>> I'd like to welcome you to the session, Facilitating Fluency and Automaticity through Purposeful, Planned, and Targeted Practice Activities. And to share his knowledge with us today is Dr. Paul
Riccomini. He's an associate professor in Education at Penn State University. He also has the experience of teaching mathematics to diverse learners and inclusive settings. As well as being an
educator and a mathematician he's also an author of two books, Response to Intervention in Math and Developing Number Sense. So, welcome Dr. Riccomini.
>> Thank you. Thank you and welcome. So, obviously from the title we're going to be talking about fluency and some ways to practice fluency and fluency has reemerged as a major focus after going away
for many years and said, "Don't worry about fluency." As a mathematician I don't really understand how that argument actually had any legs. In every aspect of our lives, no matter what we're
learning, there is something that you become fluent with and that fluency facilitate--better allows you to do that activity. So, in mathematics we're coming back with fluency and we have the common
core which is filtered down to the states and we have Pennsylvania core and one of the things that we're seeing throughout the core is there is fluency embedded at almost every grade level. And one
of the things--one of my big goals for today's session is to get you to realize that fluency extends beyond single digit arithmetic facts in the new standards and there's fluency recommendation all
the way up to algebra in terms of getting kids to become fluent or automatic. So, we're going to sort of go through why--one of the things after you leave this session hopefully when the kids say,
"Why do I have to memorize something?" Or a parent ask you that, we will no longer ever say, "Well that's because I had to do it that way. "We have to stop saying that. There is actually
reasons--cognitive reasons why we want our students to become fluent and automatic with certain things. So, we'll no longer answer with that but we'll answer it more from a learning perspective. Then
hopefully you'll have some--give you some ideas in terms of how to make practice more purposeful and targeted and that's going to be a big emphasis. So, as fluency has reemerged, I'm working at a lot
of different schools across the United States, I'm seeing teachers that are saying, "I'm doing fluency activities," But in reality they're not really fluency, they're more accuracy type of
activities. So, we're going to talk a little bit about that as we go through the course of the day. Now, I do have a couple of activities. I'm just curious, elementary level, raise your hand if
you're elementary level. Okay. Middle? Some middle high school? Okay. Good. Good mix. There's fluency at all levels and it's not--no longer can it just be an intervention time, fluency has to be in
the core of your math instruction but this is kind of the agenda that we'll go through today in the--in the time that I have. But one of the things that we're seeing is, algebra is really the
pinnacle of what we're really trying to get kids to and there's a lot of reasons why that is the case. It increases the likely of graduation. It increases the post secondary success rate and so
forth. But one of the things that we're seeing that is a big barrier for many students to get to algebra is fractions. And that's one of the areas that test--that we test out--we, the United States
test out as one of the weakest areas in our math and, you know, why are fractions such a problematic area and then why are fractions so important to overall success in math. Well, basically in sort
of a very simplified explanation your K-3 elementary math program is really focused on whole numbers, all whole numbers. Starting mid third grade but really moving into fourth grade, the focus is now
turning to rational numbers or fractions. And fractions is sort of the, you know, what people point to but from a--from a--sort of a broader perspective fractions is--are rational numbers so that's
decimals, percents, and when you start thinking about how a lot of things we do in our real life is so--are solved, it's usually through proportions, ratios, things of that nature. So, fraction is
sort of an important broader skill concept which moves into the application piece, but what we really see kids having problems with is algebra. When you don't have a good grasp of fractions, good
grasp being conceptual but also the manipulation, then you really have a hard time in algebra. And that is what is--what we're--many of us are saying as the arithmetic to algebra gap. It's the
whole--it's that rational number piece which is why you have seen a major emphasis. Is anybody in here are fourth grade teachers? The new fourth grade standards have really slammed fourth graders
with rational numbers and I had a personal experience that with my daughter the--this year in fourth grade and based--the first unit was on fractions and it's really problematic for kids if they have
not done what with whole numbers, multiplication. And it's really--fourth grade has really gotten a lot of fractions to put into it. So, this gap that we see in math is a little different than in
reading. In reading we often hear the phrase--if we can get kids reading or get them on the right trajectory by what grade?
>> [indistinct chatter]
>> Third grade. They tend to what? Stay on that trajectory, no guarantees but they tend to stay on that. Well, that's--we're not seeing that in math, and in part, math as a content area is very
different than reading. Math, every new unit or chapter or year is new concepts or extensions of previously learned concepts. So, it's not like in reading where once you can get in there in third
grade, they tend to stay on that trajectory. Math is always evolving, always extending, but what we see is with this gap is the kids that are doing poorly in elementary school, they tend to do poorly
in what? Middle school and math unless we really do something differently which enters in RtII but we're also seeing kids that do okay in elementary school and math, when they hit middle school, they
begin to struggle and that's that rational number piece. So, prior to the rearranging of the standards, it was around fifth grade that kids really started to struggle in mathematics. But right now
it's probably going to be bumped down to fourth grade where we start to see that same kind of challenges. So, at the macro level, whether you're looking at this through the lenses of an RtII
framework and I'm violating the oath of RtII trainers and that I will do this entire 90-minute session without showing you a triangle. Okay. I'm violating the RtII oath. Now there could be a triangle
up there but there are no vertices, it is not a triangle. But this is really what you're trying to do. You have to look at the math content, the curriculum, the materials through which you're
delivering the content and you have interventions that are all in the that aspect of it. Then you have the green circle which is--which is the assessment peace, and obviously there's a lot of
different levels of assessment for, you know, separate from RTI you have your end of year assessments, districts could have district benchmarks, plus then you have teacher test, but also the big
piece with the assessment is what are you doing with that data. And the most important thing is that you're making instructional decisions. So, if you're assessing and there's no instructional
connection to that data or those assessments, you really have to begin the question why are we doing this assessment. Now, of course the end of your assessments that has to happen in terms of the
school. Then you have this circle which is the teacher content and instructional, that's where--that's the circle that I tend to live in, in terms of the instructional piece. Now, what we see in math
is it--it's both the content knowledge of the teacher and the instructional techniques or knowledge of the teacher that make them an effective math teacher. And again this is where reading in math
for those of on an RTI team or sort of--and that is very different than reading and sort of what I highlight--the way I highlight this as is if I go to a 12th grade English class and I pull out
whatever book they're reading and it--let's say it's a Shakespeare book, and I bring it down to the elementary level. Every teacher at the elementary level can what? Read it, comprehend it,
understand it. They may not like it but they can read it. No problem. But if I go next door to that English class and the calculus class, and I bring down a calculus text to the elementary level. How
many are comfortable in calculus? Well, how about algebra to trig? Geometry? Algebra one, you see the big differences there and we really have to recognize that especially as the content is being
pushed when it was generally middle school is now being pushed down into elementary school that the content knowledge of the teacher is something that we have to pay a little bit more attention to in
math than--let's say in reading. All right. Instruction matters that's the big thing. There are things that you can do as a teacher, as a team that can impact students learning. Now, we're going to
do one small sliver of fluency today--in today's session and things that we can do there. Now, before we get in to the what to do, we have to understand why we are doing fluency or why fluency
automaticity has reemerged. Well, we have the national mathematics advisory panel report which came out in to 2008. So, over five years ago and in it, it has a lot of recommendations. Well, the one
recommendation that I'm point--picking out of it for today's session has to do with the content. So, in math the math panels says in order to get kids ready for algebra we have to have curriculum
that is teaching conceptual understanding, computational fluency, factual knowledge, procedures in algorithms, as well as problem solving skills. So, those are--have to be an emphasis at every grade
level. But the second bullet is one that is very, very important to our focus for today which is fluency and automaticity. What the math panel found is students that struggle in math tend to have
greater limits in their working memory. Now, you're remembering back to your first psychology class that working memory unit. So, you have working memory, short-term memory, long-term. When learning
takes place, when information is moved to where? Long-term learning--long term memory and then it is retrieved. Well, what we're seeing is they have greater limits. So, the math panel made a couple
of recommendations. One of which is get kids to levels of automaticity with certain skills in math. One of which is single digit arithmetic, but it extends beyond that in the new standards and when
you have kids that are automatic with their facts, they're using less cognitive capacity on the basics which freeze up more processing for problem solving, conceptual understanding, and application.
And one of the things that we have to recognize, when we see a kid sitting in a fourth grade class doing this with their fingers, that is a major problem. Actually third grade, this is a major
problem or this or whatever they've learned to do it up. Right. Whatever. Or than nines--whatever that is. That's a major problem you--we should look at those kids the same way that we look at a
child in fourth or fifth grade that's decoding every single word. You all work with kids that are very poor readers, right? Students with learning disabilities or even struggling learner--readers.
How do they read? Slowly, choppy, broken, because why is it broken because they're at either the word level or in some cases they're at the what? Sound level. So, let's take one of those kids who
reads like that, but let's say they read with 100% accuracy. So, they decoded every single word it was very slow, very choppy, they knew just enough site words that they read 100% accurate. When
you--when they get done reading it you say, "Tell me what you read?" What do they say?
>> They have no idea because they used up all of their what? Working memory on figuring out the words. So, when we have kids doing these strategies to all that kind of stuff at a certain point, that
>> I don't know.
becomes a problem. Now, it's not a problem in kindergarten, it's not a problem in first grade. So, counting on your fingers is okay, but when you're starting to extend that up into third, fourth, and
fifth grade depending on the operation that can be problematic, and to highlight this issue with working memory, I'm going to test your working memory. So, in 10 seconds I'm going to read some
numbers. When I'm done reading the numbers you write the numbers down in the order that I said them. Everybody got that? Okay. The--relax don't be so nervous it's not even to determine to TIER2,
okay? Just relax. All right. All right. Here we go. So pencils down. Here's the first set 38, 17, 82. Go ahead. Everybody get that 38, 17, 82? If you did not get all three of those do not volunteer
that information to anyone, just pretend like you got it right, okay? All right. Pencils down here we go. Next set 30, 11, 83, 3, 27. Go ahead. Not so easy. Here are the numbers 30, 11, 83, 3, 27.
How many of you did not get those--all those right? Raise your hand. TIER2, I'm telling you. No, no, I'm just kidding.
>> [indistinct chatter]
>> So, I lost about a third. All right. A third. All right. Well, let's--here we go. Let's do it again. Here's another set. Everybody focus. Breathe. Take a piece of peppermint candy because we know
that will help, right? That was a joke. All right. Here we go, 35, 10, 88, 9, 22, 64, 90. Go ahead. Okay. Some of your aren't event trying, like the unmotivated the students, right? Now, some of you
aren't even getting three right. Working memory is a funny thing, just because you could do three the first time, once working memory gets overloaded sometimes you can't even remember three or any
pieces so here--let's see if anybody got them right, 35, 10, 88, 9, 22, 64, 90. Did anybody get them all right? Now, you see what just--I--what I just illustrated with, three pieces of information,
100% recall. This is very important to our instructional activities. Five pieces of information, I lost a third of you. Seven pieces of information, I overloaded everybody's working memory in here,
and you could not remember what? The numbers. Now, how many of you were able to remember maybe the first number I said and the last number I said? That often happens in math. Teacher--the kids--the
teacher gives instructions, the kids processing that and the teacher what? Moves on and then the kid picks up at the end. So, working memory is a--an incredibly powerful tool for a teacher to
understand. Now, we're talking about memorizing or becoming fluent. So, why did I do this activity? Three, five, seven the biggest mistake we teachers are making with fluency practice is we are
practicing too many different facts at once. And kids are not able to what? Remember them. So, seven was too many for all of us, five was something that was even--for many of us was too many. Three
though, 100% recall. So, when you start thinking about activities for fluency or automaticity, we're going to think about this term called chunking. Putting information into manageable pieces. And
just keep this in mind three, five, and seven, the majority of your were overloaded at--well, all of us at seven, third at five, all of us could handle three. So, that's where we want to start
thinking about in terms of our practice activities. So, instructionally different that's kind of the question--the essential question that I like to ask teachers. What are you doing instructionally
different? So, if you're at a school right now, you're getting towards the end of the year, besides the fact we had to add 10 days of school on this year. My kids are in school until like June 13th
with all the snow days. And yet--teachers need--and should look back and sort of ask yourself this question. How many kids are not automatic on their facts? Depending on what great level of course.
And sort of what you have to think about is then what you've did worked for a percentage of kids but it didn't work for others. And now they need something different. Now, it's better if this
question is asked more in December than now, but that's kind of what we're trying to do. Is there's--there's a sort of common ways we do things. So what is fluency and automaticity? Well, this is
something I'm finding that there's a bit of confusion out there or different interpretation. In reading you hear fluency all the time, right? You don't really hear automaticity in reading, is that
correct? But you are doing something to automaticity in reading especially in middle of kindergarten, first grade, and second grade. What is it that you're trying to get automaticity with? Sight
words. That--so the kids look at the word and what do they do? They say it automatically. That's automaticity. Fluency has two parts to it, it is accuracy and speed. So fluency has two components,
accurately or accuracy, quickly, speed, two components. Students who posses fluency can recall facts with automaticity, we're looking at about two seconds. So you cannot count out facts in two
seconds. Now, I keep saying facts but this extends to middle school with equivalent fractions, simplifying fractions, perfect squares, recognizing all kinds of things, to a level of fluency or
automaticity is important. Now, I do believe my stance is that there's a different because fluency and automaticity. I will generally refer to automaticity when you want a kid to just to respond
automatically. So obviously facts are in that. Fluency, it enters in another area of procedures. So you have fluency, procedural fluency. So you can work through a problem quickly. You all know what
four times five is automatically. Do you know what 27 times 17 is automatically? No, but could you procedurally solve that problem quickly? That's the procedural fluency. So there's a--there's
a--there's subtle differences. But don't get too confused, as soon as you hear fluency or automaticity, it's accurately and quickly. Now, there is a progression that must occur when you're learning
operations. And I want to take you through this because I don't anyone to walk out of here and having their kindergarten teachers drilling their kids on multiplication facts. All right? Because you
could teacher, you know, make them--and they'll memorize them just like they memorized Brown Bear, Brown Bear, but that's not really going to help them because they don't what? Have no any conceptual
understanding. So there's a progression, there's a time and a place. So understanding activities, this is when the teachers are using a lot of manipulative and pictorial representations. They're
teaching the understanding of the operations. So with addition, there's a lot of putting things together, counting, for subtraction, they have a large set, they take some away or they have two sets,
what's the difference. For multiplication, you're getting into groups, and then arrays, division, you're putting into equal groups. They're understanding the operation. Most core math programs do a
good job with this. From that stage you move into relationships, and this is where you are trying to build relationships first within operations and then across operations. So you get into things
like the doubles, the doubles plus one, commutative property, fact families, the identity properties, plus zero times one, those types of things. So you're building operate--or relationships within.
Most programs do a pretty good job there, but this is kind of where programs begin to make some assumptions, and one big assumption that was very fatally flawed in the United States was many felt
that if you do a good job developing understanding and relationships, kids will then automatically become what? Fluent, and that just that doesn't happen. The way you become fluent or automatic at
anything is with some practice. So from that, those two stages, you then flow into fluency and fluency kind of has a couple of steps. When you're talking about fluency, now you're wanting the kids to
solve or answer a fact correctly. So now you've entered in accuracy initially, fluency is about accuracy, getting it right. So their kids are allowed to do all kinds of things, number lines, tallies,
drawing pictures, counting on their fingers, singing songs, whatever, that's the fluency. And kids initially, it's very slow but as long as they're getting it right, we're happy. As they continue to
get better with their strategies, they begin to get what? Faster, more fluent. And if kids keep practicing on their fingers, they will get what? Faster at practicing on their fingers but they will
never get to automaticity and that's kind of what we want. Automaticity has to be done with practice and it's at a level almost without thought, and kind of the way I illustrate that is I'll have you
blank out your minds, I want you to blank out your minds, I know you're thinking it's May blank, then blank for a month. But blank out your minds and I want you to not answer the question I'm about
to ask, do not answer it, okay, try not to answer it, okay, are we ready? Here we go. Five times four, the problem is you all what? You all answered it, 20, right? You got--I'm just making sure you
got 20. There was no counting on fingers, not even under the desk, in the pocket, that's automaticity. Now why is that important? This is--this is--this is what you need to leave here with as
educators. It is not--it's not important because you knew it, it's important because it frees up your cognitive resources to do other things and that's the --that's the number one reason why we're
doing fluency and automaticity is when you're automatic at something, you're not having to use up your working memory, and if you're not using up that limited capacity, you're freeing up resources,
cognitive resources to problem solve, understand, apply. All right. So, this is sort of a cumulative [inaudible] of what we have happening here in the sense that some fundamental things we really got
wrong. Calculators were going to save the world, right? Well, what we're finding out is, jeez, in order to use a calculator effectively, you really have to know your facts. How do you know when
you're working on a calculator you get an answer to it? That doesn't make any sense. Well, how do you know that doesn't make any sense? You've been doing mental calculations, estimation, very common
strategy in math, what really implies some factual knowledge. Now three, four, and five are what really starts to impact kids in third and fourth grade is kids that are slower at their facts take
much longer to work through problems. So a simple 10-problem homework assignment for kids that are counting on their fingers or using charts, takes forever. And then homework becomes very what?
Negative, with the parents, the family, everything. And then if the parents aren't good at math or they don't like math, then it becomes something that does not happen. From there, multiplication is
really what you need to master the manipulation of fractions, not conceptually but the manipulation. And then if you don't have fractions, you can't get algebra. So, here are some standards, fluency
standards that are in the course. So even in kindergarten, you start to see fluently show up and then fluently add and subtract within five. Second grade, now this is where it gets a little bit
interesting. So fluently add and subtract within 20 using mental strategies at the end of second grade. So think about--those of you in second grade or an RTI team, adding and subtracting at the end
of second grade is what we want all kids to be able to do it fluently which is quickly and accurately, but then you have this sub part of this. By the end of grade two, know from memory all 2, 1
digit numbers, sums of 2, 1 numbers. So by memory means five plus seven, twelve. It continues. Third grade, fluently multiply and divide within a hundred using strategies and so forth. By the end of
third grade, know from memory, all products of 2, 1 digit numbers. So where are my middle school teachers in here? Do all of your students have their multiplication facts at a level of automaticity?
High school, no, it's a problem? So by third grade now, we want multiplication by memory. But now, this is where you see fluency extending beyond a single digit. You have fluently add and subtract
within a hundred using strategies and algorithms. Here are some fifth grade, fluently multiply multi-digit whole numbers, fluently divide, fluently multi-digit numbers and find common factors in
multiples, that's an actual standard. So find fluently common factors why would we want that to be something we want our kids to become fluent with? Anybody?
>> Factoring.
>> Factoring, fractions, simplification, the whole nine yards. Here's a seventh grade one and this is one that I find, I still find it very interesting. So seventh grade use variables. So look at a
word problem right in equation with variables. But then they sneak in this solve equations of these forms fluently. So we want kids to be able to solve one in two step equations quickly so that
they're not using resources, so they can see how to apply it and so forth. Here are some algebra one fluency recommendations. So solving characteristic problems involving the analytic geometry of
lines so, given a point in an equation--or a point in a slope, right the equation. Adding, subtracting, and multiplying polynomials, fluently transforming expressions and chunking, that's the whole
factoring thing. So you see that fluency goes way beyond single digit arithmetic facts. So it's something that we got--we have to pay attention to it. And again, the whole reason we are trying to get
kids to fluency is it will maximize all their cognitive resources. And when we have kids counting on their fingers or doing elaborate strategies, yeah, doing decomposing numbers to get answers for
multiplication, they're doing repeated addition, that becomes a problem later on and we have to do something about it so now it enters in sort of practice. So in this slide here there's this bullet
here. Mass practice or Drill-n-Kill is not how we need to do this. I'm going to say that again. Mass practice, Drill-n-Kill is not how we need to do this, but we need activities that are Purposeful,
Planned, and Targeted, and that's the key, Purposeful, Planned, and Targeted. So these are things that I am seeing in schools that teachers are saying, "This is what we're doing for fluency." So I
see Drill and Kill, I see Games, and I'm going to highlight that one because that's a biggy, we're going to come back to that. I see Mad minute so they have like five minutes to answer sixty problems
on a page or something. I see Flashcards, I see Do at Home, right, and then I'm also seeing Computer games or Computer programs, give them calculators and others. And what I want to focus on is are
games. This is what you have to ask yourself. Are games, Purposeful, Planned, and Targeted? You have two minutes with the people sitting next to you, yes or no, Purposeful, Planned, or Targeted. A
second and I'm going to let you go back to your conversation. I'm going to highlight a game. Do you remember the game? maybe you did it as a kid, I remember doing it. I see it all the times still,
Around the World, you know, and--okay. Around the World, kids love it, teachers love it, it's the most and structurally, flawed, fluency game ever. Do kids like it? Yes. Do teachers like it? Yes.
That does not mean that it's Purposeful, Planned, or Targeted. Now why am I saying it's flawed from an instructional design standpoint? Too--and the way it's done is two kids stand up, they flash a
fact, whoever says the fact gets to what? Keep going. So in this activity, the only kid that's practicing the facts is the kid that doesn't need to practice the facts. Everybody got it? Now, you
could adjust it if you gave every kid a dry erase board and when the fact was shown, everybody had to what? Write it down. So it's not--you can adjust it, all right. So with that position for me back
to your discussion, Games, Purposeful, Planned, and Targeted, go ahead. Okay. Now, I'm hearing yes, or, you know, the easy way out well it could be, right? Non-committal, one way or the other, and
here and, you know, yes it could be but here is my take. Any game that is intended for fluency, that involves spinners, dice, or pulling a card, are not in any way, shape, or form purposeful or
targeted because you have entered into the activity randomness. You do not know which fact they are going to get. So it is in no way, shape, or form targeted. So they have two dice and they roll them
and they could one and three, one and four, one and six, two and three. If it's six sides, they never get what? Sevens, eights, nines, all right. Now there are ways you can make it a little more
targeted by masking tape and writing certain numbers so like write six and seven on one dice and write five and four on another so you know they're just going to get those combinations. But any time
that you have randomness, you have lost all instructional control, and that is what the most struggling kids need, very focused and very targeted. So the double-edge sword with games is do kids like
playing games? Yes. Can kids do that at a level of independence? Yes, which means then the teacher can do what? Other stuff, small group, so I'm in no way saying get rid of games, all I'm saying is
we got to make sure they're more targeted and more focused. Or we--what happens is it is not efficient practice. All right. So that's true on my take on it. Parents do it at home. Well, do you have
that support of parents? You know, that's a question you have to ask. If you're in elementary school, you know, and you've had parent conferences with parents. One of the big thing, even in the
middle--well, certainly by middle school, I experienced that middle school and high schools. You got parents coming in and you want to talk to them about their son or daughter's math, and what's like
the first thing that comes out of their mouth? I want to help but I can't, or I wasn't very good in math. So, you know, parents, they don't necessarily know how to do this effectively or not. So
these are kind of the things they start thinking about, Purposeful, Planned, and Targeted. Now there are six steps to fluency practice. Now, these six steps should be used to evaluate whatever it is,
whatever activity it is that you're doing for fluency. And a good example of this--so when my son was in fourth grade, his teacher said--wrote a letter home. It's important that they memorize their
multiplication facts so make sure that you are paying attention to this. So, my son comes home and I say, "I got a letter from your teacher," and he said, "Well, I didn't do it." No, no, he didn't
say that. But he --I said, "You're supposed to memorize your multiplications facts. What are you supposed to do?" And what does a typical fourth grader say to his dad? "I don't know." "Come on now.
What are you supposed to," "I'm supposed to write them down." "But what are you supposed to write down?" "We're on the fours today." Okay. Now, cognitively there should be some things turning, if
he's on practicing the sets of four, how many is he trying to memorize at once? Zero to ten, or zero to twelve is thirteen separate facts. How many pieces of information did I overload all of you
with? Seven, five, some of you? So that's the first issue, from a cognitive perspective, it's not really targeted. Second, how he was practicing it. He said he's supposed to write them down. Now my
son is a good advance--he's in advance math. He does really well in math so I said, "Write it down." And this is what he wrote down four, four, four, four, four, four, times, times, times, times,
times, times, one, two, three, four, eight, twelve, sixteen. He's not getting automaticity with the facts, what is he getting automaticity with? Skip counting. So we got to pay attention to how we
are practicing these activities. So here's kind of the question you want to ask. Now I just have this up through eight grade and for the purpose of conversation we're going to be focusing
on--initially on single digit arithmetic facts and I'll just do multiplication. So, the question you want to ask yourself is what percentage of kids--let's say in fifth grade in your school are not
automatic on their multiplication facts? So, if they're doing multiplication facts you begin to see what? Fingers, tallies, drawings, what percentage? Now, I don't know what percentage. I'm throwing
this to you. But this is the very first question you need to ask. Certainly if you're in an RtII team or just in general, what percentage? So, in second grade--at the end of second grade, what
percentage are not automatic with their addition? Third grade, which ones are not automatic with their multiplication? But I'm picking fifth grade because that's two grade levels pass where they
should all be automatic with addition and multiplication. If the percentage that you're thinking of is greater than 30% or 40%, you have a core math problem. Not a Tier two problem. That's way too
many. If you're saying, "Oh, it's only five percent." So, 95% other kids have all their fluency and automaticity. Then in that case, it's a Tier two issue. But if you're getting into the 30s, and
40s, or higher, that's not a Tier two issue. That issue needs to be dealt in Tier one first and get that percentage way down before you start addressing it in Tier two. But that's kind of what you
want to ask, "What's the percentage?" All right? Now, next question you want to ask is, "Well, what are we doing currently over this past year that is fluency?" And you're going to get things like
games or computers practice. Or whatever it is. You want to--and you want to look at that next to how effective it is and for whom it's effective for. And if you're talking 30 or 40% aren't fluent or
automatic, then what you're doing is not working for that group of kids and we need to do something different. Then, how can you sort of address these areas or can those--can those activities you're
doing be adjusted to meet these six pieces? These are the broad pieces for fluency and automaticity. So, if you're going to buy an intervention or a supplement to the core that says it's about
automaticity you want to look at that program and relationship to these six components. There are some programs out there that do pretty good at these six components. Or if you're not going to buy a
program, then you need to set up activities along these six. And it's really the first three that are most important. But I'm going to start with number four. The most important thing with fluency is
the teacher commits to it. Five to ten minutes daily. You--you're not allowed to say, "I'm going to save up my five minutes until Friday and do twenty-five minutes of practice." It's the distributed
practice that's important. Now, I have schools I work with, some of them do it the first part of the--of their math class, some of it do it at the end, some of it leave it up to the teachers, some of
it do it in elementary school at a separate time. My experience is it's best when everybody says we're going to do it and we're going to do it at this time. And then it tends to happen. All right.
So, let's go through these. The first, number one, how many facts are the kids starting out with? Now, for the--when I--when I say they don't know the fact, I'm talking about they don't now it, what?
>> Automatically. If you let them count on their fingers, they'll be able to figure it out. So, the--one of the best things to do with the program is how many new facts are the kids practicing at
>> Automatically.
once? And what should at that number be between three and five--really four, three and four? So, when you're thinking about this, they should be practicing three or four facts out--that they don't
know at once. Now, here's the question. Do this strategically. So, let's just ask--let's just do this with multiplication. Think about your kids that are in fourth or fifth grade or higher that don't
have their multiplication facts memorized. Which multiplication facts do they have automatically? Zeros, ones, twos, fives.
>> Tens.
>> Tens. So, those are facts that should not be in their practice because, what? They already know them. Let's make our practice very focused. So, now we're streamlining. They know them, so let's not
have them in practice. So, you're going to pick three or four that they don't know and that is going to be the focus for several practice activities. Everybody with me so far? Now, the second one is
also very important. It's not about practicing each fact once, it's about practicing each fact multiple times. Have--you know, these kids that--when I was a high school teacher learning support I'll
never forget this. The first couple weeks of school, I met the kids, I'm watching them do basic math and they're counting on their fingers and they want to calculate it, they do not know their facts.
Typical characteristic. Well, then as I got to know these kids, they're singing word for word songs from their favorite artists. They can tell me every level of whatever video game they're playing.
And how can they memorize all this stuff, but there's only 390 facts and they can't memorize their facts? Well, bottom line is we're not practicing it in an efficient way. Because if it was truly
their issue they could--would not be able to what? Memorize these other things. So, we're going to chunk it down to three or four and then we're going to give them intensive practice on those three
or four facts. This could be in a game. This could be flashcards. So, if you're doing it on flashcard, you pick three facts, each of those facts is written down on four flashcards. Everybody with me?
So, if you pick four facts, they get written down on four flashcards each, you now have 16 flashcards. And in this practice activity, they're going to practice each of the ones they don't know
multiple times. And then the third step is there's going to be a cycle that begins. As they memorize the three or four initial ones that were targeted, those get reduced and you add what? Three or
four more and a cycle continues. As they learn them, they drop out to the pile so you don't have 500 flashcards by December. Now, there's--this may help--I think I put it in here, a visual. So,
here's a visual of the first couple of steps. So, step one and two, I pick four new facts, fact one, two, three, and four. Those are the ones that I want them to have intensive practice on so they go
on four flashcards. Now, if I'm just starting this off to mix it up, I will pick four other facts that they know. We'll call those previously learned. And I mixed them in. So, now I have 20 fact--20
flashcards. Now, it doesn't have to be flashcards, this can be on paper, whatever, computer. There's some computer programs that will allow you to target certain numbers. It's not the mechanism
that's so important, but it's how that is set up. So, we practice this for a week and I'll get to practice here in a second. And the teacher says they know their facts. They do a little test out on
them and the kid knows all the--those four facts. So, now you're ready for the next step in this cycle. So, what happens is you pick four new facts that they don't know. Facts five, six, seven, and
eight, and those become the targets. So, those will go on four flashcards. The ones that had been the target, in other words they were getting four flashcards, now get reduced down to two flashcards.
So, you're fading. The four that you mixed in that they're already knew but you just mixed in now they are just gone. And you can have this cycle continue. Practice this and then you get ready for
the next step. You pick four new ones. The ones that had been the target are reduced to two cards. The ones from the previous step are down to one card. If you do flashcards, this is the most
flashcards a kid will ever have. And you process that so you don't--I don't overload your working memory. Now, there are a lot different ways that you can practice this. It can be individual, it can
be silent individual practice, they can silent say it out loud, they can write the answers, or the way I prefer is partner practice. So, they're practicing with a partner. Practicing with a partner
means that they will be saying it, what? Out loud. There's some research studies that are indicating verbal practice can accelerate acquisition of fluency. So, if you're verbal practicing with a
partner and you make a mistake or don't know it, what can happen? A correction can happen. If it silent practice to themselves they could practice it wrong how many times in this activity? Four
times. They could write it wrong. So, there's a variety ways you can do it. I prefer partner practice. So, the way this would go--if you're my partner, what's your name?
>> Wendy.
>> So, Wendy is my student in my class. She doesn't know her multiplication facts. I've got the card set up. So, we're going to practice. Now, Wendy, I'm going to pretend--we're going to pretend I'm
holding out 6x7 and I want you to miss it, okay? Here we go.
>> 36.
>> Okay. First thing, they should say the number sentence. So, you want them to say 6x7=. All right. So, say that but still miss it. All right. Here we go.
>> 6x7=36.
>> No, 6x7 is not 36. Higher.
>> 40.
>> Closer. 6x7 is not 40. Higher.
>> Okay. That's the worst thing to do. All right. That's a none example, okay? Now, there are two things I was doing that weren't good. What was one of them? From a teaching perspective? I was--when I
>> 41.
said no, she was just what? Guessing. So, that serves no purpose. So, that was--but I did something even worse than that. Anybody? When she said 40--when she said 36, I said no, 6x7 is not 36. The
problem is all Wendy heard was what? 6, 7, 36. All right? So, you don't want to that either. So, this is how it should go. All right? Are you ready? Miss it again. You were doing a very good job at
missing it. All right. Here we go.
>> 6x7, 36.
>> Stop. You've missed this one. 6x7 is 42. Say it with me.
>> 6x7 is 42.
>> 6x7 is 42.
>> Good. You say it.
>> 6x7 is 42.
>> Very good. 6x7 is 42. Everybody got that? Now, we're not--I'm not done with the correction yet, but the first thing is you stop them. I said it, we said it, she said it, I said it. So, four times
the correct answer was said. Now, this is where the cognitive piece comes in. That card, if you're doing with flashcards, that card did not go in a pile. No more piles. Get--stop with the piles. It
doesn't work, okay? That card doesn't go at the end. It gets moved back three flashcards. Now, why? Why did you want to see it again quickly?
>> So we can practice it
>> Yes, but why quickly?
>> [inaudible]
>> I want her to see that fact while the correction is still housed in her short-term memory. And that's what happens when we put them in pile, you know what I'm talking? When they miss it, it goes to
the miss pile? Well, they go through like 20 other facts before they get to what? The miss pile. So, this is a very focused, intensive, purposeful, targeted, and planned way of practicing it. All
right. Now, this time, Wendy, I want you to start counting on your fingers. But you go to hold it up so everybody can see. So, say 6x7 and then do everything--not yet, not yet. Okay. Here we go.
>> 6x7 is 42. Is that right?
>> No. I want you to start counting on your fingers, but don't give me the right one. Here we go.
>> 6x7 is...
>> Stop. You've missed this one. 6x7 is 42. And you go through that same correction. If you see them counting on their fingers, what if I let Wendy continue to count on her fingers? She'll get the
right answer. But in this activity, what is the desired response from the student?
>> [indistinct chatter]
>> So, anything else is treated as an error and corrected.
>> [inaudible]
>> Yes?
>> Can you just verify, are the students doing this with each other as well?
>> Yes. So, this was a good question. Usually at this point someone asks a--well, that's great. If I got 25 kids in my classroom, how am I supposed to practice? It's peer to peer. You teach your
students how to do the correction and they'll learn it. It takes about three days, fifteen minutes across those three days, and the kids will learn how to do it. Just so you know that will just--when
the kids learn this correction procedure, that will generalize into other areas. So, you'll calls--I'll call Wendy Marry and she'll say stop. You missed that one. My name is Wendy. Say it with me,
Wendy. Just be ready. They will generalize that. Does that make sense?
>> So, yeah, this is partner to partner. So, what would happen is you take about a week to teach the kids how to do this, get the activity set up, get your materials organized, they come in, get your
>> Yeah.
fact practice, they have 90 seconds. So, Wendy is practicing for 90 seconds. The bell goes off, what do we do? She takes the cards, I practice for 90 seconds. That's three minutes, throw in two
minutes of transition, how long does that activity take out of your math time? Five to six minutes. Never more than 10 minutes.
>> Have you matched your partners up with the same fact practice?
>> You don't--you can. Initially what I say is start everybody on the same facts for the first week. That way they're learning the procedures, you're learning the procedures. What will happen then
that some kids will begin to take off very quickly, other kids will make progress and then other kids will even still--that's why it's not going to work with everybody. Once you've start getting kids
going in different directions, it sometimes hard to match them. So you make sure that the answers are on the back or somehow the partner has the answers so that they can correct them with the
answers. Now, if Wendy does not answer, so if she says 6x7 and I can't see her fingers counting, you wait how long? Two to three seconds, and then you what? Give me her answer. A good indication of
their--they don't have automaticity is when you show them 6x7 and their eyes go up and to the left. All right? They're thinking. So, any response other than the correct answer is treated as an error.
And then it's I say it, we say it, she says it, or the student says it, and the grade for fluency. Now, if you're going to give any type of greater points it should be on if the kids are increasing.
Now, Wendy--so I'm doing all this, but Wendy is still not making progress and this will happen for some of your students but it should not be in the 30 to 40% range. There are couple things you can
do. One, you can reduce the number of new facts that they are targeting. So, instead of having four maybe you go down to what? Two. That's one thing you can do. You can also double dose in Tier two
if the kids were also going to Tier two. Instead of practicing with a peer, they can practice with you. Now, so I have Wendy--because usually at this point someone says, "Well, what do I do with
Wendy, the student, who continues to count on her fingers every time she sees 6x7?" So, I'm doing all this. I'm correcting her that whole nine yards, but she's still counting on her fingers. So, do
something different that she won't have to count on her fingers. Well, the first question you ask is, "Why is Wendy counting on her fingers when she sees 6x7? Why do you think? She doesn't know it so
that's her what? That's her strategy which, by the way, a teacher taught her which is fine. Now, we're trying to get her what? To go past that. So how can I set up my activity where Wendy will not
have to count on her fingers when she sees 6x7? This is problem solving, I'm asking you, what would we do? I'm generating some ideas from you expert teachers in the--in the audience. What would you
>> Give her something to do with her hands.
do?
>> Give her something else to do with her hands. Like what? Sit on her hands?
>> [indistinct chatter]
>> If she sits on her hands, then she's just rubbing her butt every time she sees 6x7, okay? Okay. This is good. So get--have her hold the flash cards, so you're trying to do a replacement behavior.
Have her hold the flash cards and then she does this, still counting. What else can we do? Those were good.
>> [inaudible]
>> So then just take 6x7 out?
>> [inaudible] develop a pattern [inaudible]
>> So if you're developing a pattern, is that--what do you mean a pattern?
>> So, for some kids, they form a pattern and they don't even know it.
>> Okay. So--okay. So we--I did that. She's still counting on her fingers. I agree. Sometimes, it's just a habit with kids, but we want to break that habit. Anybody else?
>> You can talk to this kid about it?
>> We talked to Wendy, she says, "I know," and then 6x7, what did she do?
>> [inaudible]
>> What else? Come on, this is problem solving. RtII, it's not going to work for every kid. Yes.
>> She could trace the multiplication table with her finger.
>> Trace the multiplication table with her finger?
>> Right. And write 6x7 [inaudible]
>> Okay. Maybe. I have something easier than that.
>> Try the inverse. Well, if --now, this is good. That's a good point. If you say I'll give her the inverse, so then what you're assuming is that she'll say to herself, "Well, I know 6x7 is 42. And
>> Try the inverse.
because of the commutative property, I know that 7x6 is 42." That's a relationship activity. So we don't want to go there. Anybody else?
>> Have her write it.
>> Have her write it? She still counts and then writes it.
>> [inaudible]
>> Well, if you're doing this activity, you are at the automaticity stage.
>> But what if maybe she's not?
>> How do you know that? If I let her count on her finger and she gets the right answer, to me, she's at the concept, she understands it. If she's not counting her fingers and not getting the right
answer, then that's a different situation. But if she counts on her fingers, she gets the right answer, put the answer on the card. Write 42 on the card. So when Wendy sees the--when she's practicing
now, what is she going to do? She's just what?
>> Reading.
>> Reading. And after she does that three or four days, she will eventually commit that to what? To memory. So once I think she has it by memory, take the answer off of two of the cards. Gradual
release. If the answer's on there, will she have to what? Count on her fingers? That will break habits because it could be a habit. It will take the need to count on her fingers out, but I want to go
back to this question up here about is it a conceptual issue? And, you know, there's really no good way to get that conceptual, but if they're giving you the correct answer with a strategy, then they
are probably at the--they probably have the conceptual, they've already moved into the fluency. Here is--if you have the handout materials, here is the step by step on the corrections. Stop the
student, I say, we say, you say, I say, moves back three cards. So they guarantee that they will see it while it's housed in a short-term memory. All right. So here's what I want you to do. Three
minutes. Number of students who are not fluent, that's the very first question you want to ask at the--at the RtII level, or the team level, or if you're a teacher level, how many are not fluent?
Then you will discuss the activities that teachers are currently doing. Think of this as you go back to your school. These are the questions you want to ask. Can those activities be adjusted to hit
the six steps? And in the end of this is develop a plan to address fluency. So this would be a professional learning community focus if you have PCLs, it could be an RtII meeting, a school leadership
meeting, or a grade level meeting. So what I want you to do now with the people sitting next to you is kind of focus in on this. How--what are we seeing in my school or what am I doing to practice
fluency and are they close to these six steps? And I'll go back to those six steps so you can see them. That's what I want you to discuss for about three minutes. And then if you have a question,
I'll come over and answer that. We have about fifteen minutes because I want to get to two other--two other types of fluency activities. But at this point, I just want to sort of emphasize a couple
of things, fluency means accurately and quickly so it has to be timed at some level. There's a lot of different ways that you can practice fluency. But what we're seeing is the struggling kids,
they're not going to respond to around the world, the typical games, get--by the way, the way those--when you're rolling dice and doing those kinds of games, that's more accuracy than fluency. So
they're fine, it's just it's not going to build speed necessarily. So you want to think about fluency as accuracy and speed. Accuracy comes first, then the speed, and then these six steps. To get
that targeted practice, you want to focus on three or four, they need to have intensive practice on those three or four, underneath it would of sort be a cycle. There should be five to ten minutes.
I'd be horrified if I ever walked in the classroom and I saw thirty minutes of fact practice. Just because you become automatic on your facts does not mean you are then automatically a good problem
solver. You have to teach that. But this will help with capacity of the cognitive resources. Are there any general questions along the six steps? I have two other activities I want to show you. Yes.
>> I have this question about the flash cards. Do you suggest having them go in order like 3x4, 3x5, or 3x7, or mix them up?
>> So the question is, "Should they go in order?" Some will do them by fact families, some will do them in order. I do not. I would rather do them out of order. You just randomly pick. But what you
need to know is each fact should be treated as a separate fact. So 6x7 is a fact. 7x6 is a fact. So you treat each fact as a separate. The key is, by the way, it's not about the flashcards, it's
about the six pieces or those six components. It doesn't have to be with flash cards. It can be on just regular sheets of paper in rows and columns, but it's the--it's the targeted intensive
practice. Good question. Any other questions? Yes.
>> [inaudible] practice should not be just about [inaudible] should be [inaudible]
>> It--yeah.
>> What if you switch that over to [inaudible]
>> What I--I guess what I was saying is if to the--to the question of how many kids are not automatic in fifth grade on multiplication or fourth grade. And if you say 40 percent or higher, what I'm
saying is you have a core program. It's got to be addressed in the core. Those kids still will need what? Tier two and it can be addressed there. But if you just look at that and say, "We'll address
it in Tier two," 60% of your kids is way too much. It will overload your Tier two. So if that's the issue, you got to have a two-pronged attack, one is in Tier two but the other one is the core
teacher has to be doing something about it. The R--the RtII document from IES in math recommends five to ten minutes of fluency practice in Tier two. But what I'm saying is because fluency's embedded
in the standards, fluency has to now be a part of core instruction. Yes.
>> So the answer that question [inaudible] 30% or more [inaudible] what kind of screening tool are you [inaudible]
>> So the question is, like, what kind of screening tool? You could give the kids a one-minute or a two-minute probe to see--or basically, what I'm finding is teachers' judgment are pretty good as far
as who's automatic and who's not. So you can go to that screening tool, but generally, when you're looking at 30% and 40% of the kids, the teachers know it, everybody knows it, it should be in the
core. All right. Good, very good questions. Any other questions before I move into fluent computational fluency?
>> [inaudible]
>> [inaudible] screening tool [inaudible] different expectations, you know. So are you [inaudible]
>> Yes.
>> Well, that's actually a very interesting question. You notice the standard, they say fluency fluently by memory, but they don't do what? They don't give you what that is and that is a big question
that we really - the range is from like--I've seen research in the 70s that's looking at 20 to 40 to 90 in a minute. So there is--it's a big range. Most of the time, people are looking at 40 facts in
a minute. If you can do 40 in a minute, then you're probably in an okay range. I expect to be--there'll be more data coming out about this quickly. All right. So there are--here's--so beyond single
digit arithmetic facts, there's also computational fluency, in other words, solving multi-step problems quickly. So there are a couple of other activities. So again, when you're thinking of
activities, what you want to think about is targeted multiple practice opportunities. So there's this one activity called Math Dash which is just another way to practice this, so--it's a good way to
differentiate, too. So, I'll have you do this in a--you need to get something to write with, a piece of paper. And then there's going to be two boxes that show up on the screen. One box is an
elementary skill, place value, another box is more middle school, integers. You pick which box you're going to do. All right. Only one problem will flash up at a time. All you need to do is write the
answer down. Everybody understand that? Pick your box, left or right. We're going to do this twice. Your goal is to answer more correct the second time. But because you're all geniuses in math, I
need you to take your pencil and put it in your non-dominant hand and you write with your other hand. All right?
>> [indistinct chatter]
>> Are we ready? Here we go. All right. Check your answers. Now, if everybody got them all right, then I wasn't going fast enough. All right. Remember, we're going to do this how many times?
>> [indistinct chatter]
>> Your goal is to answer more correct the second time. All right. So fold those, cover them up. Are we ready? Here we go. Non-dominant hand. Here we go. How many got more correct the second time? All
right. That's the idea. Fluency. Now, if you did this once, is that really going to build fluency? But if you did this everyday for five days on these skills, is that going to begin to build fluency
on these skills? Yes. Plus, with two boxes, it's easy to what? Differentiate. All right. So this is, again, it's not about the specific activities. Here's the steps. First thing, step one, explain
fluency to your students. What it is, connect it to something that they know about. Texting. Remember when you first saw--wrote a text, how slow it was? And now, you're what? Super fluent. I think
our world record was just broken with fastest text recently. Explain to them what fluency is. Kids don't really know what it is, what it means. Second, pick a skill that kids are already accurate on.
If I walk into your room and you're doing that activity with place value and integers, that tells me you've already developed the concept, the kids already have a strategy to get it, and they're
relatively accurate. What you're trying to do is build their what? Speed. Display one problem at a time, create eight to ten problems, display it one at a time. You can vary the rate depending on the
kids and the skill. Some--if you do order of operations, they're going to need a little bit longer for each problem. So there's that activity. Here's another activity. Now, how many of you have the
handouts? The--if you flip to the back, you should see two pages like this. Do you?
>> [indistinct chatter]
>> So, this is kind of borrowed from method of repeated reading. Do you remember--who--anybody familiar with that? It's a very effective way to increase reading fluency. You give the kids a passage
that they can read at 90% accuracy. They're asked to do what? Do their best reading, read for one minute, they put a circle or underline the word they read last then they go back and they have one
minute and they have to try to read what? Further. So let's take some of those principles and apply them into math. So, if you have those sheets, what you do is you give the kids a sheet with
problems on it. I pick fractions. So the kids have to do what? Simplify as many fractions as they can simplify in what? One minute. And you would say, begin and they'd go through it. Display the
answers, have them flip it over, same problems. Now, they have to answer, simplify what? More in one minute. Everybody see the connection there? It can be the same, but you're building fluency.
That's one of the things we don't do in math other than with single digit arithmetic facts is practicing the same things multiple times. That's how you become fluent. So this is a--an example that
you might do with fractions. So what's nice about this activity is you don't have to create your own practice sheets. There are thousands of free practice sheets online. What you have to do is
explain to the kids what fluency is, what the purpose, it is very, very, important in this activity more so and then in the one we just did is that they understand. This is not about answering all of
the problems on the page. Because if you give this to kids and you have not explained that, they are going to what? They're going to freak out because they think they have to answer what? All of
them. That's probably one of the biggest mistakes that we are making in math is we're not explaining to the kids what fluency is. And already by second grade, they have learned that whatever sheet
the teacher gives to them, they're--for homework, they're supposed to what? Answer all of the problems. And in these activities, it's not about answering all of the problems, it's about what?
Answering more each time. And I think that's one of the big things. For those of you that asses or are reading fluency with any of those measures, do any of the kids finish reading the whole story?
No. So we have to make sure to explain fluency. So in this method, you explain fluency to the kids. Again, you're going to pick a skill that they are in the 80% to 85% level of accuracy, create or
find a worksheet that has the problems more than they can finish, have the kids solve as many as they can in a minute, they should not be allowed to skip around. They can skip a problem if they don't
know it, but they have to go to the next one. Time them for a minute, do that twice or three times, but don't do it any more than that. Sometimes, you just say, "Well, if twice is good, I'm going to
do it five times." Two times. But it's got to be more than once a week, kind of a regular activity.
>> [inaudible]
>> Yes. Question? Yes.
>> Yeah. So they time it for one minute, check your answers, time it for another minute.
>> Two to three times [inaudible]
>> And then multiples times a week?
>> And then multiples times a week. So maybe they do simplifying fractions for two weeks and that's the activity they do. If you're at middle school or high school, integers, perfect squares,
exponents, I could go on an on with different things for fluency. All right. So fluency and automaticity, in my other sessions, I'm going to get into content scaffolding and some other techniques
tomorrow. Then my next session is, again, fluency. Tomorrow, I'm going to get into some other instructional scaffolding techniques for problem solving and things of that nature. Today's session was
about fluency. It is in the new standards. It is almost at every grade level. And in almost all things, there is a level of fluency that is needed, multiple ways to do it. The key piece is targeted,
intensive multiple times in a continuous cycle. Questions? All right. Great. Thank you very much. I hope you enjoy the rest of the day. Thank you.
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