The Beauty of Base Ten Blocks

I (Kathleen) was working with a 4^{th} grader years ago. She was struggling in math class, and came in before school twice a week to do 20 minutes of math with me. (Okay, her Dad had to bribe her with a donut…) One morning I put 2 rainbow cubes on her place value chart and asked, “What happens if the ‘Times Ten Fairy’ comes and waves her wand at these two *ones*?”

Nora picked them up and considered her options. “Times ten?”

“Yup, times ten.”

She turned and put the 2 blocks in the manipulatives basket. Then she counted out 10 *new* ones blocks. Then she stopped and thought, and did it again. She put all 20 blocks in the ones column and smiled like a Cheshire cat.

I congratulated her – after all, her thinking was fundamentally correct, and she knew it. Then I scratched my head, and wondered out loud, “What about the Place Value Police? Will they let you have that many squatters in the ones column?”

She rolled her eyes and palmed the whole set of twenty, put it back in the basket and replaced it with two tens, in the 10s column.

Now my pleasure was authentic. “She’s got it now,” I remember thinking.

“Show me *times ten* again.” I cleared the chart and put down 1 ten and 1 one.

I picked up my knitting. It’s a hobby I enjoy.

Nora picked up the 1 cube, turned to the basket, deposited it, counted out 10 ones and put them in the ones column. Then she picked up the ten rod , put *it* in the basket and counted out 10 tens.

She looked at me, checking for approval, but I pretended to be fascinated by my knitting. She paused, looked at the ceiling (a common gesture in visual learners) and exchanged the ten tens for a 100 flat, and the 10 ones for a tens rod.

*Then *she beamed at me, and I stopped knitting and praised her.

“She’s got it now,” I remember thinking.

“Okay, Mademoiselle Mathématique, try this one.” I put a ten and 2 ones on the chart.

She went through the whole process again. Exchanging, counting, exchanging, placing.

I knitted furiously, so that I wouldn’t help or make suggestions, or, God forbid, *scream.*

We did 21 times ten. Same process.

22 times ten.

15 times ten …

I began to make mistakes in my knitting.

*Thirteen. *Nora bulldozed her way through THIRTEEN problems.

My knitting began to look Gordion-knottish.

On the *FOURTEENTH *problem (25 x 10), she stopped suddenly in the middle. She looked at the 10 ones in her hand, and then at the tens rod.

“Wait, Ms J! You’re just tricking me! This is easy. Times ten just makes a 1 turn into a tens rod. And the ten into a 100-waffle.”

Now she was really pleased. I felt like Annie must have felt when Helen learned how to spell water. Now* I *was the one with the Cheshire grin, and it didn’t matter that I knew I’d have to unravel a few rows of knitting.

The best part? She really DID understand. She could do it the next day and the next week and the next year.

*Could* I have simply told her “Multiply by ten, add a zero at the end”? Yes! BUT – she’d have forgotten it several times, then mixed it up with division, and *in the end, *it would have cost more time than the 20 minutes that morning before school. (Okay, 25 – she was almost late to class. Actually, because of the donut, she probably *was* late to class.)

I was fortunate to teach her again in 5^{th} grade, when we learned decimals. Multiplying and dividing by 10 was no mystery to her. Nor was Scientific Notation in 7^{th} grade, or multiplying or dividing both sides of an equation by ten in 8^{th} grade.

I vowed *never *to rush a child through the concrete phase. I’ve usually kept that vow – of course I’ve occasionally caught myself backsliding. (We humans are prone to relapsing into old ways – let’s forgive ourselves.) But that has always backfired, so I go back to knitting in silence.

Here’s the thing: Memorizing is not Understanding. Just as Dorothy couldn’t rely on Auntie Em and Uncle Henry to save her, and had to do her growing up on her own, so children need the time to connect neurons *on their own *when learning mathematics.

Corrinne and I try to plan our lessons so that students who need manipulatives longer *get them longer. *Understanding concepts is more important than “covering” them.

It takes a concerted effort to train a *whole class* to respect others’ natural working speeds without judgment. But that’s half of what education is about, isn’t it?

** Footnote: Nora did the old-fashioned college prep sequence of math classes in high school (not the newly-fashionable accelerated courses, praise be). Successfully! She is now a junior at a reputable state college, with a major in Psychology and a minor in business. Go Nora!*