Math, Transfer, and Writing

Abstract is better than concrete for transfer, according to the New York Times reporting of recent research in mathematics:

In the experiment, the college students learned a simple but unfamiliar mathematical system, essentially a set of rules. Some learned the system through purely abstract symbols, and others learned it through concrete examples like combining liquids in measuring cups and tennis balls in a container.

Then the students were tested on a different situation — what they were told was a children’s game — that used the same math. “We told students you can use the knowledge you just acquired to figure out these rules of the game,” Dr. Kaminski said.

The students who learned the math abstractly did well with figuring out the rules of the game. Those who had learned through examples using measuring cups or tennis balls performed little better than might be expected if they were simply guessing. Students who were presented the abstract symbols after the concrete examples did better than those who learned only through cups or balls, but not as well as those who learned only the abstract symbols.

The problem with the real-world examples, Dr. Kaminski said, was that they obscured the underlying math, and students were not able to transfer their knowledge to new problems.

“They tend to remember the superficial, the two trains passing in the night,” Dr. Kaminski said. “It’s really a problem of our attention getting pulled to superficial information.”

The explanation of examples clouding up the concepts reminds me somewhat of the research on reading about seductive details diminishing recall of information. (There are many articles on this phenomenon, but see, for example, Processing and recall of seductive details in scientific text.)

Transfer is also a major problem in writing: Students often don't transfer what they know about writing in one situation to new situations. Somehow, the situations are compartmentalized so that the concepts don't transfer, which remains me of the research on students learning physics. David Hammer's research showed that students could compartmentalize and keep their every day notions about motion from the physics concepts they were learning.

So, although this was a small study (and one that needs to be replicated), it does fit in with what we know of transfer, that learning that is bound to a particular context doesn't transfer well--which explains why students who have learned the five-paragraph essay structure in high school continue to use it in college even when an assignment requires them not to.

What would be the abstract set of rules for writing? I've looked at that before, except I called them "building blocks." But although I can see the need for knowing the building blocks abstractly, I think mastering them abstractly is achieved through much practice of remixing these building blocks across contexts. (See Learning by Remixing and also this review/synopsis of Spiro's Cognitive Flexibility Theory.)

The problem remains determining what those building blocks are. Although they likely differ across genre (just as math concepts differ from geometry to algebra to calculus and so on), they must also have elements in common. At a basic level, there's always writer, audience, text, and purpose. For persuasion, it may come down to the formula in Graff and Birkenstein's book "They Say / I Say", in which writers join into a conversation with others and position themselves with respect to those others. It's a small book with three parts and ten chapters:

Part 1. "They say"

ONE: "They say" (Starting with What Others Are Saying)

TWO: "Her Point Is" (The Art of Summarizing)

THREE: "As He Himself Puts It" (The Art of Quoting)

Part 2. "I Say"

FOUR: "Yes / No / Okay, But" (Three Ways to Respond)

FIVE: "And Yet" (Distinguishing What You Say from What They Say)

SIX: "Skeptics May Object" (Planting a Naysayer in Your Text)

SEVEN: "So What? Who Cares?" (Saying Why It Matters)

Part 3: Tying It All Together

EIGHT: "As a Result" (Connecting the Parts)

NINE: "Ain't So / Is Not" (Academic Writing Doesn't Mean Setting Aside Your Own Voice)

TEN: "In Other Words" (The Art of Metacommentary)

As you can see, despite having only two building blocks--"they say" and "I say"--students are led into a variety of ways of analyzing, synthesizing, and evaluating what "they say," along with generating their own understanding and position among others in a conversation. And treating persuasive writing like a conversation has many connections to students' lives: They argue about their sports, clothes, cars, majors, professors, and so on.

I imagine that different sets of building blocks are possible, just as different sets of rules can be found in different fields of math. The key seems to be helping students practice using one coherent set of building blocks (i.e., abstract principles) across contexts.

Related posts:
The Five-Paragraph Essay and Building Blocks of Writing
The Five-Paragraph Essay (continued)
Learning by Remixing