Something about that decimal point really freaks kids out. So here we go back again to misconceptions and misunderstandings…

I’m working with a student right now who was taught procedurally how to subtract decimals, but gets all mixed up when regrouping.  The VERY BEST way to intervene with any student is to determine where the hangup is in the Concrete-Representational-Abstract approach. If a regrouping error is being made, you can guarantee that the conceptual understanding is missing, which comes in both the concrete and representational part of this approach.

If you don’t already know about this approach, here it is in a nutshell:

1. Concrete: When a student is introduced to a new concept or something unfamiliar, you allow the use of tools. Sometimes students become stuck here, and can be moved to the next stage by linking the two together.
2. Representational: When the student can perform the task using tools, they move on to representing the concept with drawings or pictures of their tools. Again, when students become stuck here, we link the next step in with this one.
3. Abstract: When the student can master the task with a drawing or a picture they move to using only numbers and symbols to represent their thinking.

* If you have a little more time you can read about this sequence of teaching and it’s accompanying research here, and also here.

So with errors in regrouping, we backed up to the basics of the American currency. We looked at what each piece of currency represents and why:

So in America, we have many coins, but to keep it conceptual (whole, tenths and hundredths) I used dollars, dimes and pennies.  A dollar is the same as 10 dimes, and also the same as 100 pennies.  A dime is the same as 10 pennies.  To have a reference guide we made the key above to refer back to. Later this translated really well to using place value blocks when we needed to start using larger numbers, and when I was out of money. (Imagine that…a teacher without a lot of money! Ha!)

So for a while we added two amounts of money. Addition seems to come really easily to students. From K-4, I’ve noticed a pattern that adding two whole numbers seems to be a piece of cake, but subtraction is like HITTING A WALL.  So, I decided to start with subtraction slowly. This was the sequence:

1. We subtracted over and over when we didn’t have to regroup, and we didn’t write anything down.
2. Then we subtracted where there weren’t enough pennies, and he had to trade (regroup) a dime in for ten pennies.  We still didn’t write anything down.
3. Next we subtracted where we had to trade in (regroup) a dollar for ten dimes. We STILL didn’t write anything down.

Why aren’t we writing anything down?

Well, that was the concrete stage. All we do in the concrete stage is play around with the manipulatives to be sure the student is understanding the concept behind the regrouping. It makes the next two steps SO much better.

Now that’s where we talk about noting what we did. We’ll use pictures along WITH the materials, which is how we link to the representational stage.

1. We just kept subtracting when we didn’t have to regroup, and this time we drew what we did.  Since the student already knew the procedural shortcut to regrouping, we linked that in with the drawing.
2. Then we subtracted where there weren’t enough pennies, and he had to trade (regroup) a dime in for ten pennies.  We still didn’t write anything down.
3. Next we subtracted where we had to trade in (regroup) a dollar for ten dimes. We STILL didn’t write anything down.
4. Once that felt good, we started writing, and then suddenly he wanted to try out the algorithm to see how it might match. Instant connections to the abstract stage…the procedure he had already “learned”. Now, it made sense to him, and he no longer had to memorize which number to cross off because he knew why.

Too often we are pulling manipulatives from our older students.  Whenever I’m working with a student who struggles, it seems to be the first place that I notice problems.  When we pull away those supports (or never start with them in the first place), math really is just a bunch of meaningless numbers and symbols without any connection to the real world.