It’s so rare that I get to talk about math with other people and I did today, on a friend’s blog. I wanted to capture that here, because I referenced a few things I don’t usually get to talk about.

I’m autodidactic by nature. I always have been since I was a young girl and I spent most of my free time in a dictionary or the Encyclopedia Brittannica or perusing the stunning pictures and reading the articles of National Geographic magazine.

I’m still that way. I am a collector of books, and some of my favorites in my collection are a few antique math books. It’s ironic, because I had so much trouble in school with math. Some things were hard for me to grasp, and I still kept at it, receiving mostly B’s and sometimes C’s. It was the one subject that tended to pull down my GPA.

I had lots of anxiety during test time and my brain would freeze, even though I may have known the homework fairly well. I think I had some mild dyscalculia. I can’t hold numbers in my head long, which made it nearly impossible to do mental math and slowed me down and led to great embarrassment. I felt much better after discovering Dr. Emma J. King – who was a mathematician who can’t add numbers in her head either. She said hello to me once when I blogged about her years ago on one of my other blogs. I thought that was swell.

“Dyscalculia (difficulty in learning or comprehending mathematics) was originally identified in case studies of patients who suffered specific arithmetic disabilities as a result of damage to specific regions of the brain. Recent research suggests that dyscalculia can also occur developmentally, as a genetically-linked learning disability which affects a person’s ability to understand, remember, and/or manipulate numbers and/or number facts (e.g. the multiplication tables). The term is often used to refer specifically to the inability to perform arithmetic operations, but is defined by some educational professionals and cognitive psychologists as a more fundamental inability to conceptualize numbers as abstract concepts of comparative quantities (a deficit in “number sense”[1]). Those who argue for this more constrained definition of dyscalculia sometimes prefer to use the technical term Arithmetic Difficulties (AD) to refer to calculation and number memory deficits.”

For me, I can think of the numbers and write them out, like 169 plus 243. But I’d never be able to hold those numbers in my head long enough to actually calculate them in my head. I’d even forget what numbers they were if I hadn’t written them out. They begin to get ‘fuzzy’ real quick. It’s really made me feel stupid sometimes.

But I found while I have number memory deficits, a lot of it stems from social anxiety related to it. Like trying to calculate tip in my head was impossible in a group of people or trying to figure out percents off when I went shopping when I was younger. I can now do it by myself and because I also picked up some shortcuts they never taught me (or I simply don’t remember learning) in school.

A mathematical savant I’d never be.

But it never stopped me from trying to learn math. I took the 5 credit hour Integrated Calculus and Analytic Geometry for Engineers and physics majors even though as a biology major, I could have taken the ‘easier’ course of Introductory Analysis. I don’t know if I had a glutton for punishment, but I just craved the challenge. And challenging it was. But I enjoyed studying with the cute engineering majors and ended up dating an Electrical Engineering major. I had gotten as high as Multivariate Calculus before I realized I was in way over my head and I dropped the class. However, in retrospect, I should have dropped the boyfriend and kept the class. 😉

And I KNOW don’t teach math right in school. Ever hear of The Mathematician’s Lament by Paul Lockhart?

http://www.maa.org/external_archive/devlin/LockhartsLament.pdf

I felt kind of cheated because there is so much that is beautiful in math that math teachers could have shared with us. I don’t know. The old belief was that girls couldn’t learn math, I guess. It’s still so prevalent that girls STILL believe they are weak in math.

I ran across this article from June 2008 in the U.S. News and World Report that disproves the theory that boys are better at math than girls. According to University of Wisconsin-Madison psychology professor Janet Hyde, the study’s leader:

[We sifted] through mountains of data—including SAT results and math scores from 7 million students who were tested in accordance with the No Child Left Behind Act. Whether they looked at average performance, the scores of the most-gifted children or students’ ability to solve complex math problems, girls measured up to boys. Although girls take just as many advanced high-school math courses today as boys do, and women earn 48 percent of all mathematics bachelor’s degrees, the stereotype persists that girls struggle with math, says Hyde. Not only do many parents and teachers believe this, but scholars also use it to explain the dearth of female mathematicians, engineers and physicists at the highest levels. Cultural beliefs like this are “incredibly influential,” she says, making it critical to question them. “Because if your mom or your teacher thinks you can’t do math, that can have a big impact on your math self concept.”

I became fascinated with numbers and patterns from reading some of Metamagical Themas: Questing for the Essence of Mind and Pattern from Douglas Hofstadter.

***

Somewhere in my learning journey I came across this Youtube short “Nature by Numbers” by Cristobal Vila.

I also loved a kind of obscure novel called The Wild Numbers by Philbert Schogt.

Isaac Swift is a mathematician – not an outstanding one, but a competent, unextraordinary pencil-pusher. And like all mathematicians, he’s constantly reminded that it’s the prodigies of his profession who advance human knowledge. The rest just try to understand. Now Isaac thinks he’s found the solution to “Beauregard’s Wild Number Problem,” a puzzle that has stumped savants for centuries. And Dimitri, his mentor at the university, once a near-great mathematician himself, thinks Isaac is correct. If so, Isaac will have elevated himself to the ranks of the immortals. But now accusations of plagiarism arise, and the threat of violence that may not stop at the intellectual level looms over the university.

I found that book and had to pick it up.

I sometimes get to teach middle school math classes when I substitute. I find myself really enjoying them a lot more than when I went through it myself.

I still have yet to read physicist Paolo Giordano’s The Solitude of Prime Numbers and I think it might be a great time to.

Giordano’s characters are provocative, even disturbing at times, and yet they have a fragility that evokes our sympathy. As Alice struggles to navigate the cruel and arbitrary rules of high school, she reaches out and retreats inward in equal measure, and when she is rebuked by her classmates, she turns to Mattia as her only friend. But while Alice is rejected by the world, Mattia, in turn, rejects the world itself, severing himself from any visible emotional contact with anyone else. He escapes into numbers, replacing the chaos of life with the peaceful structures of mathematics—and yet, even there, he finds Alice. Together they pass through adolescence into adulthood, and their private world expands to include a constellation of characters who love, desire, despise, and ignore them. Clinging together and yet never able to connect fully, Mattia and Alice are forced to question whether it’s possible to unlock themselves from their painful pasts and overcome their deep loneliness by reaching out to each other. With artful precision, Giordano illustrates the bitter beauty of love and loss and how the two extremes are permanently intertwined. His novel is a brutally honest yet generous portrayal of two struggling souls. Mattia and Alice are neither good nor bad people, they are simply human, but they pay a deep price for the choices they make. Complex and compelling,

The Solitude of Prime Numbersis an unsettling look at how the effects of a single moment can reverberate through a lifetime.

I just adore numbers now and have been really glad to have had my daughters, because I have gotten a new opportunity to revisit math and even teach them a few things when they were younger.

I’m also grateful to be talking with adults about one of my geeky passions too.

interesting, i just discovered a year ago, that i also suffer from a bit of dyscalculia. certain math with steps, mechanical processes with steps and especially physical directions/spatial skills, are the bain of my existence. i’ve always adapted and have done fine with challenges along the way, but i feel better knowing there is a logical reason behind it all ) beth

Yes, me too. It was pretty ironic to me, though. I have used math in my laboratory jobs just fine, but when I had to work behind a cash register for my mother once (she had a delicatessen and catering business), if I made a mistake keying in the amount they gave me and I had to make change, I would get nervous and my mind would go totally blank. It really affected my self-esteem for a while. But, I learned the ‘secret’ when I got older. Instead of trying to subtract in my head, take the amount the item cost, then use that as the starting point and then add coins and dollars until I reached the amount they gave me.

I think it was just strange, my sister, who was otherwise disinterested in math or science in school, could make change without a problem. I was in math and science all throughout high school and college, and had such trouble.

I wish I had a name for it back then, because I just felt so stupid and embarrassed.

Um…wow. I’ll throw my 2%’s worth in here.

Once upon a time, not so long ago really, when the sons of man decided that a privileged few of their sons were to be educated, there were established schools where this kind of thing might be accomplished. They were NOT schools as we envision them nowadays. Through the centuries–for various forms of schooling have existed as long as civilizations have existed–these establishments morphed into a particular structure that is especially relevant to the conversation at hand.

I won’t bore you with any in-depth discussion on this. Anyone who is interested can and will research it for her or his self. Let’s just say that in the Medieval Period there were two “levels” of education (for those privileged ones): the Trivium and the Quadrivium. Now, those terms were not used by these early educators or students, they just called it the equivalent of what we say is “university.” Some time later, those wishing to discuss the structure adopted the “Trivium” and “Quadrivium” labels to describe the seven “segments” of liberal arts in classical study or education.

The Trivium provided courses of study (that were not available to commoners) in Logic, Grammar, and Rhetoric. It could be said that this represents respectively 1) a foundation of the known and orderly ways to think about it, 2) the development of symbols that facilitate the orderly discussion of what is known, and 3) a way to teach or persuade others those things that are thought or known.

Those who graduated from this level would commence to the Quadrivium: the study of mathematics, geometry, music, and astronomy. As you may have guessed…bring on the NUMBERS! Briefly put, these disciplines represent respectively 1) the pure number, 2) the number in space, 3) the number in time, and 4) the number in space and time.

Do you see where I’m going here? Is it evident that our present ‘methodologies’ suffer because of poor or non-existent foundation? How can we learn things while we are standing on shifting sand?

I can appreciate the penchant for labeling. But I am not a big fan (to say the least). For instance, why was it not until the middle of the 20th Century that the term “dyscalculia” entered our modern vernacular? For the student of classical languages, this term is kind of laughable. It combines Greek and Latin to say “bad counting.” But it does nothing to address the underlying cause. Everyone is prone to bad counting from time to time, and some of us are more prone, more of the time, than others.

My questions are these: How many of us have been ‘taught’ the fundamentals of Logic, Grammar, and Rhetoric? How many of us have been taught the classical languages (Latin and Greek)? What if these studies are foundational and important if one endeavors to find deeper meaning in numbers? And, then, knowing that the foundation is lacking (because very few of us have EVER studied Logic, Grammar, and Rhetoric (in a systematic way), why would we be surprised that ‘we’ have ‘trouble’ in Math?

I have substituted for every middle school subject except music and art (e.g. English, Grammar, Latin, Algebra, Geometry, Social Studies). I can relate to the social anxiety element of trying to “do” Algebra with a classroom full of students. But the anxiety comes more from my lack of confidence (and years of non-practice) of the processes of Algebra than the “social side” of the equation. There were things that I COULDN’T provide these students, but there were other things that I COULD. Many of them were ‘hung up’ on terminology, and that led to a stilted ability to “do” the math. I took an interdisciplinary approach and invited the ONE Latin student in the class to do some peer teaching with me. We explained to the rest of the class that the terms were not mysterious. For instance, POLY-NOMIAL and BI-NOMIAL mean many-named and two-named. This ONE little thing brought several students ALIVE! Math is no mystery…especially when students have a firm place to stand.

And as for the auto-didactic thing (another ‘mysterious’ term…that was first used in the 18th Century), we are all self-taught. Remember the old you-can-lead-a-horse-to-water thing? Education is about teaching AND learning. It is up to the teacher and the learner to effect the process of learning. I think that’s the ‘trick’…effectively inviting learners to learn. When they accept the invitation, THEN we can teach (with success).

And one other thing (then I promise, I’ll shut up)… There are a whole bunch of math tricks that we can use to make tip-figuring (percentages), conversions (e.g. metric/English), discounts, etc. very easy things–even if we have struggles with dyscalculia.

So, yeah. What if it’s not about dyscalculia, or dyslexia, or attention deficit disorder? What if the LEARNER is NOT disordered? What if the TEACHING is disordered? What damage are we doing to learners by labeling, stigmatizing, and doping these young persons rather than ‘treating’ the didactic process? [Don’t jump down my throat about this. I don’t doubt that there are a FEW people who have organic causes for their struggles, and these persons need proper attention.]

Our current ‘educational system’ arose to meet the needs of the Industrial Revolution, and it is based on industry (a mechanical, factory, assembly line, standardized process). It is ill-equipped to provide personalized care to individual persons. It squelches the two characteristics that are absolutely essential if we have any hope of progress and survival: creativity and adaptability.

I respectfully submit that these struggles of which you speak are rooted more in poor foundation than in disorders of learning–even if we have nice, mysterious names for them.

You have probably seen these videos, but I think they are well worth mentioning here. Ken Robinson makes good sense!

http://www.thersa.org/events/rsaanimate/animate/rsa-animate-changing-paradigms (RSAnimate animation…yes!)

http://www.thersa.org/events/video/vision-videos/how-to-change-education-from-the-ground-up (Robinson’s followup to Changing Education From the Ground UP)

Good day!

TBS

I don’t have a ton of time to respond. I have to take my daughters to my sister’s house.

I guess this means you still read my blog from time to time. =)

I loved algebra. I hated first semester of geometry with the proofs and all, but I loved the second semester of geometry that was all application of geometry. I loved trignonometry. I failed college algebra. On purpose (lots of reasons but mostly just was angry).

I can memorize formulas, and as long as I have a calculator, I’m fine.

It’s simple maths that tripped me up. I can’t do mental math.

I can memorize my times tables and I know how to add and subtract single digits in my head. I can’t hold more than two numbers AND add them up.

Regardless of what you want to call this limitation I have, it means I can’t be a cashier.

It didn’t help that my oldest sister called me stupid and wouldn’t ever let me use the cash register because I made everybody wait while I froze and was unable to make change. I was 18. And “stupid” in that moment.

She said so, I felt it, too.

I can’t be a bank teller. I’d probably screw up.

I can’t be responsible for dividing up the tab in a group at dinner, nor figuring out MY portion of the bill, plus tax.

It’s not that I can’t, without pen and paper. But do you know how many times I have hidden my ‘disability’ from my friends, and worse, professional colleagues?

The teaching is not disordered. I am. I am okay with owning that disabled part of me. 😉

Without pen and paper, and with eyes looking at me and expectations placed on me, I can stare all day long at the numbers on a bill and freeze and be unable to process what I’m seeing.

Add in all the paranoia that they MUST all think I’m dumb. In an instant, the crushing reality came into play how inferior to normal humans I am (well, I felt that at the time). Book smart and common sense dumb (which is what my step-dad was fond of telling me). Or, more accurately, book smart and number sense dumb.

Carl Sagan said,

“The simplest thought like the concept of the number one has an elaborate logical underpinning.”

Yes, well…some of us might be missing some elements of logic.

Back later…

thank you.

I’m not sure that I can say more. I don’t really refute what you are saying. Thanks for the links. I know I’d seen the first one, not sure I’ve see the other. I’ll check it out.

I have the book A Well Trained Mind by Susan Wise Bauer with the intention of homeschooling my children based on the trivium.

You told me once I shouldn’t homeschool, and made me feel that things were so bad between my husband and I that it was pretty much pointless to even try.

So, well…the book sits, unused.

And, I recognize that I am full of deficits. I don’t know if it’s really worthwhile to even try to re-teach myself. I don’t know that I have the patience or time to shore up the foundation.

Mathematics (here, in the UK) was a subject I loved/hated at school. Like you, I didn’t like arithmetic, loved geometry and trigonometry, could get by on algebra. Later, when we revisited geometry and trig’ in physical geography, I was enthralled. And I loved the skill of bisecting an angle or line using only a pair of compasses. No maths involved. I memorised all the ratios of a right angle triangle, useful in building. There is a beauty in numbers, but to my simple mind 1 over 3, 1/3, is more appealing than 0.3333 recurring, or 2/3.

How maths impinges on the metaphysical, I do not know. Perhaps only simple rules apply.

Thanks.