## 5 + 5 + 5 = 15 is wrong, because . . .

Once again, frustrated parents are posting their children’s math assignments on social media.

Insane Common Core math problems go viral, reports Tech Insider. In one, a student is marked wrong because he or she got the right answer by subtracting rather than estimating.

Estimation questions don’t work well on paper-and-pencil tests, said Frank Noschese, a high school math and physics teacher in New York. “We want the kids to do the estimating in their heads.”

In the other assignment, a third grader was asked to use “repetitive addition” to solve 5 x 3. The answer — 5 +5 +5 = 15 — was marked wrong. The teacher wanted 3 +3 +3 +3 +3 = 15.

The student’s array showing 4 x 6 = 24 also was marked wrong. The teacher wanted rows of 6 hashmarks in 4 columns.

Whatever happened to the commutative property?

Estimation and repeated addition, are part of the Common Core standards, said Noschese. However, it’s up to teachers to design lessons and tests.

Don’t assume the teacher is an idiot, writes Hemant Mehta on Patheos.

Thinking of 5 x 3 as, literally, “five groups of three” could help students learn division, he argues.

When they see a problem that says 5 x ___ = 15, they’ll be thinking “I need five groups of SOME NUMBER to get to 15.” In other words, they’ll be able to pick up division a little more quickly because they’re learning the proper way to think now.

The array problem will make sense in algebra class. “There’s a difference between a 2 x 3 matrix and a 3 x 2 matrix.”

I got through advanced algebra and trig without hitting matrixes. Perhaps they hadn’t been invented then.

Should we stop making kids memorize times tables and ban Mad Minute Mondays? asks Jill Barshay on the Hechinger Report. Flash cards, drills — and especially timed quizzes — are “damaging” for kids, argues Jo Boaler, a Stanford education professor in Fluency Without Fear: Research Evidence on the Best Ways to Learn Math Facts.

These cards promote mathematical insight and number sense by depicting numbers in different ways, argues Jo Boaler.

“Drilling without understanding is harmful,” Boaler told Barshay. “I’m not saying that math facts aren’t important. I’m saying that math facts are best learned when we understand them and use them in different situations.”

Number sense is developed through “rich” mathematical problems, argues Boaler.

Too much emphasis on rote memorization, she says, inhibits students’ abilities to think about numbers creatively, to build them up and break them down. She cites her own 2009 study, which found that low achieving students tended to memorize methods and were unable to interact with numbers flexibly.

Also, memorizing times tables is boring, turning off high achievers, she believes.

I memorized the times tables in fourth grade. It wasn’t boring, because it didn’t take very long. In recent years, I’ve encountered many students who use calculators for the simplest problems and have no number sense.

## 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.

## Old school: Teach word roots, math facts and …

Kids Should Learn Cursive (and Math Facts, and Word Roots), writes Annie Murphy Paul in Time. New researchsupports the effectiveness of “old school” methods such as “memorizing math facts, reading aloud, practicing handwriting, and teaching argumentation,” she writes.

Suzanne Kail, an English teacher at an Ohio high school was required to teach Latin and Greek word roots, she writes in English Journal, though she abhorred “rote memorization.”

Students learned that “sta” means “put in place or stand,” as in “statue” or “station.”  They learned that “cess” means “to move or withdraw,” which let them understand “recess.”

Her three classes competed against each other to come up with the longest list of words derived from the roots they were learning. Kail’s students started using these terms in their writing, and many of them told her that their study of word roots helped them answer questions on the SAT and on Ohio’s state graduation exam. (Research confirms that instruction in word roots allows students to learn new vocabulary and figure out the meaning of words in context more easily.)

For her part, Kail reports that she no longer sees rote memorization as “inherently evil.” Although committing the word roots to memory was a necessary first step, she notes, “the key was taking that old-school method and encouraging students to use their knowledge to practice higher-level thinking skills.”

I learned Latin and Greek word roots in seventh grade. It was lots of fun.

Drilling math facts, like the multiplication table, “is a prerequisite for doing more complex, and more interesting, kinds of math,” Paul writes.

Other valuable old-school skills:

Handwriting. Research shows that forming letters by hand, as opposed to typing them into a computer, not only helps young children develop their fine motor skills but also improves their ability to recognize letters — a capacity that, in turn, predicts reading ability at age five. . . .

Argumentation. In a public sphere filled with vehemently expressed opinion, the ability to make a reasoned argument is more important than ever. . . .

Reading aloud. Many studies have shown that when students are read to frequently by a teacher, their vocabulary and their grasp of syntax and sentence structure improves.

I’d add memorizing and reciting poetry as a valuable old-school skill. What are some others?

## Drill, student, drill

Despite the education world’s rejection of “drill and kill, rote learning has its uses, writes Virginia Heffernan in the New York Times Magazine.

By e-mail, E. D. Hirsch Jr., the distinguished literary critic and education reformer, told me that far from rejecting drilling, he considers “distributed practice,” the official term for drilling, essential. A distributed practice system, Hirsch explained, “is helpful in making the procedures second nature, which allows you to focus on the structural elements of the problem.”

For knowledge that must be automatic, like multiplication tables, “you need something like drilling,” adds Dan Willingham, a cognitive scientist.

“Colorful, happy apps” can make drilling less boring, Heffernan writes.

Apps devoted to specific subjects always have the right answers in reserve. They unfailingly know stuff that might elude more fallible human drillers, like atomic weights, the order of cranial nerves and African geography. And they can make almost any exercise feel like a video game.

. . . even as they profess reluctance about drilling schoolchildren, adults who themselves are looking to learn something new — from foreign languages to bar-exam material — increasingly turn to apps that animate some version of a multiple-choice or flashcard narrative.

I tutored a girl in algebra who hadn’t memorized the multiplication tables. She had to slog through the arithmetic on every problem, which made it hard to “focus on the structural elements.”

Ten years ago, I tutored a sixth grader who was an excellent phonetic reader with poor comprehension because of her limited English vocabulary. She asked me for the definition of every word she didn’t know and memorized the definitions. I just found her high school-era web page, which lists her favorite books, including The Scarlet Letter.