Writing out numbers with

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A lot of people have trouble with math. At least the kind of math that’s taught in schools. Although there are many dedicated math teachers who are very competent, most elementary schools do not send their students to a special math class where those kind of teachers are found.

Most elementary school teachers must teach all subjects, and there is no way they can be expert in all of them. This is not their fault, but it does create a huge problem. It’s the main reason why such a large portion of the population “doesn’t get” math.

Math Mojo is here for a lot of reasons. One of them is to teach you superior methods to the ones you usually get in schools or “curriculums.” But a greater reason is to help you or your children, or students understand the meaning behind the math.

Math isn’t just a bunch of formulae. It’s not just about “finding x.” If you realize that math is a kind of metaphor for life, then the lessons you learn in math can help you steer through the deeper, sometimes trickier, issues in life.

This post addresses one of them. It will seem like it’s about a simple “rule,” until you read further, and think about the possible deeper meanings for yourself.

Not everyone will get the same insights, nor should they. You may get something special out of it that no one else has ever realized.

A concerned reader recently wrote in the following:

Hello Professor!

Just found your fascinating website via Google.

As an elderly parent of a 10 y/o I am going through the usual reactionary reflexes when confronted with the teaching methods and curriculum being presented to his burgeoning and questionable intellect.

One of the exercises the boy is currently having to undertake is to write out in long hand the number represented by a series of digits

For example: Write out in long hand 19,345,678,432

He has cheerfully been writing: “Nineteen billion, three hundred forty-five million, six hundred seventy-eight thousand, four hundred thirty-two”

I mentioned in passing that, although I can see reasons, based on economy of style, conservation of paper, ink, etc. there was a notable absence of the word ‘and’ in his answer.

I was told that he is not allowed to use the word ‘and’ because it stands for the decimal point…..

What !!!??? Is this something new?

If I say “four hundred and twenty-eight” what I mean (and what people understand) is 428  not 400.28

Is this some new arithmetical nomenclature which has been adopted since my distant school days?

If so one wonders why. Saying “four hundred point two eight” or “four hundred decimal two eight” seemed a perfectly adequate way of expressing the decimal verbally and a great deal less confusing than using words hitherto used (and widely understood) to verbalize non-decimal numbers.

Thanks,

John Pointon

Professor Homunculus sez:

John,

Your question is a really good one, and one that I am sure lots of people are interested in. Also, the greater insights that can be gleaned from examining it can be a valuable lesson for any of us. Or at least an exercise in goofing around with rules.

I think I know exactly what you mean, and I feel the same way. “Reactionary” is a funny word. Could be good, could be bad, but in this case, I think it’s pretty good.

I’m hoping your 10 year-old is misrepresenting what the teacher is saying. I’m hoping the teacher just prefers his or her way, and doesn’t actually forbid “and.” Can you imagine getting “points off” for writing something totally benign? (Actually, that’s pretty easy to imagine.)

If there is no decimal, and  you use “and,” I think there can be absolutely no cause for confusion by using “and” for the tens and/or ones columns.

“…If I say “four hundred and twenty-eight” what I mean (and what people understand) is 428  not 400.28…”

Exactly.

“…Is this some new arithmetical nomenclature which has been adopted since my distant school days? If so one wonders why…”

Yes and no. I’m sure it’s gone back and forth over the years. The thing that gets me is that, if your child is correct, the teacher doesn’t even allow “and.”

Why? Probably because the teacher doesn’t understand why. Probably because there is no “why.” There is probably no reason at all to forbid the “and.”

This is like the case of the “comma.” Did you know that there is no “law” that you must represent 3456 as 3,456? It’s just a convention. A convenience. Like so many of the math “rules.”

It happens to be a good Idea, because it makes it easier to read numbers like 6793089. But it’s not “forbidden” to leave out the comma. Only pedagogues who want to insist that their way is the only way forbid it.

For me, it would be OK to tell a child that “In my classroom, and on my tests, we always use the comma, because it’s a good convention, and it helps me and you keep things straight, but it is important for you to know that it is NOT THE LAW. I don’t want you to accept the “comma law” as final and untouchable. I just want you to understand that we accept this convention for our purposes.”

This, to me, is one of the most important things you can explain to a student, for many reasons. One, is because it happens to be true. Another is because every student will eventually face the situation where some future teacher will have exactly the opposite rule, for whatever reason, and both will insist that their way is “the only right way.” Now the kid is confused, and rightly so. This often leads to terrible resentment of schools and the subjects they teach.

Of course, this need not be so, if we teach students that, unlike in the fictional “X-Files,” the truth is not necessarily “out there.” It’s in us. We make it up most of the time. Some of the stuff has to be “law,” but most of it is just “convention,” and we need to take ourselves and our Ideas with a grain of salt.

One of my favorite dicta is “Call it what you want, and use it for the same purpose.”

Of course, if you want to successfully communicate with others, you need to call it what others have accepted as the label for it. That’s why maybe the “and” rule is a generally good Idea.

For example:

If I were to represent 875.98 as “eight hundred and seventy-five, and ninety-eight hundreths” the two “ands” might be confusing because of their redundancy. Of course, the comma would help clarify, but still, it looks and sounds awkward.

While googling to try to find clarity on this issue, I found this site (http://www.instructortomoney.com/wsp/writecheck.asp.html) which suggests this “reason” for writing “and”:

“A common mistake is an amount such as $100.42 is incorrectly written as One-hundred forty-two instead of One-hundred and 42/100.  If you do this, the larger amount may be deducted from your account.”

I don’t think that is valid for several reasons. One is that I can’t believe that that is a common mistake – I’d like to see the stats on that. Another is the “and” in that example is pretty much meaningless. It could have just as well said ” $100.42 is incorrectly written as One-hundred forty-two instead of One-hundred 42/100.”

The one-hundreths are clearly written in numerals, eliminating any confusion.  The point is, that on a check, the dollars are written in longhand, and the cents in numerals. No confusion, no “and” at all.

“…Saying “four hundred point two eight” or “four hundred decimal two eight” seemed a perfectly adequate way of expressing the decimal verbally and a great deal less confusing than using words hitherto used (and widely understood) to verbalize non-decimal numbers…”

Excellent point (excuse the pun). Is anyone confused by the term “Web two point oh,” for “Web 2.0? (Web 2.0″ is a common name for social networking on the internet, like sites such as Twitter, Facebook, Youtube, etc.) Or is there anyone who prefers “Web two and zero tenths?”  It seems to be entirely a case of, “If it ain’t broke, don’t fix it.” Sheesh, some pedagogues need to get a hobby.

The entire “what is ‘the only right way’” phenomenon reverberates all through math ed, as well as education in general, and deep into society at all levels.

Another annoying example is “showing the work.” The basic math methods I teach (most of them rooted in classical speed-math methods that beat the snot out of the algorithms that are taught “in school”) don’t have any written work to show. The student gets the answer long before children who only know the school method even have a “partial product” to write.

But teachers who only know the standard algorithms insist that the students “show the work” because they can’t imagine an algorithm in which taking time to show the work would be counter-productive. At least until they finally get with the program, and see that the students that use such methods outperform the other students, and the teacher as well. If a school were to be  as backwards as to not accept superior methods when they see them, well, that school would deserve the failing School Performance Report Card it would get.

Naturally, there might be a great reason for the new “and” rule. But if it hasn’t been made clear to the student, it is useless.

It’s like the “you can’t divide by zero in arithmetic.” This rule happens to be true (well, you actually CAN, but you’ll get a useless answer, which is called “undetermined.”) But when a teacher just parrots the rule, and the students just have to accept it, the teacher is sending exactly the worst educational method in the world, which is, “Your curiosity doesn’t count. This is MY ‘Show-and-Tell’.”

OK, at this point I must admit that it generally isn’t the teachers’ fault. Administrations are held hostage to “standards” and “curriculums” and “standardized tests” which are the bulwarks of the only-one-right-way phenomenon. Even the teachers who know better don’t have the time or resources to cover things sufficiently.

Still, I think one of the major tenets of education should be to be able to explain what you are teaching. I dunno, maybe that’s too demanding for 21st century society. Maybe that’s why our schools are producing too many drones. And not even good drones.

Why does it seem like I may be blowing this out of proportion? It’s because little cases like this expose one of the biggest problems in modern teaching. Teaching like that kills the spirit of learning. It further alienates students from the deeper truths they should be learning.

(“Hey, little Picasso! Get that eye on the right side of her head, or it’s detention for you!”)

A real-life analogy would be if, say, you had a growth on your skin. If you went to a doctor and it was found to be malignant, it would be pretty clear that something must be done about it, possibly surgery, or else you might die.

On the other hand, what if it was totally benign? It would be entirely up to you what do do about it. If it was a big growth on your nose, you might want it removed. If it were on your back, you might not care. No one should force you to have it removed without a real, pressing reason. And that reason should not be for their convenience.

Can you imagine if the “rule” was, “All tumors must be removed?” That would be ridiculous. Doctors could never get away with that. They are expected to explain reasonable options to the patient.

Teachers should be expected to do the same.

As always, I am not the final arbiter on any of this. If someone knows a reason why the “and” should not be allowed, as the teacher seems to be saying, please feel free to correct me.

“…PS: Don’t get me started on the apparent necessity to teach vital skills like estimation, which are quite simple to the post-adolescent brain with a good foundation in numeracy, to children whose thought processes and numerical skills are insufficiently developed to understand or apply the concept appropriately…”

I’m trying to hold myself back from “getting you started,” because once I “get started on this” I can work myself up into a rage. It is NUTS that every child isn’t taught estimation and many other mental skills. I wasn’t, and I suffered for it. I learned these things from (of all things) some magic books, through which I realized that math is nothing like that club that schools often use to beat us into submission with.

It may seem that after all of this pontificating, this answer may be less than satisfactory. I think that’s the point – there is no “satisfactory” answer for this for all cases, that is why it is just plain wrong to “forbid” an answer that works. Maybe we can prefer one over the other in certain cases, but why make an unnecessary law? I think the only reason we do that is that we are an obsessively rule-based society, and that doesn’t help us much in the long run. Rules are fine. Obsessing on them – well, that is a slippery slope into anal-land.

So, in a nutshell:

I’m hoping your child simply didn’t entirely understand the teacher. Otherwise the teacher should either give a satisfactory explanation for the “rule” or re-assess if it needs to be a rule.