Pure Math in High School

I'm not dead yet.*

Awesome thread in Drexel about pure math in high school.

http://mathforum.org/kb/thread.jspa?threadID=1909041&tstart=0

I wouldn't come out of six months hibernation to recommend a boring thread to you! Pay attention to the "Calculus isn't really a part of math" rearing its ugly head.

*I'm not in remission yet and still have a lot of problems. Daily severe fatigue, lightheadedness, difficulty concentrating, joint pain, nausea, lower back pain are just some of the problems. I think they will switch me to a more aggressive treatment by next month and follow up on that tumor, it's inflammation not cancer. I might have another one on the other side of my body from what I can tell. There is a miracle drug (not curative though) based on adult stem cell technology that is in phase III clinical trials and it looks like it might be available to the public in the next two or three years so I am very optimistic.

My son is coming close to finishing Frank Allen's Algebra I as well as Suppes Mathematical Logic. That logic book may be the all time best book I've used in homeschooling. Next on our list is an old 1960s Moise and Downs Geometry book and Suppes' Introduction to Logic. As usual I am extremely skeptical about having a seventh grader work on college level material, but I'm willing to go slow and give it my best.

We are in Ike's path

Miles out of the reach of the surge, we are in the burbs north of Houston and if Ike stays on that red projected path it will be over us.

I didn't have enough drama this month.

(Link to Local Coverage Online)


I did get a new gastro doc that I met with once and he was supposed to figure out if I needed surgery or not for the problem in my belly. UTMB downtown Houston evacuated their patients, everything is closed, and who knows when I'll be able to get more info on my medical situation. Looks like things might be chaotic for a while. I still have the same old annoying symptoms.

I'm spending the day cooking up all the food in the fridge and washing laundry. The electricity where I am goes out if so much as an ant passes gas so I'm reckoning that I have about six hours left. I do have a battery operated police scanner, hand cranked flashlight, several days worth of junk food, and a tank full of gas. Graphing calculator, Flashmaster, and pack of UNO cards should keep us entertained tomorrow.

Husband just announced that it's gusting and time to duct tape the bottom of the doors.

Good luck to the rest of the readers in my area!

Medical Update

After an agonizing one hour delay Dr. Gastro met with us in a consult room and explained that the pathology report came back with findings completely inconsistent with cancer. It says "no dysplasia or malignancy seen" in half a dozen biopsy findings, and "cryptitis" and "crypt abscesses" and "acute inflammation" in the same half dozen. His official diagnosis is Crohn's disease.


Evidently the finding is unusual for several reasons. One is that I have none of the gastro symptoms common with this disease. Second, every biopsy from every part of my insides shows a problem and yet everything looked outwardly healthy with the exception of that gnarly tumor. Third, the mass itself was a very hard mass and not consistent with the kind of thing felt with Crohn's disease.



Dr. Gastro said that this would require long term management and went ahead and got me started on some things that might actually have me feeling better in just a few days, but is handing me off to a different doc. I was warned that surgery is very much still something that is going to be considered, but that decision will be made by the new doc, and that there might be a possibility that even that won't be necessary. Having gone through three major abdominal surgeries in the past, it would represent an inconvenience to me rather than a personal tragedy.


If I understand everything correctly, the joint pain, neuro symptoms, and fatigue are part of all this.

And now I've got some kiddos that are behind in their skoolin that I need to attend to and some ice cream weight to lose.

Tumor NonUpdate

No result about that biopsy when I did my follow up with Dr. Neuro yesterday. Saw the radiologist report which said that it was...and I forget the exact wording..probably/likely/most likely malignant but that an infection can't be ruled out. Spot on lung. Spot on liver.

Radiologist called spot on lung "chronic" ...whatever that means. Dr. Student, this is a teaching hospital, gave me assurances that anyone who has a cat scan of their body is going to have spots of this and that turn up and that it doesn't necessarily mean anything. Dr. Neuro emphatically said that it has to be looked into more carefully but he's turning me over the cancer docs at MD Anderson and I won't be his patient any more on this issue.

I've been emailing folks privately telling them that the radiologist reported the tumor was 7.5 cm, but that is not correct. The wall of the intestine is what was 7.5 cm thick. Dr. Neuro also said that lymph nodes are swollen and then gave me more mindless reassurances that this can be the case if it's all the result of an infection.

Bought three pints of Ben and Jerry's ice cream: New York Super Fudge, Chunky Monkey, Chocolate Fudge Brownie. Fresh pineapple, more sushi, and a couple of gallons of white grape juice and apple juice set aside in anticipation of having three feet of intestine removed. Maybe if I gain enough wait from overeating this will magically cause the biopsy to come back as "Psyche! It's just a nasty infection after all."

Appointment this morning with Dr. Gastro to discuss biopsy results and surgery. I'll try to pass the scar on the belly off to my seven and eight year old as a shark bite.

I sincerely and deeply appreciate your comments and emails. I wish I had more time to respond individually and perhaps when things are diagnosed and they take this thing out I'll have time to go back and do this. Some people have asked what they can do for me other than simple emotional support and there is nothing really. We met our deductible. Having the friendship and input of cooler heads has really helped. Chit chat has helped distract me as well.

I also discovered Cicero's De Officiis. Philosophically speaking Cicero may be the shoulders that Kant stood on. Wiki says that De Officiis was the second book printed after the Gutenberg Bible and that it was the moral authority of the Middle Ages.

"For no phase of life, whether public or private, whether in business or in the home, whether one is working on what concerns oneself alone or dealing with another, can be without its moral duty; on the discharge of such duties depends all that is morally right, and on their neglect all that is morally wrong in life." Cicero in De Officiis

It's not a bad time for Stoicism.

Turn of Events: Tumor/Cancer

Things seem to be spiraling out of control since I last posted.

To quickly recap: The neuro said that the original radiologist was completely wrong about the chiari malformation and then ordered up a bunch of other tests for me. One of the many things he ordered was a cat scan of my GI tract which turned up a "mass". I wasn't happy about that but then convinced myself that this was probably just a polyp since other than bizarro neuro symptoms I don't have any GI symptoms. Nothing. Zip.

Colonoscopy this past Friday to follow up on that "mass."

This is my tumor. (There are many others like it but this one is mine.) The arrows in the pic of a mass inside my intestine in the area where the small one meets the large one were originally in the report

The biopsy should be back by Tuesday, but the gastroenterologist who is on faculty at MD Anderson here in Houston is not optimistic and thinks, based on what he saw when he was where no man has gone before, that there is a good possibility that it's already extended through the colon wall and wants us to be prepared for that outcome. The extent to which it has gone through the wall of the intestine makes a big difference in if you need chemo and in your chances of survival. The date of surgery has not been decided yet. It can not stay in and they need to "stage" it.

I'm going to name it Gustav.

Then I came home took some meds for anxiety which did me absolutely no good and my significant other opted for Kentucky's finest Bourbon whiskey which did him a lot of good.

That was Friday evening.

Saturday I spent the day gorging myself on hummus, shawarma, sushi, and ice cream because weight gain isn't going to be a concern of mine when six months of chemo/radiation might be around the corner. My physical symptoms are changing. Fortunately, I haven't had those nasty painful muscle cramps but now I'm dog tired and I'm sleeping for hours in the middle of the day. and I feel like I coming down with the flu.

This is your cue to say something like "Other than that, Mrs. Lincoln, how as the play?"

And now for some free association rambling about books I've been reading.

Dedekind's Essay's on the Theory of Numbers. I ordered the dead tree version here ---Perhaps the translator Wooster was some kin to Elmer Fudd? I managed to read a good portion of it waiting around in the prep area for the colonoscopy. Or read the google book here.

I think this publication is where Dedekind Cuts were first introduced. The pleasant surprise for me was his constant reference to Euclidean geometry. Back in the day geometry and magnitudes were used to explain how it was that arithmetic and analysis worked. Dedekind didn't think this was rigorous--in other words it was sloppy thinking--so he set about figuring out how to prove a few things. There is a fun reference directed at Kronecker to publish up or shut up his poor mouthing of set theory which at the time was the innovation of the very much still living Cantor. Savor my pun of this being a kind of Dedekind Schnitt.

(Kronecker did finally do this 20-30 years later, he was the perhaps the first constructivist (this has a different meaning than pedagogical constructivism) in math--its descendant intuitionism being further developed by Brouwer and Poincare (another book I have that I haven't finished!) In the history of math it's known that Kronecker also gave his former student Cantor a very hard time about set theory, but it seems that Kronecker gave many people a hard time and was somewhat of a jackass. I am told that classical logic is safe from criticism since in 1929 Godel showed it was complete as well as consistent.

My next idea for a good book to read is the much praised Dauben book on Cantor's life.

And, by the way, I've brought up Bryan Bunch's "Mathematical Fallacies and Paradoxes" (Dover reprint) on other forums, before all hell broke loose in my life, and my interest in intuitionism began with that book. Bunch consistently brings up the central idea of intuitionist logic which is not having the law of the excluded middle. But all of the good paradoxes in that book, it turns out, are things I've found in my Suppes logic book.

I've also really been enjoying Berlinski's Infinite Ascent simply because of Berlinski's writing style, but John Derbyshire's A Real and Imaginary History of Algebra is fascinating. You do have to learn a bit of math along the way and go slow in some sections. Rather than giving dry technical definitions of important concepts in high math, Derbyshire seduces you into following his train of reasoning as he leads you through history and mystery. It's like watching a magic trick unfold. Just unbelievable. If your only understanding of what constitutes the topic of algebra comes from Saxon math, you need to get this book.

And that, dear readers, is how my Labor Day weekend went. How was yours?

Edit (Update) Diagnosis

The second radiologist has said that when he examined the MRI results he saw no chiari malformation (brain herniation/cyst). In other words there is nothing wrong with my brain. However, after a day of testing they think they've found yet something else that may be at the root of this--back to a potentially serious diagnosis, but perhaps not. They tricked me once into getting worked up over a diagnosis and I'm going to try to stay calm this time around. If they find nothing with this other thing then I'm back to step one with no explanation for the symptoms, but at least I got a script for a half decent pain killer which does seem to work on many levels.

I am tired of this. This is rapidly becoming a medical soap opera at my extreme emotional expense.

I have some blood tests that still need to get done and I think I'm being passed off to another kind of specialist for more testing that I will elaborate on if anything serious comes of it.

The personal problems have not dissipated.

Your support and comments have been very meaningful to me. Really.

***********************************




Never having a physician spend more than ten minutes with me it was a surprise the neurologist spent an hour checking me out and asking questions.

Basically the MRI showed that the back of my brain is dropping out of the hole in the back of skull into the area where the spinal cord belongs. That's the short story and I don't remember all the fancy medical terms to label it properly. At least I know why the back of my head hurts. Also, I also have a cyst on my spinal cord and the theory now is that all of this is disruptive to spinal fluid. They want a different radiologist to spend a bunch of time looking at these specific problems since the first radiologist (paid $1500) was only requested to look for MS lesions and simply made a note that he found these other problems--he wasn't paid to examine them in detail, you see, and now someone else gets to do that.

They have an entire day of testing planned in the next few weeks which will include EKG, more blood tests to double check for Lupus, they want to check out my blood circulation because there is evidentally a problem with that also, and a bunch of other tests to rule in or out this, that, or the other. Sept 2 they'll go over all the tests results with me.

Meanwhile, and independently of the medical issues, I have personal issues that need to be attended to which are sapping any concentration I have left so I think I'll be taking a short break so I can take care of things that need to be taken care of.

I will have some awesome books to review for you when I come back.

Don't forget about me.

The article that launched a thousand posts...

I can't believe I'm getting sucked into this one, but I'm not the only one. It comes down to, "Is it okay to teach kids that multiplication is repeated addition or not."



Devlin: It Ain't No Repeated Addition
Devlin: It Still Ain't Repeated Addition
And Devlin Is Still Wrong, see Neiderbergers comments to Let's Play Math

I don't know what else multiplication could be. Doesn't my computer base everything it does on repeated addition of ones and zeros? If Devlin comes up with a new paradigm wouldn't that be the greatest revolution in math in the latest 150 years?

And finally,

Text Savvy: here, here, here, here, and here.


And in my inbox this morning this email by Adrian. Who knew that after a marathon discussion on this from yesterday, (that I took notes on!) there was still something to be said?

Devlin has made a horrible gaffe. In the case of number fields, at any rate, you don't even have to make it to Peano's Axioms. Just the integers is far enough. Anything that produces the integers is forced to define multiplication as repeated addition. It follows from the distributive law and the fact that integers as an additive group are generated by the multiplicative identity. I wonder if there isn't a more general theorem in there about all fields reducing to the second operation of the field being repeated applications of the first operation of the field.

It is true that multiplication is not *just* repeated addition in the general case. But, the process of teaching it as repeated addition in the case of integers and then generalizing and extending it to other cases is not only not "just false", but it matches up exactly with the true intellectual development of the numbers. This isn't a technicality. This is the underlying truth of arithmetic.

Now, I do understand exactly where the notion of "multiplication isn't repeated addition" is coming from and do kind of appreciate the sentiment. But, if anything is "just false" it is the view that it is just false to say that it is repeated addition. Repeated addtion is the basis of multiplication and without it, there would be no extensions of it to the rational numbers, the real numbers or the complex numbers. It is not "mathematically correct" to teach it otherwise. It is more like a math gimmick. And, maybe it is a "harmless fallacy", itself, and one worth employing. But, it is not the "real" notion of multiplication.

Certainly solving problems like 5x=17 rely heavily on using the extension of multiplication to the rational numbers and the definition of multiplicative inverses, in particular (and/or the notion of division). But, that doesn't diminish one bit the fact that 5x is understood to be x+x+x+x+x or, for that matter, that 17 is understood to be 1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1+1. Without this understanding, we don't even know what 5x=17 even *means* let alone have any means to solve for x.

The only plausible alternative to this is to abandon analytic geometry and algebra altogether and return to synthetic geometry. You could define multiplication as area. And, you would have to not start thinking of area by decomposing a region into a collection of unit squares and counting up the unit squares (since that is repeated addtion all over again). Something like that is far from more accessible or generalizable than the standard algebraic approach using repeated addition.

Now excuse me while I look up the phrases that I wrote down in my notes last night, because my lecture didn't sound like that email at all, but involved phrases such as: principle ideal domain, unique factorization domain, integral domain, ring, and field theory, matrix addition, functional analytic blah blah blah.


Since Frege, Peano, and Dedekind on the foundation of Arithmetic looked too expensive, I downloaded Dedekind's Theory of Numbers in pdf and I'll be able to print that out a few pages at a time and figure something out.

My appointment with the neuro is Aug 1st. I'll know more then.