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Slide 9: v=value

The final variable in the formula for producing the students we want is v for value. (It’s tough not repeating coming up with all these variables without repeating a letter.) To get students who are engaged learners in math, we must value their effort and reinforce that their effort has value. Let’s start with the valuing their effort idea. The research on praise suggests that we focus on the work not the person. It isn’t that a successful kid is “smart.” We tend to associate intelligence, being smart, with innate ability. If you don’t have it, why do it. Fortunately, intelligence changes over time. Our brains are plastic; they change as we learn. That’s the message we need to send our kids. It takes effort, but you can do it. Even natural abilities can be squandered without practice. It was great hearing Lebron James talk about his work ethic as the key to his basketball success during this year’s NBA playoffs. Being tall is no guarantee that you’ll be a good basketball player. It was also gratifying when my son’s advisor extolled our kid’s work ethic at his high school graduation. The poor kid is a victim of the research I’ve been following. He used to be smart. Now he’s effortful. Some students will take more effort than others, but with focus and attention, much can be achieved.

The other part of the value proposition, that student effort has value, leads us to help students see that math is relevant. Mathematical and algebraic thinking are things we do all the time (check an earlier blog). And some particular mathematical skills -- arithmetic, data analysis, and statistics, among others -- have become part of being a 21st century citizen using 21st century tools, like spreadsheets. It’s worth learning math, not for the grade, but to satisfy intellectual curiosity, to improve analytical thinking, and to be an informed consumer. Math has value and the effort to learn has value too.

There you go. b+i+f+a=the kids we get in our math classrooms (born+informal+formal+affect). And e+c+d+v=the kids we want (evaluate+connect+differentiate+value). Like most models and metaphors, these formulas are not the truth. But they do help us think about our students and how to improve their learning experiences.

Slide 8: d=differentiate

Each student is different. Some will panic when faced with a math task. Some just need a little help. Some will move slowly. Some will move fast. Technology allows us to adjust both cognitive demand and cognitive resources. Based on performance we can dynamically make tasks easier or more complex, more concrete or more abstract. Keeping each student in his or her zone of proximal development, just on the edge of what is doable independently, is a beautiful thing. Few things are more satisfying than successfully completing a challenging task. Too easy is boring. Too hard is frustrating. But just hard enough is very rewarding (chemically, in the brain!). We like it.

At the same time, we can also adjust the resources, or supports, that a student has available to complete a task. We can offer alternative visualizations, definitions, and worked examples. Think back to my Verizon Wireless analogy a few entries ago. We can give students “the network” so that those scary math dead zones aren’t so daunting.

Slide 7: c=connect

Having a notion of what’s inside our students’ heads can help us provide instruction that connects old knowledge with new, informal sensibilities with formal understanding. Educators sometimes use the swiss cheese metaphor when describing student knowledge. It’s full of holes. We simply need to find the holes and plug them. Unfortunately, this plug-the-holes approach only works if the new material bonds with what’s already in place. Otherwise, the plug just falls out.

Sometimes building on to the existing knowledge base will work, but it’s difficult to know what you’re building onto with every child. Instead, it can be more efficient and effective in the long-term to rebuild the structure from scratch. Provide students with the common concrete experiences that can provide the tangible foundation out of which you abstract the desired mathematical concepts. In any case the connection is key. If students see the knowledge as isolated bits of information to remember, you’ve got a cognitive overload situation on your hands. And you don’t want that.

Slide 6: e=evaluate

What about the related equation that determines the kids we want in math? To move students toward successful math learners, we must first understand where they are mathematically. e = evaluate. Diagnosing math understanding, though, is complicated. A general math achievement test, like a state test or typical progress monitoring assessment, only provides an indicator of general health. Is the kid about where he or she is supposed to be? If not, you don’t know why. Since a general assessment covers a broad range of material, you typically only have one or two items related to a specific area of content. How much can you tell from that little bit of information?

Let’s look at some items related to fractions to dig into this question. Here’s one from the grade 8 NAEP test in 2007. Only 49% of students answered it correctly. The most common incorrect response was E, which, oddly is the only sequence in which both the numerator and denominator go from greatest to least, the opposite of what the question requires. However clever our error analysis, we will never know if the selection of E represented a common misconception (that smaller numbers, in this case numerators, means bigger quantities when it comes to fractions) or just a futile guess with the last available choice.

Even individual correct responses can be deceiving. We did some field research for a fraction program Tom Snyder Productions is releasing this fall. We found students who consistently exhibited their understanding of adding fractions with problems like this one. Note how the numerators are added while the denominator remains constant. The student even shows the answer in simplified form. You might conclude from this single item that the student “understands” adding fractions. However, the same student suddenly forgot this understanding when confronted with the task of adding fractions of unlike denominators. Where did the fifteenths come from? Analyzing this second item in isolation might lead one to conclude that the student is applying whole number concepts to fractions. But that wasn’t the case with the previous problem. To diagnose what’s really going on with this student we need to dig deeper.

What we really need, I think, are layered assessments. The high level ones -- periodic achievement tests -- act as triage. They let us know which students need more in-depth diagnosis. But what troubles should we diagnose for? I suggest targeted key concepts, like those identified by the NCTM Curriculum Focal Points or the National Math Advisory Panel’s algebra foundations. Focus on the foundations, like fluency with whole numbers and fractions. Employ targeted, deep assessments to get a true window into what’s happening inside our struggling students’ heads.

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.

Games, perseverance, and status.

The serious gaming world continues to take itself pretty, well, seriously. Kids love video games. If we can just combine those games with education, we’ll eliminate the motivation and engagement problems prevalent in schools. Students will be beating down the doors of their classrooms to get a chance to traverse a cool 3-D world where they explode adverbs with diphthong bombs.

This promise to leverage current high tech interest to cure the ills of old-fashioned schooling has a very familiar ring to it. In the first decade of the 20th century, Thomas Edison predicted that you’d need an army with swords and guns to keep excited students OUT of school. They’d be bursting through the doors to watch films, silent ones at the time, laced with educational content. In the 1980’s the early edutainment industry promised that students would be playing educational computer games, acquiring content and mastering skills without even knowing they were learning. I’ve seen that promise repeated in the last year.

The shallow notion that punctuating popular entertainment with curricular content will make learning fun is wrong, and it misses the point. The serious serious (yes, I meant to write it twice) gaming crowd knows and argues that the opportunity to leverage kids’ fascination with video games lies deeper in the experience. In part it’s about understanding why people are willing to persevere in spite of repeated failures to advance in a game. It’s also about dissecting how games provide multiple entry points and levels to accommodate many different skill levels. How can we get our students to keep trying, and how can we support the range of abilities we find in the classroom?

I want to throw some thoughts at that first question about perseverance. Why do kids, and the rest of us, keep trying at some tasks while quickly giving up on others? When do I believe that my effort will pay off? And when do I think that further effort is futile, that I’m just not good at it? Part of the answer to these questions, the research suggests, has to do with stereotype threat. If I perceive myself and believe that others perceive me as good at something, I’m more willing to keep trying at a task to maintain that perception. I don’t want to risk undermining the stereotype of my capability. If I’m “good at math”, for instance, I need to be good at math all the time. On the other hand, if I’m not good at something, for me it could be drawing, I can give up quickly without any loss of identity. Nobody, including me, thought I was good at drawing anyway.

Status clearly plays a role here. As I’ve mentioned in a previous blog, we (not just kids) value how others see us, and we want to maintain and improve our status. Stereotype threat focuses on the maintenance part. We don’t want to risk a status we already have. But what about establishing a new status? Kids who aren’t necessarily recognized as great gamers are willing to put in the time and effort to succeed. Beginning musicians struggle through endless hours of practice before they have the status of being good. The same is true for athletes, dancers, and anyone who achieves a level of expertise at anything. Even struggling students are experts at some things. Why persevere there but not in math or history?

Maybe status matters here too. Students, some of them anyway, gain social status for the success of persistent effort in some areas, like video games, but not in others, like academics. We need to give students multiple access points to learning and differentiated paths to success. Technology can help do both. Technology can also help students see the incremental value of their effort. Imagine a video game that, instead of listing only the top 10 scores, showed the improvement from your previous scores. You get the satisfaction of growth even if you’re not one of the best. And only 10 can ever be in the top 10.

However, we should also seek ways to award social status for that growth in academically desirable areas. We need to foster a school and peer culture that recognizes and appreciates the effort that leads to incremental improvements. If achieving status matters, we should provide ways for students to establish it in ways that matter to them.