- Did you know that number
lines should not be used
before second grade?
Are you like, say what?
Right?
'Cause number lines are very prevalent
in kindergarten and
first grade classrooms.
However, it's actually recommended
that in kindergarten and first grade,
we instead use number paths.
I'm Christina Tondevold, the
Recovering Traditionalist.
I hope you'll stick
around as we investigate
how to use a number path in our quest
to help us build our math minds
so we can build the math
minds of our students.
So first off, let's talk
about what is a number path.
There's so many times when I say that,
people are like I've never
even heard of a number path.
So what is it?
A number path is just
isolated objects in a path
that allows kids to count things.
A number line is a wonderful tool
and you've probably seen
some of my last posts
that are all about the
power of number lines.
But for our young kiddos,
a number line is actually
a length-based model.
You're counting the length of something.
Our young kids need something to count,
not just the length, not a measurement.
So a number path helps our kids to see
the objects in a line and be able to count
individual objects instead of having
to count a length which is weird to have
to count for our young kiddos.
So if you've ever seen a kid struggle
with number lines, it's
because there's not
a thing to count.
When they do count, they end up counting
the little tick marks on the number line
instead of the length
between those tick marks.
So they'll often, if they're trying to add
or subtract on a number line, they will
often be incorrect because
there's not a thing to count.
So a number path gives our
students something to count
because in kindergarten and first grade,
they're still in that direct modeling
or counting on phase that
I've talked a lot about
and they need something to count.
Now the first place that I ever realized
that I shouldn't be
using number lines was,
oh man, this is probably
really going to date me,
but quite a few years ago, our state was
in the process of reviewing
our math standards
and trying to come up
with better standards.
Well, during that time, and I
was on that team of teachers
who was investigating our standards
and trying to decide is
this a good standard.
What else do we need?
Does this standard go to
a different grade level?
All of that kind of stuff.
Well about after our
first or second meeting,
that is when the original talk
of the Common Core Standards came about.
And so our job as this group
became not one of creating
the standards, but it
became very prevalent
that our state was pretty
much going to adopt
the Common Core.
And so our state wanted us to review those
and give feedback to the writers
of the Common Core State
Standards of what we thought.
So one of things that I
remember seeing was a line
in one of those very
first original documents
of Common Core Standards
that said number lines
should not be used before second grade.
And I started questioning it.
What, what are you talking about?
It's like in every kindergarten
or first grade classroom,
people are using number
lines 'cause kids need those.
So I started digging
and it took me a while
but I found the couple of
books that talked about it
and they're called Focus from the NCTM.
There's Focus in Kindergarten,
Focus in First Grade,
Focus in Second Grade, there's
one for each grade level.
But it was all about teaching with
the Curriculum Focal Points.
These were before Common
Core ever came about.
Here you go.
Is that our National Council of Teachers
of Mathematics put together
their Focal Points.
These were the major standards that we
really needed to focus in
on at each grade level.
So those books talked about
what are those focal points
and how do we do it, how
do we teach those concepts.
And both of those, the kindergarten
and the first grade book,
talk about number paths.
And they have an image similar to this.
This image shows how a
number path gives kids
a count model instead
of a length-based model.
And it's not to say that that
length-based model is bad.
Kids need it.
It's just a little too
abstract for our young kids
in kindergarten and first grade.
So that brings me to the second piece
that I want to mention which is where
the number path fits in this CRA model.
If you've heard me talk about that before,
that is the concrete to
representational to abstract.
Now one thing I just forgot, don't forget,
underneath this video if you don't have
number paths, if you've never
used number paths before,
there's a link to be able to download
my version of a number
path that goes up to 20.
There's also, you can
buy them on my website.
I created the document
at first for the teachers
that I worked with
because there was nowhere
where you could get number paths.
So I created a document
and then teachers would say
well can I buy these somewhere.
So I have them specially made
that has a coating on them
that kids can write on them
with white board markers
and then they can just wipe it right off.
They're wonderful things to
be using in your classrooms.
So there's a link to, just download it,
and use it however you want.
Or if you want the pre-made ones
that come with the special coating,
you can go to my website
buildmathminds.com.
Or if you're on the Vlog,
there's a link up above
that says Shop, and you can go to my shop
and there's the number paths there.
Alright so let's dig into
where the number path fits
on the CRA model.
So again, CRA stands for
concrete to representational
to abstract.
Concrete is when children are actually
physically touching,
moving objects around.
So if you have manipulatives,
that's when they're working
in the concrete phase.
They're counting out six things.
That's concrete.
Representational is when they're drawing
a representation of those six things.
Now the hard part with
our young kids is that
they will draw very intricate drawings.
If we are talking, if the
story is about six teddy bears,
they want to draw the teddy bears
and color the teddy
bears, and make sure that
the teddy bears have ears
and eyes and you know,
all the details.
But really we want kids to
not have to get bogged down
in drawings and instead use models.
I talk to the kids all the
time about using a model
in mathematics, not a drawing.
There's a time when we
want intricate drawings,
but there's a time when
we just want a model.
And a number path is a great
model for students to use.
And so a number path falls in
the representational phase.
And then abstract is when
they just have the digit.
So if they just have the digit six,
that's very abstract.
You don't see six things there.
So the reason again that
number paths are better,
if we look at the difference
between them again,
there's the number path has
the six individual things
that kids can see.
They can see each object there.
In a number line, it's
still a representation,
but it becomes more abstract because
where are the six things on a number line.
It's not obvious to our young kids.
The six things is actually the six lengths
in between the tick marks.
And that's hard for our
young kiddos to see visually.
So the number line is
still a representation,
but it's more abstract.
So the number path is a
more concrete representation
than a number line is.
It's still a representation,
but it's falling
more towards a concrete
representation for our students.
Alright, so let's talk
about my two favorite ways
to use a number path.
How do we actually use a number path?
Well, one of my favorite ways is to help
our kiddos build number sense.
For a lot of our students,
when they think of six,
the only thing they think
of is the digit six.
We want them to picture six things, okay.
So let's say I give the
number path to a student
and I say circle seven.
Most often, the number one way that we see
kids circle seven is they just circle
where the seven is.
And so are definitely
ways I could rephrase that
to make it more obvious,
but I just want to put it
out there to see what they do,
what they think of when
they think of seven.
So oftentimes we get it where
they just circle the digit,
where the digit seven is.
But in reality, that's not what seven is.
Seven is seven things.
So I want students to understand that
when they're showing seven,
it's not just the digit.
It's seven things.
And so on a number path,
we encourage kids to circle
seven things when they're showing seven.
This also leads into you can
do comparison type problems
where kids will show an amount
and then show the other amount
and talk about you know which one is more,
which one is less, and then that can lead
into questions like how
much more, how much less,
which is the precursor
to building this idea
of adding and subtracting.
So when kids are adding and
subtracting using a number path,
the one thing I want you to know,
you don't need to teach
them how to show it
on the number path.
It's up to them.
The number path is a model,
it's a tool for them to use.
There's no specific way
a child needs to show
their thinking on a number path.
The one thing I do want them to do is
to show their thinking.
And I should be able to
look at the number path
and tell what they did.
So if we were solving a
problem like seven plus eight
and I wanted them to show their
thinking on a number path,
I would not want them to just circle seven
and then circle eight
things and tell me it's 15.
Unless they just knew that
seven plus eight is 15.
But if they had some kind
of thinking strategy,
whether they counted one by one by one,
then they would circle one
by one by one to show me
what they did.
But if they had a
thinking strategy similar
to something like okay I know seven
and then three more would make a 10,
and then another five more
gets me all the eight,
that is what I want to
see on the number path.
However they thought about the problem,
I want them to show that thinking.
And then eventually, the
cool part is we can add
in the symbolic piece as
well, that abstract numbers
and attach it to the
picture that they made
with their number path.
Now I'm gonna show you an
example with subtraction
because subtraction always
seems a little funky for people.
People are a little unsure
how to show subtraction,
but the big idea, let's say
we're doing 14 minus six.
The big idea for students to understand
that we want them to
get around subtraction,
I want them to understand that we have 14.
So somehow I want them to
notate that they had 14.
They might circle 14.
They might put a star above the 14.
They might put a little line after the 14.
Somehow to notate that
that's what we started with.
And then if they thought
about 14 minus six
as 14 and I'm gonna take away six,
somehow I want them to show
me how they took away six.
If they took away six one by one by one,
then that's what I want them to mark off
on their number path.
But if they took away six
all in one fell swoop,
then I want them to show
that on their number path.
Again, remember it's
just all about how are
they thinking about the problem.
That's what I want to see.
So once they've taken away the six,
then I need to know like
where's your answer at in this.
How do I know what's left?
And there they show that there.
And again they could show
it in lots of different ways
symbolically or with abstract symbols.
Some of the kids might see that
as six plus something gave me 14.
Or they might think of
it as 14 minus six gave
them the eight that was left there.
So it's all about helping them to use
the number path to model their thinking.
It's not about how we
would use the number path.
It's not about our strategy
for using the number path.
It's a tool.
It's a model to help them show
what was happening in their brain.
Alright, so to wrap this up, the best way
on how to use a number path is to just let
the kids use it.
Let them show you their thinking
around things about numbers and quantities
and comparing and adding and subtracting,
and then help them attach
that to the concrete
if they're actually counting out blocks,
and attach it to the symbolic symbols
and the abstract nature of mathematics.
Use the number path as
that representational piece
when you're working with
kids in kindergarten
and first grade.
Even preschool kiddos can
be using number paths.
At about part way through first grade,
we want to start moving
towards a number line
and helping kids see the connection
between a number path and a number line,
but before that point,
it's all about helping
them see the individual objects
and build their number sense
and their understanding
of addition and subtraction.
So don't forget to get your
download underneath this video
or you can go straight
to buildmathminds.com
to buy some pre-made ones and let the kids
start writing on them and
showing their thinking.
I hope that this has helped
you build your math minds
as you go out and build the
math minds of your students.
Have a great day.
