kitchen table math, the sequel: help desk - "the minus sign"

Friday, January 18, 2008

help desk - "the minus sign"

Just spoke to C. who says he got the extra credit problem right on the test except he messed up the minus sign.

Messing up the minus sign is a chronic issue around these parts.

In the old days, C. would say things like, "I just forgot the minus sign!" Then I would say things like, "Well, the difference between plus-70 degrees and minus-70 degrees is the difference between life and death" and we'd go from there.

The other day I came across this post to Math Forum that made me think back fondly on those days:

I have wondered if anyone knows of a notation for negative numbers
> > that would help students avoid the crippling error of thinking that
> > they are just slightly flawed positive numbers. Have you not heard
> > a student say somethings like this," the solution of 2x = -6 is
> > 3 --oh, I mean negative 3.

Math Forum

Any more C. seems to take minus signs more seriously. But he's still losing track of them somewhere between Point A and Point B.

Any suggestions?

46 comments:

SusanJ said...

I have no idea if this would work but what about putting the minus sign over the number instead of in front of it?

VickyS said...

Put parentheses around the negative number or expression:

(-1) or (-x)

Catherine Johnson said...

I knew you guys would have ideas!

I was already thinking about parens...I'm going to bring both these things up.

I think Steve (or someone around here) was saying that careless errors aren't so careless....they're meaningful.

So I'm going to carry on pointing out the life-and-death quality of positive & negative, but in the meantime he's definitely just losing track of the damn thing.

Thanks!

concernedCTparent said...

While this isn't exactly related to your issue, I found it on PurpleMath and could resist posting it here:

Letting "good" be "positive" and "bad" be "negative", you could say:

*good things happening to good people: a good thing
*good things happening to bad people: a bad thing
*bad things happening to good people: a bad thing
*bad things happening to bad people: a good thing

Anonymous said...

But what if they really are flawed positive numbers?!

I tell ya, there's nothing natural about negative numbers.

concernedCTparent said...

Of course, referring to negative numbers as "bad people" or "bad guys" doesn't quite communicate the life-and-death quality, does it?

SteveH said...

"I just forgot the minus sign!"

It depends on whether it's a silly mistake (like a typo), a conceptual mistake, or an indication that the student is being careless. Too many "Oh, yeah" remarks add up to bad math and bad grades. I remember teachers being quite strict about not doing too many steps in our heads and writing everything down neatly. A teacher should be able to decide how serious each mistake is if they've had the student for a while.


As for a minus sign, I like to think of it as a factor of (-1), so

-5 = (-1)*5

x - 2/3 = x + (-1)*2/3

This doesn't always make life easier, but you have to see that things work out exactly the same if you do this.

I like to look at expressions and equations as a series of terms, where the minus sign "belongs" to the term that follows. Then I look at each term as a collection of factors (things multiplied together) in the numerator and denominator. If I don't see a denominator, I can create one by putting a 1 there.

If I had this expression:

5x - 2x^2/(5y)

I see in my head:

(5)(x) (-1)(2)(x^2)
------ + ------------
(1) (5)(y)

I can move each factor to the opposite side of the dividing line if I change the sign of the exponent. If you don't see an exponent, it's 1.

If you prefer, you could think of (-1)(2) as (-2), but you have to be careful.

I worry about saying that:

-x = (-x)

which is perfectly fine, but you can't do this:

-x^2 = (-x)^2

That's why I like to change the minus into its own separate factor of -1, so

-x = (-1)(x)

and

-x^2 = (-1)(x^2)

There are so many places where you can slip up. That's why I say that it took me until my junior year in high school (Trig) before I felt that I really was good at algebra.

Catherine Johnson said...

Steve (and everyone) - do you have a sense of how to "diagnose" whether you're seeing "careless mistakes" or something more important?? (Haven't read your comment yet - jumped in after reading the first paragraph.)

Chris is now telling me, pretty routinely, that he made careless mistakes.

I always respond by saying that you don't make careless mistakes when you're fluent in the procedures, which isn't quite true, but is true enough and important enough that I'm going to carry on saying it.

We're having a bit of a Math Renaissance around here. For the first time in over two years C. is developing confidence in his ability to study and learn math; today he wanted to study for the second half of the midterm, which he'll take on Tuesday.

He's telling me, "I want to get an A on the test."

He has NEVER said such a thing before, and in fact it would have been crazy to say such a thing since it took blood, sweat and tears on both our parts just to keep him in the B range.

The silver lining in the he** we've been through with the math curriculum and teaching here is going to be that he's seen firsthand, because he's lived it, that work and persistence pay off.

AND: you can't worry about what other people think unless those other people know what they're talking about.

In that case you listen and learn.

Catherine Johnson said...

Just noticed the end of Steve's comment, about changing the negative sign into it's own factor of -1.

I discovered that (I really did - no one taught me) back in high school and used it all through h.s.

I've found I don't need to use it now, but when I was first teaching myself algebra I did. (Actually, I still do this when factoring a trinomial.)

Will read closely, then be back..

Catherine Johnson said...

But what if they really are flawed positive numbers?!

I tell ya, there's nothing natural about negative numbers.


That's what Chris says!

You probably never heard Susan S's story -- right.

When he was 4 he came into the kitchen looking dark. She said, "How's it going buddy?" (something like that).

He fixed her with a look and said, "I know there are negative numbers."

Catherine Johnson said...

That's my diagnostic for mathematical giftedness.

Kid figures out there are negative numbers when he's FOUR, AND HE'S MAD AT HIS MOM FOR KEEPING IT A SECRET.

Anonymous said...

But let me just say that he has made many, many mistakes since then by putting the wrong sign at the end. We had a lot of problems with that a couple of years ago.

However, He has finally stopped making them so much, probably due to this being his third year of algebra.

SusanS

Anonymous said...

If they're careless, he should be able to find them himself, and not just by your facial expression when he gives an answer. There are all sorts of reasons to check over work you've just done, but maybe this [in combination with wanting that A] is motivation for it -- upon finishing a problem, he should go back through it looking for sign errors, because he *knows* by now that it's something that's tricky for him.

SteveH said...

Since I work with my son, I can look at his tests and (uusually) tell the difference. As anonymous says, I tell my son that if he wants to get A's, he has to go slowly and/or double-check his work. Being methodical is a hard thing to learn.

For a problem like:

2x = -6

x = 3

it's probably a "typo" kind of mistake.


For this one:

-3(x - 5)

= -3x - 15

It could be a dumb mistake (going too fast), or it could be a conceptual mistake. My son gets annoyed when I go over his tests. He says that they are dumb mistakes; that he really knows how to do them, but I give him more problems to prove it. Maybe that's extra incentive for him to be careful on tests.

SteveH said...

Speaking of dumb mistakes, I always reread my messages and I "uusually" catch mistakes. It's amazing how I can read without seeing. At least that error screams typo, not stupid. (this time)

Catherine Johnson said...

he has made many, many mistakes since then by putting the wrong sign at the end. We had a lot of problems with that a couple of years ago.

This is your gifted son - yes?

Interesting.

Saxon always talks about people's proclivity for making mistakes; he clearly sees mistake-making as a constant peril and a natural human behavior.

I make mistakes all the time -- there are days when I'll do the same problem 5 times and still not know what I'm doing wrong.

I find that fatigue is a HUGE creator of dumb mistakes. There've been a couple of times when I had to do C's homework set (so I could check his answers & have him re-do) and I was so tired it took me 3 to maybe even 5 times as long to do it as it should have.

These are algebra 1 problem sets, and I'm reasonably proficient at algebra 1 at this point.

At least I think I am.

Catherine Johnson said...

Steve is absolutely right about how hard it is to LEARN to go slowly -- at least for boys.

I've seen this with possibly all the boys I know: they WHIP through their homework then they're done.

One friend of mine is now pretty angry with her son about it (his grades are dropping)....but my sense is that "this is boys" or "this is the age" or both.

C. is a pretty well-motivated kid and he's quite conscientious, but "going slowly and checking" is not REMOTELY natural to him.

He brought this up today, in fact. He's in the middle of a two-part midterm and thinks he didn't do as well as he liked on the first part.

Good news: "not as well as he liked" now means a B instead of an A.

Today he actually said something like "I need to go more slowly and check"....

Catherine Johnson said...

One of the private schools we went to, Brunswick in CT, still teaches boys and girls in separate math classes. (Brunswick is a boys school but now does all of its classes except math with the Greenwich Academy, a girls' school just down the street.)

The admissions officer said that one of the reasons for this is that boys want to tear through each problem, get it right or get it wrong, and tear on to the next one.

If they get it wrong, their attitude is, "Oh well, there's more."

The girls, he said, want to know what happened -- they want to MULL.

That's the problem with the female mind, of course.

WAY too much mulling.

Catherine Johnson said...

The admissions officer did not actually say "girls like to mull."

I said "mull."

I think he said girls like to understand the problem.

Catherine Johnson said...

He also uttered the phrase "cultural literacy" in passing.

Catherine Johnson said...

We pretty much felt like we'd died and gone to heaven.

Catherine Johnson said...

I uusually catch mistakes, too.

Anonymous said...

The girls, he said, want to know what happened -- they want to MULL.

That's the problem with the female mind, of course.

WAY too much mulling.


Hmmm, don't know about that one. I'll have to think about that.

Liz Ditz said...

Plus: the use of a number lineThe darling dyslexic daughter (ddd), encountering algebra, with signs and exponents. She learned to take her time in examining problems, and use several different colors of highlighers.

Color #1: reserved for unexpressed sign (+)
Color #2: reserved for written negative sign (-) and answers that must be negative
Color #3: reserved for exponents
Color #4: reserved for roots expressions.

Before even starting to solve a page of problems, the ddd was taught to examine the problems with highlighter(s) at the ready, and to predict which would require positive answers and which would require negative.

Later on, an equation might have all four colors.

The tutor scaffolded by putting a slash of each color on the top of the assignment, writing the meaning of each color within the slash of color.

I think I've forgotten the drill about addition and subtraction of negative and positive numbers -- it was a long time ago -- but it involved a number line (The one I used with all 3 kids--I made one using a strip of oilcloth-- it went from -20 to +20, and had boxes not just slashes to delineate each integer).

Plus it had a "hint" at the top, like this:

Minus (subtract, less than) always goes <--- this away

Plus (add, more than) always goes ---> thataway

Catherine Johnson said...

Years ago I read a great study by....hmmm...I've forgotten her name. She was one of the first people to do a lot of work on the psychology of women.

Supposedly they had done a study of women's brain waves during sex (and of course today I ask myself: how exactly did they pull that off?)...showing that they didn't turn off their brains even during sex.

I would take that with a large grain of salt, but....otoh I wouldn't dismiss it out of hand, either.

Catherine Johnson said...

I wonder if colored highlighters would help Chris??

I'm definitely going to bug him to do the parens.

Writing -- physically taking a pen in your hand and writing things down -- is a HUGE help.

Since I'm now the Earth Science teacher, I've been fighting my way through C's incomprehensible Earth Science text.

Last night I finally realized we're going to have to go the index card route -- first time in my entire life I will have created index cards to memorize. As I was creating the cards I realized how much of the meaning of the text I'd missed just reading AND REVIEWING the thing.

In other words, I didn't just run my eyes over the words of the text.

I read it closely, I answered all the questions, I reviewed all the terminology, and I studied the illustrations -- and I still missed meaning. (LOUSY, LOUSY BOOK)

As soon as I had to write things down I got more out of it.

Catherine Johnson said...

I'm trying very hard to see this as an opportunity to learn Earth Science.

This would be easier if the teacher AND the department chair hadn't responded to our request for help by telling us that he can't think inferentially (the teacher), he "didn't do that well on the CTBS" (the science chair, who had misremembered his score as being significantly lower than it was), and that the other kids are doing fine and can think inferentially.

It wasn't until after the meeting that we learned that C's latest test grade (81%) is in the exact center of the class which, thinking inferentially here, presumably means that other students are having trouble learning Earth Science from this teacher, too.

In the wake of this exchange, every minute I spend teaching myself Earth Science so I can teach Earth Science to my kid is causing me to seethe.

Seething is the opposite of being grateful.

Catherine Johnson said...

I became even less grateful yesterday after learning that the median teacher salary here in Irvington is $88,000.

You have to put this in context with the fact that the median teacher age is approximately 12.

The Earth Science teacher was certified to teach Earth Science the summer before last.

This is her second year teaching the course.

But the problem is in the kids.

Each individual kid, one by one.

Each individual parent who contacts the school will be told, one by one, that his or her child isn't up to the rigors of a conceptual course.

I talked to a fellow mom yesterday - I've always liked her kid - she said she's spent years being told her kid is the only one having a problem. YEARS.

Then she said, "Your kid is never the only one."

That almost has to be true by definition. There just aren't that many outliers on the planet.

Catherine Johnson said...

*good things happening to good people: a good thing
*good things happening to bad people: a bad thing
*bad things happening to good people: a bad thing
*bad things happening to bad people: a good thing


I love it!

Years ago Ed and I had a scheme to write a book called "When Good Things Happen to Bad People"

le radical galoisien said...
This comment has been removed by the author.
le radical galoisien said...

An elementary approach to vectors (a very simple one) would probably attach some greater importance to signs.

In the example of 15=-3x the sign doesn't really appear to matter, because the variable seemed negative to start out with. Sort of the way one might do physics problems -- if you're always heading in one direction -- e.g. down, then it's not important that the down direction is negative. If the negative sign is really required, take it out of the problem and add it back at the end.

Hence the natural train of thought for children might be similar: strip away the negative sign, perform the magnitude operations .... and add the negative sign back in as an afterthought.

It's hard to imagine a multiplication operation which might change that -70 degrees to a +70 .... unless you created for example, a fictional extrasolar planet where the temperature at point Q on the surface, due to the lack of atmosphere, might fluctuate along the equation 250*sin(t/p), where p would be the length of day. [For fun, one could toss in a parametric equation, I suppose, for aphelion/perihelion. I won't even get into the idea of binary star systems.]

I am unsure of the suitability of trig functions for algebra 1, but basically there needs to be examples where multiplying by a negative number has some meaning.

Another example would be simple (point-form) population growth / decline. A deer population size P decreases or increases by x% depending on the month -- e.g. 5% during February and 25% during April (or whenever the birthing season begins), 0% during June and a steady decline until winter, where it might be -15% in December. Tada! A point-form periodic "function" for rate of growth. A question might be posed like this: if a population of deer in a national park numbered 10,000 in January, what is its population by the end of March?

Suddenly the sign becomes more than an afterthought, because it determines whether you add to the population or subtract from it.

It's also a perfect percent problem. If for example, the population of 10,000 deer at the end of December declines by 5% during January, but increases by 5% in February, one would have to realise that one doesn't go back to 10,000 deer, but 9975. [And yes, I used 10,000 so I wouldn't get problems like having 3/4 of a deer.]

(Because the model isn't necessarily calibrated for equilibrium, one might also figure out whether the population increases every year or decreases, or add factors like human interference...)

Anonymous said...

Minus signs are problems for good math adults as well. The difference between -3 and 3 is just a sign, and before anyone starts yelling and screaming, which is negative and which is positive is often conventional.

Acceleration at the Earth's surface due to the Earth's gravity? 9.8 m/s/s or -9.8?

Oh, wait, I was considering "down" positive...

Anyway, I check my work, and I check the signs in my answers. Because I won't stop making sign errors.

Jonathan

SteveH said...

Jonathan makes a good point. It all depends on how you set up your coordinate system. I remember all of those physics problems where you drop a ball or shoot an arrow from the top of a building. Is zero at the top of the building with positive going down, or is zero on the ground with positive going up? It doesn't matter as long as you're consistent.

But, of course, the solution to 2X = -6 can't be +3.

le radical galoisien said...

But it's rather the same thing in terms of thinking process, isn't it?

Strip away the negative, do the magnitude (scalar) operations -- and at this point students might forget to add it back again.

The solution x=3 isn't consistent with the equation, of course, but at this point it seems the concern precedes their application. Which seems to be the case a lot in mathematics.

As a parallel, I remember when I used to add the +C in indefinite integration as an afterthought. It's not that I was careless -- or maybe I was -- but I was usually so conceptually involved in something like doing u-substitution that the constant was the last thing on my mind.

When did this habit stop? When I realised in differential equations that the +C was actually quite important to some of the solutions.

A problem where the minus sign has true meaning -- e.g. not merely how much money you owe, but whether you gain or lose money -- would induce students to look more carefully at their answers.

Ben S said...

I always used a pencil and a rubber and simply rubbed out the minus signs whenever two cancelled as I went along. Or if there was an integral with minus signs, I'd always swap the limits over so there were the least number of minus signs involved, etc
Or a similar approach would be to write every line of working twice. One the original with all the minus signs, and a second the simplified version with minimal minus signs.

Just at every possible point you can make it so there are the least number of minus signs and it's not confusing at all.

Catherine Johnson said...

which is negative and which is positive is often conventional

good point

Catherine Johnson said...

I always used a pencil and a rubber and simply rubbed out the minus signs whenever two cancelled as I went along

You know what they teach the kids to do now??

If there are two minus signs in a row they have them draw a vertical cross through each one, turning them into positive signs.

I thought that was brilliant.

It works, too.

In fact, it works in the very beginning, when a kid is having a hell of a time trying to figure out negative numbers and do operations with them.

Anonymous said...

Oh, wait, I was considering "down" positive...

LOLOL. We got in a protracted discussion last night about how or if Hilbert's axioms were necessary for a geometrical definition of greater than or less than.

So, based on an earlier topic on this board about how Saxon kids couldn't explain why something was greater than/less than I had my son define greater than/less than. Sure enough he said, "If a number is to the right of another number it is greater than the first number." (And this is how Jacob's and Foerster's "defines" inequalities) I had him illustrate this with a numberline and then I flipped his paper around 180 degrees so that all the positives and negatives were now reversed. He called me a smartass to which I retorted, "I'm pretty sure that's what math is all about--being a smartass."

PS We arrived at no good solution to our problematic definition of "greater than" and left it up in the air.

It's all wonderful to say that it has to do with where it falls physically on the numberline but in order to place the number on the line to begin with you have to know what numbers (greater than/less than) to place it between. It ends up being circular.

My interest in this is that I wanted to add something to the student's knowledge of inequality in algebra that he didn't already have in the first grade. "The bigger number is on the right!"

The algebra book that we are using more or less defines greater than as, "a is greater than b if a-b=p where p >0" (Circular definition)

It defines (actually it prove that)a negative number as the additive inverse of a positive number and trains the kids to think "additive inverse of" whenever they see a negative number. They don't learn that a negative times a positive equals a negative, they learn that "the product of a number and negative (additive inverse of) one is the additive inverse of that number" and so the multiplication of two negative numbers is thought of as a special case of that principle and not as some separate rule.

Doug Sundseth said...

Galoisien^-1's recent comments remind me of a math joke:

Two male mathematicians are in a bar. The first one says to the second that the average person knows very little about basic mathematics. The second one disagrees, and claims that most people can cope with a reasonable amount of math.

The first mathematician goes off to the washroom, and in his absence the second calls over the waitress. He tells her that in a few minutes, after his friend has returned, he will call her over and ask her a question. All she has to do is answer one third x cubed.

She repeats "one thir -- dex cue"?
He repeats "one third x cubed".
Her: `one thir dex cuebd'? Yes, that's right, he says. So she agrees, and goes off mumbling to herself, "one thir dex cuebd...".

The first guy returns and the second proposes a bet to prove his point, that most people do know something about basic math. He says he will ask the blonde waitress an integral, and the first laughingly agrees. The second man calls over the waitress and asks "what is the integral of x squared?".
The waitress says "one third x cubed" and while walking away, turns back and says over her shoulder "plus a constant!"

Joke found here:

http://www.math.utah.edu/~cherk/mathjokes.html

Catherine Johnson said...

Sure enough he said, "If a number is to the right of another number it is greater than the first number."

That's about as far as I've gotten...

Catherine Johnson said...

Hey, Doug!

What was that line you told me - it was a name for a going-away party people throw in the military when they want the person to go away.

A rotor party??

Something like that??

Doug Sundseth said...

Ack! An egregious math error in a math joke post. That should have been "Galoisien^1/2" of course. I shall now hang my head in shame.

...

OK; done now. I blame the pernicious power of the minus sign. 8-)

Catherine: It's called a "wheels-up party". The party begins when the wheels on the honoree's airplane retract into the body of the plane.

It's actually more of a "we're happy you've gone away" party than a "we want you to go away" party.

concernedCTparent said...

What do you call a "we want you to go away" party? I'd like to throw one or two of those.

Doug Sundseth said...

I think those are called "demonstrations".

Catherine Johnson said...

no, no, no!

You had a name for it - it was a military term.

I swear!

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