Episode 44: Top Ten Math Manipulatives
Make sure to check out the blogpost for this episode where I share the completed, collaborative Top Ten list, as well as links related to this episode including ways to contact Dr. Nicki Newton and Ann Elise Record.
Transcript:
(transcribed by kayla.r.fainer@gmail.com)
Melissa Milner 00:09
Hi, this is Melissa Milner. Welcome to The Teacher As... podcast. The goal of this weekly podcast is to help you explore your passions and learn from others in education and beyond to better your teaching. The Teacher As... podcast will highlight innovative practices and uncommon parallels in education.
Before I share this amazing Top 10 episode, I just want to let you know that throughout the episode, on and off, there's a very high pitched noise. We heard it when we were recording and unfortunately just weren't able to figure out what was causing it and how to stop it. So when possible, I was able to edit it out. But it pretty much is on and off throughout the episode. So my apologies for the sound issues, number one.
And number two, thank you to the pioneers of my Top 10 episodes, Ann Elise Record and Dr. Nicki Newton, for being so supportive. This is their second Top 10 episode. Their list is Top 10 Math Manipulatives. And if you didn't hear their first Top 10 episode, you definitely should check it out. It's hysterical. It was Episode 30.
So without further ado, I hope you enjoy Top 10 Math Manipulatives with Dr. Nicki Newton and Ann Elise Record.
(Soundbite)
I'm very excited to have back Dr. Nicki Newton and Ann Elise Record for another Top 10 list. Thanks for coming again.
Ann Elise Record 01:49
Oh, it's so exciting.
Dr. Nicki Newton 01:50
Thank you for having us.
Melissa Milner 01:52
The topic that you're gonna come up with a Top 10 list for is Top 10 Math Manipulatives. So Ann Elise, why don't you talk about what you were wondering about with this topic?
Ann Elise Record 02:07
Yeah, I want to talk about what defines a manipulative. I was doing a presentation last week, and I'd asked the participants their favorite manipulative. And some people had put in the chat box: dice and cards, and a ruler. And it kind of took me back. I don't think of those as manipulatives.
I think of a manipulative as being something that I can see the number relationships and things in the manipulative itself. I hadn't thought about dice or cards, but I don't know if I'm right. It's just what my interpretation was. So I'd love for us to figure out before we do our Top 10 lists, how do we define what a manipulative is?
Melissa Milner 02:45
Dr. Nicki, what are you thinking?
Dr. Nicki Newton 02:47
Well, I'm thinking, I think of dice and cards more as number generators that you can play really cool math games with. I would consider a ruler a tool you could use as a manipulative.
Ann Elise Record 03:01
Okay.
Melissa Milner 03:01
Are you thinking, Ann Elise, that for this draft, that manipulatives should be directly related to making sense of math concepts?
Ann Elise Record 03:14
Yes.
Dr. Nicki Newton 03:16
I don't think that, because lima beans are a manipulative.
Melissa Milner 03:21
Right.
Ann Elise Record 03:22
Okay. All right.
Dr. Nicki Newton 03:23
I use-- I remember back in the days of Math Their Way, you'd spray paint those lima beans. And you'd have one side green and one side white. And then those were called counters before we were using or they were really popular for colored counters.
Melissa Milner 03:41
Which means counters are for math concepts, understanding math concepts.
Ann Elise Record 03:46
Okay.
Dr. Nicki Newton 03:47
Right. So you know what I mean? If you're using it to understand a math concept, you're using it to manipulate the numbers. Because I would consider an elapsed time ruler would be a manipulative. Especially, even in a different way than a regular ruler, they measure different things. But do you know what I'm saying?
Okay, so you have a ruler and then you have an elapsed time ruler, which has the time on it. But you also have-- what I use a lot is seamstress tape, the ones that you can get from Amazon that go up to 60 inches. And I use that as the elapsed time ruler.
Melissa Milner 04:26
Okay, don't give away too much, Nicki.
Dr. Nicki Newton 04:28
No, I'm just saying. So it all becomes very complicated.
Ann Elise Record 04:32
Well, it does. Yeah. It's not clear cut. So then I think of an open number line. Do manipulatives have to be physical?
Dr. Nicki Newton 04:39
It's a model.
Ann Elise Record 04:41
That's true. That's true.
Melissa Milner 04:42
So what you're saying is, do they need to be in your hands to be a manipulative?
Ann Elise Record 04:47
Yeah. They can be physical or they can be virtual.
Dr. Nicki Newton 04:50
But do they need to be something that you can touch at home?
Ann Elise Record 04:54
Yeah, that's what I'm wondering.
Dr. Nicki Newton 04:55
But this is what I would say. If you're trying to think about concrete, pictorial, abstract, there are a lot of manipulatives you want to also have in paper form so that kids can draw it out. So you have the two colored counters, but then you record that onto colored counter paper. Or you have the Rekenrek, but you record that on Rekenrek paper.
Ann Elise Record 05:18
Gotcha. Yeah, okay.
Melissa Milner 05:19
So Ann Elise, are you any more clear?
Ann Elise Record 05:24
Well, I guess we'll just have the conversation as we go.
Melissa Milner 05:28
And you can always veto if you don't think it belongs on the list. Remember your vetoes.
Ann Elise Record 05:32
That's true. That's true.
Melissa Milner 05:34
Let's get the audience caught up, because they might not have listened to Episode 30. Go listen to it. It's hysterical. Dr. Nicki and Ann Elise are working together. They have their own picks about what they think the top math manipulatives are. And they are working together to create a collaborative Top 10 list. So what happens is, there is a player A and a player B.
So player A will have pick 10, 8, 7, 6, 4, and 3. That means they have two more picks and they have one veto. Player B has picks 9, 5, 2, and 1. But they have the top two picks, and they have two vetoes.
Let's talk about what a veto is before we make some decisions here. So there's basically two reasons why someone might veto a pick. One is if Dr. Nicki, let's say, gives a manipulative and Ann Elise loves that manipulative and wants it higher on the list. She can veto that manipulative. She or Dr. Nicki can play it again higher on the list. But then Dr. Nicki, now for her turn, has to come up with another pick.
The other reason for a veto would be Dr. Nicki throws out an idea for a manipulative pick and Ann Elise says, "that shouldn't even be on this list." So that's the other way that you might want to veto a pick. And in that case, she would veto it, and Dr. Nicki would have to pick a different thing. If vetoed, you have to come up with another pick.
So the hope is that you have come today with lots of options. And you can change your mind about the order as the picking continue or as the draft continues. I call it a draft. Are there any questions before we flip the coin to see who gets to pick A or B?
Ann Elise Record 07:59
I don't mind letting Dr. Nicki pick whatever she'd like to be.
Melissa Milner 08:01
Okay, so you want to skip flipping the coin and just have Dr. Nicki pick?
Ann Elise Record 08:05
Yeah.
Melissa Milner 08:05
All right, let's do that.
Ann Elise Record 08:06
She can choose.
Dr. Nicki Newton 08:07
Well, what was I last time?
Ann Elise Record 08:09
You were Player B last time, because you had picks number one and two.
Melissa Milner 08:12
Good memory.
Ann Elise Record 08:13
So I'm happy to have you do that again. Or if you'd like to be the other player, so you get more manipulative picks, I'm happy with that, too.
Dr. Nicki Newton 08:21
Well, I'll be the other player this time.
Ann Elise Record 08:23
Okay.
Melissa Milner 08:23
So Dr. Nicki is number 10, number 8, number7, number 6, number 4, and number 3. All right, I got it recorded so that I at least can try to remember who's doing what. Dr. Nicki, let's get started with your pick number 10.
Dr. Nicki Newton 08:33
Number 10, I would say the geoboard. I just like the different ways that kids can explore area, perimeter, shapes, figures on the geoboard. I like what that has to offer.
Ann Elise Record 09:00
Yeah, I agree. That's wonderful.
Melissa Milner 09:03
All right. I think there's not a veto there, right Ann Elise?
Ann Elise Record 09:06
Nope. No, no, no. I'm not vetoing yet.
Melissa Milner 09:10
Do you use geoboards, Ann Elise, in your work?
Ann Elise Record 09:13
Actually, I use them in the virtual world. I love Brainingcamp as a set of virtual manipulative apps. And I love using the geoboard to be able to use fraction concepts and to be able to use the highlighter to shade amounts. Because in the virtual world, you're not having the benefits of in person paper folding and shading.
And so the geoboard allows me to do that online. But I've not used them as much in person. I was a little afraid in my fifth grade classroom that the rubber bands would go flying around the room. So they were in my classroom, but I didn't use them.
Melissa Milner 09:48
And Dr. Nicki, how do you use them?
Dr. Nicki Newton 09:50
I use them a lot for area and perimeter. But also, they're good for symmetry. And they're good to show rotations and translations and things like that - so right angle, different kinds of angles. I use them for a bunch of stuff like that. I mainly use them in the upper elementary grades. But there are things you can do in the primary grade levels, too, counting and stuff.
Melissa Milner 10:15
Wow, well, that was awesome. Pick number nine is Ann Elise.
Ann Elise Record 10:20
So I'm gonna go with pattern blocks. So the ones that have the yellow hexagon, red trapezoid, blue rhombus, the green triangles. And I also love to get the brown and purple ones, which are, if you define the hexagon as the hole, then they're the 4th and 12th as well. So I love pattern blocks for exploring fractions.
And I remember there was a consultant that came to my building, our school district, years and years ago now. It was the first time my brain was introduced to math visually. I grew up learning rote procedures and memorization with math. And I did not see and wasn't exposed to visual ways of thinking about math. And I remember sitting there with my colleagues in my fifth grade team, learning about these pattern blocks for fractions.
And it was a struggle. My brain did not take easily to it. I had to keep looking back at which one was a third, which was a half. I just didn't have that part of my brain developed yet. And so I remember a colleague of mine doing super easy with it. And I was getting frustrated with myself, because I thought I was a good math person. And I'm like, why am I not getting this?
But then the more I did it in my classroom with all my kiddos over the course of years, I changed the way I think about math. I see it so much more visually. And so when I think of some fraction problems, like 5/6 + 1/3, I now visualize the pattern blocks. And I can picture that third blue piece filling in the hole of the hexagon with those 5/6. So I know it's 1 and 1/3.
And I didn't have to go through a process of common denominator and multiply. I can visually see those amounts. So I just think they're so powerful to reach all learners and that all of our brains can learn math visually. And it's really powerful. So I love those.
Melissa Milner 12:02
Very cool. Dr. Nicki?
Dr. Nicki Newton 12:04
I'll go with that, because I like pattern blocks. And it's actually on my list. It's a little higher, but I can go with it being number nine. And I definitely-- I had specifically fraction pattern blocks.
Melissa Milner 12:16
Fantastic. All right, this is how last time started. I don't want to get too relaxed here. Okay, pick number eight goes back to Dr. Nicki.
Dr. Nicki Newton 12:30
Oh, okay. I think number bracelets are really powerful, and I think they're underutilized. And so I would say number bracelets. Because number bracelets, you can teach kids early skip counting stuff. You can teach them composing and decomposing numbers. And I use number bracelets in the upper elementary grades, in fourth grade, when they're learning to compose and decompose fractions.
So say you have six pony beads on a pipe cleaner. And then you have the kids and you say, okay, well how can we decompose this? We can say 1/6 + 5/6, or 2/6 + 4/6, or 3/6 + 3/6, etc. So you can teach composing and decomposing and also adding and subtracting. It leads right into that, of fractions with like denominators. So I really like number bracelets, because I think you can get a lot of mileage out of both grades with those.
Melissa Milner 13:31
That sounds great. What do you think, Ann Elise?
Ann Elise Record 13:33
Yeah, I love it. Yeah, I think that's great.
Melissa Milner 13:37
So would they have more than one bracelet to do some of that work? Or they could do it with one bracelet?
Dr. Nicki Newton 13:42
Absolutely. You're supposed to have a number bracelet for each number, starting at 3 all the way up to 10. And then in fourth grade, you'd have one for each set of denominators that they learn. So fourths, fifths, sixths, whatever it is.
Melissa Milner 14:00
Very cool.
Ann Elise Record 14:01
I would just add in that one of the major things that I think we can do with kids to help with the math journey and make it easier, and more accessible, and be more flexible with thinking is decomposing numbers. I don't think the kids are able to decompose all the numbers up to 10 fluently. And I think if we did that, that would fix the fluency within 10 for adding and subtracting.
And then it would translate to the fractions, as well, as Dr. Nicki had said about decomposing fractions. I think when we explore our calculation work with flexible strategies and not limited to just the procedural type things, we need to have that skill in place in order to do those efficiently. And so I really think we cannot do enough of helping kids decompose all the numbers up to 10. Because I think many of them now are doing well with the 5 and 10. But 6, 7, 8, 9, they don't know the combinations that make that.
So when we ask them a fluency question on the math running record that Dr. Nicki created, a lot of them are in the counting phase of reasoning. They count on. But I think that if we have decompose an eight often enough to know that it can be a six and a two as one possibility, then when I ask someone 6 + 2, they know it's 8. Because they've learned that as a pair. I absolutely agree that number bracelet is an excellent tool for that.
Melissa Milner 15:16
All right. I think we're actually, again, copacetic. So Dr. Nicki, pick number seven.
Dr. Nicki Newton 15:24
I don't know. I kind of want to pick-- time is so important. So I kind of want to pick the elapsed time ruler. And I'll tell you why. I have elapsed time rulers and seamstress tape. Those are two different things. But the seamstress tape, though, really only allows you to go up to 60. It really only allows you to tell time within the hour.
So if I had to pick for the list, I would pick the elapsed time ruler, because they have different versions of it. And you can get an elapsed time ruler that just has hours on it. Or you can get an elapsed time ruler that has the hour and a half on it. Or you can get an elapsed time ruler that has 15 minute increments.
You can get an elapsed time ruler that's just the hour. Or you can get an elapsed time ruler-- actually, I don't know that you can get one that’s just the hour. But you can get an elapsed time ruler that goes from 12AM to 12PM. So if you're in fourth grade, you can go across the hour.
So I really like the elapsed time ruler for third and fourth grade. And there's fifth graders who can't tell time either.
Melissa Milner 16:35
Sure. Ann Elise, what are you thinking?
Ann Elise Record 16:39
I'm thinking that the more I learn about elementary mathematics, the more that I realize I have to learn. And I love that part of this journey is that I learn new things every day. And I have known Dr. Nicki for years and years, and I have never heard her mention or I've never even heard anyone ever mention an elapsed time ruler. And I think that is fantastic.
Melissa Milner 17:03
Yes, it's so cool. Where do you get them, Dr. Nicki?
Dr. Nicki Newton 17:08
You can Google them. You can buy them.
Melissa Milner 17:12
Like Amazon or?
Dr. Nicki Newton 17:14
Yeah, I'm sure they're on Amazon. But I think EAI sells elapsed time rulers. Didax might. So you just Google elapsed time ruler shopping, and it'll come up. But Didax also has elapsed time rulers, so you can buy them. They're plastic, but you can get them online.
There's a million places that you can print out an elapsed time ruler. But who really has one that I like is UEN, the Utah Education Network. That one's hard to find now, but they had it online as a PDF for years. It's still there and nothing goes away. But it is hard to find.
Melissa Milner 17:50
Yeah. This is so awesome. Okay, I guess we keep going. So Dr. Nicki, just so you're aware, you have the next pick, which is pick six. And then you have four and three, and then you're done. So just be thinking, you probably want your three to be one of your really higher, because you're not going to get to do it two or one. So all right, number six, Dr. Nicki.
Dr. Nicki Newton 18:16
How many vetoes do I have?
Melissa Milner 18:17
You only have one. Ann Elise has two, because she has less picks.
Dr. Nicki Newton 18:22
Okay. Well, first of all, I already know Ann Elise and I are gonna battle about the number one.
Ann Elise Record 18:30
You only have one veto. I have two.
Dr. Nicki Newton 18:33
I know. But I have a veto that I haven't used.
Ann Elise Record 18:36
Oh my god. If you veto my number one and it doesn't make the Top 10 list, I cannot live and exist in a world in which the Top 10 list would not have my number one which you know what it is.
Melissa Milner 18:49
It might start with a C. And then maybe there's an R in the second word.
Dr. Nicki Newton 18:49
Well, right now we are at number-- what is it?
Ann Elise Record 19:04
Six. We're at six.
Melissa Milner 19:05
We're at number six. Yeah.
Dr. Nicki Newton 19:07
There's no battle here. It's either got to be snap cubes or base 10 blocks. And I think I'm gonna say base 10 blocks, only because you can use snap cubes for a lot more things, I think. So I would want that higher on the list. I would say base 10 blocks. Base 10 blocks are wonderful, and they have to be on the list. And people rush to use them.
Van de Walle talks a lot about that, levels of manipulative and phasing kids in and out of different kinds of manipulatives. We shouldn't rush kids to base 10 blocks before they have already built their economy, established what it means to have tens and ones. And we rushed them to a pre-made tool.
So I would say base 10 blocks, because I think they're really important. I think they're introduced too early. Because the research says that kids should build their understanding of base 10, right, that they should-- using beans and sticks, you have loose beans, and then you have 10 beans on a stick for a 10. Then you make a raft of 10 popsicle sticks of 10.
So the kids can see, this is-- and then they know, these are 1, this is 10, and this is 100. And after they really know that, then you start introducing pre-made manipulatives. I would say base 10 blocks should be on there. It's definitely a phenomenal manipulative, just with that one caveat that it's introduced too early.
Melissa Milner 20:44
All right. Ann Elise, what are you thinking?
Ann Elise Record 20:45
Well, I absolutely agree that they're on the list. I will certainly not veto that. But I really would make a push for-- I certainly think they're used too early in the kids' journey. But I think they also are used in a way that really unintentionally encourages counting. Because when you're trading in a rod for 10 cubes and it's joining 5 cubes already there in a pile, it's a pile of 15 cubes, right?
And you can't subitize. You can't tell how many that is without counting them. And then if you're doing subtraction algorithm, then you're counting out the cubes to remove, and you're counting the ones that are left. And so using the ones cubes with the base 10 blocks, I think people are very well intentioned, because they want students to understand why the regrouping-- the meaning of what's happening in the regrouping.
But I think in the way of doing that with the base 10 blocks and not, say, having a cuisenaire rod for the ones place to be able to kick in strategic number relationships, keeps them in the counting phase of reasoning, even in the contents of a multi-digit number.
Melissa Milner 21:49
Yeah. And even if they set up their little ones to show rows of five, it's still like you're saying. So they can kind of subitize it. But it's still, like you said, not going to the next step.
Ann Elise Record 22:04
And I think the unitization to know that in our number system, the base 10 is 10 of those units, I think that can be done in the context of other ways, like counting collections, or as Nicki had mentioned, the sticks and lima beans and things like that. But I would prefer the manipulatives that they're using to encourage the strategy thinking and number relationships. And the base 10, particularly with the subtraction when there's regrouping that has to happen, the kids are just counting. But they're absolutely a wonderful manipulative.
Melissa Milner 22:34
Yeah. I think six probably is a good place for that, considering there's some caveats that you both have added. Interesting.
Dr. Nicki Newton 22:43
Well, and one of the things if you look at the research around using those, and even in the core in their progressions, they talked about doing sketches along with the actual algorithm so that kids can actually unpack what they're doing. There's that cycle of engagement - concrete, pictorial, abstract. The kids would build it, draw it, and then do the algorithm and be able to explain what they have done to talk about the algorithm with meaning.
Ann Elise Record 23:14
And in the standards, in the core standards at least, the subtraction algorithm is not until fourth grade. But in my consulting work, I run into a lot of first and second grade teachers who are doing the regrouping algorithm with the kids.
Melissa Milner 23:25
They are.
Ann Elise Record 23:26
If we were seeing an entire class the age of kids in fourth grade masterfully doing the algorithm, I'd have no problem with it. But it's an epidemic. I think subtraction is the achilles heel of our kids.
Melissa Milner 23:38
Yes.
Ann Elise Record 23:39
And I think that we can fix that if we take our time and not introduce the procedural algorithm, the traditional one, until later in their journeys to give them time to think flexibly about the numbers and whatnot.
Melissa Milner 23:52
I absolutely agree. But unfortunately, a lot of the parents jump on it and teach it way too early.
Ann Elise Record 23:59
Which is why I think it's really important to get parents involved in the journey, as well. Because the emphasis when I was growing up was getting right answers. It wasn't about the why. It was just you want to get the right answer quickly, too.
Dr. Nicki Newton 24:11
Well, and you know, on McCallum's website, I emailed him one time about the second grade dilemma of teaching the traditional algorithm. And he said, read the blog and see what it says. So I went back and I read what is written in his blog about it. And there is a lot of discussion. It's a wonderful post to read. You really should read that post on McCallum's blog about the Common Core.
And he talks about how the traditional algorithm is one of many methods. And so we can introduce it to the kids and show it to the kids, but not as the queen method of all methods in the world, just as like, this is one of many strategies. And when I introduce it to the kids, I say, oh, this is what your mom's gonna do. This is grandma's way.
Because they're gonna see it. So now at least they have a context for-- instead of, my momma didn't say to do it like that. I can say, your mom is gonna tell you this way. So here's grandma's way, and then here's some other ways that we do it. One other thing is Van de Walle talks a lot about this. Instead of getting kids-- as you say, subtraction is the achilles heel.
So Van de Walle just says, then teach the kids to add up instead of subtract. Because subtraction is so error prone, whether you're doing it as an algorithm, whether you're doing it as a mental-- it's error prone. So if you can teach them to count up, it's much easier.
Ann Elise Record 25:39
One more thing to add in there-- subtraction can be fixed so easily, I think. But the interpretation of subtraction, both as removal but also as that missing add in that I can use addition to solve subtraction, is not that we want to use one way over another. We want to let the numbers help us-- the numbers lead us toward an efficient strategy for those numbers.
So for example, 90 - 12. I want to remove 10, remove 2, right? But 92 minus 89, I don't want to remove 80, remove 9. I want to think about using addition. So when the numbers are far apart, I tend to use the removal thought process. And if the numbers are close together, I think addition.
But you have to have that flexibility of thought. And I think the algorithm that is done with those base 10 blocks and all of that locks the kids into the systematic way of the algorithm. So I've seen kids literally stack 14 - 8, and cross the 1 out, and make it a 0, and put the 1 next to the 4, and then say and now we have 14 - 8. There's no number censor thinking.
Melissa Milner 26:42
Oh my gosh, oh my gosh. So all right, we're gonna move on. This is one of the few Ann Elise picks, number five.
Ann Elise Record 26:51
All right. I'm going to go with fraction tiles. And I would actually suggest, the fraction tiles, that we use them upside down and not having-- well, both. Having the numbers be visible, but also turning them upside down and having the numbers not be visible, so that we can have the flexibility of defining the whole, whatever we want the whole to be.
But also, I think the conceptual understanding of fractions being that unit fraction and that they're equal sized pieces as the iteration of that unit fraction that makes up the whole, I love how the kids are able to see the unit fractions. So if I build 3/4 with fraction tiles with the fractions visible, I can see that it's the same as 3 1/4, right? So I like that as the beginning understanding with fractions.
But I also want them upside down. Because I want the students to have to think, well, what is this piece worth? What's the whole? And if that's the whole, how many of these equal sized pieces can fit in that whole to make them constantly be thinking about equal sized pieces and the number of pieces in the whole.
Melissa Milner 27:55
Yes, very powerful. All right. Dr. Nicki, what do you think about fraction tiles?
Dr. Nicki Newton 28:00
Well, I only have one veto, so I can't use it right now obviously. If I could, I might be tempted. I like fraction tiles. I might say I would want one inch tiles. But I can't say that, so I won't.
Ann Elise Record 28:16
That's a different manipulative! I love how each of your choices, you mentioned two different manipulatives.
Dr. Nicki Newton 28:27
We'll just keep it moving. I won't say one inch tiles. You can do a lot with one inch tiles if I might might say to myself.
Ann Elise Record 28:35
Oh, I totally agree. That was on my list of things, as well. It is. I love them. They're just different than fraction tiles.
Melissa Milner 28:35
I think you're in the clear for that one. All right, so we're back to Dr. Nicki for pick four and pick three. Pick four. What do you think, Dr. Nicki?
Dr. Nicki Newton 28:56
I think my number four would be snap cubes. Because you can use those for so many things. I think that snap cubes and bears. They've got to be two different manipulatives. But you could do the same thing with them.
So I want to stand up for the bears, as well. It's like when I go into a fourth grade class, I'm going, where's the bears? In a third grade class, where are the bears? Nobody wants to hang out with the bear. But the bears are how you make an array. You can do missing number with the bears. There's so many things that you can do with the bears.
But we're not talking about bears. So I'll go on. With the snap cubes, you can do so much. You can work in kindergarten for counting and cardinality. You can work in kindergarten for conservation. You can work in third grade for a raise. You can work in fourth grade with fractions, fourth or fifth grade depending on what state you're in, a fraction of a set. You can use them for a lot of things. So I would say snap cubes.
Melissa Milner 29:59
Ann Elise.
Ann Elise Record 30:01
Yeah, I love them. Yep, absolutely. And I also love using them with a template of the 10 frame. So 10 frame is not a manipulative, right? It's a model. But it's so powerful when we use the snap cubes with the model of a 10 frame to help them-- again, students move past the counting phase of reasoning and into seeing number relationships and quantities.
Dr. Nicki Newton 30:26
Yeah. And that's why I like the snap cubes, because they really snap together. They're also rectilinear figures. And you know what I mean? In third grade, looking at area of rectilinear figures.
Ann Elise Record 30:39
I also color code at the 5. So if I have 8 snap cubes connected together, I do like to have the 5 be color coded so that we can always go to that benchmark of 5 and 10 to help us know how many we're seeing whenever possible.
Melissa Milner 30:55
So pick three, are you sticking with Rekenrek, Dr. Nicki?
Dr. Nicki Newton 30:59
Yeah. I'm sticking with Rekenrek, because you can do counting. You can do skip counting. In kindergarten, when I'm teaching the kids to count by 10, I like to move over 10 at a time. And that way, you move from a chant to an actual idea that we are counting something in a group of 10. So as they count, I'll move it over on the Rekenrek, 10, 20, 30, etc.
So I like it for skip counting. I like it for problem solving. I like it for decimals, the one that's 100, because then you can use it as decimals. So I really love the Rekenrek in all of its iterations.
Melissa Milner 31:41
Wonderful. Ann Elise?
Ann Elise Record 31:43
Absolutely. Yep, absolutely. They're on my list, too. So then that 100 beaded number line that I know is your favorite manipulative, is that included in with this topic?
Dr. Nicki Newton 31:54
No, absolutely not.
Ann Elise Record 31:55
It's not? Okay, I'm going to think strategically of my last choices here then. So we already, full disclosure, we both already know what our favorite manipulatives are. Because we had discussed this many times and said, if you were to go to a desert island and you'd only bring one manipulative-- you had to teach a wide variety of ages of students, what manipulative would you bring? And I know she'd bring that 100 beaded number line. And I'm bringing mine, my cuisenaire rods.
Melissa Milner 32:26
So you're sticking-- Ann Elise, you're not vetoing Rekenrek?
Ann Elise Record 32:32
No, I love Rekenreks. They're so wonderful for decomposing numbers, as well, having a hiding game where I can tell students to slide over eight beads. And if you imagine two rows of each having 10 beads, color coded at the 5, you can have them slide over 8 beads using both rows if they want to. And they have to guess the way that I did it. And I have them always guess every single time. So they're just constantly decomposing the number I've chosen over and over again.
Melissa Milner 32:58
That's cool.
Ann Elise Record 33:00
Yeah, and it's a fun game. The kids love it. And the kids can be the one that get to decide how many get slid over and the different combinations. So decomposing I love it for. I also love that particular one, the two rows being for related double stacks. So the idea that I can see six beads and seven beads on top and bottom row. And I asked the students, how many are there and how do you know?
Well, I can see whether they're in the counting phase of reasoning or the derive strategy of additive thinking by how they figured out how many there are there. Some kids count each bead one by one. Other kids count on from the six and count on seven more. Others know they can begin with a seven and count on six.
But then still, others are like, well I know 6 and 6 is 12, and it's just one more for 13. So I can see exactly where they are on that trajectory of reasoning with using that. So yeah, I do love Rekenreks.
Melissa Milner 33:46
Cool. All right, so we are down to-- well, let's just go through this before we get to the top two. We have at 10, geoboard. 9, pattern blocks. 8, number bracelets. 7, elapsed time ruler. 6, base 10 blocks. 5, fraction tiles. 4, snap cubes. And 3, Rekenrek. Ann Elise, pick number two?
Ann Elise Record 34:23
Oh, and she saved her veto the whole time. So I have to say this. My choice for number two would be a 100 beaded number line. So you have a shoe string, and you tie one end of it. And then you have the kids put on the beads, 100 beads, color coding it every 10 beads.
So in fact, what I do is I have Dixie cups. And I have the kids put 10 beads in each cup. I have them put the beads in the shape of a 10 frame. Because I'm constantly trying to have the students move beyond counting. So they're going to count one by one for those beads.
I've had it happen, before I thought to organize it, that I would get to the end of the 100 beaded number line, and the kids would have like 11 beads in the middle. And we'd had to take them all off and put them back on. So what I did is I had them put, on the table, the beads in the shape of a 10 frame, so I knew there'd be 10. And then you dump them into Dixie cups and put the Dixie cups in the shape of a 10 frame.
That way, you know you've got your 10 groups of 10. And you didn't have to have one be off or not. And then the kids just string it on the number line. But it's fabulous. It's such an incredible tool for so many things, adding and subtracting.
I mean, Dr. Nicki wrote a book on it. She and Dr. Alison Mello wrote a book on all the different things you could do with a 100 beaded number line. I love it, absolutely, as well. And it would be my pick for number two.
Melissa Milner 35:46
And I'm sure Dr. Nicki has nothing to say about any of this.
Dr. Nicki Newton 35:49
Well, I'm not sure if I would veto that
Melissa Milner 35:57
.Good.
Dr. Nicki Newton 35:59
Well, no, because then I still have a veto.
Ann Elise Record 36:03
Oh my goodness. We're good friends, Nicki. You don't want to do that.
Dr. Nicki Newton 36:09
I still have a veto. And I could pull out something that you've never heard of before.
Ann Elise Record 36:19
That's true. That's true. Well, actually, no, because if you veto me, I get to choose what's in its place.
Melissa Milner 36:26
Correct. You don't get to.
Dr. Nicki Newton 36:27
Oh. I don't get to pick what's in its place?
Ann Elise Record 36:30
No. I'd have to think of something else to put it in its place.
Dr. Nicki Newton 36:35
Oh. Well, I just don't know what I want to do.
Melissa Milner 36:47
You can bargain. You can say, I want to veto this so that you put it at number one. And then Ann Elise can say yes or no.
Dr. Nicki Newton 36:55
Oh, I wouldn't get-- well, actually, if I vetoed beaded number line, she would have to go beaded number lines as number two.
Melissa Milner 37:03
I don't know. Maybe she has something else.
Dr. Nicki Newton 37:05
Well, maybe she does. But that'd be so random. Well, I really do like cuisenaire rods. And I think they're totally underutilized.
Melissa Milner 37:21
If it makes you feel any better, if you come on again, I will let you roll over one veto, each of you since maybe you're not using a veto in this. If you keep everything copacetic, I'll let you roll over a veto into the next episode.
Ann Elise Record 37:39
I see.
Dr. Nicki Newton 37:40
Okay. Then because I think there are different things that you can do with a cuisenaire rod than you can do-- I mean, I love the number line.
Ann Elise Record 37:55
I know how much you love it. So can you share what you love using it for?
Dr. Nicki Newton 37:59
For all the things you said. I think you should have a 20 beaded number line, and a 100 beaded number line, and a 10 beaded number line. And the 10 and the 20 are the same as-- well, they're not really the same as the Rekenrek, the ten is, but the 20, it can be. But it's not usually the Rekenrek. And most of the time, 20 is two rows. Except on Brainingcamp, they made the Rekenrek where you can make it one row of 20.
Ann Elise Record 38:30
Well, it has both ways on Brainingcamp.
Dr. Nicki Newton 38:32
Yeah, I love the beaded number line because you can teach rounding. You can teach equivalencies when you're talking about tenths and hundredths. You can look at composing and decomposing numbers within 10, numbers within 20, numbers within 100.
You can do missing number. You can do multiplication. You can do division, division with remainders. Yeah, I'm gonna veto cuisenaire rods.
Melissa Milner 38:59
Wait a minute, wait a minute, she didn't even play cuisenaire rods! You can't do that. You can't do that. So go ahead. She played 100 beaded number line. And you have to say whether you're vetoing it or not. But you don't know if she's gonna play it again at number one.
Dr. Nicki Newton 39:21
Yeah, I've gotten lost. I've forgotten what we were going. Can there be two queens?
Melissa Milner 39:32
All right, so are we keeping the 100 beaded number line at two?
Dr. Nicki Newton 39:37
I feel like the moderator is pushing the contestant.
Melissa Milner 39:44
I don't care either way. I think they're both amazing.
Dr. Nicki Newton 39:47
You're pushing the contestant into a choice.
Melissa Milner 39:50
It's just getting late.
Dr. Nicki Newton 39:51
Oh, OK. Yeah. Let's see. Okay, okay, okay, okay, okay. I'm gonna have to veto it.
Melissa Milner 40:04
Okay, knowing that it might not be on the list at all?
Dr. Nicki Newton 40:08
That would be a shame.
Melissa Milner 40:11
I'm just letting you know. I don't want you to be upset and say you didn't understand how the game is played. You have no guarantee that she would play it at number one.
Dr. Nicki Newton 40:26
I'm gonna go ahead and veto it.
Ann Elise Record 40:28
Okay, I respect that. I would do the same thing if I were in his shoes.
Melissa Milner 40:33
Veto, you do not like it in second place. It is vetoed. So Ann Elise, what is your new number two?
Ann Elise Record 40:44
So my new number two has to be cuisenaire rods, because it has to be on the list. Actually, wait a second. Well, no. See, if I were to-- she only has one veto. So I could say my number one. But just to let the listeners know that cuisenaire rods are my favorite manipulative.
Everyone knows I'd put it at number one. But I love it, and I also love the 100 beaded number line. So since the 100 beaded number line has been vetoed, I will put that at number one. And my cuisenaire rods can be at number two.
Melissa Milner 41:20
So you're being nice.
Ann Elise Record 41:23
Well, because I can't not have the 100 beaded number line on the list. It's a phenomenal thing.
Dr. Nicki Newton 41:27
I knew she couldn't do it.
Ann Elise Record 41:29
And Nicki did save her one all powerful veto in order for her-- clearly she believes that to the death. And so I have to honor that. She's my mentor. I have learned-- I continue to learn more things from her. So I will respect that. And my cuisenaire rods are number two on the list.
Melissa Milner 41:49
Very, very well played, Dr. Nicki.
Ann Elise Record 41:54
I didn't see that happening. I was like, hey, I got picks one and two. I'm golden. Well played, ma'am. Well played.
Melissa Milner 42:05
Can we talk about cuisenaire rods and just give why it should be way, way up here on the list?
Ann Elise Record 42:10
Yes, absolutely. So George Cuisenaire was a music teacher years and decades ago. So this is not a new tool or anything. And he wanted the students to learn about the intervals between notes. And Caleb Gattegno had seen this and these wooden colorful rods he had made. And he's the one credited for bringing it to the rest of the world.
And they're typically now in buckets in closets in schools collecting dust, because people don't know what to do with them. But the thing that I love about them is they allow students to see the numbers existing as a group and not just a collection of ones. So they can use number relationships to help them.
So the colorful rods, there are 10 of them, and they are in centimeters. So you can use them in the ones place together with your base 10 rods and flats to be able to model, again, decimals and whole numbers as well. But you also can use them for fractions. But you can decide on what you want the whole to be.
So again, you're thinking about, well, how many of these equal size rods are there? And how many of them make up the whole, that kind of thing. But I love using it for everything, starting from the youngest kiddos doing decomposing of numbers. But you can visually see when you build-- there's a story problem with both add ins unknown, part, part, whole, where I might say there are five pieces of fruit on a table. And they could be either apples or oranges. What are all the possibilities?
And I can build that with cuisenaire rods. You can see the underlying pattern that it could be five apples and no oranges, and four apples and one orange, and three apples and two oranges. But you can see the commutative property of addition kicks in. You can see how first, the apples go from 5, 4, 3, 2, down to 0. And then the other add in starts at 0 and goes up to 5. So it allows the kids to see the patterns underlying in the number system. But also, in addition to that, doing let's say a 15 minus an 8.
If I put the eight rod on the right hand side underneath, it can encourage the students to remove that eight as two chunks, remove five, then remove three. But if I slide it to the left hand side, it encourages the additive thinking of going 2 more to 10, 5 more to the 15. So with one tool, I can be modeling all this flexible thinking.
In addition to multiplication and division, so your equal sized groups. You can see the numbers existing as a group. When I do division, if it's quotative division where I know the size of the group, I can put that size of the rod beneath the total amount and see how many can go into it. And even if that wasn't enough, you also can do the problem type structures.
So it's a perfect concrete representation of tape diagrams and part, part, whole. So for all 15 problem types that involve single step addition and subtraction, the cuisenaire rods are a perfect model when you use numbers within 10 for that, to build that foundational sense.
Melissa Milner 45:07
Wow, sold.
Ann Elise Record 45:08
Yeah, that's just some of the reasons why I love them.
Melissa Milner 45:12
Yes. Dr. Nicki, did you want to add anything about cuisenaire rods?
Dr. Nicki Newton 45:16
Well, actually, I've known about them forever. But Ann Elise has really taught me a lot in terms of what you can do with them. And so I am much more of a fan of cuisenaire rods now than I ever had been before. I love them for problem solving. Because you can take the eight rod and when you're introducing division, you can say, Miss Betsy had an eight foot piece of wood and she cut it into bookshelves that were two feet long.
But then when you go into fourth grade, you can say she cut it into bookshelves that were three feet long. How much did she use? How much did she have left over? And it's visual. The kids can see it completely. I also like it for a multiplicative comparison problem, because it's right there. She had two times and he had three times more than or whatever it is. So I really, really like it for problem solving. I think it provides a really good visual.
Ann Elise Record 46:12
And I want to give a shout out to Dr. Mahesh Sharma. I had taken a one week class with him. We actually were in Kennebunkport, Maine, beautiful hotel right there on the water. And we had-- and it was an amazing week. It was really another one of my moments of just going deep into math understanding. One of those is when I had a class with Dr. Nicki.
And then this other time was with Dr. Mahesh Sharma where he had cuisenaire rods. And I had used them in elementary school. I went to Henry Bonnard on the campus of Rhode Island College. And so I had cuisenaire rods. I remember that being in fourth grade and using them. But then I went to public school after that, starting in grade five. And I didn't see them again until his class.
And I had been a teacher already for many years. And it was he that shared with me the power of what these rods can do. And I was just sold from that point on. So I definitely want to give credit to him for showing me the mathematical applications for cuisenaire rods.
Melissa Milner 47:06
Yeah, we were lucky that our district actually had him come and do these intense workshops. Yeah, really very, very amazing mathematician.
Ann Elise Record 47:17
Yes, absolutely.
Melissa Milner 47:19
All right. So let me just repeat the list one more time. At 10, we have geoboards. At 9, pattern blocks. 8, number bracelets. 7, elapsed time ruler. At 6, we have base 10 blocks. At 5, we have fraction tiles. At 4, we have snap cubes. And the top 3, we have Rekenrek at 3, cuisenaire rods at 2. And number one is the 100 beaded number line. Wow! So you want to talk about discards, things that you had on your list that you didn't get to play?
Dr. Nicki Newton 47:57
Well, I do want to talk about one of the manipulatives just very briefly. The balance-- Didax has that virtual balance. But you can buy those, and they're a manipulative. And although it didn't make our list, people should definitely have those. Because for a missing number and for equivalencies of equations, I think they're wonderful.
Melissa Milner 48:21
Great. Any other discards?
Ann Elise Record 48:24
Well, yes. One of the ones I was going to mention that Dr. Nicki mentioned was the number balance, because that is such an important thing for equality. I think very often, our students don't think relationally. They think that an equal sign means I have to calculate something. Because we don't typically put answers on the left hand side of the equal sign, even though we should.
And so if you give them a missing number, such as five plus five equals something plus six, they want to put a ten in that something. Because it comes right after the equal sign. And they're not paying attention to that plus six also being on the right hand side.
So the balance allows the students to hang those weights and be able to see whether it balances or not. So I think that's a really important tool. I would want every classroom to have that for sure.
Melissa Milner 49:06
So all right, anything else before you get to plug what you're doing? I know there's courses, and books, and so on.
Ann Elise Record 49:14
There's just one more manipulative I want to make sure we mention. Now, Nicki has mentioned several, because she had mentioned two at several of the spots. But I do want to reiterate those one inch colored tiles. I love getting them in foam.
But for particularly the area model multiplication, I think that the heart of multiplication and to develop fluency is partial products and be able to decompose a factor and distribute it to the other factor, to get the kids moving past skip counting, which is the additive phase of reasoning. And so I love using the square tiles to build a rectangular model of multiplication, but then use different colors to show how I want to break it apart into a variety of ways and build that flexibility. So I do love them. I had to mention the square tiles for that.
Melissa Milner 49:59
That's great. Ann Elise, I know you have plenty of-- I just got your email. So you're doing lots of amazing things this summer. And I know that, Dr. Nicki, of course, probably has books coming out and all that kind of thing.
Dr. Nicki Newton 50:17
Oh, always.
Melissa Milner 50:18
Yes. So what's coming up that you can promote?
Ann Elise Record 50:22
Well, I do have my on demand workshops on my web page, anneliserecord.com. And I currently have - surprise, surprise - a class on cuisenaire rods K-2 and another class on cuisenaire rods 3-5 where I show how we can explore fluency, problem solving, and the core standards with cuisenaire rods for all those variety of grade levels. And I have one on place value K-2 and another one on fractions, fraction essentials.
My goal this summer is to create three more. I want to do one on creating parent math nights for schools and districts, which would include a video of me you could show at the parent night to parents so they can understand why we're teaching math the way that we are. And also problem solving K-2 and one on 3-5. So that's my goal this summer, to get those up and online.
I'll be appearing in the Build Math Minds Virtual Summit. So Christina Tondevold, this is her fifth annual Build Math Minds Summit. It will be for free. It's July 29 and 30th. And all the sessions will be available. And Dr. Nicki has a session as well. That will be available until August 8th. And then at the end of August 8th, they all go into her for pay Build Math Minds membership site.
But it is open enrollment in her membership site, which I think is a phenomenal investment as well. Because you have all the previous summers of Build Math Minds sessions, plus lots of videos, hundreds of them, that she and I and Rosalba Serrano have been making for years on elementary math.
Melissa Milner 51:45
Yes, I highly recommend it. It is worth every penny. Fantastic. And your website is?
Ann Elise Record 51:52
anneliserecord.com
Melissa Milner 51:54
Perfect. And Dr. Nicki, what are you up to?
Dr. Nicki Newton 51:58
I've been writing. I have a new series of guided math in action coming out by grade level and also the second edition of the book. And then I am doing a second edition of Math Running Records. And I think-- oh, well, the daily math thinking routines, that's coming out. And I'm gonna be a part of that institute this summer. I'll be talking about dyscalculia.
Ann Elise Record 52:34
The Virtual Math Summit.
Dr. Nicki Newton 52:36
And that's kind of it.
Ann Elise Record 52:38
You also have your academy of 20 classes.
Dr. Nicki Newton 52:40
Yeah, I do. I have a math academy. Just look up Dr. Nicki's Math Academy, and there's 22 courses right now, from math intervention, to guided math, to running records and different courses.
Melissa Milner 52:55
Amazing. You guys are incredible. And I know Episode 30 was very popular. And I know that people listening to this are just gonna want to buy up these manipulatives now. It's really so inspiring to listen to both of you talk about math and helping kids learn these concepts.
So I just want to thank you so much, Ann Elise and Dr. Nicki, for taking the time out on a random Wednesday evening to do this crazy Top 10 list.
Dr. Nicki Newton 53:31
We are honored that you have invited us.
Ann Elise Record 53:34
Absolutely. Thank you so much for having us. I just have a ball every time.
Melissa Milner 53:38
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I am sending a special thanks to Linda and Lester Fleishman, my mom and dad, for being so supportive. They are the voices you hear in the Zooming In soundbite. And my dad composed and performed the background music you are listening to right now. My intro music was "Upbeat Party" by Scott Holmes.
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