Zandra de Araujo: Hello, and welcome! Nice to see everyone today!

Give it one more minute. While we’re getting settled in, some of you are familiar names on the list, and some of you are not. So, if you want to introduce yourself in the chat, maybe your role in education and where you’re joining us from, that’d be great, so we can see who’s in the room today.

And everybody should be able to see the chat, I believe. All right.

Thanks for kicking us off, Jane. I appreciate that. Instructional coach, awesome!

​If other folks want to share where you’re coming from and what your role is, that’d be great. I’m joined today by two members of the math team here at the Lestinger Center, Amy Strickland, who handles partnerships and is a research coordinator, and Jose de Leon, who is the math manager here, who many of you have seen and met.

​So, nice to see all of you today. I’m Zandra de Araujo, I am the Director of Mathematics at the Center. Today, during this lunch webinar, I think we’re gonna keep it brief. I always say that, and then we go kind of close to time.

Today we’re gonna talk about connections, which is maybe not the most exciting topic after we got to talk about discussions and tasks, but when making this and putting this webinar together, I got more and more excited as we went, so I think there’s a lot of good stuff in here that you’ll enjoy today.

​Okay, so as per usual, I am going to start with brief recaps of these webinars. There are five in this series. This is the fourth of five. They are all being recorded, so you will be able to see them later if you wish.

​And so, in webinar one, the very first one, I think it was in January, we talked about math proficiency. We said it’s not just about procedures or doing drills, it’s also about feeling good about math, feeling like math is useful, understanding the why behind math, being able to problem solve, explain your reasoning and all those kinds of things. So that was the first one.

​And we looked at this rope from the National Research Council in 2001. The Adding It Up report is freely available online. It’s still good, even though it’s 25 years old, but you can see a lot of details there.

And we talked about the characteristics of a brilliant mathematician, and we talked about how those coordinate with the ropes.

So that was the first one. Then we moved to Webinar two where we talked about math discussion. So, a lot of math proficiency is through discussion. We have to talk about math, listen about math, read and write. And so we talked a lot about math.

Discussions, and we talked about the teacher’s role, and we really focused on getting kids to explain their thinking and then questioning.

We talked about some math moves, hopefully you’ve been trying some of those. If you’ve tried any of these since the last time we met, put them in the chat. Jose and Amy are monitoring the chat, and they’re going to interact with you guys today. So, maybe you’ve tried adding it on or wait time or revoicing. So, if you’ve tried any of the things from any of the webinars, share with us, because we’d love to hear what went well. If you have any questions, they can manage the chat and chime in as well.​

We talked about closed questions, which is our tendency if we’re not aware of ourselves, we might ask a lot of closed questions, like, what’s the answer? And we said, maybe occasionally we should ask more open questions, so tell me about your process to get to the answer, things like that.

And then we talked about process and product, so getting a good mix of all the different iterations of these open and closed and process and product questions to really spur mathematical thinking and proficiency.​

Webinar three, which was the last one we did. I don’t know if you remember what it was on, but it was on tasks, which are my favorite topic. It was the centerpiece of my dissertation study back in 2011. And so math tasks are something I’m very passionate about, if you know me.

And we talked about rich mathematics tasks that have a high cognitive demand, so, like, a lot of thinking is required of students, they’re not just routine procedures or naked problems.

They have multiple solution strategies, kids can solve them multiple different ways. They can get into the problem in different ways, so they don’t have to do one particular way, and it fosters reasoning and creativity. And we also talk about how not every task has to be rich.

But some of the tasks, some of the time, should be rich. They shouldn’t just get procedures. So that was last time.

And that brings us today to connections. So, this is the fourth of five, like I said. Connections today, and the last one will be habits, which I think a lot of you are really excited for.

Okay, so we try to make connections seem as undaunting as possible. And so, the way we’re going to talk about them is to kind of categorize them into two camps.

One is long-term connections. So, long-term connections are connections that span across the years as students move between grade levels. It’s kind of that vertical connection piece that we have.

They focus on the evolution of content over time, like across grade levels and ages. The development of tools is not just content, it’s also how do we use representations and manipulatives across the whole grade span? And then also connections to other disciplines in the real world, possibly.

Short-term connections might just be within a single school year, or a unit, or a task. They’re shorter. They’re contained within probably the course or the grade level you’re teaching.​

They might be connections between representations, different components of the lesson, maybe a context for a single task and that kind of thing. So, as teachers, we really want to think about longer-term connections, and also these short ones, and maximize both to the extent possible.​

So, for example, if you taught something yesterday and there’s a connection to what you’re doing today, it might be clear in your brain as a teacher that these two ideas are connected. For your students who are learning math for the first time, those connections might not be as clear, so part of our job is kind of bringing them up and saying, like, hey, remember what we did yesterday? How do you think that looks compared to what we’re doing today? Do you notice any connections? And that helps support students in thinking in this way that we know is important. 

So, we’re going to dig into each type. I’m going to give you very practical, quick instructional nudges for some of these things, so that you can try them out and see if you can build connections.​

Like I said, long-term are more challenging, often, because you really have to have a deep understanding of the curriculum to foster how what we’re doing in kindergarten connect to fifth grade, connect to middle school and high school, that’s harder to do and you don’t have to do it all. You just have to at least start with, like, where is this going next? And that’s a good place to start. 

So, I’m going to show you some very deep connections, but again, I am an incrementalist in a lot of my approaches to education. Start with something you can do tomorrow, some connection you know, long-term or short-term, for the content, and hopefully he takes away some ideas from this today.​

Okay, so long-term connections. Here’s, like, a little mind map of how you might perceive this. (If you’re familiar with the math matrix, you might have seen this before.) But we’re seeing this web of knowledge, and I like webs. I’ve talked about this last time. If we have knowledge connected to lots of points, if you drop off one point, then you still have connected knowledge or ways to get from one place to another. 

If they’re linked like a chain, as soon as one of those links is broken, like, it’s not connected, you can’t get there. So, as we work with students, we want to make lots of connections to ideas so they can make sense of it for themselves and build stronger conceptions of what they’re learning.

So, you can see this mind map. This is kind of what long-term connections kind of look like in my brain.

So, you have kindergarten, and there’s ways in which kindergarten is connected to calculus, and it might be through different checkpoints along the way. And first grade, second grade, and seventh grade, there might be things outside of mathematics that are connected to their kindergarten experience, etc.

So everything is woven together in such a way that kids are making these connections, and we’re helping them do so, because if you’re 5 years old, you only know what it’s like to be five years old, four years old, three years old. You’ve never been older. We’ve already been there. We can help them see where things are going and understand some of these connections and anticipate them.

Alright, so here’s some examples, some concrete takeaway examples of long-term connections. So let’s take fractions.

So, in elementary school, we start to understand what is a fraction? Actually, quite early, around first grade, even before they go to school, they kind of have an intuition for what a half is, because we talk about sharing half with your brother or sister often, or something like that.

But we start by understanding fractions as parts of wholes, or knowing that it’s A 1/B sized pieces. In middle school, we build on that. We do more operations with fractions, we bring in algebraic expressions into our rational functions or fractions, and then in high school, it just builds further. But there’s still that base idea of fractions that is the underlying current throughout all these years.

So that’s kind of an example of fractions.

Manipulatives can also be long-term connections. We think about base 10 blocks early on, then eventually we move into, like, manipulatives like algebra tiles, and there’s some shared traits among those things.

And then we might move into computer simulation. So it’s different ways of representing things that get more complex over time, but they’re connected to one another. If you understand how to use base 10 blocks, you’re kind of abstracting an idea from a concrete representation in some kind of way, and you take that knowledge and build on it as you move to middle school and high school.

​Measurement, another great one. So, if you’ve ever taught, or you are currently teaching elementary school, measurement is one of my absolute favorite things to do. And with really small children, we will measure things with non-standard units, like how many Zandras tall is this room? Or, you know, how many lollipops long is this table, and things like that. So we start with some of those kinds of ideas in elementary school.​

We move into middle school, and we might do solids with volume and surface area, things like that, and then in high school, we move into more complex ideas. So, it’s building across, which is what is so cool about math, and also what is so hard about math. Because if you’re not understanding some of these early ideas, you have less things to connect with.

​So your knowledge is already a little fragile, and so when we try to add, it’s less and less and less sturdy over time, which is not what we want.

​Okay, I’m gonna show you one idea, and I’ve shared this before, so if you’ve seen this before, hopefully it still resonates with you.

​I’m gonna take the idea of an area of a rectangle,  and I’m gonna bring it from kindergarten, pre-K, all the way up to calculus, okay? So, this is the idea of, like, how I see one idea really connected across the curriculum.

​So, these are so cute. In the early grades, you might see rectangles that look like this.

They’re not all the best representations, I will just say, because sometimes it’s hard for students to know what part of this is a rectangle, what part isn’t, but you see a lot of this in early learning centers, in coloring books, things like that, early on.

So these are rectangles. Early on, we’re like, yep, this is a rectangle. Students start to get an idea of what a rectangle looks like.

And then we’re like, okay, we know there’s the classic-looking rectangle, but actually they can look different kinds of ways. Like, they can be tilted, they can be long and narrow, they can be squares, (which blows kids’ and adults’ minds,) you know, all kinds of different ways. So we’re like, yes, rectangles, these are also rectangles, so we’re expanding your idea of a rectangle.​

And then, eventually, we can describe their differences by things like area, and measure them with area to say the coverage, how much coverage it takes to cover that rectangle, or how to draw it, things like that.

​And then we can say, can two rectangles have the same area and look different? And we expand our notion a little deeper.

​We build in arrays, and the underlying idea behind an array is connected to the area of a rectangle, so if I have five rows and there’s six stars in each row, how many stars are there? And we start to see, oh, this you know, skip counting, or this notion of sets of things is connected to the area shortly after, where we kind of simplify representations to be arrays like this. And then students are like, oh, it’s 24, that’s great.

​And then we bring in this formal idea of the area of a rectangle. So we see this, and they’re like, oh, there’s actually a formula for it, and you’ve known it for years, because we’ve been doing arrays for a long time.

​And then we make them more complex, and we say, hey, this relates to all kinds of things, like base 10 blocks we can use to get the area, we can break them down by place value and decompose in different ways, which is great. Again, to get here, we kind of had to understand what rectangles are, what the dimensions are, area, but now they’re at a place where they can use it as a tool for reasoning.

Then we get into really interesting ones like this, complex shapes that we can cut into rectangles, and then add together the area of those rectangles to get something interesting, like the final area.

And then we use that length times width to derive all kinds of other area formulas for geometric figures, like parallelograms and triangles. So these are all coming back to, what is a rectangle, what’s the area of a rectangle and how does it grow?

And then eventually, we’re like, hey, there’s these weird shapes that are just areas under the curve, like this shaded area here. It doesn’t really fit a formula, so we have a challenge. How do we find how much area is underneath here?

And then we say, what if we go back to those rectangles and we put rectangles under the curve.​

And you’re like, oh, that’s pretty good, but there are some gaps. And you’re like, okay, I know how to get rid of those gaps. Let me make them skinnier and skinnier until they fit better and better underneath that curve.

And then if I add up all those rectangles, I probably get the area under the curve pretty well.

​And that’s calculus. Half of calculus, anyways, right? Like, a big idea of calculus is these sums where you’re finding the area under a curve.

​But that idea, the core idea, starts very early with students, where they have to understand what a rectangle is, they have to understand the area, and they have to understand it well enough that they can see why it makes sense to find the area on the curve if I make really infinitesimally small rectangles underneath that curve.

​So, as a former calculus teacher, and somebody who came to love elementary math very deeply a little bit later in my career, I see the connections very deeply, and I would tell my middle school students where it’s going and sneak in some of that stuff, because I had a really good, long view of mathematics. 

I’m not saying you need to do that, but I am saying, maybe sometimes when you see this array, you can start linking it to representations of rectangles and stuff to kind of forecast where it’s going, or talk about where it’s going.

​So, some questions to keep in mind around long-term connections. You can think about these if you’re trying to foster them with your students.

​How do you explicitly connect whatever they’re currently learning to what they might learn in the future? 

So if you’ve never taught the grade level beyond your own, maybe go to the standards for the next grade level and just see where it’s going. So, say we’re counting today, and next year they’re counting to a higher number. Okay, like, that connection’s clear in your mind where it’s going. What strategies are underlying both counting to 20 and counting to 100? You know, things like that.

​What strategies do you use to help students make connections between different mathematical topics? Like, just asking the question, have you seen this before? Have we done something like this before? Do you remember something like this from last year? Those kinds of things will help students kind of prepare their minds to think for and look for connections.

​And then, how do you incorporate real-world applications of math to reinforce them? So, not all of school mathematics is super applicable in the real world in a genuine way. Sometimes it’s kind of hokey.

​In elementary school, there’s a lot of real rich connections because we do a lot of adding and subtracting in our daily life, we measure things. When you get to algebra, sometimes those connections are not as clear to me, because I was trained as a teacher, and also they’re not as real. So, factoring. I’ll just pick on factoring, because I feel like I spend a lot of time teaching Algebra 1 and working on factoring.

​There’s not a ton of good, authentic, real-world factoring tests that I’ve found. If you have one, share it with me, because I would love to see it.

​It’s an important topic, it’s interesting and it helps us solve for roots of quadratics. But, you know, I don’t have a lot of real, genuine connections. And I would tell my students, I actually don’t think people factor a lot in the real world, or at least I haven’t noticed it, but it is important to do this or this. And I was just really upfront with them about those things, but when I found the connections, I definitely shared them, because they like to hear how they were used.

​Okay, if you have any questions about long-term connections, put them in the chat. Like I said, Jose and Amy are on tap, ready to go to answer your questions, or if you have other ideas that help you, just put them in the chat, because the more we share, the more we learn.

​Now I’m going to go to short-term connections. These are usually easier for us because we’re more familiar with them and we’re more comfortable with this.

​So, short-term connections are kind of lesson-level, task-level within the year. They’re things that are right there in your purview because they’re within the grade level, often, or within a task.

So, middle school geometry, for example. Within one year of middle school geometry, probably students are introduced to some shapes, they learn how to classify them, they measure them with different tools and then they learn about some basic transformations, perhaps. That might all come in one year.

​So if you think about those connections, when you get to the middle of the year, you need to connect back to the beginning and forecast what’s coming. When you get to the end of the year, connect backwards. Probably a lot of you are doing these connections right now, because you’re probably prepping for testing season as we speak, but these connections are really important so they kind of understand how it all fits together.

​Arrays, I already talked about arrays some, but arrays in a single year could be used in multiple different ways to decompose numbers, to connect to real-world situations, etc.

​Systems of equations within one task, or set of tasks or a unit.

​You might solve them in different ways with different representations. Having students connect all those representations is how you build the connections. So, a big disclaimer, and I’ll talk about this more later, is if you just put a symbolic representation and a graph together, that’s not really connecting. I mean, they’re connected in terms of they’re both on the board.

But you have all this knowledge about, like, oh yeah, I see where this point is and this point is. I see how they are similar and different. Students are learning for the first time. They are probably not seeing what you’re seeing and making the same connections.

So, a part of our job as teachers is to remember what it’s like to learn something for the very first time, what is not immediately obvious and helping students to uncover what is not immediately obvious, because it’s only obvious to us because we’ve done it for a long time, they’re learning it for the first time, and that’s really important to keep in our brains.

So here are some questions to ask yourself about short-term connections:

​How do you ensure that students see connections between different components within a single lesson? So, we started the lesson here, we ended here. What’s the thread that connects those two parts? Can you help students make that connection?

How do you connect the current lesson to the prior and the future? So, yesterday we did this thing. How does this connect to that thing? And tomorrow, this is where we’re going. Just road mapping with kids like that is very helpful.

​And then what strategies do you use? Maybe your connection is, we used the same strategy yesterday and today, or this task uses the same strategy as this one. That’s a nice connection to bring out with your students.

​All right, so now we’re to the very clear takeaways. Here are some quick strategies you could do tomorrow in class, when you get back from lunch today in class, that should help you build some of these connections with your students.

Okay, so one is called Pair and Compare.

You choose two problems from a set of problems that the students have done. These are short-term connections, by the way.

​So, two problems that you find are similar or different in some kind of way, and the students partner up and try to guess why they thought they were similar or different.

​Okay? So, for example… Here’s a set of problems.

​And I’ve never said do not give problems like this. Kids do need fluency practice, and part of that is repetition and doing some of these types of problems. However, say they had these problems. A really easy connection to foster so that they can see the structure and think about it and connect these ideas is, which two problems did you find were the most similar?

​And then you could just have them say, you know, maybe somebody said, I thought A and J were the most similar. And then their partner would be like, huh, A and J, why did you pick A and J? And their partner would have to guess. I think you said those were the most similar because they have the same digits, or something like that.

And then you could do the same thing for which two are the most different. So you’re trying to foster connections by having students work and look across problems to see what’s similar, and that is helping them to think about connections that exist, even if they’re not obvious at first.

So that’s the first strategy. You could do it in upper grades with division. I like this task, I’ve probably used it before, I just like it because when I ask students to tell me which two are most similar, I always hope they say three and five.

Because three and five are exactly the same, but they often don’t say it, because one has a whole number and the other one has a fraction as the first part of the expression. So, it’s always interesting what they have to say about this one. 

​Give them some practice problems, we all give practice problems, and then just pause and ask them which two do they think are most similar, why? Which two are the most different? Why? So that’s fostering connections across problems.

​So, having a context in mathematics is an opportunity for students to connect mathematics to a context, to the real world, to their prior lived experiences. It’s very important.

We do this with word problems, we give a context in word problems in the hope that students will make a connection to what’s going on in the word problem and use that to reason and make sense.

So here are my commandments, I guess, of using task context. So first, you should try to connect to students’ interests. If you’re going to make a word problem or find one, it’s nice to use something they actually care about. So that requires you to talk to students.

I mean, word problems that would be interesting to me as a 45-year-old woman probably aren’t as exciting to most of my students.

​But also, not every student is going to connect with every context, and that’s okay. But imagine I’m very into gardening and dogs. If you made a problem about gardening and dogs, I would be so into it as a student, because those are two of my favorite things.​

Students are similar in that you connect to something they’re very interested in, Roblox or whatever they’re into these days, and they will engage with it in a different way because it shows that you care enough to connect to something that they care about.​

And most of you talk to your students every day, they talk about things that they’re interested in, so, you know, getting into that with them is always fun. So, that’s the first thing, is try to connect occasionally to things that they actually care about. Not things you care about, necessarily, and you can do some of that, but things that they might actually be passionate about.

And then, try to make the task meaningful. So we see a lot of tasks like this, where they’re kind of nonsensical, or the math just doesn’t make sense, and it’s clear that you’re just making hokey problems for this.

​So, an orchestra of 120 players takes 40 minutes to play Beethoven’s 9th. How long would it take for 60 players to play the same symphony? That doesn’t even make sense!

If you’ve ever played an instrument, the reason why it doesn’t make sense is just because you have more players, doesn’t mean you play the music quicker. It’s probably actually harder to do quicker with more people.

But it’s the same thing with cooking in an oven. Can I cook my croissants in the oven in half the time if I double the temperature?

No, you’re just gonna burn everything, right? And then we get mad at students for not taking things seriously, but they are taking it seriously. They’re just like, this is silly, and it doesn’t make sense. So, you know, think about what we’re giving, make it make sense so that the students are actually reasoning about a real situation, which is helpful.

Engage students with the context. The number of times I see teachers have a cool word problem, it’s an interesting context and then they never talk about it. They just pull the numbers and start operating. It’s a missed opportunity. 

So, for example, say you have this Mokey task.

​My dog Mokey is a schnoodle. He’s adorable, except when he barks at squirrels. He’s so loud! One day, I looked out at the yard and saw eight squirrels playing. Mokey went outside and barked and barked and barked, and I saw five squirrels run away.

I want to talk to the students about this task. We could just start pulling numbers and operating, but I hope that the students are like, huh, what’s a schnoodle? And I would have this picture ready, because he was the best good boy, and I’d be like, well, he looks like this, a poodle and a schnauzer. 

And then we might say, are there any words you don’t know in there, right? Especially for teaching elementary school, it’s a literacy opportunity to kind of bring in new vocabulary, connect to their prior knowledge.

Have you ever seen a dog bark at a tree and seen squirrels run away? Can you have a mental image of this? That is helpful to solving the problem. So you want to engage with the task. Squirrels! That’s a funny word. Did you know it was spelled like that?

​Okay, does everybody have an idea of what a squirrel looks like in their brain? Sometimes kids don’t, especially if they grow up in the middle of the city and they’re not thinking about squirrels, right? So things that I think are very obvious may not be obvious to students.

For example, I was working in a school in Missouri, and we talked about chickens, and students did not know how many legs a chicken had, because they thought the wings were legs.​

And I was like, no, they actually just have two legs on a chicken. Those are wings, they’re not considered the same thing. Which is weird to kids, right? If they didn’t grow up with chickens, but we just don’t make assumptions. We can talk about these things to make sure we’re all on the same page.​

So, don’t lose the context. Actually engage with the context. If you make the trouble to go through and make a context, use it.​

And the other thing I see is sometimes we do a really good job at the beginning, and we’re like, oh, who’s Mokey? What are squirrels? All those things. And then by the end, we just give a number that’s the answer. And it’s like, well, what does that answer represent?​

So here’s an example task. Cindy has two-thirds of a cup of flour in one bag, and another one-half of a cup of flour in another bag, and you can talk about why Cindy didn’t just combine them? Kids will ask these things, you can talk about it and say, like, I don’t know, maybe there are different kinds of flour.​

But to bake her famous cookies, she needs one and 1/2 cups of flour. How should she adjust her recipe to use up all the flour? So, we might have a great conversation about this context. Maybe I have some very passionate up-and-coming bakers who watch the Great Baking Championship, or whatever, at home.​

And then we solve it, and they just get a number like two. I don’t know what the answer is here, but, let’s say two.​

But, we never go back to what does that two represent in this problem? What did we actually solve? And so we need to use the context all the way to the end. One of my favorite ways to keep it in mind is, before they even start, anticipate, what kind of answer are we getting?

So, in this instance, our answer is going to be what? Is it going to be a number, an integer, a rational number or a fraction? What kind of number do we think it might be? Are we gonna make a sentence? Are we gonna make a graph? What are we actually looking for? What is this asking us to do? And then, what units format?​

In this instance, it’s gonna be an adjustment to her recipe, so probably, how many batches she has to make, or how she adjusts her recipe. I don’t know what units you would call that, but, the adjustment for her recipe, or the factor that she multiplies or divides it by, something like that. 

But you talk about that, and then when we’re done, remind students, like, what did we actually answer? Don’t just let them get away with just giving you a number and moving on. But ask them, what did this actually answer? What did we solve?​

Okay, so that was my four commandments for context, and then I have four commandments, I think, for representations as well.

So, make an effort to connect between representations. Representations are so important in mathematics, they’re so vital and yet sometimes we just assume things that are not true. 

Here’s a great example of connecting across four different representations. Often what we’ll do is we’ll show the representations, but we won’t connect across explicitly with our students, and it’s nice to connect explicitly. 

So in this example, students had to connect the initial value in each representation. I’ve never seen where kids really have to dig into, do I see the initial value in each different representation? How do I see it?

It’s a really valuable skill, because they’re kind of sharing the same information, just in different formats. So what can we pick out of these and what can we not pick out of these individually?

So, connecting explicitly across, is very difficult to do for novices. As teachers, we can foster some of this by encouraging them to say, “Hey, do you see the initial value in the graph?

​Can you also see it in the table from what we have there? Can you see it in the equation? Etc.”

​Let’s not assume that they see it, because, again, they’re learning for the first time. It’s very different when you’re learning for the first time than when you’re an expert.

​Okay, so we’re gonna connect across representations explicitly.

​We also should note what is being shown when relevant, and what I mean by that is this.

​I have walked into many classrooms where elementary teachers, if you’re middle or secondary, you could think of any representation, show something like this. Say we’re adding these quantities. There’s the top 123 and the bottom 130, the kids put it all together, and they say there’s 253, which is great. It’s a very concrete representation if they’re playing with the manipulatives and using those to reason.

But, we want to make connections, and we also want to make connections across the concrete, representational and abstract.

So, one enhancement that I’d encourage you to do is draw a representation of what this looks like. So, say I’m at home and I don’t have base 10 blocks, which is a very good chance you don’t have base 10 blocks at home.​

What would I do in my brain that would represent those base 10 blocks as if I were playing with them, or using them? I say play, I think math is fun, so I say play, but it is math thinking.

​So, you might encourage students to say, like, how could we represent this on paper and pencil so that your brain is representing exactly what you did with the concrete things, even when you don’t have them? 

So, I like to have drawn representations to show this and this are exactly the same thing in my brain. My brain thinks of it in the same way, it’s just a way to represent it quickly on paper.

​Because guess what? When they have… (I was gonna say the FCAT, but that was a long time ago). When they have an EOC or the FAST test, and they don’t have the concrete manipulatives, having some way to represent them and drawn representations is very valuable. So we can help scaffold them to that.

And even better, if you connect it to abstract ideas. Like, this is 123. You could even write the word there, one hundred twenty three, so they can see what it looks like written out.

​And then, you can even write the number sentence, or the equation, 123 plus 130 equals 253. The more you put these opportunities to connect and show how you see the same idea in each representation, the better the students will be able to do so.

​The other thing is, as teachers, I promise you all have a bias towards your favorite representations. Mine are symbolic. I was a math major, I love symbolic representations. If I’m not careful, I only use symbols.

​If I’m doing fractions, I have a bias towards area models. I always show area models because it’s just how my brain works easier. I am conscious of that because I’ve found this out about myself.

​You, as teachers, have to break your own bias towards some representations to make sure kids get access to multiple representations. 

So for fractions, do you sometimes use set or discrete models to show things?

​Some representations are better for some ideas than others, I’m not arguing that, but, you know, if you’re trying to show, you know, two-sixths of these animals are dogs, you could do it like this. You could do it with dots, you could do it with an area model.

​Like I said, area models, even within area models, are you always showing rectangles? Are you sometimes showing circles, or using pattern blocks, different types of area models, not just always rectangles? It’s nice for them to see it in different ways.

​Linear models, are you not just using number lines like that? Can they make their own number lines? Can you use linear, like, Cuisinere rods, things like that as well. 

So when you’re doing a particular concept, think about opportunities to bring in different types of representations, and don’t just default to what you always default to, because then students will default to that, and then when they see something that looks different, they’re gonna be like, I don’t know how to do this.

They just haven’t thought about different representations, because they haven’t had access to them. So, keeping that in mind is important.

And finally, make sure they’re providing and getting the opportunity to draw multiple representations. And they’re not just gonna probably do it, you probably have to encourage it. 

And so, here’s a quick way to do it. It’s called For the Win, one of our instructional nudges.

And you make a game board, we call it a game board, it’s like a 2×2.

And you have two matching items.

And then one item that doesn’t match.

And then one blank space.

So what you tell students is you find the matching pair, and then create a new match for the one that doesn’t fit.

​So you can see in this example, there are two numeric expressions, there’s one word problem and then there’s this blank space.

​And so the idea is you ask students, which of these two match?

And they could say why as well, which is always good.

And then you say, okay, that one that didn’t match, can you make a match for it with a word problem in this instance, or story problem? Or maybe it could be a graph or something, you know, you can keep it up to you what you want to do.

Similar to this, if you taught calculus, you could do derivatives, integrals, whatever. Algebra, you could do something like this, where you have two other break expressions, or equations in this instance, and then literal equations, and then you ask them to match.

Just two more for fun. They’re not our original ideas. You guys probably do a lot of these, but I just want to share that they are quick. It doesn’t take a lot to build connections, it just takes an awareness and a desire and a remembering to do it, because it’s not obvious to us.

​So, problem strings are one of my favorite things to do, number talks, they have a lot of different names, but problem strings, so say you have 18 times two, then the next one you might show is 18 times 20.

​How do those two ideas connect? Can you use 18 times two to help make sense of 18 times 20?

​And then, could you use those two prior facts to make sense of 18 times 22? So you’re leading them toward connecting ideas.

​And then, what about this one, where you do 18 times 23, can you make sense of it? So, trying to get students to use connected reasoning to make sense of things, look for structure, and also reason.

And then the last one, which one doesn’t belong? There are lots of books that are available and beautiful on these, and there’s a website that’s which one doesn’t belong? Just Google it, and you’ll find the website, but it’s great, I use them all the time.

Again, they drive connections. For each of the two cards below, find a reason why each of the four parts does not belong. So, you know, why does this donut not belong, or why does this cup… any of them could not belong for some kind of reason, and you’re just saying, why?

​So it’s a great activity, it’s encouraging connections and thinking across ideas, which is the big idea of mathematics.

​So, with that, I will stop.

​I hope you got some good ideas for bringing connections to your classroom. I think if you take away nothing else, all I want you to think about is could I make a connection? And if so, just bring it up. Like, try one of these strategies, do something.

​Start the lesson with, hey, does this look familiar? Anything ringing a bell from this? Does this look like anything? Do you understand this? Just bringing in the connections, because remember, they’re learning it for the first time. 

They don’t have the connections yet. You’re trying to help them. You have the connections. You’ve already learned this stuff.

​So you’re trying to help them uncover those connections so that they can be stronger reasoning and sense-making students.

​So, with that, I will open up for questions. In the meantime, if you want to join us for the final of these five webinars, it will be on May 27th. Same time, same place.

​But we will talk about math habits, like productive struggle and productive dispositions, that kind of thing. So I hope you’ll join us. If you have any questions, put them in the chat.

​Otherwise, thanks for spending your lunch hour with us. We always appreciate it.