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It's not you. Algebra IS hard.

When I was a child in Baltimore there was one excellent public high school with a Science and Engineering curricula, but it was restricted to boys. One day, a girl (KS) sued the school district to force them to allow girls to study math and science. Because of her, Baltimore Polytechnic Institute (Poly) opened its doors to girls, and the next year I was admitted.

Poly has a long established tradition of sending its students to some of the top engineering universities in the country, and I was no exception. I was one of 79 students in the school of over 1,000. In the Advanced College Preparatory track, there were less than a dozen women in the class of 360+. Few made it through the vigorous program, but even the girls who dropped down to the College Preparatory track have done well in life. With my Poly education, I was admitted to Carnegie Mellon University while still in the tenth grade, though I did not attend until after I completed the eleventh grade. At the time Carnegie Mellon was rated #7 in engineering overall and #3 in electrical engineering.

Thanks to my gift of mathematics (which was handed down to my from my oldest brother, who has degrees in both electrical and mechanical engineering and who is also a Poly-grad), I am easily employable, have enjoyed a wonderful life, and find few things difficult. Moreover, when I applied to Harvard University, although I was quite sure I would not be accepted due to my lack of experience in the area I was applying (public policy), Harvard selected me based solely on the strength of my math and science.

The kicker --- I HATED ALGEBRA!

Worst course ever!

Still the only class that made me cry.

Why? Algebra is hard! Algebra teaches you to think differently. Algebra is so crucial in higher mathematics that even if it takes three tries or more, it's worth it. As used to tell my students over at the Tiger Woods Learning Center, if you can master algebra, you can master mathematics.

(Calculus is much easier than algebra, especially if you understand algebra and trigonometry.)

I am a John F. Kennedy Fellow, Science, Technology and Public Policy and former Navy Missile Engineer (rocket scientist) thanks to mathematics, physics and chemistry.

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Karen Rozier's picture

Best advice my brother gave me

Keep working the problems until you understand the concepts.

Mauri's picture

And as I see, you did a good deal of it!

Merry welcome, Karen!

Your post made me remind my first day at university, as a student. Milan, 1982 (late Permian, or something alike).

"Let's define a Composition Rule as a mapping of the Cartesian product of a set with itself to itself"

??

The professor, an aged lady, looked perfectly serious hen saying so.

I vividly remember that I turned back (I see no better than a mole, so I got a place near the blackboard hoping the signs written there, seen from a close distance, would have got a somewhat more intelligible meanings - they *did* eventually, but after no less than one year!). I glanced to my coursemates, and remember did see a void expression not different from mine.

Meanwhile, mrs. Marchionna was telling us something like "Oh, no, don't be afraid. It's so simple! It's almost automatic. You just have to learn the trick, the simplify this with this, then consider that ..., then afterward notice that... . Simple, isn't it?"

Incidentally, prof. Cesarina Marchionna Tibiletti was a genius. She was one of the founders of abstract algebra. Maybe not so well known outside of Italy, but in the same class than Emmy Noether, and three or four in the world.

I was so lucky!

But also terribly, irreversibly stupid in comparison.

Don't know why exactly, but I too did never had a good feeling with algebra, even with a teacher like Cesarina.

But even today, almost 30 years later (!!), some of her kind suggestions return to my mind.

"But, guys and gals", once she said just in the meanwhile of a complex lecture, "do you know how to negate an expression?"

She didn't wait for an answer. Our lost expression was enough, I imagine. And she told us. Twenty second, more or less. Now, when I have to invert some complex boolean expression in my awful Fortran programs, invariably I see her in front of my mind, sweetly and calmly explaining what to do, step by step, in absolute clarity.

We get many gifts, during our life. And sometimes, we don't even realize the seeds we had been given.

Later on, however, they germinate. Sometimes they become just some nice weed. In few cases, they evolve in huge baobabs. But they change us.

It's very interesting to learn how a very gifted young student, given the right opportunity and the stubborn determination to proceed, may grow in a fulfilled adult.

I'd like to know from you how it was. Not so many people had to pass this - I feel it may be an ordeal. I mean: you begin very successfully, but life asks you to change. And then, you have to "learn to unlearn"... I feel how difficult this might be.

(Fortunately for me, I wasn't that so brilliant in my early Devonian, so I guess was spared this experience ;-) )

Yo raise another very interesting point. Analysis looks somewhat "simpler" than algebra.

As I've seen with some of my friends, this is very personal. Some people, like me, felt in deep love with discrete mathematics and graph theory. Some others, as you, had a much easier grasp with analysis. I even know one, who was a born algebraist.

Yet, it so often happens there is something in the whole realm of maths which attracts you deeply. Which, in a way difficult to explain, seems "congenial".

What's sad, you know of this your personal space only very late, during college, or university.

How wonderful would it be, discovering this earlier!

But this isn't so simple. At least here, in Italy, when I got my high school maths courses the sequence of arguments was very rigid and, now I realize, terribly narrow. In the first two years, I hated maths cordially. Got very good scores in geometry, mainly because I encoded all graphical-type problems in terms of logical, verbal rules (a strategy which, it turned out without any merit from me, proved much more effective than just trying to fidget with rotating and shrinking triangles). But, I wasn't touched.

I'd like to learn from you what do you mind by "understanding". My definition, I realize, is quite emotional. If I get a set of assumptions and check out logical passages until some conclusion, I feel not having understood anything. My concept of "understanding" is more akin to becoming able to "speak to the problem", and being able to follow its answers. An assimilation, at intuitive level - which requires more time than the usual bosses tolerate.

And so, we end up feeling a strong connection with some very specific subjects which, surprisingly, "move" us.

I wonder what might happen, if young people is exposed to both the normal math learning programs and some glance of what mathematics really is. To teamwork, too!

Maybe, giving some of these glances could be a valuable thing we may say, and others might find interesting, too.

Last: one of my (latest) appreciations on maths is it's a highly connective discipline. It happens to me, in my actual professional life as a geologist, to see directly how strongly the Nature is connected, and how many of these connections occur behind the scenes. A bind threads together the evolution of life forms, the properties of the atmospheric boundary layer, the water/oxygen/carbon dioxide/nitrogen cycles, and many other things... Very similar connections manifest within mathematics, and among maths and "the world".

Sometimes surprisingly.

Quoting your nice image, I remember my astonishment when first seeing a simple formula:

i * pi
e = -1

Four of the most important constants in maths all tied beyond (my own) intuition... Isn't it cozy? ;-)

So, mathematics is not only an effective model-building tool to interpret the physical world. It also reminds some nice properties of it.

Love.

Mauri

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