Hello and welcome back to What in the World is Dyscalculia presented by Educalc
Learning and hosted by me, Dr. Honora Wall.
Thank you for joining us today.
I want to recap a very interesting article for you that talks about some of the ways
in which we learn math and some of the problems we run into.
And I hope that this will illustrate the many pieces and parts that go into mathematical
thinking.
It's not as easy as we think or as straightforward as we think it is.
And it's very complex.
There's a lot of processes involved.
So let's take a look at a couple of them.
I'm going to be discussing some findings that were published in the Developmental Disabilities
Bulletin in 2004.
The article is titled Learning Math, Basic Concepts, Math Difficulties and Suggestions
for Intervention, published by Das and Jansen researchers at the University of Alberta.
I'm just going to hit some of the highlights today and talk about why these findings are
important, how they matter to us as math educators, as parents of students with dyscalculia, or
as people who have dyscalculia.
Do better understand yourselves as mathematical thinkers and learners.
But I will put a link to a PDF of the full article on the dtri.org so that you can read
it for yourself.
And the dtri is the Dyscalculia Training and Research Institute, a 501c3 non-profit
organization dedicated to disseminating information about dyscalculia and getting that research
out there and conducting new research.
That is definitely the goal.
But that is in the future.
Let's talk about what's already been published and where it leads us, what it tells us today.
So in this particular article, Das and Jansen reviewed a couple of research studies and
the key points that really popped out for me.
One is what makes someone quote unquote good at math.
And one of the things they talk about is flexibility in strategies, the planning that goes into
approaching a math problem.
Now this planning is instantaneous.
It requires flexibility.
It requires a non-rigid thinking.
It requires taking feedback, correcting your approach, coming at a problem from different
ways.
And it happens when we look at a math problem.
It could be an equation, 2x equals 14, could be an elementary math problem, adding decimals
or multiplying fractions.
It could be reading a word problem and having to figure out what we're being asked to do.
In each case, there's a process of planning that the mind does automatically on its own.
And these researchers found that people who we call quote unquote good at math, the high
achieving students, they're very flexible and quick with that planning.
They can look at a math problem and come up with lots of different strategies for how
they might solve that problem, lots of different approaches.
And they can move from one to the other quickly.
So when we're working with students or when we're talking to our children or our students
or we're doing work on our own and we look at a problem and we have to pause and think
ah, what do I do?
Hold on, I've seen this before.
I know what I've got to do.
I know these steps.
I can figure this out.
Just give me a second.
Or when we look at a problem and think, okay, I have to do this first, do this second, do
this third.
No shortcut, no secondary way of doing it.
I learned it this way.
That's the only way I ever want to know it.
That kind of rigid approach or that kind of stuck in one method and that kind of pausing.
Those are signs that we're just not very confident with that particular math skill or we might
be struggling within that math skill.
For teachers, that's where your conversation is going to come into play.
The only way you can know if your students are flexible or if their planning is to talk
to them, you're going to have to ask them, how might you solve this?
Don't tell me the answer.
Tell me what you think you might do.
Anybody else have a different approach?
Anybody else have something different they might do and have those conversations?
You cannot see a mental process.
You cannot tell from the final answer what the mental process was.
You're going to have to talk with your students to really dig into what they're thinking,
what they're planning, what kind of strategies they have so that you can gauge skill set
and their strength in this area.
And if you find there's a lot of pushback, there's a lot of resistance, then you know,
okay, we're going to take some different methods.
We're going to find different support tools.
We're not going to show every single strategy, every single possibility because that's overwhelming
to some students.
And that's fine.
We don't all have to be excellent planners.
We don't all have to be flexible thinkers.
We don't all have to be great at taking feedback.
Some of us are very linear, very direct, one foot in front of the other.
One step after the other.
And it takes all different people and approaches to make the world go round.
But the more you talk to your students, the more you're going to know their personal strengths
and weaknesses.
Okay, another really interesting point in this article.
I thought this was great the way these researchers explained two fundamental concepts in learning
math, magnitude and value.
And I thought they did a great job of really explaining these two in a way for us to break
it down from the point of view of the learner, especially young learners.
So we can see differences in magnitude and value from very young ages, preschool to second
grade.
But we can also look for lingering issues with magnitude or value in our older students
and it can give us some insight into where we want to support students where they're
struggling.
Okay, so magnitude.
We'll take that first.
What is, according to the researchers, the way they explained it, size, bulk, quantity.
They use adjectives like big, medium, little, large, small.
That is magnitude.
And we can see that even infants have a sense of magnitude, which things are larger, which
things are smaller in objects.
At this point with magnitude, we are not necessarily talking about digits.
We're not talking about number, written numbers.
But it is a part of number sense, which is very interesting.
So if we have a sense of magnitude that the distance from our house to the moon is larger
than the distance from our house to the grocery store or our house to the mailbox, those are
all part of understanding magnitude.
If we have students or just children just playing with children, not everything has
to be a lesson.
Just having a good time lining things up, putting them in order from larger to smaller
or vice versa, making pyramids out of different shapes, making things that connect based on
their size, grouping by size, anything like that, we're discussing magnitude.
Which animal is larger than the other?
Which book is larger than the other?
Who sits in a smaller or a larger chair?
Adults or children?
All of these conversations increase a sense of magnitude.
And there is some question about why we can see this even in infants.
It seems to be something present at birth and innate sense that we are born with, part
of that innate number sense that we have from the beginning of time, our time.
This is totally different than value.
The concept of value has no inherent meaning on its own.
Value has to be taught and it has to be learned.
It has to be an agreed upon condition.
It is not something that is inherently true or false.
And this is where we can really see a big difference between natural mathematical thinking
and what we're learning in school.
So value and the value that we tie to digits, to written numbers, these are very unique concepts
that we have constructed.
And when we see students having a problem with that, we can see where they're struggling.
And for people with dyscalculia, those struggles don't really go away.
That value piece doesn't connect or click at the same time.
And sometimes not in the same way.
Now the authors in this study say, and this is a quote, the difference between magnitude
and value is analogous to the distinction between speaking and reading.
While speaking occurs naturally and perhaps a blueprint exists in the human brain for
speaking, reading has to be learned.
Reading is acquired.
That's the end of their quote.
And I thought that was such a great way for us to think about this.
We understand that very well, whether or not we teach reading.
We need to understand the same concepts in the world of math.
When we teach math, magnitude is one thing, value is another.
And we need to not assume that value is going to come from counting from one to five on
a number line or one to ten on a one to one hundredth chart, simply repeating these words
or simply writing down digits doesn't necessarily automatically connect to value.
So if you're trying to really strengthen that, I would definitely be as tactile as possible.
First your objects to your counting.
I know you're already doing that in elementary ed, but if you're seeing that by the end of
kindergarten, the end of first grade into second grade, students are still kind of struggling
at putting that piece together where they still are relying on the tactile piece.
I'm not using the word manipulative because you don't have to go out and buy a bunch of
fancy things and the special rods and that's going to magically make value make sense.
So count, organize, structural work with literally anything in the room is going to really make
that connection.
So stop thinking only in terms of manipulatives.
Think of anything that's around you and really bring that math piece into all parts of the
student experience within their actual real world experience.
Okay, so keep working on that piece and if we're seeing that piece is still like there's
a disconnect.
So when the way you're going to see that in the classroom is a student who can count with
objects and count on their fingers, maybe they're not counting in their head.
Maybe they're not adding without that number line or without those tactile objects that
can be assigned either of low numeracy, a little slower development or past second grade,
probably an indication of dyscalculia and we need to look further at a potential math
learning disability.
So we want to look at the difference between magnitude and value.
And the last piece I'm going to pull out of this article, which I definitely want to talk
more about in a future podcast, I want to get some more information to really flesh out
this idea.
But one of the last things that the authors talk about is how much work goes into one
problem.
And this kind of goes back to that planning piece and the flexibility we started talking
about in the beginning to kind of flesh out that idea.
The authors talk about addition and say you have to add.
Now math teachers and parents and adults, we consider adding like the very first thing
you do, the very basic thing.
So easy.
It's just adding.
That's totally false.
Here's all of the different things that can happen when the student is learning about
adding and they're figuring out what kind of strategy they can use to solve this problem.
First of all, they've got to have cardinality and ordinality.
They've got to understand a number line.
They've got to understand counting from one number to another number.
We're not even talking about subtraction and working backwards.
Just going forward.
And the strategies could be counting on their fingers, either visibly where you can see
it, or sometimes older students will hide their fingers.
They're still using them to count, but they don't want the shame and the embarrassment.
So they're hiding.
Make sure you don't allow that kind of shame or embarrassment in your classroom.
Let people use whatever tool is helping them out.
Next, they could be representing numbers with their fingers, adding without you seeing that
they're counting or kind of in their head, they're thinking about counting on their fingers.
They might be using something like a touch math system is a very popular one for early
elementary interventionists.
Older kids are not going to be able to use that system very well for a bigger, longer
problems.
This is something that impacts time and extended time.
So they might be hiding that and doing it mentally whenever possible.
There's also verbal counting.
You could count out loud.
All right.
If I have five and I'm adding three, so it's five, six, seven, eight.
Now of course, students with dyscalculia are going to start at one, one, two, three, four,
five, one, two, three.
They may or may not get that they're supposed to continue past the five.
Even if they do, they're going to start at one, count to five, then count three more
to eight.
So verbal counting, totally different strategy.
And the last strategy, the one that's kind of the gold standard for classroom teachers
is retrieving that answer from long-term memory, not having to do the counting, having memorized
some basic facts or being able to count so fast, you're not even aware that you're doing
it and having those answers pop right out.
Students of students with dyscalculia, people of any age who have dyscalculia, teachers
who have been working with this population for a while, we know that strategy is not
going to happen.
That's highly, highly unlikely.
In the same way, if you asked me to read a paragraph and took my glasses away from me,
I would not be able to do it.
I'm a pretty good reader and I have pretty good reading comprehension.
But I still need to wear my glasses if I'm going to read something.
So I would need a different strategy if we were talking about reading strategies in this
analogy.
So all of these strategies get you to a final answer.
So as the adults in the classroom teachers, we're used to judging which strategy is appropriate
at what grade level.
And that might be something for us to all reflect on.
What kind of judgment are we putting on student work?
And what kind of student work do we give value to?
And what kinds do we not?
And what does that say about what we're valuing?
The right answer or speed or being a good memorizing person who has that grade skill?
Or are we valuing the work of combining quantity and coming up with a final answer?
So let's think about that in our classrooms.
Okay, again, to wrap up, this was a very recap of some highlights that really popped out
from me.
When I was reading the article Learning Math, Basic Concepts, Math Difficulties, and Suggestions
for Intervention compiled by researchers Das and Jansen at the University of Alberta.
And I will make this PDF available on www.thedtri.org.
And I'll try to get it up on educalclearning.com as well for whichever website you're using
to find dysCalculia information.
I think the non-profit is going to be more focused on that research dissemination piece
and the educalclearning is really moving towards books and materials and math courses that
go directly to students and teachers.
So different focus areas for those two things, but they're both a great way to reach me.
If you have any questions or comments about what we've talked about today, you can find
me at Honora, H-O-N-O-R-A at educalclearning.com.
If you want to know more about our teacher training courses, having me come to your school
for an in-person training or if you're looking for some of our books, you can also find our
books at Amazon, BarnesandNoble.com, ask about them at your local bookstore, or on our website.
educalclearning.com.
You can also email me Honora at thedtri.org.
If you have a question about our research, about any of the research articles we talk
about on the podcast, or if you'd like to get some free materials for yourself or for
your school, educating people about the math learning disability, Dyscalculia.
Thank you so much for listening to What in the World is Dyscalculia.
I'm Dr. Honora Wall, and I will speak with you next time.