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May 29, 2009

In Praise of Mathematics

Via Bird Dog, this interesting quote by Harvard economist Greg Mankiw:

Your math courses are one long IQ test. We use math courses to figure out who is really smart.

Not sure whether I agree with this or not, but more on that later. Full disclosure here. My Dad majored in math at Dartmouth. My brother has a PhD in number theory which, although it has practical applications, is more what I think of as pure or theoretical mathematics than applied math. His eldest daughter did quite well in math in high school and is currently contemplating majoring in finance.

The blog princess, on the other hand, had little use for math in high school. But then she had little use for most subjects, the allure of boys and generalized mischief having exerted a greater pull upon her time and attention than academics of any kind. But if she could have been said to display aptitude for any one thing, that subject would have been language: English composition and grammar, foreign languages (she took 4 years of French, 2 of German, three of Spanish in high school and leapt into a mad infatuation with Russian during her brief flirtation with college).

I was ahead of my grade level in most language arts classes but behind my private school peers in mathematics. Somehow I managed to graduate without a single Calculus class, but did quite well on the SAT Math section despite consistently having taken the easiest math courses offered.

It wasn't until I returned to finish my undergraduate studies at the age of 30 that I discovered I rather enjoy math.

I spent a good 3 months before enrolling in my first classes working my way through a College Algebra refresher course at the dining room table. At first it was quite difficult. I'd been out of school for 12 years and had mostly coasted on my natural problem solving skills in high school. As a result, I had learned few of the formal rules that govern algebra. It was acutely painful to discipline my wayward brain to go painstakingly, step by step through problem after problem without indulging in intuitive leaps to the answer that breezily dispensed with the bothersome necessity of showing my work.

Consequently I was not only an intuitive, but a lazy thinker. Years of prodigious reading and a near perfect memory left me weak in formal reasoning skills. All too often, I "leapt into" the answer, only to find myself utterly unable to explain how I'd gotten there (much less why that particular answer was correct).

When I embarked on my adult studies, I planned to major in the liberal arts - my natural strength. But by the end of my first semester, I found myself drawn to more quantitative subjects. There were several reasons for this:

1. English and political science didn't challenge my mind the way math did. Both subjects played to my natural aptitudes, and so neither was a stretch.

2. In the humanities and social sciences, the answer never seemed to be the answer. Oddly enough for someone who normally values the open-ended over the concrete, I found this squishiness annoying.

3. A large part of the reason I returned to school lay in the idea of becoming a more well rounded thinker. I felt lopsided, and college would force me to take subjects I would never have bothered with on my own.

Interestingly enough, I ended up tutoring and leading supplemental courses in College Algebra, Elementary Probability and Stats, and Business Law. All three were interesting subjects that imposed a disciplined reasoning process over the analysis of information. One of my economics profs led a series of after class sessions that covered the same macro- and micro- theory, but from a math based rather than chart driven perspective (i.e., we examined the formulas behind the theories rather than taking the more traditional approach that illustrated economic concepts by way of graphs).

I was mildly shocked to find that I understood most theories better after having gotten into the math. Part of this may be because I am one of those morons who (like the blonde who puts WhiteOut on her computer screen) often had to place her fingers on the axes of a graph in order to grasp the relationships between two variables. I could "see" that relationship better when it was expressed via an equation than when it was depicted on a chart. Low structural visualization. So much for my nascent interest in engineering.
Anyway, interesting post. I think we concentrate far too little energy on good mathematics instruction because it's "hard" and in the warm/fuzzy politically correct atmosphere of academia, teachers are somehow supposed to perform the Vulcan Mind Meld on lazy students -- anything, rather than asking them to exert themselves to understand difficult subjects. Despite having some aptitude, math never came easily to me. Or perhaps I was just so good at language that math seemed comparatively difficult.

Either way, there is something to be said for struggling to understand and fully master a subject. I'll never forget my first Calc class. The prof was an egotistical ass, but he ended up making me so angry that I would have died before giving him the satisfaction of watching me fail (and that was his stated expectation - that most of us would fail the class).

I completed the class with a 96 average. Smart man. Not terribly likeable, but extremely shrewd :p He also turned out to be quite willing to help me when I got stuck on differentiation with respect to two variables. All I had to do was persist.

What a concept. At any rate Mankiw's observation on the correlation between math competence and IQ probably holds true on his level. But in lower level courses, I think success in math has more to do with the willingness to put in the time and effort needed to succeed. I tutored Algebra, Stats, Probability and Calc I and II in college. The people who succeeded were nearly always the ones who didn't give up. Even if it took them two tries to pass Calc. I also found that because math knowledge relies more upon the accumulation of learned theory/skills, it's damned hard to keep up unless you have the right foundation going in. Passing Calculus has far more to do with your basic Algebra skills than it does with being a huge, pulsing math brain and if your Algebra is weak, you'll have a tough time keeping up. For this reason, I often advised students who were failing to drop the class and sign up for a good, basic Algebra course (even if they'd already taken it) to refresh their basic skills.

Preparation and effort can trump natural ability, at least in lower level math. Most male students in my classes showed far more natural aptitude than I did. I nearly always outperformed them, but it was a matter of effort and not ability every time.

Posted by Cassandra at May 29, 2009 05:58 AM

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Comments

GEZACKLY!!! OMYGAIA! I stayed up most of the night forcing my mind through the sequence of a particular accounting step and once I got it, the rest of it came easily. Duh.

I did get an A in college math...and I am rather proud of that simply because majoring in fun is what I did in high school too. After all, it was the easiest subject...but I also enjoyed literature and languages.

Can the female mind be taught to reason?
Yes, but we won't like it. It hurts. There are synaptic misfires and hormones involved, not to mention blank stares at walls while mumbling to oneself. It scares the CLUs.

heh.

Posted by: Cricket at May 29, 2009 07:20 AM

*snort*

re: Can the female mind be taught to reason?
Yes, but we won't like it. It hurts. There are synaptic misfires and hormones involved, not to mention blank stares at walls while mumbling to oneself.

Interestingly enough, I tutored mostly women in math.

I think women understand math differently than men do. I had to use different techniques to explain mathematical concepts, but found that if I was versatile enough, women had no more trouble with math than men do. In fact, they often excel at actually understanding what the helk they're doing, as opposed to a lot of guys, who don't care why something is true - they are more results oriented.

As I've gotten older, I have wondered whether training women to be good at math doesn't involve teaching us to actually turn part of our minds OFF rather than learn new skills. IOW, our natural thinking style is more integrated (think the men vs. womens' brains vid I linked a while back -- for women, evvvvvverythang is connected, while men put things into isolated boxes that don't touch). So math success for us involves narrowing our focus to exclude unhelpful thinking styles and suppress the desire to relate math to, say, shoe shopping :p

Anyway, interesting idea.

Posted by: Cassandra at May 29, 2009 07:34 AM

Another interesting male vs. female math tidbit.

Men nearly always waited until it was too late to ask for help. And they didn't much care for asking a woman to help them. I had to spend way more time stroking their egos and assuring them I respected their intelligence even though they were having trouble mastering this particular subject.

But men were also more independent and less clingy. I spent *less* time trying to build their self-confidence: in most cases they had plenty of that. It was just that idea that someone else might look down on them seemed troublesome.

Women, on the other hand, needed more propping up of their self-confidence and didn't feel threatened by asking for help. But they tended to undervalue their ability and intelligence, where men often overvalued the same attributes.

Of course these are general observations and won't fit every person. Nonetheless, people are fascinating.

Posted by: Cassandra at May 29, 2009 07:39 AM

Hmmm, I would think relating math to things like shoe shopping would be helpful.

Say, teaching intersection of lines by teaching them how they can choose a better cell phone plan so they can pay less and still talk more. :-p

*running away*

Posted by: Yu-Ain Gonnano at May 29, 2009 10:41 AM

I hate math. I love intuition. I hate dealing with people who don't see the answer and I have to reverse engineer it for them.

Yet I was very good at Gunnery.

Probably because I got to blow stuff up at the end of the equation. I like that kind of feedback.

Posted by: Come see the violence inherent in the system! at May 29, 2009 10:55 AM

See what I mean? You just have to find the right motivator.

Posted by: Yu-Ain Gonnano at May 29, 2009 11:00 AM

Integration of the spheres? Hm. Logic being what it is, I have found that in accounting, most accountants, bookkeepers and office managers for doctors and dentists are women. I write left-handed, so my intuition is strong (intuition and creativity being a more right-brained function than feminine), but I tend to be either right handed or ambidextrous when it comes to other things, which argues for a left-brain, or logic application.

Is that what you mean?

There was a woman who was in the Navy, a mathematician, who worked with computers. I daresay she designed some of the ones used on the battleships of the late 1940s. I am sure MathMom prolly knows who she is. Some of the most notable contributions to the understanding of computers, math and science were made by women.
I am not denigrating men, but making a point that I think you are right about the overvalue/undervalue aspect.

Posted by: Cricket at May 29, 2009 11:15 AM

Grace Hopper! :)

Watch it on the praising women, Cricket. You'll stir up the women-are-evil-stupid-wrong/bad crowd.

*sigh*

I keep thinking that some time in history some woman must have done something right other than to bear children (which I don't for one moment denigrate) or take her clothes off for folks she's never been formally introduced to.

There's a rumor we even have brains, though personally I don't believe it :p

Posted by: Cassandra at May 29, 2009 11:28 AM

My first college roommate used to try and help me with Algebra. But I would always frustrate the begeezus out of her when I'd ask, "Why?" I wanted to know why you were supposed to do the problems 'that way only'. It felt like I was being programmed like a computer, and I certainly didn't like that. She called me one day -- ten years later -- and said, "I can tell you 'why' now....." She had just finished her Masters in both Computer Progamming and Math. I told her if it took her that long to find out (she is extremely gifted in the math department)......I didn't want to know.

Oh, and I'm putting your number in SWHNOB's cellphone....she'll be calling you next year when she has a math question.
heh
0>;~}

Posted by: DL Sly at May 29, 2009 11:51 AM

Grace Hooper was one heck of a lady...

IIRC she coined the term bug as it relates to computer hardware/software malcontusions due to a moth gumming up the mechanical relays in an early computer on which she labored.

She was also a senior consultant for a big-arsed computer company I worked for long ago and far away.

May her bits be in a common block and at peace.

Posted by: The-Manchurian-bubba_hun at May 29, 2009 11:57 AM

Do to scheduling flukes, in the eighth grade my

daughter had a all female math class (including the

teacher). I think that she learned more that year

than all the other years combined!!!

Posted by: Sine Nomine at May 29, 2009 01:56 PM

I actually think Mankiw is wrong (words that have possibly never before passed my keyboard) or at least only half right. Doing a lot of math tests both intelligence and industry (or whatever term we want to use to describe what you're talking about, Cassandra). No, you can't do math if you're not reasonably intelligent but you also can't do if you are only reasonably intelligent and don't work at it.

As for women and math, I have two theories about that, both of which help explain the whole "girls can't do math" thing that engulfed Larry Summers. First, I suspect that the average woman's math ability matures later than the average man's. I don't know why this would be - testosterone levels maybe - but I offer two very flimsy pieces of evidence to support it. In Summers' remarks he referred to a study that looked at math and science tests for 12th graders and found more men at the "tails" than women. I wonder what would happen if you looked at the same information for men and women at, say, 30 years old. Furthermore, my own experience bears this out. I could not learn calculus to save my soul when I was 17 but at 25 I could master it with some - but not really brutal - effort.

Which leads to my second theory. When I took calculus at 17, I simply took calculus. Not only could I not figure it out, I wouldn't imagine why I would want to. (There's that female "why?" again.) At 25, I needed it for my graduate level econometrics course which I desperately wanted to take because I had fallen madly in love with computer programming. So perhaps women do better with math when they can see a practical use for it.

I often think of an IQ test I took in my early 20s. When I got to the spatial perception part, I looked at the diagrams, looked at the half-red, half-white blocks I was supposed to use to reproduce the diagrams, and I literally felt my synapses start to fry. I was amazed smoke was not pouring from my ears. But give me an assortment of leftovers and an assortment of different-sized plastic containers to store them in and all the leftovers end up in the most efficient container.

Furthermore, I was once honored by a theoretical physics major telling me I had "good spatial perception for a girl" after I stared at his Winston hard-pack for a while and remarked that it was interesting they'd managed to make it from a single, multi-folded piece. And please don't tell me my efforts there had no practical use: he was a very cute physics major.

Posted by: Elise at May 29, 2009 02:00 PM

That's not surprising. Girls (and boys, too!) generally do better in single sex classrooms. But girls in particular are far more likely to excel in math and science when they're in an all girl class. As far as I know it doesn't make a bit of difference statistically whether the *teacher* is male or female. But the student composition makes a huge difference.

All other things being equal, I'd guess a large part of that is that girls suppress "unpopular" interests when boys are around. I remember being annoyed to find that I sort of dumbed down my persona around boys when I was in school. It was an unconscious accommodation - didn't realize I was doing it most of the time.

Boys aren't immune to this nonsense either. Many view smart as equivalent to geeky. Weird, huh?

Posted by: Cassandra at May 29, 2009 02:03 PM

So perhaps women do better with math when they can see a practical use for it.

Bingo. Interestingly, spatial perception is passed down via the mother. My Mom and MIL have it in spades. So does my spouse.

It skipped over me entirely (my brother had it, though).

One of my sons has it, the other doesn't. Genetics is interesting to me. Both our kids look like neither of us particularly. They're really a blend of our features. My youngest looks the most like me, but he has his Dad's eyes and skin and hands.

My oldest boy has my eyes and the red hair. That's about all he got from my side though. The older he gets, the more he looks like his Dad, though the coloring really throws one off (my husband is darker haired and eyed than I). My eyes are brown, but so light they are close to being hazel - sort of golden brown. My husband has those really dark, dark brown eyes. His skin is far lighter than mine though - I have olive tones, though my hair and eyes are lighter than his.

Posted by: Cassandra at May 29, 2009 02:09 PM

When I got to the spatial perception part, I looked at the diagrams, looked at the half-red, half-white blocks I was supposed to use to reproduce the diagrams, and I literally felt my synapses start to fry.

*snort*

I think I scored in the *bottom* 10th percentile in structual visualization :p

I scored high (95th %ile or above) on somewhere between 7 and 9 aptitudes (can't remember precisely). Most people score 75th or above on 2 or 3. Probably explains my randomness - too many aptitudes is not a helpful thing.

Posted by: Cassandra at May 29, 2009 02:12 PM

re: whether Mankiw is correct or not.

Not sure. I think it may depend upon the level of math instruction they are screening for.

I think at the lower levels, industry will substitute for intelligence. But I think at some point one hits a brick wall with math and then, if you haven't the mental wherewithal, you can't brute force it with mere effort. So he may well be correct in using transcripts to infer intelligence. Not sure.

But I also thing that so many other things could also account for success (good preparation, starting early, diligence, etc.) that I'd view the claim with a bit of skepticism. On the otter heiny, if you pass the classes, does it really matter whether you're naturally smart?

The proof, arguably, that you have what it takes to succeed in econ is in the pudding: having successfully completed the math courses.

Posted by: Cassandra at May 29, 2009 02:30 PM

I am to decide on a minor. I am leaning toward math or true insectoid specialization: Actuarial Science. You know, those people that lie awake at night thinking about how to make your life more miserable...uh, manage risk.

Posted by: Cricket at May 29, 2009 06:52 PM

*still waiting for someone to incorporate the tried and true,if not mandatory, on the other hand... into their commentary*

Rote, diligent, hard work and intelligence all factor into the mix...

I wonder how many folk have had Eureka moments occur in the middle of a dead sleep after a problem has had its way with you for hours or days. Elise, as a former applications programmer I suspect that you know what I'm talking about. Anyway, that moment will make you wonder about the power of the subconscious, of Divine revelation, former incarnations or the decision to eat too many habaneros with your Kung Pow Chicken...

Yup, the mind is a mysterious thing. =8^}

Posted by: bt_needs-batteries-for-his-sliderule_hun at May 29, 2009 07:11 PM

I took Algebra I in 8th grade. It was brutal. I scrounged a 'C' out of that class but only because I camped out in that classroom 24/7, had a tutor, and busted my hind end. Geometry in 9th grade was a walk in the park and I passed it with 125% thanks to the extra credit that I burned through like a hot knife through butter.

I was worried about taking Algebra II in 10th grade - because of swim team and other extra curricular activities (and my discovery of a social life) I didn't have the time I had in 8th grade to dedicate to slogging through it. Amazingly I found that it was relatively easy. Not a walk in the park like Geometry but easier than 8th grade. No tutors or long hours necessary.

And then I got to PreCal and life was WONDERFUL. I'm not sure if it was the teacher or the subject matter but life.was.good. I thrived. I soaked it up like a sponge and couldn't get enough. And then it was on to Calc and life just got better.

To this day, I have an AP Calc study guide on my nightstand for those evenings when my brain is craving it. I tutor math - mostly algebra because that's what people seem to have the most trouble with. I helped MacGyver through PreCal and Calc. I'm itching for my own kids to get to the age where they need my help to understand the concepts in their text books.


I. Am a Nerd.

Posted by: HomefrontSix is a nerd at May 29, 2009 09:52 PM

Ok, Cass, you're off the hook.....I'm changing that number to my Wardrobe Mistress.
heh
0>;~}

Posted by: DL Sly at May 29, 2009 11:45 PM

Heh.

Posted by: Sly's Wardrobe Mistress at May 30, 2009 04:54 AM

Yes, bt, I do know what you're talking about. I find the same thing now with crossword puzzles. I drag myself off to bed in despair at how I've become so stupid I'll never figure out another crossword puzzle clue and wake up not only knowing the specific word I couldn't get but able to finish the rest of the puzzle in five minutes.

Posted by: Elise at May 30, 2009 09:46 AM

Sheesh... I missed the summation in your 02:30 PM comment yesterday M'lady, not to mention the On the otter heiny remark that is so appropriate in any discussion concerning economists or when one group or profession uses a set or their set of data to support a who is really smart conclusion. Since I'm not having to fuss with unprofessional insurance adjusters and agents who would rather be playing golf, today I'll attempt to say something serious on this topic.

Not being particularly smart myself, just willing to do a lot of hard work over the years, I have little more than my observations. So please bear with me. Your patience is appreciated.

I have been privileged to have worked with a lot of really smart folk over the years in DC, particularly in our nation's DoD, at Goddard and later consulting for my company with undeniably smart folk from all over the globe. I have seen the smart, the super smart and the average perform in ways that are inspired and in ways that will amaze. I've also seen the same folk act in ways that will make you wonder how they managed to live as long as they have.

In one instance I can say that I've had a fairly close and long term working association with a very bright mathematician. He was brilliant in his areas of study. He was one heck of an analyst. Methodical, focused, attentive to detail and often inspired in new ways of attacking problems. But this same fellow could, as often as not, be seen wandering the parking lot in search of his car. He would, on occasion, pace the halls wondering what he needed to do that had caused him to leave his office. There was a time when, while attempting to introduce some members of his family to strangers, he could not remember his children’s names. And this was not a fleeting moment, it was more of a lost in the weeds for a few moments incident. I've known and worked with more than a couple of folk who were just like him in this regard.

Another fellow you might take on first glance to be anything but really smart could read hexadecimal dumps like I read the newspaper. He could unwind the stack, analyze the problem and often have a code fix outlined before most of the rest of us, including the folks who wrote the code, could find the initiating thread. And just so my point is not missed, most of us could find the initiating process/thread real quick, usually as soon as the debugger could initialize and read in the data. The man may be an average, good old boy in most areas, but in his profession he is unbelievable. The word savant, when applied to this guy, is no exaggeration.

I suppose what I’m driving at is that when anyone attempts to quantify intelligence or stake out the data set that encompasses the really smart person, I trend towards the skeptical position. My experiences with and observations of some pretty smart people do not always square with the data. And as Elise points out, a person who might be tagged with the not so smart –I’d chose adept as the more appropriate adjective- label of today may, with more effort, more maturity, or a different environment, be the really smart person tomorrow.

So, I suppose that I too quibble with Mankiw's comment in point # 4. Your math courses are one long IQ test. We use math courses to figure out who is really smart. I think it’s unfortunate that his view is such, but maybe it’s necessary for a person who has a finite amount of time and resources to lend to the endless stream of people who are seeking his assistance in their quest for knowledge. Performance is the practical, rubber meets the road, gauge of intellect. So when a student does not perform for whatever reason...

I'll hush now by decree of Walkin' Boss and the outstanding items on the task list.

*looks around for proof reader widget, gives up and hits post*

Posted by: bthun at May 30, 2009 09:51 AM

My father loved math. If you shared his interest, he didn't care (or even notice) if you were male, female, human, or Martian. He had 3 daughters, no sons, so no chance to do the bonding thing only with the boys. I don't remember a time when I didn't love math, too, I assume because I just absorbed the attitude and liked the bonding -- though I must say the same wasn't particularly true for my two sisters.

Anyway, when you're a child and the most important person in your life loves math and finds it perfectly natural that you do, too, and assumes that you'll be good at it, it's surprising how little your teachers or classmates will have to say on the subject. I would have been distantly curious, no more, if I had noticed a belief on the part of others that I would find math difficult or unpleasant because I was a girl. And a guy who was troubled by my being good at it couldn't have held my attention even long enough for me to be very irritated.

Posted by: Texan99 at May 30, 2009 02:43 PM

Now I get a bit of time to proof my previous and see that I typed chose instead of choose... not real smart! Ahh well, I'll blame my undocumented proof reader. =8^}

Posted by: bthun at May 30, 2009 05:05 PM

As I've gotten older, I have wondered whether training women to be good at math doesn't involve teaching us to actually turn part of our minds OFF rather than learn new skills. IOW, our natural thinking style is more integrated (think the men vs. womens' brains vid I linked a while back -- for women, evvvvvverythang is connected, while men put things into isolated boxes that don't touch). So math success for us involves narrowing our focus to exclude unhelpful thinking styles

Interesting. Algebra (1 and 2)and Geometry were the only classes I struggled with in high school. They about made me crazy--I couldn't seem to do the work until I understood the "why" of it. I could follow a list of instructions, but I couldn't memorize it or understand when/where the list should be used--my brain just wouldn't stay on the path; I had to understand it globally with all its interconnections before I could break it down to the small problem in front of me.

It usually took me about a week to figure out the why and all the interconnections, but meanwhile I get a week behind in homework and suffer the consequences to my grade. I never understood why I had that problem until I read the paragraph I've quoted here--I think that's why!

Posted by: FbL at May 30, 2009 09:42 PM

I am getting that silver stake...

Or wolfbane.

FuzzyB, don't 'feel' bad. Logic can be learned when you reason it out.

Ferexample: FIFO and LIFO are two of several methods used in accounting to track inventory.

They are acronymns for 'first in, first out', and that means the first inventory that comes in is the first sold.

The second is 'last in, first out' and it means you sell your older inventory first.

Where it gets fun is how you calculate what gets sold and for how much; and how many remaining units you have in inventory based on which method you use.

It does make a difference. I swear someone lies awake at night to figure this out.

While it theoretically has nothing to do with algebra, algebra is essential to understanding accounting because what you do to one side of the equation, you have to do to the other.

Next up: We watch Cricket's head explode as the Accounting Gnomes explain 'Debit' and 'Credit.' It doesn't mean what you you think it means. It depends on what your definition of 'is, is.'

Posted by: Cricket at May 31, 2009 04:10 PM

One day the oldest guy in the Accounting Department retired. He'd worked there forever and had never once screwed up his ledgers: they always balanced perfectly. The whole company turned out to give him a big going-away party, thanked him for his years of perfect service, and told him how much they'd miss him.

The next day his replacement started work. He sat at the old accountant's desk and opened his top drawer. It was completely empty except for one piece of paper which said:

Debits are by the door. Credits are by the window.

Posted by: Elise at May 31, 2009 07:09 PM

"Debits are by the door. Credits are by the window.
Eighteen months later, as the new accountant was testifying at the company's Chapter 11 proceedings, Ralph the night custodian, wondered whether he should have rotated the new fellow's desk 180 degrees so the morning sun would shine in over his left shoulder? Or not...

Posted by: Rod Serling at May 31, 2009 11:55 PM

Bwahahaha...

I used to do a minimal amount of it when I worked at the video store and then at the bakery. What I had to do at the bakery was train the manager (yes, the manager!) and the rest of the employees (about six people in all) how to properly ring up a sale, make change and do a paid out for supplies/ingredients. The deposits had been off, wildy varying between 50 and 100 dollars, either over or short.

I then had to teach the manager how to count out the drawer (none of this 'rounding off the change' nonsense...we had exactly X in there each and every time, regardless of the amount of each denomination). Lo and behold, the sales and deposits started to balance.

I like fixing things up...but math is one of those skills you need, but how far you want to go in it is determined first by what you need it for and your desire to learn it, regardless of your career/job.

Posted by: Cricket at June 1, 2009 01:56 PM

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