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.