Friday, January 27, 2012

Montessori Math: Building Big Numbers with Base Ten Blocks

As we've casually, slowly, and happily meandered through our first couple of homeschool years, we've gradually evolved into ways that keep us happy, engaged, and productive. We're no longer the unschoolers that we were for over a year, and yet we still do not follow a curriculum, and, believing that play is still the most important work of my so-big seven-year-old and five-year-old, I still strictly limit the amount of formal, structured work that I ask my girls to perform each day.

In other words, no matter what you're doing with your own kids, you can feel free to agree that I'm doing it all wrong.

The girls' schoolwork is almost entirely guided by their interests--Ancient Egypt, werewolves, the desire to be in the next community fashion show, the desire to earn exactly enough money to buy herself a certain ipad app. I see my role as to surround the girls with all the resources at my disposal involving their interests, to guide them through formal study of their interests to deepen their knowledge (and thus appreciation) of these areas, and to set up and moderate as many hands-on, context-deepening, multiple-intelligence activities as is desired. We do this until their knowledge is, for the time being, sated, adding other areas of interest and focusing and re-focusing and coming back to former loves as the little ones wish.

Our number-building study came about when, for some reason, it kept coming about that Willow needed to add multi-digit numbers. She needed to add the tax to the list price of an ipad app to calculate how much money was going to come out of her piggy bank for it, then she had a lot of change to add up to see if she had enough, and when I did the math in front of her, talking her through the carrying the ten and such, it was mysterious and fascinating and something that must be learned!

We watched the Khan Academy video on addition with carrying for background, but obviously when it comes to actually mastering the skill, you've got to back that train on up. Here's the progression:
  1. Before she learns the shortcut, Will needs to understand the concept of carrying tens and hundreds to the next place value.
  2. Before she understands the concept of carrying tens and hundreds to the next place value, she needs to understand how numbers are built from hundreds and tens and units.
We've played with that second concept plenty, so that it was a comfy review for Willow before we moved onto the first concept (where we'll be for a while), but such regular review is very important, because not only does it continue to cement the concept, but it also aids contextualization--Will sees that multi-digit addition with carrying is built upon the concept that numbers have place value, and when we go back to this review again before we start subtraction with regrouping, she'll see it again.

To build numbers in a way that highlights their place value, in a way that internalizes the basic fact of each number, in a multi-sensory, hands-on way, you need two things: Montessori-style number cards, and a BIG set of Base Ten blocks. Base Ten blocks consist of one-centimeter-square unit blocks, ten-bar blocks that are ten centimeters long by one centimeter wide and represent "ten", hundred flats that are ten centimeters long by ten centimeters wide and represent "one hundred", and a thousand cube that's the size of a stack of ten hundred flats and represents "one thousand." We're happy with one regular set of Base Ten blocks, and an extra purchase of eight more thousand cubes, on account of I wanted nine of them total.

To start the number-building unit, Will cut out a few of the following math journal prompts, and glued them, one to a day, in her math journal:
Building Numbers With Base 10 Blocks Math Journal Prompts

She didn't do all of these prompts, but Sydney, when she finishes the patterning math stuff that's currently fascinating her and starts on number building, probably will.

Each day, Will reads the prompt in her math journal and gets out the appropriate supplies. To build the number 487, for instance, she first gets out our Montessori number cards. She chooses a 400 card, an 80 card, and a 7 card, and lays them out in her work area left to right. Next, she gets out her Base Ten blocks--

--and builds the number, with four hundred flats next to the 400 hundred card, 8 ten bars next to the 80, and 7 units next to the 7:

To finish, she stacks the number cards from biggest to smallest, and behold! The number appears:

We also own a set of Base Ten stamps, so that Willow can write the number and record how it's physically built right in her math journal:

On days when we didn't feel like dragging out all the blocks and the stamps and doing some elaborate math, Willow played with this Montessori number-building app to further reinforce her skills:

I like the step-by-step, physical building work involved in this activity because I want the concepts, and the ones that come beyond, to be something that the girls can mentally visualize. I think it helps them to actually, physically see what a thousand looks like, and to see that a big number is made up of so many thousands, and so many hundreds, and so on.

Here are the manipulatives that we're using:


Tina said...

Thanks for this post! Math scares me, and I think you just made it a little less scary. Of to order some stuff from amazon!

julie said...

When I watched the Khan Academy video of subtraction with borrowing with Will recently (I'm ashamed to admit that I had forgotten exactly how to do it when you have to borrow from more than one place value at a time), I realized that the video was the first time, in all of my schooling, that I remember somebody explaining the CONCEPT behind borrowing--you know, WHY it works! If all I was taught was the shortcut (borrow a 1 from the next place value), then of course I forgot how to subtract, because all you have to do is forget the context-less shortcut and the entire skill is lost. If you understand the physical concept of how borrowing from a higher place value works, then even if you forget the shortcut of borrowing a one, then you can still figure your way through subtracting big numbers.

Basically, what I'm trying to say is that I seem to be learning math right along with Willow.

Malke said...

I've just finished a first read of "Children Continue to Reinvent Arithmetic: Grade Two" by Constance Kamii. There's a lot in there about how mindlessly learning an algorithem for multi-digit adding and subtracting can essentially wipe out any prior understanding kids had of place value. So, you're totally on the right track with all the hands-on stuff. Your kids will actually know what the numerals stand for! :)