Fluent on facts, weak on abstraction

Fluency in addition and multiplication isn’t everything, writes Education Realist.

. . . plenty of solid math students don’t have fluency and—here is the important part—many exceptionally weak math students have strong fact fluency.

Ed Realist’s “math support” students, who are trying to pass the exit exam and graduate from high school, tend to be very literal and easily thrown by symbols. Ed Realist  asked students to read a simple equation as a sentence. When a student turned x + 6 = 14 into “what number do I add to six to get 14?” the answer was clear to most of the class.

One student, Gerry, still didn’t get it.  He said he could only do math if it doesn’t have letters.

“You need to look at these problems from a different part of your brain.”

. . . “X + 6 = 14. This is when you have to do stuff to both sides, right? I can’t do that.”

“What plus 6 = 14? 8.”

Gerry said he couldn’t do fractions. But when he turned x/5 = 9 into “what divided by 5 is 9?” he got 45 right away.  “I feel like a math genius,” he said.

“You know a lot more math than you think you do,” the teacher said. ” You just have to figure out how to ask the question in a way your brain understands.”

Not everyone is capable of understanding abstractions to the same degree, Education Realist concludes.

Some people do better learning the names of capitals and Presidents and the planets in the solar system. They’d learn confidence and competence through interesting, concrete math word problems and situations, and enjoy reading and writing about specific historic events, news, or scientific inventions that helped society. Instead, we shovel them into algebra, chemistry and literature analysis and make them feel stupid.

She quotes psychologist James Flynn on why IQ’s have risen steadily and significantly since the start of the 20th century (the “Flynn effect”).

Modern people . . .  are the first of our species to live in a world dominated by categories, hypotheticals, nonverbal symbols and visual images that paint alternative realities.

. . . A century ago, people mostly used their minds to manipulate the concrete world for advantage. They wore what I call “utilitarian spectacles.” Our minds now tend toward logical analysis of abstract symbols—what I call “scientific spectacles.” Today we tend to classify things rather than to be obsessed with their differences. We take the hypothetical seriously and easily discern symbolic relationships.

Well, some of us do. Flynn has a new book out, Are We Getting Smarter?

1. One of the things I’ve noticed in many college classes is a struggle between the professors and the students over the level of abstraction. The professors want more, because they have come to understand the power of those abstractions, and the students want less, because they have trouble wrapping their minds around those abstractions, without the examples the professor learned, years ago.

2. It is questionable which of the two versions is more abstract, but I accept that some people find one version more clear than the other. But only one version generalizes and is easier for everyone to manipulate: the one with letters. For example, try to say in the “what” form:
x^2 + xy= 1/x.

• Roger Sweeny says:

For example, try to say in the “what” form:
x^2 + xy= 1/x.

This suggests to me that the ability to solve such an equation should not be a requirement for high school graduation. Nor should it be part of the basic high school curriculum.

Perhaps it would be worth trying to force it on students if many of them would use it after graduation. However, very few of them will.

• But the reason we use such notation is precisely to solve (or simplify) certain equalities (or inequalities). And for many people (not all), such notation is even useful for solving “what number when multiplied by 30 yields 90?” One writes “x times 30=90”, and then one observes that is suffices to divide both sides by 30.

In general, it is a normal sign of rigorous thought that if one is seeking information about one or more unknowns, one writes down the things that one knows about those unknowns. This is the core of what high-school algebra is, and it is an abstraction that is vastly useful in life.

• Roger Sweeny says:

For some people it is vastly useful. Not for most.

It is vastly useful for me. LTEC, I suspect it is useful for you. But it is factually wrong to generalize from our experience and morally wrong to require everyone to be like us.

• J. D. Salinger says:

While it may be morally wrong to require everyone to be like us, it is the purpose of educators to optimize choices for students. Educators attempt to provide a minimum math background for students so that they may decide to pursue interests that neither students nor teachers may be aware of.

• Roger Sweeny says:

For a frighteningly large number of students, requiring them to try to learn math like this does not optimize their choices. It condemns them to failure. It is not a nice thing to do.

3. momof4 says:

I don’t consider forcing kids who are cognitively unlikely/unable to be able to master classes like (real) algeba, its successors, chem and physics into those classes to be optomizing choices. Allow a wide variety of voc ed options, spec ed skills training and, perhaps, what was called in my day a general option – for those going directly into the workplace. In my day, the latter kids became farm workers, cashiers at local businesses, gas station employees (changing oil, tires etc. but not mechanics) etc. However, even the latter group had basic literacy (read newspapers), numeracy (balance checkbook, calculate interest etc) and general knowledge.

• momof4 says:

Optimizing – sorry about the typo.