# What Is The Concrete Representational Abstract Approach

– The Concrete Representational
Abstract approach to teaching mathematics
has a very long history. However, there’s a lotta people
that don’t know about it, and I believe the people
who do know about it, are probably doing it incorrectly. I’m Christina Tondevold, The
Recovering Traditionalist and I hope you’ll stick
around as we investigate The Concrete Representational
Abstract approach to teaching mathematics as we
try to build our math minds so that we can build the
math minds of our students. So let’s start off with what is concrete,
representational and abstract? I’m gonna shorten this up for us. I always call it C-R-A,
it’s just easier to say. But you might hear it is CRA,
other people might say it as Concrete Pictorial to Abstract,
so you might also hear it as CPA, it’s another common way to hear it. The general idea is the same. The first stage is concrete,
that’s what people talk about. We need to make it concrete,
get it hands on for our kids. They need to physically
be doing the mathematics. This is often the case where
we bring in manipulatives. We want kids to have
physical objects to use. The representation stage is when we start doing away with the manipulatives and we just have kids draw,
just drawing a representation. And then the final stage
is this abstract stage, where we’re just writing digits. This is also known as when
we’re doing the algorithms, as seen as very abstract for kids. Now, why is this so important? Well, number one, we have kids who are just various learners. We know kids who can jump
straight to the abstract, but we also have kids who
just need the visuals. They need that hands on learning. We have very different learning styles and so varying the way that
we approach the mathematics with the students is
helpful in that respect. The other thing to think about is that in so many other areas,
we don’t expect kids to just jump straight to the abstract. We attach a lot of visuals to it. We have kids that have
experiences that help them be able to make connections. So I once heard Marcy
Cook talk at a conference and she gave this example. If you close your eyes and think of cat, what conjures up in your mind? It is not C-A-T, it’s an
actual image of a cat, and yet, with mathematics,
when we think of 37, what conjures up instantly
is a three and a seven. It’s the abstract form of
it and we need to allow kids to have the different
visuals, different connections of these abstract concepts in mathematics. This is typically the way that this approach is seen is that
representational phase, and then we move them
to the abstract phase. We see it all the time in the things that show up in our textbooks. Here’s an example from
a particular textbook, where it laid out what you’re
doing lesson by lesson, and you can see they’re
using manipulatives. Then in the next lesson, they’re
using those manipulatives and relating it to a written method, which, that’s a really
big jump, let’s just say. But at least they’re
doing some relationships. And then another lesson they’re
doing the math drawings, which is the representation
and they’re attaching that to a written method. So they have all these
different lessons where kids are doing each different
phase and for our kids where it’s a difficult
to make connections, by the time you’ve done all
of those different phases, they seem them as three different ways to approach the problem, instead of seeing them as connected. To me, seeing this as a linear path, where we must start kids
in a concrete phase, then move to representation,
then move to abstract is the wrong way to view it. Instead, I want to
encourage you to view it more as this Venn diagram. These overlapping circles that if we can hit an activity in what I call the sweet spot. If you get an activity
where you can help kids see that what they’re doing
concretely, with manipulatives, connects to this drawing that
we’re doing right over here, and that connects to
these written symbols. When you help kids make all
three of those connections within the same lesson, it’s those lessons where at the end of it we’re
like that was so awesome. But we don’t really know why
it was so awesome sometimes. Was because kids could see
all of these connections, things were flowing,
they started having those light bulb moments. When you’re in that sweet spot, it just makes a world
of difference for kids. And the cool part is, is
that if you allow kids to work wherever they are at any activity, so if I go back here to this visual of this lesson plan, basically, you have a whole class of 30
kids and you’re doing a lesson using manipulatives to represent addition, what about those kids who are past it, who don’t need the manipulatives? And then maybe in the next couple lessons you’re jumping into the written method, but you still have half your kids who need the manipulatives. Being able to do an
activity where you allow all three of these phases
to happen at the same time allows your students to be able to work where they are at within these phases. Allow kids to be doing stuff with objects. Have ’em always out, it doesn’t matter if you’re lesson plan calls for it or not. Letting kids always have objects available so that they could work
the problem out concretely. Then have visuals, help kids be able to do drawings and models. Now, here’s one thing, change
your wording from drawing to modeling because
when a kid wants to draw a representation, especially
in the young grades, they wanna draw a very
detailed drawing of it. If it’s about butterflies they
wanna draw the butterflies and color it, so in the
representational phase is a time where you can
teach kids the difference between drawing and
modeling the mathematics. There’s a time and place for
we need detailed drawings, but there’s a time and place
where we’re just modeling what’s happening in the problem. And then we can move them
into attaching those symbols. That’s one of the downsides that I see is when kids are working hands on and they’re doing manipulatives, we aren’t connecting it to a drawing and we’re not connecting
the abstract symbols, and so when they get to those phases, they don’t connect it
back to the hands on stuff that we did five lessons ago. So, here’s a little example,
just starting out with letting kids build a certain number. Here’s a couple examples of
moving them from building it concretely to
representational to abstract. We want them to be able
to see those visuals, and connect it to the abstract
because there’s so many more ideas around that number that they build besides just seeing it as
a four and a three for 43. When they can build it hands
on, they see the drawings, and they connect the representation to it, the abstract representation to it, so much more number sense gets built. And then as they start moving
into working with numbers, doing operations, we still want to do concrete, representational to abstract. We want kids to physically
build and model the problem. Here’s an image of a rekenrek. They’re building seven
plus eight on a rekenrek. And then they can use the number path to represent what they
saw on the rekenrek, so it’s the same visual
that these kids created. They both had seven on the
top and eight on the bottom, but one kid saw the fives within there and so on the number path, they’re circling the fives and then the group of two and three. While another kid saw the
groups of sevens within, that seven plus eight is just
like having seven plus seven. So on their number path,
they modeled what they did on their rekenrek there. And then we attach the symbols to it. That’s all it takes is when
you’re doing the concrete stuff, layer in the representational,
layer in the abstract piece so that we can help kids
start making those connections and go quicker throughout these phases. We want them to go from
concrete and representational and finally get to abstract,
but it doesn’t happen in a linear path. We need to be doing it together
so that then they eventually get to the point where maybe
I don’t need the concrete, I can jump straight to the drawing piece and connect that to the written piece. And then when they get
comfortable with that, they will naturally not
want to do the concrete, they won’t wanna do the representation because they see it just
takes way longer than that. I see far too many
worksheets that have kids draw out all of the 10s and ones when a kid could do it quickly. If a kid is at that abstract
stage, and they’ve built that representation, they
understand the representations, they understand the connections
and now they’re just able to do it abstractly, that’s okay, even if your textbook says they have to do it with
a drawing, they don’t. And even if your textbook
says they should do the written method and
you’ve got kids who need to draw it out, let ’em draw it out. The concrete,
representational and abstract, kids will be working in
those phases in a messy way. It does not happen the
way that our textbooks want it to happen. Here’s another quick example
using multiplication. We do a lot with building area model when it comes to
multi-digit multiplication and we use base 10 blocks to model that. So the concrete phase we’re
modeling with base 10 blocks. Then we move into the
representational phase of drawing an area model and
then we move kids into what’s known as a partial products or even the traditional algorithm. But we don’t let kids see the
connections between those. How often have you done a lesson where kids do all three of
those within the same lesson and then, the key point
here is, at the end, is you ask how these are connected? If somebody just did the concrete and they did just the base 10 blocks and they didn’t make it all the way to the partial products algorithm, can we help them see
connections between those? Or do they see these as
three separate algorithms, basically, to do? We want kids to see that you’re
doing the exact same things, it’s just one was physical,
you’re doing the concrete. One was a drawing and one we’re doing it without any pictures at all, but we’re doing the exact same mathematics in every single one of those. Now, take a look at your standards, because I can bet at
almost every grade level, you have at least one standard in there that has it specifically
called out that kids need all three of these stages. This is one example from second grade where it’s talking about
adding and subtracting within a thousand and they
need to have concrete models. They need to do drawings and
they need a written method. It doesn’t say these need
to be separated though. We can do all three of
those at the same time. That’s my reminder to you is the concrete, representational and abstract is a very powerful thing in mathematics, but it’s not a linear progression. These things overlap and
there are definitely times when you might just work
in the concrete phase. Or you might just be doing
drawings or you might be doing a drawing and the abstract, right? That’s the cool part
about these Venn diagrams is that sometimes you’re in that spot where just two of those overlap. Or sometimes you’re at the
spot where you’re just doing abstract, you’re only working
on the written method. That’s okay. They’re all spots that work for kids, but when you get all
three of them together, it helps build some magical
connections for students that are just so, so, powerful. I hope that this helped
you build your math mind, so that you can go build the
math minds of your students. Have a great day.