Core opens door to ‘garbage’ math

You’ve seen the viral “Common Core” math problem and the letter from the engineer father who thought it was idiotic. The “stupid” problem predates the Common Core, says Brookings’ scholar Tom Loveless. But it’s only “half right” to say you can’t blame the Core for this. The new standards are  opening the door to “garbage math,” says Loveless.

One of the Core’s messages is: “Kids need to be doing this kind of deeper learning, deeper thinking, higher-order thinking in mathematics,” says Loveless. This is a blast from the past.

“It gives local educators license to adopt a lot of this garbage, this really bad curriculum . . . under the shield of the Common Core,” says Loveless. “And that particular problem is just a terrible math problem and should not be given to kids.”

Phrases such as “mathematical reasoning” are like “a dog whistle to a certain way of approaching mathematics that has never worked in the past,” says Loveless. It’s been tried in the 1960s and again in the 1990s and “failed both times.”

Simple math made complicated — for a reason

The Common Core makes simple math more complicated in order to teach understanding, writes Libby Nelson on Vox.

In the past, “students had this sense that math was some kind of magical black box,” says Dan Meyer, a former high school math teacher studying math education at Stanford University. “That wasn’t good enough.”

Students will learn different ways to multiply, divide, add, and subtract so they can see why the standard method works, writes Nelson. “They can play with them in fun, flexible ways,” says Meyer, who blogs at Dy/Dan.

Using a number line for subtraction lets students visualize the “distance” between two numbers. A father’s complaint about a confusing number line problem went viral on the Internet. Nelson provides a clearer version. 

Students put the two numbers at opposite ends of the number line.

Screen_shot_2014-04-17_at_5

It’s 4 steps from 316 to 320, 100 steps from 320 to 420, 7 steps from 420 to 427.

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Then they add the steps together: 4 + 100 + 7 = a distance of 111. LearnZillion, a company that creates lesson plans for teaching to the Common Core standards, has a 5-minute video explaining this technique.

“Students should be able to understand any of these approaches,” said Morgan Polikoff, an assistant professor of education at the University of Southern California who is studying how the Common Core is implemented in the classroom. “It doesn’t mandate that they necessarily do one or the other.”

“A key question is whether elementary school teachers can learn to teach the conceptual side of math effectively,” writes Nelson.

If not, number lines and area models will just become another recipe, steps to memorize in order to get an answer, Polikoff says.

This is a real risk: Many elementary teachers are strong on reading and weak in math (and science). Perhaps we need math/science specialists in elementary school who understand their subject deeply and can teach kids to understand too.

Is your kid getting reform math?

Here’s how to tell if your kids are being taught reform math by Robert Craigen and Barry Garelick.

“In the past students were taught by rote; we teach understanding.” First, ‘rote’ literally means ‘repetition’ — and this is a good idea, not a bad one. Second, it is simply false that teaching was without understanding — by design, in any case — in the past. There have always been teachers who taught math poorly or neglected to include a conceptual context. 

. . . Under reform math, students are required to use inefficient procedures for several years before they are exposed to and allowed to use the standard method (or “algorithm”) — if they are at all. This is done in the belief that the alternative approaches confer understanding to the standard algorithm.  . . . But this out-loud articulation of “meaning” in every stage is the arithmetic equivalent of forcing a reader to keep a finger on the page, sounding out every word, every time, with no progression of reading skill. Alternatives become the main course instead of a side dish and students can become confused — often profoundly so.

If you hear references to “drill and kill,” “the guide on the side not the sage on the stage”  or “just-in-time learning,” it’s reform math, they write. Praise for ambiguity, flipping, group learning and “making students think like mathematicians” also are danger signs, they write.

“We use a balanced approach”  means “go away.”

Many educators are interpreting Common Core to mean fuzzy math, says Garelick in a Heartland interview.

From Parents Against Everyday Math:

Photo: Its like this.

K-5 teachers: Homework = 2.9 hours per week

Elementary teachers assign an average of 2.9 hours of homework per week, middle school teachers assign 3.2 hours and high school teachers expect 3.5 hours, according to a Harris poll for University of Phoenix.

A high school student taking five courses could have 17.5 hours of homework per week. (When my daughter was in high school, she averaged three hours a night.)

Teachers say homework  helps them see how well their students understand the lessons (60 percent); helps students develop problem-solving skills (46 percent); gives parents a chance to see what is being learned in school (45 percent); helps students develop time management skills (39 percent); encourages students to relate classroom learning to outside activities (37 percent) and allows teachers to cover more content in class (30 percent).

Understanding why algorithms work

Children need to learn algorithms to understand math, write Alice Crary, a philosophy professor, and W. Stephen Wilson, a math professor, in the New York Times.

. .  it is true that algorithm-based math is not creative reasoning. Yet the same is true of many disciplines that have good claims to be taught in our schools. Children need to master bodies of fact, and not merely reason independently, in, for instance, biology and history.

Mastering an algorithm requires “a distinctive kind of thought,” they write. It’s not “merely mechanical.” In addition, algorithms are “the most elegant and powerful methods for specific operations. . . . Math instruction that does not teach both that these algorithms work and why they do is denying students insight into the very discipline it is supposed to be about.”

Some commenters claimed math reformers advocate a “balanced” approach that includes algorithms, writes Barry Garelick in Education News. He is dubious.

I am reminded of a dialogue between a friend of mine—a math professor—and an public school administrator.  My friend was making the point that students need basic foundational skills in order to succeed in math. The administrator responded with “You teach skills. But we teach understanding.”

. . . The reform approach to “understanding” is teaching small children never to trust the math, unless you can visualize why it works. If you can’t “visualize” it, you can’t explain it.  And if you can’t explain it, then you don’t “understand” it.

According to Robert Craigen, math professor at University of Manitoba, “Forcing students to use inefficient procedures that require ham-handed handling of place value so that they articulate “meaning” out loud in every stage is the arithmetic equivalent of forcing a reader to keep his finger on the page and to sound out every word, every time, with no progression of reading skill.”

The power of math, however, is allowing for exploration of concepts that cannot be visualized.  Math is what takes over when our intuition begins to fail us.

Garelick, who’s launched a second career as a math teacher, links to a 1948 math book’s illustration of different ways to do mental multiplication:

Figure 2 (Source: Study Arithmetics, Grade 5)

Thinking deeply about … um … what?

Students will read more short informational texts under the new Common Core Standards and have less time for complete books — fiction or nonfiction — writes Will Fitzhugh, editor of the Concord Review.

Among the suggested texts are The Gettysburg Address, Letter from Birmingham Jail, Abraham Lincoln’s Second Inaugural Address, and perhaps one of the Federalist Papers, but no history books, writes Fitzhugh.

In the spirit of turnabout, English teachers could stop assigning complete novels, plays and poems, Fitzhugh writes.  Instead of reading Pride and Prejudice, perhaps Chapter Three would do.  “They could get the ‘gist’ of great works of literature, enough to be, as it were, ‘grist’ for their deeper analytic cognitive thinking skill mills.”

Teachers will have to “to wean themselves from the old notions of knowledge and understanding” to offer “the new deeper cognitive analytic thinking skills required by the Common Core Standards,” Fitzhugh writes, perhaps with a touch of sarcasm.

In 1990, Caleb Nelson wrote in The Atlantic about an older Common Core at Harvard:

The philosophy behind the [Harvard College] Core is that educated people are not those who have read many books and have learned many facts but rather those who could analyze facts if they should ever happen to encounter any, and who could ‘approach’ books if it were ever necessary to do so….

That’s the idea, writes Fitzhugh.

The New Common Core Standards are meant to prepare our students to think deeply on subjects they know practically nothing about, because instead of reading a lot about anything, they will have been exercising their critical cognitive analytical faculties on little excerpts amputated from their context. So they can think “deeply,” for example, about Abraham Lincoln’s Second Inaugural Address, while knowing nothing about the nation’s Founding, or Slavery, or the new Republican Party, or, of course, the American Civil War.

Students will learn that “ignorance is no barrier to useful thinking,” Fitzhugh predicts. “The current mad flight from knowledge and understanding . . . will mean that our high school students [those that do not drop out] will need even more massive amounts of remediation when they go on to college and the workplace than are presently on offer.”

Via Jim Stergios of Rock the Schoolhouse, a Common Core skeptic.

Among Common Core exemplar texts are Evan Connell’s Son of the Morning Star about The Battle of Little Big Horn and Dee Brown’s Bury My Heart at Wounded Knee, RiShawn Biddle points out.

Math teacher: I don’t know enough math

After earning an applied math degree and teaching math for years, Darren realized he doesn’t really understand math as well as he should.

In college,  he “could calculate my butt off, but so often didn’t fully understand what I was doing.”

As an example, in differential equations I could calculate eigenvalues all day long, but to this day I don’t know what an eigenvalue is or what it does for me or why I need to calculate it. I’ve taught myself plenty –sometimes just days before I had to teach it to my students.

He told a student bound for Cal Poly “not to make the mistake I made; ask the questions, go for the deeper understanding.” And Darren decided to go for it himself.

I’ve never understood the Fundamental Theorem of Calculus. Why, exactly, are an integral and an antiderivative the same thing? I’ve followed the steps in my calculus books, and understood each step, but never really understood how they all fit together. So today I pulled a different calculus book out of my closet and I started studying. I found one that provided a very user-friendly explanation, which then allowed me to understand the very rigorous (read: dry and difficult) proof in a second text. It took a few minutes to replace decades of deficit.

In the fall, Darren will start a masters program in Teaching Math through the University of Idaho’s Engineering Outreach Program.

Most of us never have the courage to face our eigenvalues, much less blog about it.

 

The myth about traditional math education

“The traditional method of teaching math has failed thousands of students,” claim new math proponents. That’s a myth, writes Barry Garelick in Education News.

Garelick looked at math books and methods used in the ’40s, ’50s and ’60s.

Mathematical algorithms and procedures were not taught in isolation in a rote manner as is frequently alleged. Concepts and understanding were an important part of the texts.

Then and now, nobody argues for memorization without understanding, he adds.

Traditional math education was working reasonably well, Garelick argues. In Iowa, test scores rose steadily until about 1965, and then declined dramatically for a decade.  This pattern was repeated in Minnesota and Indiana.

 

Source: Congressional Budget Office (1986)

Some researchers blame increased drug use and the rise in divorce and single-parent families for the decline. Garelick blames progressive education which called for student-centered, needs-based courses.

After taking not-so-early retirement, Garelick is now a student math teacher at a California junior high school.

 

 

The secret of Singapore Math

Last week, I asked if the New York Times story on Singapore Math described the program accurately. Barry Garelick, co-founder of the U.S. Coalition for World Class Math, answers the question on the Core Knowledge Blog: No way.

(The Times) described a program that strangely sounded like the math programs being promoted by reformers of math education, relying on the cherished staples of reform: manipulatives, open-ended problems, and classroom discussion of problems. 

. . . Those of us familiar with Singapore Math from having used it with our children are wondering just what program the article was describing.  Spending a week on the numbers 1 and 2 in Kindergarten?  Spending an entire 4th grade classroom period discussing the place value ramifications of the number 82,566?

Singapore Math books provide pictures, examples and problems, but doesn’t tell teachers how to teach, he writes.  If a kindergarten teacher is spending a week on the numbers 1 and 2, that’s the teacher’s choice.

Singapore Math uses traditional approaches to math education, such as “explicit instruction and giving students many problems to solve,” Garelick writes. This is not what math reformers advocate. Nor does Singapore Math rely heavily on manipulatives.  It does use “bar modeling” to help children solve problems.  

 Singapore’s strength is the logical consistency of the development of mathematical concepts. And much to the chagrin of educators who may have learned differently, mastery of number facts and arithmetic procedures is part and parcel of conceptual understanding.  Starting with conceptual understanding and using procedures to underscore it is an invitation to disaster—such approach is making profits for  outfits like Sylvan, Huntington and Kumon.

The underlying message in articles such as the Times’ is that math education is bad in the U.S. because it is not being taught according to the ideals of reformers—and the reason it is successful in Singapore is because it is being taught that way.  Never considered is the possibility that the reform minded methods and textbooks written to implement them are one of the root causes of poor math education in this country.  Katharine Beals in her blog “Out in Left Field” does an excellent job describing this.

Garelick plans to start his career as a math teacher next year, after he retires from the Environmental Protection Agency, where he’s an analyst.