Keep following their thinking!

In the summer I wrote a post about Charlotte's addition algorithm with base-10 blocks.

I sort of thought this would be her algorithm and she would just move away from the blocks.

However, lately I've noticed lots of work with the abacus (her favorite) and base-10 blocks again!

However, her strategy has changed.

Today I started exploring HOW it has changed, asking her questions as I watched her solve.

She told me that she knew she had 7 tens because 3 and 4 tens make 7 total.

Okay, I'm following.

Then, she says that 8 only needs "2 more" to make another ten.

So, she grabs another ten stick.

She throws it in the pile to make 8 tens.

She immediately grabbed 3 extra units and throw those to the side.

 

So I ask her "How did you know to grab 3 units?"

Her answer: "Oh, well since you took 2 away from the 5 to give to the 8, there is only 3 left there".

So, I'm totally following her logic and it makes mathematical sense.

and….ya know what?

She is doing the "traditional algorithm".

Is this a bad thing?  Absolutely not!  She totally has a grasp on what she's doing.

So I show her how to record her work on the board with numbers and no base-10 blocks.

She giggles and does it correctly with 8 more problems.

Kids' brains are awesome.

We….well, we just have to make sure we keep up ;)

Following their thinking...

 Today was another wonderful day in the neighborhood….

No, seriously.

It was one of those days as a teacher/homeschool mom 

when you get the warm fuzzies because you are that sort of nerd.

I don't even fight the inner-nerd anymore ;)

Today I planned for Charlotte to do a worksheet of problems.

I chose this particular worksheet because the problems are horizontal instead of vertical.

Vertical placement of problems sometimes rushes kids into a procedure without understanding 

the concept of place value within adding two digit numbers.

I was going to have her do the same sheet of problems 3 times using...

Base-10 blocks 

A basic abacus

A 100s chart from her desk

Mostly, we do a ton of word problems in real-life scenarios.  Every unit I create a new set of word problems following the Cognitively Guided Instruction (CGI) problem types.  We are almost finished a set of problems I created for her as we studied money.  Below are the type of basic problems she has been working on alone, with the help of manipulatives of her choice.

So anyway.

Today I thought she could use a few different tools and maybe she would get to the point she might be able to make some generalizations about when we put together (add) two digit numbers.   

If not, no biggie.  I just wanted to see where we were.

So, away we go.

I start with base-10 blocks because she uses those a ton.

I ask her to add a problem like 13 + 25.

I expect to see what I've pictured below. 

I expect to see a 13 and a 25.

I did not see that at all.

Instead I see her count out 3 ten sticks, then 8 units. (like below)

I watch and she does this about 4 times.

When I ask her what she's doing, it takes her awhile to put it into words.

I keep digging, asking how she knows to just "grab 4 ten sticks".

I ask her to point to where she is getting that information on the page.

Eventually, she explains that she looks as the 10 and the 20 first. 

She adds 10 + 20 and just grabs 3 ten sticks.

Then she adds 3 + 5 by counting onto the larger number

(like …. 5… 6 , 7, 8) and then writes 38.

At this moment, I realize my child is no longer in the concrete phase of 

understanding and is completely adding the numbers in an abstract way.

Do I then STOP and show her the traditional algorithm?

No.

See?  I'm a "new math" person.  Only "new math" really isn't new.

It's the way mathematicians have been thinking for thousands of years, but that chat is for another day.

Why am I an advocate of "new math"?  

New math isn't about teaching your kids a more complicated way of

 doing something that we learned another way.

Nope.

Instead, it's about listening to a kid.

Listening and watching how a child is already thinking about a problem or type of problem.

Then?

Follow their thinking.

Don't push our thinking on them.

So today, instead of the plan I had for Charlotte, I taught her how to record her thinking.

She was naturally adding tens first, then ones.

So…I show her how to record just that.

I show her that she can write down that answer so she doesn't get lost.

Then, I show her how to record her answer to her ones.

She was thrilled and even asked me how she might go back 

and do that if the answers were written horizontally.

I showed her that she could write her work vertically or horizontally, 

whichever way made the most sense to her, no matter how the problem is presented.

Why?  Because this isn't about me and what I know.

I already passed second grade math.

Instead it's about her understanding how she is making sense of the problem.

So today….was a good day.

It was a good day because I had the time and opportunity to notice and listen to her thinking….

Then,

Follow it.