Webinar: Assisting Struggling Students with Mathematics


STEPHANY BROWN: –works
for tiered interventions in elementary and middle school. I welcome you to
today’s online seminar. I’ll begin with an overview
of the webinar goals and objectives, today’s agenda,
and a little information about the Regional Educational
Laboratories serving in the central region, who is
sponsoring today’s webinar. And then we’ll introduce
the presenters. When you entered
the seminar today, your audio was muted due to
the number of participants. However, as Joe mentioned,
you do have access to the chat box, which you
can use to ask questions anytime during the webinar. The facilitator and panelists
will see the questions, and will have several
minutes at the end of the webinar for responses. The online seminar
or will be recorded, and will be posted
within the next few days to the Institute of Education
Sciences’ YouTube site. A link to the site
will be provided at the end of this webinar. So our goals and objectives
for today’s session are to increase understanding
of research-based strategies, to improve mathematics
instruction for struggling students, and to acquire
a greater understanding of available resources and
actionable knowledge that can be effectively implemented
to meet the needs of students who struggle in mathematics. Our agenda today features
Dr. Russell Gersten providing an introduction
to the IES practice guide, Assisting Students
Struggling With Mathematics, our participant
activity, discussion of fractions intervention
by Robin Schumacher, and Introduction to Data-Based
Instruction and the National Center on Intensive
Intervention. And again, several minutes
at the end for questions. If you post your questions
during the webinar, we will capture
some of those and be able to answer them at the end. And then we’ll also
provide the information on participant contacts if you
have any additional questions. The Regional Educational
Laboratories, or REL, work in partnership with
educators and policymakers to develop and use
research that improves academic outcomes for students. The 10 Regional
Educational Laboratories work in partnership to conduct
applied research and training with a mission of supporting a
more evidence-based education system. REL Central at Marzano Research
service the applied education research needs of Colorado,
Kansas, Missouri, Nebraska, North Dakota, South
Dakota, and Wyoming. We welcome our colleagues
from the REL Central states, along with everyone
from other REL regions and states to today’s webinar. Our presenters today
our Dr. Russell Gersten and Dr. Robin Schumacher. Dr. Gersten is the executive
director of the Instructional Research Group in Los
Alamitos, California, professor emeritus in
the College of Education at the University of
Oregon, and a subcontractor for the Regional
Educational Lab Southeast. He served as the chair
of the panel that developed the What Works
Clearinghouse practice guide that is being discussed
in this webinar. Dr. Gersten has
authored the book Understanding RTI
in Mathematics, as well as multiple chapters in
books on mathematics screening efforts and conducting
rigorous group studies. Dr. Robin Schumacher
is a research associate with the Instructional
Research Group, and currently manages an
NSF grant investigating a fractions intervention
for struggling mathematics students. She has coordinated several
other large scale research studies funded by
IES and NIH, all focused on increasing
outcomes in mathematics. Dr. Schumacher has
written on topics related to fractions intervention,
intensifying instruction for low performers, and
analyzing error patterns in mathematics, and has
authored multiple articles in peer-reviewed journals. Welcome, Dr. Gersten
and Dr. Schumacher. And Dr. Gersten, the
panel is yours now. And Dr. Gersten, if your
phone is on muted, please– RUSSELL GERSTEN: Oh, I’m sorry. I’m very sorry about that. Yeah, OK. Good afternoon. STEPHANY BROWN: We can hear. RUSSELL GERSTEN: Yeah. Can you hear me now? Correct? I’m Russell Gersten,
and I’m going to quickly begin by introducing
the practice guide and who helped create it. It was done a decade ago,
when RTI in math was new and rarely implemented. I’m going to, at the end,
give a quick update on some of the research since then. But in my view–
and I think a belief of some of the leadership at
IES is– an update of a guide like this would
be a nice project to begin in the near future. In any case, I chair it. We have, intentionally,
a research mathematician, Sybilla Beckmann, whose
text is used heavily in pre-service training courses. on the group. We had
two folks with more of a school of psychology
background, Ben Clarke and Anne Foegen, who knew a lot about
screening progress monitoring; Laurel Marsh, who was then
a math specialist, beginning some implementation of
interventions for kids struggling in math, and is
now an assistant principal or principal; Jon Star is more
of a cognitive psychologist; and Brad Witzel, who had done
some pioneering work, really, in algebra instruction for
kids with learning problems. Okay. What we tried to do here– and given the fact that 10 years
ago, when the panel met and we did the research
searches, there was not a lot out there on RTI in math,
there was not a lot going on, we had to borrow pretty
extensively from the special ed research at that point. Because that was pretty
much all that there was. So our goal was to
take all of this and figure out
what can we create that’s practical and coherent. And these guides are
downloadable for free. And Stephany can talk a
little about that at the end, and how they may be used. But basically, the idea
was to try to make, in clear, comprehensible
language, ways to improve practice in this
kind of neglected area, helping kids who struggle
in math, before teachers or parents consider an
actual special education possible placement. So we wanted things
that were not crazy, but that took some risks, that
didn’t have the typical thing that we read and hear about– I just came from a
conference where– on the other hand, we can
overgeneralize from this one study in whatever. We were encouraged
to take some risks. And most importantly, to develop
something that was coherent. Because I think all of us have
read literature reviews, where there were all these
things in there, but you don’t really
know how to fit, and how to get a
hold of what they’re trying to say about improving
teaching, or improving those folks who supervise
or train interventionists to improve intervention
instruction. This document has been
pretty consistently, over the past decade, the most
frequently downloaded document from IES, which is the research
branch of the Department of Ed. And that’s how to download it. And you can– what I usually
do is just Google RTI practice guide and math, and you get it. What each of these guys have–
and I imagine quite a few of you are familiar with
them in some area or other– is specific recommendations,
and then some how-to steps, or just action steps,
and levels of evidence. And this took a
lot of work of us. And this was one of
the early half dozen to figure out how
to give relatively clear, sensible information
on the level of evidence. And we also talked
about roadblocks, because anything you do in
any field, any walk of life, you run into roadblocks– some ways how to overcome them. The levels of evidence
at that point– and they really have not
dramatically changed– were strong, meaning there were
at least three high quality, rigorous studies that met really
rigorous standards that IES initiated 15 years ago
when Congress decided it’d need to radically
change the nature of educational
research so that it had the rigor of
many other fields like engineering, public
health, economics, workforce studies, etc. Moderate means there’s one
or two studies suggesting that this is a good way to go,
but we’re not totally sure. Minimal means the panel
thought it was a good idea, they’d maybe seen anecdotal
evidence of it being effective, but there is no hardcore,
rigorous evidence. Now, in some cases, people
believe more strongly in the minimals than
they did in the strong. But we wanted this
to infuse principles of the scientific
method and rigor. And this is a quick overview
of the eight recommendations we made. And these are quite
truncated, but you can download the practice guide. Some of you, in
advance, have maybe taken a look at it and all. And what I used to do is walk
through all eight of these. And then I realized
I was starting to get bored by the time I
got to about number four, and thought, if
I’m getting bored, what about the folks out there. So we’ve arranged for more
either interactive– well, some level interactive
way to do this. And what we’ve done
is split them in two. And what I’m asking you to do
is to look at each set of four– and many of you may be watching
this together with colleagues, you can have a
little bit of time to chat with your
colleagues about them– and then we’ll poll you and see
which is the most surprising. So I’m going to
turn things over now to Joe, who’s going to
oversee this polling process. But essentially, the idea is
to look at the first four, see which surprises
you the most. OK, Joe? JOE: Indeed. Here is the question. Which levels of evidence
are most surprising? Please choose one
from each group. And I believe you’ll
now see the poll. And just go ahead and click
on which one you think is the most surprising when
you’ve come to a conclusion. RUSSELL GERSTEN: And remember,
the one minimal one here– I don’t know, Joe,
if we can go– well, we probably shouldn’t go back. But the one with
minimal evidence is that the core
should be instruction in number, basically–
whole numbers more or less. You can see that here, that
that had minimal evidence. And screening was moderate,
the other two strong. Yeah. That’s perfect. JOE: All right. People are still polling. All right. It looks like everybody
has chosen one. I’m going to go ahead
and end the poll. And here are the results. RUSSELL GERSTEN: So
I’ll talk these through and get a little context. Then if questions come
up, please enter them in the chat area. And at the end, Stephany
will go through those and ask me and my
colleague those that can fit into this
eight or so minutes. It seemed like you were
surprised about the importance, as we could imagine,
of using word problems. And one issue that we
felt very strongly about, we didn’t have a lot of hard
evidence on, is integrating word problems with computation
work– be it with fractions, decimals, whole numbers,
geometry, et cetera– doing more what Singapore,
Korea, and hopefully, increasingly, as
some American texts are doing to integrate them. And the idea that instruction
needs to be systematic, and kids need adequate practice
and feedback surprised you. Focusing heavily on number, that
was an idea that was pushed– it was pushed very hard by
the National Math Panel, which I also served
on and chaired– the Instructional
Practices Committee– it was pushed very hard in
that period about 10 years ago, the idea being that in
these earlier grades, if kids don’t– and Sybilla,
the research mathematician, said if kids don’t
understand number, especially rational number,
the whole idea of geometry, which relies so heavily on
proportionality and visualizing relationships among
and between numbers– it’s just not going to sink in. It’s not going to be meaningful. It’s not going to be understood. So that’s why we focused
so heavily on math. And there is an
evidence update, which I’ll share with you
in a few minutes. But at that point, it
was simply a belief. Universal screening–
I’d be curious why people were surprised. But I think when we looked at
the realities of the evidence on the screeners and how well
they did, it was pretty good. It wasn’t great– could
be better, we think. Maybe there’ll be
newer ones out– I haven’t necessarily seen
them– that may do better. They tend to very often be time
measures, looking at just one dimension of the numbers. So it seemed an important
thing to do, but something that the evidence– it would be nice if
there were measures that were better at predicting
who might need help, and especially– what the tendency is
for young students, especially– is to say these
kids require intervention, when studies have shown many of
them would do fine without it. So that’s a little feel
for the evidence for these. And let’s do the same
with the second set, Joe. JOE: Great. So here is your second set. Did you want to
talk these through? RUSSELL GERSTEN: This
is the first set again. JOE: Whoops, I’m sorry. Sorry about that. RUSSELL GERSTEN: Yeah. So quickly, this is
cryptic, but it’s really stressed that kids use
both concrete objects but semi-concrete, or
visual representations, like tape diagrams,
strip diagrams, and various other visuals
to reinforce things. I think the others are
pretty self-explanatory. So again, in the poll, which
is the biggest surprise? JOE: All right. The polling is up. All right. I’ll give it a couple
more seconds here. I saw a couple
stragglers coming in. All right. Closing the poll, and
here are the results. RUSSELL GERSTEN: These usually
are big winners, the two that came in the highest. Progress monitoring–
beginning when MTSS or RTI was implemented,
progress monitoring was stressed extremely heavily. This is the reason why
the level of evidence was minimal at that time, and
still probably is minimal. It may get into that moderate
category, but not really. Almost every study we read,
and reviewed, and re-read, and looked into every detail,
and sometimes inquired by the researchers what really
happened when it was unclear– they did not do the progress
monitoring that many of you are familiar with from AIMSweb,
STAR, DIBELS Math, easyCBM. That’s not what they did. Those measures, which
tend to have better psychometric properties, tend
to look at the big picture, be based on the
year’s objectives, or sometimes the
semester’s objectives. Or, sometimes, even
a test like STAR, or some of the math
measures, include material from lower grades
and upper grades, but they tend to be
big picture measures. And they tend to do a
better job at correlating with various state assessments
or other standardized tests. What they did in these studies
was either very often daily, but if not, weekly– is the kind of probes or quizzes
that people have been doing in math since time immemorial–
and in special education type interventions,
and often in Title I interventions for
about 50 years– that they did daily, that
kids mastered this material, or do I need to
spend more time on it during our next lesson
tomorrow or the day after. So with no
evidence that this was– people who did not give an
AIMSweb or DIBELS or anything, they had the kids
for four months– no evidence that there was
any change, you know, due to this. So the kids did
fine when they had the visual representations,
and the explicitness, and the systems. So that’s the reason
for that. Motivation. It kind of was upsetting many. We looked at the
National Math Panel, and they said motivation
is very important. And they cited the work of
Carol Dweck and others, but none of it was in math– not, zero rigorous
studies in math. So it’s something
to think about. It does seem to be key, as
well as ways of moderating– basically, helping
kids focus, ways to help kids increase the amount
of time they persist and attend with tasks. But it has not really
been studied in math. Now, I don’t know if
this guide were updated. There may be some new studies. I hope there would be. So that is the reason there. I do want to say this. One thing that we
have experienced– and Robin might be able to talk
better about this firsthand from our research– is you
don’t necessarily want to overdo this constant praise, which
goes back historically, of praising everything positive. Because more contemporary
techniques that really try to build a sense
of self-regulation, or now called
executive function– that kids develop inner
motivation seems important. So it definitely seems
important, but minimal evidence, unfortunately. And I just have one more slide
before I turn things over, which is more or less on time. So, Joe, maybe it
might be as easy for you to just click
onto the next slide. The only real update
of the evidence is– and this is really
quick– it needs to be done more exhaustively. This is just me, informally– that the evidence of the
importance of teaching number– now, notice I didn’t
say not to also spend time on data
science, statistics, which is increasingly
stressed, especially in an upper elementary
and middle schools– but a lot of time on
number and making sure kids understand numbers. It’s not so
important that kids– and you’ll notice in most
contemporary state standards, if not all, that
some of the problems we did when we were younger
about converting 9/51 into 17ths, and a lot of
these more obscure fractions, are probably unnecessary. But understanding fractions,
that we’re doing things with fractions, even if an
instruction involves primarily– and Robin
will give some examples– halves, quarters, thirds,
fifths, 10ths, and lead into then decimals as a
tool for computation– but understanding
number, and especially using the number line. And you will get that
a lot more in-depth. But there have now been
about a half dozen studies. Robin’s been involved
in almost all of them, so it seems perfect time to
turn things over to Robin. ROBIN SCHUMACHER: So
just to revisit the eight recommendations that we
polled on and discussed– and that Russell gave
such a nice overview for– today, because of the
reasons he stated earlier, really want to highlight
two, three, and five. And I bring up two again mainly
because that recommendation of the way that whole numbers
and rational numbers were divided with K-5 and 6-8– now with the more
contemporary state standards, rational numbers is
really also becoming more included in
grades 4 and 5 as well. And so some of the material that
I’m going to talk about today are two separate fractions
intervention programs that focus on grades 4 and 5. And we’re also going to look
at some systematic construction in terms of the design
of building skills within fractions
within an intervention, and then the use of visual
representations within that. So those are really the
three recommendations that I’m going to focus
on for a few minutes. So I’m unable to
advance my side. There we go. OK. So the two fractions
interventions that I’m going to speak
about, the first one is called Fraction Face-Off! And it was researched over five
iterative years in a study– a fraction center funded by IES. And that was focused
primarily on fourth grade. And then TransMath
is a curriculum that includes many
content areas, but the approach to teaching
fractions is really wonderful. And it spans fourth, fifth,
sixth, through middle school, but the fraction material that
I’m going to talk about today is levelled for fifth grade. So some of the big ideas
within Fraction Face-Off! is to build understanding
of a fraction as one number, so really helping
students internalize that a fraction is its own
number with one magnitude rather than two
separate whole numbers. And the approach that
was taken in this program is the use of linear
representations as a primary focus, which
builds on the measurement interpretation of
fractions, and that they are all a distance from zero
on a number line or a ruler. Fraction tiles were used as a
concrete visual representation, as a way to transition
from something that can be manipulated
with hands and seen in 3D to number lines that are
represented, 2D, on papers. And really, linking those helped
to build students’ magnitude understanding, and to help
form some abilities to reason about the size of fractions with
a focus on benchmark numbers. A more secondary
focus was looking at part-whole understanding,
which is really the more traditional way
of introducing fractions to young students, often talked
about in the form of how many slices of pizza. And that’s really
been the hallmark of fraction instruction
across the US for many years, whereas Asian countries
and Asian curricula really stress the measurement
interpretation and number lines. So in terms of thinking about
the systematic construction and visual representations, the
Fraction Face-Off! intervention started off with having
number line representations, building on unit fractions, and
adding unit fractions together to end up with fractions
that have numerators larger than one, and connecting the
different representations together, marking fractions
on the number line. So you can see how
the visuals here link the understandings together
to various computations. So showing the 1/8 plus
1/8 plus 1/8 equals 5/8 can be shown using the number
line and moving across. Or, those different fractions
are each added, and then tying it into the part-whole
understanding– so really, linking part-whole
and measurement. And so this was
how fractions began to be introduced in the program. And then moving toward
understanding magnitude through comparing two
fractions using inequality symbols or the
equal sign, and also ordering fractions from
smallest to largest or least to greatest. And in terms of
thinking about this the systematic instruction,
this was sequenced intentionally to start off with two
fractions of comparison, then moving to three fractions
that would be compared, and integrating similar
thinking strategies for solving these problems. So students learned
about when fractions have the same
denominators, and how you might think about
those fractions first to evaluate magnitude,
and then fractions with the same
denominators, and then how to use 1/2 as
a benchmark number. And so you can see on those
cards on the right side, it goes through the steps more
clearly and more step by step. But that’s essentially
the process, is moving towards
benchmark numbers, and then other methods
for evaluating magnitude without doing cross multiplying,
which is the way that I was taught in elementary
school, and I know is still sometimes included. But that then precludes students
from having to understand the magnitude of a fraction. So then, to extend
on that, number lines are continued to be used, not
just as a representational tool but as a way to compare two
or more fractions’ magnitude, and really build on the
magnitude understanding. So number lines are taught next,
first with zero to one number lines, then with zero
to two, and again, building on the
idea of magnitude very intentionally
and sequentially while including these visuals. Then, after all three
skills are taught, there’s time devoted
intentionally about building
management understanding and how all three
of these activities are essentially
doing the same thing. The students are thinking
about how big or small their magnitude is to try
to order them, place them on the number line, or
compare two fractions. And so this tying
in together is also very intentional in pulling
all of those understandings together. So this study was
researched over five years. This is results
from three years. All the results that
are grouped together, I didn’t have all in one place. So these are effect sizes
for comparing fractions. 1.82 is quite large. An effect size of 0.8
is considered large, so a 1.8 was a very
impressive effect size between students who
received intervention and those that did not. 1.14 was for students’
ability to estimate fractions on a number line. The 0.94 was on an assessment
of released NAEP items. And then the 2.51 was on a
procedural calculations measure that looked at addition
and subtraction. Again, this was
for fourth grade. So then, I’m now going to
transition to TransMath. It also introduces things
very systematically and with intention,
and relies on visuals. That number line is
also very, very much highlighted and centralized
in this curriculum, and so are Cuisenaire
rods, which are different than fraction tiles. You can see them on the right
side of your screen there. They are wooden bars of
different and increasing sizes so that different fractions
can be represented with various colors representing
the unit or the whole, and so different comparisons
can be made in that way. And so here you can
see on the number line, as a tie in to the
Fraction Face-Off! program, they used thinking
about relative size as an approach for comparing
fraction magnitude, or assessing fraction
magnitude rather. So a fraction of 2/3
would be closer to 1, because the 2 is relatively
large compared to 3, whereas 1/5 is closer
to 0, because the 1 in the numerator– 1 part is relatively
small compared to the denominator at 5. And then building
on number lines, TransMath uses them to
teach all four operations. And because I’m
focusing on fifth grade, We also– I included a whole
number times a fraction, and a whole number divided by
a fraction– how the number line can be used there. So you can see
addition, starting with the first fraction,
moving to the right. And then with subtraction,
similar procedures starting with the first
number in the equation, and then moving toward the left. And these can also be
used when fractions have different denominators. There’s just the extra step
within showing on the number line what an equivalent
fraction with that denominator would look like. And then you can see with the
multiplication and division, when you’re multiplying
3 times 2/5, students learn that it’s
similar, three groups of 2/5, and so moving on the
number line that way. And then similar with
division, if you start with 3 and you divide into 1/2,
you have 6 as your answer. So that was just
a quick overview of different curriculum that
really systematically presents information and builds
sequentially in a smart way so that students
can connect ideas, and then also uses visuals in
a nice, smart, mathematically correct way. And so what we’re going to do
next is view a video that– it’s a short clip. And what I want you to
think about and look for during the video is
about foundational skills, and whether or not
those were included, and if there was
anything else regarding the design or
explicit instruction that you noticed or
want to comment on. One thing that I forgot to
highlight as I was talking is part of the
systematic instruction in both of those programs is
that foundational skills were included throughout as needed
to build new understandings, and to remind students about why
multiplication and basic facts might be important for
finding equivalent fractions. And so earlier
skills are embedded to help students have
access to later skills. So with that, move to the
video, and I’ll release control. [VIDEO PLAYBACK] – 2/3 minus 1 times– – Does everybody see mine? [INTERPOSING VOICES] Everyone should write
on their boards. – Are we doing the boxing thing? – So we’re subtracting. So would we draw
boxes in subtracting? – Oh, no. – No. Instead we followed other
steps with making sure that what are the same? – The denominator. – Good. Denominators have to be the
same when you add or subtract. Does that matter
with multiplication? – No. – No. – OK. So let’s look here. We have 2/3 minus 1/2. Are these denominators the same? – No. – No. – No. OK. So when you see that the
denominators are not the same, what should you do? – You should find
out each factors. – OK. So factors is what we use
when we’re simplifying. So instead, we need– – Multiples. – Multiples. – Multiples. – I was going to say that. – OK. So 2 and 3. So the first thing
that we would always do is look at the smaller
denominator– which is? – 2. – 2. – 2. And is 3 a multiple of 2? – Yes. – No – No. – No. – No, it’s not. Because you can’t mult– – [INAUDIBLE] – OK. Just a second. You can’t multiply 2 by
any number and get 3– by any whole number, I
should say, to get 3. So instead we have to change
both of these fractions, right? – Yes. – OK. So we need to figure out what
the least common multiple is for 2 and for 3. And Dawson had his hand up. What do you think it is? – I think it’s 1. – You think it’s 1? Could we change 1/2 to be an
equivalent fraction with 1 in the denominator? – Well, wait. 1/2, that would be [INAUDIBLE]. – Then they wouldn’t be
equivalent, would they? – No. – No. OK. So we need the least
common multiple. Remember the multiples for 2– 2, 4, 6, 8, 10. Let’s list multiples for 3. – 3. – 3. – 6. – 6. – JJ, stop. – [INAUDIBLE] – 9. – 9. – 12. – So which is the
least common multiple? – 6. – 6. – Very good. So what we need to do is
rewrite each of these fractions with 6 in the denominator. – I already knew this. – [INAUDIBLE] – You’re making it harder
than it needs to be, huh? OK. So what number
times 3 gives us 6? – 2. – So 2 is the factor we’re
going to use for 2/3. So what is our new
numerator for 2/3? – 1. – Who said that? – I did. – Can you say it again? – 1. – Dawson, please talk normally. It’s 4. – 4. – It is 4. Because 2 times 3, we
move down to the 6. 2 times 2 is 4. OK? What fraction is equivalent to
1/2 with 6 in the denominator? What factor do we
need here, Willow? – Oh, 3. – So what is our new numerator? – 3. – I know it, 3. – 3, because 1 times 3 is 3. So what is the new
problem that we have here? Marley? – 1/6. – That’s the answer. What’s the problem? [INTERPOSING VOICES] – 4/6 minus 3/6 equals 1/6. – Good. So you were right, Marley. 1/6 is the answer. But the new problem
is 4/6 minus 3/6. OK? So I want to– [END PLAYBACK] OK. So we’ve seen the case study. If you have any
questions specifically about the case study, you can
pose them in the chat box. And the questions that
I had posed before were thinking about which
foundational skills might have needed to be
reviewed or included, or was there anything
else about the instruction that you noticed. So we can wait and hold some
of these video questions for the end as
well, because I want to be sure to get through
the rest of the slides before we have
time for questions. So one thing I
want to mention is that both of the programs,
Fraction Face-Off! and TransMath, include
ongoing cumulative review. It’s not sporadic and
short-lived but included throughout lessons– sometimes just
briefly, but usually included– as you
saw in the video, with reviewing least common
multiples for the students when it was relevant. Also a key element of
including cumulative review in a smart way is
having problems embedded that caused students to
discriminate between problem types and procedures. So to keep from doing
multiplication of fractions over and over and over, and
then have students forget about procedures
for subtraction, is one reason to keep all the
different types of problems that have been taught
included, so that students learn that discrimination. Also, all of them include
very systematic learning progressions, as I went over,
and have foundational skills to support grade level content
like the multiplication facts for common multiples. So also, both programs
offer the opportunity to give immediate
feedback, which is more attainable in a
small group intervention setting than it is in a large
group intervention setting. But it’s also
extremely important, because teachers are
then able to eliminate false assumptions or
other overgeneralizations before students become
ingrained in thinking of them that way, which could
potentially lead to longer inaccurate understandings. Also, both programs are linked
to the grade level standards. So Fraction Face-Off! was teaching fourth
grade material. The material that I showed you
from TransMath was fifth grade. However, they also
include other time devoted to difficult
essential concepts more often than is typical. So other features
of both, going back to the eight recommendations,
both programs do also include word
problem instruction. Fraction Face-Off! also includes
embedded motivational systems. So Russell mentioned that
that goes beyond praise. The motivation system that’s
included in Fraction Face-Off! includes trying to
solve problems correctly and thinking about
how many they solved to get better and improve each
time they solve problems– so a self-regulation
type of activity. And students are also
motivated with a activity by staying on task
and being focused. What the programs are
missing, as far as the eight recommendations go, is a
progress monitoring tool and built in individualized
components, which leads me to give a quick
overview of the National Center on Intensive intervention– NCII. Many and most of
their materials, in fact, maybe all
of their materials, are publicly available. They focus on databased
individualization. And a lot of their
resources talk about how that can really
be a key for some students to progress to the
next level of learning. They have a lot of resources. This is their primary website. They have a large focus on
intensifying instruction and using progress monitoring. Their databased
individualization framework looks like this, in that
students start with a validated intervention program at Tier
2, are progress-monitored, and then might– if they’re not
responsive– then might look at more diagnostic
or academic assessments or a functional assessment,
adapting intervention, continuing to monitor
progress, and going through to try and get
learning to accelerate. So these are also some extra
links for mathematics resources that are free to you and
anybody, available from NCII. So some cover counting
basic facts, place value, and computation. Others look at college and
career readiness standards with math instructional guides. There’s also specific resources
for progress monitoring, diagnostic assessment for
planning intervention, and then also
intervention design, and then also math
interventions, and assessment tools
for progress monitoring and for screening. So I think I got us for
10 minutes of questions. I’m going to hand
it back to Stephany to pose some questions for
Russell and I to answer. STEPHANY BROWN: Great. Thank you so much,
Russell and Robin, for a great presentation. We have a couple of questions. First is, going back
to the practice guide, one participant asked, we were
surprised by the whole number and rational because it
had minimal evidence. We thought it would be strong. Can you give an
explanation for this? RUSSELL GERSTEN: I can do
that, because I was hoping to. I think there are two problems. One is that– maybe
a major one is– not many studies or researchers
think about comparing, let’s say, a fifth grade
curriculum or an intervention that focuses primarily
on rational number and whole number like most
state standards to do, versus one that covered a
wide array of things including geometry, and
measurement, and some data and probability, which would
then be less intense on real– the kind of in-depth understanding
of rational number that Robin’s two curricula had. So as I say that,
I think it would be a quite important study
to do for that reason, that it’s a critical issue. It’s a common belief. And so it’s good to have
evidence to support belief. And there are a lot of
intervention programs that are linked to– I know big programs are GO Math! in the elementary grades,
and Everyday Math is big. And to compare those
intervention programs, which tend to cover whatever
is done that week, provide support in
the week’s topic for the fourth or fifth
graders, versus something that really hones in on rational
and whole number concepts, and use of the number line–
it would be a good study to do. Who knows? It’s for another
generation of researchers. So it’s just lack
of hard evidence. So I was surprised, too. But I will say this, that
the big belief, going back to something like the ’90s,
was that math curricula should show that number quantity,
geometric concepts, are all around life. They’re in our
everyday experiences, and what you learn in
your science courses related to astronomy,
physics, biology. So that idea is quite
different than what we see in other curricula,
where there really is– no one says that isn’t true,
and that it’s a great way to augment your curriculum. But the key thing,
especially in intervention, is making sure kids really
understand what equivalence is, that it’s not something that
you focus on for two or three weeks, and then
you drop, and then you come back to it when
you do adding factions, but then other times
you ignore it when you’re doing multiplication. That’s really what you need to
understand in grades 3, 4, 5. So I think it will be
a good study to do. They haven’t really been done. STEPHANY BROWN: Great. Another question was, in regard
to the motivational strategies, what about Jo Boaler’s work– if
I’m pronouncing that correctly. It seems that she’s done a
great deal of work on mindsets. RUSSELL GERSTEN: I
think what we did– and it’s not just our
particular panel, Institute of Education Sciences does– is try to differentiate between
rigorous experimental research, where we control for
confounding factors, and other kinds of
observational work, case study work, for actual hard
evidence to rely on the latter, on the rigorous studies. I believe a lot of
Dr. Boaler’s work has been more in the
qualitative case study mode. I know she had done one
quantitative evaluation. I’m not familiar
with all the details. But that certainly would have
been eligible for review. And if it didn’t pass the
standards, I’m not sure. So people are definitely
working, exploring. There’s a lot of scholarship
on mindset and motivation. But we really– we’re looking
for controlled studies that showed, if you had a kind of
motivational system in place and compared it to just more
traditional teaching, where every so often you may praise
a child, or challenge a child, is it effective. We did not find such studies. STEPHANY BROWN: And there
was a little confusion about the strategies that were
shown to have minimal evidence. Do you still suggest
these strategies as ones that should be used? RUSSELL GERSTEN: I
think the answer is yes. And that comes up every time. And I’ve presented on
practice guides and RTI in reading and English learners,
and a little on algebra as well. Yeah, the panel did believe
these were important. It’s just there was no evidence
as of, in this case, 2008. And as time goes by, they do
need to be updated, especially in an area where there is– so much research has been
conducted in the past decade. So the answer is yes. We do believe them. But you do have to be
aware that people– expert opinion has been
wrong on everything from the rotation of the Earth
to a thousand other things to Newton’s laws, which are
not scientifically valid. So we need to keep that in mind. But we do recommend
those as practices still. STEPHANY BROWN: And
one last question. Marjorie Petit has some
valuable insights for all of us around whole number reasoning
applied inappropriately. Can you speak about the impact
of this issue for struggling learners when fractions,
2/3 for example, are portrayed as
two sticks or marks on top of a horizontal bar over
three or marks under the bar? Do these programs ease this
way of portraying fractions? ROBIN SCHUMACHER: Do you want
me to take that one, Russell? RUSSELL GERSTEN: Yes. Yes, Robin, that’s you. ROBIN SCHUMACHER: OK. The programs do not portray
them with two sticks over three sticks with 2/3. And there is, in both
programs, a large focus on a fraction being
one number and not two separate whole numbers. So I’ve read several articles
about whole number bias. I think it’s another– is a term I’ve seen often used
for whole number reasoning applied inappropriately. And so speaking to
that a little bit, part of why many mathematicians
believe that the number line is a superior way to
represent fractions is that it helps students
consolidate and integrate the whole number principles
and understandings with rational number
principles and understandings. Because it’s really
the best representation where you can include both
of those kinds of numbers, and really start to
integrate and consolidate rational numbers being part– especially between zero
and one, and then moving to fractions that
have quantities larger than one involved– part of why the measurement
understanding is becoming more emphasized in many
curriculum now. So to answer your big
question, no, the sticks are not used in these
program at all– and in fact, really try to build
the understanding of them being one number, and the relationship
between the numerator and denominator as
determining magnitude. STEPHANY BROWN: Great. We had a couple more comments
come in on Dr. Boaler’s work, but we will forward
those to the panelists, since we need to
close out our webinar. But thank you all
for those questions. We really appreciate it. These are just some additional
references on today’s topic. Somebody asked if a copy of the
PowerPoint will be available. And yes, it will be. I will show that
in just a minute, but wanted to thank
you all for your time today, and really thank Russell
and Robin for a great session, and you all as
participants for joining us for this webinar focused
on Assisting Students Struggling with
Mathematics: What Works for Tiered Interventions
in Elementary and Middle Schools. This is the contact
information for our presenters, in case you would like
to get in touch with them with any additional questions
or for more information on the work that they’ve done. And this gives some information
on the PowerPoint presentation materials, where
they’ll be available. There will be an archive. The recording of
this will be posted on the Institute of Education
Sciences’ YouTube channel. And you can also
follow our website, or follow REL
Central on Twitter, for more information
about our events, and to access many of
our free resources. So thank you again,
everyone, for participating. And we hope you
have a great day. ROBIN SCHUMACHER: Thank you.

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