Fractions

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

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

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

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...