>> Good morning.
>> [indistinct chatter]
>> Good morning.
>> Thank you. Okay. We're going to get started. Okay. Hi, my name is Kim Jenkins. I'm a PaTTAN consultant in the King of Prussia Office. And we have just a few housekeeping things before we get
>> Good morning.
started. We're piloting two new things this conference. The first one is the online application. I hope all of you either have a flier or have had the chance to go online and download the mobile app.
It's very exciting. It's something new this time around. And you can access a lot of really good information. You can--your schedule, the speakers. You can set your own schedule. There's a lot of
exciting new things. We're hoping--and this is the first time we're trying this out and hopefully for future conferences, we'll be able to add and build up on that. So, take a look at that. Something
that's also new this year, reporting for your professional credit, Act 48, LSLS, ASHA, all the different professional credits. You will not be handing in a paper form to us. In your folder, you have
a schedule and there's a place for you to write the session codes that will be provided at the end of each session. I want to make sure you write them down but you're not--you're going to keep that
form, you're not handing it in to us. You will be using those codes going online either through the mobile app or on the front page, the direction, there's a link that will take you to the PaTTAN
page. And from there, you'll be able to load--download each daily link. At that link, if you're applying for Act 48 questions, you'll answer Act 48 questions for each session you attended. That's
something new this year with the online apps, I mean, with the online verification portion. So, you'll answer five content questions based on each session. You will also be able to put your session
codes in online. If you are not seeking Act 48 questions, you will be directed to the program, to the section just to enter your session codes. But you must do this in order to get credit. We will
not be accepting any other paper format for your professional hours. So, any questions related to that? Brief questions? Quick questions? Okay. This session runs--yes?
>> How long do we have to answer those questions after the conference?
>> You can do it daily at the end of each day. The link will open for--that day, or if you procrastinate and you want to do it at the end, just remember, if you attended Monday and you're looking
Friday, you'll have to remember the content for Act 48 questions. But it will be open until August 15th. They'll close off business on August 15th, you have to get online. And do you have a question?
>> [indistinct chatter]
>> Oh, okay. Okay. If you haven't signed in at registration, please make sure you do that. When you sign in, you'll be given your nametag, and this is your ticket to get your lunch today. It is a
buffet lunch with the keynote speaker and there--you do not have a lunch ticket for today, each of the other mornings you were given--you would've been given a ticket for your lunch, for the box
lunch or the bag lunch. Today, you do need to wear your nametag to get in to lunch. And this session will end at 11:30. Make sure that if you haven't registered before lunch, you'll need to do that.
So if you get a break, you might want to go up and check the registration.
>> [indistinct chatter]
>> For the--yes, and Project MAX, if you haven't signed in, make sure you sign in as well. Okay. Please make sure your cell phones are on silent, vibrate. And at this time, I'd like to introduce
Sharon Leonard. She'll be getting us started.
>> Good morning.
>> Let's try that again. Good morning.
>> Good morning.
>> Good morning.
>> Awesome. There we go. Day two, huh? This is exciting. Each year at the conferences, it is very exciting, and the team that I've been working with--we're actually doing all the conferences. We're on
the tour this summer. We're selling t-shirts in the parking lot outside the back of the car. No, only kidding. But, uh, it's been really exciting. I want to situate this day. This day is really
different than all the other conferences. So, you're really in for a treat and an opportunity to gain knowledge that we're very, very excited about. This session is called Accessing PA Core Standards
for Students with Significant--with really Significant Cognitive Disabilities. And we know that with Pennsylvania adopting PA Core back in March, that this makes a difference for all students. And
we're talking about each and every student in Pennsylvania at school age between the ages of five and twenty-one who are in school, have access to PA Core Standards. So, what this is meaning for
students with significant cognitive disabilities is really a change in the content where the expectations, the expectations of what's assessed for Pennsylvania, and the expectations of what we teach
in the classrooms. So today, what we're going to do for you is, this morning, is we're really going to land really hard on math. And we're going to be looking at math and math instructions for
student with the most significant cognitive disabilities, and we have with us Allen Muir and Jared Campbell, PaTTAN consultants, math experts extraordinaire. I have to tell you, you're going to
be--they are going to keep you engaged, they're going to keep you moving and learning, and thinking about what math could look like and what students need to know and be able to do who have
significant cognitive disabilities. Then this afternoon, we're going to come back. We will have John Machella from the Bureau of Special Education with us. And he's going to walk you through what
some of those changes are going to look like from the Special Education Bureau and state level for students with significant cognitive disabilities. And then we're going to spend time looking at and
examining what is that alternate eligible content going to be. What we used to call PA Alternate Standards are going away. And there's going to be something new called Alternate Eligible Content. And
very excitedly, you, by being here will have a chance to give input on that as well. So, that's what we're going to do this afternoon. This morning, we're going to really immerse ourselves in what
it's going to look like in the classroom especially when we look at math. This afternoon, we're going to look at Alternate Eligible Content, Math, and English Language Arts. So, put your seatbelts
on, this is going to be an exciting day. We're really excited to be here. So I'm going to turn it over to Allen and Jared.
>> Thank you, Sharon. My name is Allen Muir, and as Sharon said, I'm the consultant at PaTTAN Harrisburg. We also have Jared Campbell up here, and Sharon, Jared, and I will all be, kind of, you know,
leading this and interjecting from time to time. Before we get started, let's just get an idea of our audience. Show of hands, how many are parents? Okay. How many teachers do we have in the
audience? Okay. How many administrators do we have in the audience? Okay. How many IU consultants, or PaTTAN consultants, or consultants otherwise, do we have in the audience? Okay. Good. So, we have
a pretty good mix here. What we want to do is provide you an opportunity to experience CRA, okay? It's an instructional strategy, Concrete Representational Abstract. It's one instructional strategy
that can be used to help students access the PA Core Standards. So, we're going to model it, and then we're going to let you as a student, participate in some activities. So, like Sharon said, this
will be a time where you're going to be engaged in some things. And we want you to keep an open mind, okay? And just think about this from a student perspective and then think about how you'll be
able to, you know, turn that around in the classroom. Before we get started, let's talk about where we can find some more information about CRA or about anything, you know, around math. We have an
iTunes course, okay? Jared, I'm going to let you talk about that iTunes course real quick.
>> Am I on? Can you hear me? Hi there, way in the back. So, the iTunes course right now is locked. And you're going to get the code in a minute. For us to get the ball rolling, we've had to get the
course started immediately. So, we do not have an institutional account with Apple yet, but we are working on getting that institutional account with Apple. When we do that, we'll be able to make
this court--course searchable, so you'll be able to find it in the iTunes U store or just like you would any other course whether it's from hack, or Stanford, or whatever. It's self pay, so while
it's called a course, it's not really a course. It's continued professional learning for you. So, there may be parts of that you never look at, there may be parts of that that you revisit, you know,
a week before you teach that content. It's a resource for teachers to help remind you of what CRA instruction looks like in different areas so that you can stay connected to the content even after
the sessions. So, just as a reminder, right now, they are only available on iOS devices because it's a private course, okay? But you'll get enroll code, and if you have an iOS device, you'll be able
to get on to that. As soon as PaTTAN is able to secure an institutional account, then it will be searchable and open to the public and you won't even need an enroll code anymore. So, you'll see the
enroll code in a minute. It's in the PowerPoint. If you have a copy of it, if you get connected to us on the Wiggio site, we'll also make sure that we post that there and these materials will all be
posted there for you to have access to as well. If you don't have an iOS device, whether it's Mac or PC, you cannot right now, access this course from--directly from iTunes on a computer. You have to
be on an iOS device. So, if you're waiting to get it from a computer, make a note for yourself, we will definitely make sure through Wiggio that we put out announcements that let people know when
this course is accessible on all platforms.
>> Where would we establish our position? [inaudible]
>> You'll be able to get a paper copy of the PowerPoint on Wiggio, and I'm not sure--no?
>> We did not print copies to bring today.
>> But I mean, you can get a copy of the--of the PowerPoint on Wiggio. So, no, we don't have any paper copies that we're handing out, okay?
>> The math Wiggio or the...
>> Well, let me continue through, and I'll show you a little bit specifically what that is. I just wanted Jared to point this out about iTunes. Other sources, like I said that we have, are Wiggio, our
PaTTAN Math website, and then we have a QRT code here. Wiggio is where we have information, okay, about--what is this, CRA and about other math initiative things that we're leading. And if you want
to join us, what you would do is you could click down here, not a member of PaTTAN math yet. So, you can go to Wiggio, you would put in your email, you would put in your password. Now, it's going to
ask you--and this is the site address. It's going to ask you for a password, okay? In order to join this. So, go to this site, okay? And Jared, are the--is the PowerPoint there now for this?
>> It will be there before the close of business today.
>> It will be there before the close of business today, okay? But this is where you can go and you can join, so like I said, at the bottom you click join, you put an email in there, you put a
password, and it's going to ask for a password to access it. And this is what you'll, use, PTN Math, okay? So, that's Wiggio. Again, Wiggio is a place where there'll be some materials and resources.
Right now, the PowerPoint is not there, but it will eventually be there. The iTunes course, Jared already gave you a lot of the background information. Okay. This is the code, but can you access this
course through a PC or a Mac computer? Give me a head nod. Not at this time, okay? In order to access iTunes, we need to use an iOS device. It's FHN-J8M-HJ2. Okay. Again, you can only access that
right now through an iOS device. On your tables, okay? You'll see little blue cards. And if there's not enough blue cards for everybody at your table, just let us know during the break and we'll make
sure that everybody gets those. But that's a source of this information as well, okay? This is the blue card that I'm talking about. Twitter. There's your hashtag for Twitter. Okay. Here's your web
address for PaTTAN, and you go on the PaTTAN site, you can get it to the math initiative. You'll find some information there. Here's the Wiggio, okay? Here's your QRT code. So, then, you have all
that information in one place. So, we just wanted to let you know that here's where you can find more information about CRA. I think they're just about math in general. Makes sense? Another source of
information will be CRA days. And what we're doing--excuse me. This is for teachers and this is for, you know, math leaders. What we're going to do is get in to more depth about using this CRA
strategy. Okay. Concrete, to representational, to abstract. So you'll see that we have a day in Harrisburg where we'll talk about addition and subtraction. What we're going to do today is, we're
going to provide you an overview of what CRA is, to let you see and experience that as a student. But by no means would we expect you to be able to walk out of the room right now and be able to turn
this around and start using this in the classroom at this point, okay? So if after today's presentation, you're starting to think about this and say, hey, CRA is something that I want to use in the
classroom, it's something I want to learn more about, here is where you can learn more about it. So you can attend these days in Harrisburg, the King or Prussia, or Pittsburgh, okay? This addition
and subtraction, multiplication, division, fractions, integers, and equations, you do not have to attend all four of these, you can attend these and, you know, if you're not able to go to the
addition, subtraction, you want to go to multiplication, division, that's fine. Another question we hear is, "Is this only for elementary teachers?" Okay. No, this is not--it's not grade specific.
What will happen is, as teachers learn more about CRA, then the hope is that they can transfer that to whatever content their teaching. Makes sense? Okay. So you can go on PaTTAN website, go to the
calendar and you can find out how to register for these CRA dates. Everybody okay with that? So, we're going to introduce CRA, then we'll focus on addition. Okay. Jared's going to take us through,
he'll model it as a teacher, I'll be a student. He'll teach you a little about place value and how we use this thing called CRA. Then we'll go to the integers. The whole time, okay, it's going to be
interactive, Jared's going to be asking questions, you know. So we want you to keep an open mind as you're doing this. On each table, you'll see sticky notes. So if there's a question that you have,
okay? Place that question on the sticky note. Now, what we're going to caution you against is, sometimes when people are hearing something, you want to get in to the nuts and the bolts right away,
okay? And we're going to caution you and say, "Hey, let's look at the big picture first and really see what CRA is, see--experience it as a student, and then start thinking about it, then you can get
to some specific questions." So we're not going to go through, like, specific accommodations or modifications or things like that, those--that would be some of the things that we do during those CRA
days. Makes sense? So you put the questions, any questions that you have on the sticky notes, we'll collect those during the break, and then if we have an opportunity to respond to any of those
questions. If we don't respond to all of them, then we'll have, on our Wiggio, there'll be a place there where we respond to some of those questions. Makes sense? Jared, anything to add to the sticky
notes? Okay. So that's a lot of preliminary information. Everybody ready to rock and roll?
>> Uh-hmm.
>> Okay. Sharon, I'm going to pass this over to you for a second and let Sharon tell the story.
>> Okay. I'll just stand here. What we're going--what is probably not new to many of you in this room, but just to situate when we start looking at math content and we think about some of the math
that Jared and Allen are going to talk today, I want you to keep this in mind, this is the journey of where we've been. And I can tell you, I've been on this journey, I was telling some folks today,
I ran into parents of a student that I had that I taught years ago, and this young woman just turned 49, so it's kind of, like, frightens me. So way back in the day when kids with significant
cognitive disabilities first started in school. And remember, they weren't even allowed to come to school before 1975. So it hasn't been that long. We had a developmental model, and we have always
done the very best that we have known based upon the research that's out there. So we had a developmental model. And then, we learned more and we learned the capabilities and what students with this
disability, what they could learn. And we moved and we began teaching really functional skills and daily-living skills and moving students towards independence. And we've been successful with
students. But things are shifting, the tides are changing, and this isn't just in Pennsylvania, this is nationally. We're changing and shifting, we're looking at students with significant
cognitive--significant cognitive disabilities and we're really looking at an academic model. But we're not throwing the baby out with the bath water. There are independent, and individual, and daily
living, and functional skills that are very, very important, and IEP teams are determining what students need to know. So we're looking at a blend, we're looking at ways that we can take where we
know functionally, kids need but we're going up the ante and we're going to be upping the radar and looking also at academic standards and looking at that PA Core and digging deeper with this
population of students. So as--keep that in the back of your mind. We've come a long way, baby. You know what I mean? For lack of a better word, we've come a long way and we're learning more and more
about the capabilities of kids who fall within that one percent of the one percent, the one percent of all population of students.
>> So you have an idea of where to get information. This is kind of the background, okay, to what we want to talk about. So some other foundational information, PA Core Standards. In Pennsylvania,
okay, we had the PA Core Standards. Over here on the left, you can see that the four domains, okay, numbers, and operations, algebraic concepts, geometry, measurement data, and probability. We're
going to focus today on numbers and operations and some algebraic concepts, okay? Most of the material and resources that we produced at PaTTAN that you'll find on the Wiggio site and you'll find on
the iTunes course are around, you know, these two particular domains. Also, with E Core Standards, okay, the standards, that's the content, that's what you teach, it's what the students are supposed
to know and be able to do. But also, with those standards, we have what's known as the mathematical practices, okay, and there are eight mathematical practices. These are behaviors that we want
students to engage in when they're in the math classroom, okay? It's different now than when I was in school. When I was in school in a math classroom, I would sit there, the teacher would be up
front, put a couple problems on the board, and then they would say, "Turn to page 27." Okay. You're going to do questions one to ten, I'll walk around help you. That works for some students, okay?
But there's large gaps that are created and it doesn't work for all students. So one of the things that--when Common Core was created and NCTM, The National Council of Teachers of Mathematics,
they've been then a very big proponent of this. They said, these are behaviors that we want students to participate in as they're learning math, okay? It's getting them actively engaged because the
more actively engaged you are on things, the better chances are that you really truly learn. You notice the first one says, make sense of problems and perseverance solving them, okay? You know, what
does that mean? How do you persevere? These are all conversations that teachers and educational leaders K through 12 need to--you know, have and figure out how they can teach the students how to
persevere. Not all eight of these will happen every single class period. Not al eight of these will happen everyday, but over the course of the year, a teacher should look back and an
educator--educational leader should look back and say, "Hey, did I have the students engage in these eight behaviors?" Okay? As we do CRA today, think about how we're engaging you. And you'll notice
that we're really going to focus on two, three, four, and five, and Jared will be a little bit more specific when we get there, but these will be the mathematical practices that we're engaging you
in. So we have the content which is the PA Core Standards, and then we have these mathematical practices. Here are some of the research behind, in this case, CRA and the mathematical practices. Okay.
There's an IES practice guide that deals with RtII. And one of the recommendations that this guide says is to use visual representations when we're teaching mathematical ideas and skills, okay? CRA,
Concrete Representational Abstract makes use of this specific recommendation. The verbalization of problem solving, the verbalization of showing your understanding is also very important. So you
notice that when Jared takes us through this, he'll be asking questions and the expectation is that the student which you will be the students today, will be responding, okay, and verbalizing what
you're thinking. Now, we know that when we talk about students with severe cognitive disabilities that sometimes verbalization is a hurdle they have to overcome. So when we talk about accommodations
and those are things that with CRA days and with other intense consultation and training, you will learn how to handle. So like I said, today, we're focusing on verbalization, but we're not--we're
not going to talk about these accommodations because we just want to give you a big picture. Makes sense? Here's some more research that talks about using concrete materials to develop more precise
and more comprehensive mental representations. In the core, PA Core, one of the big pushes besides using the mathematical practices is for students to understand conceptually, why things happen the
way they happen. Okay. In the United States, we've had a tendency to teach from procedures only, you know, just generating answers. And again, that works for some students, but there's a large
majority of students where those gaps start to get created because students are wondering, you know, why things work the way they do. And what the research says is that if students understand the why
and they can show the why, then the chances are that they're really going to learn it, okay? The other thing you're going to notice is when a student is working with concrete manipulatives, the
thought process will be the exact saying as if they were drawing pictures which will be the exact saying as if they were doing it abstractly. Now, when I talk in terms of abstractly, I'm talking
about using numbers and using symbols. I mean, math is a language and that's what we go to, those numbers and symbols, but we don't have to live there all the time. So with this CRA progression,
you'll see that we're using concrete materials and that it develops this picture, this idea, and lets the students really see what's going on. I mean, they can see it and start to understand the
whys. So I'm going to pass this off to Jared now. He's going to give some back--more background information on CRA and then take you through some of this experience. Jared.
>> Thanks, Allen. Can everybody hear me okay?
>> Yeah.
>> Good. Good. So in a moment, I'm going to say, "Hi, everyone," and then I want everybody to respond with, "Hi, Jared." Okay? Hi, everyone.
>> Hi, Jared.
>> When they edit that, everybody is going to think I'm famous on this video. That's my plan. So what I don't want you to do right now is panic. And I know a lot of times, we start talking about math
and we've talked about CRA, we're talking about concrete and abstract and this is how we transition, this is what we're going to do and its visual models and its multiple representations. Everybody
just kind of just like, hold up, you didn't even really define CRA for us yet, okay? We know. I'm going to do that with you, okay? You're going to engage in this as a leaner, you're going to get to
see me teach my student up here, he's not the model student, but he's pretty good. So I'm going to be teaching Allen, so you're going to get to see what this instruction looks like. We have a doc cam
set up, you're going to get see him using the blocks for, you're going to get to hear the language that I'm using. And then, you're going to have a chance as a student to see that as well. I will
preview content with you, then I will teach Allen, then I will teach you, and then we'll summarize that content together, okay? The only time I won't do that whole sequence is the first time, I'll
just go straight to teaching you because I know you guys are going to pick it up really, really quick. So I don't want you to panic if you're overwhelmed. And there's one big thing you need to take
away from the first part of this morning, how will you stay connected? And the answer is to go to Wideo. How are you going to stay connected?
>> Wideo.
>> How are you going to stay connected?
>> Huh? I got to teach you something too. You got to wait for my hand signal, okay, because I'm going to do a lot of this corresponding, so when my hand's up, you're thinking, how are you going to
>> Wideo.
stay connected?
>> Wideo.
>> I got a quick room. Okay. Are you ready? So CRA stands for Concrete Representational and Abstract.
>> Concrete Representational and Abstract. And this will be up on slides as we go through. When we talk concrete this morning about addition, we're going to be using place value blocks. And the first
>> Say it again.
thing I have to tell you, you need to teach the blocks. Textbook materials, we'll use place value blocks. They have pictures of them, they'll tell you use them. And if you just put blocks in front of
kids, then all you've done is put blocks in front of kids. They'll build with him, they'll throw them, they'll find places to stick them that they shouldn't. But what they're not going to do is
intuitively use them to model mathematics, so we have to teach the block. So I'm going to teach you the block right now. This is a unit, it's worth one. What's it worth?
>> One.
>> Wait.
>> [indistinct chatter]
>> What's it worth?
>> One.
>> Okay. Good. If I take ten of those, ten of those is a lot to really move around, so if I take ten of those, I can put them altogether, and you're going to see they're the same length as a ten,
okay? I call it a ten, some people call it a rod or a rod of ten, or there's different names for it, but--so that's a ten. So the second one is worth how many?
>> Very good. And if I take ten tens and put them together, I can create a new one. Ten tens are worth a hundred. So instead of using ten of these individual tens, I can pull from another block and
>> Ten.
this one is worth how many?
>> A hundred.
>> That's how you do blocks, okay? Now, obviously, when you work with kids, there's a lot more that you would have to do with those blocks. You wouldn't introduce the hundredths right away, you'd be
introducing just the ones initially, especially if you're working in the single-digit stuff. The tens come in when you get into the team numbers. As you actually get into place value, then you're
going to have all those blocks come into play. One of the things that I don't want you to think is you cannot alter those blocks for yourself, write on them, put tape on them. Help children remember
what they're worth, okay? Do what you need to do there when you teach those blocks to help kids use those. We use those for addition, subtraction, multiplication, and division. We can also use them
with decimals. You're not going to see that today. So, we need to teach the blocks because that's how we're going to do all of our modeling. So, Concrete Representational and Abstract. I'm going to
run you through place value, and what we're going to do with place value with this, and then you're going the have the chance to experience this as a learner, okay? So, first, when we're talking
about place value, we're going to start with an abstract. There's always some level of abstract with CRA. It's not just going to be getting the blocks and doing something with the blocks, there has
to be intent there. So, if I gave you the number 342, that's abstract, it's symbols, it's digits. So how we make sense out of that is going to determine whether we're in a concrete level of
understanding, a representational level, or an abstract level. If all you did was look at 342 and you knew everything about the number 342, then you're thinking abstractly. If you start drawing
pictures that represent that, you're thinking representationally. And if you're using some sort of physical manipulative that you can alter, then you're thinking concretely. So, let's talk
concretely. If I'm trying to model three hundred and forty-two, the first question I'm going to ask you is, how many hundreds are there? How many hundreds are in three hundred forty-two? How many
hundreds?
>> Three.
>> And then I'd say, show me. And you would pull three hundred blocks. And then I'd say how many tens are in three hundred forty-two? How many tens?
>> Four.
>> I would say show me, and you would pull out four tens. I'd say how many ones?
>> Two.
>> Then I would say show me, and you would pull those out. And one of the things you're going to notice over here that I didn't address right away was this actual decomposition based on place value.
What we were doing is modeling this number in what's called expanded form. Expanded form says, if you have hundreds, tens, and ones in your number, that it's hundredths plus tenths plus ones. So
three hundred forty-two written in expanded form is three hundred plus forty plus two. And that's what we see right here, three hundred, four tens, two ones. So, if I went through the full line of
questioning as you're going to see me do with you in a minute when you're doing your representations, I'm going to say, how many hundreds? And you're going to say...
>> Three.
>> And then I'll say, show me, and you'll draw. And then I'll say, what's three hundredths worth? And you'll say...
>> Three hundred.
>> Three hundred. And I'll say, write it. And you'll write this three hundred in your--on your paper. I'll say, how many tens?
>> Four.
>> I'll say, show me, you'll draw four tens. I'll say, what's four tens worth? And you'll say...
>> Forty.
>> I'll say write it, and you'll write the forty. How many ones?
>> What's that worth?
>> Two.
>> Two.
>> And you'll do that as well, okay? So, that's pretty much the sequence. It's very explicit, it's very direct. If you need to ask less questions for your kids, that's fine. If you're going to help
them model, then you can do that. Those are the accommodations that you'll make along the way. We know one of the things we--are asked of us right up, what about the kids that can't move the blocks?
You know, then help the kids select the blocks, help the kid count the blocks if you need to do that. With the drawing aspects, if there's some fine motor skills, find other ways to engage the kid to
that. Maybe they can trace blocks at the beginning, maybe they can use stamps along the way. There are other ways that we can make this accessible. But when we talk about CRA, what it does is sets
the groundwork for us to begin to look at more accessible ways than just having kids always deal with digits and abstractness, okay? So, here's what the drawing looks like. Instead of you having all
place all your blocks on the table, you're going to use the representation for me in a minute. So instead of these hundredths blocks, you're going to draw a square. So, if I said, how many
hundredths, three hundred, show me, you draw that. Then I say, what's that worth? You'd say 300, I'd say, write it. You'd write 300 underneath. For tens, we're going to do lines, and for the ones,
we're going to do circles. The reason we switch up and do the circles instead of the squares even though they would look more like our ones up here is because little squares and big squares are going
to get confusing for students. You ready? Okay. So, you're going to be working in which concrete representational or abstract? You're going to be working in the representational. Everybody you're
going to be working in...
>> Representational.
>> You're going to be drawing them for me, okay? Remember, there is going to be some level of abstractness though because you're--still have numbers that you're working with along the way. So, here's
our first problem. Three hundred fifty-seven. Three hundred fifty-seven.
>> [indistinct chatter]
>> Okay. First question, for three hundred fifty-seven everybody, how many hundredths?
>> Three.
>> Show me. What's three hundredths worth?
>> Three hundred.
>> Write it. At this point, you should see that you have three blocks drawn and three written below it. Everybody, how many tens?
>> Show me. What's five tens worth?
>> Five.
>> Fifty.
>> Write it. How many ones?
>> Seven.
>> Show me. What's seven ones worth?
>> Seven.
>> Write it. Now, we want expanded form, so we're going to put plus signs in between those numbers. Three hundred and fifty-seven written in expanded form is three hundred plus fifty plus seven, all
right? Three hundred written in expanded form is three hundred plus fifty plus seven. Everybody's going to say this with me, okay? When I drop my hand. What's 357 written in expanded form? It is
three hundred plus fifty plus seven.
>> Three hundred plus fifty plus seven.
>> Always repeat that problem at the end, okay? We need to make sure we do that with kids. Otherwise, there's no sort of conclusion, there's no closure to that problem. All of a sudden, they've got
something written and they're not sure why they wrote it. So, we always want to come back to that. A couple of questions that will come up right away. Your pictures are going to look different than
my pictures. It doesn't mean your pictures are wrong. You may have a different way of organizing this for students. Some teachers like to keep everything lined up vertically, and if that works to
your kids, then do that by all means. Some teachers we were working with yesterday said, "Jared, why are you doing five of these? Why not do one, two, three, four and then a bundling mark with it?
Why wouldn't you do that?" My argument for it, and I'm not saying it's wrong to do it, my argument though is if you are modeling this concretely, you have five of these. There is no way to bundle
these unless you're starting to lay your fifth one across. So, if you wanted to lay fifth one across for kids, I'm fine with that. See how it works, okay? Otherwise, I would keep all five
individually like this. The other reason I do that is because we know the ten frame is a great structural support for kids especially in early numerously scales. The five frame and then ten frame.
So, in the five frame, there are five across, in the ten frame, there's five across and then a second row of five across. So, students should be used to seeing groups of five. Groups of five in my
mind are very, very important because groups of five are one of the first ways that we can break one-to-one correspondence, and we know that kids that are eventually labeled learning disabled in math
tend to have an issue with one-to-one correspondence. They're stuck there. No matter what numbers they get, I have to count one, two, three, four--so you just start doing twenty-seven plus five, and
the kids go, okay, one, two, three, four, five. And they finally hit twenty-seven and they're like, "Wait, what was I adding?" Their working memory is just taken up trying to count up instead of
being able to count on or think more flexibly about those numbers. So, one of the ways we can remediate some of that here, is to make sure that we've worked with kids with groups of five and seeing
groups of five because if I can see five, now I can count on. So when I see these blocks lined up this way, I know five and then, six, seven. It's a little bit easier for me to see seven if I'm
lining them up this way. I know a lot of us have the tendency, myself included, to think dice and we do six, we two groups of three and then another one. And that's okay. If you're working on
subitization and kids being able to see different patterns and know how many are in those pattern, and six and one is a pattern that you're working on, then by all means, go ahead and do that. I
think in the big picture, groups of five are going to last longer for a group because two groups of five are going to make a what, everybody?
>> Ten.
>> Ten. And our number system is base ten because we have ten digits, zero through nine, and the arrangement of those digits has significant value because every time you're in the next place over,
you're ten times larger than where you were to the left, right? So that set of ten is very, very important for our number system. So, that's one of the reasons that we arranged that this way. Now,
you'll also see me break my own rule because it's very hard for me to line up ten--or five one hundredths all on a row on the screen. So, working space we know is going to be a concern. You're going
to need to have space for kids to use the blocks, you're going to need enough space for them to complete problems when they're drawing their pictures. Any questions about the problem? Ready for
another one? Okay. We're going to move quicker this time. Here's problem two, four hundred eighty-three. Four hundred eighty-three. Now, don't just do it, work with me. Four hundred eighty-three, how
many hundreds?
>> Four.
>> Show me. Four hundredths is worth what, everyone?
>> Four hundred.
>> Four hundred. Good. Write it. How many tenths?
>> Eighty.
>> Show me. Eight tens is worth...
>> Eighty.
>> Write it. How many ones, everybody?
>> Three.
>> Show me. Three ones is worth...
>> Write it. What sign are we going to use between the four hundred, the eighty, and the three everyone?
>> Three.
>> Plus.
>> Plus. The addition sign. So four hundred eighty-three written in expanded form is four hundred plus eighty plus three. What's for four hundred and eighty-three written in expanded form everyone?
It's four hundred plus eighty plus three.
>> Four hundred plus eighty plus three.
>> Any issues? Are you feeling good? I'm not going to run through a whole bunch of these. I think you got it. That's the representation. Now, you don't want kids drawing that until they've done the
blocks with it, right? Because drawing is a different thing that they have to work with especially younger children who struggle with drawing or anybody who's struggling with drawing with fine motor
skills. So the blocks are going to be a lot more accessible. I can pull from my piles, right? I still have to know what those blocks mean and what those piles mean. The other thing I want you to key
in on is the language that we've been using. There's two different ways that you experience numbers. There's the name and the meaning, and we do this great in our alphabet. We have A, B, C, D and
then we have /a/, /b/, /k/, and we learn what the name of the letter is, and the sound that it makes, so the meaning of that letter. We don't teach numbers that way. We don't. We use the names and
then we expect children to inherently understand their meaning or we wait until place value shows up 37 times in the standards, and then we think that place value means which ones in the tenths and
they can say eight, and that's all that place value is important for, it's not. Place value is important for understanding the meaning of the number, all right? And that's modeled in the language.
Eighty is the name of the number 8-0. Eight tens is the meaning of eighty. Eighty-three is the name of a number, eight tens, three ones, is the meaning of 83. So, we're using the meaning of the
numbers here to model place value and to help keep that language consistent so that it can then be condensed back down into the number name. So, 317, we move through it real quick. Three hundred, one
ten, seven and of course, after you've done all this work for kids, then they get here and they go three hundred, ten, eighty-seven. As it follows the pattern, right? Because we struggle with names
and meanings when we teach numbers, the teams are extremely difficult in the English language. In other languages, they're not necessarily that way. They teach kids to count one, two, three, four,
five, six, seven, eight, nine, ten. Ten and one, ten and two, ten and three, ten and four, ten and five, ten and six, ten and seven, ten and eight, ten and nine, two tens. Two tens one, two tens two,
two tens three--that's how they teach your children to count and that's how their language is structured. Our language is not structured that way. We get in all sorts of crazy numbers in the teams.
This is not going to fix that, okay? We're still going to have to teach that language, we're still going to have to work through those [inaudible] but in terms of the meaning of 317, it still fits.
Three hundred and seventeen is three hundredths, one ten, seven ones. And we can model that abstractly in expanded form, we can show that with representations when we draw it, and we can use our
place value blocks. All of our computation, addition, subtraction, multiplication, division, all of our computation is based off of expanded form. So, you are going to see as we move in to addition
and I show you what subtraction looks like, we're going to keep coming back to these same skills in expanded form and seeing numbers that are written out in these hundreds, tens and ones, and they're
going to get recomposed and all of a sudden, you're going to end up with the something and go, "Wow, it's really just place value in expanded form," and it is. One of the important things for us to
consider is we have to have a way to each kids conceptually or procedurally that is going to last. Place value isn't just important for this day. Place value needs to be important and continuously
drawn upon so kids really understand place value and they always have that strength to draw back to. We start changing it up down the road and we're teaching brand new things down the road. But if we
can stay consistent with our language and our modeling, we're just going to help students that much more. You don't have these, but if you connect with us on Wiggio or if you are teaching in a
classroom, come see us afterwards. We have a couple sets up here that we can give teachers. We're not going to have enough for everybody in the room because we weren't necessarily prepared for
90-some people but we think we have enough for every teacher in the room. If you join us on Wiggio, you will definitely have the PDFs where you can print your own, all right? And place value cards is
what we're talking about. So, here's place value cards. Three hundred and forty-two is three hundred, forty, and two. I can see expanded form written here, I can also see it in the numbers on the
top. When I stack those cards, guess what I see, 342. This is the cool part about place value cards. If I can't write digits, if that's a struggle for me, place value cards can be a support that we
can put in place for kids so they can still engage in the content, and we can decrease some of the needs for writing, okay? So, when they're separated its expanded form, when we put them together,
it's regular notation, 342. But notice even at 342, it still helps with the reading, 342. They can still see expanded form. It's 342, it's helps with the language. But of course, the only thing
that's an issue is when you get into the teams, right? Because it's not going to solve that every problem, it's just going to help us bridge some gaps. Five hundred ninety-seven together, five
hundred ninety-seven, kids create them, kids separate them. Now, I've got expanded form. So, it goes both directions whether they were composing or decomposing these numbers based on place value. And
you can use these cards to support the operations well. It doesn't stop with place value and expanded form, all right? So here's a real quick question. I just want to engage the whole audience before
we move off of place value. With these cards--would these cards be considered concrete, representational, or abstract? I want you to think about it for a second. Would these cards be considered
concrete, representational, or abstract? Think about it. Ready? Answer?
>> [indistinct chatter]
>> Yes.
>> I got a yes. I like yes. I don't know. Don't you love it when people get up and get, "Hey, I know math." And they go, "Yeah, I don't know." Here's what we like to believe. Don't argue with yourself
about what definition stuff exactly fits into because kids' thinking is messy. Our thinking is messy. Most likely, the kid will always have some sort of hybrid going on. We always said there's going
to be some level of abstract, but mainly, it's how the kids making sense out of it. I see kids getting stuck in the abstract somewhere along the way and starting to draw themselves a picture, and
they have half of a picture, and then they can't finish completing the abstract. So, were they abstractly making sense of that or making sense out of that representationally? At the end of the day,
I'm--like, after the kid's been through this sequence, I'm not really stressing. I'm more concerned about the kid was able to make sense out of it. And the kid was able to think about their own
thinking, enough to know that they could go back and draw some pictures to help them work in the abstract, okay? I would probably argue that this is abstract. Why is it abstract? Because everything
contains numbers. There is some physical manipulation, but it's not--like, it's still digits that they're manipulating, so, they're still making sense out of everything through the digits. But where
it exactly fits, I don't know. Here's what the map looks like. The map--the place value map. You'll see language over here, how many hundreds, what's it worth, how many tens, what's it worth. We've
already changed this map actually since this PowerPoint was created because the question came up, "Should we really be saying worth or should we say value?" That maybe we should say, "What's the
value?" So, if you get online, you'll probably see value pop up there. But again, you'll see what it looks like concretely, you'll see what it looks like with the drawing, and you'll see what it
looks like in the abstract. You'll notice that there's some sort of component across all three of those, okay? This is about the simplest map, and if we showed you this upfront, you would go, "I
don't know what they're looking at." But as soon as you experience this as a learner and you do some of your own drawing and you see--you can make sense out of it, this map is a great way to refresh
yourself about what these models look like and potentially help you practice your language, which by the way, you need to practice with a colleague before you sit down in front of kids, okay? Because
there's going to be changes to that language that you want to make and you won't know until you sit down with a colleague and you hash through that. We've given you a starting point here to work
from. So CRA is not in a day. Everybody say it with me. CRA is not in a day.
>> I just made that up, but I feel really good about it. It rhymes, so I went with it. It's not the void of the concept, okay? I'm not having kids take blocks and line up blocks, and all of a sudden,
>> CRA is not in a day.
they're doing addition. That does not guarantee they have any idea what addition means. I still have to work with the concept of addition. It means combining, adding on, increasing, those types of
definitions that you work with. There is still concept development that happens. But I am going to have to teach concretely and get them to practice concretely. Then, I'm going to have to transition
them into the representation. So what that might look like, one way to think about it is start modeling concretely, and then have them trace the block, line it, circle it, remove their blocks and
look at the pictures that are left over. It may be even creating a little space for yourself where they have the picture right beside this or the drawing taped right to the block. So instead of
pulling the block, they could see the picture, but you're going to have to transition kids across into the representation and then you're going to have to transition kids again to the abstract. And
at some point, you're going to have to make connections across all of these because you want kids to be able to work in the abstract and fall back on representations or potentially the concrete to
support themselves when they need it. We talked about the standards for mathematical practice really quickly. I can make an argument for eight of these and how they fit into CRA, but the ones I think
that stand out the most are reasoning abstractly and quantitatively, constructing viable arguments, and critiquing the reasoning of others and people always like this one like three, right? How do
you know three hundred and fifty-four written in expanded form is three hundred plus fifty plus four? Because you go, "Well, you taught me." Yeah. But how do you know? "I don't know. That's what you
told me how to do." Now, you could say, "How do you know?" "Here's the blocks. Look, it's three hundredths, four--you know, five tenths, four ones. That's what it is." So that you're giving them
access to mathematical proof essentially and they can use that to converse with other students and engage with other students. We're definitely asking them to do a lot of modeling and we're thinking
about using their tools strategically. So, whether they're using place value blocks and how are they organizing those place value blocks. Those really fall into the ideas of reasoning, explaining,
and modeling, and using tools. So, you're going to see some commonalities there. Are you ready for addition? Okay. We're going to do addition pretty quick because we got out about 30 minutes for it.
So remember, the goal isn't that you can go back and you can do addition. The goal is for you to see my student up here and what the teaching looks like for that student and then the next step will
be for you to practice some. He'll work concretely. You'll work representationally. Okay? Get that lined up. Okay. So, we're working on addition. Now, we're showing three digit by three digit
addition, so two hundredths that are--two numbers that both have hundredths. So, the first number is 312 and you don't have to write this. You can just follow along. Two hundred and fifty-six, two
hundred and fifty-six. I'm not telling you to tell the kid that problem every time. You may be handing that to them or it might be written for them or you might be writing for them, but that's going
to be our addition problem. Okay? So, why am I showing this to you? Because this is about the most by hand addition kids are going to do. If you've just got tenths to two digit numbers that you're
adding like twelve and fifty-six, then that's fine. You don't have hundredths blocks to deal with. You just focus in on the tenths and the ones at that point. So you can always work down. So, 312,
256. Now, you're going to see my hand going, but remember, Allen is my student, so he's the only one that's responding. We're going to put him on the spot here and see if he knows his addition. Three
hundred twelve plus two hundred fifty-six, Allen, first number, three hundred twelve, how many hundredths?
>> Show me. Second number, Allen, how many hundredths?
>> Three.
>> Two.
>> Show me. Allen, how many hundredths are there altogether?
>> Five.
>> See what he did? One-to-one correspondence, right? He started pointing and counting. That's how he's going to make sense out of addition for me. I can already see it. You're going to have a lot of
kids do that. If you say, "What's 300 plus 200?" They're going to go, "I don't know." And hopefully they wouldn't try to count to 300 and then try to count 200 more. But you would have kids do that
because that's all they know how to do is one-to-one correspondence. Here's what we've done to reduce this complexity. It's three blocks and two blocks is five blocks, and what's the meaning of those
five blocks? So Allen has told me that he has five hundredths. Allen, what's five hundredths worth?
>> Five hundred.
>> Write it. Allen likes to write across the bottom. I don't have an issue with that. Some teachers will say, "Allen should be writing over there on the abstract. If you want to keep your kids
organized that way, then that's fine, but we're still doing all the same thinking. Allen, first number, how many tenths?
>> One.
>> Show me. Second number, Allen, how many tenths?
>> Five.
>> Show me. Allen, how many tenths altogether?
>> Six.
>> Allen, what's six tenths worth?
>> Sixty.
>> Write it. Allen, first number, how many ones?
>> He's a good student, isn't he? Show me. Second number, Allen, how many tenths?
>> Two.
>> Six.
>> Show me.
>> Twenty-one.
>> How many ones? Did I say how many ones? I [inaudible]
>> Too good.
>> [inaudible]
>> I know. How many ones in the second number, Allen?
>> Show me. Allen, how many ones altogether?
>> Six.
>> Eight.
>> What's that worth? I'm sorry. What's that value?
>> Eight.
>> Write it. Okay. Allen, we're in expanded form, so, what sign are we going to put between five hundred, sixty, and eight?
>> Good. Allen, what's five hundred plus sixty plus eight?
>> Addition.
>> Five hundred sixty-eight.
>> Write it. Partial sums. He added his hundredths, he added his tenths, he added his ones. Here's what we were maybe never told in school. Algorithms, whether you do in partial sums, left to right,
right to left, carry as you go, however you wok them, algorithms are all about single digit computation. Kids get confused on algorithms because they can't keep everything lined up and they don't
know where stuff goes anymore. There's too many steps to remember. So, what we've done is we've helped them with the addition by keeping the blocks and the one-to-one correspondence. If they need it,
remember there's still going to have to be instruction that helps break that grouping of fives, five frames, ten frames, that kind of stuff. The other thing we've done is we've taken the kid off the
algorithm right now. I didn't only have Allen work on the algorithm and look at the hundredths, the tenths, and the ones and keep everything lined up and write this and move this, right? I let him
make sense out of this with the blocks. It slowed down what was going on abstractly and let him do all of his reasoning concretely with those blocks. He's still got that partial sum and he wrote
it--I'm sorry, ma'am. And he wrote it in expanded form and then we put it back. See, why place value is so important and expanded form is so important because we can put it back together. One of the
things that some teachers do to help kids that can't keep stuff lined up is instead of three hundred twelve, they have them write three hundred plus ten plus two and then two hundred plus fifty plus
six and then add the hundredths, add the tenths, add the ones. And what they're doing is putting each number in expanded form and then writing the answer in expanded form. We've don't the same thing
here. We just haven't forced Allen to write this in expanded form. He's modeled it expanded form. Okay? Question.
>> Why wouldn't he start with the ones? We always teach kids to start with the...
>> Yeah. Okay. So, here's my quick argument for that and then we're going to keep moving because this is one that always comes out. We as adults are brainwashed. What are we everybody?
>> Brainwashed.
>> We were taught left--right to left. We think everything should work from right to left and they should do it the way we did it. What we did was not necessarily the best way to do it or the only way
to do it. Here's my proof. Talk to adult, ask them how many enjoyed math, ask them how many feel successful in math, ask them how many people feel prepared to teach math, and you will--just from the
smiles in the room. Okay. If we were taught math perfectly, everybody in this room would say, "We're capable of this because I know you're capable of this." You might not feel like you're capable of
this. So the way we learned it is not necessarily the only way to learn it nor is it necessary the best way. Different people are going to like and attach themselves to different ways. The reason the
right to left algorithm has survived is because it's very generalizable. Meaning, I'm always going to have ones to add. So then when I get in two digit numbers, I move over and I add my tenths. And
if I have hundredths, then I can move and add my hundredths and I can go the whole way out to thirty-seven places if I really wanted to. It would still work. This is generalizable, but it becomes
cumbersome at some point. So if we're starting with thousandths, now I've got four numbers for my partial sums. If we've got ten thousandths, I've got five partials sums that I have to deal with. So
at some point, we do want this to go away. I'm not saying this is necessarily end result but then there's this balance of this works by hand and very efficiently to the point that maybe the students
using a calculator to go along with it for those very big numbers. The other reason that we stress this one at the beginning is because it promotes place value and estimation. So here's the big issue
with the right to left algorithm. I teach you to add this way, right? Add the ones, add the tenths, do the carrying, add the hundredths, and then I say, "And basically, you know, if you were to
estimate 312 plus 256, what do you think it is?" And the kid's first response is, "I don't know." Add the ones, add the--and we like have to force kids to try to estimate, right? Because they want to
just regurgitate the algorithm. When we add hundredths first, we have started with estimation. So instead of changing and having a new pattern or a new algorithm for estimation, it's just a truncated
version of this algorithm. Your mind works this anyway. You've taught yourself this over the years. Three hundred and twelve, two hundred and fifty-six is about what? The first thing you would do is
go 300 and 200 is about 500, so it's going to be in 500 somewhere. If I say, "Get a little bit more specific," you might think 310 and 250 and, you know, it's going to be over 350, maybe it's 360,
something like that. Your mind works from left to right when we do computation mentally because we've just adapted. Maybe we had some good instruction along the way or maybe our parents taught us or
whatever, but we can use this left to right algorithm to help with estimation and it's still going to be generalized with what we have to do by hand. The right to left algorithm exists because it's
the most generalizable form, because it works for everything. It never gets more complicated. You just keep tacking steps on to the end.
>> What happens if you have some [inaudible]
>> Oh, that's a good question. We're going to do some carrying. We're going to do some carrying. Ready--Allen, are you ready for some carrying?
>> Sure.
>> Okay. So, here's our next problem. What happens when you have to carry? Allen, let's work representationally on this.
>> Okay? So instead of the hundredths, he's going to draw hundredths. Instead of the tenths blocks, he's going to draw tenths. Instead of ones, he's going to use the circle. Allen, here's our problem,
>> Okay.
265 plus 126. Still some level of abstractness here, okay? Allen, how many hundredths in the first number?
>> Two.
>> Show me. Allen, how many hundredths in the second number?
>> Show me. Allen, how many hundredths altogether?
>> One.
>> Three.
>> What's that worth, Allen?
>> Three hundred.
>> Write it. Just added the hundredths. Allen--he's advanced. Allen, first number, how many tenths?
>> Six.
>> Show me. Allen, how many tenths in the second number?
>> Show me. Allen, how many tenths altogether?
>> Two.
>> Eight.
>> What's the value of eight tenths, Allen?
>> Eighty.
>> Write it. Allen, how many ones in the first number?
>> Five.
>> Show me.
>> [inaudible]
>> It's gross.
>> No. Good correction, guys. We just need to check and make sure you're still with us. Allen, how many six is in the second--or how many ones in the second number?
>> See what you did. Allen, how many ones altogether?
>> Six.
>> Eleven.
>> Write it. Okay. You can go ahead and put your plus signs in because I want to address this. So you've got two options here, okay? Depending on how you want to deal with this with your kids and
we're going to address this again when you practice. So we're--I'm not just going to skim over and leave it alone. Eleven can be written that way and then you can go ahead and add them up. That works
for him, okay? Or we can decompose that right now. Allen, eleven, what's the meaning of eleven, Allen?
>> One tenth, one ones.
>> Write it. Put your plus sign in between and cross out our eleven. Ten plus one. Allen, you have eight tenths and nine--or eight tenths and one tenth, what's eight tenths and one tenth, Allen?
>> Nine tenths.
>> What's nine tenths? What's the value of nine tenths?
>> Ninety.
>> There's the 90. So now go ahead, Allen, and cross out your 80 and your 10. You can put it 90. It's still a little bit messy, right? He just--he keeps moving on. He's making sense out of this. Now,
he's working abstractly. If he's struggling with that, I'm going to go back to the concrete. And maybe I would do that initially when I'm introducing this type of regrouping. Get me eleven ones,
Allen. And he gets eleven of those. And I say, "Can you make a set of ten out of those?" And he goes, "Well, I don't know." I said, "Well, add them up. Can match the length of a ten rod?" And he does
that and he goes, "Oh, okay." So not let's get rid of those and let's exchange ten ones from one tenth and he starts working on those exchanges concretely. So then he's modeling one tenth and one and
then he's putting this ten over with his other tenths to say nine tenths which has a value of ninety. You can do that after you write it abstractly. You may want to do all that while you're still
working with the representation or you're still working concretely. It's up to you based on the needs of your kids. It's going to be the same thinking along the way. Are you ready to practice? I
didn't hear a big excited, "Yeah." Are you ready to practice?
>> Yeah.
>> Do you want to address this quick, Sharon, before we go on? Before we go on, Sharon wants--we're going to connect what you're doing right now with what you're going to see this afternoon. So you
can keep that in focus.
>> Okay. What you see up here--don't worry about writing it down. You're going to get a copy of it in your packets this afternoon. What we've done is we've taken a look at the PSSA Anchors that align
to the PA Core. And that's your assessment anchor up there in the reporting category of numbers and base--Numbers and Operations in Base Ten third grade. Then we look at what at the PSSA Eligible
Content is. And from that, we have what--we have some already what's called draft Alternate Eligible Content. There were some teachers that came together in May in addition to some work that we
pulled from some of the consortiums that have been working nationally. And from those, there were some draft Alternate Eligible Content. And that's what you're going to look at this afternoon and
>> Oh, yup. Sorry.
>> What we've given you is what might this look like in an IEP goal and objective for a student with significant cognitive disabilities. So we've given you some examples of what that could look like.
So you're going to see that all in one page. You're going to get that this afternoon. So when you think about what Allen's talking about, this addition and subtraction, and then breaking it down
using place value blocks. We actually put the place value blocks in there or concrete manipulatives, however, you want to write your goals and objectives. Is that it?
>> Yup.
>> Oh.
hand. But my question for you to think about is, did you really understand what it meant to say use place value understanding to perform arithmetic? I'll be honest with you and I was a Match teacher,
I don't really know what that meant. I thought place value was, there's the tenths place, there's eight, like what's the important thing? Line them up, do the addition kind of thing. Because I was
pretty good at those algorithms. I didn't really know what they meant though until the last couple of years. When we talk about using place value, we're taking about all those things that we just
talked about, okay? So all those things that we were just doing with the expanded form, with the modeling aspects, the drawing aspects to help us with our addition, that's what we're doing. We're
using place value. That's the assessment anchor. That's tied to the PA Core. Now the limits to that, right, sums and differences. One piece that's in draft right now that you're going to look at is
estimating sums and differences which is a great one, I think, to keep as eligible content. Because if I'm estimating sums or differences for three digit, I've just taken that left to right algorithm
and truncated it to one step, just do the hundredths, you know, or just do part of that algorithm. Just do the hundredths and the tenths. So it's not something different that we're teaching the kid.
We're teaching the kid to stop short because that's how we estimate, is to do part of that process. Okay. We already talked about this. We already talked about the exchange. I'm going to go ahead and
run you though this real quick just to kind of give me this overview again. If you we had five hundred twenty-five plus one hundred thirty-six, we model the first number, five hundredths, two tenths,
five ones. And then this time I'm going to show you the algorithm from right to left because I want to prove to you that it still works, the modeling still works no matter which algorithms you're
responsible for teaching kids. If I add on six ones, I put my six ones there. And my very next question is, can I exchange ten of those to make a ten or do I have ten ones that I can exchange for a
ten? In this case, there's eleven. So do I have ten ones?
>> Yes.
>> Yes.
>> Yes. So I can take ten of those and switch them out for a tenth. We do that physical movement with the kids. Now, I've got three more tenths that I can put with that group. And what we've just done
over here is demonstrate to the kid why that one gets carried in the eleven and then we added to the two and the other three. So one plus two is already done when we do that movement over here. And
then there are five hundredths and we're supposed to add another hundredth and so that's going to give us six hundredths and we've got it over here modeled in expanded form. The kid can write it in
expanded form. The kid can be writing it over there in the algorithm. However, you're--or wherever your student is at in terms of that learning curve is what you're going to focus on. It's going to
be the same thing with the representation. So as we do this process, whether I ask you questions from left to right or right to left, you're going to have work that looks like this. You end up
modeling the first number. Then in this case, I would ask you to include six ones. You'd identify that and write it and I'd say, "Do you have ten of those to make it one--to exchange from one tenth?"
And in this case you'd say yes and you'd scribble those out. Now, here's the one thing I recommend. You need a history of what that child has done on this problem. If you've ever done math and you've
seen kids get frustrated, they start erasing and then they go, "I need help." And you walk up and go, "I don't even know what to help you with. You erased all of your work. Like, I don't know." So we
don't want kids to erase. You're going to see some stuff disappear on slides simply because I need to keep the slide kind of clean, but you want to keep a history of what your kids have worked on. So
I recommend scribbling those out. On the slides, you'll see that we also circle them sometimes and maybe circles work. A question came up yesterday. What if I have a kid that absolutely refuses to
scribble those out because that's a messy paper and he will not allow messy papers? Then create new sets of papers, right? Let them work on Post-it notes and he can have a clean Post-it note each
time and you could see the progression that way. I don't care what you do as long as when you think about how to help the student overcome those issues with scribbling, that you're able to preserve
the history of the child's thought process, that's the most important piece with this. Okay. We add on the tenths. We add on the hundredths. Same concept as before. Are you ready to practice?
>> Yes.
>> Okay. Here's our first problem. I want you to write this down. Two hundred thirteen plus three hundred and thirty-five. Two hundred thirteen plus three hundred and thirty-five. How many hundredths
in the first number, everyone?
>> Show me. How many hundredths in the second number, everyone?
>> Two.
>> Three.
>> Show me. How many hundredths altogether?
>> Five.
>> What's five hundredths worth?
>> Five hundred.
>> Write it. I don't need them to write two hundred. I don't need them to write three hundred. Okay? I just need them to use that sum. How many tenths in the first number, everyone?
>> One.
>> Show me. How many tenths in the second number?
>> Three.
>> Show me. How many tenths altogether?
>> Four.
>> What's four tenths worth everyone?
>> Write it. How many ones in the first number?
>> Forty.
>> Show me. How many ones in the second number?
>> Three.
>> Five.
>> Show me. How many ones altogether?
>> Eight.
>> What's eight ones worth?
>> Eight.
>> Write it. We've got partial sums and we're in expanded form, so what operation do we put between those numbers, everyone?
>> Addition. Put in your plus signs. Five hundred plus forty plus eight is expanded form for what number, everyone?
>> Addition.
>> Five hundred forty-eight.
>> Perfect the first time. I'm done. I feel like I got a pay raise after that. And we're done. We can quit early. Here's what's going on in the algorithm. If you wanted to line it up from left to
right and keep everything in line, you start with your five hundredths. So 200 and 300 is 500. They'd write the five and then they fill the rest in with zeros because it's in the hundredths place.
One tenth and three tenths is four tenths, so they're writing four tenths, 4-0. Three ones and five ones is eight ones. Okay? Now, they've got their expanded form written vertically. It's still the
same concept. It's just how they're arranging those numbers. So I think that this arrangement really lends itself well to working into the abstract nicely because now they've already practiced
arranging their numbers this way too. And you can just kind of remediate the blocks--remove the blocks I should say. Remove the block and you've still got the same thought process. And I want to draw
you in on that whether we've got Allen with blocks or I'm working with you with presentations. Has my language changed? Everybody, has my language changed?
>> No.
>> If my languages, answers changes. Answer had changed, and language changed, through processes have changed. The key to this sequence is whatever you're doing in the abstract, it has to be the same
language for the concrete and for the representation. Because if the block language is different than the picture language is different in dealing with the numbers language, then you have just
created three separate things that you're trying to teach this child and that's confusion. It has to be consistent. Now I've been talking left to right. If I'm doing right to left, I'd change my
talking. I stay consistent with that talking, okay? You could teach whatever algorithms you're supposed to teach and each ever ones you think are going to be benefit your kids the most because
they're going to work, but make sure your language is consistent across those. I'm teaching you, so I change things up a lot. I want you to see different ways. Like remember before I was saying,
"Keep a group of five," and then--and now I'm grouping my five this way because I want you to see there are different ways to show five. The point is, which way are you going to use with your kids
and stay consistent with that. Okay? We're not going to do this one because you guys are moving really quick. So I'd just take you through it and I'll do all the drawing. How many hundredths? How
many hundredths?
>> Four.
>> How many hundredths in the second number?
>> Four hundredths and two hundredths are how many hundredths?
>> Two.
>> Six hundredths has a value of?
>> Six.
>> Six hundred.
>> How many tenths in the first number?
>> Two.
>> How many tenths in the second number?
>> One.
>> How many tenths altogether?
>> Three.
>> Three tenths has a value of?
>> Thirty.
>> How many ones in the first number?
>> Five.
>> How many ones in the second number?
>> Four.
>> How many ones altogether?
>> Nine ones has a value of?
>> Five.
>> Nine.
>> Three hundred plus sixty plus nine is what number--I'm sorry. Six hundred plus thirty plus nine is what number, everyone?
>> Six hundred thirty-nine.
>> Do you see how quick that is? That's the exact same thinking the child has to do in the algorithm. I have to identify what's in the hundredths, it's four and two, and four and two is six and that's
in the hundredths, so it's six hundredths. That's the meaning of place value. When I get a six in that third spot, it's worth six hundred. Okay? And that language is consistent from left to right or
right to left. All I have to do is switch up that order. Let's work on this one together. Two hundred eighty-three plus one hundred seventy nine. Two eighty-three plus one hundred seventy-nine. First
number, everyone, how many hundredths?
>> Two.
>> Show me. Second number, how many hundredths?
>> Show me. How many hundredths altogether?
>> One.
>> Three.
>> Three hundredths is worth?
>> Three hundred.
>> Write it. First number, how many tenths?
>> Show me. Second number, how many tenths?
>> Eight.
>> Show me. How many tenths altogether?
>> Seven.
>> Fifteen. What's fifteen tenths worth?
>> Fifteen.
>> One hundred and fifty.
>> You changed your language on me. See? We're brainwashed. We have to be careful. These are some of the biggest arguments that Allen and I get into is we're trying to script stuff out for everyone
because we're brainwashed. We're used to saying fifteen tenths that's one hundred and fifty. Fifteen tenths literally [inaudible] fifteen tenths, 15-0, just like five tenths is 5-0. Seven tenths is
six--or 7-0. Okay? Fifteen tenths is written is 15-0. I would keep that. If you switch fifteen tenths to one hundred and fifty, what you're telling the kid is all of a sudden fifteen tenths, I'm
supposed to call something different. We know they're equivalent, but we have to be prepared to model that when we're ready for it. Okay? So right now, we're going to say fifteen tenths is worth
what? And you're going to say fifteen tenths, Fifteen tenths is worth what?
>> Fifteen tenths.
>> Write it and you would write 150. How many ones in the first number?
>> Three.
>> Show me. How many ones in the second number?
>> Show me. How many ones altogether?
>> Nine.
>> Twelve.
>> Weird language, right? Because now it's 12. There are twelve ones. That time the language didn't have to change for us. You see how sometimes with the language number it changes and sometimes it
doesn't. We have to make sure that we're focused on what we're actually meaning for the child to write. So here is what's going to happen next, right? You don't have to do this. I just wanted to show
it to you. The issue is we've got sets of tens. We've got a set of ten ones and we've a set of ten tenths and we need to account for that work. So when I--the question is, can I take ten ones and
exchange it for a ten? And the answer here is obviously yes.
>> Yes.
>> Because I've circled it. I've circled ten of those. You actually saw my movement too. Okay? So if I exchange those, I'm going to get one more ten. Now remember, we want the kid to scribble that out
or do something to remember that wasn't there. We wouldn't necessarily want them to erase because now they're getting rid of their history. Now I've got one more ten. So the next question is, do I
have ten tenths that I can make a hundred? And the answer, everyone, is?
>> Yes.
>> Yes.
>> That's why I like my groups of five. It's easy for me to circle two groups of five. Ten tenths are equal to one hundred. So I can give myself another hundred block. Now I've got four hundredths,
six tenths, two ones. Right away you're probably already going, "Well, that's 462." Here's what's happened in terms of our numbers. What we've just told the child is decompose fifteen tenths to one
hundred plus fifty. That was the language you even gave me upfront. Fifteen tenths is equal to one hundred plus fifty. Twelve ones is ten and two or one ten, two ones. So our language sometimes
intuitively does this decomposition. We just have to be careful of what language we're using and is it modeling what we want it to model. If I want the child thinking that there's fifteen tenths
there, I'm going to have them write fifteen tenths. If I want the child to think 150, I want them to write 150, okay? What's that look like when we do our partial sums? Here is your justification for
when you write twelve, why the ones place or the one in the twelve ends up in the tenth column because ten ones is worth one tenth. So now that one is in the tenths column. It helps to see that
that right to left algorithm if that's your ultimate goal is to get the right left algorithm. If that's where you're starting, then you wouldn't necessarily do this with kids but you can see how the
modeling is still going to fit that mold. I'm not going to ask you to write this one. I just want to show you a little bit different way to handle the problem. Because remember, we're giving you a
basis. We're giving you a starting point. You're going to be the one that has to figure out exactly how I wanted to model those problems so that I can keep my language consistent. First problem has
it up. Four tenths, five tenths, six ones, five ones, now I go back through. I have--do I have ten ones that I could put together and make a ten? Absolutely. I'm going to do the exchange. Do I have
ten tenths that I can put together to make the hundred? Yes, I'm going to do the exchange. I haven't even thought abstractly in this version. Now I've got three hundredths, no tenths, and one ones.
And then you'll get crazy stuff like three hundred none-ty one as they're trying to figure out the middle. And it opens up that conversation for you about the language of numbers. No, it's not three
hundred none-ty one even though it looks that way. It makes sense that way to us. What do we actually have here? Three hundredths, one, three hundred one. The manipulatives, the drawings are
reinforcing our language of 301. Place value cards come back into play here. The child's still [inaudible] they're still struggling with it and you're trying to find another way. You get them to the
modeling point, the three blocks and the one circle. So pull that from your place value cards is three hundred and it's one, compose that, three hundred and one. The language is still up top, 301.
Okay? So you start to see how these things are fitting together. Different ways that we can show them. And this one, this was a right to left model as we go, all right? Or carry as we go. Here's
place value cards. So you've seen partial sums, okay? One hundred twenty-three, you could do this with addition, one hundred, twenty, and three, I build. Two hundred, thirty, nine, I built. I
decompose. I focus in on the cards, my blue cards, my hundredths cards. One hundredths and two hundredths is three hundredths. I introduce that new card. Two tenths and three tenths is five tenths. I
introduce that new card. Three ones and nine ones is twelve ones. "Wait, Mr. Campbell, you lied to me. I don't have a twelve ones card." No, you don't have a twelve ones card. So how can you show me
twelve? One tenth, two, right? That's how we showed twelve, one tenth, two ones. Now watch what happen since you've modeled it that, you can put those cards together. Now it understands again another
way to reinforce why we're carrying or why we're regrouping based on place value. What's five tenths and one tenth? Six tenths. Recompose, 362. Here are the child making sense very abstractly, but
it's a way for them to engage in the abstract. We can remediate some of that writing and help reinforce that expanded form. Okay? So place value cards aren't just for place value. Those place value
cards can be used to help support some of the abstract stuff that's going on. If you start here with kids though, they're still not going to understand necessarily what those cards mean. The language
could still be an issue. So a lot of the concrete and the representation along the way is going to help us clean that up. Here's what the math looks like. If you join Wiggio, you get on the iTunes
course, these are accessible right there. They can download them and print them. You're going to see my language from what I used with you is a little bit different than the language on the card.
Remember, this is the starting point and you're going to have to modify. I like to be very, very direct with my questions. How many in the first one? How many in the second one? How many altogether?
What's the sum? That's four questions. Here, you're going to see it too. How many hundredths does each number have? How many hundredths does the sum have? Okay? That's going to depend on the kids
that you're working with, how explicit you want to be with that. Again, this is left to right, okay? So you're going to see that abstract. You just have to make sure that the language from what the
abstract works looks like is going to be consistent whether they're using the blocks or the pictures. Cool?
>> Okay. Remember, the goal isn't there would be turn around and go do this. Really, for me, the goal is for you to see how this is a great foundation to start making all those accessibility choices
>> Cool.
come to life for students. So you've got a new way to engage them in math besides just looking and memorizing digits and facts and things like that. I'm just going to show you subtraction. Here's
what it looks like it. 312 minus 261, so I model the 312 just like I did before and then I start removing things. Addition was putting together, adding blocks on, drawing more pictures. Subtraction
is the removal process. I've got two ones and I want to subtract one one. Can I remove one of those two without using negatives? And the answer would be yes, I can remove it. So I want to stop for a
second. Everybody says, "Why are you talking about negatives? These are third graders, all right? Or our kids don't get exposed to negative numbers enough." And the rest of the world, they do because
of temperature in Celsius, anytime it hits freezing, they've got a negative number. So from the time they're children, they're inundated with negative numbers at different points in the year. It's
not like a magical thing that all of a sudden they hit middle school and now they can handle negative numbers. I'm not telling you to work with negative numbers. The reason I use that language is
because I need to be correct in my language as much as possible. So what I don't want to say is, "Can I do two minus one?" Yes. And then my next one is, "Can I do one minus six?" And you'd say no. "I
can do one minus six. It's negative five." So what we're doing is we're trying to clean up that ambiguity along the way. It might not make sense to the kid right away what you're saying about
negatives, but when you get to negatives, it will make sense why you have been talking about negatives. Instead of--and you've all been in here, how many people were lied to in math class, you were
told, "Don't--you can't do that." And then all of a sudden one day they said, "Oh, you can do that. You just weren't ready for it." And you're going, "How many other rules have you people lied about
and I don't even know whether or not it works? How long can I rely on this sort of math problem? So it all falls apart. So that's why we use that negative language, okay? That's a decision that
you're going to have to make with your kids, but we recommend it. So the question then is, "I need to subtract six tenths from my one ten, can I do that without using negatives?" And the answer would
be no. I can't take six away from one unless I could represent negatives. So now I'm going to have to regroup. I'm going to have to borrow, do an exchange. Where am I going to borrow from? If I need
tenths, I'm going to take them from my...
>> Hundredths.
>> ...hundredths. So one hundred is worth how many tenths?
>> Ten.
>> And make that switch. One hundred is worth ten tenths. A great equivalency proof, stack them up on top of the block, same area. Now what I've done, as I've said, if I borrow one of my hundredths,
there would be two hundredths left and my one ten just went to eleven tenths. If you've ever had children confused over here with what was going one, this make sense. Now I understand why I have
eleven tenths. Now, eleven-tenths, remove six of them, I can see that I'm left with five, two--or two hundredths, remove two hundredths, I have no hundredths left. My answer therefore is 51. It's the
same thing with the model--or the representation. We're drawing the same figures. The only thing that's going to change is what we're doing. I scribble out my 100. Draw myself ten more tenths and
then I scribble at my one. I scribble out my six tenths. I scribble out my two twos and this what you're left with. A little bit messy when you're looking at student work. So it's really great to be
able to see the kids doing that along the way and hear them modeling that thought process out loud. If I get rid of all that, you could still see those five tenths and one. Five tenths is worth,
everyone?
>> Five tenths.
>> Five tenths is--has a value of?
>> Fifty. One one has a value of?
>> Fifty.
>> One.
>> Plus one is?
>> Expanded form, all right? It's still expanded form because the expanded form is what sets up using place value to perform multi-digit operations. Here's the mat. So here are some questions that
>> Expanded form.
we've already gotten and we want to address these upfront. We promise you that we would do that. The other thing that I want you to know is that if you get on Wiggio, you can post questions there
with us as well and not just us. We are not the givers of all information. I want to be honest with you. I love PaTTAN. I love my job. I love working with teachers. If I had all these answers, I
would be a millionaire. I would still be working with teachers. They would pay me a lot more to work with teachers. So what I'm trying to say is we're giving this base of information for us to build
off of collectively, okay? It's a collective knowledge base. It's going to help move the field forward. Not what Allen and Gerard are standing up here talking about. All right? The other thing is, in
Wiggio, don't feel that you need to; A, only ask questions or B, have all the answers. It is a community group. So while Allen and I and other consultants will try to stay in engaged with that and
provide some things to think about, you can answer each other's questions as well and feel free to do that. That's what we want. We want this group to survive in Wiggio. Now for the PaTTAN Math, it
is not specific to this group. Our PaTTAN math Wiggio is going to be across the state. So we work with teachers of students with autism. We work in the gen ed setting. We work across the multi-tiered
system of supports. We preach the same thing in terms of good instruction. How that good instruction looks is a little bit different depending on the students that you're working with, but CRA is a
type of instruction that should be able to support the kid from the individual setting getting access to Core Curriculum and being involved in Core Curriculum as much as possible in the gen ed
setting. Those things should all look the same, okay? So one thing that has popped up is people already part of Wiggio and they're joining or they're going into their Wiggio accounts and then they're
trying to use the Pattanmath.wiggio.com and they're getting messages like this content doesn't exist, something like that is happening. I didn't know that would happen if you're already part of it,
but what I can tell you is that if you are already a member of Wiggio in the upper right-hand corner, there's going to be a little tab that says "Join a group."
>> Yeah.
>> If you click "Join a group," it will say, "What's the group name? What's the password?" You already have the password. It's PTN math and our group name is just PaTTAN Math. So if you just do "Join
a group" and then type in PaTTAN math and then the password PTN Math, that will get you onto that group if you're having trouble with Wiggio. And if you do try to join today and you can't get on, you
can talk to us before you leave or you'll have our email addresses and things. You can always email us if you have some more questions with the Wiggio site. Okay. Next question? The question came up
and I might unintentionally answer some of the other ones. Accessibility issues. So remember what we're trying to do today. I have to and Allen has to, we have to get a common knowledge of content
what this instruction looks like with this content. If we can do that, then we can get kid specific. We can start to look at accessibility options. We can start to troubleshoot. If we start there, it
will be devoid of content and we could lose the original intent of the content, right? Like all of a sudden it becomes putting blocks in front of the kids and that helps or putting a calculator in
front of the kid and that's supposed to solve everything. So we have to get a common understanding of what the content looks like and then we can start to look at the issues. One issue already I can
say is core vocab. We have students that have limited language capabilities and they utilize core vocabulary. We have not addressed that. The answer is, no. We have not and we are not prepared today
to address that, but here is what I can tell you. We have a speech and language pathologist that works with core vocabulary and works with assistive technology and she is getting ready to start
looking at the language that we're creating consistent, when we get that consistent language, now she can look at it as a speech path and say, "What's the core vocabulary here and what would this
language look like for a student on an assistive device?" Okay? So those are the things we're thinking about. Motor skills, another issue. My student can't draw. My student can't arrange blocks.
These--you know, this becomes a distraction. I can't scribble out because they will not allow mess on their paper. Those are things we have to move along with as we look at--once we have the content,
how do we make it more accessible for kids? If you're--my immediate response would be, if your kid can't select from the group of objects, maybe you're giving them the objects they need and they're
arranging them, okay? Maybe you're practicing the language together. Maybe they're not a--they're nonverbal. They're not able to respond. Is there something they can do with eye gaze? How do they
communicate and take the way that they communicate and use that to check for understanding with this content, okay? The--and I know not everybody in here is with Project MAX, but I know a lot of
people are. From my understanding, Project MAX is about studying the individual child and really making content accessible to that kid and that what I want you to see is this common foundation that
we're laying right now should set the stage to continue work in your Project MAX work, okay? So that now you have--you know what math instruction could look like or you've seen some ways math
instruction could look and you think, "Okay. I think that would work or I think these pieces would work and this is what it would have to look like for my student because these are some of the
barriers that I need to overcome." Since the IEP goals are supposed to be written at grade level, how do we use a third grade level standard for an eleventh grader? We need to look at the
instructional level of the student as well as the grade level content. You're going to see an example of lessons that we've done in life skills classrooms, a life skills classroom in particular that
you're going to get to see this afternoon. I'm sorry, this morning. And what you'll see or what you should know about the background is these students were being assessed at a second and third grade
computation level. They're sitting in middle school. They're trying to get ready for high school and the teacher is freaking out because now all of a sudden they're going to have Algebra and she's
testing them on a second grade, third grade level for computation. How am I going to do that? Okay? One of the things that we did with her was reduce the complexity of the numbers so that we can
introduce other complexity. Meaning, we quit looking at three-digit and three-digit addition and long algorithms that kids were getting lost in and we started saying, "What are your kids capable of
doing? Where are they comfortable?" And she said, "They're really good with small numbers. They're solid there." "Okay. Let's take those small numbers. We're going to work with some middle school
content which would be integers but we're going to restrict it, negative 10 to positive 10, okay?" So because we didn't just blow the doors off integers and try to do all this computation with
integers, we kept the numbers are smaller based on the strengths of the kids. We were able to hit grade level content with some reduced breadth and depth of that complexity. For a student with a
severe disability, if they're never able to get to abstract, is it still worth teaching? Here's--I guess this is Jared's answer. If you don't do this, are they going to get to abstract any other way?
If you can get them farther to abstract or closer to abstract this way, then I would say, "Absolutely, it's worth doing it." Does every child everywhere need to be in the abstract? That's not a
question that I can answer. That's a question based on what kind of future goals do you have for your individual students. I will say for myself as an adult, there are very few times where I have to
compute algorithms in my head or on paper somewhere. I do a lot of estimation. And if I need to be perfect, I'd pull out a cell phone or other people are doing computations on machines for me and I
need to have in head that they ring me up right or did I--am I overpaying for something? Or if I'm purchasing items, I need to be able to do some basic summations and some estimation now. So those
are the things that I think become more important as we generalize this into this abstractness. If you don't fully get to the abstract algorithm, what I'll say is at least the kid is thinking of the
algorithm. The language doesn't change. The thinking doesn't change. So if the kid only ever deals with concrete or only ever gets to the representation, they're doing the same exact thinking as any
child that only ever does the abstract. We have been taught that abstract is the best. It's not. I love Brad Witzel who's done a lot of research behind this. I admire the guy. We have the same
wristwatch. That was by chance, but I like to point it out. It makes me feel a little bit cooler. He said one time, "You know what, if you get kids that get stuck in the concrete and the
representational, congratulations. You've just created an engineer." And I think that's kind of powerful because what happens is we--in Math, as general and math teachers, we are so abstract all the
time and then it comes to the practical nature of things and how to build things and construct them and organize them and that is just lost on students. And now it's like a whole another discipline
that we're trying to teach them. So there is a lot of power in understanding how things fit together in the arrangement of items and the thinking is still consistent. So in class, kids can be working
on the same standard, some that's concrete, some that's representational, some that's abstract. Is it still the standard? Absolutely. Standards do not tell you how the kid is being taught the
content. They tell you when that content is to be mastered. So if we're talking about accessing grade level material and it's saying you're doing multi-digit computation, it's not going to say right
to left algorithm, left to right algorithm, partial sums, carry as you go, use blocks, use picture. It's not going to tell you any of that. It's going to say, "We want the kid doing multi-digit
computation." And so you need to help the kids do multi-digit computation. I'll tell you that if the language is consistent and you've got a group of three kids, two of them could be doing blocks and
one could be drawing pictures and there's not going to be any issue because your language is the same. Or one kid's doing blocks, two kids are abstract, not going to change. The language is
consistent and the thinking is consistent. You can have kids working on all different levels. Now it's a management issue, so we have to deal with that. So you may want to think about that in your
classrooms as well, but it can be done. The instructional piece is not going to be a barrier to kids working at different levels. It's how we manage that instruction. Can you go from concrete
to--or--yeah, to abstract without the representational? It's a question that Allen and I asked ourselves a lot. My answer for you right now is no. I would not skip the representation. The child does
not going to have place value blocks in their back pocket for the rest of their life. So if they take an assessment, are they going to need blocks for that assessment? I mean, if they're still
concrete then maybe but should our goal be concrete to abstract? No. Our goal should be able that kids can draw models for themselves because I can do that on paper wherever I'm at. If I'm taking a
test, I can draw my models on the side, if it's PSSA and I have to write something in there that says explain, I can explain based from that picture I've drawn. I can explain on PSSA as easily if I
have blocks that no longer exist and the reader can't see them. That type of reasoning is the same on paper even with pass, right? Kids have to do with things with pass and explain them. They're
still going to be explaining them, they're just going to be videotaped in the process so we still want to work with those explanations and the verbalizations as much as possible or however students
communicate. Are there times where you might be able to move quicker? Absolutely. So if you're sitting there and you're going, "This kid really gets the block stuff like they're just moving." However
you made that accessible to kids. They're flying through and you're going, "I could probably skip the representation." I would say, don't, do the representation and then just be prepared to move them
at much quicker to the abstract because when kids get it, we want to move them along. We don't want to spend six days doing concrete, six days doing representations, six days in the abstract. We want
to move kids across as they're ready to move across and there's going to be back and forth fluidity because you know one day they're going to get it, the next day they're not, the next day they're
going to have to get it, the next day they'll be in between as we--as we get kids to understand the concepts that are going on. So I would say move them faster through that progression but I wouldn't
really skip steps because if you skip the step, now the kids in the abstract in struggles, what do they have to do? They have to go find place value blocks, what if your place value blocks are being
used up by the other kids at that point? Here is a kid that didn't need them, it could've transition off for them but now they're eating up materials for the other kids or they have to walk across
the room and get the blocks and now it's an issue of behavior concerns as they cross the room and, you know, mess with other kids in the room. So anything that you can do to get kids off the blocks
into the representation before the abstract is going to be beneficial, that doesn't mean that you won't may be have some kids somewhere that you need to move them from concrete to abstract because of
maybe some accessibility things going on with the representation. You see how that works for your kids. You're going to be the one that ultimately makes that decision. I just want to caution you, be
careful not to do that, okay? I will tell you in some of the step we do this afternoon, we've looked at the drawings and said the drawings are so cumbersome. We're not sure if they're really worth it
and maybe you can go straight from concrete to abstract but we don't have an answer to that yet and so our--it's just hesitation. I wouldn't necessarily do that. Wouldn't plan on burying it, speed up
the progression if you feel that you could skip it. All right. Thank you.
>> Just a quick question. You've talked about consistency of language, how important then is it as a teacher to say 347 versus 342 is [inaudible]
>> Yeah. I--I'm going to answer this in 30 seconds because we're already down to an hour. We say as teachers and as students we have been taught only say and with the decimal place, one hundred and
four-tenths. We have to research this but we're not sure why that is because if I said, "What's three and two?" You would say, "Addition." How did you know to add? Because I said three and two, so it
seems to me that and is inherently talking about addition. Okay. Let's think expanded form. One hundred and thirty and two is--everybody?
>> One thirty two.
>> It's addition, right? One hundred thirty-two. One hundred and thirty and two. Why don't we put and there between every single place value because it's cumbersome and messes kids up. And then when
we get to the decimal places. I mean even if we say two and three tenths it's implied two plus the fraction three tenths. That's what it is, two plus three tenths. It's a mixed number even with the
decimal. So I'm not sure why we say--only say and with a decimal other than it probably created issues when kids start to say and all over the place. And so as for teaching efficiency we probably
narrow that down and say just use it with the decimal and let's forget about it everywhere else. I don't really have an answer for that though because we haven't research it. But that's kind of what
we're thinking.
>> One other thing that we do a lot is--we call it an O instead of a zero. It's 6:08. That--there's inconsistency there as well.
>> Yeah. Absolutely. Okay. Integers, number line, operations, here's what happens. This whole thing at the beginning is setup first in the number lines and then we'll move up with some more of a CRA
type model with this. This was work that was done in the middle school classroom. We'll show you the students actually doing it. You'll get to see the teacher instructing it. But again, we're going
to engage you in this instruction as well. So what pops up and you'll see this again later this afternoon is to look at working with rational numbers, okay? Operations with rational numbers, so that
includes addition and subtraction of negatives and that's what we're talking about right now. We're going to see a pop up here potentially with the positives, and negatives in that draft Eligible
Content. So when we look at some of those goals and some of those objectives, we're talking about real world situations using prompts, representing things in a bunch of different ways for students
and this is what we did. This is a grade 6th through 8th life skills classroom. We win it with the teachers said where your kids being assessed at. What are your scores telling you? She said it's
basically second and third grade computation for middle school students. Some of them maybe a little bit better in problem solving but mostly it's second and third grade computation. So we looked
out, okay. But these are middle school students. So we want to think middle school content and it was second and third grade computation. She's going, "They're probably stuck there because the
algorithms are longer and they're getting confused along the way." She said, "Because what I've notice in my classroom informatively assesses are pretty good with smaller digits. Smaller digits are
fine. Algorithms are the issue." So we said let's tackle some integers, grade level content but we're going to restrict them from negative 10 to positive 10. The first thing we had to do then was
structure the language exactly what language are we going to use with them so that when we model on a number line that is not going to change whether they're walking the number line, whether they're
putting their finger on the number line it's not going to change. It's going to be the same thought process along the way. And the importance of the number line is because when we get to Algebra we
use two of them and now it's a Cartesian Plane, it's a grid system. So the original intent was to model the instructions for the teacher and then she would takeover. What happened was I did one
problem and then she kicked me off and taught her own kids and I was really upset by it. And then there was some data collection that went on as well, and we'll show you what that look like. But we
really want to get you into this content. So here's a summary of the teachers plan. The teachers plan was to say, okay. What am I going to do to get kids to be able to do three plus negative five?
That's difficult. How am I going to explain this to kids? How are we going to model it? And what she said is--what we decided we're going to get sidewalk chalk. We're going to go outside. We're going
to draw all these things on the parking lot and we're going to have kids walking. We're going to the keep the language the same and they're physically got to move themselves, and work with the
partner to do that. Then in the classroom she had lizards. Today, you'll see some little bears on your table. You're going to use some little bears. She chose a lizard because they had them and the
kids like them, fine. Here's the important part. The lizard and the bear both have a head because they do that, it fits our language which included facing certain directions or focusing your
attention on a certain direction. So if you don't have something with a head then you can't say face that direction because which way is it facing, right? We have to be able to keep track of that for
kids. So she was going to start then in the classroom with the number line, and with lizards, kids following the directions. Okay? Then kids starting to give each other directions, kids giving their
lizard the directions, children do all those things to really try to solidify that language with students. Then she was going to try to remove the lizard and just go to a finger and the kid give
their finger directions, or can they follow directions with their finger? Now, that we solidified some of that which direction are we facing and how are we moving. And then if we can do that we've
got computation down for addition and subtraction from negative 10 to positive 10. You can start to use that to generalize rules but typically what we do with integers is we teach the rules and
expect kids to memorize them and they're not sure why they're getting the answers they're getting. So this way making connections to the number line is going to support them not only with the work
with integers. It's going to support them into Algebra as they continue moving forward in their careers. So you have some white paper or you can do this on your notes wherever you're writing. I need
you to do a number line from negative 10 to positive 10. I ran out of room so I just put the negative 10 on one side and the positive 10 on the other. But I need you to draw one of those for me.
You're going to be using that. Do you want to work in the back? And I need two volunteers.
>> Easy. Easy.
>> And you're going to stay up here with me. You can actually--yup. You can actually stand at zero. Perfect. You want to go back with Allen? Thank you. Good. Some negative signs with the one, two,
three. Okay. You may or may not have been able to see this depending on where you're sitting. But in the back of the room and up front, we've laid index cards down to cut and mimic a number line in
here because they would have been really upset if I came in and drew on their carpet. So you're going to see the language and you're going to get to see two of our wonderful assistants will be
modeling this language and what the kids were physically doing with it, okay? You don't have to do anything yet but if you want to pick up a bear and do it with this you can. You are going to have
additional time to use your number line and a bear to practice these movements with this language, all right? So here's what it looked like with students. If you want to watch the--what are our
names, ma'am?
>> Cathy.
>> Cathy and?
>> Megan.
>> Cathy and Megan, okay? So Cathy is in the back, Megan is up front. Here's what I want you to know. Negative 10 is on this side, positive 10 is near the wall. Right now, Cathy and Megan--here's our
language. I want Cathy and Megan to stand on zero and face the positive direction. The first piece of language orients the student and we have to do that on the number line every time. It's that
starting point whether they're facing their bear or they're standing themselves. You can see our individual stand on zero facing positive. Okay. So the next thing is we have to determine based on the
first number where you're going to move. You're going to see absolute value written here because the sign is going to indicate whether they walk forwards or walk backwards. Okay. So in this, four
plus three, the first number is four. Everyone move four numbers forwards. You're going to see them walking one, two, three, four. And right now ladies, what number are you at?
>> Four.
>> Four.
>> Four. It's always going to work like that, right? Because the first thing we have to do is locate that first number on the number line. This sign says to add so everyone face the positive
direction. This is the positive direction. We're facing positive 10, right? This direction is the what direction?
>> Negative.
>> Negative. This direction is the...
>> Positive. Because it's addition, they have to be facing the positive direction. The second number is a three. So everyone move three steps forwards. Positive moves forwards. What number are you
>> Positive.
standing on?
>> Seven.
>> And then you'll see the students writing, you'll see them carrying white boards. We'd say, so four plus three is seven. Everybody say it with me.
>> Four plus three is seven.
>> Four plus three is seven. I modeled it with the number line. I had them move to the first number. I reset for the operation and then I created that second transition for the second number. Okay?
You're going to get a chance to practice this. Let's reset. Ladies, stand on zero, face positive. Here's three minus two. The first number is three. Everybody move three numbers forwards. The sign
says to subtract. Everybody face the negative direction. See the change? Now, watch this. The second number is a two. Everybody move two steps forwards. Notice their forward, and this time it's
taking them in the negative direction. What number are you on, ladies?
>> One.
>> So three minus two is one. Everybody say it with me.
>> Three minus two is one. We've modeled it. The addition subtraction is this change the positives and negatives will indicate whether you're walking forwards and backwards. Let's reset. Everybody on
>> Three minus two is one.
zero, face positive. Same set of directions every time, right? The only thing that's changing is based on this positives and negatives. So first number is six. Everybody move six forward. It sign
says to add. Everybody face positive. The second number is negative two. Everybody move two steps backwards. What number are you on?
>> Four. Six plus negative two is equal to--everyone?
>> Four.
>> Four. We're not doing this but this is a great time to say, why is adding a negative the same as subtracting a positive? Because six minus a positive two still takes me back to four whether I turn
>> Four.
and walk forwards or I just walk backwards. So again we've got some proof in here. Everybody stand on zero, face positive. I know we're moving quick. You're going to get time to practice this too. If
you want to use your finger or your bears along the way you can do that. Five minus negative three. Okay. Everybody stand on zero, face positive. The first number is five. Everybody move five numbers
forwards. The sign says to subtract. So everybody face the negative direction. The second number is negative three. So everybody move three steps backwards. What number are you standing on?
>> Eight.
>> Eight. Five minus negative three--everybody is...
>> Eight.
>> Why is it the same as five plus three? Because I would move five plus three more would be three steps forward. So we can start to show equivalency and expressions this way as well. Let's reset.
Everybody on zero, face positive. Here's another option. We've got a negative plus a positive. Negative three plus two. First number is negative three. Everybody move three steps backwards. What
number are you standing on right now?
>> Negative three. That was just a little check from me. The sign says to add. Face positive. That was tricky. We're waiting for kids to turn every time we ask the question. They did really well with
>> Negative three.
that. The second number is two. Move two steps forwards. What number are you standing on?
>> Negative one.
>> Negative one. Negative three plus two is one or negative one. So everybody negative three plus two is...
>> Negative one.
>> Negative one. Ma'am?
>> Are you all--when you reset and you're on zero, are you always facing positive? Always?
>> Positive. Yup. Always. It's in the language of the...
>> When it's a negative you're stepping backwards?
>> Negatives walk backwards. Positives walk forwards. Addition I face positives. Subtraction I face negative. So those are the rules for the kids as they learn. You see why it's important to do a lot
of these walking and then transition to them with an object with a face on it. It's going to be them following directions than them giving directions and responding to those directions. So there's a
lot of different steps as a teacher is looking at to try to build independence here. But it was a consistent model in terms of the language for kids. Negative one minus two. Everybody stand on zero.
Face positive. First number is negative one. Move one step backwards. Sign says to subtract. Face negative. Second number is two. So take two steps forwards. I'm sorry. Move two numbers forwards.
See, I said steps. We originally we're talking steps because we thought every kids would step on a number what we didn't think about was every kids has different size of legs. There are some kids who
are going to take big steps and we had to focus kids on the numbers which was good because that numbers are what they're focusing on, on a number line when they're transitioning as well. What number
are we at?
>> Negative three.
>> Negative three. So negative one minus two is equal to negative right three and we will repeat that process. Okay. We're not going to do the last two. Here's what it looks like. Thank you, ladies.
You can sit down. Weren't they great? Volunteering for math work before noon. Negative two plus negative three, right? Negative two plus--I'm facing forward, negative three, three more back. I'm at
negative five. Negative five minus four, negative five minus--I change direction, negative four. I'm walking backwards, I end up at negative one. We can start to use this to develop the rules. So
kids under see why adding a positive or subtracting a negative the same as adding a positive and adding a negative is the same as subtracting a positive--all that stuff that we always get confused
with we can model them on the number line and if we never remember the rules we can work through the number line as a representation to do the computation here. Okay. Are you ready to do some
integers practice with your bears? Grab a bear.
>> [indistinct chatter]
>> All right. Here we go. First step everyone. Your bear needs to be all in...
>> Zero. And should be facing--which direction?
>> Zero.
>> The positive direction. Make sure your bear is on zero, facing the positive direction. The problem is negative two plus three. The first number is negative two. So we're going to move our bear two
>> Positive.
numbers backwards. Right now the bear should be at what number?
>> Negative two.
>> Negative two.
>> Facing positive.
>> Facing positive. So the sign says to add. So we want to make sure our bear is facing positive, so no change there. And now the second number is three. So everybody move your bear three numbers
forwards. What number is the bear at?
>> One.
>> One. So negative two plus three is equal to one, and you repeat with me. Ready?
>>Negative two plus three equals one.
>> Good.
>> Yes.
>> Reset your bear. It's on zero facing positive. The first number is five because it's positive. Everybody move your bear five numbers forwards. What number is your bear on?
>> Five.
>> Here's what's going to happen. Eventually kids are going to start going why don't I just start at the first number? Okay. Right? They're moving forward, it's counting on. What we've done with this
language is scripted to the point that it's one to one correspondent. So as kids are able to get passed that or kids are able to count on and count back, you can transition them quicker to that, and
make their process more efficient. The sign says subtract, so everybody, make sure your bear is facing the negative direction. And we're subtracting a negative two so move your bear two numbers
backwards. Make sure the bear is moving backwards away from his face. Think about yourself walking on that number line. What number are you at?
>> Seven.
>> Seven. So five minus negative two, when you turn his back in the--or turn him to face the negative direction, it move him backwards, you're continuing on in that positive direction. Okay? Here's
another one. Negative three minus negative four, everybody reset. Your bear should be on zero should be facing the positive direction. The first number is negative three so everybody move your bear
three numbers backwards. The sign says to subtract. Your bear should be facing the negative direction. We're subtracting a negative four. Move your bear four numbers backwards. Okay. Remember
backwards is according to where the bear is facing that's why it's so important that kids walk this so they can put themselves in that position of the bear and remember what they did. What number are
you at?
>> One.
>> Negative three minus negative four equals one. Okay? Question? Allen's got you. Okay. I'm going to move past the next couple of examples so we could keep moving on because we want to make sure you
get a chance to see the kids doing this. So negative three minus two again you move three back in subtraction you change direction and then you walk forwards too, and negative one plus three, you
move one back and then you face positive and move three forward. So what we're doing is we're setting up those directional changes on the number line so that we can then remove the object and the kid
can use the finger and I can count with my finger where I'm going. We can generalize some rules. Kids can create some more efficient strategies because they will get sick of moving bears and using
their fingers potentially. And if not, they have access to the number line. They're locating positives and negatives on the number line which is going to be a reinforcement for them when they get to
Algebra 1. All right. We got a couple minute video. I need you to tune into this. I'm not exactly sure where the volume is at so I may have to adjust up here. One of the efforts that we're making in
the iTunes course and or Wiggio in general is to collect more evidence of kids doing this. It's one thing to say that you could do this and you could make it accessible. It's another to see kids
doing this. This is a life skills middle school classroom. The classroom that we talked about where the planning happen, and the kids wrote a second and third grade computational level. You're going
to see them in the parking lot, you're going to see them working with pairs with whiteboards. They're working abstractly and they're modeling concretely. You're going to have to kind of listen past
some of the birds and some of the trucks in the background as there is noise outside.
[VIDEO BEGINS]
>> [indistinct chatter]
>> What's up? Everybody is on zero and you're facing positive. The first number is five. So everyone move five numbers forward.
>> We could see Allen chair there taking notes, they're transcribing some data for us.
>> Okay, great. The sign says two--make way, so everyone face the positive direction, positive direction, this way, all right? Now, the second number is negative two. Everyone move two numbers
backwards.
>> This is a count two numbers backwards facing on the back, right?
>> What number are you standing on?
>> Three.
>> So there was positive help yourself walking backward where there was no backward, turn around actually walk themselves and then reposition, and you let it all happen for them.
>> So we got five plus negative two equals three. Read it with me.
>> Five plus negative two equals three.
>> Five plus negative two equals three. Good job.
>> So that was one model in cognitive. She went [inaudible]
>> I just one positives and negative there's adding and subtracting.
>> I learned about the number line.
>> [inaudible]
>> Right, I learned some--I learned something about positives and negatives.
>> Keep and allocate to kind of to get the book and we'll have to do the math.
>> I learned subtraction and adding.
>> It's integers, it's math, I guess.
>> [indistinct chatter]
>> Thanks.
[VIDEO ENDS]
>> They are great. Weren't they great?
>> Those are my students.
>> Oh really? Isn't that great to see them? So again, this is their first day doing this. She did a little bit of a preview lesson trying to locate numbers in the number line, introduce the word
negative to them so they would check for the negative number but they had no integer instruction other than a little bit of vocabulary before this. We didn't do all addition, subtraction, all the
different combinations they got into adding positives and negatives and that what was where we finished up with them for that day. So there is some continuation of this but she said she was probably
taking them into the classroom for a while and let them start to work with the lizards and the manipulative is on paper before she introduced subtraction with positives and negatives. It's how she
decided to work it. I want to show you data that's collected and we want to get you into some other ways that we can model with integers. So what we did for these students was we took our language
and we scripted it and said what's the problem, what's the name of the student, and then were they able to stand on zero and face positive, locate the first number, deal with the operation, with the
directional shift. Second number, determine location and read this with us at the end. We wanted to check all these things. What we didn't want to look at was could the student complete it, right?
Let them worry about the answer. We want each individual think. So if they were having problems following directions along the way, we wouldn't be able to isolate where that happened and how are we
going to fix that across the bottom you'll see an X, a slash, and a blank. The X represent a complete independence from the student. A slash represented that they needed some kind of support and if
we left it blank, they weren't able to do it unless we had to say stand here, right? Like stand onto or walk with me. If it was a slash you might say, you know, somebody might have had to reiterate
like two steps backwards, guys or something like that if there were some prompting there. Here, this is all between the students when you see an X. Here is our data obviously names are out. Do you
see all the Xs on there? We were stunned at all the Xs on there, absolutely stunned. We took those, and coded them and said arbitrarily because I'm--love math, I'm not a statistician. We code them
and said a two will be the X, a one will be the slash, a zero will be the blank, a two then represents independence, a zero represents no independence and a one represents some independence. We coded
all that for every type of problem for every kid and then we looked at some stuff and said overall, across these positive plus and positive, positive minus positive, positive plus the negative and
negative plus a positive. How did our kids do if a zero, a one, and a two represented not able to complete, completed with prompting or completed with independence then the closer we get to two, the
more independence the kids were showing. There's our averages, overall and in each individual problems at. And we added up those averages to get kind of a big picture view. Is it good math, good
statistics, probably not but we wanted a way to collect some data in that classroom and see what was going on. And here's what we can say without a doubt. Kids were able to complete this task with a
high level of independence. We're not claiming they learned integers. We're not claiming they learned the number line. We're not claiming they mastered any content, it was the first day of
instruction but because of how we designed it, because of Mrs. Ferrari's work with those students, they were able to access and it's the first step forward to being able to prove that they did learn
it and they were doing it independently.
>> Okay, Jared. In your packets this afternoon, you're going to get a copy of the lesson plan, you're going to get a copy of the language sheet that we showed up there that we used with this lesson
and you're going to get a blank copy of the data collection form. So that will be in your packet that you have this afternoon.
>> You're ready to do some other work with Sharon in integers?
>> Uh-hmm.
>> That was just one example. That was using a model line or a number line as our model but we can also do it with manipulatives. This is the one I said I'm not sure if the representation is that
strong that it's cumbersome or not cumbersome. So you all want to think about that. We're going to work concretely with this one. So you have a bag with some pieces in it. You don't need to get them
out right now. I'm going to show it to you first. I'll give you a chance to pull them out. They're going to be two sided. I want to show you what it looks like and then I'm going to give you a chance
to get those out and practice them with me, okay? So here's the big issue with addition, subtraction and this is why this looks a little bit different at this point than those maps that we've created
it's because there's decisions that have to be made here. All those other things are pretty much a linear process. Any decision was yes or no and if it was no or yes, you kept moving down the line,
right? Here, your decision changes future questions. It's why integers are difficult for students because if I'm supposed to add I follow this set of procedure. If I'm supposed to subtract, I have to
follow this other set of procedure. So now we have two different things that we have to deal with. So what you're going to see is I'm going to say stuff like show me egg or what's the first number,
you'll model that. Decide whether you add or subtract. If you add, you have to introduce B which would be some number. You'll introduce the second number. You will move it in zero pairs which I'll
teach you about and then you'll find the right--the final answer and answer the question. If it was subtraction then the first question is can you remove what you're supposed to remove? If you can,
follow yes, good, take it away. What's the answer? If you can't, you're going to have to introduce zero pairs and again I'm going to talk to you about what zero pairs are. Then you can take out what
you need to take out. Okay. So here's what you need to know, one plus negative one is equal to zero, that's a zero pair. If I have a positive one and a negative one, they equal zero, they cancel out,
it has no impact on the problem anymore. So if you're modeling this and you don't have to right now. If you're modeling this with the blue and the red, blues, I have to find this positive, reds as
negatives. If you have the chips, the reds are going to be the negatives, the yellows will be the positives. If it's another two sided coin, you define those values for your kids, okay? It's your
choice. Stay consistent though. Don't change the model up on them. So if I want to do nine minus five, the first number is nine, how would I represent nine? Nine positive ones, so I show that. The
second--or the sign says to subtract so I'm supposed to remove something. What am I removing? Five. Do I have five there that I can remove? Yes, take them away. What am I left with? Four. Nine minus
five is four. Groundbreaking. It's important for us to always start there because as we get more complex with the numbers, the rules won't change. Negative five plus nine. Now, if you're with me
already, you're going to go, "Wait, Jared, that's the commutative property, you change the order of the numbers, right? You got a nine and a negative five and you just switch them around so we should
get the same answer right away. Let's prove it. The first number is negative five, how would I model that?
>> Five reds.
>> Five reds. The second number says to add in nine, how will I introduce nine?
>> Nine blues.
>> Nine blues. What have I created?
>> Zero pairs.
>> Zero pairs. Remove them, they're worth zero, what am I left with? Now it starts to feel more like magic, doesn't it? Like wait, you mean integers weren't that hard? Okay. It demonstrates the
commutative property, either way it's going to work, it's that same modeling throughout. Negative seven minus negative two, how do I model negative seven?
>> Seven reds. I'm supposed to take away two negatives, can I do that?
>> Seven reds.
>> Yes.
>> Take them away. The answer is?
>> Negative five. Negative seven plus two, how do I show negative seven?
>> Negative five.
>> Seven reds. I'm supposed to add in two positives.
>> Seven.
>> Two blue.
>> Two blue. What have I created? Zero pair. What's going to happen? Cancel out. What am I left with?
>> Five.
>> Why is adding and positive the same as subtracting a negative? We've shown it, we've proved it. Negative three plus negative four, how do I show negative three?
>> Three reds. I'm supposed to add in negative four, what am I going to do?
>> Three reds.
>> Put down four more reds. What do I have?
>> Put down four.
>> Negative seven.
>> Negative seven or seven reds which would translate to negative seven. Here is negative four minus three. Other variation, how do I show negative four?
>> Four reds.
>> Four reds. I'm supposed to take away or remove three, can I do that? I'm supposed to take away three, that three there is positive. The subtraction says take away, I'm taking away three. I can't do
that. I don't have three blues there to take away. So what can I do? I can put in zero pairs, great. I can introduce zero pairs when I need them, right? Just like when I couldn't do my subtraction
without using negatives? I could exchange things. I can put in as many values as zeros as I want, plus zero doesn't change, plus zero doesn't change, that's why I could eliminate them. I can put them
in too. Here's what happens. We've put in three zero pairs. That gives me the three that I need to take away. What am I left with?
>> Negative seven.
>> Negative seven. Okay. So I can introduce zero pairs along the way if I don't have what I'm supposed to subtract. What's this look like representationally? Probably plus seven or, you know, three
plus signs, three minus signs, you know, do some more plus signs, do some more minus signs, scratch those out when you get rid of them. Questions about adding and subtracting integers? I'm going to
give you about five minutes. Choose any of these. I want you to work at your table, talk with each other, we'll circulate, answer some questions, and then we'll kind of run through to make sure that
everybody's comfortable with these, okay? Try some of these problems.
>> [indistinct chatter]
>> We need to come back together quickly.
>> [indistinct chatter]
>> We've got about 15 minutes left and I want to get you out of here on time.
>> [indistinct chatter]
>> I'm a former high school math teacher. I will keep you here during lunch.
>> [indistinct chatter]
>> That was my mean face. You weren't supposed to laugh.
>> [indistinct chatter]
>> Kind of help, some people were like, "Yeah, he's clapping, I get it. Whatever, I'm finishing my stuff anyway." That's all right. Listen, first of all, I want to say, I was really happy to see as I
walked around every single person went to this stuff was having conversations about math, was testing this stuff out and I got into some conversations with people and heard some conversations
reflecting back on what we already do, how it marries to this, how it fits in core instruction, how it aligns across a tiered system of support, really, really good stuff. Thank you for engaging in
that. Because we have 15 minutes left, I'm gonna have to move a little bit quicker towards the end but I want to remind you, Wiggeo, CRA days. CRA days will get very, very specific, addition,
subtraction together. We'll spend half the day in addition, you'll get time to teach another adult in the room. Subtraction, you'll get time tomorrow and then teach another adult in the room. Okay?
Sign up for the CRA days.
>> Jared, can I have just one more minute? Just one real quick point. The [inaudible] test that we use the bears, okay? These probably are not appropriate for middle school students. Just be aware of
that. We use that today, not only because we had easy access to this. Not worth to [inaudible] we want to make sure that you don't walk away from here thinking you have to use bears as the
[inaudible] to work on the number line. Does that make sense?
>> [indistinct chatter]
>> Yes.
>> Okay. I need everybody to come back together because we got to close this out. We got a little bit more content to look at, but I need you to stay focus with me. It's an important point.
Personally, I'd like to use little green army men. I think they're much cooler than bears but you may need to cut the guns off and that kind of stuff or maybe use revolutionary war figures and tied
into some history stuff. George Washington can march around. I don't know. Be cool with it. Any problems here that gave you issues?
>> [indistinct chatter]
>> The zero pairs. That's always the one that comes up and need some practice. If you have to take away something but it's not there to remove it, you're going to have to introduce zero pairs. It's
why you need to practice beforehand. I'd also argue, it's why you need to practice with us at CRA days. Remember, the goal is not that you can do all of this when you walk out of here and go teach
kids. The goal is that you understand what is capable of being done and you could start thinking through a little bit different lengths in terms of content so that you can make it accessible to kids.
Here's multiplication. I've got to show you multiplication and I got to wrap this up in the next 12 minutes. So we're going to move pretty quick. Multiplication really comes down to two things, the
first factor and the second factor. The second factor is telling you what you're making. The first factor is telling you how you're making it. Two groups of three, right? Two groups of four, five
groups of six. We use that language, we use arrays but we can do it with negative numbers as well, and you're going to see that. What happens is if it was a negative in the second number, then
instead of making groups of positives, we're making groups of negatives. If it's the first number, then it's going to be the opposite of whatever we made. Let me show you what it looks like. Same
chips, same idea. Three times four, literally translated as three groups of four. That's how we're using it right now. So that's a group of four, group of four, group of four, count them up, that's a
group of...
>> Twelve. Okay. Three times negative four is three groups of negative four. That's negative four, negative four, negative four. That gives you what?
>> Twelve.
>> Negative 12. Negative three times four is the opposite of three groups of four. Okay. It sounds cumbersome at the beginning. Here's what we've done mathematically, we factored the negative out of
>> Negative 12.
the three. And we've said instead of thinking about negative three, think about it as negative one times three. I'm showing you that works mathematically, I'm not saying you have to do that proof
with your children. It's the opposite of three groups of four. So here's what it looks like modeling. What's three groups of four? What's the opposite of that? Flip them. What do you get? Negative
twelve. Okay. It's a lot easier model that the language is almost the barrier in this one. So we have to introduce that negative sign as meaning the opposite. It works because when we talk about
mathematics vocabulary, what's the opposite of four?
>> Negative four. What's the opposite of negative ten?
>> Negative four.
>> Positive ten. We do that. We talk about opposites in math. It means multiply times negative one. Negative three times negative four. This is about as complicated as you're going to get. It's the
>> Ten.
opposite of three groups of negative four. So we're pulling that negative off of the three and we're saying, just think about the three times negative four. Three times negative four was what? Three
groups of?
>> Four reds.
>> Of negative four or three groups of four reds. So lay them out. But you don't want that. You wanted the...
>> Positive.
>> Flip them. You got it. Positive twelve. Ma'am?
>> So I think we did a lot of flipping integers and a lot of the--this concrete, do you have any suggestions for direct representational...
>> ...in positive and negative?
>> Yes.
>> Do you remember my comment--the question came up, can we skip the representation and I was going, no, don't, really, don't, like it can create a breakdown. You need to marry that to the abstract,
kids need to fall back and then you get here and what you could do is draw plus signs and I would certainly be drawing plus signs. So if I need in this one the opposite of three groups of negative
four, I'm going to do four minus signs, I'm going to do that three times. That's going to give me my negative twelve. Would you going to have kids do after that, maybe they turn them on the plus
signs and that's the flipping aspect of doing the opposites, maybe that works. Maybe they could draw a circle around them and just a put a big plus sign to indicate that it was supposed to be
positive because they flipped it. We haven't work that part out as to exactly how we would want kids to model it but you can have kids use plus signs and minus signs instead of reds and blues or reds
and yellows. Put colored pencils with it as well, put plus signs in blue and negatives in yellow, you know, if that helps the kids make that transition or maybe they could use the color to, you know,
circle it in the color that they want kind of thing. So there are ways to model that but when you get into multiplication, you get very, very big very, very quickly. When we're talking about CRA, big
numbers aren't a concern. Ten times ten is a hundred. I can show that with three blocks. Ten, ten, hundred, easy, right? Three blocks. What's nine times nine?
>> Eighty one.
>> Eighty one. That's nine blocks and nine blocks creating an array of eighty one blocks. I would not ask a child to model multiplication of nine times nine concretely nor would I do it
representationally unless they were using graph paper and drawing a nine by nine square but then even then I would want them to have a more efficient way of counting than counting up eighty one
things. It's not big digit or big numbers, it's big digits, right? Eleven times twelve is not bad. It's one ten, two ones, and one ten, one ones. I can add them, I can multiply them. It's not bad.
But if I'm going to do a number like nine times nineteen, not very big numbers but big digits. Those nines in there create nine individual concrete pieces that kids have to move or nine more drawings
and when we start multiplying, they get very big. So when you're working with this whether it's concrete or representational, if you keep your digits small, you can drive home the concept and you can
work with larger numbers like hundreds and thousands and that's okay but you want to keep the digits small or you're going to get an abundance of blocks laying out on the table and you can lose the
material because of the pieces that are out in front. So we're not going to practice this because of time. I just want to show you two times four means two groups of?
>> Four.
>> Four. There it is. Two times negative three would be two groups of?
>> Negative three.
>> Negative three. There they are. I've arranged them in arrays this time, nice and neat instead of that grouping aspect. Six times negative four, how many groups?
>> Groups of what?
>> Six.
>> Twenty four.
>> Okay. You can see twenty four pieces is a lot, right? Would I do six times negative four? Maybe depending if the kid can handle that number of pieces, or draw that number of plus signs and minus
signs. Negative three time six. So because the first number is negative, it's not three groups, it's the...
>> Opposite.
>> Opposite. We'll handle that in a second. So I need three groups of what?
>> There's three groups of six. I don't want to stay there because I want the...
>> Six.
>> ...opposite. Now, if you love the commutative property as much as I do, you'll start to teach kids, that might cumbersome, right? So if you see negative three times six, maybe you want to write six
>> Opposite.
times negative three because six times negative three would be six groups of negative three and that's what we're showing here in our column, six groups of negative three. Negative four times
negative five. The first number is negative so I'm going to need the...
>> Opposite.
>> ...opposite of four groups of negative five. So here's four groups of negative five. Here's its opposite, positive twenty. You work with this, kids will start to generalize, you could start introduce
some of those rules like a negative times a negative is a positive and negative times a positive is a negative, why? Because it is flipping aspect. Okay? Instead of these rules being arbitrary, they
start to have meaning. They can reflect back on when they flipped and how they flipped and what that meant. A couple things I want to kind of sum up with, remember CRA is not the void of the concept.
It is not the only type of instruction that's needed for students. There's fluency acquisition. There's summarizing activities that are going on. This is not happening everyday all day with every
kid. This is as appropriate as we need, we find places where our kids have struggled and we need to put in more supports to get them to the abstract. We've shown you with some stuff with algorithms,
we've shown you some stuff with integers, we've got stuff with equations, we've got stuff with fractions. It doesn't mean that every kid needs all that every single day. They had--we didn't even talk
about geometry yet, right? But there's still geometry instruction. There's measurement instruction. There's money instruction. Money is interesting because it's kind of inherently concrete even
though those concrete pieces are very abstract for kids because they're all different sizes that aren't necessarily proportional. So remember, it's a progression. CRA is not done in a...
>> CRA is not done in a...
>> Day.
>> Day.
>> ...day. You don't do that. You don't say, here's a problem, do it concretely. Here's a problem, do it representationally. Here's a problem, do it abstractly. You can't do it abstractly, this stuff
don't work, right? You can't do that to a kid. You're going to live concretely until there's understanding. You're going to marry that to a representation to get them to draw pictures to still make
sense out of it. And you're going to have those things tied into that abstract so you can start to remove the pictures and just leave the same thinking in the abstract. Remember that you need to stay
connected with us. One way is through the training CRA Days, they are not sequential and that you have to attend all four and you can't come to the fraction stuff if you haven't been to the addition
stuff. Obviously, the more you come to the more comfortable you will be with a wider range of content but you can just show up the integers and equations if you're a high school teacher working with,
you know, population students, that they're saying, "You need to teach the kids Algebra," and you're like, "I'm not really sure how to do some of this Algebra." We can help you with that. Remember,
PaTTAN.net, you can find that information. I'm going to leave email addresses up there for you. Remember, you can access these materials through Wiggio and you can also join the iTunes course. A lot
of the stuff in the iTunes course is also organized in Wiggeo so there's multiple places for it depending on how you'd like to engage in that content. Any last minute questions? Okay. Before I turn
this over then, remember that you're welcome to come up and talk with us afterwards but what I need right now is a quick summary from everybody. I need to know how I did. Don't judge Allen or Sharon,
you--no, don't clap. We save that. We'll plan that, we'll edit it, and then we're going to look good, so it's popular and well-liked. Here's what I need, in front of you, not up in the air, so
everybody can see, this is personal, I need a measurement from zero to five, zero being I have no clue what you're talking about, five being I get this content not that you could turn it around but
that you're comfortable with it, okay? I'm not asking you to teach it. Are you comfortable with it? Zero through five. Where would you put yourself with what we talked about today, CRA, addition,
subtraction, integers, zero through five, give me that rating.
>> [indistinct chatter]
>> Okay. Yes.
>> No. Okay. One more time, zero through five, how do you think I did in terms of presenting this content to you and teaching you? Zero being get off the stage, five being you're going to take the
>> Ten.
time to write my boss a letter and ask for a large, large raise for me. So zero through five, how well did this presentation go for my end, just me. You don't have to hold that up either. In fact,
it's on camera so can you put it down? All right. You, guys, have been a great audience. Thank you for engaging in math today.
There are no comments currently available