The Dock Spot

I've Moved

2009-09-07T07:46:00.000-07:00

My blog has a new home. Hope to see you there soon!
http://blog.tomsnyder.com

Slide 9: v=value

2009-06-12T05:23:00.000-07:00

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

2009-06-12T05:22:00.001-07:00

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

2009-06-12T05:20:00.000-07:00

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

2009-06-12T05:02:00.000-07:00

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

2009-06-12T04:58:00.001-07:00

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

2009-06-12T04:57:00.000-07:00

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

2009-06-12T04:55:00.000-07:00

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

2009-06-12T04:52:00.000-07:00

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

2009-06-12T04:51:00.001-07:00

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...

2009-05-04T10:05:00.000-07:00

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

2009-03-22T13:50:00.000-07:00

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.

2009-01-02T08:52:00.000-08:00

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.

Blogs are for old people.

2008-11-16T19:36:00.001-08:00

Here are I am trying to be technologically hip by thinking of something insightful and interesting to share with an unknown audience on a fairly regular (maybe not so regular; it’s been a while) basis, and I keep reading that blogging is dead. Okay, “dead” might be a little strong, but it’s certainly not hip or cutting-edge anymore. The latest nail in the coffin came from The Economist. The magazine noted that blogging has gone corporate. Every company has at least one, and they use their blogs as another way to reach and influence customers. I’m not blazing trails; I’m just one of the pack.

The early blog adopters have moved on to twittering and other emerging means of communicating. A similar migration is happening with Facebook. Once mom and dad join, the kids flee (my son recently unfriended me). Who can blame them (I’m not really in the 17-year old peer group)? As Walmart, Proctor & Gamble, and other staid corporate entities develop their Facebook presences, I may follow. The problem is, wherever the trailblazers go, mom and dad and the corporations will follow. You can’t really escape for long.

So what happens when new becomes old, when cutting edge turns into mundane, when the digital natives leave and the immigrants (like me) move in? Will enough people with sufficient energy stay in the neighborhoods to keep them vibrant? Which neighborhoods will survive and which will whither? It’s going to be very interesting to watch what happens. The speed with which new services arise, grow (or not), and disappear reminds me of biology experiments with fruit flies. Their lifespan is so short, the passing of generations so quick, that you can see the effects of evolution unfold in real time. Keep your eye on the web, but don’t blink. You might miss the life and death of a transformative service.

I love standards. I hate standards.

2008-09-21T13:10:00.000-07:00

I guess it’s clear from the title of this blog that I have a love/hate relationship with the standards movement. Breaking down curricular content areas into grade level objectives -- in 3rd grade students in Texas are expected, among 40+ other math standards, to “use fraction names and symbols to describe fractional parts of whole objects or sets of objects” -- have done much for clarifying expectations for what students should know when. Enforcing these standards on a state, or even national, level insures that all students should be receiving a rigorous education. Classrooms, schools, states, and the nation can test students on their mastery of these objectives to identify where the instructional system is faltering. Curriculum developers and publishers can create instructional materials to support the expectations. When the system works, it should be beautiful.

Unfortunately, the reality rarely matches the plan. In fact, in some ways, the standards movement may have been more harmful than helpful. Part of the difficulty rests with the lack of a common national standard. It’s expensive to maintain a different set of standards for each state, along with different high quality assessments matched to those standards. Local control (and local standards) is not a bad idea, so long as the funding is available at each local level for successful implementation and support. It’s not particularly efficient for each state or district to develop standards, the particular assessment items to evaluate the acquisition of those standards, and the training and curriculum materials to support the implementation of those objectives. Funding shortages lead to short cuts and misalignments between what we say we expect and what we actually test.

But there’s another important problem with the current standards movement; it has led to the atomization of content and instruction. The list of grade level standards has become just that, a list to be checked off lesson-by-lesson. Instructional lessons carry a list of the individual standards they address. Check, check, check. Next lesson. The coherence that connects and makes sense of the path through the standards gets sacrificed in the urgent need to cover each individual objective. The learning expectation above about fractions cannot be taught successfully in isolation. Students need a coherent understanding of fractions and of number that unites the individual ideas, concepts, and skills described in the standards. Jumping from number lines for whole numbers to pizzas for fractions to place value grids for decimals may meet individual math objectives, but it can leave students feeling that each number form is a different number system. We’re just setting them up for later confusion. Standards are great but not at the expense of the path connecting them.

Good news. NCTM has recognized that 40 to 60 individual, equally-valued math objectives per grade level is a recipe for speed teaching. Each standard on the list gets 5-minutes. Better pay attention. The organization’s Curriculum Focal Points document attempts to identify the core standards upon which others are built. The Report of the National Math Panel reiterated the need to focus on the critical ideas and to build coherent curricula. Last year’s National Research Council report on science education moved in a similar direction. Hopefully, we can sustain the momentum, but there’s plenty of work to do to reconcile my rocky relationships with standards.

What Before How

2008-09-04T19:23:00.001-07:00

I’m spending a lot of time these days thinking about multiplication. Keith Devlin, a Stanford mathematician, author, and NPR Math Guy, has created a swirling controversy through his Devlin’s Angle column on the Mathematical Association of America website (http://www.maa.org/devlin/devangle.html). Last summer Devlin sparked a debate by exhorting teachers to stop defining multiplication as repeated addition. His follow-up articles this summer fanned the flames and ignited a raging firestorm in the blogosphere. 4x3 can readily be rewritten by the repeated addition of 4+4+4. But what does repeated addition look like for, say, 3/4 * 5/8? It’s a challenge to articulate a definition that is both accessible to kids in elementary school and still true as the numbers become more complex.

I’ve enjoyed reading the unfolding arguments. It’s a healthy and important discussion, because determining what we should teach is essential before we decide how to teach it. Designing technology to more effectively instruct and engage students in a misconception or limited explanation shouldn’t be the plan. I fear that too much of the “innovation” and promise of technology doesn’t go deep enough into the roots of why our kids aren’t succeeding. It’s far too easy to take the existing curricular canon and put it into a glossy technological wrapper and be satisfied that kids “like” using it. We need to push ourselves further.

So, the search for a new way to define multiplication continues. When we have it, we can devise ways for technology to help visualize it, explore it, practice it at appropriate levels, and connect it to the content that came before and follows. It’s rigorous work, but if we get it right we can truly make a lasting and meaningful difference with kids’ understanding. It’s worth the extra effort.

First Pitch

2008-07-29T09:45:00.000-07:00

Yes, I’m throwing out the first pitch at tonight’s Red Sox game against the Angels. My wife won the prize at a charity auction and gave it to me for my 50th birthday. It’s an incredible gift, and one that allows me to reflect directly on some of the research I’ve been reading and writing about -- particularly the research on anxiety and performance.

Ever since my wife presented me with the gift last fall, friends, family, and strangers have inquired about my fitness and ability. “You know, it’s pretty far from pitcher’s mound to home plate. Can he throw it that far?” “Has he been practicing?” My cousin sent me a link to a YouTube video of a mayor throwing the first pitch into the dugout. “You don’t want to be like THAT guy.” Thanks everyone for the pressure.

I’m no athlete, so I’ve been playing a little catch. Sixty feet, six inches. When I’m just throwing the ball with my wife or son, it doesn’t seem like a problem. But practice is different from high stakes performance. Throwing in an empty field is different from throwing under the gaze of 35,000 fans. Okay, so none of those fans finding their seats and buying their hot dogs will actually be watching me; they’ll still be there. Will anxiety about not screwing up dominate my thinking and make me forget how to throw? Is this how some kids feel when they’re getting ready for high stakes testing?

I’ve got it easy. Whether I bounce the ball into home plate or throw a looping strike, I’ll give a satisfied smile and a fist pump. They only take pictures of me, not where the ball goes. Wish me luck.

Digital nomads with their heads (and data) in the cloud

2008-07-15T12:22:00.000-07:00

My son Jake, a rising high school senior, doesn’t have his “own” computer. Instead, he seems to have everyone’s. At school he moves from the computers in the computer lab to those in the student center to those in the film studio. He emails documents to himself or carries work on a flash drive. At home, well, at home no computer is safe. I’ll open my work laptop and find FirstClass open, my son’s Facebook page on the browser, and a couple of homework documents on the desktop. “Do you need any of this stuff?” I ask. “No, I got it.” My computer was handiest; he used it and pushed his work into the Internet cloud to be retrieved at the next available machine. When we travel Jake always manages to find someone or someplace with a computer, iPhone, or some other cloud-friendly device. He connects with friends, moves pictures to Facebook, and checks tennis scores. Welcome to the world of the digital nomad.

Don’t get me wrong, Jake wants his own laptop. He needs a base (we have an aging iMac at home that is “his” in general), but his ability to exist in the cloud frees him from that base (making anything “his” at any moment). And the shrinking size of computers and growing power of cell phones offer the promise that maybe the base can move with him as well. Or at least be available wherever he goes. For him, with the exception of a few specialized applications, school work has no boundaries. Any computer in our house, at school, at a public internet station, or anywhere will do.

As more and more work, student and teacher, exists in the cloud, the opportunity to do something really interesting with it grows because we can all access the cloud and what it contains. Despite my own experiences, direct and vicarious, with mash-ups and remixes, I’ve been slow to see the potential of the collaborative Web 2.0 promise. I’m finally beginning to see the light, but it’s fraught with questions and issues we still need to figure out.

Let me give a specific example: We are about to launch a new program, Timeliner XE. The original TimeLiner 1.0 in 1986 turned Apple IIs and early PC’s connected to dot matrix printers into simple systems for generating banner time lines with ease. Type in events in any order, and the software created a proportionally-spaced chronology that could be printed out sideways on the scrolled paper of those early printers. Simple and sweet. Over the years and releases, the program added the ability to print posters, save as html, create slide shows, and add graphics, links, and other media. This new version takes a huge leap forward. Timeliner XE, built with Adobe Flex and Air, contains a built in browser that enables users to gather, organize, and present information within the same application. In addition to time-related events, any sequential information -- like the life cycle, the steps in a research project, or the plot of a book -- can be managed in the program. It’s very cool.

So what happens to the time lines and sequences that students and teachers create? While having thousands of students create their family histories or story arcs for Tuck Everlasting each year is nice, can’t we take their efforts further by putting the work into the cloud? Imagine a Day-in-the-Life time line that students from around the world contribute to. Each contribution gets its own category and color code so that they can be distinguished and turned on and off. How about collaborative time lines highlighting the flow of information and technology around the world as it happens? What might it look like for users to stand on the shoulders of previous work rather than simply repeating what others have already done? We’re having fun imagining the possibilities.

We’re also straining to understand the implications. What student information is private? Is it okay to share a student’s day in the life in the cloud? Can other users change someone’s time line? What about ownership and attribution? How do we make sure that copyright is respected in what gets posted? It is exciting, but it’s also complicated.

Many of our students are already living in the cloud. It’s happening even as the rules are being created and understood. We’re working on it. I hope you’ll be part of the unfolding story.

Brain Doping

2008-06-20T13:20:00.000-07:00

A recent edition of Economist magazine had a very interesting editorial and article about the coming wave of cognitive enhancement drugs that augur the possibility of some tough ethical dilemmas ahead. As research uncovers chemical paths to improving memory and mental processing power for those afflicted with Alzheimer’s disease and other cognitive disabilities, what happens to those who seek to use those chemicals for a brain boost rather than just a neurological repair? Many of us already use over-the-counter drugs, like caffeine, to enhance general alertness or to help study for a test. Where do we draw the line?

Already, it seems, a surprising number of intelligent folks are stepping well over the caffeine line. The scientific journal Nature surveyed its readers, a pretty knowledgeable crowd, about their use of cognitive enhancers (Nature 452, 674-675 (2008)). Of the 1,400 respondents to the poll, one in five “said they had used drugs for non-medical reasons to stimulate their focus, concentration or memory.” Does that mean that 20% of Nature readers are cognitive cheaters?

Maybe I’m overreacting, but if we found that 20% of Sports Illustrated readers used prescription drugs, like steroids, for non-medical reasons to enhance athletic performance, I wager many of us would feel that those folks were doing something unfair. How should we think about “natural” intellectual ability versus one that is artificially enhanced? Is it cheating to use drugs to help you study longer or more readily recall what you’ve learned? What about the student who uses a beta-blocker to reduce the effects of anxiety before a test? Are non-prescription boosters okay but not ones that require a doctor’s permission?

I’m not sure I know how to answer these questions yet. If only the well-informed and well-to-do have access to these brain boosters, then I do think there’s a problem. The rich get richer and the poor get poorer. On the other hand, improved intellectual ability seems like a good thing in general. Shouldn’t we encourage it whenever we can? Except when it’s unfair or unhealthy. But who decides that? It seems like a good debate is brewing. I look forward to participating in it. I may need a cup of coffee to keep me awake to read all the relevant research and opinions.

Fewer is More, too

2008-05-12T14:00:00.000-07:00

My son, Jake Dockterman (he likes it when I use his name; it raises his Google hit count), is a bit of a stickler when it comes to the use of the words “fewer” and “less”. He’s not alone, as a web search of “fewer vs less” will reveal. You see “fewer” refers to how many, and “less” refers to how much. If you can count it -- marbles, tortilla chips, brothers and sisters, or Manny Ramirez home runs -- you use the word fewer. If you can’t count it -- applesauce, happiness, sand, and milk -- you use the word less. You don’t say “I’m fewer happy” when something upsets your birthday. And you don’t say “I have less siblings than my friend.” At least, you shouldn’t say you have less siblings. One last note: the signs above the express line at the grocery store should read “10 items or fewer.”

I’m a big believer in the less is more philosophy, and I enthusiastically embrace the sentiments of the NCTM Curriculum Focal Points and the final report of the National Math Panel. Both documents advocate focusing instruction on the essential content, the core ideas that students need for success in math, Algebra in particular. A report from the National Research Council last year recommended a similar emphasis on big ideas in science. Taking Science to School offers a very accessible review of the research on science teaching and learning and recommends: “The next generation of standards and curricula at both the national and state levels should be structured to identify a few core ideas in a discipline and elaborate how those ideas can be cumulatively developed over grades K-8.” The trend (hopefully) in both math and science is a welcome targeting of the curriculum on what really matters.

So, are these reports calling for less or for fewer? The distinction is important. Fewer topics means dropping some of the content that’s crowding the curriculum. Maybe we don’t need to cover probability or tessellation or the orbits of the planets. Covering the lengthy lists of state content standards provides little time for depth of learning. Cutting the list in half doubles instructional time for each learning objective. Fewer instructional goals could mean more time for truly learning those that remain.

On the other hand, devoting less time to some of the items found in state curriculum guides could also lead to more time for other, more critical, items. The big ideas certainly merit more focus than the supporting skills or concepts. Maybe the issue is one of emphasis. Not all content objectives are equal. The NCTM Curriculum Focal Points document makes this case very clearly. NCTM recommends “areas to emphasize” rather than to slash. Focus on the core ideas and use the other objectives to support those crucial concepts.

From my perspective, I think we need both less and fewer. The list of learning objectives in some states has simply become too long and too atomized. The pressure to cover each standard has turned them into a checklist of disconnected items. It’s time for some judicious winnowing of the curriculum. We need fewer standards to give teachers tangible evidence that they have the freedom (maybe mandate is a better word) to focus on what matters. Even so, we still need a re-emphasis in the curriculum. We must identify the core ideas that merit more time and show how the remaining objectives can support them. Less time on some content can lead to more understanding overall. And fewer topics in the list can help make that re-emphasis possible. Less is more, and fewer is more too.

Arts and Smarts

2008-04-29T07:11:00.001-07:00

In March The Dana Consortium released a compilation of research dealing with the arts and cognition. How does studying music or dance or the visual arts affect brain development and learning? I’ve been fascinated with this topic ever since the governor of Georgia, Zell Miller at the time, proposed providing the parents of every newborn in the state with a CD of classical music to play for their infants. The Mozart effect promised to boost performance, particularly in math, simply by listening to classical music. Music could make you smarter.

Well, maybe not. A small study that showed improved performance on a paper folding task after some of the subjects had listened to Mozart became headline news in a media hungry for big stories. Other studies that showed temporary, but not lasting, boosts to IQ (which itself raises interesting questions about what IQ really measures) fueled the media flames. The fact that subsequent studies showed that similar increases in performance could be sparked by other relaxation techniques did little to dampen the media-whipped excitement around the Mozart effect. Naturally, those follow-up studies received fair exposure in academic journals, but the popular media had little interest. Bold stories sell papers (or clicks on the Internet). Nuance is too complicated.

But nuance is often the true story behind the research that makes headlines in the popular press (as opposed to the academic journals). Whenever I see a story, even in Education Week, about a dramatic research finding, I track down the source article in the academic journals. How was the research conducted? What do the results really tell us?

The Dana Consortium report does a wonderful job of describing the bits we know and the many questions we still need to answer about the arts and cognition. Learning music intensely, for instance, does seem to make a difference in academic performance. Just listening to music or the occasional weekend music lesson doesn’t reveal any meaningful differences. But serious music study does appear to have a positive spillover effect on academic tasks.

However, the report cautions that it isn’t clear that music is what makes the difference or just the intense study of a subject. Learning music teaches students how to focus their attention, and that habit may be the key underlying skill for success in other areas. Or maybe there is something about music in particular. Of all the activities studied by brain imaging techniques, performing music lights up the most parts of the brain. While the question lingers, there’s certainly no harm in encouraging music study among our children. It does seem to make a positive difference for whatever reason. I see it in my teenage son, who is a serious music student and a successful school student. Like chicken soup, it couldn’t hurt.

Science Dinner

2008-04-07T07:26:00.000-07:00

Boston hosted the NSTA (National Science Teachers Association) annual conference at the end of March, and Tom Snyder Productions hosted 50+ science educators from the conference for an open house at our office. A mix of science teachers, district-level science coordinators, state consultants, and other science education specialists from across the country joined us for an evening of demos and dinner. It was great fun and a wonderful opportunity to connect with the people doing the hard work at the front lines.

I was particularly impressed with the number of TSP folks who surrendered a Friday night to hang out with a bunch of science educators. Engineers, quality assurance specialists, producers, IT personnel, customer service reps, and others mingled and chatted with our guests. Most of these TSP employees don’t get a chance to meet the people who actually use the products they create. There were fabulous conversations going both directions. I look forward to the next time a relevant conference is in town.

I was invited to say a few words at dinner. The last thing I wanted to do was interrupt the flow of the evening (and the meal) with a boring speech. So I kept it short. I offered three bits of research-based advice for how to behave at dinner.

1) Talk to your neighbors. Research is very strong about the value of sharing what you’re learning to build your own understanding. In fact, recently published research on problem-solving transfer among young children concluded, “The general lesson might be that if you are having difficulty in understanding something, you should try explaining it to your mom.” (Rittle-Johnson, et.al., 2007). I like that.

2) Be careful what you say. Last year’s report from the National Research Council called "Taking Science to School" offers a very nice summary of the available research on science learning and instruction. The report notes that past science instruction paid little attention to the informal background knowledge that children brought with them to school. Kids’ have some well-entrenched notions about the workings of the natural world, notions that can help or hinder the acquisition of accurate scientific concepts. If we don’t take students’ existing understandings into account, then what we tell or teach them may well reinforce a fundamental misconception.

3) Have fun. Affect, the way we generally feel, has an impact on how well we learn. Engagement and happiness tend to reinforce retention. We remember what we enjoy. So have a good time. Learning is fun.

National Math Panel

2008-03-07T06:37:00.000-08:00

The final report of the National Math Panel is finally out, and I have to admit that, overall, I really like it. I completely agree with their conclusion that the tension between conceptual understanding and procedural fluency is a false battle. It isn't the standard algorithm OR flexibility in solving problems. Conceptual understanding, fluency, and problem-solving all work together. Quick recall of basic math facts is very important, but it isn't everything. The opposing sides in the Reading Wars eventually accepted a truce; kids need to know how to decode, and they should love and understand what they read. Let's hope the antagonists in the Math Wars have reached a similar accord.
One area noted in the report should readily be embraced by everyone. Effort and attitude matter. They summarize the research of Carol Dweck (see one of my previous blogs) about shifting the learner's attitude from one focused on innate ability to one that recognizes growth through effort. It's critical for students and teachers to acknowledge incremental improvement and the effort it takes to achieve it. I wouldn't be surprised to find a relationship between these attitudes and math anxiety. The panel recognizes the reality of math anxiety and recommends more research to uncover its source. Indeed, the report makes many recommendations for further research. We need it.

The panel's report did, though, leave me wanting in a couple of areas. While the report talks about the importance of problem solving, it never describes what "problem solving" means or how it should be developed. Maybe the research isn't robust enough to illuminate clear directions. I would also have liked more clarity about why the panel members singled out particular areas in measurement and geometry for instructional focus. I don't disagree with the importance of the selected areas of content, but I would welcome more elaboration about how they fit into an algebra trajectory.

Hopefully, NCTM's Curriculum Focal Points and this report of the National Math Panel provide enough guidance for states to review and revise their curriculum standards in math. The list of learning objectives on state curriculum frameworks tend to be long and without emphasis on what's really important. The objectives get treated as separate, isolated teaching and learning events. Fitting them all into a school year inevitably leads to shallow coverage trumping real mastery. Let's concentrate on what's important and make sure kids really get it before moving on. Now, if only we can get the testing establishment to reflect this focus, but that's the topic of another blog.

Being a struggling learner

2008-02-28T08:19:00.000-08:00

I received a helpful reminder at TSP’s annual holiday party about what it’s like to be a struggling learner. Our “holiday” party really needs a new name, unless the holidays we’re celebrating are Martin Luther King Day and Valentine’s Day. Each year we seem to edge the date closer to President’s Day. There are plenty of holidays to celebrate, but the traditional Christmas and New Year’s holidays are well-behind us by the time we party. Anyway, for the last couple of years, we’ve had dance instructors at the party. Last year the instructors taught salsa; this year they did swing. You get a sense where this struggling learner story is headed?

One of the party organizers came to me during the party and asked me to help get folks participating in the swing instruction. I long ago learned that you can’t ask people to do something that you’re not willing to do yourself. Fortunately, my wife likes to dance. Unfortunately, she married someone who doesn’t have a lot of natural rhythm or coordination. Nonetheless, we took center stage, and I loudly invited others to join us. The student-to-dance teacher ratio at the outset would have been the envy of even the most well-endowed private school. Gradually, though, we gathered a good-sized crew of dancing students. It was fun, but I clearly struggled. At one point I noticed people looking at me and realized that my lips were moving along with my feet -- 1, 2, 3...5, 6, 7. I’d like to think my awkwardness was endearing, but I felt embarrassed. At the end of the night the dance instructor told me, “Well, not quite, but you have something to work on.” Not exactly the kind of robust praise I prefer to hear.

Without the obligation of recruiting participants, I’m sure I would have abandoned the effort after just a few missteps (and I had many). The experience got me thinking about Carol Dweck’s work on motivation and self-efficacy. Dweck’s research has received a lot of attention and press in the past year, and her book Mindset has been well-received (for more about the book and links to relevant articles go to http://mindsetonline.com/). The basic premise is that those who view intelligence or other abilities as fixed -- I’m good at these things but bad at those -- tend to gravitate toward activities they’re already good at. Those who see intelligence as something malleable -- if I work hard, I can get better -- are more willing to try and stick with things that are initially difficult.

I was living the research during the party. Dancing doesn’t come easily to me. (Neither does skiing. I got the award for most perseverance after a week at ski school in Aspen.) I’ve never stuck with dance lessons for more than an evening. The initial failures to succeed have led me to believe that I just can’t dance. I prefer to avoid it so I don’t look foolish. For me to stick with it, I need to believe that eventually, with effort, I can succeed. And the instruction needs to be structured in a way that gives me a sense of progress, that reinforces my belief that I can get better.

While my story is about dancing, students everyday are experiencing similar senses of “I’m not good at this” in math, reading, history, science, music, physical education, and all the other school subjects. Why, they might think, should they even try if they’re just not good at it.

Recent neuroscience research has revealed that our abilities are not fixed at birth. With focused effort and attention, our brains can change. It’s called neuroplasticity, and we should help our students understand it. We also need to adjust our instruction to reward effort and help students see incremental progress. Adaptive technology that challenges students at the edge of their competence, that isn’t too easy or too difficult, can play an important role.

The morning after the party, I shared with my son that I was caught moving my lips with my feet while learning to swing dance. I told him that I’m not much of a dancer…yet. He said that he wasn’t much of a dancer either. I pointed my finger at him and said, “Yet.”

How different are kids today?

2008-02-22T07:43:00.002-08:00

I’m writing this blog on the flight back from FETC, the Florida Educational Technology Conference. My visit was very brief. I flew down, cruised the exhibit floor, did a session, and flew back. In that day, though, I did feel a much higher energy level than I expected to find. With all the dire news about the economy and ongoing grumblings about NCLB, I anticipated a somewhat depressed sensibility among the conference attendees. Not so, at least around the Tom Snyder Productions booth. It was constantly crowded and full of enthusiasm. Participants were excited, curious, and scrambling to learn more about lots of products. It felt good. But that’s not what this blog is about.

Several of the sessions I breezed in and out of and a number of the informal conversations I had reflected the underlying assumption that today’s kids are different. And I often wonder what that really means. Certainly 21st century children are using tools and interacting with each other in ways that are completely alien to old folks (like yours truly). I can’t imagine as a teenager being willing to share my personal history and preferences for the world to see like my son does on Facebook. I remember in graduate school scoffing at the notion of computers and the Internet threatening the existence of libraries. How could you do research without wandering the stacks and feeling the heft and integrity of the books that contained the knowledge. Wow. What a drag it is now when a journal article isn’t online, and I actually have to open a book. Many kids never had that kind of old-fashioned library experience.

My son and graduate students text each other without a thought and flit from one computer to the next without a care. Their work and identities travel with them or wait patiently in the ether until accessed from whatever machine is at hand. I just didn’t grow up that way. I’ve had to relearn how to do some things, and the pieces of this emerging world that I have embraced don’t look the same as those growing up in it initially. (My son thinks it’s just wrong for me to be on Facebook.) So, yes, it’s clear that today’s kids look and behave in new ways; but didn’t I look strange to my parents?

My parents couldn’t understand how I could spend so much time on the phone. How could I do homework while watching TV? And that music on the stereo…. Yes, every generation is different from the preceding one, but isn’t the pace of technology outpacing our ability to keep up? These 21st century kids REALLY are different. I wonder.

I had a brief conversation with a year ago with the recently deceased Peter Lyman, a cultural anthropologist from Berkeley who had been studying digital youth culture. He gave me some sense that underneath the surface today’s kids are a lot like yesterday’s. They still value status and being part of a group. They still feel anxieties about who they are and will become. The venues for exploring those feelings have dramatically changed, but maybe the core remains familiar. Maybe. It's a good question. I’ll keep looking for the answer.

A moment to brag…but not about myself

2008-02-04T12:24:00.000-08:00

I recently finished evaluating student projects from the software design class I teach at the Harvard Grad School of Education. With all due respect to my past classes (which were all fabulous, each one better than the others), this year’s group really clicked, and the range of high quality projects was impressive. To complete the projects students work in small teams on a project of their choice. They gather evidence to define an educational problem they are tackling, review relevant research for clues about effective approaches, devise a plan, construct a prototype, and test and revise it. I learn an enormous amount; hopefully they do too.

I had a number of former, current, and prospective teachers in this year’s mix. They helped maintain a grounding in reality that sometimes gets lost when folks start playing with the possibilities technology offers. One group of students, who are in a mid-career teacher transition program, took on negative numbers. They wanted a program to help them with a problem they were encountering in their classrooms. Digging into the research gave them a depth of awareness about a core learning issue that few teachers have the luxury to plumb. They found no easy answers, but I suspect the knowledge and experience they did gain will serve them well down the line.

So too for the group that focused on teaching the physics of sound, the one that hoped to use a program on the Pilgrims to help elementary students see outside themselves to better understand history, and the one with fabulous activities to build reading comprehension skills. The physics group developed a pre-assessment on student conceptions of sound that was incredibly enlightening. The drawings and descriptions revealed a great deal about kids’ thinking and provided a powerful reminder about the importance of respecting the knowledge students bring with them to class. The Pilgrim project worked to push second and third graders developmentally, and the effort uncovered the edges of what young students can comprehend. The reading comprehension project simply radiated a graduate student’s passion about what she teaches in her high school English class in Florida. Her project introduced me to some very intriguing ideas about using comics and graphic novels to teach comprehension. More importantly, it allowed her to focus and deepen her thinking about some very powerful instructional lessons. Wouldn’t it be fantastic for all teachers to be able to take a semester to really grapple with some element of their teaching?

Three projects gave me hope that recently neglected parts of the curriculum may soon recover. One was a project that engaged students in using primary sources scaffolded reading and interpretation and helped turn what is too often a “read and remember” subject into one that is dynamic and alive. History is being made and interpreted everyday; we should teach it! When the “Culture Shock” team presented its project to the class, they started with a National Geographic geography online quiz. While the performance of my graduate students was pretty solid (this is Harvard, after all), the national results were pretty sorry. We are a geographically ignorant nation. That’s sad given how interconnected the world has become. The team’s simulation puts students in the role of manufacturers looking for the best resources and most responsible working conditions from around the globe. That’s real and relevant. Can’t we squeeze a little geography and culture back into the curriculum? The third project dealt with decision-making, civic responsibility, and conflict management. They developed a whole class prototype that had my students engaged in rich conversation and perspective-taking for two hours. Shouldn’t we devote a bit of school time to learning how to see and work with others?

Two groups of students gained experience by working with real clients. One team of international students partnered with a professor at the University of Athens in Greece. They aimed to design an online support environment for a course that’s part of a program training future teachers of English to native Greek speakers. The team that developed a technology-rich program for a media literacy unit in a local urban high school also had to revise and rethink based on the input from the teacher who would ultimately be using the materials. We all relearned the lesson that designing something cool might be different from designing something that will work somewhere real.

And speaking of contextual constraints, one group turned to cell phones as a delivery platform for teaching French to children and adult learners in Ghana. That team had to think through a whole new infrastructure that leverages a number of emerging technologies with solid, proven pedagogy. This mobile learning project, along with one working to incorporate some web 2.0 features into an online vocabulary-building environment, particularly highlighted the promise of new technologies. All of the projects had solid research foundations, but they each took creative paths to applying that research with the aid of technology to meet important educational needs.

I think I covered all the projects from this fall’s course. I hope I didn’t embarrass any of my students or leave anyone out. I just wanted to brag about their efforts a bit and tease out some of the general lessons that are easy to overlook in the details of each project. Nice work!

Fractions

2008-01-31T10:26:00.000-08:00

Dennis Deturck, a noted mathematician and the Dean of the College of Arts & Sciences at the University of Pennsylvania, has provoked a little firestorm within the math education community by suggesting that schools consider delaying fraction instruction until students are dealing with higher level math. In a 60 Second Lecture a few years ago, Deturck said: “I have a simple suggestion when it comes to teaching fractions in elementary school: Don’t.” Decimals are sufficient. With a new book offering these and other ideas coming out next year, the UPenn dean’s thoughts about reforming math instruction have been making the news.

Critics of Deturck’s suggestion argue that fractions are a fundamental part of our daily lives, unless, of course, you live with the metric system. Some argue that his suggestion of pushing fractions higher up in the curriculum is elitist. Then again, we delay a lot of content until students are better prepared to handle it. Frankly, I welcome the conversations sparked by this controversy. Math instruction in the U.S. is failing a lot of kids. We should be challenging it.

I met Dennis about a year ago when we were both playing advisory roles for the PBS show Cyberchase (a good program), and we talked about fractions then. Dennis does a lot of work in the Philadelphia area schools. He has a good deal of direct experience with struggling kids, and he feels we push them into finding common denominators and computing with fractions long before they have an understanding of what fractions are. I agree.

We’re doing some work ourselves now at Tom Snyder Productions with fractions. We’ve found kids in upper elementary grades who don’t know that 3/3 is 1. They don’t know how to compare 0.6 and 5/10. And they don’t believe that a fraction can ever be greater than 1; after all, we tell them that fractions are parts of a whole. How could it ever be more than that whole? These students, who don’t get fractions, are being asked to add and otherwise manipulate them. The arcane rules they’re learning for these procedures are meaningless, confusing, and readily forgotten.

We’re seeing what we can do to build a better foundation, to help kids make the tough transition from discrete to continuous quantities, from counting how many to measuring how much. A rich, intuitive sense of fraction quantity and equivalence can provide a much stronger base for learning and understanding rational numbers. We’re working on it.

The Effect of Affect

2008-01-22T11:21:00.000-08:00

It seems like one of those "duh" statements: emotion and attitude matter in learning. I certainly expect that students who are excited and happy about school perform better than students who are emotionally down and dour. In cognitive psychology these emotional states are called affect, and they can be positive or negative. Now, even though we (or at least I) have assumed that a student's emotional state has an impact on learning, the cognitive science research historically hasn't really incorporated affect into the way it looks at teaching and learning. Remember Mr. Spock from the original Star Trek (or Data from a more recent edition). Spock, a Vulcan, had no emotion. His thinking was clear-headed, logical and rational. Many theories about teaching and learning reflect a Spock-like view of the world. They're logical and rational. Affect is in another category.

Fortunately, more recent research from cognitive neuroscience to neuroeconomics (an up and coming new field) has begun to respect the role of affect in learning and development. Decision-making, for instance, isn't just the result of an emotionless, cost-benefit analysis (look at the work of Antoine Bechara among others). Affect plays an important role (that's why Kirk was captain and Spock second in command). The affective part of the brain also seems to matter in memory (happy experiences are more memorable) and in working memory. Stress, for instance, releases chemicals that impact brain function. Anxiety eats up working memory. Your feelings about yourself affect your performance on tests (check out the article on stereotype threats in the October Education Week).

It's time we started accounting for affect explicitly in instructional design.

Brain Metaphors

2008-01-08T10:12:00.000-08:00

New technologies, like functional MRIs, have made brain research more accessible to neuroscientists and cognitive psychologists, and they've made it more accessible and interesting to curious novices like I've become. I'm now an avid reader of neuroscience books and articles. I've got a lot to learn, but I'm really into it. I've even got multiple neuroscience news feeds on my iGoogle home page.

This reading has prompted me to reflect back on the assumptions I had as a beginning social studies teacher in the late 1970s about how my high school students' brains incorporated what I was teaching them. As I recall, I pictured the brain as something of a filing cabinet with a complicated cross-referencing system. That metaphor had a substantial impact on how I structured my instruction. I figured that each new bit of information I taught my students got filed somewhere in their brains. It was easier, I thought, if they already had a file under which to add something new. That led me to emphasize themes and narrative that might become headers for file folders. It also led to me imagine each bit of information residing in a solitary place in the brain.

My image of the brain and how it works is very different today. Now I picture an incredibly complex network of distributed information and skills. I have much greater respect for the role of emotion and affect. And I acknowledge that as much as we've learned about cognitive neuroscience in the last two decades, we still have a long way to go to fully understand how the brain works as we learn.

In any case, I'm curious how other teachers past and present imagine the brain at work and how those models may influence their teaching. I've started searching for research in this area, but so far I'm not finding anything. I think there's a good thesis topic in here. I don't think I'll have the time to pursue it, but I'd be happy to advise.

Schema

2008-01-04T08:28:00.000-08:00

So I'm driving in the car with my 16-year old son (actually, he just got his license and he's driving), and we're listening to music, Grounds for Divorce by Wolf Parade. My son says, "Nice hemiola." I say something intelligent like, "Huh?" My son, who is a musician (saxophone) taking Advanced Harmony this year, explains that a hemiola takes two standard 3-beats (1 2 3, 1 2 3) and plays it like three 2-beats (1 2, 1 2, 1 2). It's about the way the beats are accentuated. I concentrate on the music, and I can begin to pick out the hemiola as well.

I tell him that I think what he's just done is an example of what the cognitive neuroscience literature I'm reading calls a "schema." He's curious. "What's a schema?" Explaining it to him is a good exercise for me.

Putting my thoughts into words helps me clarify my own thinking. Indeed, it often reveals how far I am from really getting it. A schema, I summarize, is like a generalizable pattern or model that helps you make sense of new information or situations. The hemiola pattern is something that my son can recognize in music he's never heard before. In fact, music and the arts are full of schemas. Those of you (not me) who dance, for instance, can readily pick up a waltz or salsa or disco beat in a novel tune. Genres of literature and art follow patterns that allow readers and viewers familiar with the genre to anticipate the flow of the story or to look for particular aspects of color or shape.

In fact, schemas are everywhere, and they don't have to be narrow and technical like a hemiola. Neuroscientist Daniel Levitin, in his wonderful book, This is Your Brain on Music, offers the example of a kid's birthday party schema. We know the pattern -- games, cake, presents -- and we recognize it from one party to the next. The children, the setting, the games, the cake, and the presents may all be different at each party, but our brains don't get overloaded with the uniqueness of each situation. Instead, we take comfort in the common underlying structure.

We relied on schema research for a program we created called GO Solve Word Problems. Our goal was to help students see the underlying patterns in arithmetic problem solving situations. Rather than treating each problem as unique or applying a weak schema (like focusing on key words or automatically dividing when one number is a factor of the other), students should focus on the mathematical patterns. For instance, is the problem about something changing or a comparison?

I think we've just scratched the surface with how schema theory can guide improved instructional strategies. Schemas help make new information and situations familiar and manageable. We all use them all the time.

Understanding the ones that struggling students use on academic tasks may provide some very useful insight. I'm curious about the overlap between schema research and the work on student misconceptions. More to come on this topic...

Darius Goes West

2007-12-14T08:45:00.000-08:00

Logan Smalley, one of the students in my course at Harvard has been making the rounds on the daytime talk show circuit. A contingent of fellow masters students raised signs saying “Harvard Loves Darius” on the street outside the studio of The Today Show, and Ellen Degeneres embraced Logan and his crew on the Ellen Show. Why has the 25-year old Logan become such a media magnet?

Logan directed an incredible documentary film called Darius Goes West that tells the story of Darius, a 15-year old young man from Athens, GA with Duchenne Muscular Dystrophy, who travels cross-country to LA in the hope of getting his wheelchair pimped on MTV’s Pimp My Ride. Duchenne Muscular Dystrophy is an incurable affliction that typically takes the lives of its victims by the time they reach their early 20s. Logan was a friend of Darius’ older brother who, before succumbing to the disease himself, asked Logan to take care of his younger sibling. To raise awareness of this disease Logan and his friends plotted this trip with Darius. The younger generation, they realized, knows nothing of Jerry Lewis and his annual efforts to raise money for muscular dystrophy. But this generation does know Pimp My Ride!

Raising money with a door-to-door bar-b-q sale and by selling movie credits for $10 and up (possibly the longest credit list in movie history), Logan and his crew rented an RV and took Darius on the trip of a lifetime, his first out of the county. The documented story is wonderful, funny, and heartwarming. It’s also won more than two dozen film awards and got mentioned in The Wall Street Journal as one of the notable documentaries to qualify for the Oscars. And Logan has developed materials to support the educational use of the film. He has hundreds of interested schools who want to share the lessons of the story. Keep your eye out for Logan and Darius; they are worth seeing.

Board Games, Bowling and the Risks of Technology

2007-12-07T09:50:00.001-08:00

I’ve always loved board games, and I’m happy to see new research demonstrating their educational value. The new study (Siegler & Ramani, 2007) looked specifically at the impact of playing a simple numerical (versus color-coded) board game on the number understanding of low-income children. Earlier studies (notably Case & Griffin, 1990) found that the number knowledge of low-income pre-schoolers significantly lagged that of their middle- and high-income peers. The difference most likely comes from the informal mathematical experiences kids receive at home. Just as exposure to books and vocabulary helps prepare a pre-school child for more formal reading instruction in school, so too do informal mathematical tasks like counting place settings, measuring ingredients, and playing board games get a child ready for learning math in the classroom.

From simple games like Chutes & Ladders to more complex ones like Monopoly, these fun activities contain lots of math lessons. Rolling dice reinforces sets and addition. Moving a playing piece along the ten squares on each side of a Monopoly board highlights number quantity relationships and a make-ten strategy (how many spaces to land on Free Parking?). The new research suggests that playing these games can significantly close the number knowledge gap among pre-schoolers. That’s a cheap (and fun) way to make a difference.

And then there’s all the social benefit that comes from playing board games. Children learn about turn-taking, following the rules, and self-handicapping (to keep things competitive among unevenly matched players). Players have to monitor each other for mistakes and cheating. Disagreements must be resolved and fairness maintained. That’s all really good stuff!

Sadly, technology can put many of these lessons at risk. Putting the games on the computer typically means robbing children of the opportunity to calculate the dice or count spaces on the playing board as the software does it for them. I felt robbed of these chances myself when I went bowling over the Thanksgiving holiday. I hadn’t been bowling (big ball, not candlepin for my New England friends) in over 20 years. I stunk, but that’s beside the point. Horrible low-tech graphics on LCD displays hung over each alley, and a computer counted the pins and did all the scoring automatically. The families with young children on either side of our “older” group missed out on some valuable educational moments. Sometimes technology can make things easier without making them better.

NCTI Conference

2007-12-07T09:19:00.000-08:00

I'm just returning from a conference hosted by the National Center for Technology Innovation. A major part of this government group's charter is to foster assistive and universal technologies to make learning accessible to all students. I was invited to moderate a panel titled Commercialization: From Research to Market Reality. Hopefully, the panel was valuable to the audience, but I much more enjoyed being a participant in some of the other sessions. Jim Fruchterman, the morning keynote, was engaging and inspiring. Fruchterman recounted his journey from rocket scientist to entrepreneur with two successful OCR (optical character recognition) companies to founder of Benetech, a social focused technology nonprofit. He is a dynamic advocate for the power of technology to tackle critical social problems around the world. And he has certainly put his money where is mouth is. Good stuff.

And Yong Zhao, from Michigan State, has to be the most entertaining academic researcher I've ever heard. His insights and analogies were wonderful, including his call to measure the success of faith-based initiatives by measuring how many members of the religious group ended up in heaven or hell. His presentation echoed the demand from many other speakers for better measures of the range of educational and life outcomes we hope for our students.

(written 11/15/07)

Flexible Technology

2007-12-06T10:02:00.000-08:00

Debates in education often remind me of the old Saturday Night Live skit: "It's a floor wax." "No, it's a dessert topping." "It's both!" (I'm sure you can find the routine on YouTube.)

Unfortunately, the arguments in the education world often fail to get to the punchline. Whole language OR phonics. Conceptual math OR skills and procedures. Direct instruction OR constructivism. Hey folks, it's both!!!

In the educational software design course I teach at the Harvard Graduate School of Education, I restrain my students from thinking about the technology until they have a solid understanding from the research about what problem they are addressing and what success looks and sounds like.

There is no one instructional approach that works for building fluency, acquiring deep conceptual understanding, and prompting behavioral change. Only when they know what they want to accomplish, with whom, and in what context (classroom, home, museum, etc.) can my student designers bring their creative energies to the task of figuring out if and how technology can help.

There's nothing cookie cutter about this approach. So it's not surprising that a project focusing on vocabulary acquisition will look very different from one targeting improved student engagement in a civic society that's different in another way from one building deep conceptual knowledge of negative numbers. And technology can take a supportive, but very different, role in each of those instructional endeavors.

UDL

2007-12-04T09:29:00.001-08:00

I had the honor recently of attending a Summit on UDL (Universal Design for Learning), hosted by CAST and The Tremaine Foundation. The main tenets of this approach to reaching all students reflect a significant shift in the way we think about instruction. Traditionally, we have considered the curriculum and the means to deliver it (textbooks, filmstrips, worksheets, and the like) as fixed. The students we teach though are not. Some are better readers than others; some don't come to school speaking English; some have hearing or vision problems; some come well-prepared by their parents; some are drilled in the multiplication table by their parents; and so on. All those fixed curriculum materials may not provide accessible pathways (physically or pedagogically) for this incredible range of student backgrounds and abilities So, historically, students who suffered from either learning disabilities or learning difficulties have had to learn to cope, catch-up, and accommodate the materials.

UDL, in the words of David Rose, co-founder of CAST, hopes to co-locate the disability between both the student and the curriculum. What can we do in the design of the instructional materials and lessons to carry some of the load? How can we create materials that adjust to the student as well? UDL proposes three main design principles to guide this effort:

* Multiple representations of the content and instruction.
* Multiple means for the student to express his or her learning.
* Multiple pathways to engage students.

Technology, of course, makes it much easier to incorporate these principles into instruction, and we've worked to infuse them into our software products. The concept of UDL is incredibly appealing and a straightforward way to think about differentiating instruction. To learn more about it, visit the CAST website at: http://www.cast.org/.

Who am I?

2007-12-04T09:12:00.000-08:00

I’ve spent my entire life in the field of education in one way or another. I went through the public school system in Rock Island, Il before heading to Yale, where I studied history and received my certification as a social studies teacher. I volunteered and did my practice teaching in the New Haven, CT schools in the late ‘70s. It was quite an experience. After Yale, I taught for a few years in suburban Connecticut. I loved the teaching, but I needed a change of venue and in 1982 entered the doctoral program in what was then called Teaching, Curriculum, and Learning Environments at the Harvard Graduate School of Education (HGSE).

I had dabbled with technology a bit as a teacher. A housemate taught me how to write a grading program in BASIC on his TRS-80 Radio Shack computer. The homemade software was a real timesaver, but I was less than thrilled with what the publishers were offering to support instruction. Drill and practice software on the states and capitals just didn’t seem worth the heavy investment the school was looking make in computer hardware. I was barely making $10,000 a year. I thought paying me more would be wiser than buying computers. It didn’t happen. So I entered Harvard with a heavy dose of skepticism and maybe even a bit of resentment about educational technology.

My opinion shifted dramatically after a college friend introduced me to Tom Snyder, a revered teacher at Shady Hill Academy, a highly-regarded private school just down the street from Harvard. I visited a lot of classrooms using technology during my first year at Harvard, but Tom’s stood well apart from the rest. He used one computer to engage students in wonderful group activities. His classroom -- with students focused, talking, thinking, and learning -- mirrored the one I always wanted for myself. And the computer was helping him do it.

I was hooked. Tom and his partner, Rick Abrams, had started a little company on the 2nd floor of a 3 family house in Cambridge. I joined as an intern and within a year was pretty much working fulltime while still pushing my way through the Harvard program. The work slowed the doctoral progress, but I remained productive, churning out a dozen or more educational software programs for home and schools. My doctoral research focused on historical efforts to integrate technology into schools. I looked at everything from chalkboards to overhead projectors, from film projectors to televisions, to better understand the process of change that is supposed to accompany technological revolution but often doesn’t. I finished the doctorate in 1988, long before the educational technology revolution was complete (I think we’re still waiting).

Although I guide product development as the Chief Academic Officer at Tom Snyder Productions, I retain a healthy skepticism about the use of technology for technology sake. I share that skepticism in the design course I teach within the Technology, Innovation, and Education (TIE) program at HGSE. And I respected the balance between technology’s promise and the daily demands of life in the classroom during my nine years on my town’s school committee. I have the best jobs. I get to work with passionate people and curious students in the pursuit of making a difference in the lives of children in school. That’s pretty good.