions and focus on the process, and we’re going to talk more about that today as well.

Then, the third webinar. We talked about math tests, my favorite topic, and we talked about rich mathematics tasks. Those that require more thinking, which means they have a higher cognitive demand, they might have multiple solution strategies, so there’s more than one way to get the right answer.

They allow students to evidence and engage with reasoning and creativity.

And there’s lots of different ways to get in, so even if you don’t know exactly one strategy to do it, you probably can figure it out or reason through it in different ways. So it has, like, a low floor and a high ceiling, basically. We talked about strategies for making existing tasks more rich as well in that one.

Then, the very last webinar we did, Webinar four in April, was math connections. And I said this in the webinar last time, and, I’ll say it again, that was not my most excited, kind of like, I was looking forward to that webinar, but it ended up being one of my favorites when I, was thinking it through, because math connections are so important. And I think giving names to some of them helps me. 

So we talked about long-term connections. Sometimes you can think about these as vertical connections. Where they span years or grade levels, potentially. And then we talked about some short-term connections or horizontal connections, things that can be connected from the prior day, the prior week or the prior lesson.

And you can connect to context, to real-world applications, to other mathematical ideas. There’s all kinds of ways to make connections between representations, etc.

And so I showed a couple of these graphs, like, here’s some example of long-term connections that you can see across all these different grade levels.

And then some shorter-term ones that you might think about. So if that’s of interest to you, you can go back to the April webinar and rewatch that one, or watch it for the first time.

So today, we’re gonna end with math habits, and I think these are the kind of things that make everything else possible.

If you don’t illustrate and exemplify math habits that are productive, then all the other stuff is going to start to fall apart. So, this is the last one, but it doesn’t mean that it’s less important. This is the glue that kind of holds everything together. You really need to help foster and say and evidence strong math habits with your students in order to make all the other parts possible in your classroom.

Okay, we’re gonna start with a video, and so I’m gonna have you watch this video. We’re just gonna start with the first part, and then you can share in the chat what resonates with you. But these are people talking about what it means to be good at math.

So, if you’ve ever asked your students that, or another teacher that, what does it mean to be good at math? I bet you have some assumptions about what people would say.

This is gonna have some children saying it, and some adults saying what they think it means to be good at math. So again, your job is, as you watch, hopefully the sound works great, you can put in the chat what’s resonating with you, or how this aligns with your experiences. We’re just gonna watch a couple of minutes.

[LINK TO OUTSIDE VIDEO HERE]

Zandra de Araujo: All right, so that’s just, like, a sampling of some students we worked with, and some adults that we invited to the studio. So, what resonated for you all about what they were saying about what it means to be good at mathematics?

So, I, for one, when the teenage boys, the blonde boys, were saying, it’s not, like what you do, it’s like how you get over the stuff that you don’t do, that really resonated for me, as somebody who struggled in college-level math and beyond. But I thought that was neat. Asking questions, being curious, yes, that resonates well with me, problem solving.

Yeah, Jane, you similarly said the one that I did, yeah.

That was really impactful to me. Also, there is a perception among some children that it’s also just being good at some procedures and things like that, but we want to broaden their conceptions of that over time.

And it is interesting to say they need to know numbers, yeah. And we asked this in the listening tour that we did around Florida, too, about, like, you know, what mathematics do you need, and why do we need mathematics? And a lot of people said for everyday basic skills, like, financial literacy and things like that.

But as we’ve seen across all of these webinars, and I’ll reiterate again today, it’s more than just that. It’s also, understanding problems, asking questions and quantifying the world around you so that you can get really interesting insights into issues and problems that make sense of the world.

So, math is pretty cool. In the first webinar, I said I’d come back to this, and I did.

I asked ChatGPT what are the characteristics of a brilliant mathematician, and people that were on that webinar anticipated what it was gonna say, and these are the kinds of things that they said on AI, and that I think there’s a lot of agreement among mathematicians for. 

Like, this deep curiosity and passion for understanding, not just answer getting, but understanding.

The persistence, which I would say those teenage boys were saying, the creativity, active engagement, and deep thinking. There is a lot of deep thinking in math, and in a world where things are instantaneous, sometimes it’s hard to get back into the pattern of deep thinking through something for a while.

And then this idea of patience, which goes with deep thinking and sustained focus.

So it’s kind of like digging into a novel, right? Like, you’re kind of doing, like, more long-form thinking and reading, not just all quick, like, tweets and things like that. So mathematics takes time, it takes persistence.

So if you think about the habits that your students have to build to be good at math, if this is what we’re saying it means to be good at math, then that’s what we’re going to talk about today.

And at the Center, when we were thinking about this, we’ve kind of distilled it down to three habits that are really, really important for students, and so we’re going to go through each of those three habits.

I’m gonna give you two quick practices you can do to foster each of those habits, and then we’ll wrap it up, so it’ll be a nice quick one. You’re gonna get some instructional nudges again today, which, you know, are my favorite, so I’ll do a lot of those.

Alright, so habit number one.

So these are habits that students should develop as learners to promote the kind of mathematics we care about.

Do students that you work with focus on strategies on the way to solutions? So they’re not just focusing on the solutions, on the answers, but they’re focusing on how they got there.

So, often, math instruction focuses on getting answers, which we know, and when we only focus on correct answers, it can limit our critical thinking. So we’re only focused on, like, what’s the right answer, but we never talk about what happens with the wrong answer, why is it wrong, how do we work through the situation that got us to the correct answer, that kind of thing.

So we want to emphasize strategies.

That’s really, really important, because that helps foster deeper conceptual understanding, and we want to know the why behind what we’re doing.

 We want students to develop as stronger problem-solving students and reasoning students, and we also want strategies that promote long-term success in math. So answer-getting does not promote long-term success in math, and I’ll just share my own personal experience in mathematics.

I’m a really good memorizer, and that got me through K-12 school just fine. I did very well, I got a five on my AP exam, I was, like, top of my class, you know, all that stuff.

I got to college here at the University of Florida, and I got to Calc 2, and I almost failed it.

Because I try to memorize proofs. And if you’ve ever taken a higher level math class, the purpose of proofs is not to memorize them. It is to logically think through a situation or a theorem and prove that it is true. There’s no need to memorize, you’re just kind of stating why it’s true or how you arrived at a particular outcome.

So, I tried to memorize all these proofs. I had no sense in terms of, like, how I was trying to memorize them, so they were all just jumbled messes, and I didn’t understand why I was failing at math when I had always been so successful. 

And I think that’s the outcome of a lot of students, and it wasn’t until I started teaching mathematics and really had to explain why things worked for students that I learned it very deeply, and I think that’s the experience of a lot of teachers, actually. And so, we really want to focus on the strategies, not just the answers, because the answers only get us so far.

Okay, so here’s some simple, (I’m gonna call them simple. They’re, you know, they’re complicated, but they’re, like, not super complicated.) We built them with the hopes that you could implement these quickly and try them out.

So they’re nudges, instructional nudges, and I think we talked about this in an earlier webinar. So they’re quick routines that we think could help foster this habit, where you’re trying to focus on strategies. 

Okay, so the first one’s called rate and review.

So, if you’re familiar with our nudges, I’ll put the link in the chat later, so you can get all access to all of them. But we made these nudges, we call these instructional strategies instructional nudges. They all fit on one page, because we figured people don’t have a lot of time to do a lot of things, but, like, we can communicate the idea in a page.

There’s a lot of text out here, but basically, what you’re gonna do is give students a worked example which is important. When you have a worked example, you’re seeing the strategy. And then they rate and review it. So, kids and adults are used to rating and reviewing at this point in our lives. This is part of our life, I think, in the world right now. And so, they’re gonna rate the strategy, not the answer, the strategy from one to five stars.

And then they have to explain why they gave the rating, so they also have to review it, like, why is that the rating?

Okay? Now, I will give this disclaimer. Just like with Amazon, students might pick up on very superficial aspects. So, like, you can see this one, “It’s definitely a bucket.” That’s not a great rate and review. So students initially will give you some poor examples of ratings and reviews, or reviews more specifically, and you want to talk to them about what makes a good review, what are you explaining, so that you don’t get stuff like this.

There’s a lot of research that says there’s a lot of value in worked examples. So doing error analyses, like a lot of you probably do, is very important, and showing worked examples, so that students are analyzing and picking apart somebody else’s worked example, not always producing the worked example themselves.

Alright, so here’s an example:

Here are four different work solutions, each of which is for a different problem. We’re just going to treat them each separately, so there’s A, B, C, and D.

So, you would, and you can do this in the chat, I’m going to pick one for us to do together. So you would look at either A, B, C, or D, and you would say, huh, I think that this is a one-star solution strategy, or I think this is a five-star solution strategy, and then you would say, why? 

These are kind of like a middle school example, mostly, but let’s do A. That’s a nice common error that I saw in middle grades and high school math. So A, if you were to rate and review A, if you want to put it in the chat, what would you give it? Like, would you give it one star, five stars, three stars or four stars, and why?

So, if you want to have a minute, I’ll try to have good wait time for A, How would you rate and review A? Based on whatever criteria you want.

I see two stars, two stars, okay. Anybody different than two stars?

One star. Okay, and now, if you were working with students, you would say, great.

Why one star? Why there? So Aaron said three, while they did not combine like terms correctly, they did remember to divide both sides by seven.

Karina, two stars.

16/7 is correct for X, but the pathway is incorrect.

Oh, so maybe they did that last step correctly, but the prior step was incorrect.

So, lots of different types of answers here. Now, what you will learn with students is that they will rate it differently than you, and differently than each other. And a teacher that teaches the same exact class as you might rate it differently than you do, which is cool, because it gives you an opportunity to talk about something interesting.

The reason why this is nice for this habit is because it’s focusing on the strategy. So what about the strategy is a five-star, or a four-star, or three-star, is what you’re trying to focus on? Not just the answer, because students do focus on the answer, but what about the strategy? So it’s a way to get them into worked examples and focusing on strategies rather than answers.

Here’s another example here. These are all the same problem in this instance.

So you can kind of see, some people don’t love B. Or A.

I think A is five stars. Some people might disagree with me, and I would say it’s five stars because it is absolutely a valid strategy to get common denominators and divide straight across, even though it’s sometimes less efficient than inverting and multiplying, or multiplying by the reciprocal, but it is about strategy. 

And you might disagree with me, and then we’d have a great conversation about it. So it’s not only a good way to get them talking about the strategy, it’s also a good formative assessment to see what students are picking up on.

So, your students, when you do this, they might write it based on how correct it is, how clear it is, how elegant or efficient it is.

But you want them to think about the mathematics of it, not just, they didn’t like how I wrote my sevens or my fours, or something like that. So they might need a little bit of encouragement to talk about the mathematical strategy itself, not just superficial features.

And I always give one caution with this one: don’t use actual student work, because we don’t want them evaluating another student’s work and them feeling bad or hurting their math identity. 

Just make up a fake student. Say, like, last year I got this one, what do you guys think? You know, the fake student that you had last period or last year. But just don’t use your own student’s real work.

Because, it could lead to them feeling bad about themselves, or, you know, just, like, a lot of comments from their peers that you don’t need.

Okay, so that’s one way. The second one, and I may have shared this before, but I really like it. It’s, it’s, again, quick. It’s called And the Answer Is. So again, we’re trying to get students to focus on strategies rather than answers.

Now, this one’s gonna seem a little counterintuitive, because you are focusing on the answer for it. But I’m going to tell you why it actually works in a minute. So what this means in the answer is, what you do prior to solving a task.

You have the students anticipate the format, like, what kind of answer is it going to be in a unit? So, they have to say, like, it’s going to be a real number, or it’s going to be an expression, or it’s going to be a whole number, fraction graph or a sentence.

And then if it has units, they should say what units it’s gonna have. Maybe it’s meters, inches, apples, dogs or something like that.

So that’s what they’re gonna do before they solve it.

Why I think that this fits in this section is that if you think about a problem like this. And all I tell you is you’re not solving it yet, you’re gonna think about the answer type, like format and form. Students have to read the question carefully, which is important if you’re gonna focus on strategies, and they have to think through all the quantities that exist, and they’re focused on what the target is, but now it frees up some brain space to kind of think about, like, how am I going to get there? I kind of know what I’m aiming for.

So, like, how do I make sense of this and solve it? So it’s a way to take some pressure off of, like, I don’t know what I’m aiming for. Now they know what they’re aiming for, and they can kind of think about the strategies more carefully. So that’s why I think this still fits in this section.

So, Cindy had ⅔ of a cup of flour in one bag and another 1/2 of a cup of flour in another bag. (I’ve done this many times.) To bake her famous cookies, she needs one and one half cups of flour. How should she adjust her recipe to use up all the flour?

So in this case, you might say, it’s probably going to be a fraction, some kind of fraction, and it’s going to say, like, how she should adjust the thing. So the unit’s going to be hard to describe, actually. Like, it might be the number of batches, or she should half the recipe, so the recipe is probably the unit here, but it leads to a good conversation. 

You’re trying to figure out some factor of the recipe that will allow her to adjust the measurements of the different components. Now, there’s different ways to say that, but you’re really focused on what you’re aiming for, and then you say, like, okay, like, how are we going to make sense of this, and how are we actually going to figure this out? And then you can focus on the strategy.

Here’s another example, in case you’re curious, for smaller children.

So you have three pennies, five nickels, and six quarters.

You want to buy a candy bar that costs $1.86?

Again, this is intentionally having a lot of units and a lot of quantities, just to make sure we focus on the right thing.

Can Marvin buy his candy bar? How do you know? So the answer is gonna be a yes or no answer, actually, right? Like, what they actually provide is gonna be, like, a yes or a no. You have to answer the question, can he buy the candy bar? Yes or no? But it also is going to say, how do you know? It’s gonna be an answer, a yes or a no, with an explanation, with a little bit of backing on it.

So you might do this orally, you might do it written, but this is kind of the answer type that you’re going to give for this one.

And students will be confused, because a lot of times they’ll be like, but there’s pennies, the answer’s gonna be money. And you’re gonna have to use the money to get the answer, but the answer itself is not the money. It’s gonna be yes or no with a reason why, which might include some money in the reason, but it’s not the actual answer.

Relatedly, if you really want them to focus on, like, what the problem is saying without thinking about the question, the answer yet, just take away the question.

So, if you have this same question from before. Just take out the question sentence, like, the last part.

And now you’re really just focused on the context and what it’s saying, and so you don’t have to worry about the question, you can’t actually answer it. But you’re anticipating the question and thinking about the quantities and what’s happening, so you can kind of get a feel for what the strategy might be and where it’s leading you without knowing exactly what it is. Often math problems lead you in one kind of direction. 

Even though you don’t know the question, it’s probably pretty obvious where it’s going. And if it’s not, maybe there’s different questions you could answer, which is also interesting. So you could do that for that one. Or you could do it for this one, where you just leave out the last part.

Okay, so that’s the first habit. We have two more to go.

Before I go into the second habit, this is just a summary of things that you could do as a teacher to support that habit. So, these are questions you could ask, and these are practices you can engage with with your students. And I showed you two specific, or three specific nudges, but these are other things that you can do, like providing feedback on their strategies, not just their answers.

Give them the answers, then they have to focus on the strategies, because they know the right answer, they’re just solving it.

And then you can ask them questions that align with strategy focus, like, what did you use for that strategy? How did you solve this problem using this strategy? Was there a different strategy you could have used, etc.

All right, now, habit two. This one’s a hard one, because I think a lot of teachers think that kids already make a lot of errors, and we don’t want to emphasize errors.

However, if you’re not making errors from time to time, you’re not learning. If you never have friction, you’re not learning. It’s just not possible. And I use this example a lot. I’m an adult who learned how to play the piano and is still learning how to play piano as an adult.

 And I like to do the pieces that sound really good and are easy for me, but because I usually play things that are fun and easy, I don’t get better. Because you get better when you do hard things, and when things aren’t a straight path. 

Learning’s not a straight path. It’s not a new thing, get an answer, new thing, get an answer. It’s like a wiggly path of errors and mistakes along the way, which is important. That’s necessary for learning.

So, we want to encourage students to make errors, and use them to learn. So, like, hey, I made an error, what did I learn from making that error?

Learning to ride a bike is similar, yes, that kind of thing.

So, mistakes are a necessary part of learning mathematics, and if you’re using rich tasks, kids are gonna make errors, and that’s okay. We have to normalize it. As a teacher, when you make an error, you shouldn’t be embarrassed, you should be like, oh yeah, I did make an error, let’s fix that. Normalizing errors, I guess, is the name of the game.

But you have to have a classroom that supports this.

So if the only kids that make errors that you notice are the kids that struggle, that’s gonna make the perception that only struggling kids make errors.

So, we have to teach everybody, including the students that are higher achieving, to celebrate their errors and say, I learned something from that. And you as the teacher are seen as the highest achieving student in the room often, or the most knowledgeable person, and so you should also normalize your errors and be like, yep, I made an error, good job catching it, I learned that I should pay more attention to X, Y or Z.

Again, when you make errors, it strengthens your conceptual understanding, your reasoning, it promotes a growth mindset, which we want kids to do. We don’t want kids to think that they’re either good at math or bad at math. We want everybody to think, hey, I can be better, and I am getting better because I am learning along the way.

And, this notion of productive struggle, which a lot of us have heard about before, helps students say, like, yeah, you’re struggling, but you’re struggling in a good way towards something. You’re not just, like, spinning your wheels with no progress. It’s helping you learn and grow.

Okay, so here’s a couple of nudges for this. So, my favorite one is wrong two ways.

I called it Goldilocks at one point, but I think Wrong Two Ways works, too.

So, for a numerical task, you ask students to provide two answers that are clearly wrong, clearly incorrect. One that’s clearly too big, and one that’s clearly too small, and then this is the Goldilocks part. You can also ask them, do you know one that’s just about right?

It emphasizes number sense skills and estimation skills, but it’s also just saying, hey, there are wrong answers. Let’s bound our answer for a numerical computation-type problem. Let’s bound it and say this is clearly going to be too big, this one’s going to be too small. Our real answer, we know, is going to be somewhere in the middle there, but now we have kind of boundaries that we think are reasonable, and a good estimate is, like, around here.

So here’s one that you could do. 

So, what is an answer that is clearly too large for this? Like, clearly, it’s not going to be this big. You should have students justify it, because that’s, like, the good part. It’s like, why do you know it’s too big? Because they could say a million, but how do they know a million’s too big? They have to give me a reason, mathematically.

And then, what is an answer that is clearly too small, and how do you know?

So, did anybody have a too big answer they want to share with a reason? One is too large.

And 1/20 is too small. And then you would say, like, how do you know? So, like, two-thirds of five-halves, so it’s like two-thirds of about half of two and a half?

Right? So I think five is definitely too big, because you’re taking two-thirds of something that’s only two and a half to start. You only have two and a half to start, and you’re taking, like, about a little more than half of two and a half. It’s not going to be five, right? It’s going to be something smaller than two and a half.

Or you could say zero.

I know zero is too small, because I’m starting with something, and I’m taking some fraction of that thing, and they’re both positive, so I know it’s zero. It’s gonna be bigger than zero, there’s something there. You know, that kind of thing.

It takes a lot of reasoning to do that, a lot of understanding to do that, but these are good questions for kids. You could do it with problems like this. Which is another nice one. And you can think about, if you teach third grade or fourth grade, this is a great problem to give them. 

Like, can they bound things just like, you know, thinking number sense? So it’s kind of like number talks, but, like, a different version. Like, it’s a more estimation take on it, I guess, and you’re really just trying to think through this, and think of answers too big, too small, but the justification piece is pretty important in these.

So that’s the first one. And then the second one here, this is another one of my favorites. I really like wrong answers and focusing on wrong answers, because teachers seldom do, and we tell kids, it’s great to make errors, and then we never ask them for wrong answers. We only ask them for right answers, so this is a way to ask for wrong answers.

So, the intention behind this is you are intentionally, on purpose, making a wrong answer.

But you’re justifying how you know it’s wrong. So instead of saying, this one’s too big, this one’s too small, this one’s just about right, this one’s like, just give me a wrong answer. And there’s two different kinds of wrong answers in this activity.

One that’s super wrong, like, it’s way off base, and one that’s, like, close to right, kind of right.

A lot of people will say wrong is wrong, and I agree, but I feel like there’s also different degrees of wrong, and kids like this because you can have really wrong answers, but you have to explain, of course. I’m big into explaining, as you know.

And then you can have ones that are, like, so close to being right, but they’re just not quite right. So, here’s an example.

16 puppies is two-thirds of the largest litter of puppies on record. True story. How many puppies were in the full litter? 

I could say, give me a super wrong answer to this problem. How do you know it’s super wrong? And, I’ve shared this before, but, like, you know, a teacher said to me one time, two kittens.

And so I’d say, how do you know two kittens is super wrong? Well, first of all, we’re talking about puppies, so if it’s a different unit, it’s gonna be wrong.

And 16 is less than the full amount, so two has to be wrong, because it’s got to be bigger than 16. You know, so that kind of reasoning is really important, and it lets kids come up with the wrong answer, which is very comforting, because they know it’s wrong.

When we say there’s a right answer and you have to give me the right answer, there’s only one possibility, often, but when I say, create a wrong answer, and explain how you know it’s wrong, that gives them a lot of freedom and creativity, which I think will open access to more kids in your classroom.

Here’s another one. Tell me a number that is less than three-fourths, but more than two-thirds. 

They can give you a super wrong one, or they can give you a kind of right one. You could also do something in between, but it’s a nice one. You could also have all the kids tell you them, and show all the different ways you could go wrong in this problem. It’s kind of fun.

So, if you want to emphasize using errors and learning from those errors, which we think is important, make sure you follow up questions not just to right answers, but also to incorrect answers.

Like, treat them both well. So what happens oftentimes is when a kid gets the right answer, there’s no further conversation. When they get the wrong answer, the teacher’s like, oh, tell me more. We want to treat both of them equally in terms of following up on them.

We also want them to revisit approaches and say they learned from the wrong answer. That’s really important. So, like, hey, you got a wrong answer, what’d you learn from that? I learned not to do it this way, or this wasn’t a fruitful approach. Great.

And then questions. I gave you two strategies, but you could ask them, what’s a strategy you shouldn’t use? How did you realize you made a mistake? Did you see any patterns in your mistakes, etc. So those are good after a reflection on an assessment or something like that.

 Last one: I use resources and tools to learn.

 So, a lot of times, we only use manipulatives or tools when we’re not doing good at math. So, I’ve been in many elementary and middle school classrooms, and the only time the manipulatives or the drawings come out is when a kid is struggling.

What that is saying to your students is these are only appropriate if you’re struggling. If you’re not struggling, you don’t need those. That is not the case.

Everybody should be able to model things, draw pictures, etc. So, it’s not just for kids to struggle, it’s for all the kids.

So, it’s a big part of math, how do we foster this habit that I use resources and tools to learn? Because we want students to use them consistently to make sense of mathematics and explain mathematics, not just as a tool for struggle.

Here’s two ways to do it, and these are pretty quick.

Alright, so the first thing is, here’s where I’m at.

And this one’s pretty straightforward, and you may have already done this, but you give a problem. 

So, here’s a problem.

If anybody on the call is a Portuguese speaker, then you can put it in the chat, because then you’d probably read this just fine.

If not, if you’re a Spanish speaker, then you probably can read a lot of cognates in here.

But say I gave a problem to your students, it didn’t make sense to them. So I’m assuming that most of you do not speak Portuguese, because most of the United States does not.

Say I give you this problem, and you’re like, I don’t know how to make sense of this.

Instead of just saying, like, hey, do this thing, or don’t use tools or look back at your notes, a good thing to do that’s a strategy for resources. Resources doesn’t just mean technology or manipulatives, it also means their peers. Their peers should be learning resources for them. Collaboration’s an important part of mathematics.

What you could do is you could say, hey, I gave you this problem, I’ve given you a couple minutes, you guys are stuck, find a partner, and talk about where you’re stuck.

Find a partner and talk about a question you have. Find a partner, and just talk about, like, what makes sense and what doesn’t make sense. Often, we only partner up around answers and the end of problems.

It is very valuable to partner up around the early part of problem solving, where you’ve had enough that you see the task, you kind of have tried some things and now you’re ready to just talk to somebody about it. So that’s what this one is about.

So when students are working independently on a problem set.

So they just got it, they’re working through. You pause them, and you have students pick a problem they wanted to discuss, and find a partner. They don’t just have to compare answers, and in fact, you should encourage them. You can ask a question, you can tell them where you’re stuck. You can say, hey, I did this wrong, don’t try this, you know, that kind of thing.

Let them discuss the problem for a couple minutes, and then you let them go back to work. So now they’ve used a resource, their peer collaboration, and then they’re going back to work. They’re not just talking about answers, they’re actually talking about things that they need support with, with their peers, which is pretty valuable.

The second one is gonna sound kind of similar, and this is one that I thought was very obvious, but it actually was very helpful to some teachers that we worked with in our PL project.

And so, we call it getting unstuck.

Sometimes, I go to classrooms, and every student has their hand up. And the teacher, I feel so bad for, because there’s only one teacher. Sometimes there’s a paraprofessional or another help, like, a special education teacher or somebody else in the room, but often it’s just one teacher against 20, 25 or 30 kids.

If there’s a lot of hands up, it’s impossible to make your way to everybody. And if they all have the same question, you can just pull them together, but sometimes they have different questions.

So, what we found helpful is to remind students that there are lots of tools that you have access to and you can use as you’re working.

So, for smaller kids, I tend to just always have out manipulatives and encourage them to use it no matter what.

For older kids, you may still have that, but you might have other tools, some digital tools, etc.

But just reminding kids before they go to work, hey, if you get stuck, you have resources you can use. It’s not just me, the teacher, I’m not the holder of all the information. You have a lot of resources you can use to get help yourselves before you ask me.

And sometimes teachers do this, where you have to ask three before me of students and things like that, but just reminding them that we have math manipulatives, they’re helpful and we use them to help us learn. Or, I have technology, and you can use that technology to help you learn. You have notes! There’s a reason why in middle school and high school, we often take notes: to go back and refer to them later, not just to, you know, warm up a notebook with them.

 You might have some stuff on the board that would be helpful as they work.

They might have peers that they can use, you know, whatever it is, just reminding kids of all the resources they have in addition to yourself is often sometimes helpful. It seems very obvious to us. I know for new teachers, sometimes this is not obvious, and also sometimes it’s not obvious to kids, and we just have to remind them, so it’s a quick way to do that.

How do I engage students in this habit? How do I support and underscore this habit of using resources and tools to learn?

So, like I just said, you can make a list of the tools and resources they have access to, and encourage them to use it. Don’t just give it to the kids when they’re struggling, say, hey, you did a great job there, do you want to show me how you could have done it with these tools? Or, do you want to use tools to do this as they’re working? That kind of thing.

Praise students who do use tools, and encourage them to show other kids how to use tools. That’s very helpful.

You can also directly ask them, what tools could we or should we use to solve this problem? And then check in. Did anybody use any resources to solve the problem? How did they help? That kind of thing. So we want to normalize it. We don’t want to just attach it to struggling learners. We want to say, hey, you’re working on math. Tools are a normal part of math work. You should use them when appropriate. It might be a hundred’s chart, it might be counters, whatever it is. Use them regularly.

So, with that, it brings me to the end of today. So today, we talked about habits that are important to doing mathematics in the way to build the proficiency that we’ve talked about. So, what are those key habits that we might want to foster and think about? One is to focus on strategies, not just answers.

If we only grade tests, and only put a check or an X on it, we’re not really doing this. We have to talk about the strategies, encourage strategies, give feedback on strategies.

We also have to encourage students to learn from mistakes and normalize the making of mistakes. Again, if we only ever mark things as right or wrong, and we never talk about making errors, or we never celebrate errors, then probably they’re not going to do this, because they know that even though she says that errors are important, she never wants to hear about them.

So we want to normalize that.

And we also want to normalize tool use as a way of reasoning and sense-making and learning mathematics, not just for when we struggle. So, make sure it’s clear what resources they have access to, encourage them to use their peers to reason through things from time to time, etc.

All right. Now, with that.

You can rewatch any of these webinars or share them with a friend. They will be posted on this website. I think Hope put the link in the chat as well.

I also will put in the link to the website that has all of our instructional nudges, and I think she put that in earlier. If anybody has any questions, you can put them in the chat. I will hang out for another minute or two.

Otherwise, I hope you all have a brilliant end to your school year, and a lovely summer of rest and relaxation, or summer school that goes well, for those of you that are still working.