What associations do words like “meaningful, joyful, relevant, human, with thinking” invoke in your mind? Of all the possibilities, I bet one of the least likely associations is “mathematics”. And yet, if you ask James Tanton, the Mathematician-at-Large for the Mathematics Association of America with a PhD from Princeton, what his goal is in teaching math to students from kindergarten to college, those are exactly the words he would use to describe how he wants his students to feel about math.
When going into the K-12 world after teaching college, James deliberately chose grades 9-12 because that’s where he saw the least joy happening with mathematics. “I tried to find the wiggle room. If I had to teach grade 9 Algebra 2 quadratics under a textbook I didn’t choose, with a midterm exam I don’t get to write, with standard lockdown dreary questions to be on the exam, how can I find the wiggle room to still teach the story of symmetry and the wonderful interplay between geometry and algebra, and still have my kids answer 50 questions that involve the quadratic formula in 5 minutes? That was the challenge, in that horrible rigid structure to find space for thinking like a mathematician.”
Here are some more thoughts from James on the challenges and opportunities for improving math curricula for the youngest students.
What should be the primary goals of an elementary school math curriculum?
“It’s got to maintain that natural sense of joy and play that young kids associate with curiosity and wonder; a way of being about exploring the world and being curious about the math world as much as anything else. Not getting locked down with “math is either right or wrong” – that very binary state of thinking about mathematics. Unfortunately it’s usually a message they get early on.”
In terms of curriculum content, James believes that the most important one is giving kids a sense of place value and what it really means. “There are subtle tricks in there. For example, using base 10 blocks, these give the subtle message that hundreds are physically bigger than 10’s than 1’s. We give them one model where size matters and then write it in a way that size doesn’t matter. That’s confusing. Or we give kids little discs with 100, 10, and 1 written on them to get them used to size not mattering, but then it doesn’t matter where you place the 100 disc, it’s still worth 100 so writing 216 is still a mismatch for them. Place value is really hard to get right for young kids and it’s the bulk of the work for K-3.” James adds: “My colleague Greg Tang describes these problems and does a lot of work to help students and teachers handle them.”
How should a curriculum be evaluated?
James is very troubled by the nature and impact of the standardized assessment of students. However, when pressed about how a math curriculum should be evaluated, James says we should use the Math circle approach. “Do the kids naturally ask for more? If so, you know you’re doing the right thing.” He also cautions against doing too much assessment. He hopes that “the culture is that kids naturally ask why something works, that they’ll want to play with mathematics and numbers as much as with anything else, draw the shapes, ask questions about the shapes, that sort of thing.
What are James’ thoughts on memorization?
Whereas he believes that at some point a student needs to know their number facts, he thinks that it’s a terrible idea to have a fixed timeline for that. In fact, there doesn’t need to be much memorization – kids just need to remember a few holding points and then we need to give them the strategies to figure out everything else from there. For example, “7×7 somehow sticks in kids heads very easily whereas 7×8 is the universal stumbling block for everyone. Teach them the strategy of figuring out “if I have just one more 7, I can figure this out”.
Why is there so much math anxiety?
“The general perception of mathematics in the US is that math means computation. That’s the expectation, so anything that deviates from that is foreign, weird and scary. Who in their right mind does long division by paper and pencil in this day and age? If the goal is to get the answer, just pull out your phone, and that’s the right, intelligent thing to do. Yet, all the parents still expect their kids in grades 4-5-6 to still be doing the long division algorithm, with the understanding that the point is to get the answer. Well, if the point is to get the answer, then you wouldn’t even be doing that in the first place!
So why are we even teaching it? It’s for the thinking behind the process. But since parents weren’t taught to think about these things, the thinking is very scary. They equate the familiarity of doing the algorithm with understanding the algorithm and don’t realize there’s no understanding of the algorithm in the first place. Therefore, when you start doing things that are about the thinking, it seems very foreign to parents and it seems wrong to them. They survived the system, it worked well for them, they’re functioning adults, and we’re deviating from it. That’s very scary.”
And what about the Common Core?
“By and large I have to say yes to it if only for the simple fact that it’s the first time ever some sort of curriculum is written down that talks about how to be a mathematician. I can only say yes to that. The content is just the same old stuff, probably rearranged a bit.
So I have no objections to common core.
Now how it’s implemented is another story. And I think it’s really the assessment that’s the disaster. The culture of assessment, how you measure it. Here silly things happen and it’s why parents have every right to object. For example, the common core talks about looking for alternative ways to do simple addition formulas and might list different ways of doing addition. An administrator sees this and thinks, oh, they just listed 5 ways to think about this addition problem; therefore we must teach all 5 different ways; and worse, we must test all 5 different ways. Oh, and we must give each of the 5 different ways names, so we know what methods to tell the kids to use.
The common core has done a poor job of supporting the administrators and the parents on how to interpret it. Listing the 5 different ways doesn’t mean teaching every strategy and testing every strategy! It means giving kids the flexibility of thought – THAT was the spirit of that! But I applaud the common core for attending to thinking.”
What are James’ favorite activities to do with elementary school kids?
He coauthored a whole book about that called Avoid Hard Work! One thing that James really likes to do is mess with kids’ minds, show them things that look paradoxical. For example, chop up a an 8×8 square and put it back together into what appears to be a 13×5 rectangle. Another thing he suggests is to “Leave stuff on the walls and say nothing.” Kids will stare at them and ask their own questions. For example, he once put up a 5×5 grid of squares. After counting all the 1×1 squares, kids started wondering about the number of 2×2 squares, and then 3×3 squares. And then he’d ask them why the counts of them are always perfect squares.
Final word of advice from James.
“Keep the kids’ curiosities alive because the content will come swiftly through play.”