In our last blog, we asked, “How do we humans react to confusion and difficulty?”
- Amelia claims grumpily “I don’t get the multiply and divide by 10 thing.” She tends to get grumpy when confused.
- Michael withdraws into silence. “Shall we get the blocks for this, Michael?” Silence. “Which part is confusing?” Silence. Sigh.
- Tomas minimizes his struggles. “I’m fine now. I was just confused on yesterday’s quiz. I’m good now.”
- Nicole writes notes to us on her quiz. “I need more instruction in this concept. It makes no sense at all.”
Today, we’ll look at Michael’s silence.
If Michael decided to talk, he’d say, “I’ve been failing math for 5 years, why should I keep trying? Because even if the teacher tries to help me, it doesn’t work; it doesn’t make sense, and there’s too much to remember, and I get it all messed up on the quiz. I just stink at math. If I don’t say anything, the teachers leave me alone. They just find someone else who hasn’t given up.”
Don’t we all know how this feels? Don’t we all feel it at times? Everyone’s tried to learn a foreign language in middle-age and given up. Or tried to learn to drive a stick-shift from an overly-critical teacher, or tried a recipe that failed, or a new knitting pattern that failed, or come in last on their first 10k race. Now multiply that times five years, and you know how Michael feels.
We can’t change the past. That damage is done. But children can be resilient. Can we leverage that? One thing we think we know about human psychology…
Nothing succeeds like success.
Okay, success… It’ll have to be baby steps. And – it has to be based on how children learn math, so Concrete > Pictorial > Abstract.
A couple years ago we started using LEVELED classwork and assessment. Level 1 is Concrete/Pictorial. Level 2: transfer to the abstract. Level 3: Challenge work. See our blogs from October titled “What About Assessment, Part 1 Part 2 Part 3 “
Now comes the hard part: convincing students that success at Level 1 is something to be proud of.
This is hard because our society is competitive, because schools value speed, because assessments value the abstract over the pictorial. It is hard because math is more than just the gatekeeper to academic progress; it has become the definition of intelligence.
So we have to teach to a student’s strengths. For many, that means stay at the concrete/pictorial for as long as needed. We even choose challenge problems that challenge our abstract thinkers to try a visual approach when it works better. See a recent quiz for examples of student work at 3 levels.
And we reward success at the concrete/pictorial level with the same respect and praise that we give to success at the challenge level.
Exploding Dots: a window to base-10?
This month we’ve been working on place value and multiplication/division by powers of 10. Every year we notice that students’ understanding of base 10 is broadly algorithmic and based on tricks and mnemonics. Even their conceptual understanding of multiplication and division is shaky.
James Tanton to the rescue! His website Exploding Dots is phenomenal. Play around with it for a while. It’s an amazing combination of creativity – visuals, videos, fantastic software coding. He’s also the author of wonderful books like MATH WITHOUT WORDS and MATHEMATICAL THINKING, and teaches an intriguing course online: Visual Thinking.
We started with a hands-on investigation using blocks, so that our students’ first exposure to other bases is concrete. We made this Power Point: Base Six Investigation and this accompanying worksheet: CW 12_6
We boil down the first slide to a metaphor: “When there are SIX blocks in a column, the house is too crowded – the place value police come and enforce a house-cleaning. Every six blocks become ONE block in the next-larger column. (The rods are actually 6 units long, the flats are 6×6. Do you see how well this is going to transfer to base-10?)
It went well – here are 2 photos:
We had a few green rods SIX units long (from our Cuisinaire set) but we had to cut more flats and rods from cardboard.
Then we got our computer cart and went on to the Exploding Dots website. Most students stuck to Islands 1 and 2, but a couple students moved on to the 
optional Island 11,
which is challenging and historical -Napier’s Chess Board.
We spent about 2 and a half days on this website. Here’s a possible pathway for student exploration: CW 12_10 What’s really great: You can illustrate MULTIPLICATION (lay out repeated groups of blocks) and DIVISION (share blocks 
equally between a given # of groups, then un-explode any ungrouped blocks to the next smaller column and repeat.
Here’s a photo of 4 x 14 in base six, before “cleaning house”.
We don’t quiz students on other bases (except maybe a few challenge problems). It’s not integral to 5th grade. It’s fun. We spend time on it because it remediates shaky concepts around place value, multiplication, and division in base 10.
Here’s what’s fascinating: Our traditionally struggling students excelled at these activities! Our traditionally high achievers struggled. The thing is, Exploding Dots is so visual, it plays to the strengths of what we thought were our weakest students. They’re not weak after all – they’re visual thinkers. And our algorithmic, abstract math lovers found all their hard-won procedures, mnemonics, and tricks rendered useless. Talk about empathy!
Michael is a case-in-point. He totally mastered the visual concepts around exploding dots. He was explaining concepts to top students, he was explaining at the board. He glowed. On a problem a few days later – a question around normal base-10 multiplication, he asked “Can I draw the dots?”
(Duh!) Of course.
He’ll draw until he’s ready to let go of that tool and use numbers (which we call squiggles — what’s seven-ish about a 7?) and algorithms (which save time, in the end). Don’t worry that your 5th graders will head off to high school with blocks hidden in their backpacks; the human brain loves efficiency. Eventually, we choose squiggles.