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>>JESSICA LOGAN: Alright, we're going to go ahead and get started. Thank you
so much for coming to the First Friday Colloquia, although this is actually
rescheduled First Friday-a last Monday Colloquia?-for the Crane Center for Early
Childhood Research and Policy. My name is Jessica Logan; I'm a research scientist
at the Crane Center, and I'm very excited to introduce Dr. Battista today,
who is a professor in Mathematics Education here at Ohio State University.
His major research interest, as I understand, is in student learning in math, with a focus
especially in geometry and geometric measurement. He's also really interested
in the use of technology by teachers in mathematics education. His current project that
we're going to hear about today is looking at elementary school mathematics
teaching, with technology-that was the word I was looking for-
development and instructional use, and we're excited to hear from you.
[AUDIENCE APPLAUSE]
>>DR. MICHAEL BATTISTA: OK, so it's hard for me to imagine talking about
learning progressions and squeezing it into 45 minutes, but I've tried.
I've really been working on learning progressions for over 25 years, although they weren't
really called learning progressions early on. And I think as you work with learning
progressions, there are learning progressions that you develop for other
researchers, and then you can sort of rewrite those learning progressions
so teachers can understand them, and then you can try to integrate those learning
progressions into curriculums for students. And basically I've been involved in all
three of those. So these are sort of curricula for students. This is a series of books
that came out a year ago, that's written for teachers and helps them use learning
progressions in teaching students in elementary school mathematics.
And this is my current project, and almost all of this was supported by the National Science Foundation.
In this particular one, it's a computer-based geometry curriculum for grades 3-8,
and it incorporates learning progressions into its branching and linking.
In a sense, it's an attempt to try to make learning-progression-based teaching
available to all students, not just those that have master teachers
who can handle this kind of thing.
So what I'm going to try to do is talk about what are learning progressions, why use them,
instructional use of learning progressions, and also something about using them with teachers.
So here's just my definition of learning progressions, which is not
that different than many people. So learning progressions are descriptions
of the successfully more sophisticated ways of thinking about a topic that follow one
another as students learn about and investigate the topic. For me, thinking is this
conceptualizing, reasoning, procedure strategy implementing.
OK. So all the learning progressions that I'm talking about are empirically based.
They're not: somebody sat in an office and decided that logically this is how things should go.
So I think all the learning progressions I'm talking about were empirically determined.
So, why use learning progressions? The research foundations...
basically it's sort of built on this idea that students construct meaning
based on their current meanings. Ok? So, if we're going to teach students in maximally effective
ways, we have to figure out where are they in their reasoning, and their conceptualizing,
and how can we build on that in instruction. That's sort of what the ten-second
theoretical framework is. But I think the other part of this, for me, is that if you look at all
the research done in mathematics education over the last 30 years, I'd say the strongest
finding is this: that instruction that is based on and continuously adjusts
to the personal meaning the students are constructing for
specific mathematical topics improves student learning.
So there's a number of related concepts that I thought I would talk about, and these
will make some issues pop up, I suspect. Learning progressions are not
developmentally inevitable. They're not like Piaget's stages. Learning progressions
always depend, to some extent, on instruction. Ok? A level of sophistication, which is
basically the concept I used before people decided to call it 'learning progression',
is a distinct type of cognition that occurs within a hierarchy of cognition levels.
So, for me, that's the level of sophistication. Some researchers are viewing levels
in a learning progression sort of like Doug Clements and I did many years ago,
that in some sense it's sort of like a configuration of problem-solving strategies that
has certain activation weights. Somehow I don't think that adequately captures how
I think of students at a level, but it is certainly a way that people think about it.
So, for me, an ordered set of levels of sophistication is basically a learning
progression, and I sort of view them like this: there are these cognitive plateaus that
students can move through. And I think there are a lot of people talking about
learning progressions, and the sort of grain of learning progressions differs widely.
For instance, in science education, often the grain is very wide, whereas in math
education it's narrower. But, for me, I think that one of the big problems we have in
teaching students mathematics is that students at this level, you want to get them
to the next level, and you can't figure out how to do that. I think about my early
teaching career, and I think about my failures at times. The successes I don't remember so much,
but I can remember some of the students I failed with.
And I can remember certain students that I kept explaining and explaining and explaining
the idea and they could just never get it. And now I know why, but I just couldn't understand then.
So, in some sense, the kind of learning progressions that I try to come up with
are ones where the jumps are reasonable for students. And, for me, reasonable means within
a day or two, working intensively with the student, I can probably get them to make that jump. Ok?
So, you have to get vacation pictures in your presentations.
[AUDIENCE LAUGHTER]
>>DR. MICHAEL BATTISTA: In many ways I sort of think it's the next picture.
I sort of think about when I went rock climbing with my son when he was in junior high,
and he didn't have very much experience climbing. It wasn't this particular one,
this one was a little too steep for me, but it was actually sort of easy for me to figure out
where the footholds were, and where I should step next, and where I should put my hands.
So I could try to help him, I could try to point out the footholds, but ultimately it was him,
he had to pull himself up. And I think that's exactly what happens with students in learning mathematics.
You can sort of help them see the footholds, but ultimately you can't force them to move up.
They have to do it themselves.
So another term that's out that's similar is 'learning trajectory'. And for some people,
learning trajectories mean the same thing as learning progressions. But for other people,
there's a difference. So, Les Steffe talked about an actual learning trajectory for a student,
so a model of children's initial concepts and operations, an account of the observable changes
in those concepts and operations as a result of the children's
interactive mathematical activity in the situations of learning.
So basically trace out everything that you do with the student.
And if you've ever read any of Les Steffe's stuff, you'll know that that's exactly what he does.
Doug Clements and Julie Sarama talked about learning trajectories as descriptions
of children's thinking and learning in a specific mathematical domain and a related
conjecture route through instruction tasks. So, both Steffe and Clements and Sarama
were talking about progressions through pretty much specified instructional treatments.
For me, learning progressions are more general than that.
They're not tied to a specific curriculum.
So, for me, that simple definition of learning trajectory is an instruction-dependent
path through a learning progression. So you can sort of see my picture of learning
trajectories for two different students. I don't think all students necessarily
go through the exact same sequence. Some students jump through, or jump past.
And you can also think about a learning trajectory as this hypothesized route
that you're going to try to engender in instruction.
So, why use learning progressions? For me, it's pretty simple. Once I know
where the students are in the learning progression, I know what to do with them instructionally.
In the work I did with teachers, once we identified where students were, I knew exactly what to do
and they said, "We have no idea." So that's why I spent a lot of time working with teachers
to figure out what they need in order to use learning progressions.
So I wanted to just give you a quick couple of examples. So here's a learning
progression for addition and subtraction. And I'm going to zoom in on just one part of it.
And we're just going to talk about those three different ways of adding numbers, ok?
So: the count all strategy. A student might count 1, 2, 3, 4, 5 without raising fingers,
then 6, 7, 8 while raising three fingers. Count on: a student might say 5,
then count three additional numbers 6, 7, 8. So count's on 'from'.
And then a derive-a-fact might be they know 3+3 because it's a double, then add 2 more.
So those are the three strategies.
And you can see Tom Carpenter and colleagues sort of traced students through
the first three years of school in a particular curriculum, and you can sort of see how
things progressed. And it's important, when the number facts are really
memorization of a fact, and a derived fact. So they clump those together.
Ok. So we have these levels. The first step in using learning progressions is to determine
where students are. What level. In some sense, individual interviews are always best.
But teachers always ask, "Well, how else can we deal with this." So I'm going
to give you an example of what one teacher did. She spent a lot of time
with her students, first graders, not only getting them to talk about their strategies,
but talking to them about how they would write their strategies down on a sheet of paper.
I think she got more information from her first graders than when many fourth and fifth grade
teachers do it and don't say anything other than, "Tell me what you're doing."
So she spent a lot of time with students on that. So here's some of the stuff
that her students wrote. You're probably familiar with this kind of stuff.
So, "I counted in my head, 4, 5, 6, 7." The important thing to sort of recognize here is,
can we figure out the strategy and map it to the learning progression. "I counted by 1's,"
and basically counted all, right? "I thought a little bit, and I counted like 1, 2, 3, 4, 5, 6, 7."
[AUDIENCE LAUGHTER]
>>DR. MICHAEL BATTISTA: I mean, it's really interesting: the spelling that kids use.
There are times that I think I had to ask my wife, who taught at that level for a number of years.
And the numerals, obviously, are not exactly what you would expect.
It's important here that she had this-I'm not going to show you what all
the students wrote-but I think it's important to recognize that there were a few
students in her class that she knew couldn't write their strategies.
So she would intentionally go over to them, ask, "How did you do it?"
and then she would write on the student sheet.
"I started at 8, 7, 6, 5." OK, so you get the idea. Now, let's sort of map what
the students said to a chart for the whole class. So we have a whole bunch of students
who are counting all, and we have a bunch of students that are counting on or down.
So what I'm suggesting is in a learning progression based instructional approach,
those two groups of students need very different kinds of instruction.
I thought I should illustrate what that might look like.
For the students, we want to get students from counting all to counting on.
So there are some sub-skills which I have listed here.
Students can correctly produce the sequence of counting words beginning
at an arbitrary number in a sequence, they don't have to start at one.
And they have to know that if I count a set of objects and the last number
I say is seven, that the number of objects is seven.
For a lot of students, young children, you say, "Count" and they just start saying numbers.
But it's not really related to the cardinality of the set.
OK, so, moving from counting all to counting on.
One way that we might help students is with this task.
So there are five squares in the configuration below. And ask how many squares there are.
Then cover the squares, and write five below it, put three more squares,
and ask how many squares in all. So the idea is to sort of get the student to not start at one.
And it turns out that it's actually more effective to not start with the task that looks
just like this, and just say that there are five squares here. For a lot of children,
they have to see them, and then they believe that there are five under there,
and then they can visualize it. If you sit with children doing this, they'll actually count
in the same configuration that they saw the squares.
At some point, with some repetition, and especially if you keep starting with five,
they get that they don't have to count one through five. They can just start with five.
And we can push that. OK? So that's students moving from counting all to counting on.
What about students that you want to get to derive facts?
One of the fairly effective ways of getting students to develop
this kind of reasoning is with what people call 'quick images'.
I'm going to show you a picture of some dots for 3 seconds then, I'll hide the picture.
I would like you to tell me how many dots there are.
Okay, so how many?
>>AUDIENCE: Ten.
>>DR. MICHAEL BATTISTA: Ten. Okay. Alright so there's ten dots.
Now we're going to do the task again.
>>AUDIENCE: Eight.
>>DR. MICHAEL BATTISTA: Eight right? But how many of you counted eight?
You didn't count eight right? Okay , the strategy is; the whole is ten and you took two away.
Here's one more.
Okay, So whats the strategy that students use on that?
Well basically, they sort of envision moving these two here; that's ten.
And then four more here. Okay?
It's a common kind of strategy with young students is to say something like eight plus six.
Well I take two from the six put it with the eight, that's ten and then the four more.
And it's just a quick image to sort of build those kind of mental model in students,
alright? So the um - I guess the moral of the story is - at least from my perspective -
is that the students at level 1.1 need very different kinds of instruction than
the students at level 1.2 and this teacher in fact did sort of teach in that way - she would group students.
Not every day - but maybe two or three days a week or something like that, okay?
>>AUDIENCE: Can I ask you a question?
>>DR. MICHAEL BATTISTA: Yes go ahead.
>>AUDIENCE: It'll be so much easier to do it now.
Let's say you have students that are at 1.1 on your progression there - and you
taught them 1.3 and you skipped counting on. If you went for the more complex
place on the progression - would they generalize downward and get 1.2 along the way.
>>DR. MICHAEL BATTISTA: Um . . . probably not.
>>AUDIENCE: Okay.
>>DR. MICHAEL BATTISTA: I mean - it's not clear. I mean I think in general,
what happens with the learning progression system if you try and skip a step -
if you as a teacher tries to skip it - you lose some students.
Some of the students can't make the jump. And so they - they're not as successful.
Now that doesn't mean - you know some students might go through from one level
to the next really fast - especially if you're working with gifted kids. They can move
up really fast sometimes. And sometimes students will move up just by hearing
another student tell about their strategy. But in the kind of classrooms that I envision,
if I listen to Susie and she comes up with this strategy, then I have to
decide if I want to use it or do I not want to use it. If the teacher shows me the strategy,
I have to decide do I want to use it or not use it. As soon as we get into 'you must use",
then what happens is a student's sense-making starts to fade, okay?
And that's what we're trying to avoid.
Okay - um - how much time do we have left.
Well - to skip ahead quickly. So one of the issues that people have with learning
progression sometimes is, "Are students 'at' levels. Should we talk about a level one student?
And my answer to that generally is no. I'm more interested in what kind of reasoning
is this student using as this point in time. And usually, just saying a student
is at level three isn't completely accurate. I mean sometimes that happens.
But students are constantly in transition so . . . it's a little too . . . it's over-simplified
to say a student 'is'. And I also don't like the idea of classifying a student.
I like to think about the student's reasoning because that's what I can build on -
not well this is a level one student.
Okay. So how do you deal with that? Well one way we've dealt with it -
and so I can get into some additional stuff - is thinking about profiles.
A C-gate learning progression profile. So here's another learning progression.
This is for place-value. And here's a student in grade 2 -
and I'm not going to show you all his work but I'll show you just a little bit.
So Mary has 24 cookies - she eats six of them - how many cookies are left?
24 subtract six. He's at 24 and he counts down - but he misses a number. 20.
So in the learning progression, he's still having difficulties with counting.
Now this is counting backwards or down - not counting up - which the two . . .
you get one before the other. Here's another problem: Here, he used an algorithm,
so he said it was 11 so I put a one right here under 4 and 24 and a 1 up here
above 4 and 47 then I put 1 plus 4 plus 2 and that will equal 7.
What does this one stand for? To add with the 4 and the 2 is it just a one?
Yes. For us, it's an indicator that he doesn't really get place value in the algorithms.
The students who did get it, would say: "No, it's 1, really is a 10.
So he's still sort of level 1 and I probably should mention that if a student can use
an algorithm to do something, it doesn't tell you a whole lot about their conceptual development.
So we always try to go beyond that.
Here's another problem, but now he's using base 10 blocks. Okay?
I've got 36 and 28 base 10 blocks and he put 3 tens in one hand and 2 tens in the other.
30, 40, 50 and he touches ones, 51 and counts on.
So he's actually jumped up to level 2 with the level base 10 blocks.
On this problem where he has kind of pictorial material to operate on he's still at level 2.
So the profile, I mean there are a whole bunch of tasks and we keep track of that.
But the profile is actually, basically this, with no perceptual material he operates
basically at level 1 with perceptual material he jumped up to level 2.
So that actually helps us think about what does he need?
And I'm not going to - I had more to say about it but I won't, to get on.
But basically we have to sort of build on what he's doing with the pictorial material
and get him to be able to internalize that okay.
So there are specific instructional techniques to use to get students
to sort of internalize that, that image.
Okay so the first two examples were primary grades now I'm going to jump up
a little bit to give you a good example of what happens when teaching a learning progression,
or the learning progression isn't sophisticated enough.
Okay here's one learning progression for shapes okay.
It's been around for 30-40 years something like that.
And here is an elaborated learning progression.
So certain parts of it were expanded.
Basically expanded it using some fairly long term teaching experiments
where we watched students everyday.
So what happens when teaching is inconsistent with learning progressions?
So here is an example.
So one is sort of quantitative alright.
So (?) found students who start high school geometry at a (?) level zero have little chance
to learning to write proofs. Students at level one have less than a 1 in 3 chance
and students at level 2 have a 50/50 chance.
Okay so you know get students to level 2 they only have a 50 percent chance of
succeeding in high school geometry. That's not really - that's not my vision of
a successful approach to this. At the end of the school year 4 percent, 13 percent,
and 22 percent of students at level zero one and two. But 57, 85, and 100 percent
at levels three, four, and five had mastered proofs.
These levels, stand hill levels, were at the end of the year
and those were the ones at the beginning.
So we've known for a long time we have this huge problem in mathematics,
high school mathematics, students don't do well in high school geometry.
At one point it was the most hated course of all courses in high school.
You know you just talk to someone and they say oh yeah I remember proofs and then they frown.
Okay so here is a slightly different assessment but fairly recent,
I don't know how many years ago.
So here is the problem that tends to distinguish pretty well between students
at level one and students at level two, okay.
So you look at those two shapes, it looks like you have a rectangle and a square but
if you look at the measurements then it's not the case.
In a fairly good school district only 22 percent answered that item correct
and to me even more distressing is that the students at the end of high school geometry
in 8th grade, this is basically the upper 5 percent of students in this particular school
district only about half of them got the problem correct.
And there is no doubt in my mind they had the capability to think about this.
Okay so that's the quantitative so now I want to show you the sort of qualitative.
To me it really helps to understand what's going on.
So here is our task and we have a teacher sitting
with a student who has completed high school geometry.
So she looks at those two shapes and she says her choice is A,
it's a rectangle and a square and the teacher says I'm curious
as to how you knew A was a rectangle B is a square to the student.
It's pretty obvious. You can just tell by looking.
So you didn't need to use these numbers at all?
I guess I could have, but I didn't think I had to.
If it weren't so obvious, then yeah, I'd look at the numbers.
So you get this idea. That's a classic level one response. Remember,
this student has been through high school geometry.
So, let's go a little bit farther. The teacher asked the student, "How do you know if a
shape is a parallelogram?" "I think it's just 4 sides and a set of parallel lines."
"OK, how do you know if a shape is a square?"
"All the sides are equal in a square, and parallel. Oh, and the angles have to be 90."
"And what about a rectangle?" And the description...
So, she sort of jumped up in level, unprompted. OK? So, what does learning-
progression-consistent instruction look like? The teacher comes back to a similar
task, and the student says, "That's a square." And the teacher says, "Well, what
is a square and what is a rectangle? Can you define them?" So the idea
the teacher has in mind is, can you look at the definition compare what I see
to what I've defined, and that should help the student move forward.
And then the teacher does some other good stuff that's consistent with the learning
progression. What he does is he sort of looks at the definitions and he tries to get
the student to clarify, what do you mean by those definitions? So, pointing to two
neighboring sides in the figure, are these two sides parallel? Because remember,
the student said that for a rectangle, for a parallelogram, and for a square,
all sides are parallel, OK? That's not what the student really meant.
"You said in a square all sides are parallel?"
"Well, I meant the ones that are across from each other. These two would be parallel."
"So do you want to change your definition?" And so the student clarifies. OK?
How about the rectangle? Same idea. "For the rectangle, you wrote that two sides
are the same and the other two sides are equal. Do you want to specify which sides
you're talking about?" And then the student clarifies. OK?
So now, using this information, look at this picture again. What is the shape? I don't know.
[AUDIENCE LAUGHTER]
It looks like a square. These definitions are right, right? I guess it's a rectangle
because the sides are different, but it looks like a square.
Not real enthused about the classification.
So now the teacher is getting into where I think it's learning-progression-inconsistent instruction.
"So I wanted to go back to what you said about there possibly being a rule
about squares and rectangles. So, you have to think about, what did the teacher just think?
The student agreed that, according to the definition, the thing that she saw was a rectangle.
OK. So the teacher might think, aha, I got the student to say the right thing;
I'm ready to push them farther. And the learning progression approach would be to say,
no, you just don't know enough to say you should go forward.
So, at some point, the students always-when they're sitting
with someone who seems to know something about geometry-so many will say something like,
"Is it all squares are rectangles or all rectangles are squares.
I remember we talked about that." But they haven't the slightest idea which is right,
or how to figure it out. So they had talked about this before, and the student,
through high school geometry, still didn't know what was what.
"So I want to go back to what you said about there possibly being a rule about
squares and rectangles. Based on your definitions, can you decide whether there
might be one?""I don't know, I can't remember." Draws a square and labels each
corner 90 degrees and each side length 5. "OK, let's go through your definition of
a square and test this. Then go through the definition of a rectangle and test this."
So,the student says, "For rectangles, the sides across from each other are the same.
"But here they're all the same, so can it still be a rectangle if they're all the same?"
"Yeah. That's what it is. That's what I was trying to think of yesterday. A square is a square
and it's a rectangle. Like this is also a rectangle because it fits the rules."
So the teacher got the student to say the right thing.
OK. So then the teacher comes back to this. And the teacher writes a definition of a rectangle:
a quadrilateral with four angles equal to 90 degrees. Now, the difference
between the teacher's definition and the student's definition is that the student
listed all the properties, which is sort of level 2 kind of reasoning. The teacher lists
just one sufficient property, which is level 3 kind of reasoning-later level reasoning.
So, what's the student think about this? "Is this right,
or am I supposed to tell you why it's wrong?"
[AUDIENCE LAUGHTER]
>>DR. MICHAEL BATTISTA: "Well, do you think it's wrong?" "I mean I guess
probably all this stuff is true, but for the rectangle you don't say anything about
parallel or equal sides or anything. You don't say anything about the sides,
so this wouldn't be the definition." Level 2.3 response. And then they keep talking.
So the teacher is talking about whether all parallelograms are trapezoids. "No." "Why not?"
"Because then they would be called trapezoids and not parallelograms!"
[AUDIENCE LAUGHTER]
>>DR. MICHAEL BATTISTA: "But what about squares and rectangles?
Remember, you said all squares are rectangles. Yeah, I guess,
but I don't really see how that makes any sense either."
[AUDIENCE LAUGHTER]
>>DR. MICHAEL BATTISTA: So, in my mind, this is a clear example of a teacher
trying to push a student to a higher level before the student is ready. I mean,
I would have never tried to push the student into level 3. I would have done a whole bunch
more at level 2.3, trying to make sure they really understand the properties.
And there are sort of steps up, to get the student there. I mean, the teacher's really trying
to get the student up to level 3.3. or 3.4, and the student was sort of operating at 2.2
or 2.3 It's just too big of a jump. The student just can't make it. And whenever you put
students in that situation, they start having...they start to abandon personal sense-making,
and to me that eventually means they're going to end up failing in mathematics.
It happens to people at various places: second grade, sixth grade,
I've seen it happen junior year in college as a math major.
All of a sudden it just starts to unravel.
OK, so back to the picture. We don't have too much time, but I'll try to say a little
about the cognition-based assessment and teaching project. Basically a 10-year
project with NSF support. In the first phase I developed assessment tasks,
learning progressions, and instruction tasks for elementary school mathematics.
And in the second phase I basically investigated how elementary school teachers
would use these learning progressions.
I'm supposed to say something about how you construct learning progressions...
that's a whole topic in itself. But basically we started with what the research said,
we tried to consolidate what the research said, we tried to derive some initial levels
based on what the research said, and then we just spent four or five years testing
them, giving them assessment tests. Like, for place value, we probably gave 50 or 60
different tests over a five-year period. A lot of learning progressions are actually
developed based on giving just 2 or 3 tests. And so we found that if you just looked
at that, you didn't get a big picture. Suppose you're looking at all the possible tasks.
So I'm going to skip that. So the first phase, developing.... It's interesting that these
books came out just recently, but at the end of the 10 years. The first five years
I had pretty much figured out where are students were in these learning progressions,
what they looked like.But then I spent another 5 years talking to teachers,
trying to figure out how we should write these, how we should illustrate these
so that the teachers could use them. For instance, I told you that when I know
where a student is in a learning progression, I know exactly what to do with them instructionally.
The teachers told me, "We don't know what to do. You need to help us.
You need to say, if a student is at this level, what do I do?" And these are good teachers.
I mean, I enlisted teachers that I thought really knew what was going on.
When they told me things, I paid a whole lot of attention to what they said.
So I actually went beyond geometry in this particular project, into numbers.
It's probably worth just saying for a minute what we put in these books.
So first you have to identify what the core mathematical ideas and processes are.
So, place value-OK. That's a core idea in elementary school mathematics.
Core reasoning processes sometimes aren't so familiar. So, iteration, like in skip counting,
or if you want to find the volume of a box, and you find out how many are in a layer
and you iterate that. I mean, iteration occurs everywhere.
The distributive property actually occurs all over elementary school mathematics.
Then we provided these research-based learning progressions and sets of tasks that
would help teachers figure out where their students were.
And then really important examples of students reasoning at each level,
or each task that they gave, and its suggested instructional activities.
So, with the teacher, basically.... What, 2 minutes? OK.
Basically we followed the teachers for 1 to 5 years. Originally I was going to get one
set of teachers, work with them a year, then get another set and work with them a year.
And the first set of teachers wouldn't quit. I mean, they just wanted to keep going.
OK? So it's an interesting progression from teachers who were skeptics- at the very
beginning I can think of one teacher-to, at the end, years 4 and 5,
they were teaching other teachers how to use the learning progressions.
We had teachers basically doing things at all three tiers in the RTI levels.
So what are some of the things the teachers say? "Well, I think that in some cases
I gave my students credit for knowing more than they actually did, which hampered my instruction.
In other cases, I gave them less credit for knowing what they did.
Knowing exactly where students are in mathematics isn't such an easy task.
I had one teacher, a good teacher, pick the person for her teaching experiment with
a student, which I required all teachers to do, she picked the student which she
thought was best in mathematics and the student which she though was worst,
and after the interviews she switched. The one she thought was the worst wasn't very
articulate but had really good thinking, and the one that she thought was best was
really good at saying the few things that she knew.
Well, there's some other stuff, but in the interest of time, let me just
say these two things. So, helping teachers understand learning progressions.
What are the difficulties? One, the amount of information in learning progressions.
It's fairly complicated. That's an obstacle. Two, not only in mathematics, but there's a logical
complexity to students' thinking. Three is time management. Once I get what these
levels look like, how do I integrate it into my curriculum, so I can make it useful.
And basically the teachers resolved these issues, and it took at least two years.
In year one, they just figured out what the levels were and felt comfortable with them.
In year two they sort of revised their curriculum and figure out, ok, in this unit I'm
going to do this from CBA and in this unit I'm going to do this in CBA.
So they had a plan and then they implemented that plan.
OK, I think I'll stop with teacher 13.
I have a lot more to say, but I think I'll open it up to questions.
>>AUDIENCE MEMBER: We want to know about teacher 13.
>>DR. MICHAEL BATTISTA: OK.
>>AUDIENCE MEMBER: Yeah, who's teacher 13?
>>DR. MICHAEL BATTISTA: OK, so this is just an example of learning the levels.
So I asked her, in year one, how important is it for you to be determining the exact CBA
levels for students? "It could be, but not right now. I'm still trying to get my head
around and think about, do I want to group these children in math groups?
But I don't know if I want to do that yet. I think it could be very useful and very important,
but I don't know how I'd work it, so I'm scared to try." OK? Year 2.
She just blew me away when I walked into her classroom. She said,
"I have 18 kids in my class. I have one that's at level 0,
four that are probably at level 1.1 independently and always,
probably ten of my students are at 1.2 independently, but they probably will go back
and forth depending on how hard the problem is, and then I probably have an
extraneous three that are at 2.1, and then I have one that's at level 4.2,
and I think that covers it. And she didn't look at any notes.
She just already knew where her students were. So it's just this huge change,
from not even being sure that she wanted to use this stuff.
This is the first grade teacher that I showed you with the written work.
And she did group at times. So, another teacher basically totally integrated
cognition-based assessment into his teaching.
And he teamed with another CBA teacher, and so they actually integrated
some of the CBA assessment tasks into their grade-level tasks for their school districts,
so they could actually screen students. They worked in what they called a "quad",
so four teachers would work with four classes and they'd group the kids based on that.
And generally the CBA teachers would take the students
who were struggling and try to move them forward.
Probably the greatest success story that I can think of for CBA is teacher 19,
who was a third-grade teacher, and at the beginning of the year he said one of his
students was at the kindergarten level. He started working with that student with CBA,
and by mid-year he said she was at third-grade,
so she actually fit in with the rest of the class.