Episode 203: Are Calculators Good or Bad For Math Class?

Oct 17, 2022 | Podcast | 0 comments



In this episode Jon & Kyle discuss whether calculators are good or bad for mathematics learning.

Dig in with them as they chat about their past calculator #classroomfails and how they now position their mathematics lessons to more strategically consider when and where it might be helpful for students to leverage the use of a calculator.

You’ll Learn

  • How I can optimize my learning intentionality so that my lesson does not hinge on calculator use; 
  • What to do if you think your math lesson will be “derailed” if students don’t have access to a calculator;
  • What portion of my lesson is better than others to promote the use of a calculator; and,
  • How to design your lessons so that strategies and models emerge to lessen student dependence on their calculator.


Start your school year off right by downloading the guide that you can save and print to share with colleagues during your next staff meeting, professional learning community meeting or just for your own reference!


Kyle Pearce: Welcome to the Making Math Moments That Matter podcast. I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr. We are from makemathmoments.com and together-

Kyle Pearce: With you, the community of math moment makers worldwide who want to build and deliver problem-based math lessons that spark curiosity-

Jon Orr: Fuel sense making.

Kyle Pearce: And ignite your teacher moves. Math moment makers everywhere, you are back with a little bit of time here with Jon and I. We are recording this fresh off the heels of NCTM and NCSM. We've had a lot of opportunity to do some learning with other math moment makers from around the world, and a discussion came up throughout the week and we started asking a few people about their thoughts about calculators. And actually, I believe on the DebateMath Podcast, they actually had a pretty good debate about calculators. So Jon, you and I were thinking, I think it's about time we really dig into this one here because I feel like sometimes people take a hard stance between yes, they're great or no, they're horrible. And I think it's about time that we share our thoughts, where our heads are at. And of course we always reserve the right to shift our thinking as we learn more. But based on where we are now, we thought it'd be a good idea to dive into this discussion with you all.

Jon Orr: Yeah. And I think you've hit the nail on the head already, Kyle, with this false dichotomy here between the two extremes. It's thou shalt use a calculator to make life easier for your students when you're learning different skills. Let's say you're learning a skill that's not multiplying and you're trying to get through your lesson. Should I be using a calculator then, or should I just tell my kids to break out the calculators or put them away? There's that versus the other side of the coin. It's like, I'm not going to use a calculator ever because I want to build number sense and numeracy and fluency. So I think it's good for chat about this, and I think we'll probably even start right now by saying that the purpose of using that calculator has to come down to some intentionality of which you're going to choose. And we'll talk about what that means through some examples here in this episode. But I think if you are clear on what you want to get out of your lesson for a learning goal, that can help dictate whether you choose to use a calculator or not. Like most things we talk about here, there's a gray area and it's not either or. It's about choice and what you value and what you're trying to showcase in that lesson.

Kyle Pearce: Yeah. And you know us, we are not going to say it's all in or all out by any means as Jon just mentioned. But I will say something. If I had to tell you right now ... If I had to just go on a limb and just say what I would recommend for you, based on what I see happening in so many classrooms and based on what we hear, based on what I know, my own experience as a teacher, as a student, I would say if you had to do one of two things, if you had to use the calculator more or you had to use the calculator less, I would be pushing for using it less. Now, the word less now, there's lots of gray there still.

Jon Orr: Right. Why is that, Kyle? Why are you saying less when somebody's like, "Wait a minute. I don't know. I thought he was going to go more, especially in a certain grade level."

Kyle Pearce: I think it's almost ... And again, it's this human nature that we have. This all in or all out approach and it's almost like ... A question we get a lot is, what grade level should we introduce the calculator? And then the question is interesting because it's like, well, you could present it pre-kindergarten if you wanted to. It's not going to hurt anybody to do that. I wouldn't hide it from them until a certain point. But the part I don't like about the question or the part that worries me about the question is it's almost as if we're saying at what point is it okay for the kids to just use it as a tool moving forward? And that to me, it almost suggests as though it's like we have this permission to go, all right, just use the calculator and rely on that and everything else will work it itself out. When in reality it's more of a when's using a calculator maybe appropriate?
When might you suggest a student use a calculator and when might you maybe nudge them to push it away or maybe just keep it in their desks for now? And this is something I have to say. I want to say, I think one thing we do a really good job of you and I, Jon, is we always share all of the things we've done. And this is one of those things that I did for so many years. I had my students so reliant on the calculator. It was like if they didn't bring a calculator to class, I was upset. I was like, "Why didn't you bring the calculator?" It was like we couldn't do any useful math without a calculator. And now I realize it's like, holy smokes. I'm sure I'd like you to bring it for when we might use it, but the reality is more often than not, I think I want it to be maybe pushed to the side, especially during the lesson portion of the day. I feel like that opportunity, that scenario, that experience can be made so much more rich by allowing students to wrestle through problems. Not painfully. And I think we need to dig into that a little bit too. But to craft the experience so that maybe the use of the calculator isn't a necessity and it isn't going to cause students to completely shut down either.

Jon Orr: All right. So let's talk about the moment. When is it appropriate to use the calculator in the lesson and when it's not. You're saying push it to the side, but people are listening going, "Well, when do I push aside and when do I not?" So for example, Kyle, we talked about intentionality at the beginning. In your experience working with teachers and also working in classrooms as a high school teacher and also as the K to 12 math consultant in your district, what do we gain if we ask kids to push it to the side? And what does that look like?
Let's say I'm in a middle school class and we're working on ... I don't know. We've already done multiplying. We're working maybe with fractions or we're in that land and you're working on a lesson or a task and you're teaching maybe through problem-based lessons like we've talked about numerous times here on the podcast. Let's say you're doing that. What do we gain by asking kids to not use the calculator? Because I know there's teachers out there going, "If I don't let them use the calculator, my lesson will be derailed." So let's first go with what do we gain and then we'll talk about the derailing after.

Kyle Pearce: And you know what? To be 100% honest, Jon, if I taught my lessons the way I did when I had students relying on the calculator, I'd gain nothing. And the reason why is because I actually didn't have math models and I didn't understand the importance of different strategies when we're solving certain problems. So I want to be really cautious. It's like if you're just grabbing any problem with massive numbers or very unfriendly numbers, and you're working through, let's say a ratios problem, or you're working through proportional relationships or something along those lines where you're going, "Okay, I don't want to derail the kids when the number is $17.46 and I need to divide it seven ways." Or something like that. Just in that, I feel like those selected quantities, even though they're not huge quantities, they're not friendly. So you had mentioned this intentionality idea. It's like, okay, what is my intentionality? And if I truly ... Like you and I, we have in these, we'll say past decade, realized that there's so much to be learned about the behaviors of the mathematics that intentionally selecting hard numbers to work with actually hides the behaviors. Because it's like in order to see a behavior, you need to see the pattern. You need to recognize what's happening. So if let's say it's a ratios problem, but it's not friendly ratios-

Jon Orr: Yeah. Let's talk ratios for a minute.

Kyle Pearce: You might almost be hiding the behaviors. And of course, if I plug it into a calculator that hides the behavior even more so. So what I'm thinking is, okay, if I want them to understand that ratios really are multiplicative comparisons and that you know what, some ratios actually scale in tandem and I can scale them up or I can scale them down. If I intentionally select really challenging numbers or unfriendly numbers, I might not recognize that as we look at a ratio table or a double number line or any of the other models that we might be leveraging to try to help students really gain a good sense of how ratios and later on rates actually interact.

Jon Orr: Got it. So if I'm, let's say doing, like you said, a ratio problem. I'm asking my students ... We're going to solve a proportion and we're going to use a model like a double number line or a ratio table to do that. You're saying that it's very unlikely that students will see the multiplicative relationship by saying, "Look, I've got this many eggs and this many spoons of whatever." Spoons of ... I don't know. What do you put with eggs?

Kyle Pearce: Yeah. I guess you put the eggs in the spoon. I don't know. Is it a spoon race or an egg race or whatever?

Jon Orr: And so when you're comparing, it's like now when I double the eggs, it's like all of a sudden, if I have four eggs, then you're choosing these careful quantities so they can go, "Look, if I double that, or if I have quadruple, I can easily quadruple this." But if I have 17.5 times of them, then students aren't naturally going to go, "Oh, then do I need 17.5 of those?" Because that's a hard number to see all of a sudden if I have this number down my number line that I don't actually see a multiplicative relationship between them. So to be clear here, we're talking about in our problem based lesson, we're saying put the calculator away as long as our learning goal here is to use strategies and models to make sure that we bring out this learning goal we're looking for and make sure the numbers make sense to elicit those models and strategies. Because you can derail your lesson by just choosing the wrong numbers and it has nothing to do with the calculator.

Kyle Pearce: Absolutely. And we had briefly chatted about this before we hit record was like, if you know ahead of time that the reason you want to have a calculator on the table is because the lesson will derail otherwise, then I feel like that's a point in time where you have to hit the pause button and go, wait, is this the most appropriate lesson then? Has the lesson been designed so that good deep learning can happen? If we pick really ugly or unfriendly numbers to work with here, then it's like those patterns might not be apparent and it might be very difficult other than ... And this can also lead to us rushing to algorithms. Where we go, okay, well every time we just do this. You write this over this equals X over that, and then you just cross multiply. That's what we end up doing. And then we're like, and you can use your calculator to do it.
So you're like okay, great. That'll take the weight off this hard thing called cross multiplying. When in reality you go, wait, what if we could solve this problem eliciting models that actually use reasoning? Where students can actually reason through the problem in order to go this makes sense because it just makes sense. It just feels right. It feels natural. And I would say that is a great place for us to start when we're starting to evaluate when might the calculator come up? Now, you and I were chatting about this as well. It's like we're not saying every time you do a ratios and rates lesson that a calculator can't be involved or that we never explore problems with unfriendly numbers right? When does that happen, Jon?

Jon Orr: Yeah. I was just thinking about that comment that you just said about using the calculator a little later. But I even still think if you're actively learning a goal ... And I know that this is true for you, Kyle, but in my classrooms, if I'm teaching proportions, I'd rather spend the time teaching students how to be fluent along the number line with that scaling than go to any super ugly numbers right away. So for example, I would spend time ... And I spend time with my students scaling forwards and backwards, especially down to a unit rate or a one. If I'm looking at 10 of these and five of those, well, if I divide that by 10, I still should be able to do that somewhat fluently without the use of a calculator. And then we can scale up from one and then that makes it real easy.
So spend time building the skills you want to see in your students, which means, let's think about using the strategies along the number line. Like halving and doubling. An additive relationship can also work. And this is obviously just in ratios that we're talking about here. But I mean, we've got this really, really nice tool that you want to spend time with to build those skills up. And then yeah, later on, and you're working with some purposeful practice and you're going to put in, all of a sudden you've got some money that you want to look at, and all of a sudden there's cents in the money you might say, "Hey, on these problems over here, you can use a calculator, but on these problems over here, let's use our strategies along the number line." And I would even still encourage using the number line or the model that we're developing to solve this problem and use a calculator with the model itself.
If I know I need to find a scale factor that goes up by 17 and a half, then I will draw that on my model and go, "Hey, I'm just going to quickly multiply to get up 17 and a half times as big." We can use that calculator then. So I think what I've learned so much from Jenny Bay-Williams' book, Figuring Out Fluency was that when we ... And we did this on a one of our webinars recently about worksheets and how to do purposeful practice well. One of the things, her book ... I think John SanGiovanni was also an author on that book. They were recommending that when you do your purposeful practice problems and you're enforcing it or you're encouraging a strategy to use with this particular learning goal, then have problems where the strategy makes sense to use and then problems where the strategy doesn't make sense to use.
So it's over here out of these six problems, which problems from this list can you use your number line for and no calculator? Because think of that skill right there. Kids are now selecting based off their multiplicative principles that they're going, "Okay, I can see a double there." They're looking for that strategy, which is a great skill to build. And then they can recognize, "Oh, over on these ones that I didn't select, I'm going to need a calculator for those because there is no nice multiplicative principles showing up right in front of me."

Kyle Pearce: I love it. I love it. And something that just popped into my mind as you were chatting about that is even just this in between as well. When the numbers aren't friendly, encouraging students, "Hey listen, I want you to model it using friendly numbers. So round those numbers. And I'm not judging you on how close or how far, but let's practice the use of the model without making your life miserable." Because that's the other thing. You do not want to be promoting students to use models and strategies with really unfriendly numbers that are just going to cause more frustration. That's an unproductive struggle. Whereas if you selected the numbers nicely or with intentionality, then maybe go all at it. But when the numbers aren't so great is encouraging students, "Hey listen, I'm totally cool with using a calculator, but I do want to see you model with friendlier numbers."
So if it's 9,848, maybe it's 10,000. Just use 10,000 and let's see where you're at. And then you could even ask them, "Hey, is it going to be bigger or less than the number that you came out based on how you estimate it?" So if I round it up and I'm dividing, what is that going to do? Or if I'm rounding up and I'm multiplying, what's that going to do? Is it going to be a bigger number or a smaller? Is it going to be way off? You could use your spicy, medium, mild. Like, is it going to be like a spicy estimate or is it going to be a really mild estimate? And just giving students that skill, not because you want it and you want to evaluate them, but because you really want to promote them building that fluency I think is so key. And every opportunity we have, especially during a lesson, during a lesson with the time that we are together as a learning community, every opportunity we have to do math without just relying on punching numbers into a calculator, I think is a win.
Now I didn't say it can never happen.

Jon Orr: Like the black box to black box.

Kyle Pearce: Exactly, exactly. And we're not saying it's a never thing, it's just a, if you can hold off on that, I think I would. And I think that it's worth the while. So if we think about division for example ... Hey, I'm playing with the two types of division. I need to start with really friendly numbers so it's obvious to students that, wow, there's two types of division. In this scenario, I reveal a rate. In this scenario, I'm revealing how many groups. Oh my gosh, those two types of division are there. Then we're going to practice it, model it, and we're going to do those things. But eventually there's going to be a time where I'm dividing big ugly unfriendly numbers. And unless the learning goal is learning how to divide three digit by two digit numbers without a calculator, well guess what? Now they have a use for the model. We do that, we practice that, and then maybe we move on. And when those come up again, maybe once in a while you say, "Hey, listen, this one, I selected these numbers. I want you to model it just so that you don't lose your familiarity with it and you can build on this skill when we get too more complicated or complex concepts I should say."

Jon Orr: Yeah. So it sounds like we're on the same page there for sure on thinking about how can I build the understanding of the skill we're looking at or the learning goal that I want to elicit? How can I make them flexible and fluid with that same learning goal or skill that we're emerging? And then once a student feels comfortable with that particular fluency, whether it's mental skills, skills, they've worked on strategies and the models, then it might make sense to go, okay, now that I can represent this thing in lots of different ways, I might be able to now move on to use a calculator because it doesn't prohibit me from understanding this particular skill.
Kyle, that reminded me of the way that I've been teaching trigonometry, which so often has been viewed from my students in the past ... My old self. I'm going to teach using the sine ratio, and I didn't even say it back then that. I would say, we're going to teach the sine of sohcahtoa. We're going to do sine cos and tan. And the sign is you say the opposite over the opisthotonus because you're talking about labeling the triangle and being able to do that, but then it's like hit the sine button, put the value in, put the angle in, hit your sine button, and then use that number here to do this calculation. It is such a black box moment when you bring up trigonometry that people would freak out if you didn't use a calculator on trigonometry. But thinking about the way that I've pivoted over the years to actually not use a calculator to do trigonometry with my 10th graders, because we want to do what you said. We want to understand what's actually happening. We want to build a model and strategies around that. And then if that's the case, if we feel flexible and fluent with that model and those strategies, then hey, we might can go over here and speed this up a little bit.
So for example, instead of just saying it's a button on our calculator, it's going to spit it out, we're going to actually going to talk about the ratios between ... This is why we call them the ratio, right Kyle? It's like we're going to look at the ratio between the opposite side and the hypotenuse and why don't you build me a triangle where the ratio is a half. Why don't you build me a triangle while the ratio is three quarters? What does that look like? If the angle is this, let's find the ratio. And what we do now is we will look at different triangles of different sizes and build a trigonometry table.
We'll build, say, nice benchmark angle increments like 10 degrees, 20 degrees up to the 90 degrees and go, well, what is the sine ratio there? We've done a lot of measuring and realizing that it doesn't matter the size of the triangle, it's the ratio that we're looking at, right? This side divided by this side always gives us this number. And so we'll build that table out. And then once we use that table with these easy benchmark angles, it's like, okay, you know what, I've built this other table, I'll speed this up for you guys. We'll use this model, which is what the table is. It's a model that says, okay, now I've filled in the gaps. I've said, you know what? Well, let's look at what one degree, two degrees, and then a per degree table that shows the ratios. And so we've basically eliminated the use of the black box calculator because now the table shows the ratios.
It's a one pager that shows the ratios for the three ratios. And I don't even call them sine, cos and tan until later.

Kyle Pearce: I love that.

Jon Orr: It's the opposite over the hypotonus. It's the adjacent over the hypotenuse. It's like that's the column we look at. Hey, if I need to know what that divided by that is, I'm going to look it up in my model, which is my table, and then I can reverse look up as well. If I divide these two sides, I'm going to get a ratio and I know what that ratio is. I'm going to go look that ratio up in my table and I'm going to reverse look it up to figure out what angle that has to be. So it's like eliminating the use of a calculator in trigonometry helps to build the understanding so that we can be fluid and flexible and use strategies and models to make it happen.

Kyle Pearce: I love it. I love it. And I'm so happy that you specified that you don't name it all up front. And actually last week you and I were presenting, and that came up in another context and a completely different topic, but we were just saying we weren't using any of the terminology yet. We were letting the students experiment and get exposed to what was going on. We were dealing with circle measurement, and it was like we weren't saying diameter, we weren't saying circumference, we weren't saying area. We were just talking about the stuff related to the context. And you could do that with trig as well, or you could go completely ... I mean the context is triangles. It doesn't have to be a really cool context from the real world. It's like the triangle is something from the world that students can relate to. They know what triangles are.
And when you talk about different characteristics of the triangle, that is the exploration that you're having. And when you have that conversation, it's almost like you can get kids envisioning. And for those who are teaching the younger grades, you're thinking like, "Whoa, trigonometry." Maybe you haven't done this since high school yourself. Totally cool. But as you explore it, it's almost like you can help kids develop a visual in their mind of what's going on. If I'm picturing this right angle triangle, where the opposite side or the angle that you're focusing on is getting bigger, what's actually happening? Oh my gosh, the opposite side's getting bigger and the hypotenuse are getting bigger. What does that mean for the ratio? It's like, wow. The opposite side's actually getting a lot bigger, a lot faster than the hypotenuse and therefore my answer's going to get closer to one. That relationship is going to be like, wow, they're almost going to get as close to the same as possible.
And these are all things that kids can develop. They have this understanding and then we formalize it and we go, "Okay, okay, here's going on it. It does have a name." And then eventually you're like, "And there is this button and the calculator that basically has this entire table you've been playing with that's in here. They just took it all, they put it in here so that you don't have to carry the table around with you, you just carry the calculator around with you." And you're like, oh, okay. I get it. It's almost like they can envision it looking it up in the calculator to find that number. It's not just the black box anymore. So great example there, Jon.

Jon Orr: Yeah. And I think that so many kids already see mathematics as a big black box. Tricks to memorize, things to pull out. Okay, I got this box over here, I got these steps and procedures and algorithms. Let's not make the calculator one more thing that'll be a mystery to kids. Let's teach the proper use. Like NCTM's position on calculator use in elementary grades is a strategic aid so that you can use it to recognize and extend numeric and algebraic and geometric patterns and relationships. That's one of their positions. It's a tool to use to help with higher order thinking, not a replacement of this black box. Another trick to add to our bag of tricks, the way that we used to teach math class.

Kyle Pearce: And something else we were talking about WolframAlpha. So the founder, Conrad Wolfram, has mentioned in his keynotes and a TED talk, I believe it was where he said, "It's more important to know when to do something than to know how to do something in today's world." And while we would agree with that, totally agree with that, if the purpose in our math classes to help kids better understand how numbers and operations work to become more fluent and flexible with them, then I feel knowing how to do them can be really helpful. I think if I had to guess, if we had Conrad Wolfram on the show right now and we asked him to jump in on this conversation, I think what he's trying to get at is we don't want kids to blindly memorize long division or blindly memorize the steps to solving an equation. That's not the important part. He wants them to know when to use something, but then also when you know when, it's also typical that you understand how it's working, not necessarily that you can just do what the calculator does. So I think there is a difference there, what he's saying versus what we're saying. But maybe if he's listening to the podcast, hey, come on the show there, Conrad.

Jon Orr: One of the examples he gives ... And I'm sure that you'll come up with the counter to his metaphor here, but basically in one of his talks, he talks about a long time ago when the automobile showed up on the market, it was really important for any purchaser or owner of the automobile. We're talking in the very first ones. You had to know how that car worked from top to bottom. You needed to be able to fix it if it broke down. You needed to be able to change its oil. You needed to be able to fix the engine. If you bought a car, there was no mechanics lying around ... Or not that they're lying around. But there's no mechanics hanging around ready to fix it. You had to know how to operate that thing at a higher level than we know how to operate cars now.
So he's saying today, you buy a car, you don't know how it works, you just drive it. You drive the car to where you want to go. You use it to do other things. You use it for a tool, you use it at work to drive if you're a delivery person. You use it to get to work. You might use an automobile in this way. You don't know anymore how it works because you don't need to. So his argument is eventually we're not going to need to know how to factor a trinomial. He's going to say, we can ask very smart calculators, computer algebra systems to factor for us. It was more important to know when to factor. I need to factor this thing so that I can figure out its X intercepts or I need to factor this thing so I can graph it. These are important skills versus the actual how. That was his argument. So that whole morphed it also to the calculator. Do I really need to know how to multiply? Well, we think so-

Kyle Pearce: Yeah, exactly.

Jon Orr: But the calculator can do it for us. There's a point where it's like, wait, there might be something to that statement, but we definitely want to know how this works in this sense so that we can build on that flexibility and fluency as well.

Kyle Pearce: Yeah. It's a great point. But as I'm thinking about this and I'm going, yeah, I don't know how to fix my car, any car, but that makes me very ignorant. Especially when we think about society in general, I think as a society, I think we are becoming more ignorant with pretty complex things when in reality, even just my dad, my dad knows how to fix I feel like everything, and it's great, and I think it's wonderful. Does he have to? No. But I have to pay a lot more money in order for those things to happen. And so coming back to this math context, at some point it becomes very difficult to know when to factor when you're not really sure what factoring is. I think that's the balance that we need to strike.
I can appreciate his thinking that, hey, listen, don't overemphasize sum and product. The method of factoring over here when students don't know why they're doing it. I would almost argue it's almost as if he's saying, listen, we need kids to understand how factoring works, not necessarily to be the best factorer of all time, but to be over here going, "I understand how factoring works and oh my gosh, when I'm factoring, holy smokes, I'm making a rectangle if I use an area model." Wow. I never knew that. And I could factor any problem that was factorable. Anything that could be factored nicely I could do that, but I did not know that I was making a rectangle. And I feel like that's what he's trying to get at is that my old me is not so great.

Jon Orr: Right. And he's saying that thing that you said earlier about if the goal here is to just blindly factor, then yeah, it doesn't matter if you have a computer do it for you because you're just blindly following an algorithm that you don't know why it works.

Kyle Pearce: You just waisted 10 minutes.

Jon Orr: Right. If you've spent more time on the reason that we need to factor because it's useful this way, then it makes more sense to being able to build that model so that we can factor.

Kyle Pearce: Yeah. I love it. And I think this is a great example to end on here and I hope people are still hearing that message that we had mentioned in the beginning that there is no cut and dry answer here. However, our main message is, I got to assume you're thinking maybe I've got the calculator out too much in my class. Maybe you do and maybe you want to just try pushing it back a little bit. We're not saying throw it out. We're not saying disallow it in your classroom. But if students are just mindlessly using an algorithm to calculate, then the calculator is probably a better tool. So if you're just mindlessly factoring like I was, then go for it. But if your real goal in math is to try to help to emerge big ideas, get students to understand that strategies can be really helpful, some more clever or more efficient than others in certain situations, and that I can actually model a lot of what I'm doing, I think pushing that calculator away at least a little more than maybe we have in the past in general, is probably a good move moving forward.

Jon Orr: Good summary, Kyle. Good summary. And with that, hey, we are so glad you joined us here for another episode of The Make Math Moments That Matter podcast. And as we always learn so much from the discussions that we have here together, like we did here today and also from any of the guests we talk to and people like you when we interact through social media, we always learn and it's awesome to reflect on that. So what are you going to do right now to reflect on your practice? Like Kyle said, and suggested, thinking about your lesson and the intentionality of your lesson. What are you trying to emerge or what are you trying to bring about in your lesson? Is it strengthening numeracy? Is it strengthening number sense with your students? Can we do that a little bit more in certain ways? How are we choosing the problems we work with? Super important stuff. Don't forget to reflect on that and then share that with a colleague. Share that learning, that understanding, that new change in learning with someone else. Maybe it's your partner at home, maybe it's a trusted friend. Make sure you share it so that the learning you're doing here doesn't wash away like footprints in the sand on the beach.

Kyle Pearce: I love it there, Jon. I love it. And friends, listen, we are at episode 203 and right around the corner, as of the date of recording, we are getting ready for our 2022 virtual summit. This year, we're starting on Friday night. Friday, November 18th. We've got Saturday, November 19th and Sunday, November 20th. And it is a 100% free live event for you to attend over those three days. Maybe you can stick around for one session, maybe all the sessions. But if you head over to makemathmoments.com/summit, you can get yourself signed up and you will reserve yourself a seat. Something else that might be really interesting for you is that we have a giveaway going on with almost $5,000 in prizes to give away, including white book flip charts. We've got academy memberships for Make Math Moments where you'll be able to see the summit replays all year round. And we've got a few goodies like our favorite professional development books for you to check out and much more. So if you want to get in on the giveaway, head over to makemathmoments.com/giveaway. That's makemathmoments.com/giveaway. And once you enter you'll see some awesome opportunities to get more entries so you're chances of winning go way, way up.

Jon Orr: Yeah. You can have more than one entry.

Kyle Pearce: Absolutely.

Jon Orr: That's right. You can double them up, triple them up. So show notes and links to resources and the complete transcripts to this episode can be found over at makemathmoments.com/episode203. Again, that's makemathmoments.com/episode203.

Kyle Pearce: Well, my math moment maker friends, until next time, I'm Kyle Pearce.

Jon Orr: And I'm Jon Orr.

Kyle Pearce: High fives for us.

Jon Orr: And a high five for you.

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