Why Counting Is the Hidden Foundation of Math

Segment 10
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[00:00:00] Hey everybody. It is Cheri Dotterer here at Tier One Interventions podcast. It's great to have you here with us while we talk about the core of the math classroom. Today we're gonna be talking more about. Pizzas, rectangles, lockers. Oh my.

Cedric:  Back on March 10th, 2026, we started a 10 segment series Here is segment 10 from our October Workshop. If you are enjoying what you are learning, please subscribe and leave a comment. Look for another series coming up next session.

I think that was the right order. Over to Jonily

there's always a third way we think as educators that it's either this way or that way, and that those are all my options. There's always a third way. In the mastery math method. The third way is to support students through reference tasks. So when my kids struggle, even my high school kids don't know how to get the factors of [00:01:00] 24 or don't know what factors mean, I can now say what dimensions?

I'm not gonna type this 'cause we're outta time, but what dimensions of rectangles? Give me an area of 24. Or I can say which students touch locker 24. Or I can say, where is 24 in the dimension chart? And because my high schoolers have experienced these reference tasks, hopefully almost every year, by the time they get to me and I've launched my year with all of the reference tasks, I can now use reference task dependent questions for intervention.

And that my friends, was the goal of mind today for you as the live group to see this in-depth, deliberate, [00:02:00] intentional process of the power of reference tasks. Now, I didn't make the explicit connection back to counting, but if you go back and listen to this again, or you go back and look at all the samples I gave.

It's the underlying groove is getting kids to be better counters, counting the blocks in the rectangle, counting the length of the side, counting how many dimensions, counting which students touch, locker 24 in the locker problem. So counting to add, counting to multiply, counting to get area, counting to get.

So we are focusing on counting at its bare minimum through reference tasks to then use as extension and scaffolding to intervene and accelerate and make all of these connections based on the essential math content.[00:03:00]

That's it. Natalie, we're one minute over. I went one minute over. Okay. If you guys need to jump, I love you, Sarah. Go ahead. Okay. So I know I do this a lot, but I, as I was watching you with the dimension chart, it like, popped into my head, and this is, I think the point of reference task is that how they can connect to other things, that we may not see right away. When you do like three. By eight, they're three tall and eight long. I was thinking about the fact that like we can also have the eight tall and three long and the idea of commutative property, but that it changes how the rectangles oriented. And then that made me think about rotation.

And in eighth grade, our kids struggle so much with rotation in the coordinate plane. And now I'm like, I'm thinking it's really that commutative [00:04:00] property of the numbers. And then the fact that we're like, we're rotating from one quadrant to the other just means we also have to pay attention to sign.

But I'm like, now I'm just trying to like, think about connections between, the making rectangles and having some conversations about what they see and notice as we, change the or interchange tall and wide. And then how that kind of translates to what we're trying to do in those rotations.

I wanna cuss so bad right now. 'cause when I get excited, all that trucker top comes right out. I'm not gonna, I'm not gonna, I see I put 20 5:00 AM why am I doing this? I'm, it's 'cause my brain is going, Sarah. No, that's perfect. That's perfect. That's the whole point. And I'm gonna make the point. I don't know if I did this the right way.

I probably didn't because the eight, no, I did it the way that I saw the rectangle. So [00:05:00] as far as like height and length and how that multiplication goes, I don't know. And I don't care. Okay. The way that I do it is when I draw the rectangle, that's eight long, I go across the top for eight long. Because for me that transfers.

I don't know what the multiplication prob, I don't know if this is the eight times three or the three times eight. I don't know. And honestly, I don't care. I probably should care, but I don't. You can school me on that. Somebody can, if you're listening to this, you can go in the comments of the podcast and be like, you should care.

And this is how it is and this is how I remember it. Please school me on it. Okay? But I don't. But the point is if you look at, and Sarah, you make this point, I didn't make the point, but I should have made the point. When we have the eight by three, we go eight across and three down. But the flip of that is it's the same factors.

But now I go three across and eight down. So for 11, remember I said about 11, we have to do the same thing with 11. And that's, so I count these as two different [00:06:00] rectangles when I talk making rectangles, because with 11 we wanna say that there's only one rectangle. There's not, there's two rectangles because how many people touch locker 11.

Two. The same number of people that touch the locker. This is important. The same number of people that touch the locker is exactly the same number of rectangles that can be created. Withhold number, side lengths. Lemme say it again. The same number of people that touch the locker are exactly the same number of rectangles that can be made for that locker number with whole number side lengths.

So for 11 two people touch locker 11. There must be two rectangles. There's an 11 by one and a one by 11. The point that you make, Sarah, is a point that must be made and is absolutely essential to make these connections as well. But then you extended that point, which is beautiful. I can't say it any better.

It's all about transformations and I'll just let it be with what you just said.

I see a big extension in counting. I was with one of your groups yesterday, you had a [00:07:00] diamond problem with four x and five and they were struggling with the multiplying and just wanting to put 20. And I was talking to 'em four x and five was, what's that really saying? You ha, how many four Xs do you have?

And we were counting, Y have one, four x and another four x and another four x, and they're like, oh, it's gotta be 20 x. So that just clicked for me, like they're counting by groups of four x. How many groups do you have? So yeah. Anyway,

I should have done that yesterday, Natalie. I should have done that. I was trying to, I was trying to help them out with this problem and I was like, we got to like the adding part at the bottom and why like you, you're just adding, and we pulled out algebra trials and yeah, green and yellow.

They're not like terms, I can't combine those. And then I was trying to help them with the multiplying and I was like, so I had to how would I explain this to a sixth grader? 'cause I couldn't, I wasn't getting there with them. And I was like, okay, you have four XI was [00:08:00] like, five groups of four x.

That's just counting. So it was me not being able to like communicate at an algebra two level, but it's the same. And they're like, holy shoot. Yeah, that makes sense. I just, how many groups do I have? But Natalie Sugar Muffin, you saw what I did with first period. Yes, I did. Yes. I was trying to do commun, which, which wasn't bad, but, oh my gosh, Joly.

Because with first period they were struggling to understand that you can combine this. So the way that I taught it, the way that I told it was that multiplication is commutative. So we only have one term and it's all multiplication. And they did know that there was multiplication. So then I said because multiplication is commutative.

And then we, I did a little mini lesson on commutative. I said, we can change the order and do four times, five times x. We know four times five. I did not, I'm gonna go back and I'm gonna do, that was important too, but Sugar [00:09:00] Muffin Natalie, but I was trying to remember what you did. I'm like, oh, I know.

She said something really good, but it wasn't coming back to me. And I was like, but wait a second. We can just think about it like this. So that is beautiful. Yay. But I was like, yay, I haven't touched Algebra two in a long time. And to be able to apply that, like I was feeling pretty good about myself.

So it was fun. It was so good. It was so good. So fun day. Okay. Anything else for the good of the order?

Where do I find this note doc

on my computer? I didn't know. I know, I was like, did you share that with us? And I just, Nope. I have the Mastermind one. So I went in, was like, oh, does that hook in with Mastermind? It doesn't, it, it should, lemme just tell you about my life right now. No I can't. 'cause it would take another [00:10:00] hour.

I'm with you. So there are so many productive things that are happening right now that I'm missing all of the detail stuff. Like I have no detail. Resurrection at all. So this is bad and it's terrible. So if you guys don't have something, I'm writing this one down right now. The notes, doc, if you guys don't have something, email me and Cheri both on the same email because I have bailed on Cheri for months, okay?

Because I do not have a sliver of time in my day. Like I literally am working

18 of the 24 hours in every day right now. Okay? That's not an excuse. I'm not saying it's an excuse. And then I have some personal things that are flaring big time. The reason I say that is I have, there are a lot of gaps in my details, a [00:11:00] ton. So for you guys to help me if you can, today, tomorrow, in the next week, if you can send an email to Cheri and I both with just bullet points of things, I'll know, it'll trigger me just email.

Don't tell me right now. Just email and say notes, document from tier two, tier one, level two with the locker problem. Gimme a little description. I've got to get these details solidified. There's so many other great things happen happening, but at the expense of all of these minute details. So Amy, you don't have this and I'm a disaster.

But I don't apologize for it because there has been so much progress in my professional life this year. That would've never happened if I would've attended to details. And then I got some personal stuff that's like just, oh, I get it. You can't do everything. So that is your answer to that. So if you guys could do that would be [00:12:00] absolutely amazing.

And then I think what we should do, if you guys have just a few more minutes, I think everybody needs to give a good testimonial takeaway from today with camera on Please, Kirk. Okay. So if everybody can give a good testimonial takeaway from today, I think that will wrap us. Now. If you have to go, please don't let us stop you.

If you've got somewhere to be, just jump off. Don't feel like you're being rude at all. But if you've got a good aha or addition to something, or testimonial or whatever it is, we'd like to hear that. Before we wrap up today, if you could please Kirk. I was just saying, I'm on my way out the door right now.

Ah, can you tell I'm going to have lunch. You go with Vanessa. What? I'm going to have lunch with Vanessa. I gotta go. No, all, bye. All right, bye. [00:13:00] On Tuesday, October 21st. That's two, three days from now. I'm doing a masterclass on writing skills. And so the registration is in school, so please come and join me on that.

It is brand new. What time? So while Jolee has been off doing her professional development, I've been doing my own and I have a brand new program and like a big realization about writing and learning. And I am going to be revealing that on Tuesday. And it's free.

And at what time is that though? Cheri? 7:00 PM Okay. I got parent-teacher conferences that night sign up anyway, or send me some kind of thing. The the way it's gonna work in school is if you like enough things and you get to level two, you'll get access to [00:14:00] it. So just go in and everything from today so that you'll be able to get access to the replay when I get it put up.

Okay. So I'm done. Who I, who wants to share, take away Aha. Or extension for today, or any connection you've.

I'll go. I think that this today's session was especially beneficial to show how everything is connected. And for years, teachers have looked at it the wrong way. They've looked at it as a backwards model like the high school teachers have, if the students aren't where they're supposed to be, rather than figure things out the high school teachers tend to blame the middle school teachers and the middle school teachers blame the elementary teachers and the elementary blame the preschool teachers.

And what today's session did was totally upend that and say. How [00:15:00] important those fundamental skills are for all levels. And we went through so many how counting is fundamental in math achievement and how there's all the different kinds of counting. And I think that elementary and middle school and high school teachers don't see the value in counting.

And what we talked about today was, we count without meaning. Wrote counting, without meaning, wrote, counting with meaning and how we can, we've talked before about how we can count with fractions and count with decimals for place value and everything else. And then today the epiphany was counting with Xs and how important that is to lead into the algebra string.

So everything in math, we know, we, we typically have added before we've subtracted and things like that. So it is cumulative, but it all starts from counting. And that was such an epiphany for me today yay. And I think the challenge for me, for [00:16:00] you all in level two this year is I know that you all are deep in mastery math method.

You are deep in every program that Cheri and I offer. However I know, and I knew that you were all still missing so many pieces. And that is what I, that is what I think about to plan tier one, interventions level two. So just know I'm doing all of this for you because I know what pieces you're missing and I have not effectively connected those pieces.

And that is my goal for you, that are participating in tier one interventions level two this year. Beautiful. Kathy other takeaway? Who's next? Natalie? Yeah. My, for our specific session, I say if we want that long-term achievement, we have to let them struggle. They have to be able to work [00:17:00] through it and live through it and stop stealing that from them.

But personally, like I just had, this whole big boom in my head. Our district has brought in Elevate Lori, God, her last name just got away from me, but her, they're all about mindfulness and wellness for te not just teachers, but they have a big education presence. And oh my God, my brain is full of so many.

I know. You and me both, I can't even articulate anymore. But when they first came to our school, like this would be two school years now, and one of their things is not even that you can't pour from an empty cup, but they want your cup to be so overflowing that you're pouring from the saucer.

Okay. But when they first started they were like saying these little things and I was like, that's cool, that's nice. I like that. I can do that. But what's the big thing? What's the, the big piece. And then the next time we saw 'em, it was another little piece. And I was like, that's cool. I like [00:18:00] that.

I, I'm on board. I like, but maybe we'll get to the meat and potatoes in the next one. And then next time is another little piece. I was like cool. Where's, when do we get to the big thing? And it wasn't a big thing. It's their thing is micro changes over time. You make a little change over time and that keeps going.

And I sometimes even say nano changes over time, personally. And I'm like, it isn't a big thing. It wasn't one big piece. It literally is micro pieces that you make over time. And I just made that connection to our exposures over time. And it's not, maybe I, and then I said, I haven't given myself enough credit.

I think I've been waiting for what's the big thing. And I just realized with today, I've seen locker problem, I've seen rectangles, I've seen pizzas. Oh my. And I haven't given myself enough credit. I was like, everything you just said, I'm like, I have made all those connections. I know this. And I haven't given myself enough credit to [00:19:00] say I've got I know the big thing is the little things over time.

So I, I am like just, I'm swimming in the struggle right now. But it is, I Wow. I don't think I've ever felt like one of, I'm like, how are we done? We just got started. There's so much going on in my brain right now. I am blown away. So there's that. All I can say is welcome. Not that I don't still want all of the pieces, but I gotta give myself more credit.

Natalie, you should give yourself a ton of credit. And Natalie, that statement, all of those statements were an epiphany for me. You said it in a better way that I think any of us ever could. What you said was so profound. I think that maybe will probably be a soundbite that Cheri will grab and go with some of the short clips because I wrote it down as you were [00:20:00] talking.

It was absolutely beautiful. I need to swim in it a lot because I need to. I'm, see, I can't even articulate because my brain is racing so much. Based on what you just said. Yeah. Blown away. I don't even know that I articulated everything the best. 'cause my brain is swimming so much. But I way overthink things 'cause I did that with them.

I did that my first couple years is like that all or nothing? If it's not all, then I just way overthink things. But that's a me thing. No it's not. A lot of us are like that. Yeah. We gotta stop. Do it anyway. Do the micro change. Do the nano change. Whatever's smaller than that, do that. 'cause that's better than nothing.

If you're doing nothing, that's, that is one thing my admin says. Doing nothing is not okay. But if you're doing something, little changes, go do something. There it is. Go do something. Perfection is the barrier of progress. Perfection [00:21:00] prevents progress. Sarah? I wanted to say, my takeaway is definitely, like Natalie said, the the importance of productive struggle and students persevering and like us stepping back.

But setting up a structure, that allows that. But then what we've all said about the importance of the repetitions over time and just what we're learning as adults with continued exposure to things and how that parallels with students. And then also what, Natalie was just saying about herself, like getting students to realize that too.

Especially when they are like behind in math. And that as they get those small gains, like each one of them is really important to celebrate for themselves. And then my last piece is just I've definitely gotten a lot out of, I appreciate your reflection on what has been missing from some of the conversations.

And I do feel like these have been in a sense maybe more linear than usual. [00:22:00] But like I, I've appreciated just the structure of these conversations. 'cause it's definitely boosted my growth and understanding

to respond to that real quick, since nobody jumped in, like really quick to make a point on what Sarah said, the linearness of the level two could only have happened with the non-linear ness of level one.

So think about this, think about Beth. Beth is, Beth has been participating in level two. She doesn't have any of those non-linear chunks to be able to make the connections. Do you see once she gets more of those non-linear chicken and egg chunks, she'll go back and listen to level two [00:23:00] and be like, but see, this is my point.

You can't have the linear first. Okay. All right. Other takeaways, testimonials. I always just enjoy that I made a lot of connections. I feel like today being a first grade math teacher, having it brought down to the number 24 also, then I feel like I can make better connections with numbers that I'm familiar with versus the fractions and the exponents and things.

But you saying like the last time we talked about the X problem, it was product sum. And so I like that this time you talked about area num. Either it being the array number or area at the top for vocab to use with the kids.

Beautiful. Fantastic. And those left and right numbers, those side numbers, [00:24:00] those are the dimensions. Those are the dimension. They always are because that is the phenomenon of mathematical multiplication. When I get a product, it's the result of two dimensions of a rectangle.

Jonily, I wanna close out Tier One Interventions podcast, and that is, if you are listening to this podcast and you are thinking, how do I get involved, you wanna head over to your show notes and click on the link. We are offering you to come to a session like this where you get all two and a half hours at one shot for $47.

You can then join another session for another $47. Or you can buy the whole year for $497 plus you have to buy the. [00:25:00] When if you're going to go buy the whole year, you're gonna have to buy level one as well. But the coaching and these conversations that we're having are $497. That sounds to me like a really good deal for to, to really think the way you're thinking about not just mathematics, but occupational therapist delivery, speech therapy, delivery, special ed delivery, and how we're helping these kids rethink, reregulate, relearn and think about life as whole.