Writing math

From the U.S. Coalition for World Class Math:


Interdisciplinary teaching is the big thing now.  In my most recent assignment, the principal had all of the teachers talk with each other to figure out how to integrate their lessons--i.e., math with science with history, with home ec with English, etc.  I wish I had had this cartoon to "share" with everyone:

Drill and skill

“Drill and kill” — practicing math skills taught by the teacher — works best for struggling first graders, concludes a new study. Yet teachers with the most math-challenged students are the most likely to use ineffective “student-centered” strategies, researcher George Farkas, a UC Irvine education professor, found.

. . . “routine practice or drill, math worksheets, problems from textbooks and math on the chalkboard appear to be most effective, probably because they increase the automaticity of arithmetic. It may be like finger exercises on the piano or ‘sounding out’ words in reading. Foundational skills need to be routinized so that the mind is free to think.”

Hands-on activities that use manipulations, calculators, movement and music may be fun, but they don’t improve first graders’ achievement, according to Farkas. It takes a teacher explicitly teaching facts, skills and concepts with plenty of time for practice.

“Teacher-directed instruction also is linked to gains in children without a history of math trouble,” writes  Maureen Downey. “But unlike their math-challenged counterparts, they can benefit from some types of student-centered instruction as well – such as working on problems with several solutions, peer tutoring, and activities involving real-life math.”

A friend who teaches in a Title 1 school lamented that her students didn’t do as well in the math CRCT as the classroom next door where the teacher used worksheets all the time. My friend’s classroom was a beehive of fun activities around math, but the worksheet class continually outperformed hers.

The study was published online in Educational Evaluation and Policy Analysis, a peer-reviewed journal of the American Educational Research Association.

Common Core math: deep or dull?

According to a New York Times article by Motoko Rich, parents and students are finding Common Core math not only confusing but tedious and slow.

To promote “conceptual” learning, many Core-aligned textbooks and workbooks require steps that may be laborious for students who already get it. A second-grade math worksheet, pictured in the article, includes the question: “There are 6 cars in the parking lot. What is the total number of wheels in the parking lot?” To answer the question, the student drew six circles with four dots within each. (Actually, this doesn’t seem new; it reminds me of “New Math” and “constructivist” math.)

One nine-year-old, apparently weary of this kind of problem, stated that she grew tired of “having to draw all those tiny little dots.”

Students with good understanding may be put through steps that seem redundant to them. If they skip those steps, they may be penalized.

“To make a student feel like they’re not good at math because they can’t explain something that to them seems incredibly obvious clearly isn’t good for the student,” said W. Stephen Wilson, a math professor at Johns Hopkins University.

One reason for emphasizing “conceptual” learning is that employers apparently are demanding critical thinking. Several questions remain to be answered, though: (a) whether Common Core math–in its current forms–really is promoting conceptual learning; (b) if so, whether it also promotes math proficiency; (c) whether the current approach is benefiting students at the upper and lower ends–and those in between, for that matter–or holding them back; and (d) whether this is the kind of “critical thinking” that will serve students well in college, the workplace, and elsewhere.

I will comment briefly on the first question; I welcome others’ insights.

Tedium and depth are not the same. One can go through a long explanation of a problem without gaining any understanding; one can solve a problem quickly and come to understand a great deal.

In sixth grade, in the Netherlands, I learned mental arithmetic: I learned to add, subtract, multiply, and divide double-digit numbers in my head, using all kinds of tricks that the teacher taught. Those tricks enhanced my understanding of what I was doing. I enjoyed the swiftness and ingenuity of it; I would have detested it, probably, if I had to write it all out, step by step, and illustrate the steps with circles and dots.

Detailing and explaining your steps is a worthwhile exercise. But part of the elegance of math has to do with its mental leaps. Sometimes, when you do steps in your head, or when you figure out which steps in a proof are assumed, you not only understand the problem at hand, but also see its extensions and corollaries. Sometimes this understanding is abstract, not visual or even verbal.

There seems to be an unquestioned assumption that one comes to understand math primarily through applying it to real-life situations; hence the Common Core emphasis on word problems. While word problems and practical problems can lead to insights, so can abstract reasoning, and so can models that bridge the abstract and the concrete, like the multiplication table.

Yes, the multiplication table–horrors, the multiplication table!–abounds with concepts. If you look at it carefully (while committing it to memory), you will see patterns in it. You can then figure out why those patterns are there (why, for instance, any natural number whose digits add up to a multiple of 3, is itself a multiple of 3). (Something similar can be said for Pascal’s triangle: one can learn a lot from studying the patterns.)

In other words, conceptual learning can happen in the mind and away from “real-life situations”; it need not always be spelled out at great length on paper or illustrated in terms of cars and wheels. Nor should students be penalized for finding shortcuts to solutions. Nor should memorizing be written off as “rote.” Yes, it’s good to understand those memorized things, but the memorization itself can help with this.

In ELA see a similar tendency toward laboriousness (that likewise long predates the Common Core). Students are required to “show their thinking” in ways that may not benefit the thinking itself. For example, they may be told to explain, at great length, how a supporting quotation or detail actually supports their point–even when it’s obvious. Students with economy of language (and, alas, clarity of thought) may lose points if they don’t follow instructions. Instead of being at liberty decide whether an explanation is needed, they receive a message along the lines of “Explain, and explain again, and then explain that you have explained what you set out to explain.”

Critical thinking is important–and one should think critically about how it is conveyed and taught.

Savor the mystery

More school, less summer?

Top-performing South Korea requires 220 days of school, “22 percent more than our measly minimum of 180 days,” writes the New York Post. Are the lazy days of summer too lazy in the U.S.?

“More advantaged families . . . travel to Civil War battlefields, visit foreign cities and their art museums, and learn about the geography of the Grand Canyon,” says Jay Greene, a University of Arkansas education professor. “I’m convinced that my own kids and those of many other upper-middle-class families learn far more from those summer experiences than they do during the rest of the school year.”

But low-income kids lose a lot of learning over the summer, says Robert Pondiscio, a senior fellow at the Thomas B. Fordham Institute.

That’s why high-performing charter schools like KIPP, Democracy Prep and Success Academy have significantly longer school days and longer school years.

“When it comes to learning math and science,” Pondiscio says, “more is more.”

If school isn’t working well, more may mean more boredom. I’d prefer to see fun, educational summer programs for kids who aren’t going to be visiting the Grand Canyon.

The gender gap is TEM-only

Here’s the percentage of Bachelor’s degrees conferred to women, by major (1970-2012) courtesy of Randal S. Olson.

percent-bachelors-degrees-women-usa

More than 80 percent of degrees in health and public administration are earned by women, he notes. Nearly 80 percent of education and psychology degrees also go to women. In biology, women earn 58 percent of degrees.

Even in math, statistics and physical sciences, women earn more than 40 percent of degrees. Business is close to 50-50.

He flips the chart to show that men are lagging in everything but engineering, computer science, physical science, math and statistics. Women are close to parity in everything but engineering and computer science.

STEM gets broader — and shallower

In a vain attempt to make STEM appealing to right-brained students, educators are ignoring and alienating the left-brained math and science guys, writes Katharine Beals in Out in Left Field.

Efforts to Inspire Students Have Born Little Fruit, reports the New York Times. The story cites President Obama’s Educate to Innovate initiative and the lack of improvement by U.S. students on the Program of International Student Assessment (PISA) tests.

Beals sees it differently.

. . . our schools, and our society more generally, are no longer encouraging and educating the kind of student who is most likely to persevere in STEM careers. These are the left-brained math and science types, more and more of whom face a dumbed-down, language-arts intensive Reform Math curriculum, and a science curriculum that increasingly emphasizes projects over the core knowledge and quantitative skills needed to succeed in college level science courses.

At the expense of encouraging this type of student, K12 schools are trying to broaden the appeal of math and science—by making them even less mathematical and scientific. And so we have algebra taught as dancefraction muralsphotosynthesis as dance, and science festivals featuring showy displays of gadgetry as well as theater, art, and music.

“The kind of student who finds these approaches engaging and enlightening” isn’t likely to persevere through a STEM major, she predicts. Those with the potential to be STEM specialists want to learn math and science.

At Auntie Ann’s school, the science fair used to require students to conduct an experiment. Now they can make a Rube Goldberg machine or a robot or research an environmental issue. “This year they’ve also connected it to an art exhibit to make it the full STEAM experience.”

It used to be the only time students did a research project and wrote a “serious paper,” she writes. Now students get full credit for writing 30 sentences. “The kids who did Rube Goldberg machines had nothing to write a paper about, so they had to write a biography of Rube Goldberg.”

Remembering the old new math

peanuts

Common Core isn’t the first attempt to teach students to understand mathematical concepts, writes Mark Palko, a former math teacher, in the Washington Post. If we remember the old new math, perhaps we can learn from its mistakes. But Core reformers suffer from collective amnesia.

A 9-year-old faces the Core

Chrispin Alcindor was a star student in the early grades, but he fell way behind in third and fourth grade, reports the New York Times in Common Core, in 9-Year-Old Eyes.

Is it the new curriculum’s shift from rote learning to understanding concepts? (The Times assumes that no teacher tried to teach understanding in the pre-Core era.) Or is it the Haitian-American boy’s subpar reading skills?

A pet store has 18 hamsters. The shop owner wants to put 3 hamsters in each cage. How many cages does the shop owner need for all the hamsters?

Math had always been Chrispin’s favorite subject. Wherever he went, he was counting: Jeeps, pennies and basketball scores. He liked the satisfaction of arriving at a neat, definitive answer and not having to worry about things like spelling and grammar.

But as he worked on practice questions one day, the hamster problem stumped him:

Draw a model using equal groups or an array to show the problem.

Write a division equation for the problem.

Write a multiplication equation for the problem.

How many cages does the shop owner need?

Chrispin scribbled aimlessly in the margins. He hated word problems, a hallmark of the Common Core. Ms. Matthew had once told him to act like a detective and look for “clue words.” If a question referred to a “border” or “outside,” for example, it was asking for its perimeter. “Math is very, very, very, very logical,” she had said.

But Chrispin did not see any clues before him. After a few minutes of intense reading, he settled on an answer: 6. But he still did not fully understand the question. He could not remember what an array even looked like.

At Chrispin’s school in Brooklyn, producing the right answer isn’t enough. Students “had to demonstrate exactly what three times five meant by shading in squares on a grid.”

The Times prints Chrispin’s letter to Carmen Fariña, New York City’s schools chancellor, about standardized testing. If he only he really wrote this well . . .

Scared of math

Math anxiety — fear that prevents learning — starts young, writes Dan Willingham,  a University of Virginia psychology professor, in RealClearEducation. Half of first and second graders feel moderate to severe math anxiety. By college, 25 percent of university students — and 80 percent of community college students — suffer from math anxiety.

Anxiety distracts. It’s hard to focus on the math because your mind is preoccupied with concern that you’ll fail, that you’ll look stupid, and so on. Every math problem is a multi-tasking situation, because all the while the person is trying to work the problem, he’s also preoccupied with anxious thoughts.

“Children who have trouble with basic numeric skills — counting, appreciating which of two numbers is the larger—are at greater risk for developing math anxiety,” he writes.

But math anxiety also is learned from anxious adults. If an elementary teacher is nervous about her math skills, her students are more likely to be anxious.

They conclude “it’s hard not because you’re inexperienced and need more practice, but because lots of people (maybe including you) just can’t do it.” They conclude they’re just not “math people.”

Teaching children basic skills is the first step to preventing math anxiety, writes Willingham. In addition, teachers can be traind on “how to talk to kids who do encounter difficulties; how to ensure that kids see their setbacks as a normal part of learning and problems that can be overcome, rather than as evidence that they are simply no good at math.”

A third strategy — giving students 10 minutes to write about their emotions before an exam — can raise scores,  recent studies show. Writing may help students put their “upcoming confrontation with math in perspective, and so feelings of anxiety will not be consuming the student’s thoughts and attention during the exam.”

Willingham has more on math anxiety in American Educator‘s Ask the Cognitive Scientist.