Uh-oh…the secret’s out. If you want your child to do well in math, teach ‘em long division at the kitchen table after school. Traditional formulas have been supplanted, the Associated Press has discovered (long after the horse has departed the barn) by concept-based curricula aiming to “teach the ideas behind mathematics.” This is leading “renegade parents” to teach basic math formulas on the sly at home.

Renegade teachers too, as Matthew Clavel described in a terrific piece in City Journal some time back:

If school officials knew how far my lessons would deviate from the school district-mandated math program in the months ahead, they probably would have fired me on the spot. But boy, did my kids need a fresh approach….Not one of my students knew his or her times tables, and few had mastered even the most basic operations; knowledge of multiplication and division was abysmal. Perhaps you think I shouldn’t have rejected a course of learning without giving it a full year (my school had only recently hired me as a 23-year-old Teach for America corps member). But what would you do, if you discovered that none of your fourth graders could correctly tell you the answer to four times eight?

You’d teach them the algorithms, like Clavel did, I did, and countless others. The idea that teaching for understanding precludes automatic recall and traditional methods of instruction–that children haven’t learned unless they ”construct” their understanding of math–is one of those mindless orthodoxies that have squeezed out common sense and strewn failure in its wake. Watch a 4th or 5th grader struggle with partial sums addition and lattice multiplication and you’d quickly revert to time drills and memorization too.

If I run into one of my 5th graders even 20 years from now, I will ask him or her, “Do you know how to divide?” I’d bet my rent money I’ll get the answer, “Does McDonalds Sell Cheese Burgers?” Sue me. Take away my teaching license. But I’ll bet they can divide.

I’ve done the same in high school English for years – teaching grammar and spelling covertly. The funny thing is, I’ve never had an experience where kids didn’t want to learn it. Usually, they have a thousand questions they’ve had for years and never had answered concretely, like where does the apostrophe go and how do I know, and how do I use commas? Seems students actually like to know how things work – not to have to figure it out themselves through trial and error.

Comment by Redkudu — July 16, 2008 @ 6:27 pm

Ditto, and amen Redkuku. One of the most popular routines we had in my class was Daily Oral Language. Every day when the kids came back from lunch, there were several sentences written on the board, chock full of errors, which we would correct out loud, whole class (whole class?? The horror!). Within weeks, everyone could find at least some errors. One of my personal highlights was having my class correct errors found in an official Department of Ed publication, and write letters to the Chancellor offering to help. The New York Daily News picked up on it for a story, and my students felt incredibly powerful.

Comment by Robert Pondiscio — July 16, 2008 @ 7:03 pm

I’m afraid I’ve gotten to the point of being grumpy and contrarian about the “I’m not politically correct, I teach math skills, not this goofy concept stuff.”

Yes, I’ve pointed out to my kid that the math problems she finds boring would go faster if her times tables tripped little more easily off her finger tips. But that doesn’t give me hives in the same way that having her completely stumped when asked to explain why you divide fractions by “turning the second one upside down” does.

There’s a place for facts, buts there’s also place for understanding something well enough that you can figure out the fact you’ve forgotten. Many of the college students I teach treat math as a random collection of tricks; for every “type” of problem you memorize the trick for it, and if you forget, you’re just stuck. The idea that thinking about the problem might tell them what to do with the numbers, and that it might not be something that I’d put a box around in lecture, seems totally alien to many of them.

Comment by Rachel — July 16, 2008 @ 11:33 pm

There’s a place for facts, buts there’s also place for understanding something well enough that you can figure out the fact you’ve forgotten.Why is it assumed that learning facts and algorithms precludes understanding concepts? I’ve worked with many textbooks including older ones, as well as Singapore’s. The concepts are laid out quite clearly. As for understanding why one inverts the divisor, even Singapore doesn’t explain it, but rather leads students to see tha inversion of the divisor is part of a pattern in fractional multiplication that they’ve been working with over and over. When it comes time to divide a fraction by a fraction, they can easily extend the pattern so it’s a natural leap to invert the divisor. As for understanding why, once one gets into algebra, it is easy to see why it works. In fact, the proof of it can even be posed as a problem for students to work, either in class or as a homework assignment.

Comment by Barry Garelick — July 18, 2008 @ 11:05 am

Where “teaching for understanding” leaves the rails, in my opinion, is in the false orthodoxy that there is no value in computational competence in the absence of understanding. This is patently false. When I was a little kid, I got a calculator for Christmas. Playing with it, I stumbled accidentally on the formula for figuring out baseball players’ batting averages. I couldn’t explain at gunpoint why it worked, but it worked every time. It was magic! I didn’t understand percents, ratios — or even that a batting average WAS a percentage — until a few years later. But knowing the formula first certainly solidified my understanding later.

In the meantime, it made me popular on my little league team.

Comment by Robert Pondiscio — July 18, 2008 @ 11:17 am

It is easy to explain why invert and multiply works when dividing by a fraction. Children need to know that the rule has a logical foundation and it should be explained to them. They do not need to remember why it works unless they aspire to teach arithmetic.

Other methods of dividing fractions are taught that are perhaps more intuitive but they do not transfer well into algebra.

Comment by Charles R. Williams — July 18, 2008 @ 1:51 pm

The basic concepts of elementary math are few. Understand them and then learn to “Do the Math” using the common methods (algorithms) of arithmetic. see below: “PASS IT ON”!

“A Two Page Algebra Book” By Carl M. Bennett, BEE; MS(3)

Mathematics is a language for expressing precise, logical ideas. The basic language of mathematics is common Algebra.

Algebra is based on the definitions of two rules of how to “operate on” or combine two real numbers to form another real number, and five other definitions of the characteristics of these operations needed to make Algebra logically consistent and practical as a language of logical thought.

The first rule is called addition or the “+” operation on two real numbers.

For two real numbers a, and b, a + b is defined as the real number c which is equal to the combined value or “sum” of the numbers a, and b. For example 3 + 5 is defined as 8 or 3 + 5 = 8.

The second rule is called multiplication or the “x” operation on two real numbers.

For two real numbers a, and b, a x b is defined as the real number c which is equal to the combined value or “sum” of the number b added to itself a times. For example 3 x 5 is defined as (5 + 5 + 5) = 15.

To be logically consistent and practical, both of the above operations, addition (+) and multiplication (x) must have the four basic characteristics defined below.

1 – Both addition (+) and multiplication (x) must be “associative” in character.

This means that the order in which we associate and add the numbers a + b + c, gives the same real number. That is to say, if we first associate and add (a + b) and then add c, or we first associate and add (b + c) and then add a, we get the same real number. For example, (3 + 5) + 7 = 15 gives the same result as 3 + (5 + 7) = 15.

This also means that the order in which we associate and multiply the numbers a x b x c, gives the same real number. That is to say, if we first associate and multiply (a x b) and then multiply by c, or we first associate and multiply (b x c) and then multiply by a, we get the same real number. For example, (3 x 5) x 7 = 105 gives the same result as 3 x (5 x 7) = 105.

2 – Both addition (+) and multiplication (x) must be “commutative” in character.

This means that the order in which we add the numbers a, and b, gives the same real number. That is to say, if we add a + b, or we add b + a, we get the same real number._For example, 3 + 5 = 8 gives the same result as 5 + 3 = 8.

This also means that the order in which we multiply the numbers a, and b, gives the same real number. That is to say, if we multiply a x b or multiply b x a, we get the same real number. For example, 3 x 5 = 15 gives the same result as 5 x 3 = 15.

3 – Both addition (+) and multiplication (x) must have the “identity” characteristic.

This means that for addition (+), there must be a real number, I, when added to any real number a, always gives, a + I = I + a = a. For addition (+), this “identity” is I = 0. For example 5 + 0 = 5.

This also means that for multiplication (x), there must be a real number, I, when multiplied by any real number a, always gives a x I = I x a = a. For multiplication (x) this “identity” is I = 1. For example 5 x 1 = 1 x 5 = 5.

4 – Both addition (+) and multiplication (x) must have an “inverse” characteristic

This means that for addition (+), a real number b is the addition (+) “inverse” of any real number a, if and only if a + b = 0. The addition (+) “inverse” for any real number a, is the “negative of its real value”, defined as “minus a” or -a, since a + (-a) is always equal to the addition (+) “identity”, I=0, that is to say a + (-a) = 0, for all real numbers a, of both positive or negative value. For example (-5) + (-(-5)) = (-5) + 5 = 0.

For simplicity we often write –5 + 5 = 0 = 5 – 5, but 5 – 5 is mathematically, actually 5 + (-5). The “minus (-) notation” only tells us that -a, is the negative in value of a. Thus minus (-) is not actually a valid algebraic operation on two real numbers like addition (+) is, since the minus (-) operation is neither “associative” nor “commutative” as defined above. For example, 7 – 6 = 1 is not the same as 6 – 7 = (-1).

This also means that for multiplication (x) a real number b is the multiplication (x) “inverse” of a real number a, if and only if a x b = 1. The multiplication (x) “inverse” for any real number a, is the reciprocal of its real value, defined as the real number (1/a), since a x (1/a) is always equal to the multiplication (x) “identity”, I=1, for all real numbers except zero = 0.

Zero has NO multiplication (x) “inverse”, since (zero) x (any real number) = zero, thus not the multiplication (x) “identity”, I=1, as required for zero to have a multiplication (x) “inverse”.

This is why “division by zero”, for example, a x (1/0) or a /0, is NOT allowed in common Algebra.

For simplicity we often write, for example, 25 / 5 = 5, however; 25 / 5 is mathematically, actually 25 x (1/5) = 5. The “divide (/) notation” only tell us that a number is the “inverse” of a real number. Thus divide (/) is not actually a valid algebraic operation on two real numbers like multiplication (x) is, since the divide (/) operation is neither “associative” nor “commutative” as defined above.

For example, 8 / 2 = 4 is not the same as 2 / 8 = 0.25 or the rational, real number, one quarter, (1/4).

To be logically consistent and practical, both of the above operations, addition (+) and multiplication (x) must have the one additional basic joint characteristic as defined below.

5 – Multiplication (x) must be “distributive” over addition (+), that is,

a x ( b + c ) = (a x b) + ( a x c ) and ( a + b ) x c = (a x c ) + ( b x c ) for all real numbers.

The above “rules and characteristics” define common Algebra. Every logically and mathematically correct manipulation of any algebraic equation involves these “rules and characteristics” or a composite of them.

© Carl M. Bennett, 18 August 2006. May be reproduced only for educational and research purposes.

—————

Comment by Carl M. Bennett — July 19, 2008 @ 9:39 am

Just a note from a regular parent of a 5th (now 6th) grader… My daughter spent this past year under a newly adopted math program by Houghton Mifflin which was based on “sequencing” math concepts. It seemed like a decent program BUT I quickly realized that, with all the work “mastering the concepts”, there was very little actual crunching the numbers going on. And, speaking of doing long division, you should see the crazy processes she was taught to accomplish this! I find it difficult to even understand what she’s doing, and she certainly isn’t too interested in my arcane way of doing division!

To top this off, her school opted to teach the entire year of math in 8 months so the kids would be “ready” for the CRT (important to No Child Left Behind grading of schools). Then, they were “done” with math for the year!!! I was livid.

Anyway, that’s my soapbox rant for the day. Sounds like most of the commentators here are educators. Hope you are advocating for sane education practices for our children…lord knows we need it.

Comment by Terry Jiron — July 21, 2008 @ 1:56 pm

I am a parent also. My daughter’s teacher told me they are supposed to teach grammar through “modeling.” Her teachers covertly throws in grammar rules but are in fear of the principal “catching them.”

I send my daughter to school then I have to teach her math, history and grammar at home, which leaves little time for us to have a relationship.

I am so saddened by the “group think” that surrounds the new “teach them to learn.” Once they are done with high school and have been taught to learn for 12 years, is it then up to them to spend the next 12 years in a public library teaching themselves content?

Comment by Nancy Quintana — July 22, 2008 @ 8:21 am

After reading all of the other posts, I can understand why many students, teachers, and parents like Garden Math. It is Traditional Math but allows a student to work on what they need along with allowing time needed to accomplish. Garden Math provides concepts in a Literacy (LanguageArts)approach which stress Reading, Writing, and Talking Math. Finally, it provides necessary Standards to be competive with countries in a 21st Century World.

http://www.itws.org/TMTTofCA.htm

Comment by Tom Love — July 22, 2008 @ 8:55 am

I am a middle school remedial (Title I) math teacher and I have taught 4th, 5th and 8th grade “regular” math in the past. I agree with almost all the emails sent in response to the “renegade math” article. However, I would like to add (no pun intended) that ALL the previous math teachers, from pre-K through 8th AND the parents at home, MUST ensure that the students learn the so-called “basics”. This is because once they move on to the next grade, or even the next chapter in the same grade, the math teacher cannot successfully teach the new “concepts” if they haven’t mastered the old ones or the basics like multiplication facts. It is next to impossible to teach fractions if the student doesn’t know the multiplication facts completely. I can re-teach the concepts if they haven’t been mastered previously, but I cannot teach long division without holding up everyone else in the class.

So I say to all previous math teachers, PLEASE focus on basics first! Even the so called crazy way of teaching long division (partial quotients) cannot be successful if the student doesn’t know multiplication tables. By the way, if taught properly, partial quotients will innately teach the concepts to which we have been referring. I pride myself on being able to remediate students in the concepts of math, but there truly is no way I can remediate in basics once the student is in middle school! I love teaching math! Many of my students who hated it previously (including the principal’s 8th grade daughter) now not only understand it, but think it is fun! PLEASE elementary teachers! PLEASE teach basics

so we middle school teachers can teach what we know best: concepts!

Comment by Sandi Eichler — July 22, 2008 @ 11:36 am