Same old new math

Common Core’s Newer Math is a lot like like the old new math, writes David G. Bonagura Jr., a teacher and writer, in National Review Online.

In 1961, New Math “was supposed to transform mathematics education by emphasizing concepts and theories rather than traditional computation,” as this article shows.

Flash forward 50 years, and Common Core is today making the same promises:

The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

New Math, Sequential Math, Math A/B, and the National Council of Teachers of Mathematics Standards also “promised to transform (America’s children) into young Einsteins and Aristotles,” writes Bonagura. It didn’t work out that way.

Despite claims that Common Core doesn’t tell teachers how to teach, the new standards come with a flawed pedagogy, Bonagura charges. “Common Core buries students in concepts at the expense of content.”

Take, for example, my first-grade son’s Common Core math lesson in basic subtraction. Six- and seven-year-olds do not yet possess the ability to think abstractly; their mathematics instruction, therefore, must employ concrete methodologies, explanations, and examples. But rather than, say, count on a number line or use objects, Common Core’s standards mandate teaching first-graders to “decompose” two-digit numbers in an effort to emphasize the concept of place value. Thus 13 – 4 is warped into 13 – 3 = 10 – 1 = 9. Decomposition is a useful skill for older children, but my first-grade son has no clue what it is about or how to do it.

Students can’t just solve math problems and show their work, he writes. They need to provide a written explanation.

With sons in first and third grade, he’s seen Common Core math books that “devote enormous space to word problems that have to be answered verbally as well as numerically, some in sections called Write Math.”

That makes math much harder for students who aren’t proficient in English or in reading and writing, Bongagura points out. (Handwriting was very difficult or both my brothers.)

With so much focus on teaching students the “why” of math, teachers will have little time to teach the “how,” Bonagura predicts.

Mathematical concepts require a high aptitude for abstract thinking — a skill not possessed by young children and never attained by many. What will happen to students who already struggle with math when they not only are forced to explain what they do not understand, but are presented new material in abstract conceptual formats?

“Instead of developing college- and career-ready students, we will have another generation of students who cannot even make change from a $5 bill.”

Instead of algebra, ‘citizen statistics’

Is Algebra Necessary? asks political scientist Andrew Hacker in the New York Times.

A typical American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.

My question extends beyond algebra and applies more broadly to the usual mathematics sequence, from geometry through calculus.

Inability to do math — specifically algebra — is the major academic reason so many students fail to complete high school, Hacker writes. He proposes “citizen statistics” as an alternative.

. . . it would familiarize students with the kinds of numbers that describe and delineate our personal and public lives.

It could, for example, teach students how the Consumer Price Index is computed, what is included and how each item in the index is weighted — and include discussion about which items should be included and what weights they should be given.

This need not involve dumbing down. Researching the reliability of numbers can be as demanding as geometry. More and more colleges are requiring courses in “quantitative reasoning.” In fact, we should be starting that in kindergarten.

I think it is dumbing down math — so far down that it will close the door on many careers. But it’s better to teach some math than stick unprepared, unmotivated students in dumbed-down classes labeled “algebra” and “geometry.”

Frustrated by huge failure rates in remedial math, some community colleges are teaching “quantitative reasoning” rather than algebra to students who don’t have STEM ambitions. That makes sense. But it’s an admission of failure.

Hacker also wants to see classes in the history and philosophy of math, which he thinks would draw more math majors.

Why not mathematics in art and music — even poetry — along with its role in assorted sciences? The aim would be to treat mathematics as a liberal art, making it as accessible and welcoming as sculpture or ballet.

Maybe more people would major in math if it didn’t require learning math, but what would be the point?

A commenter recommends The Number Devil: A Mathematical Adventure, which sounds like a cool book.

Here’s how Times readers responded to Hacker’s essay.

Yes, algebra is necessary, responds cognitive scientist Dan Willingham.

First, it’s not true that otherwise talented students are dropping out because of algebra. Motivation, self-regulation, social control and a feeling of connectedness and engagement at school are as important as grades, and a low grade in English is as accurate a predictor of failure as a low grade in math.

Second, “the difficulty students have in applying math to everyday problems they encounter is not particular to math. Transfer is hard.”

The problem is that if you try to meet this challenge by teaching the specific skills that people need, you had better be confident that you’re going to cover all those skills. Because if you teach students the significance of the Consumer Price Index they are not going to know how to teach themselves the significance of projected inflation rates on their investment in CDs. Their practical knowledge will be specific to what you teach them, and won’t transfer.

Well-educated people can learn on the job, Willingham writes. “Hacker overlooks the possibility that the mathematics learned in school, even if seldom applied directly, makes students better able to learn new quantitative skills.”

Kids who can’t understand math usually can’t read well either, writes RiShawn Biddle on Dropout Nation. “The very skills involved in reading (including understanding abstract concepts) are also involved in algebra and other complex mathematics.”

Brain calisthenics

Brain calisthenics” — such as computer-based exercises in quickly linking graphs to equations –  help students internalize abstract ideas and see patterns intuitively, say cognitive science researchers in a New York Times story.

Now, a small group of cognitive scientists is arguing that schools and students could take far more advantage of this same bottom-up ability, called perceptual learning. The brain is a pattern-recognition machine, after all, and when focused properly, it can quickly deepen a person’s grasp of a principle, new studies suggest.

In a 2010 study, UCLA and Penn researchers used perception training to teach fractions to  sixth graders in a Philadelphia public school.

On the computer module, a fraction appeared as a block. The students used a “slicer” to cut that block into fractions and a “cloner” to copy those slices. They used these pieces to build a new block from the original one — for example, cutting a block that represented the fraction 4/3 into four equal slices, then making three more copies to produce a block that represented 7/3. The program immediately displayed an ‘X’ next to wrong answers and “Correct!” next to correct ones, then moved to the next problem. It automatically adjusted to each student’s ability, advancing slowly for some and quickly for others. The students worked with the modules individually, for 15- to 30-minute intervals during the spring term, until they could perform most of the fraction exercises correctly.

In a test on the skills given afterward, on problems the students hadn’t seen before, the group got 73 percent correct. A comparison group of seventh graders, who’d been taught how to solve such problems as part of regular classes, scored just 25 percent on the test.

Notice how few students understand fractions.

Reading the comments reminded me of the parable of the six blind men and the elephant. Every reader seems to think the research proves their theory: Kids need more practice; kids need to construct knowledge, kids need real-world examples, kids need visuals.

I’m not doing well with abstract ideas this week, due to a horrible cold and a racking cough, but here’s UCLA’s graphs ‘n equations module.