Ep 19: Figurative Counters: Stages of Counting Series Part 3

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Moving right along with our counting constructs, today's is figurative

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counting and this one can be a little bit tricky and is often called the

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forgotten construct. So I'm going to tell you all about it in today's episode.

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Hello and welcome to Making Number Sense Make Sense, a podcast for elementary

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teachers, specifically early elementary teachers, looking to really make an

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impact in the number sense of their students. We had talked about emergent

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counters, perceptual counters, and today's focus is figurative counters. Now this

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one can be tricky because it is very similar to perceptual counters, which is

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why it's usually called the forgotten construct because it's kind of in the

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middle and it's kind of tricky to get kids through this construct to the next

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one in a way that will help them in the future because we don't want to miss

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anything. So kids in this stage of counting will be able to count our

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create or keep in their head a collection of items, a relatively small

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collection of items, that is hidden that they can't actually see but they're

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gonna have to recreate it. So if I said there are seven counters here and then I

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covered them, they wouldn't necessarily be able to use that seven and count on.

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They would have to recreate the seven. So if I said there's seven and then three

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more over here, they'd have to start counting from one, one, two, three, four, five,

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six, seven, and then add on whatever number that you said. So it can seem redundant

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to grown-ups because counting on would be a more efficient strategy but they're

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not quite there yet and that's really what the focus is for a figurative

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counter, having them be able to keep numbers in their head and then use other

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numbers in relation to that number. My previous example was just one but they

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can even do that with two collections. So if I said there's seven here, covered it,

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and there's three here, covered it, and they're both covered, they can get, okay,

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seven and three, but they wouldn't start counting from seven or even from three.

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They wouldn't go seven, eight, nine, ten. They would start from one, one, two, three, four,

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five, six, seven, eight, nine, ten, and count that way. So what are you looking for in

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this kind of counter? You're looking for them to be able to count beyond ten to

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thirty or more. They will have one-to-one correspondence down. They can usually

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correct their errors or recognize their errors and self-correct if asked. They

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understand cardinality, that the last number you said is the number that they

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have. They can keep track of, like I said, screened or covered collections of

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counters even if they're in different arrangements. Their number writing, they

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have to at least 20. They're able to write bigger numbers in the number

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system or starting to get there. They can tell you the number after with a

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running start, but they might have to count up to get there. They might not be

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able to just tell you yet. They can count backwards from ten or from 20, but like I

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said, they're having trouble keeping that in their head, like manipulating that

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number in their head. They'll have to recreate it and make it real with their

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fingers or taps or whichever. In this stage of counting, how can you help them?

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You're gonna want to keep working on that number sequence, especially the tricky

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parts. You're gonna encourage centers or you want to start to use centers with

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them or small group instruction that really has them counting on and pushes

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them towards counting from one to counting on. They'll also start to

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recognize and understand common number relationships. So this would be

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decomposing numbers like decomposing ten, the friends of ten, but also other

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numbers too. Like know that seven is five and two, it's also four and three, but with

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every number so they can really start to see those relationships and really

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understand it. Because if they understand those tens, then they can apply it when

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the numbers are bigger. So if they knew that nine and one is ten and they have an

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understanding of that 90 is nine tens, that the numbers will still work the

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same way. So nine and one is ten, so 90 plus ten is 100. So they'll be able to

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make those connections and if they have a really good grasp on the numbers to ten,

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they'll be able to apply it to bigger numbers. So that's something else that

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you're really working on in this construct. Not to say that you wouldn't

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work on decomposing numbers in perceptual counters in the previous

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construct, they kind of work together because they definitely overlap and

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you'll see kids kind of showing different strategies at this point and

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really kind of bouncing back and forth. Another thing that will help kids in

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this construct is number talks with number strings to support counting on

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strategies. You might start with what's one plus two? Three. Then you might start

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with what is ten plus three? Then they might start to see ten and three is 13.

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If they know how the tens work, start to understand how the number system works,

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they'll be like, oh that's easy, ten and three is 13. And then if your next

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question is what is nine plus four, they might start to see that nine and ten are

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really close, similar to the problem that you just had. And some students might

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start to understand that you can take one of those numbers from the four away,

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add it to the nine to make ten, and ten and three is 13 that the answer is the

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same. So you're really wanting to encourage kids talking about numbers, how

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they relate to each other, and refining their strategies for efficient mental

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math and using numbers in that way in their head. So I talked about some things

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that you can work on, some things to keep in mind when you're working on centers,

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but I'm going to give you some more specific centers that you can do. One

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that's very popular for decomposing numbers is called shake and spill or

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toss the chips. So the way that that game works is that students have a certain

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number that they're working on the target number and the same number of

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double-sided counters. So if I'm working with the number six, I'd have six

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double-sided counters. And what they'll do is they'll shake it and then drop it

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gently, shake it gently, and then drop it and figure out how many are of one color,

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how many are of the other color, and write those numbers down. And since

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they're doing it over and over, they'll start to recognize the similar patterns.

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That three and three is six, that two and four is six, that one and five is six. If

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they continue working on that number and then once they're good with that number,

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they can move on to the next one. Another game they can play that will really

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force them to keep numbers in their head is called hide and count. And this is a

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fun partner game. And the way that they play this game is they have their two

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dice and they'll, one partner will roll the dice and then give the student about

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three seconds to see what one of the dice is and then hide it with their hand. So

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hide it underneath their hands. The second player will have to then figure

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out, okay if I have how many are under the my partner's hand and how many are

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on this other dice, how many is it all together? So they'll have to really start

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counting on from the number that's hidden and add it to the other dice. So

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if I rolled two dice that were five and three and I covered the three, the kids

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would have to think about, okay there's three under there and then count on four,

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five, six, seven. And they can keep going with that game. I do find that sometimes

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with a game like that students will be like, okay I'm kind of done with it. So

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you can add a really simple game board. I'll have examples in the notes that you

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can see or a recording sheet so they can write down how many they rolled and then

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their partner gets a turn to hide the die and then go back and forth until one

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of them finishes their game board or finishes their recording sheet. And

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throughout all of your learning it's really helpful to include tasks that or

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rich thinking tasks that students can work on together to have more

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opportunities to discuss and figure out problems. So for example for a story

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problem, story problems are a great low floor high ceiling task especially for

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decomposing numbers because there are so many different answers and you can

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really modify the range to fit the needs of your students. So one story problem

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that you might give is that there's two soccer teams, the Red Foxes and the Green

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Frogs and they played in the championship and all together both teams

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scored eight goals. So then you have the question, well how many could have been

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scored by the Red Foxes? How many could have been scored by the Green Frogs? And

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that way they can think of all the number combinations and if you've been

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doing math centers that have a lot to do with decomposing numbers like shake and

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spell, like things like how many more, snap it and hide it, and they have

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opportunities to practice all of these different ways to decompose numbers and

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think about numbers, they'll be able to pull from that knowledge and those

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strategies to solve this problem. As a recap, students in this stage of counting

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figurative counters can count collections up to 30 or more, their

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number writing is coming along, their number sequence is coming along, but

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they're having trouble counting on and keeping numbers in their head in a way

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that they can easily manipulate to solve problems. For figurative counters, you're

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gonna want to work on decomposing numbers and number combinations. You're

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gonna want to keep practicing on number writing and it could be, it doesn't have

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to be worksheets, it could be you're doing this math station and part of it

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is recording the numbers that you're getting so that they still get that

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number writing practice but it's not just sitting there and copying numbers.

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And I say that because number writing can be something that is so tricky for

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the kids and it's really just need practice. Like for all of these counting

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stages, to help students move on to the next stage, yes you need you know centers

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and it's helpful if they're fun and engaging so they'll want to practice but

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they really need time and opportunity to create these understandings of what each

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number means and how each number relates to each other. So back to decomposing

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numbers, again I'll have all of these listed and you can check the blog post

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to see pictures of all of these games as well as games that will force them or

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encourage, I will say, encourage students to count on instead of counting from one.

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And if they do these games enough they'll start to see that counting from

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one is not an efficient strategy and kind of refine it on their own and that's

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why it's so helpful also to have kids talking to each other and talking with

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you because hearing different how different kids solve different problems

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can really help them grow their own understanding. Math talks, number talks,

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and opportunities to really discuss their thinking in a problem-solving

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situation in a low floor high ceiling rich math test situation will be very

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helpful for students in this stage. Get ready to move to the next one. As always

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make sure to follow the show. I would love it if you left a review and I have

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a Google form in the notes for anything you'd like me to talk about further or

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any topics that you would like me to discuss in general. Until then you'll

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hear me next time.