>> Okay. I would like to introduce our presenter. An experienced classroom teacher, author, mathematician, and leading special education expert, Dr. Paul Riccomini has taught math to both general and
special education students in inclusive settings. He is coauthor of the bestselling book, Response to Intervention, in math and recently reduced--released Developing Number Sense Through the Common
Core. Paul is an associate professor of education at our own Penn State University. His teaching experiences required him both to have a strong content knowledge in math and to develop and maintain
strong collaborative relationships with both general and special educators. Please join me in welcoming Dr. Riccomini.
>> Thank you and good morning.
>> Good morning.
>> All right. So, today's session, we're going to primarily focus on two strategies or techniques. Now, if you're viewing this through an RtII framework, these are perfect for Tier 2 but actually they
are best done in Tier 1. And in particular, any type of homework is one of the strategies. And these two strategies may run a little counterintuitive to math teachers out there. I know when I first
was learning and reading about the strategies, I was a little apprehensive, I didn't want to accept it until I got into the research but what--the one strategy, if I had known this strategy when I
was a teacher, I would have changed many aspects of my classroom. So, we're going to kind of go through some background information and then, we're going to get into the strategies. So, one of the
things, again, this is the--sort of the framework that we're all trying to do in--whether it's an RtII model or just trying to improve the math instruction. There's three primary areas, and they are
inter--they are interconnected. What I tend to see is schools and districts spending a lot of time in the blue and green circle and not getting down into this pink circle. And when you're talking
about this pink circle here, you're talking about the teacher content, knowledge, and instructional strategies knowledge. And of the three circles, this one here is the most difficult circle to
change, so, Tier 1, Tier 2, and Tier 3, all are within that circle. But when you're talking about that circle and instruction, what are you trying to change?
>> [indistinct chatter]
>> Teacher's behavior. And that is a bit of a challenge especially with math teachers, especially at the high school. Are there any high school math teachers in the room? Now, I was a high school math
teacher, so, I'm allowed to say this. We tend to look at people through, "Do you know calculus or you don't?" And we tend to have that kind of view and some of--some of us call it content arrogance,
because we know--we know math that a lot of people don't have. And we're very reluctant to change things. Elementary teachers, on the other hand, they often will lack self confidence in the content
but they're great because they'll try any strategy that you possibly could imagine and they'll get down on the floor and they'll do all kinds of things. So, it's a--it's a unique situation in terms
of the spread. Now, middle school, if you're a middle school--any middle school folks in here? You're kind of like the kids. You don't know whether you're elementary or whether you're high school
yet. So, if you come from an elementary background, you're more into the strategy stuff, if you're more from a secondary, you're more focused on the content, but it's very important to kind of
recognize that when we're talking about instructional change, we're talking about getting the teacher to try things that are out--maybe outside their comfort level. Now, the big thing that's driving
everything right now or the--is the PA Core Standards, which is coming from the Common Core. But within the standards, they also have delineated these mathematical practices and there are eight of
them. And these practices are what we want the kids to be able to do with the math after they learned the math. And what I'm finding is, they're very overwhelming or intimating to teachers because
there's eight of them, any one lesson could address multiple ones, so, it's kind of a--it's kind of a little bit of a challenge for teachers to kind of wrap their heads around what's happening. But
with these eight, and I'm not going to get into, like, discussing all eight of them but if you sort of take a step back from these eight, what you will see is they really clustered down to three
broad areas that we want the kids to do. So, with--and this is a little bit easier to get our--get our heads around but we have reasoning and explaining. We need our kids to be able to reason through
and explain what they're doing. And the explanation can no longer be, "Well, I did what you told me to do," right? It has to get into the why, the how, the when. Then, you have modeling and using
tools. So, we want kids to model math. That could be with concrete manipulatives or pictorial representations or diagrams, but we want them to model what they're doing and use tools. And then, the
last one is seeing structure and generalizing, and that's probably one key ingredient for those that become proficient in mathematics is that they really are looking at structure and then, they're
able to generalize that structure. And that's one of the things our kids don't do a good job of is see the structure and generalize. They either undergeneralize or overgeneralize, and the classic
example of overgeneralizing is when you teach a student to subtract with regrouping and you're developing that concept, that skill, so then what does he do from that point on with every single
subtraction problem, he regroups. That's overgeneralization, and they're not--what's happening is they're not seeing the critical attributes. And those that are proficient in math, mathematicians, we
don't just solve problems a certain way because we feel like it, the problem and the characteristics of that problem are dictating our solution path. And our solution path is going to be the most
efficient path, even though if you ask us solve it in another way, we can solve it multiple ways, but we will generally go to the most efficient solution path as dictated by the problems. Now, I want
to highlight this today because all three of these clusters, these behaviors, these skills, what we want them to do are very, very difficult for students with disabilities. It does not happen
naturally, it does not happen automatically, the only way it will happen is if it's facilitated by the teacher, purposeful instruction with a lot of scaffolding. And, you know, one of the things that
we got to remember, in the day and age of testing, we're tending the focus on academics, reading, math, science. But we got to remember, our kids aren't just poor problem solvers in math. That poor
problem solving extends through their whole lives. And now, we're trying to get them to do it in mathematics. So, it's--this is a problem or a challenge in general and it's not going to happen
quickly or easily and has to be facilitated. So, the strategies that we're going to talk about, the first strategy, Interleave Work Solution is really going to help in this seeing structure and
generalizing. And that's something that can be very, very advantageous if we can get kids to do that. Now, the other strategy has to do with retention. And we all--if you work with kids, we know that
they have problems with what? They can't remember what we did at the beginning of the year, right? Well, they can't remember what we told them 10 minutes ago, let alone at the beginning of the year.
So, we have this retention piece. And there are--there are things that we can do to help increase retention. Now, if we increase student's retention of the important mathematical concepts and skills
throughout the course of the year, what will we increase by accident towards the end of the year? Their test scores, because they will have remembered more. So, we want to try to do things working
smarter not harder. Sometimes, we work hard but not very smart. And so, those--these two strategies can really play into that. The other thing that we have to take into account is the type of
learners. Now, I'm sort of moving away from the labels, specific learning disabilities, ADHD, struggling, ELL, ESL, and really, in the two camps, you have kids that are strategic learners and then,
there are kids that are non-strategic. The strategic learner is the best way to summon up without going through point by point is these are the kids that at the end of the lesson or in the middle of
the lesson, you want to hug, right? Because they say things like this, "Oh, this is just like what we did yesterday," or, "Oh, I see. We can also do this." They are making connections, they are
activating prior knowledge, they are in tune with what's happening, all through these characteristics. But then, you have the non-strategic kids. So, these are the kids you don't want to hug or you
want to hug and say, "Oh, bless his heart," okay? These are the kids that they just approach learning in no-organized fashion. And those first two bullets, they're unorganized, they're impulsive,
they do not know where to start. And then, the second bullet, they can't break things into steps. That leads to a very common problem or error, or I should say a flawed strategy in solving word
problems, that we see all the time with our kiddos that struggle, which is they look at the word problem and what do they do? They pull out all the numbers regardless of what they are and they will
do whatever operation you've been doing most recently and slap a dollar sign on it, with at least they get the label right, okay? They are really not problem solving in any way, shape, or form
because of those first two characteristics. They have issues with memories. So yesterday, I did a--two sessions on fluency. So, this working memory piece kind of placed into this as well. They lack
persistence, they give up quickly, they experience feelings of frustration and when they do struggle, they tend to blame it on something else. These are challenging learners. The strategic learners,
they will learn. The non-strategic learners, they are only going to learn if we specifically pay attention to their needs. Now, one of the things to keep in mind with this is not all students with
learning disabilities are non-strategic learners. Some are strategic. Not all gifted kids are strategic learners. Some are non-strategic. So, don't lump them in to that type of category. So, a good
example of strategic and non-strategic, I'm going to ask you a question and I want you to show me with your hands what's the first thing that you're going to do with this question. Okay, ready? Here
we go. What's the first thing you do when you put a puzzle together? What do you do? Show me with your hands. Okay. You all need to go back to kindergarten. Okay. That is not the first thing you do,
or you're telling I'm square, is that what you're saying? That is not the first thing you do. If I gave you a puzzle, you would, one, take the pieces out of the box, two, turn them all right side up,
three, look at the picture, four, begin to sort for the what, edges and corners, look at the picture again and then, begin to put this. Now, how come you all skip five things? I know that if I gave
you the box, if I gave you that box, you would do everyone of those. Now, maybe not at the exact order but you do everyone of those. How come we skipped it? Because you don't--it's an automatic key
thing. You're strategic. It's automatic, but you're so automatic with it, you tend to not even think about it. So, sometimes, in our Math lessons, because we're good in math, we are starting our math
lessons three to four steps ahead of where the kids are. And they're lost before we even open our mouths is what it ends up being. So, we really--in math, you really have to take a step back, what is
it that I am wanting my students to do in this lesson and what are the pieces, because it's very easy to skip ahead because we just are so automatic at certain things. So, these are the kiddos that
we're really earning our paychecks for is--are these non-strategic kids and they're the ones that they can't--they have horribly organized notebooks, how about their lockers, if you're at that level
where they have lockers or their book bags, their papers come back, there's footprints on it, it's all crumpled up, they're just not organizing their daily lives. So, when we provide instruction,
we've got to do things to help organize their thinking. So, just keep these kids in mind and--as we go through. The strategies that we are going to address today are specifically trying to address
these red characteristics here. So, we're trying to help the kids figure out where to begin problems. And not just where to begin but why, that's a big piece of seeing structure and generalizing is
the why. Second, when--we're going to try to help them see the steps. The other--the other piece in here is if they do start working through problems and they get stuck, so, in other words, their
plan hits a barrier. Strategic learners are able to adjust. What happens with our struggling kids when they hit a barrier, they don't know what to do or they're stuck or it's not working, what do
they do? They stop, and if they're in middle school or high school, they will circle or box that answer, wherever it's at and they don't--they're not able to adjust. And experience feelings of
frustration and anxiety. That's another big thing because that is directly linked to motivation. Everybody says we got unmotivated kids in math. Well, one reason they're unmotivated is they are
experiencing failure daily. Now, what do I mean by that? Well, very common activity in math class and starting in about fifth grade and up is something that's called a warm up or a bell ringer or a
do-now, everybody has some secret name for it, right? And I haven't been in your classroom but generally, the structure of that activity is there are three to five problems on the board, when the
kids come into the room, they're supposed to what? Sit down and do those problems, almost always individual sometimes independent. Or, I'm sorry, individuals sometimes in a--in a group, or with a
partner. After about five minutes, then what happens? Teachers go over the answers and I'm kind of saying--and that takes five minutes, right? On a good day because a lot of times, they're going over
the answer, ends up being what? Re-teaching. Now, here's what I want you to think about. In that structure, the purpose of that is basically accumulative review. Those problems or things that kids
should already know, things that might show up at the end of the year, the purpose of that is accumulative review. But in these classes, we start to see groupings. So, there's usually a group of kids
that are getting those problems right 90% at the time. Then, there's another group of kids that it's 50-50. They get them right some of the time. And then, there's another group of kids that we're in
May and they haven't got any of those problems correct at all through the whole course of the year. So, you have differentiated classes into three groups. What do we make them all do? The same
problems, and that is where we--at the lowest level of differentiation, we should be giving kids different problems, different--at different levels of difficulty, because these kids that are not
getting those problems right everyday, think about them for a second. They're usually the struggling kids. Do they like math? No, and we're starting our math class off every single day confirming to
the kids, once again, that they aren't very good in math, and that's the problem. So, they're--we've got to take a look at what we're doing and the outcome, too much--too much of what we say as well,
that's what we're supposed to do, and there's really no connection to the learners. If that's not--if kids aren't experiencing success in that, they will become unmotivated, get frustrated and give
up, and that's one of the things that happens very quickly in math. The other piece working memory, the strategy that we are going to do today is really going to address this piece here. So, working
memory, if you were in my sessions yesterday, one of the things we're seeing is the capacity of students that are struggling in math in terms of their working memory is generally more limited than
whatever capacity they have. They tend to be using it on low-level calculations and computations, which then interferes with reasoning, explaining, seeing structure and so forth. So, today's strategy
is specifically going to put into this one, learning is more effective when practiced is combined with instruction. Now, in this strategy, the instruction is built in to the materials. And it's
trying to provide more guidance without the teacher having to do what? Chicken teach, right? Do you know what chicken teaching is? That's when the kids are trying to do their independent work and
they start raising their hand for questions and then we ended up running over here like this and you're--and running around the room like chickens. And when it's in a co-taught setting, it's even
more funny because there's two of you running around in the room. So, that--those are kind of the things--so, today's strategy, the main strategy is trying to build in guidance into the materials.
Also, both strategies today, very much well aligned with the instructional recommendations based on the National Mathematics Advisory Panel. So, the panel did a lot of things, one of which they
looked at, what does a--the research--they did not do new research, they looked at what research was available, they looked at it through a critical eye in terms of how was the research designed.
They wanted experimental, well-designed controlled studies. And what they found is there are two general approaches to math. You have student-centered, which is generally more associated with
discovery or inquiry-based learning. And then, you have teacher-directed, which is more explicit. What the panel's recommendation was--and this is very important for an RtII framework, Tier 1, Tier
2, and Tier 3, is all kids should have a balance. So, Tier 1 should be balanced. There are--and unfortunately, we have programs that are not balanced. They say we are this, we are that, but there
should be a balance for all students, so, that's Tier 1. They need to be a balance. Then, for these subgroups of kids, those that have disabilities, those that are struggling in math or those that
are chronic low achievers in math, they need, on a much more regular basis, more explicitness, teacher guidance, clear problem-solving models, carefully orchestrated examples and sequences, and
that's going to be a key to one of the strategies today, it's the sequence of examples and practice problems that we are giving students. Carefully orchestrated or--I'm sorry. The use of concrete
representations to help kids understand the abstract process and then, thinking aloud and those are what our students need. Today's strategies are really addressing the red bulleted, they'll be very
explicit, we are paying close attention to the sequence of examples, and that's something that textbooks don't really do. The--and if you think about your books, the problems that are given to the
teacher to start the lessons, whether the teacher is modeling it or it's more an inquiry, are generally the simplest occurrence of whatever you're teaching. They're the easiest. Then, the next set of
problems tend to be a little bit more what? Challenging, then, where are the most involved problems in the homework? So, if you think about it from an instructional standpoint, we're giving the most
instruction and guidance on the simplest problems and we expect the kids to what? Transfer. And then, they get home and they've got these problems that don't look anything like what the teacher was
doing, so, they ask their parents and then, the parents go, "I don't know what this is. I don't like Math." Even with my son when he ask me stuff, I have to think about it for a second because I'm so
far removed and if they don't have any help at home, what happens? They don't do it. And then, they're missing their opportunity to practice, become more fluent, become more comfortable with that
content. So, the strategies we're going to talk about today are really fit--fitting in those things. The other piece is where to do these strategies and that's the other thing. The strategies today
can fit in multiple places but not necessarily multiple content. The strategies today, one of them is retention. So, as soon as you hear retention, that's a strategy that's used after the kids have
what? Received the instruction and learned. The interleave strategy is a little bit more focused at multiple-step problems. So, you don't, you know, there's a lot of strategies out there and
sometimes teachers are just doing a strategy and they're not matching the content and the kids to that strategy. So, today's strategies, they can be used in any of these four areas. So, you're
opening, warm up, to do now, could be is--a place, certainly guided practice, certainly independent practice, any time that's independent. So, homework, although homework is really not--not in my
house, homework is not independent, as well as weekly and monthly review. So, those are the main areas. So, what I want you--thinking about is the structure of your math class and where these pieces
can fit. Now, in Tier 2, you're going to have a similar structure. It's usually not as long as a regular math time. So, today's topics are going to be in these areas and these are instructionally
different topics or strategies. One of them is very different. It's been around for a long time but it's not penetrated the math classroom yet. All right. So, what I want you to do now is I want you
take about five minutes with the people sitting next to you or behind you, in front of you, and I want you to sort of discuss right now what is it that you do to help kids in these following areas,
homework, independent tasks, how do you help kids study for tests, study guides, what do you do, and what do you do to review important--to help kids review important skills? So, five minutes, right
now, have that discussion.
>> [indistinct chatter]
>> That was a very lively discussion. The coffee must had kicked in. That was good. So, now that you're thinking about what you do for--to support students and so forth, in these activities, because
these are all regular activities in math class. And what we have to think about is how are we going--how are we trying to support these activities and I'm sure you are having all kinds of discussions
about--some of you may give study guides, some of you may review the problems for the test, some of you go over the homework, typical types of things. Some of you probably discuss why I do the
homework problems in class and then, they have the same problems at home, I give them study guides then, the problems on the study guides are exactly like the problems on the test and just kind of
the traditional types of things that we do to help students. So, the two that we're going to talk about today will be very similar in nature to some of the things you talked about, but I'm going to
go to an additional level of instructional support. So, when we look at these things, this is a good time for me to give you a really good resource that is also free. It's--as--in the world of higher
ed, it's referred to as IES or Institute of Education Science. This is a governmental organization and they're really--they have a lot of different things they do but two main responsibilities of
this group is one, they [inaudible] out the research dollars for education, excluding special ed but--and the other thing that they do is they are--form groups to look at research very critically to
look at best practices for teachers. So, they produced these documents called practice guides and there are many. I think at last count, there may have been 16 or 20, and they are in all different
content areas, and there are some in math, so, there's one on RTI in math, there's several on problem solving, there's one on fractions and so forth. And the one that the two strategies that I'm
talking about today are coming out of this one, Organizing Instruction and Study to Improve Learning. So, they're coming out of this guide. They're free. If you Google this, IES Practice Guides,
they'll come up. They're very, very useful because basically, it's a short description of what the strategy is, the--all the research references and citations are behind it, and then, what's really
important is they give it a rating of high, moderate, or low in terms of the research that is supporting it. Unfortunately, I guess how you look at it in math almost nothing gets the high rating
because we have really not developed our research based in mathematics. Now, the other thing you want to keep in mind in order to get into one of these guides, the research has to have been done at a
high quality and they're usually have to be an experimental control. So, they give ratings of strong, moderate, and low, as I said, a lot of things in math get low because there's not a lot of
research done. Today's two techniques get a moderate rating and that means it's been researched multiple times in different settings across several studies and they're finding the same results. So, a
moderate rating is pretty good in our world in terms of math instruction. So, I highly recommend if you're not familiar with these guides that you take a look at it, they're all free. You download
them as a PDF and they have a checklist for implementation. They have possible areas of mistakes that can come up in terms of challenges with any of these strategies. So, the two strategies we're
going to do are Interleave Work Solution Strategy and Space Learning Over Time. So, IWSS strategy is I refer to it is one of my favorite strategies. If I were a math teacher, man, would I--I would
use this strategy in a lot of different areas. So, in this particular strategy, interleave means to alternate and like the first bullet there, Interleave Worked Example Solutions. So, essentially,
what you do in this strategy is you alternate worked-out solutions, unworked out, worked out, unworked out. And what happen--what you do is you tell the kids, and we're going to--this is--I'm going
to elaborate upon this. You tell the kids to study the solution, the worked-out problem. It will help you solve the next problem. And this is built into--initially, the research was doing this with
homework. So, essentially, what they did is one group got the traditional homework. In math class, the traditional homework looks like this. Here's your 20 problems. I tell you, if there's anything
math teachers can do and that is give some homework, and a lot of it. And it's basically 20 problems, whether it's a calculation problem like this one or word problems or a combination, it's 20
problems that the kids have to what? Solve. Almost no guidance. Where are the kids going to get their guidance? Well, they can look at their notebook, right? Well, we know that students with learning
disabilities and their notes are horribly inaccurate and incomplete, so, if they do study their notes, they're probably studying something that's wrong. They could ask--they could look in their book,
although the textbook in math is becoming--for the kids, it's becoming less and less frequent or they can ask their parents, or maybe they search on YouTube or Google for help. But other than that,
they've got no guidance. So, that was your typical homework. Then, the other group of students got homework that involved the interleave. Now, the research on this was primarily done at the sixth and
seventh grade on but I think it has a lot of application in elementary school when you're getting to multiple-step problems. This is a strategy for multiple-step problems, not one-step fact problems.
So, when we look at this, the other homework basically was this, all of you guys, you got homework where every other problem was solved. And you were told, "Study that solution. It will help you
solve the next problem." So, that's how the homework was done. They pre and post tested the kids. Which group of kids performed better on the assessments? Traditional homework or the interleave
homework? The interleave homework. Now, if you think about that for a second, if you had 20 problems to solve, you had 20 problems but every other problem was solved, you practiced half as many
problems as these kid--as this group but you learned more. Now, is anybody--is anybody a little uncomfortable with this? From a high school perspective, I was like, "What? Less practice but it was
better learning? How was that possible?" And when I went back, I went in and read all the original studies clear as day. Giving kids guidance like this produce better learning effects. So, why? Why
is giving kids a solution helpful when they are asked to study it? And this changes the whole way we do math homework. This also should change how we do study guides. My son brings home study guides.
They are what? They are problems that he is supposed to what? Do, and there's no guidance on them. So, this is--I would--all my study guides were a joke when I looked back at what I did because
basically, if I put my study guide next to my test, it looked the same. So, interleave could absolutely be applied to study guides but for homework. Why? I want you--give you two minutes with your
group, why could this strategy be so effective? Go ahead. So, remember now that kids were told to study the solution. So, really, what you're wanting the kids to do is a think aloud of how the
problem was solved. So, what you want to see from the kids is something along this. So, solve 12+2x=15. I want to solve for x, okay. So, 12+2x=15, they subtracted 12. Why did they subtract--okay,
because I'm trying to isolate my variable and it's plus 12, so, the inverse of addition is subtraction. So, they subtracted 12, okay. And they subtracted 12 and now, they get 2x=3 and it looks
like--okay. So, they divide it by two. Why did--okay, because we're trying to isolate my variable. It's 2 times x, the way I undo or inverse of multiplication is to divide, so, now I get x=3/2 and I
get x, one--okay, because they changed the 3/2 to a decimal. That's what you want the kids to do. Now, question is, why does that--why is that an effective strategy? What did you guys discussed? What
do you think? You're studying? You're studying a correct problem, one
>> Promotes self-efficacy.
>> Promote self-efficacy because they've got--there's ways to look.
>> If you're doing it at home and you think you have it right, and you keep reinforcing the wrong process, then, you're not...
>> So, practicing--this will help reduce--hopefully, practicing it wrong. So, let--okay. So, those were all right, but let me ask--let me phrase it this way. When I'm studying that problem, what am I
not doing? I'm not actually solving the problem. I'm not doing any calculation. I'm not writing anything down, so, I'm lowering the load on my working memory so that I'm better focused on what? The
process. You're essentially, verbally rehearsing the process here while not having to do tedious calculations. So, you reduce the load on working memory, which better let you focus on the problem.
When you are solving problems, you are blending multiple things together. You're blending the--any calculation or computation, writing, alignment, at the same time you're trying to remember the
process. We rarely separate that in math. A good thing to ask your kids is how do you solve this problem without having them do the problem? A lot of times it's the problem that's prompting the kids
and they cannot just verbally tell you how to solve the problem. Like, you ask your kids, "How do you--how do you solve for a variable?" And you would want them to say things like, "Well, I have to
isolate the variable by doing the inverse operations. In other words, I want to get the variable by itself. And I do that by using the inverse operations. I have to do anything with addition or
subtraction first then, I move to multiplication or division, and I get x equals this." A lot of kids cannot even tell you that separate from looking at a problem. So, with this strategy does is it
focuses them on the process. Now, is anybody--are we all all right with this or anyone still not buying the technique?
>> If that's about the alignment, the question that we had is how do we motivate kids who make that [inaudible] work through it rather than just them to [inaudible]
>> Yeah. Good. That is--so, when we started implementing this in schools, we began to see some challenges. First off, the above average and gifted students love this strategy because when they can't
solve a problem, they're not--they don't like that and they had--they want to figure out how to do it. Now, with the absence of this, the way those kids would do it is they might look at the answer
in the back of the book if the problem happens to be what? Odd, and that answer can help them what? Work backwards, all right? So this is--the answer is in the back of the books on steroids, all
right? This is the solution. So they love being able to look at it. The average kids and the struggling kids, well, they kind of broke into a couple of groups, the average and below average and
students with disabilities that were, for lack of a better term, the conscientious students, in other words they were trying would look at it. The other students maybe non-conscientious would just
skip the solutions. Therefore, they were not having any benefit to them. So what we begin to realize is, one reason they're skipping the solution is, they don't know how to study the solution and
this is really that non-strategic kid but especially the kids with learning disabilities. They don't know how to study problems. So what we saw them doing is, they'd look at the solution and then
they would just skip it, go to the next problem, and kind of do this hunt-and-peck back and forth. That defeats the purpose. So, how did we get the kids? So, schools that were doing this, what I
began to see is--well, in the same school, one teacher was having incredible results with this. She said, "The kids are doing it. They're studying it. They're talking." And then another teacher,
"Well, they're not--they're just ignoring it." So, why was that case? So, when we went in and started looking at the teachers that were having good success with this versus the ones that were not,
there was one glaring difference between the two classrooms. The teachers that have success with this, upfront teach the kids how to study the solutions. In other words, they model it multiple days,
have the kids do guided think-alouds, partner think-alouds, and give feedback to those kids. So, this strategy needs front-loading for our struggling kids and that front-loading is the discussion
about, "Hey, you're going to get a solution. This is what you need to do with this solution. Study it, verbally talk through it, try to solve the next problem. If you get stuck, look back."
Front-loading was key for this strategy. Then even with that there are still kids that, what? Skip. Does any strategy work with all kids? No. That's the one thing that research clearly shows us. So,
then the teachers started doing things like high--making the kids highlight the steps or check the steps or verbally or write down what was happening. So they put all these mechanisms in to try to
get the kids to look at the steps. Now, in the--in the--in the end of the day, there were still some kids that were, what? Not looking, but guess what? Those were also the kids that probably weren't
doing the homework anyway and we should not use those kids as an excuse to not to do these, the strategy. So, now in the research, these--the solutions that were provided varied. Some were just
skeletons like this. Others provided some annotation, so description of what was happening. So, it was varied. But the point that you want--the big point of this strategy is getting the kids to study
the solution. So, obviously homework is a place this can happen, but also in class. A lot of--there are a lot of opportunities in class for kids to work together with a partner in a small group to
solve problems together, right? So, you could use this technique by instead of giving them a problem to solve together, you give them a problem with the solution and they are studying the solution to
make sure they understand how that problem was solved. So, this has application in whole group, homework, as well as in warm-up. Did you have a question over here?
>> Yeah, and I think you may know the answer to it. I would be trying to put in so much more, and I want her to believe in your saying not to [inaudible] you don't want to overload.
>> So...
>> Like I wanted to show the divided--what you do to one side would really [inaudible]
>> So that includes--I would put in about two more sets [inaudible]
>> Yes.
>> So, you want--you want to be careful about that, you want to be--some of the teachers reported that some of the kids were getting--especially in the upper levels, we're getting a little overloaded.
There's nothing that say you can't, the research is not that refined, but--so you can add in--you could actually write subtract 12 from both sides and show every step. The solutions varied in the
research. So there isn't a one way but don't overdo it, okay? And pay attention to that. The key is, you want the kids studying that solution and they need to learn how to do it. Now, when I had sort
of my aha moment, after reading this research, still not really accepting it, it sort of clicked on me, "This is how I studied for tests when I was in college for math." I would go into my notebook
and I would find, what? Problems that were worked out. What would I do? I didn't just practice them. What did I do? I studied them, why, why, how this, this, then I covered them up and tried to,
what? Solve them. And if I got stuck, what could I do? Refer back. So, essentially you work--this strategy is teaching kids a method for learning or studying multiple step problems. The other thing
that happens is, generally what I see in classrooms is about three examples from the teacher, and those examples are generally the simplest occurrence, and then the kids have homework. And in their
textbooks, if they have a textbook, not many problems are, what? In kids' textbooks worked out. So, by adding worked out problems into the homework, you're significantly increasing the access to work
out problems. Now, if--so, you know, homework is a juggernaut in math, a lot--a lot of kids' grades is dependent upon homework. And the problem with that is, most homework is not graded for accuracy,
except maybe at the elementary level when the teacher only has 20 kids or 25 kids. You get into the periods, six periods with thirty kids, you have a hundred and eighty kids. Most of the time
homework is just graded on, what? Whether it's completed. So, as far as I'm concerned, this would not interfere with that in any way. But the other thing, separate from this strategy if they were up
to me, I would have three homework rules that guide all homework in math class. Number one, the teacher--the math teacher cannot assign any problems unless they themselves have actually solve the
problem. Unfortunately, we assign problems and then the next day we realize, "Oh, I shouldn't have assigned this because out of my hundred and eighty kids, all hundred and eighty missed it, and I got
all these nasty letters from parents saying, what is this is, I've never seen this before." So that would be the first thing. Second thing, attached to every homework is a URL that says, "If you
can't do this at home, here's a video link that shows you how to do it, that technology is there and it's free." Third, the answers to all homework problems are accessible at home, all of them. We're
not grading it on accuracy. So, all of that should be provided to kids, it's practice, and some kids need extra guided practice. So, you take kids--so for example, my son comes home, he has homework,
he doesn't know how to do it. I'm at home or my wife's at home and we're able to, what? How do you think we help him? We--"Well, okay. This is how you do it." And we're basically giving him, what?
Guidance. You got another kid sitting in that classroom, does not have any help at home. They either practice it wrong or don't do it. They come back to class. Who's now ahead? The kid that got help.
So, this is a way to build this in to your homework. So, let's look at the example. So, this would be a typical homework problem assignment, a bunch of fraction problems. So, all you would need to do
is write in the solutions to alternating problems. And when they don't have that, they can look back in their notebook, they can ask parents or siblings, they can look on the internet, or they
complete homework wrong or not at all. And what they end up doing is D. Now I want to talk about C for a second. I think we as math teachers are missing a big golden opportunity for kids to learn to
be self--to have self-help strategies, so--and I'll use my own kids. And you--one of sons--my son is not a struggling learner. My daughter is a--struggles a little bit. They can find anything on the
internet related to skiing, fishing, Minecraft, or whatever video game they're trying to find. Then my son comes to me and says, "I can't do this problem." I said, "Well, did you look on the
internet?" Because I want him to begin to look and he looks at me like, "What? The--did you Google that?" "Why?" So--and if you know--if you Google any math concept, there's unlimited 10 minute short
videos. So, why--so my kids advanced, both of them struggling advanced. They can find anything on the internet related to games. How come they can't look for math stuff? Well, my conclusion to that
is, they don't know what terms to search. So, one of the things that I would do is the first week or two of school, I would spend some time teaching the kids how to search math problem terms,
self-help, because they can find just about anything on the internet related to games and so forth, but math, they--I don't think it's--I think it's because they can't search the terms. So, you--and
there's all kinds of videos. So, without any guidance, it's--they're left up to their own. So, in an interleaved homework assignment, you see that the problems are provided. So, this solution helps
the kids solve that problem. This solution helps the kids solve this problem and so forth. Now, here's some other things to pick up on. The problem that follows the solution should be similar to the
problem with the solution. Now, they can be different, you can vary that, but for struggling kids initially, it should be, what? Very similar. So, this is a mixed number times a fraction, this is a
mixed number times a fraction. And again, you want the kids to study. Okay. So eight-fifths--okay. So they took this mixed number and changed it to an improper fraction. So now they have eight-fifths
times one-sixth. All right. So they're multiplying the denominators and they're multiplying the denominator and the numerators. They get eight-thirtieths. And then they simplify eight-thirtieths to
four-fifteenths by dividing by two. That's what you want the kids to do. You do not want the kids to skip this, come here and do hunt-and-peck, hunt-and-peck, hunt-and-peck. That kind of defeats the
purpose. Another interesting sidebar add--that we--benefit that we came out of this was parents, teachers started getting notes from parents, "Thank you. This helped me help my kids," because there
was additional problem solved. Here's another example for algebra. So, in this example here--now, this is doing a little more steps over here. But again, verbally directions, verbally talk through
each solution problem, solve each unsolved, use the solution problems for guidance. You really have to lead the kids to the strategy. And then the problems are solved, so they study then they try
one, notice they're very similar, not exactly, they're--but they're similar, one variable both on the left side. Down here, you got two variables on both sides. Down here, it's a subtraction. Down
here, it's two variables. Down here, it's two variables. So, these solutions solved. So, you know, teachers always say, "Give me a strategy. Give me a strategy." Well, this is a strategy that
actually is well documented and could be very beneficial. Now for students with learning disabilities, they need the additional guidance. So in an RtII model, if you can get the tier 1 teacher doing
this, the tier 2 should reinforce this, giving them more instruction on how to study the problems. If you're a co-teacher, this is a good way to modify homework. You know, I don't really like to say
that because really all kids can benefit from this strategy, but this is a good way to modify for it struggling kids. So, there's a lot of applications for this in terms of the classroom. I think I
have another. So, in this one, the solutions were whole solutions through the whole homework. The other thing you can do is fade the solutions and that's what this one did. So, here is the problem
completely solved and then the next one is similar. Now, what do you begin to see in this solution here? They're beginning to fade parts of the solution. So the--this is going to--it's going to be
hard for kids to skip this because they have to actually do something. Then here it's faded more. Here, just the first steps provided and the last three are done independently. So you can--some of
the research faded the solutions. Yes?
>> Yes. So, I have a question and I don't know if this should be--like if it's overload. Would you recommend--or did you see any of the teachers, like, with the homework accompany with the steps so
that at home, they were also checking off, or is that too much?
>> Like on a separate sheet?
>> Some--none of the teachers that I work with did that. Some of the research did provide some annotation. I don't see anything wrong with that. But I haven't done that or work with teachers that have
>> Yeah.
done that.
>> I don't know if that was too much or is that...
>> No. I--you probably would have to see it--for some kids, it could be too many steps, but I would play around with that. I think we don't show kids enough, the verbal steps and isolation. All right.
So--all right. So it's kind of just rehashing some of the things. The amount of description varied. Some provided a lot, some did not. There was some fading, so the solutions begin to be faded like
that one I just showed you. And then the problems can vary great--the great--they can vary more from the solution. That's when you really begin to have kids seeing structure. So, remember back to the
mathematical practice, seeing structure and generalize? That's what we're really trying to do with this strategy. It's going to allow the kids to see the structure and generalize it. If you have
really advanced kids, another good technique with this is you give them the solution, they have to study it, and then they have to come up with another solution, a different solution, and then they
have to say which one is more efficient. Now, that's how math people think. We don't--like I said earlier, we just don't solve problems because, "Oh, I feel like solving it this way today." We're
going to solve it in the most efficient way. And the only way you know it's the most efficient way is you know other ways as well. So, during whole class, so talk about homework, study guides, I
would absolutely change study guides. I would give two problems of whatever the big idea is. One would be what? Solve this. Study it, try one on your own. Study it, try one on your own. That's how I
would do all my study guides. Now, in class, anytime that the kids have opportunities to work in class, you could do this in class where they study a solution then they are given a problem to solve,
study a solution, give a problem to solve. So it can be done in whole group. So, so far we're covering independent practice, homework, study guides, whole class, and then in the opener. Now, the
openers have to be adjusted a little bit. Right now the warm-ups usually consist of three to five different problems. So, you can't provide a solution and then the next problem is completely
different. So, the warm-up could be adjusted where you give one problem solved, the kids study it, and then they have to what? Solve another one. Teaching the kids what you want them to do with these
solutions. So how you would do this is, you would think about which part of the lesson this best fits. So this could be a PLC meeting, a common planning meeting, or a team-based meeting, something
where there's teachers together. Where could we apply this to the lesson? Oh, I want to do it in my warm-up. I think I want to do homework this way or I want to do study guides. You identify where
you're going to do it. Then you talk about how you're going to do it and then you plan right there and develop the materials that you'll use for the next week or two, or for that unit. This takes a
little bit of planning in terms of doing this. All right. Any questions on interleave? We got about 20 minutes to finish up the last strategy. So, we're good on time. But any questions on the
interleave technique? Certainly tier 1 for all kids but absolutely tier 2 technique as well. Questions?
>> This is not a question but a statement. This is with the Khan Academy [inaudible]
>> Yeah, I think--I mean--so he said Khan Academy. You know, Khan Academy puts their videos on. I really think we're missing the video piece in math. Now, Khan wants to flip the classroom. I
don't--you know, I don't know about that, but what I like is, you have a video of somebody solving a problem and if you start to get lost, other--in other words, what's happening up here? Overloaded.
You can, what? Pause. Did you know if you pause and replay it? It will say the same words, the same tone a hundred thousand times. Unlike us in our classroom, '"One more time for Johnny because he
was not listening," right? We all have been there, right? We've all been there. But you're right, that pause and then they can process it. But again, here's what we see with struggling students or
students with disability, they watch a video and they watch it from start to finish. You got to teach them to self-regulate. And when they start to get lost, pause and listen to it again. Remember,
non-strategic children do not know how to approach learning in an organized fashion. So, you have up front-load this with those kids. Any other questions on this? My biggest piece of advice for you
on this, working with a lot of teachers implementing this is, if you just show up one day and there are solutions embedded in the homework, you will get an awful lot of kids that are simply skipping
those solutions. You need to front-load this and--in class, showing them how to study the solutions. And I think the comment up here, about giving them the steps written down may be a good way to
initially begin that process. Yes?
>> I did [inaudible] intervention so I support [inaudible] teacher and he is so concerned with time in getting through the curriculum so that they have everything in by Keystones that we're not
getting to, like, the depth of knowledge that's on the actual Keystone test. So you're just tapping the surface on everything. Could this be like a method to maybe up the depth of knowledge by using
the examples and applying the skills that [inaudible]
>> Yeah, absolutely. You know, when you're verbally rehearsing something, the--and the--with this strategy, you're basically reasoning and explaining why it is happening. So, yeah, the--absolutely,
you could go to adapt the knowledge piece there. A lot of times our kiddos are never getting to the reasoning and explaining piece because they can't even solve the problems. So this is a good--this
is actually a material scaffold. My next session, we're going to talk about instructional scuffling. This is a type of scuffling. Okay. The last--the next strategy that I want to talk about, I call
it SLOT and I've also called it the SIR Strategy, Space Instructional Review. So, this is a strategy designed for retention, all right? Now, what do you do at the end of a chapter or unit in math
class? What happens? Test, right? What do you the day before the test? Review. Now, let's be honest. What's the purpose of that review?
>> [inaudible]
>> Very--we have some honest people in here. To pass the test. How come no one said to promote long-term learning of important and essential mathematical concepts and skills? Okay. The purpose of that
review is to help them pass that next test. That's a short-term review. If you review something, you're going to get an immediate bump the next day in the test, but then what happens? Falls off. All
right. Now, what I'm going to see in your classrooms the month before the PSSAs or the Keystones? A mass cram review session. You're reviewing everything from what? The whole year. And you hope some
of it sticks and then they take the test. Again, what was the purpose of that review?
>> No, not--promote long-term retention of important mathematical concepts and skills. So that's what this strategy is. You can continue to do those things, but this is something that you do. Now,
>> Pass [inaudible]
this strategy is best done at the beginning of the year. And what I found, it takes about 75 minutes of planning to map out the plan for the year as long as the teachers are in tasked. So, just
giving you an idea about 75 minutes. This is best done by either grade level or class. So algebra 1, teachers are together. Geometry, teachers are together. Fourth grade, teachers are together. It
should be done by grade level or class content. So, in this--in this technique and I believe you have this document--this form in your handout. There's nothing magical about this form. I made it in
word in like 15 seconds. So you can make it--most of the teachers I work with are putting this on a legal piece of paper to give them more space. So in this document, what this strategy showed in the
research is, it is best to wait four to six weeks before you revisit something. So, the purpose of this is to revisit, refresh, and reawaken. Forgetting is natural, all right? I have a math degree, I
took a lot of upper level math classes, but if someone were to give me a calculus problem right now, there's three things that I can guarantee, probably one, at some point someone taught me how to do
that and I could do it. Two, not only could do it, I could teach someone to do it. But three, not having done calculus in a long time, I probably, what? Couldn't do it. I would do have to, what?
Revisit, refresh, reawaken. So this forgetting is natural. Now, there are some things in math that once we teach it, we never revisit it until the end of the year. Probability is in there, certain
geometry and measurement pieces are in there. Now, there are some things that once they learn, they are doing from that point forward. Those are not areas you want to target. So the way you do this
strategy is in the first column is you blank, you chunk up your year in the four to six week interval. So, day one, six weeks, six weeks, six weeks, pay attention to holidays and all that good stuff.
So, you chunk it in six weeks. Row two--or, I'm sorry, column two, you begin to list what are the big ideas that are covered in each of those six week segments. Now, realistically and in most cases,
the first two columns are already done because the districts have their scope and sequence and their curriculum maps. But the first two columns, a lot of times if there's a math coach in your
district, we'll have those completed when the meeting starts. The most important columns are three and four. So, you--for columns three and four, you are going to use all available data that you
have. So you're going to use PSSA results from previous years, Keystone results if you have a district benchmark test, you have progress monitoring, or teacher test, all available data including your
teacher judgment. If you've taught the same level grade three years, you basically know which areas kids are going to struggle with before you even teach it. So, you're going to revisit, target those
areas, and you list the big ideas that the kids have problems with. Now, in this third column, they have to get a little bit more specific. You can't say fractions, what is it about fractions or
proportional reasoning. You got to get more specific. Now, it's the fourth column that I find drives the OCD teachers absolutely bonkers, okay? You're going to--I don't--do you have anybody, anybody
in--that fits that category in here? You cannot revisit everything, all right? The OCD teachers, like they want to write ditto, ditto, ditto, ditto. You cannot revisit everything. This is selected
and targeted. How do you select and target? Well, you look at, of the areas that are problematic which are very important in your grade level, which are emphasis points in your grade level, or which
things are just very important mathematically to the future. And you begin to select two. Two things out of each row are identified as priorities. Now, you're not just looking at the struggling kids,
you're looking at across the board. This is tier 1 intervention. But it can be doubled down in tier 2 once it's completed. They can get double dose. Now, the fourth column, if you see up there, I
have 30 to 40 minutes, that's because I was working with a school that had 90 minute classes. If you only have 48 minute classes, then it's reduced to maybe 15 minutes, 10 to 15 minutes, and it would
replace your warm-up for that day. So you don't do the warm-up, you do this. And you identify the date four to six weeks after it was taught, you write down October--the week of October 15th and when
that's written down, that means every teacher in that grade level or for that class will revisit that topic during that week. Now, when you get more advanced at this then you can begin to
differentiate, but first, go through everybody gets the same. Now, this is what you do. This is how you do this strategy, "Class, six weeks ago, we learned this. Do you remember?" Now what are the
kids going to say? No. And then what is--what are you going to do? You're going to get mad. "What do you mean you don't remember this? We spent a week on this." And then the teachers I work with,
they come back to me and say, "They did not remember." And I just refer to them, "What was the purpose of this technique?" We predicted that they, what? Would not. If they all remember then we picked
the wrong thing. So, after you get over that, you then say, "Class, this is what we did, this is why we did it, this is how we did it. Let's do one together. You try one with a partner, let's do one
more. Okay, class, this is what we did, how we did it, and why we did it, and that's it." The purpose of this is to revisit, refresh, and to reawaken. It is not to re-teach. Now, some of the kids,
you'll see through these activities that they do need re-taught. But then let's say for another time, this whole purpose, revisit, refresh, reawake, it's what--it's the best way to promote long-term
retention. Now--so I've had some schools that take this and run with it and they do things like throwback Thursday, so--or--yeah, throwback Thursday, Friday--or flashback Friday. So they build a
structure that every other Thursday, the kids know the first 15 or 20 minutes of class is they're going to be what? Revisiting something from six weeks ago. So, it becomes a structured class
activity. And they come up with Friday so--sometimes Friday is not a good day to do it if you have high absent rates on Friday, so if there was Thursdays, Wednesdays, Tuesdays, but the whole idea was
it became a structure. Now, a lot of times, math teacher say, "Well, I do this, I do cumulative reviews every once in a while." You do, however, how are those cumulative--how are those cumulative
review days structured? Usually it's giving the kids problems to what? Try. And if they can't do it then what would--what do we do? We then help them. This is reversing it because we're predicting
they won't be able to do it so we don't want them to try on their own. We revisit, refresh, and reawaken. This is really best done from the beginning of the year on. Now, from my experience, some
schools say, "Well, let's just plan the first six months because we don't know what the kids are going to be struggling with because we don't have them." I would tell you to avoid that line of
thought because what happens is they plan the first three or four months and then what happens? School starts, they get so busy, and then they don't the strategy. So it's really best to predict--now
you can adjust but I'm telling you, kids will struggle with the same things year in and year out for the most part. So here--three to four weeks, these are all things that I said. So we already tried
to do this in in-class reviews. We do this on homework. We do this on midterms and finals, but those are all generally done in, try it first then I'll help. And this one is the reverse because we
know they're going to probably not be able to do it or they forgotten it so we're going to revisit it, refreshen it, and reawaken it before they try it on their own. So, there's a lot of different
ways that this is--can be done. It's generally four to six weeks now. Is anyone on a semester class schedule? So they have 90-minute blocks everyday and they only run a semester class. That is a
little bit different. These were done across the course of the year. So, if you're on that situation, instead of six weeks, you're probably want to--going to want to do three weeks or four weeks
increments. So, how do you do this? Best way to do this, like I said, it takes about 75 minutes in a group, team working together on task as teachers are always on task, right, like our students? You
need to have your calendar, you need to have a scope and sequence, and you need to have available data at the meeting. That's where it will go, 75 minutes. It won't go 75 minutes if you have to run
and get this data somewhere. It should all be ready to go. This is a great activity for a coach if you have that in a situation. Arrange the schedule so you're re-exposing about a four to six week
period--oh, there's one thing I forgot to mention. Let me go back here. Things that are identified in these first two rows, they could be revisited in October or November. They should be also
revisited one additional time in the spring. Everybody see that? So, you may revisit it in October but then the thing that was identified here, you revisit it again in February because it was--you're
revisiting it so early in the year. The way we typically are doing this is homework, quizzes, and exams. And that's why I call this initially the SIR strategy because I wanted to emphasize Space
Instructional Review. The way we do these other things is more practice, cumulative practice, and that helps kids that basically still remember how to do it. Now, for the tier 2 framework, once the
tier 1 does this, you could also double dose the topics in tier 2. Take 10 minutes out of your tier 2, revisit. What would it be great is if you revisited it prior to the tier 1 teacher revisiting
it. So then those struggling kids show up to that class and the teacher says, "Hey, six weeks ago we did this. Do you remember?" All of a sudden, the tier 2 kid can go, "I remember. I know how to do
it." And then they are more engaged. So, that's that pre-teaching scenario. But again, at six weeks, my OCD friends, double dose your medication that day, so you don't ditto everything, selective
targeting of certain things. All right. So, I think we're about ready to wrap this up. But again, these are both strategies that have a good bit of support. What we find is for the students with
disabilities, you have to front-load that interleave strategy, really teaching them how to study, the questions to ask. The SLOT strategy is a specific technique to promote long-term learning of
important mathematical concepts and skills. Are there any questions? Yes.
>> So--because I'm a little OCD.
>> Okay.
>> So, you mentioned only take one or two, so do you still do them, those two in one day or...
>> ...it's being set out?
>> Oh, no.
>> Good point. When you--so she asked about, do you--those two things that you identify, are they revisited in the same day? No, they are separated. You revisit one of those ideas which is also very
different than the way we usually do reviews. Cumulative reviews usually have what? A whole bunch of problems. So this is very much a focused targeted review. So, good question.
>> So would you say--so after those, that's where we [inaudible] with period, you schedule in one Thursday and then with the following Thursday, you keep on doing the next test?
>> And then move on?
>> Yes.
>> You could do it that way. I've also--some schools, they revisit it everyday for a week. So, there's a lot--it's--there's a lot of variations in how you do it. The key is, you're waiting a four to
six weeks to revisit it. Good question. I like that. That's implementation question. That's good. Any other questions? Yes.
>> The topic that you take, that's only based on data, right [inaudible]
>> Yeah, the data--you're basing that on data plus your teacher judgment. Now, if you're a first year teacher, you're not going to have a lot of experience but if you've taught four or five years, you
know where kids tend to struggle and forget, so data driven plus teacher judgment.
>> And with all the preparation, we would pick the same topic?
>> That's right. So, it--there's some discussion, some wrangling data, but then what happens is, if everybody does it through the course of the year, you have a tier 1 intervention. All right. Great.
Thank you very much. I hope you got some ideas. Thank you.
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