Slide 5: a=affect

The final piece of the formula reminds me of Verizon Wireless commercials where a cell phone user is confronted with the prospect of getting lost in a “dead zone.” The road ahead is scary. It might overwhelm most cell phone users, but the Verizon Wireless customer has a network of resources with him. He can prevail. That balance between the demands of the task and the resources we believe we can bring to it is a critical part of students’ affect, the state of mind they bring to learning math (or anything for that matter). I’ve written about affect, the work of Carol Dweck, and neuroeconomics in other blog posts, so I’ll keep this one short. Emerging research continues to show that students’ willingness to participate in the learning contract in the classroom along with their belief that their effort will matter pays huge dividends in their performance. It’s tough to do well if you don’t try. Do kids feel daunted at the task of learning math? Do they believe that it’s pointless because they’re just not good at it? Or do kids feel that they can meet the challenge? And do they think that meeting the challenge will bring benefits, immediate in the satisfaction that comes with meeting any challenge and long-term in their future academic and occupational success? This variable, often overlooked in when we assess our students, is a huge part of the formula that determines the kids that enter math classrooms everyday.

Slide 4: f = formal instruction

Unless you’re teaching preschool or kindergarten, your students already have a history of formal instruction in math when they enter your classroom. Did the prior instruction build on the child’s innate mathematical capacities and informal experiences? DId it treat each state standard as a separate, isolated and atomized bit of content, divorced from the other objectives. Or did it provide a coherent and connected progression of instruction with new instruction explicitly building on what students already had learned? Did the prior teaching protect students from cognitive overload by constraining new learning demands and providing time for practice and mastery before moving forward? Did the students fall victim to the math wars with overemphasis on either mindless procedural mastery or unformalized invented algorithms and approaches?

I could ask more questions, but the point is that past instruction matters. Many of the struggles students have with math come from what and how they were taught. Confused ideas about equivalence and the equal sign (=), overgeneralization of whole number algorithms into fractions, the lack of a unified number system across integers and rational numbers, and so on are really instructional issues. Without a good understanding of the models and approaches students have accumulated in school, it’s tough to make the kind of connections that can move them forward sensibly.

Slide 3: i=informal experiences

The variable i stands for our informal experiences and actually has two parts, i(sub1) and i(sub 2). i(sub1) refers to our informal mathematical experiences. Children have variable informal opportunities to count place settings, divide up Halloween candy, play numerical board games like Candyland, or share continuous quantities like pudding or juice.

Sadly, the variation often falls along socio-economic lines, with kids from poorer homes experiencing fewer early number experiences. It matters, just as it does in reading. Children who grow up in a home full of books, who are read to and enjoy rich language exchanges with family members come to school familiar with the alphabetic principle and the structure of books. They have a growing vocabulary and a head start in learning to read. Similarly, kids who play board games (the research is very strong here), read thermometers, tell time, count anything, share equally, and do all kinds of other informal numerical stuff have a richer number sense when they start school. They are ready to learn math.

In addition, children have all kinds of informal experiences that have nothing directly to do with math but a lot to do with their attitudes and abilities as learners. Chronic stress (like hunger or fear) early in life, for instance, may contribute to a reduced working memory capacity that in turn hinders the acquisition of certain math skills, like math fact automaticity. Early responsibilities, actions, and interactions can influence the development of self-regulation and executive function, the ability to control and manage one’s actions. Children who can monitor their own behavior are highly correlated to academic success. Put simply, students with the skills and attitudes tuned to school culture and formal learning increase their likelihood of classroom success across the content areas.

Slide 2: b=the math ability we are born with

Even though we frequently hear people complain (or maybe apologize is a better word) that they just aren’t good at math, we are all, in fact, born to do math. Studies with infant humans (and many other animals) show that they can recognize small quantities, like one, two, or three, without counting, an ability known as subitizing. Google the term and you’ll likely find a little subitizing game that let’s you compare your ability to subitize versus a chimpanzee’s skill.

We’re also born with the ability to make comparisons. We can tell the difference between a lot and a few. From an evolutionary point of few, it’s not surprising that we (and, again, other animals) can quickly make quantitative comparisons. Knowing whether the odds favor running away or staying to fight helps improve the chances of survival. We’re generally not as good when the ratio gets closer to 1 to 1, but some of us are better than others. In fact, you can once again poke around on the web and find a game to test the edges of your ability to determine which of two sets is bigger at a glance. Some researchers found that students who performed better at comparing sets as the ratio got closer to 1 to 1 had histories of better performance in math in school. Interesting.

Using puppets and watching how long babies stare at something unexpected, several research groups have found that we also seem to be born with the ability to add and subtract, at least up to the number 3. Show 3 puppets and then show 1, and the baby looks puzzled. Show 2 puppets, and the baby is still puzzled. Where’s the third one? We’re pretty amazing even before we’ve had any formal education.

There’s even some research suggesting that we have some innate ability to recognize fractions and ratios. It’s tough to explore this ability, but very young children do look surprised when something like a book is hanging more than halfway over the edge of a counter, and it doesn’t fall. Maybe all the times that my son pushed his sippy cup or food bowl off the edge of the table, he was actually exploring his concept of proportionality (and gravity).

These born-with-it math abilities are variables. They are not the same for each of us. Some of us have more robust spatial awareness than others. Some can subitize larger quantities. But the natural math capacity we bring into the world is part of the equation for our mathematical identities.

NCSM Talk - Slide 1

I gave a brief breakfast talk at the NCSM (National Council of Supervisors of Mathematics) national conference this spring in Washington, DC, and I was gratified to receive numerous requests for the slides. The slides by themselves, though, are not that useful. I just use them as prompts for what I want to talk about. I figured I’d give a shot at trying to capture the talk in my blog. We’ll see how it goes.
I figured that since I was at a math conference, I’d work with a math metaphor. Here are 2 equations: one captures the variables that determine the students we get in our math classes; the other captures the variables we want to control to turn them into the math learners we want. Let’s take the variables one a time.

Thinking like...

So I’m focusing heavily on math these days, learning the content, working with math educators, and talking with mathematicians. The inevitable question arises: “Why do kids need to learn this stuff?” I’m not talking about arithmetic -- adding, subtracting, multiplying, and dividing whole numbers. We count stuff all the time. We keep track of batting averages; how much money we owe, spend, and save; how much more we need; how much extra (hopefully!) we have; what it means to double or halve a recipe; how many zombies we need to shoot, points to score, or crystals to capture to get to the next video game level. Arithmetic is part of our lives. But what about the math that comes after arithmetic? Why do we need to learn algebra? How often do we solve equations with exponents in daily life? It often feels that the reason we learn math beyond arithmetic is to do more math in school. No wonder kids find it boring and struggle to see the value.

When I talk to mathematicians, they don’t get it. Math is so exciting; it’s arithmetic that’s kind of boring. In fact, it’s a bit embarrassing when I describe FASTT Math to a mathematician, and he or she confides that she’s not very good with her math facts. How can an advanced mathematician not be good with math facts? Well, advanced math often doesn’t have much arithmetic; it doesn’t even have many numbers. What’s the deal?

The apparent conundrum got me thinking about my own training and teaching. In college I learned how to be a historian, and I learned how to teach history. They’re not the same. In fact, I never really liked history very much, but I loved being a historian. The facts of history -- the dates of events, the order of Presidents, the names and places of battles -- held little lasting interest. I haven’t used them, and, not surprisingly, I’ve forgotten many of them. However, what I learned doing history, being a historian, I use everyday. Doing history means deciphering the truth through the lenses of whatever evidence is available. What caused the Civil War? Why did Truman drop the atomic bomb? How did African tribal leaders feel about European explorers? So many witnesses and viewpoints to sort through, understand, and weigh. Lots of people seeing the same event and describing it in different ways. What really happened?

Anyone who has children exercises this way of thinking like a historian all the time. He did it. No she did. I use these skills as a husband and father, a teacher, and a manager. I collect and weigh evidence. I consider the biases of the sources and compare their versions with other available objective and subjective evidence. The skill of thinking like a historian has long out-lived the content of the history I learned. But that’s okay because the content was the vehicle to hone these skills. It served its purpose, and I know how to find it if and when I should need it again.

These historical thinking skills unfortunately frequently got lost in the obligation I felt as a teacher to convey the content. Indeed, the historical information became the end, it’s retention the measure of educational success. How do you measure thinking like a historian anyway?

I get the sense that we’ve followed a similar path in math. We don’t spend enough time helping kids see that learning the content of math is a path to a way of thinking and problem solving that they can use for the rest of their lives, long after they’ve forgotten a particular formula. You can read about mathematical thinking and rigor, but it’s tough to find in the standards. Ask an adult when they think mathematically, and they’ll typically describe an exercise dealing with numbers or spreadsheets.

But thinking mathematically thrives in the non-numeric world. The concepts of equivalence, commutativity, and order, for instance, have implications in many aspects of our everyday lives. Can I combine these recipe ingredients in any order or will the results differ if I add the egg first? Is there another way for me to get the outcome I want? How can I break down this complicated situation into more manageable pieces?

The teacher training books from Singapore, a nation that produces the highest performing math students in the world, talk specifically about the goal of mathematical thinking. But even they acknowledge the difficulty of focusing on this nebulous outcome compared to concreteness of specific math skills. Without our help students won’t see the power of thinking like a mathematician any more than they see the daily value of thinking like a historian. We’ve got to make the connections explicit and show how the exercise of learning the math (or history) makes those skills stronger. That’s a good design challenge. I’ll keep you posted.

Lessons from Behavioral Economics

Charles Darwin celebrated his 200th birthday in 2009 (so did Abraham Lincoln -- in fact, Lincoln and Darwin were born on the same day). This year also marks the 150th anniversary of the publishing of Darwin’s The Origin of Species. It seems an appropriate time then to delve into the evolving research in behavioral and neuro-economics. What can education learn from the search to understand human irrationality?

For generations economic theory was based on the premise that people make rational decisions. In reality, however, they don’t. For instance, a rational economic decision-maker would always choose to maximize gain and minimize loss. And we may think that we do. Our actions, though, prove otherwise. Imagine you’re one of two participants in this little experiment. The other participant is given $10. She can give you as much of the $10 as she wants, a nickel, $5, or even all of it. If you accept what you’re given you both get to keep the money. If you reject what you’re given, neither of you gets anything. So you have the power to get something or nothing. Let’s say the other participant only gives you $1. Do you take it and let her keep the remaining $9? Or do you reject it, so that neither of you has anything? What would you do?

Rational economic theory assumes you take the money, whatever the amount, because you’re better off getting something than nothing. If you’re like most subjects in the actual study, though, you’d reject any uneven split. You’d rather punish the other subject (and yourself) for not being fair than get a little money you didn’t have before. Here’s an interesting twist to the story -- how much the other participant would offer depends on whether or not she can see you. Most subjects actually offered a $4 to $5 split if they could see the other participant. However, if the other participant was unknown, the amount offered dropped dramatically.

Experiments with how we choose among several items, which kind of tasks we procrastinate about, and how different triggers (like smells or large numbers) affect our decisions, among many others, have begun to reveal what on the surface appears to be a very quirky brain. Neuroscience is helping to identify patterns of brain activation in that apparent quirkiness. And evolutionary psychology is offering explanations about why these irrational behaviors may actually help us survive as a species.

Marketers have long been exploiting our behavioral tendencies. For example, the fresh produce is typically at the front of a grocery store because we’re more likely to buy junk food if we’ve already committed to something healthy. (A study revealing our greater tolerance for unethical acts when we have clean hands -- as in just washed -- versus when we have dirty hands highlights this behavioral oddity from another perspective.) In fact, our behavioral buttons are constantly getting pushed. Can education play the game too?

Many teachers likely already are playing the game. They’ve learned through experience what prompts desired actions, and they use whatever tricks help get the job done. I wonder if we can be more systematic and systemic about it. The research is fairly new, and I’m still swimming through it. But I think there’s potential. We’ll see.