Subtraction made complicated

One page in his daughter’s third-grade math book explains the “counting up” method of subtraction, writes Erick Erickson, editor in chief of It’s one of four methods taught.

The traditional method of subtracting, borrowing and carrying numbers, is derisively called the “Granny Method.” The new method makes no freaking sense to either my third grader or my wife.

We send our child to a Christian private school. We thought our child could escape this madness. But standardized tests, the SAT, and the ACT are all moving over to Common Core. So our child has to learn this insanity.

Parents and kids feel helpless, writes Erickson. And, in his wife’s case, homicidal.

Alternatives such as “counting up” are supposed to supplement traditional methods, not replace them, responds Andy Kiersz at Business Insider. Common Core’s fourth-grade standards say students should “fluently add and subtract multidigit whole numbers using the standard algorithm.”

The “counting up” method . . . captures some of the underlying aspects of subtraction and place value that allow borrowing and carrying to work.

. . . The student starts counting by ones from the smaller number up to the nearest multiple of 10. Then she counts by 10s to 100, then by hundreds to the first digit of the larger number, then takes the remaining two-digit part of the larger number. These are all just a different way of subtracting in different place values. Adding these intermediate steps together, the student gets her result.

The point of these alternative methods is to provide a different perspective on a problem, which is often useful in learning math at any level.

Ideally, these alternatives build students’ understanding of key concepts,” making it easier for them to work with the standard algorithms later,” writes Kiersz.

Ideally, perhaps.  What does it do in reality?

2+2= deep learning

In The School of Deep Understanding, Diana Senechal satirizes the gee-whiz discovery of “deep learning” in Common Core math classrooms.

The teacher, Gideon Pelous, buzzed about the room like a shimmering dragonfly while the children—second-graders from the deep inner city—discussed the essence of numerals in small groups.

Before the Core, students would be taught that two plus two equals four, but they would never know why.

Now, everything had changed.

“I just had a realization,” said Shelly Thomas, arranging four rectangular blocks in front of her. “I used to think that numerals were quantities. I was trying to figure out what the curve on the 2 meant, and what the double curve on the 3 meant. I even tried measuring them with my ruler. Then I had the insight that numerals aren’t quantities, but rather symbols that represent quantities.”

“You mean to say—“ sputtered Enrique Alarcón as he seized a crayon.

“Yep,” she continued. “This 1 here represents a unit of something. It can be a unit of anything. Now, when we say ‘unit,’ we have to be careful. That’s another thought that came to me, but I haven’t figured it–.”

“I have,” interrupted Stephanie Zill, banging on her Curious George lunchbox. “We use the word ‘unit’ in both a contextual and an absolute sense. That is, a unit is unchanging within the context of a problem, but it may change from problem to problem. Also, certain defined units, such as minutes and yards, have a predefined size that doesn’t change from one context to the next—until you consider relativity, that is.”

“Oh, I get it,” said Enrique. “So, this numeral 1 represents one unit, which could be a unit of anything, but within a given problem, the word “unit” does not change referent unless we are dealing with more than one kind of unit at once. Hey, what color crayon should we use: magenta or seaweed?”

“Magenta,” said Shelly.

Stephanie and Orlando go on to discuss the “sockiness” of socks to conclude that two pairs of socks equal four socks in the real world.

The right answer does matter

Math Curmudgeon is listing “things we need you to stop saying.” Number 1: “The right answer isn’t important. It’s knowing what you’re doing.” The right answer is the whole point of doing the problem … has always been, is now, and will always be, the Curmudgeon argues.  The “knowing what you are doing part” leads to the right answer. If it doesn’t, then you don’t know what you are doing.

What we should be saying is “The right answer is vitally important … so important that we also want students to explain the method and how we all know the answer is correct; they must be able to detect an error if it occurs and describe how to fix it so that the solution IS correct.”

“You can’t detect errors unless you know the right answer, or at least have a sense of what that right answer should be,” the Curmudgeon writes.

Same old new math

Common Core’s Newer Math is a lot like like the old new math, writes David G. Bonagura Jr., a teacher and writer, in National Review Online.

In 1961, New Math “was supposed to transform mathematics education by emphasizing concepts and theories rather than traditional computation,” as this article shows.

Flash forward 50 years, and Common Core is today making the same promises:

The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy.

New Math, Sequential Math, Math A/B, and the National Council of Teachers of Mathematics Standards also “promised to transform (America’s children) into young Einsteins and Aristotles,” writes Bonagura. It didn’t work out that way.

Despite claims that Common Core doesn’t tell teachers how to teach, the new standards come with a flawed pedagogy, Bonagura charges. “Common Core buries students in concepts at the expense of content.”

Take, for example, my first-grade son’s Common Core math lesson in basic subtraction. Six- and seven-year-olds do not yet possess the ability to think abstractly; their mathematics instruction, therefore, must employ concrete methodologies, explanations, and examples. But rather than, say, count on a number line or use objects, Common Core’s standards mandate teaching first-graders to “decompose” two-digit numbers in an effort to emphasize the concept of place value. Thus 13 – 4 is warped into 13 – 3 = 10 – 1 = 9. Decomposition is a useful skill for older children, but my first-grade son has no clue what it is about or how to do it.

Students can’t just solve math problems and show their work, he writes. They need to provide a written explanation.

With sons in first and third grade, he’s seen Common Core math books that “devote enormous space to word problems that have to be answered verbally as well as numerically, some in sections called Write Math.”

That makes math much harder for students who aren’t proficient in English or in reading and writing, Bongagura points out. (Handwriting was very difficult or both my brothers.)

With so much focus on teaching students the “why” of math, teachers will have little time to teach the “how,” Bonagura predicts.

Mathematical concepts require a high aptitude for abstract thinking — a skill not possessed by young children and never attained by many. What will happen to students who already struggle with math when they not only are forced to explain what they do not understand, but are presented new material in abstract conceptual formats?

“Instead of developing college- and career-ready students, we will have another generation of students who cannot even make change from a $5 bill.”

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.

Social studies follies

There are no Common Core social studies standards, nor even a framework for standards, but there is a “vision” of a “framework for inquiry,” reports Ed Week.

Welcome to the social studies follies, writes Checker Finn on Education Gadfly. The “vision” of a College, Career and Civic Life (C3) Framework will “focus on the disciplinary and multidisciplinary concepts and practices that make up the process of investigation, analysis, and explanation.” The document goes on:

It will include descriptions of the structure and tools of the disciplines (civics, economics, geography, and history) as well as the habits of mind common in those disciplines. The C3 Framework will also include an inquiry arc—a set of interlocking and mutually supportive ideas that frame the ways students learn social studies content. This framing and background for standards development to be covered in C3 all point to the states’ collective interest in students using the disciplines of civics, economics, geography, and history as they develop questions and plan investigations; apply disciplinary concepts and tools; gather, evaluate, and use evidence; and work collaboratively and communicate their conclusions.

The C3 Framework will focus primarily on inquiry and concepts, and will guide — not prescribe — the content necessary for a rigorous social studies program. CCSSO recognizes the critical importance of content to the disciplines within social studies and supports individual state leadership in selecting the appropriate and relevant content.

Nowhere is there a mention of “knowledge,” complains Finn.  “When was World War I, why was it fought, who won, and what were the consequences?” Dunno.

Of course, “content” is mentioned, but Finn isn’t impressed. “This could turn out to be simply awful.”

American students don’t know much about civics and aren’t prepared for citizenship, writes Rick Hess, who’s co-edited a new book, Making Civics Count, with David Campbell, political scientist at Notre Dame and authority on civic engagement and Meira Levinson, education philosopher at Harvard and author of No Citizen Left Behind. In a 2006 survey of college students, “more than half of seniors did not know that the Bill of Rights prohibits the establishment of an official national religion.”

New math: Concepts precede skills

Under new math standards, students will be asked to explain why procedures work before they’ve mastered the  procedures, writes Barry Garelick in The Atlantic.

Under the Common Core Standards, students will not learn traditional methods of adding and subtracting double and triple digit numbers until fourth grade. (Currently, most schools teach these skills two years earlier.) The standard method for two and three digit multiplication is delayed until fifth grade; the standard method for long division until sixth. In the meantime, the students learn alternative strategies that are far less efficient, but that presumably help them “understand” the conceptual underpinnings.

Students are expected to:

Make sense of problem solving and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning

These are “habits of mind that ought to develop naturally as a student learns to do actual math,” Garelick writes.

True habits of mind develop with time and maturity. An algebra student, for instance, can take a theoretical scenario such as “John is 2 times as old as Jill will be in 3 years” and express it in mathematical symbols. In lower grades, this kind of connection between numbers and ideas is very hard to make. The Common Core standards seem to presume that even very young students can, and should, learn to make sophisticated leaps in reasoning, like little children dressing in their parents’ clothes.

Teachers will need to adjust Common Core guidelines, Garelick writes. But will teachers have the freedom to do so?

Some people are misreading the standards, responds William McCallum, a University of Arizona math professor and leader of the math standards team, in a comment.  “The standards do not settle the debate on how fluency and understanding should interact in the curriculum,” he writes.  The phrases “critical thinking” and “collaborative learning” do not occur anywhere in the standards.

Math needs a revolution too

Math Needs a Revolution, Too, writes Barry Garelick in response to The Atlantic story, The Writing Revolution. He first encountered reform math when his daughter was in second grade.

. . . understanding takes precedence over procedure and process trumps content. In this world, memorization is looked down upon as “rote learning” and thus addition and subtraction facts are not drilled in the classroom–it’s something for students to learn at home. Inefficient methods for adding, subtracting, multiplying, and dividing are taught in the belief that such methods expose the conceptual underpinning of what is happening during these operations. The standard (and efficient) methods for these operations are delayed sometimes until 4th and 5th grades, when students are deemed ready to learn procedural fluency.

Students are expected to “think like mathematicians” before acquiring the analytic tools necessary to do so, Garelick writes. Procedural skills are taught on a “just in time” basis.

Such a process may eliminate what the education establishment views as tedious “drill and kill” exercises, but it results in poor learning and lack of mastery. Students generally work in groups with teachers who “facilitate” rather than providing direct instruction.

As reform math has become the norm in K-6 classrooms, high school math teachers are trying to teach algebra to students who “do not know how to do simple mathematical procedures,” he writes.

In math, as in writing, learning the fundamentals may not be fun or engaging. It may require practice. But students who skip the basics rarely develop the ability to “think like mathematicians” or write like “authors.” They’re confused. And bored.

Calculators: Useful or not?

In response to Konstantin Kakaes’ Why Johnny Can’t Learn Without a Calculator, math teacher Paul J. Karafiol argues that Calculators in the classroom are useful.

Teaching math requires actually understanding math, and people who understand math have always been in short supply, in and outside of the teaching profession. So a different, simpler explanation for the failure of students to learn math is that there aren’t a lot of excellent teachers out there teaching math. Technology doesn’t enter into the picture.

Where it does enter the pictures is in a new and completely unexpected change in mathematics education. Excellent teachers who use technology can increase access to higher mathematics for students with poor computational skills, by allowing these students to reason about concepts without getting bogged down in computation. This year, my AB Calculus class included some students who couldn’t reliably add fractions. By the end of the course, almost all of them could explain what the derivative of a function means (in abstract and contextual terms), how it is calculated, and what it could be used for. They could do all this because they used calculators with computer algebra systems—calculators that give algebraic answers, not just numbers—to do the heavy lifting.

Finally, Kakaes never engages what is, to me, the central question that technology poses to the mathematics teacher, namely, what of the traditional pencil-and-paper mathematics is worth teaching?

“The argument should be about when and how often students should be taught to use their calculators,” Karafiol writes.

Kakaes responds here.

Do timed tests cause math anxiety?

One third of students end up in remedial math in college and “the level of interest in the subject is at an all-time low,” writes Jo Boaler, a Stanford math education professor, in Ed Week.  She blames timed math tests — solve 50 multiplication problems in three minutes — for causing math anxiety that cripples learning

Math has become a performance subject. Children of all ages are more likely to tell you that the reason for learning math is to show whether they “get it” instead of whether they appreciate the beauty of the subject or the way it piques their interest. The damage starts early in this country, with school districts requiring young children to take timed math tests from the age of 5.

Common Core State Standards, which call for math “fluency,” may encourage timed testing, Boaler worries.

Stress caused by timed testing can lead to changes in the brain, permanently hurting children’s ability to learn math, she writes.

There are many good teaching strategies for encouraging fluency in math, but the ones that are effective are those that simultaneously develop number sense—the flexible use and understanding of numbers and quantities—without instilling fear and anxiety. Strategies that involve reasoning about numbers and operations, such as the pedagogical approach called “number talks,” are ideal for developing fluency with understanding.

Beyond the fear and anxiety, timed tests also convey strong and negative messages about math, suggesting that math ability is measured by working quickly, rather than thinking deeply and carefully—the hallmark of high-level mathematical thinking.

Children can learn math skills and concepts in tandem, writes Barry Garelick on Education News.

Reformers criticize traditional math instruction as “skills-based,” implying “students who may have mastered their math courses in K-12 were missing the conceptual basis of mathematics and were taught the subject as a means to do computation, rather than explore the wonders of mathematics for its own sake.”

Students have struggled with math for a long time: If one dinosaur eats two cavemen per hour, how many cavemen can four dinosaurs eat in 30 minutes?  When I was in elementary school in the ’50s, before calculators or timed tests of math facts, many kids were anxious about math because there were right and wrong answers. We didn’t tackle the lowest common denominator to appreciate math’s beauty or explore its wonders. We though the point was to “get it.”

“New math” came in a few years later, when my brother was in first grade. In trying to teach concepts, it made kids even more anxious.

My daughter did timed tests of addition and subtraction problems in first grade — 25 years ago! They probably did multiplication in second grade.  She thought the tests were fun. Of course, she was good at it. But Boaler says math anxiety is worst for high-ability students.