Teachers study how to teach fractions

Teachers observe lessons at New York City Math Lab. Photo: Elizabeth Green/Chalkbeat

How do you get students fired up about fractions? asks Elizabeth Green on Chalkbeat. She looks at a New York City’s program that teaches teachers how to “reinvent” math lessons.

The Math Lab stresses “learning math by talking and thinking about it,” writes Green. Students preparing for fifth grade agree to “add onto each other’s thinking” and “analyze and observe each other’s work.”

(Math Lab co-founder Kim) Van Duzer led an activity called “convincing a skeptic,” where students were asked to fold pieces of green paper into squares one quarter the size of the original and then convince their partner that the new shape was, in fact, one-fourth of the original.

Some students struggled to articulate why the squares they folded where one fourth of the whole piece of paper. “Sometimes my partner asked questions I didn’t understand,” one student admitted. But encouraging students to challenge each other’s ideas paid off later that morning.

After introducing the idea of representing fractions on a number line, co-founder Peter Cipparone asked students whether eight-sixths is greater than one.

One student declared that eight-sixths is less than one, only to be told by someone sitting nearby that he had the numerator and denominator confused. The ensuing debate ended when the first student admitted his mistake and leapt at the chance to offer a correct answer in his own words.

Many of the teacher observers said they’d never been able to “watch another educator teach consecutive lessons,” reports Green.

Is this really revolutionary?

The math problem: All rote, no reasoning

Community college students placed in remedial math — a large majority — may have memorized a few procedures, but they don’t have a clue what they’re doing, according to researchers.

In one study, few could place -o.7 and 13/8 on a number line from -2 to 2. Asked which is greater, a/5 or a/8, 53 percent answered correctly, barely beating a coin toss.

“Seeing two fractions near each other apparently triggered an urge in some students to use the cross-multiplication procedure they had memorized,” writes Nate Kornell on Psychology Today. If all you’ve got is a hammer, everything looks like a nail.

Warhol’s fractions: Teaching through the arts

A Maryland middle school integrates arts with the standard curricula, reports Edutopia on Bates Middle School in Annapolis.

For example, in a science classroom you might see students choreographing a dance using locomotor and nonlocomotor movements to demonstrate their understanding of rotation versus revolution of the planets (PDF). In a math class, you might see students learning fractions by examining composition in Warhol’s Campbell’s soup paintings. (See more arts-integrated lesson plans from Bates.)

“Engagement can also be leveraged to boost academic growth and improve discipline,” Edutopia argues.

Why our kids hate math

“Our kids hate math” because they’re pushed to learn higher math before they’ve mastered the basics, writes Patrick Welsh, who teaches at T.C. Williams High in Virginia, in USA Today.

The experience of T.C. Williams teacher Gary Thomas, a West Point graduate who retired from the Army Corps of Engineers as a colonel, is emblematic of the problem. This year, Thomas had many students placed in his Algebra II class who slid by with D’s in Algebra I, failed the state’s Algebra I exam and were clueless when it came to the most basic pre-requisites for his course. “They get overwhelmed. Eventually they give up,” Thomas says.

Thirty-one percent of eighth-graders took algebra in 2007, nearly double the percentage compared to 1990, reports the National Center for Education Statistics. In California, 54 percent take algebra in eighth grade. But many repeat it in ninth grade — and still do poorly.

My colleague Sally Miller . . . is the first to warn that too much math too soon is counterproductive. When Miller asked one of her geometry classes what 8 x 4 was, no one could come up with the answer without going to a calculator. “In the lower grades, more time has to be devoted to practicing basic computational skills so that they are internalized and eventually come naturally.”

Charlotte-Mecklenburg’s eighth-grade algebra classes have a “negative effect on most students, especially those students who weren’t stellar in math background,” says Charles Clotfelter, a Duke professor who studied the effects. Doing poorly “knocked them back on their heels.”

“It is time to ensure that all kids absorb the fundamentals of math — computation, fractions, percentages and decimals — first before moving on to the next level,” Welsh concludes.

A frightening number of students never learn math fundamentals. It’s the single greatest barrier to success in community colleges, which attract the un-stellar students. Students who’ve passed high school math classes — including a class called algebra — don’t understand fractions, percentages or decimals and can’t multiply 8 x 4 without a calculator.

Fractions are key to math learning

Learning fractions and long division in elementary school leads to math success, according to a Carnegie Mellon research team led by Robert Siegler. In the U.S. and Britain, fifth graders’ understanding of fractions and long division correlated with their ability to learn higher math.

“We need to improve instruction in long division and fractions, which will require helping teachers to gain a deeper understanding of the concepts that underlie these mathematical operations,” said Siegler. “At present, many teachers lack this understanding.” By contrast, East Asian teachers can explain why math operations work.

And college remedial classes are filled with people who never “got” fractions.

Spatial thinking skills predict future math success, reports MindShift.

In a University of Chicago study students in first and second grade who chose the right shape to complete a square “also showed the most growth in their number-line knowledge over the following school year, and scored highest on a measure of mathematics ability at age eight.”

Parents can help their children develop spatial skills.

. . . Temple University psychology professor Nora Newcombe and her coauthors found that parents and children playing with blocks together were much more likely to use spatial terms like “over,” “around,” and “through,” than participants who played with a pre-assembled toy—and that it’s hearing and voicing such words that helps improve children’s spatial awareness.

Another 2011 study, this one from the University of Chicago, reported that the number of spatial terms (like “circle,” “curvy,” and “edge”) parents used while interacting with their toddlers predicted how many of these kinds of words children themselves produced, and how well they performed on spatial problem-solving tasks at a later age.

My first reaction was that kids who are good at math at a young age are likely to be good at math when they’re older. But there’s got to be more to the research than that, surely.

Teach music for music’s sake

Teach music because it’s a universal language and “the arts are our most potent means of human expression,” not because it might help kids learn fractions or raise test scores or develop teamwork, writes Nancy Flanagan, a music teacher.

Yesterday, the Learning First Alliance asked Can Arts Education Help Close the Achievement Gap? I appreciate the perspectives and data assembled by Anne O’Brien–who points out that students from high-poverty schools who study the arts are more likely to graduate HS, attend and finish college, and register to vote. But I believe the real question is:  What do the arts teach children that other subjects can’t?

Teachers defend music, art, dance and drama by arguing they help teach something considered more important, such as “enhanced brain development, spatial/visual/temporal processing, improving memory and attention, physical coordination, personal discipline and teamwork,” Flanagan writes.

But where did we get the idea that artistic expression is less useful or important than the sciences? How did music, art, dance and drama get pushed aside in our American school curriculum? I’m not surprised that studying or listening to music has beneficial effects on learning fractions or other academic skills, but those are side effects.

Kids should study music because it’s central to every human society on earth and has a vitally important role in every aspect of culture, from history to literature to media and communication studies. Music is part of what it means to be a human being.


I’ve got rhythm, I’ve got fractions …

Children are clapping, drumming and chanting to learn fractions at a California elementary school. It seems to be working, concludes a study which will be published in Educational Studies in Mathematics.

 “If students don’t understand fractions early on, they often struggle with algebra and mathematical reasoning later in their schooling,” said Susan Courey, assistant professor of special education at San Francisco State University.

Students in Academic Music scored 50 percent higher on a fraction test than students in the regular math class. Lower-performing students narrowed the gap with high achievers.

Fourth-grade math scores have soared since a San Bruno school adopted Academic Music, reports the San Jose Mercury News.

On Tuesday, 29 children at Allen Elementary School tapped out a rhythm with drumsticks as (Endre) Balogh stepped and clapped in 4/4 time at the front of the class. He stepped four times per beat. One clap equaled a whole note, two claps indicated two half notes, and so on.

“Which is larger, the whole note or the half note?” he asked.

“Whole note,” one of the third-graders replied.

“Whole note, but why?” the teacher said.

“Because it’s longer,” another student called out.

Toones Academic Music, a nonprofit, is working to expand the program to more schools.

Coach G has tips for teaching “the dreaded f word, fractions.”

A vote for new math standards

Common Core math standards are as good as the best state standards and correct common math misperceptions, writes Hung-Hsi Wu, a Berkeley math professor emeritus,  in the cover story in American Educator.

Dr. Wu, who helped write California’s math framework, praises the “mathematical integrity” and logical progression of topics in an interview with Rick Hess.

The standards teach fractions over grades three to five, giving students enough time to learn and internalize the material, says Wu. He also likes the standards approach to learning negative numbers and moving from middle school geometry to algebra and high school geometry. Delaying algebra instruction till high school is not a problem, he argues.

However, he’s not confident teachers will be able to teach the standards.

. . .  we need better teacher preparation and improved professional development in order to stay educationally afloat no matter what the standards may be. If we cannot get better teacher preparation or improved professional development, then we would be better off with a set of standards that is at least mathematically sound.

Wu is wrong, responds Ze’ev Wurman, another veteran of California’s battle for math standards and a fierce defender of eighth-grade algebra.  Wu changed sides because he concluded “American elementary and middle school teachers are incompetent to teach algebra or prepare for it,” Wurman writes.

“School mathematics in this country is a sad joke,”, comments Michael Goldenberg, a math coach. “Knowing procedures and manipulations and calculations is great for standardized tests (which drive just about every contemporary education deform scheme) but say very little about mathematical reasoning, thinking, and or understanding.”

NYC tracks grads’ college progress

New York City is telling high schools how well their graduates are doing at public colleges, including how many need remedial classes and how many drop out after the first semester, reports the New York Times. High schools are judged on graduation rates, in part, but not on graduates’ skills.

Illinois, Denver and Philadelphia also are tracking high school graduates to see how they do in college, reports the Times. Studies show many high school graduates falter in college because they lack basic reading, writing and math skills.

New York, like other cities, has made a considerable effort to improve its high school graduation rate — now 59 percent, up from 47 percent in 2005 — and push more of its students to enroll in college. But many of those students are stumbling in basic math and writing: 46 percent of New York City public school graduates who enrolled in one of the City University of New York two-year or four-year colleges in 2007 needed at least one remedial course, and 40 percent of them dropped out within two years.

At a third of the city’s 250 high schools, at least 70 percent of the graduates who went on to CUNY needed remedial help.

This is nothing new, community college instructors told the Times.

Elizabeth Clark teaches remedial writing at LaGuardia Community College to high school graduates who are unprepared to write a college essay.

“They don’t know how long it should be; they don’t know how to develop an argument,” Ms. Clark said. “They have very little ability to get past rhetoric and critically analyze what is motivating the writer, and you have to push them past simple binaries.”

There are also more basic problems, Ms. Clark said, such as students not knowing that each sentence must begin with a capital letter or using “u” instead of “you.”

. . . Susan L. Forman said that many of the issues have remained the same for the four decades she has taught remedial math at Bronx Community College, including students easily confused by fractions and negative numbers and becoming paralyzed when they are told they cannot use calculators.

What has changed, she said, is that students are often overly confident.

They don’t understand how much they don’t understand, she said.

Update: Chicago City Colleges can’t afford remedial classes, said Mayor Richard Daley, who called for limiting admissions to students prepared to do college-level work. Daley envisions offering remedial classes at alternative high schools. This would be a dramatic change, if it happens.