Drat These Greeks!

Pure Math in High School

2009-03-11T07:19:00.001-07:00

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

2008-09-12T08:25:00.000-07:00

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

2008-09-03T17:01:00.000-07:00

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

2008-09-03T04:18:00.000-07:00

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

2008-08-29T18:45:00.000-07:00

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

2008-08-02T16:15:00.000-07:00

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...

2008-07-21T07:17:00.000-07:00

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.

Why not use Singapore's NEM?

2008-07-13T07:21:00.000-07:00

This was the topic of a recent thread on the WTM message forums. The reasons given by many parents had nothing to do with the content of the book, but addressed student and parental factors. One person finally addressed content:

NEM does not provide lists of theorems and postulates for algebra and do "proofs or show tha this equals this" which I think is really improtant in algebra.

For that matter, the only American algebra text still in print that I have seen that does so in Foerster's Algebra. [I have no idea about Lial's, Chalkdust, Teaching Textbooks, or any other book you may be using. I do have an idea about Jacob's: It's not there. If you go flipping through your book to look for it also make a mental note, or even better a written list of all theorems proved in the book and make a note if the student is then required to prove theorems in the exercises] It lists out the field axioms in a list at the back of the book--something like getting assigned a basement office--but that at least is better than being completely absent.

Now that I've had time to think about it, I think what was missing from the ensuing conversation (monologue actually on my part) that these properties of the numbers, when considered together are very, very, very important in math. They aren't important because they help a 14 year old become a better rune manipulator and solver for x , they are important because they create a new universe of math called group theory. Asked one person puzzled about the need of such a list of properties at such a late date in the student's math career,

But aren't those properties all taught it most elementary mathematics programs? Or am I being naive? Certainly they've been featured and reviewed constantly in the program we've used. ... I'm not saying they're immaterial -- just that they're basic enough principles that I would expect students to know them well before they reach algebra (if not expressed in exactly the way that they are in your link). Am I wrong?

And so this is how I responded:

They aren't "basic" in the sense of being simple. They are more like foundational. Mathematicians have made careers talking about the differences between things (groups) that follow some of these rules and things that follow all of the rules. Essentially all of these various properties and what happens when they are or aren't true is the sum total of what one learns in Abstract Algebra so this isn't something that can be covered in its entirety in a K-8 arithmetic program. Usually what you might see in arithmetic is the appearance of the distributive, commutive, and associative properties and the book states it and then has the kid apply it in the simplest way possible or identify the use of it in a very simple way.

However, what you don't see (but if your arithmetic book does this I'd be interested) is go in depth into distinguishing between properties of numbers, axioms, definitions, theorems, and notation. It's all thrown in there and all mixed up. It's all ad hoc. [I should have said here is that I'm talking about algebra books, not arithmetic] And the students aren't required to use them as axioms, but instead they are taught "facts." [Axioms play the role of the premise in a logical syllogism. Think of algebra as having 11 premises, you will get the bigger picture in the long run if in algebra you start thinking of them as assumptions rather than as "facts." What follows from accepting these assumptions, if you do your derivations and proofs correctly, are all things that you use to do algebra arithmetic such as the algorithm you use to find equivalent fractions, multiplying double digit numbers together, etc). The sceptic then asks, "What happens if I reject one of these premises and instead only use these five? What sort of a math world would that create? You ask the question that open ups a worm hole and you leave the 9th grade algebra universe and go straight into group theory. Another sceptic might say, how can those 11 properties be proven? And that takes you to a different area of math as well, Back to the original post...]

That 2 X 3 = 3 X 2 is true is not "learning the commutative law". In fact, even learning that such a property is called "the commutative law" is not "learning the commutative law.""Learning the field axioms" entails being able to write down the general statement of the principle involved, not merely as a fact about the integers, but as a property that any set of elements together with any kind of operater [+ - x ÷ √] on that set may or may not satisfy.

"Learning the field axioms" also entails being able to use the list of axioms/properties that some given set and its operators might satisfy and actually derive significant mathematical results from that list through an unbroken chain of completely valid logical deductions.

What seems to happen in most algebra books (Saxon, Singapore not excluded) is that the student never is expected to make a distinction between the field axioms and other "facts" in algebra. They don't understand that multiplication distibutes over addition is axiomatic, but that it distributes over subtraction is not. Or, that the natural numbers are closed under addition is axiomatic but that they are closed under subtraction must be proved. And, they would "know" and use the rules of order of operations but they wouldn't know how those fit in with the other rules they've memorized. The rules about order of operations are not on "the list" of field axioms, for example. So, unless there is a master list ,the student is likely to go through his algebra book lumping it all together because he's never been given an indication that there is some sort of structure or hierarchy to all these facts that he's memorizing. [But then again, unless you have theorems for the student to prove or something else to say about the field axioms, nothing magically will come of simply memorizing it. ]

Once this list is memorized then there can follow discussion about what happens if one of the properties is missing. So even beginning in ninth grade algebra, with a text like Allen or Dolciani the student is getting an introductory taste of groups, rings, and field theory along with the skills it takes to prove theorems. This maturity of proving theorems, to gauge the appropriate amount of rigor for a given problem, constitute "true" math skills that cannot be acquired over night, skills that are absolutely necessary in doing in higher math (not to be understood as engineering calculus), For instance, after a diet of Allen, Gelfand, Oakley and Allendoerfer even an average student has practiced the skills needed for a Calculus text such as Spivak. Even a really good program like Singapore would leave a student unprepared for Spivak.

That's our story and we're sticking to it.

Always Look on the Bright Side of Life

2008-07-06T07:15:00.001-07:00

I finally got around to seeing a neurologist who thinks I may have multiple sclerosis and I had a battery of MRIs last week done to look for lesions in my brain or problems with my spinal cord that account for those pesky symptoms I've been having.

I don't have a diagnosis yet and the anxiety and stress of waiting for that phone call alone is causing another flare up of whatever it is that I have.

The optimism of the people around me makes me cranky, "You know there's a lot of people with MS that live to old age." Yeah, and there's a lot that don't. As far as I can tell if this were a game of roulette there is one empty chamber, four chambers with rubber bullets, and one chamber with a lead bullet. I am not amused.

I was going to post how I am just not that interested in math any more, or really much of anything since I'm having a lot of difficulty concentrating in general, and more specifically with a mystery diagnosis hanging over my head the anxiety alone is disruptive, but at any rate, a couple of days ago I did read a chapter in David Berlinski's book "Infinite Ascent" about how Galois Theory explains why the quintic is unsolvable and it got me motivated to want to learn (not to be confused with putting out the effort to actually learn) abstract algebra again.

Related to that, there is an interesting proof in Frank Allen's Algebra that I was dismissive of the first time I saw it and now, after finding out what the formal defintion of a normal subgroup is, I wonder if there might not be more to it. Allen proves that "d-x+x=d" for the real numbers. The whole thing led to a neat discussion, at my house not in Allen, about graphing the solutions to the "roots of unity". I just don't remember discussing that all in any class I ever took and it's neat stuff. I'm pretty sure that it never came up because to graph it properly requires knowing polar coordinates and de Moivre numbers which I never got around to. And this is all a round-about way of saying that I now have a more personal reason to work through Gelfand's Trig than I did before. Although I just need to see how things are going to go in the next few weeks.

Finally, for those with a classical education bent, you may enjoy "Lapses in Mathematical Reasoning". It's an English translation of a Russian book from the 1950's. It is a collection of 80 false proofs that are at the high school level and you read through the solution to some problem and spot the fallacy. Some of them just reflect run of the mill mistakes and some of them lead to interesting discussions about deeper issues, long-winded answers are at the end of every chapter. At any rate, as you can see by the table of contents the authors of the book chose to extend Aristotle's refutations to sophisms of a mathematical nature which gives it a very philosophical feel that I had a lot of fun with. The chapter on arithmetical errors was not as good as the others since they relied on pecularities of algorithms that we don't use any more.

I'm not paranoid. They really are watching me.

2008-05-22T11:01:00.000-07:00

My neighbor's camera pointing into the room where I homeschool our children:

And it's all perfectly legal since he's doing it from his property so I keep the blinds shut.

If it stays this way I guess we'll take it to the HOA.

Other ideas:
A) Put a sign in my window that says, "If you can read this you're invading my privacy."

B) Put a sign in my front yard that says, "I don't think my neighbor is a pedophile just because he has a camera pointed at our home and videotapes the children."

The background to this story is that my neighbor had his home broken into a couple of weeks ago so this weekend he was putting up security cameras and they all were pointed to catch anyone approaching the side of the house. I'm hoping the camera somehow got knocked out of place by accident and he'll notice this and fix it. Still, it's kind of creepy.

Solutions to Gelfand's Trig

2008-05-11T17:56:00.000-07:00

Is anyone interested in a solutions guide to Gelfand's Trig? Did you benefit from the algebra solutions?

Paragraph Form of Proof

2008-04-16T16:16:00.000-07:00

My sixth grader is progressing extremely slowly through Frank Allen's Algebra. A couple of days ago we ended up on page 326 which ostensibly is on the topic of odd and even numbers and divisibility tests, however, nearly overlooked by me was the first proof in paragraph form. All prior proofs were presented in two columns. And as I flip forward through the book it looks like more and more of these will be in the form of a paragraph.

Frank Allen has a nice motivating speech about teaching proofs and logic in high school math here and it was useful for me to read it.

To back up a bit, we've been nicely plodding through the two column proofs. I've been having him use the same format that Euclid used 25 centuries ago. His cookie cutter proof format is to consist of these five parts:

1. Statement in prose of the thing to be proved.
2. Declaring the variables to be used.
3. Restating the theorem in symbols.
4. Proof in two columns
5. Conclusion.

The most pedagogically useful has been 1 & 3 since it's led to several discussions about how concepts can be expressed by both by natural language and by symbols and how it is that one can come up with logically meaningful conclusions when all one begins with is imprecise and vague concepts.

To undigress, when my son came across the following in his text he didn't even recognize it as a proof. I'll give you the exact wording of the original paragraph and then tell you how it was rendered meaningful.

An even integer is defined to be an integer which is a multiple of 2. Thus if n is an even integer, then n = 2k, where k is an integer. An odd integer is an integer which is not even.

If t is an odd integer, then t ≠ 2k, where k is an integer. By the multiplication property of inequality, it follows that t/2 ≠ k, where k is an integer. Since t/2 is not an integer, it must be between two consecutive integers which we shall represent by k and k + 1. Therefore, k < t/2 < k+1, and by multiplication property of inequality, we have 2k < t < 2k + 2. Since 2k +1 is the only integer between 2k and 2k + 2, we conclude that t = 2k + 1. Thus we have proved the following theorem: If t is an odd integer, then t = 2k + 1, where k is an integer.

Okay, so the first problem that the kid had was that it all looked like "blah blah blah." Admittedly, stating the theorem in bold before proving it would have served as a nice sign that what was about to follow was a proof so it's no surprise that my kid didn't have any idea what had happened when instead the "to be proven" came after the proof itself.

The first thing I had him do was rewrite the above proof as a list of sentences rather than as a paragraph. The second step was to make distinctions with the individual assertions being made. "Is this a definition? Is this a justification or a conclusion? Why does he keep saying "where k is an integer?"

My next idea was to have him convert all of it into the friendly recognizable two column proof, but before I could do that he said he wanted to write it all out for himself in a way that was meaningful to him. He also wanted to include number lines so that he could understand the greater than and less than relationships and I requested that he make the number lines in a different color to make sure that he ultimately didn't confuse illustration with justification. So here is his chicken scratch:

In a nutshell, that is what it took to understand the paragraph on page 326 and those activities occupied the entire time we had set aside that day for math. The next day the assignment was, simply stated, "The proof of the converse is left to you as an exercise."

Grabbing his chicken scratch he decided to list all the step backwards (that paper is not shown here), convert the symbols to prose, and list them out as sentences and turn them in to Dad for grading. Dad innocently toiling away in his office at home had no idea what the assignment on his desk was even about and made a few comments before giving up and telling me it required too much effort to sort through:

There was a second draft followed by a final draft. I'm really pleased that the kid did it mostly on his own (the motivational speech to get him to do it included references to Spartans) it somehow clicked with him that he needed to write this out as if he were explaining this to someone rather than listing steps like you do in a two column proof. I told him that explicitly, but I didn't have to resort to dictating the words to him or any such thing.

My complaint with the book is that after this surprise attack of the proof in paragraph form, it looks like we will be returning to typical ninth grade algebra activities such as dividing polynomials and by the time we encounter another proof the kid will have forgotten how to write out proofs in this new way. I'm thinking I'll have him go back once a week and rewrite some of the two column proofs from prior chapters as paragraphs to keep on top of this skill.

How Math Doesn't Get Done

2008-04-02T09:24:00.000-07:00

Say I have a theorem I want to prove because its given in an exercise set. In this case show that,

│x₁ + x₂+...+x_n │ ≤ │x₁ │+ │x₂ │+...+ │x_n │

and my next thought is that I need a “good” definition. How do I decide which definition to use? I find a bunch and I think they ought to be good definitions because I looked them up in a book written by someone with a degree in math.

Definition 1: │a│ = { a, -a } ⁺
Definition 2: │a│ = max{ a, -a }
Definition 3: │a│= a sgn a
Definition 4: │a│ = sqr root of a²
Definition 5: The absolute value of a real number a is defined to be a if a is positive or zero, and to be -a if a is negative.

Day Two:

But on the other hand, what if I don't have a book and I'm online or I get a bunch of trash from wiki. How am I supposed to know a good definition when I see one?

And say I get in the middle of a logic book to figure this out and I start reading the chapter in the middle of the book on “Theory of Definitions” and it makes a lots of references to Aristotelian logic, then out comes the Aristotle and see that I have at least 350 pages to read if I want to make any sense of it. I get back online and look up what I can about the original theorem. Why is it given in the exercise set? What makes it important? It doesn't look like the rest of the problems in the exercise set so it must be special.

Day Three:

So I go back to wiki and there is a discussion about the “properties” of absolute value and one of its properties is “subadditivity” which, lo and behold, looks exactly like the problem that I have in my problem set. So is there something important about subadditivity? By the dogs, it seems there is something to this subadditivity thing. All sorts of things can have this subaddtivity property, but most importantly, it seems a distance function must have this property....(What if it doesn't?) What are the other properties that a distance function must have? Is a distance function the same thing as a metric? Is a metric the same thing as a metric space? No. it seems not. A metric space includes a distance function. What sorts of distance functions are there other than absolute value?

Day Four:
So while I'm thinking about what makes a good defintion good, it occurs to me that perhaps illustrations are not definitions. Are illustrations actual objects? In geometry an illustration of a circle is not the actual circle we are discussing, it's an idealization of some circle, but wait a second, there aren't any actual Platonic circles floating around anywhere, so what can an illustration of a circle really represent?

Day Five:

This inequality is like the triangle inequality but for a triangle with an infinite number of sides...er sort of...like a circle? No. Must stop thinking in pictures. Maybe Euclid could be done without any pictures at all, maybe circles aren't really circles but sets of points, lots of symbols, what would Euclid look like if it was written in set notation? (Answer handed down from above: Don't worry, Euclid defined a circle, you're safe, they aren't just illustrations).

Day Six:
And when I look at a kids textbook and it says that absolute value is "the" distance, is it really the distance or is it a way of measuring that distance? I mean, the actual number yielded is the distance, but the thing that got you that number, the absolute value thingy itself does not seem to be "the" distance. See what I mean? So maybe, since there are other distance measuring thingies like absolute value,which I have no idea what they are but wiki or some other math site said it so it must be true, maybe I should say, "Absolute value is one way of measuring distance." I'm having issues with the idea of measuring distances on non-physical objects.

Six days later, I'm finally finished with my little proof, but if I'm going so slow because I can't really focus on the actual problem then it's going to be forever before I learn anything.

But before I go on to problem #26 let me just check this one little thing out about this Fekete's Lemma. Wonder what that's good for? And if I could find out would I even understand? It's not in any books around the house but it looks interesting.

Teaching Textbooks Invokes National Math Panel, Crusade is Called.

2008-04-01T13:30:00.000-07:00

The original blog entry has been removed.

The main purpose of my blog is to discuss K12 math in general with an emphasis on
- the role of axiomatics and pure math in K12
- updates about my personal interest in the "theory" behind math
- a log of my 12 yo son's successes and failures at proving theorems.

If you are interested in a discussion about content in math my latest blog entry on that issue is here and I invite you to comment on how your algebra program addreses the particular issue that's being brought up.

My site meter was showing me that this link was getting disturbingly high number of hits and I don't want this post to degenerate into yet another interent thread of "skimmers, those who cannot read but only read-in. The sophists who, abusing argument, argue for the sake of argument. The ideologues, those who are out for power, not truth. The uncivil. The illogical. The politically correct. Worst of all, perhaps, are those who exemplify the anti-Socratic property: those who think they know what they don't know."

My email is Hocklemeier@yahoo.com if you have any specific questions.

Tall Ship Elissa at Dawn

2008-03-24T21:24:00.000-07:00

See the first post about this topic below.

The wind wasn't blowing nearly as hard as predicted this morning and it was chilly. No one was out at that time in the morning except the kids and the moon. With the exception of the seagulls everything was very quiet and you could hear the captain giving orders across the water. I took this pic when the crew was already aboard and they were preparing to leave.

I can't adequately convey what a good experience this was for my son.

Looking back at the hotel behind the "mosquito" fleet of shrimp boats...

Same spot looking in a different direction at the Elissa returning home...

Tired of the Snow? Read this.

2008-03-23T02:52:00.000-07:00

I thought I'd rub this in your face.

For the past six months every Saturday more or less, I have taken my sixth grader to Galveston for sail training. From 9 to 4 I sit in Starbucks and have this view while I read or do math. And that is a silhouette of my husband "working from home." Horrible working conditions but somebody's got to do it---(you'll get your chance to gloat when they transfer us to Green Bay, WI.) The masts you see in the background are of the ship on which my son will be sailing.

To the left of the tall ship in the above photo there is a restaurant that overlooks the museum's harbor. We ate lunch frequently and had this view off the balcony in the back.

If you could look more to the left (east, the harbor for the ship is on the backside, or northside of the island facing the Texas coast)...you'd see the less picturesque Haliburton, as well as that 'tiny' wooden life boat from the above pic now out on the water and full of people who are practicing...something:

So tonight we drop off the kid and tomorrow he will help sail that ship with a crew of about 20 other boys and plenty of adult supervision. The weather for tomorrow is supposed to be perfect. Sunshine with temps in the 60's and a fresh breeze (that's winds of about 20-25 mph) are predicted. Handling 12,000 square feet of sail with that kind of wind out to be an experience!

There is web page which features a live cam of the ship. I tried to link to it directly to just the cam from my blog but I can't figure out how to do it. It looks like they've already moved her from her berth to the end of the wharf in preparation for tomorrow and the gigantic white cruise ship in the background is Carnival Cruise Line's Ecstasy. Check in around 8:00 am, 12:00 pm, and late afternoon Monday and you should see a lot of action.

If you are more or less in the Houston area and have a child between the ages of 11-17, they can participate in the program for free. Email me and I'll hook you up with the details--next year's training starts in September. Math and pirate drinking songs with Myrtle, the Charles Dickens festival, and Mardi Gras are optional and not part of the official program.

hocklemeier@yahoo.com

Cool Pod Cast

2008-03-19T06:00:00.001-07:00

Thanks to Joshua for the link to this math podcast. This is awesome.

These topics are not the thing you normally find in the traditional curriculum...things that make you ask more questions. I wish my math text had had cool side bars on these kinds of things. The monologue is very, very short, very few technical details, and presents cool ideas to ponder. The first one starts with Hilbert's Hotel. I have no idea what's up with the Mormon reference..anyhow,

"Today I want to talk about Hilbert's Hotel, a famous paradox introduced by the mathematician David Hilbert, challenging naive ideas about infinity. Imagine a hotel with an infinite number of rooms. This cannot actually exist, of course, though they are trying to build one next to the Mall of America. Anyway, suppose you arrive at the desk and are told that the hotel is full. Can they still find a room for you?

Surprisingly, the answer is yes. The guest in room 1 moves to room 2. The guest in room 2 moves to room 3. And in general, the guest in room moves to room . Since all numbers can be increased, everyone still has a room, and you get one too!
Now suppose you are a Mormon, and you arrive at the front desk with an infinite number of family members in tow. Can they still accommodate you? Yes, they can. The guest in room 1 moves to room 2. The guest in room 2 moves to room 4. And the guest in room moves to room <2n>. Every number can be doubled, so everyone still has a room. But now all the odd-numbered rooms, of which there are infinitely many, are free!

As with all podcasts, I've just scratched the surface here. For more information on this topic, search for Hilbert's Hotel on Wikipedia. Until next time, enjoy your math!"

I pwned mathematical induction this week which means I can get to any room in that hotel I want to now. If you want an entire book on philosophical stumpers in math (and all sorts of conundrums in infinity are considered) I have really enjoyed "A Certain Ambiguity: A Mathematical Novel. This book gave me the insight I needed into infinite series in order to handle them.

Celebrating and Integrating

2008-03-18T16:42:00.000-07:00

"Different solutions, interpretations, and approaches that are mathematically sound must be celebrated and integrated into class deliberations about problems.
I'll bite.

"Okay kid, how did you come up with these answers?"
"Dunno, just thunk them."

Our guess is that he's keeping track of the steps in his head. By the last uncompleted answer I think he's writing a single digit, subtracting in his head, and then writing the next digit.

How did I teach him long division? I tried using the same approach that I did with my first son: I used manipulatives. But kidlet #2 can't stand the things and says things like "I don't want to do this stupid stuff, I want to do real math." So I gave up on that idea and dictated....dictated...the steps in the algorithm to him for three problems and then he told me to quit and did it on his own after that.

19th century Math Ed in the US and Euler divides by zero

2008-02-25T20:42:00.000-08:00

JD 2718 has a post up about New York resurrecting Euclid in the high schools. And that prompted me to go look up a few things.

I was somewhat surprised to find out that geometry wasn't as popular in the 19th and 18th centuries in the US. I was under the impression that synthetic geometry was the cornerstone of high school math education in the past but it turns out that isn't quite true. This table shows a survey of offerings in various towns in Massachusetts during 1834 - 1840. At that time Harvard didn't even have geometry as an entrance requirement.

The Rise of the High School in Massachusetts By Alexander James Inglis

To give you an idea of small enrollment of geometry compare it to the more than ten thousand students taking US history. Geometry is listed second from the bottom.

The Rise of the High School in Massachusetts By Alexander James Inglis

And all this isn't to say that it ought to be done this way in general, or that it "worked for us." Marcus de Satoy describes America up until WWII as a backwater of mathematics compared to Europe. I was able to glean from Florian Cajori's very detailed and amusing history of math education in the United States that even at the university level math was a matter of an engineer teaching students what they needed to know to become engineers, and wasn't anything like I imagine a modern mathematics department to be. While Cauchy's students in France were rioting against him teaching them the newly formalized definition of a limit, VMI was struggling to find even decent math texts for its students. Cajori did such a good job with math history that one of his books, the history of notation, is still in print. If you want to know the history of your math department at your university, assuming it existed in the 1800s, you can look up all the details.

At any rate, for a better idea of what classical education might have consisted of in a 19th century American high school check out these courses:

The Rise of the High School in Massachusetts By Alexander James Inglis

You can peruse the book linked above to see exactly what authors and texts were used. They are listed out in various places and I bet you can also find them online as well in Google Books. Did you notice that there was no English, just Latin and Greek? And, by the way, Euler's Algebra is listed as an algebra textbook used in some high schools. How cool is that? I was compelled to go looking through it and found some real gems. So, do you remember learning how to divide the polynomials using the long division algorithm? Euler's idea of a good long division problem is 1/(1-a):

Elements of Algebra By Leonhard Euler, John Hewlett, Francis Horner, Jean Bernoulli, Joseph Louis Lagrange

And then he says, "let a = 1" and our modern day algebra book keeps telling repeating the mantra that it's undefined, it's undefined, it's undefined, but look what neat shizzle you would have learned in high school algebra in the 1800s. See, he divides by zero and it did not cause a rip in the time-space continuum! Dividing by zero isn't just great, it's infinitely great.

Elements of Algebra By Leonhard Euler, John Hewlett, Francis Horner, Jean Bernoulli, Joseph Louis Lagrange

Finally, to respond to the idea of removing all proof from the curriculum I wanted to end with the words of Steven Krantz

If one were to remove "proof" from mathematics then all that would remain is a descriptive language. We could examine right triangles, and congruences, and parallel lines and attempt to learn something. We could look at pictures fractals and make descriptive remarks. We could generate computer printouts and offer witty observations. We could let the computer crank out reams of numerical data and attempt to evaluate those data. We could post beautiful computer graphics and endeavor to assess them. But we would not be doing mathematics. Mathematics is (i) coming up with new ideas and (ii) validating those ideas by way of proof. The timelessness and intrinsic value of the subject come from the methodology, and that methodology is proof." --History and Concept of Mathematical Proof

Tater says, "That Cletus knows him some math!"

2008-02-03T16:36:00.000-08:00

Here is the pre-calc book I said that we had found:

Principles of Mathematics by Carl B Allendoerfer and Cletus Oakley

Tater helped with this review of it. In the preface, the authors say that it is intended for serious high school students and in fact, one of the copies we were able to get bears the stamp of Cabrillo High School 1964. While the book was in print for a couple of decades we suspected that any of the 1960s editions would have the desirable qualities. Last time I checked there were a few used copies being sold in the UK as well as in Canada.

Here are the low-downs on the chapter hoe-downs:

Chapter 1, Logic and Sets - This is the typical intro chapter on set theory that any New Math book had. But, you can tell the authors assume the students have seen this material before, given the difficulty and speed at which they cover the basics. (Our New Math algebra book does in fact slowly spoon feed this and it is revisted again in more depth in Algebra II. The older Dolicianis also cover this topic). The set theory is then used as a foundation for the rest of the book.

In the following chapter, similar to any other New Math book, they then cover how the real numbers form a field. It doesn't stop there (in Frank Allen's much lower level course Algebra I, it does stop there) but actually gives the general abstract definition of a field, discusses modular arithmetic and residue classes, and axiomatically defines what an ordered field is. It ends the chapter with a definition of the least upper bound.

Chapter three is on the integers and they immediately introduce mathematical induction. The typical problem looks like this:

But take a look at problem #27 where all in one problem he defines n choose r and asks the student to prove:

He also defines what an Integral Domain and a Ring is.

In chapter 4 they cover groups, and I hope you can see by now that this is a quite serious treatment albeit for a high school level book and definitely not a token gesture juat to be able to say they "cvoered it". For instance he discusses the symmetries of an equalateral triangle and the symmetries of a square in the problem sets. He had been talking about isomorphism up until now and now he gives a formal definition of an isomorphism for a Group:

Finally, chapter four ends with subgroups and cosets. A student following this textbook closely would definitely have a serious introduction to group theory, ring theory and field theory, the contents of a typical first course in Abstract Algebra that all math majors must take at the senior college level.

Chapter 5 is a solid treatment of the rune manipulations that everyone expects students to do with inequalities. Nothing special here. For a specific axiomatic treatment of inequalities as well as coverage of famous inequalities see Beckenbach. And then in Chapter 6 he discusses functions and not only gives the typical definition that you might expect to see in a high school text, but the "ordered pairs" definition that you'd expect to see in a senior level college text. Digressing for a moment, in Calculus by Thomas and Finney they define a function as :

“A function from a set D to a set R is a rule that assigns a unique element f(x) in R to each element x in D.”

It is clear that Thomas and Finney have some sort of algebraic expression of x in mind and that the student is supposed to imagine that when they think of a function. Not only do Allendoerfer and Oakley not really invite the student to imagine a formula of x in their first definition, but, clearly, they are trying to push the student to think in terms of sets of elements as opposed to an algebraic expression of x in their second defintion. And, that is because they go on to bring up, for instance, Relations later on in their chapter. Not only does he go into more detail, explicitly and more carefully discussing the “rule” of the Thomas and Finney defintion, but the whole level of the discussion is more general, more abstract, and more powerful than Thomas and Finney. He also covers the algebra of functions, eg f: (x,y) + g:(x,z) = (f + g): (x,y +x) as well as the composition of functions, inverse functions, monotone functions, and algebraic functions.

Chapters 7, 8, and 9 provide a typical treatment of transcendental functions and analytic geometry.

The goods are in chapter 10 where they give the Archimedean property of the real number and then the formal definition of the limit of a sequence. Behold the Tater litmus test:

They also give a formal defintion of the limit of a function and continuity, and prove a number of typical theorems related to the two. This treatment covers everything from lines tangent to a curve, rates of change, and so on -- like you'd typically expect. But chapter 10 doesn't go so far as to define the Derivative which is formally defined in chapter 11 along with the Integral (and bear in mind this is a pre-calc book). They go on to prove the theorems you'd typically expect to see in such a treatment, going into maxima and minima along with some short applications.

In chapter 12, the chapter on probability, they make heavy use of set theory,

He includes joint distributions and conditional probability as well as counting techniques involving permutations and combinations. The chapter culminates in the proof and applications of the binomial theorem in a completely rigorous fashion using mathematical induction as well as the binomial probability distribution.

The last chapter is a brief is a brief chapter on Boolean algebra.

Singapore 2

2008-01-30T11:33:00.001-08:00

Bar diagrams are introduced in the supplement called Challenging Word Problems in the second grade. The word problems for each topic are grouped into easy and challenging problems. Originally, I drew the bar diagram for each and every word problem and had my second grader label these. I no longer do this unless he has done the word problem incorrectly. Ultimately bar diagrams are simply a graphic organizer and they can be dispensed with once they have served their purposes. The point, at least one point, of using the bar diagram is to illustrate visually the relationship between quantities and later in algebra the student will translate this into symbols.

In order to succesfully solve 2-step word problems the student may need some training wheels.

2-step training wheels: You will notice that the 2-step word problems have prompts, they don't just start throwing out hard word problems with kids without leading them up to that point. Here is an example of these "A" and "B" prompts:

I bought 4 red stickers and 5 blue star stickers. Eah sticker costs 5 cents.
A) How many stickers did I buy altogether?
B) How much did I pay for all the stickers?

When I had my son show his work I required him to do so like this:
A) 4 + 5 = 9
B) 9 x 5 = 45

He started doing this automatically and without prompting and I decided that he was old enough to handle the term "equation" rather than using the phrase "number story."
I would then start saying, "I am going to express these two statements as a single equation," and I'd show him how to do it. I had to explain to him that parentheses meant, "do this step first" and that it all amounted to writing the second equation but replacing the first term with the first expression. And all that sounds complicated in writing and seemed to go so much smoother in person. He did not get it until after several tries. That is okay. It takes practice.

Today I gave him a test and left the room. He did all of the word problems like this one:

I liked this one because it uses both addition and multiplication. I think our lesson for tomorrow will begin with discussing an alternate way of expressing that equation, (4 x 5) + (5 x 5). In fact, I may very well give him several problems to apply the distributive law to although I won't yet tell him the name of what he is doing.

Overdue Updates

2008-01-20T08:08:00.000-08:00

Adrian is writing up a blog entry on why set theory is absolutely essential for understanding algebra. I think it goes something like this,

-What does "greater than" mean?
-It means that when you subtract number a from number b you get a positive number.
-So what's a positive number?
- A positive number is a number that is greater than zero.

And then the Definition Ninjas jump out and hack you to pieces because you just used the term you were trying to define in your definition of that term. He wants to show me that to get around this you have to use set theory.

This was inspired by a comment left at the KTM blog in which a commenter was criticizing the Saxon math program because his incoming 8th graders were not able to answer the question,

Why is it that when you divide by a number bigger than 1 your answer is a smaller number (e.g., 8/4 = 2), but when you divide by a number smaller than 1 your answer is a bigger number (e.g., 8 divided by 1/2 = 16)? Nearly all of my students had had years of Saxon math, so they answered "invert and multiply." A truly logical non-answer if ever there was one.
These kinds of discussions get us clucking at our house. What sort of response is appropriate for an 8th grader? Does the commenter himself know what the correct answer is? Go ahead, try to discuss "bigger number" without using the word "greater/bigger/more" than to define the same.

An anonymous poster came in and answered my burning question. Here is how he would have rephrased the question for an eighth grader:

When you divide by a number bigger than 1, your quotient is smaller than the dividend; but when you divide by a number smaller than 1, your quotient is greater than the dividend. Is this statement always true, sometimes true, or false?
And as he explained further to me, "Explain by giving examples. The key here is you are training the eighth grader to think of 'the dividend' as a variable. You want them to consider all possible values for that variable – positive, negative, and zero. -- I could probably have put a period after the word 'think' and the statement would be even more profound:)"

Also, we found a fantastic pre-calc text from the 1960s. According to the authors, it was written for high school seniors and college freshman. It went through more than a decade of editions so it had some followers. The used copy that we got had too much writing in the margins, we ordered two more copies in the hopes that one would be clean enough to scan samples from and blog about at a future date.

Random Noise:
Finally concerning online politics, I was disappointed to see the new WTM boards renamed "Hive Mind." This is a pejorative term applied to group-think and I don't think the slur can be reclaimed by playfully applying it to oneself and friends. On a more interesting note, you can go to the roster of members and click on "posts" and it will give you a list of members according to the quantity of posts and you can see for yourself who the busiest bees are or aren't. There are options that allow for a threaded view which I use, and another viewing option which orders the threads according to when they were started rather than when they were last commented on. Using that option keeps the threads from rearranging themselves. Mental Multivitamin opines on the new development as well.

I think if a chat room were available the vast majority of posts to the board would simply get channeled to chat.

More random thoughts: I was casually perusing the Baytown, TX arrest log online and couldn't help notice that there were many arrest warrants issued daily for overdue library books. I am not joking.

Show Your Work!

2007-10-19T19:07:00.000-07:00

My English teacher told me today not to bother to practice writing persuasive essays because if I think that I've come up with a good argument in my head for any particular position on a given issue I'll automatically know how to write a good essay if I need to in college.

No, that's not a true story.

I am seeing several threads on message boards (plural) in which this argument is made about kids showing steps in math! I don't always use texts that have answer keys, much less solution guides, so I depend on my son writing out a coherent chain of logic that I can follow to the end. He doesn't like it and made a motion to lift the requirement of having to declare the variable in word problems. ("Mr. Gambini, that is a lucid, intelligent, well thought-out objection.... Overruled.")

This view states the obvious (apparently it's not obvious):

Primarily the reason math teachers ask you to show your work in high school is to prepare you for advanced algebra and calculus. Perhaps it does slow you down, but at the same time, if you can't prove your work, it will be thrown out. For all anyone else knows, you got the answer by magic. This is a basic tenet of science and mathematics: reproducability. If someone else can't do the same thing you did and get the same result, then you (or anyone, in the case of new research) don't understand what you're talking about, or you can't pass on that understanding to the next person, thus making your work pointless.

We've had some good discussions about what constitutes "showing your work"...it seems to be fractal. No matter how many steps you show there seem to bethat many more substeps that you could have shown. [Topic shift] For example, my latest project is showing my work for 1 + 1 = 2. But before I can do that I need to understand the following and I think I do except for the a and the b with the subscripts. What's up with that? It says on another page that all italicized letters stand for natural numbers. Why let x be equivalent to a₁ and b₁? Is it to show that there is x' has only one natural number that corresponds to it? A illustrative cloud shaped drawing would be of use right about now.

Failure

2007-10-03T11:27:00.000-07:00

Sometimes failures are more interesting than successes. I thought I would cleverly teach my second grader to add and subtract negative numbers this morning. I planning to use the same technique that I used with my older son, (which I explained in the comment box of another blog)It goes something like this:

If the kid understands that 0 - 5 = -5,
then by the defintion of addition and subtraction you can show that
If 0 - 5 = -5, then (5) + (-5) = 0
It's just another "fact family."
I'd use decomposition to show that
(5) +(-2)--->(3 + 2) + (-2)---> (the additive inverse rears its head)
--->(3)+ (2 + (-2)) = 3

Detour on what motivated this idea: I came across the blog of Timothy Gowers, a fields medalist, in which he gives his opinion on math pedogogy. I hope he does this more often and I will be buying his book on the topic of the difference between school math and real mathematics. He suggests that at some level in school math, probably not the second grade, that we cease to rely on intuition of concepts for explanations (concepts informally explained by use of rubber duckies, plastic chips, and fraction overlays and) and instead rely on formal definitions.

Back on topic: I remember the way I first introduced my younger son to negative numbers. What I did not do was show him the symbol and then try to explain what the symbol meant. Instead, I showed him a problem that couldn't be resolved by using the numbers he was already familiar with. I used this number line with only the natural numbers labeled:

(Software used to create the above illustration is paperpen 1.0)

He used the above number line to demonstrate these problems:

5 - 1, 5 - 2, 5 - 3, 5 - 4, 5 - 5, 5 - 6

When he got to "5 - 6" he pointed to the unit that's to the left of the zero and wanted to know what it was called. Sister interrupted to say it was "1", he said, "No, this is 1, and pointed to the right of the zero. I guess what I tried to do was give him a concept in need of a name, rather than a name in need of a concept.

Six months later...

It occured to me this morning to teach him addition and subtraction of negative numbers. Rather than resorting to analogies such as "negative numbers are like temperature/debt/rubber duckies/electrons" I was going to use real math concepts. In other words, perhaps if I could get him to apply the definition of subtraction he could discover a few things for himself.

The first step is for the kid to understand the definition of subtraction.

Addition: a + b = c
Subtraction: c - b = a

That should be easy enough since working with fact families and writing "2 addition sentences and 2 subtraction sentences" gives the kid a working understanding of this. At a later date, when the kid was doing word problems I had thrown in a "?" in the mix.

Now, this was the first test (for the teacher, not the kid). If he had indeed understood subtraction to be "partitioning" or "taking away" or "extracting" he wouldn't have got this correct because he wouldn't have been thinking about it correctly:

I didn't give him any explanation, I just let the results speak for themselves. He did ask me however, if a number plus the negative was always zero. Bingo, an understanding of what it means for something to be an additive inverse, only without the fancy term. He also "discovered" that subtracting a negative five from zero resulted in 5 and he told me this. He found it very amusing. This is my idea of "discovery math". Being very brave, I decided to try the next step out and here's where everything fell apart. Again, I offered no explanation, I just wanted to see what would happen.

I let all that go without comment. I think my next step will be to try to get him to realize the contradiction involved in some of those answers. What's more remarkable to me is that had I asked him prior to this exercise what 0 + 5 is he would have gotten it right without hesitation.

Edit: Forehead slap.

I can't simply get him to assume that -5 = -2 + -3. I think one would need the distributive law to get there, right? -5 = (-1)(5)=(-1)(2 + 3)=(-2 + -3)

Yeah, that's over the top. Never mind.

Algebra at Our House

2007-09-27T09:17:00.000-07:00

Here is what my eleven year old did today. The problem was from Singapore's NEM 1. I'll show you the work and then explain what it means and what it took for us to get to this point.

The problem: "One number is greater than half of another by 15. If their sum is 48, find the numbers." He begins by declaring the variable and stating the given information.

The bracket indicates "and" and the arrow is an implication as in "If a then b" "a ----> b" or "implies" Before attempting any algebra problems the book teaches logic. Note the number "1" on the arrow, at the end of the work the kid then provides the justification that was used for that implication. The use of the symbols (bracket, arrow) is simply a continuation of what he was already doing in the logic chapter, so this is not new. Here's the rest of the work:
The interesting thing to me was that Frank Allen doesn't have the kids "group x's" or "collect like terms" he requires them to justify this using the distributive law. I suppose at some time after the kid knows to the core of his bones why it is permissible to "collect like terms" he'll be permitted to skip this step and simply go straight from x + x to 2x. Before I saw this done in this manner (using the distributive law) I had no idea that there were skipped steps or that it had to be justified.

The other thing that Frank Allen does is list out what things you can use as justifications. Very handy. Very helpful to know the rules before you play the game. I'll note, this isn't done in the books that I have that claim to teach adults to prove things and that is very frustrating. Our idea of rote memorization in math is to memorize the list:

This list was almost meaningless at first to my son, but in Frank Allen's Algebra, he takes the kids through these properties one at a time and he shows them the significance that each property has. For example, the distributive property states that you can distribute multiplication over addition. But can you distribute it over subtraction? Not according to the property. Allen then shows how the rules can be used to derive "multiplication distributes over subtraction" (and he does this without the concept of negative numbers which hasn't been introduced at this point, we haven't gotten to that property yet). In Algebra I the kids aren't expected to do a lot of proofs on their own, only supply the justification for the steps; it's in Algebra II they are expected to prove all the theorems that came up in this first book. Meanwhile, they are slowly working their way in that direction.

Other permissible properties are the properties of equality: reflexive, symmetric, transitive, and then some things that go back to Euclid "equals added to equals are equal" (addition property of equality and Allen states subtraction, multiplication, and division in the same manner since up to this point he hasn't defined them as inverse operations yet).

Finally, the kid can use the definition of operations as justification if he needs to. That should should cover everything. The justifications are the properties of the real numbers, the properties of equality, and the definitions of the operations. Hopefully, if I'm missing out on something someone will let me know.

Also at the top of the list are the symbols used for the different sets of numbers. The topic of "sets", and things that can be done with sets, were covered in the chapter before logic. It was because of this that my son didn't have a problem when he came across those symbols, they are merely a reminder of the relationships.

Finally, the child provides the justification for the steps in writing. My son normally doesn't do this since it's time consuming, but we do do this orally. Writing it down every now and then is good practice though:

An interesting thing that I just figured out not too long ago is that "arithmetic facts" really aren't "facts" at all; they too can be derived from a set of properties (Peano's axioms), five to be exact, but it takes a billion sheets of paper to write down all the steps. My contribution to the prevention of global warming will be to save the trees and assume the arithmetic "facts."

Interesting philosophical questions that have come up in the past few weeks:

"You know that 0 + 3 = 3 because you memorized that as an addition fact. How do you know that 433 + 0 = 433? That wasn't memorized."

"When you add two whole numbers how do you know that you'll get another whole number?"

"When you multiply two numbers less than one, how do you know that their product will never exceed 1?"

"When you multiply two whole numbers why is it that there can't be more than one right answer?" (This led to a discussion about uniqueness proofs, Allen does the first one in proof on why any number has only one multiplicative inverse. I thought that was clever since the property doesn't say that any number has ONLY one. That isn't given at all. So you must show that it has only one by using the other properties in the form of a proof.

Math isn't turning out to be just word problems and solving for x. It's turning out to be philosophy, logic, and lawyering assertions all at once.

Sailing Pics

2007-09-26T09:08:00.000-07:00

If all goes well my eleven year old will be ready to sail the tall ship Elissa on a youth crew March 20th. I'm jealous. It will take me a bit longer to complete the requirements in the adult program. I figured out how to record video and uploaded about five minutes worth to youtube. The first half is the ship, the second half is us playing hookie from seamanship school and getting out on the water. Meanwhile...

Quick Update

2007-09-19T07:32:00.000-07:00

We are still doing the math thing. Still making progress somewhere in chapter 5 of Allen's Algebra I, proofs and all.

I thought Talk Like A Pirate Day would be an appropriate day to talk about our latest fantasy which is to buy a sailboat and homeschool the kids while living aboard.

And with that end in mind, behold our latest field trip, pirate song included:

(Edit: I removed the link)

What could be the possible connection between math and sailing? Celestial Navigation.
Best true pirate stories for grown ups that I'm reading: "The Pirate Hunter: The True Story of Captain Kidd" by Richard Zacks. It's awesome. I've got a whole stack of others to read but this one is one of the best.

If the video doesn't show try this link.

Logic Game

2007-06-07T09:33:00.000-07:00

A family member gave me an old math game that she had from the 60's. Before I uploaded some jpegs of the booklet to this particular game I searched online for some mention of the company and lo and behold they're still in business and am I the only one that's never heard of them before?

"The WFF 'N PROOF games came out of the ALL project (Accelerated Learning of Logic) at Yale Law School. This project was established in 1960 to develop materials to teach mathematical logic to elementary school students. The authors' first principle in designing the games was that they be fun to play."

Wff'N Proof

We ordered the symbolic logic game which was designed by a logician and supposedly can be played by children as young as six. We'll see. It's hasn't arrived yet.

Here is a better description of it.

And here is a blog entry on it with commentors happily reminiscing. I also found this sample from it posted on line: The Tardy Bus Problem. There's also something about Polish notation, looks like a veritable treasure trove of wisdom.

There are other games besides the logic game, the one I have ("Configurations") is meant to be a solitaire game based on finding finite geometries with ten or less points. There is one that is called "Propoganda" but the description of it seems to be of informal fallacies.

I've been occupied with health issues and volunteer work so blogging's been slow. I'll try to update soon about how that logic chapter is going. It goes fine when we do it, but with summer here everything has slowed down to a crawl.

Math and Logic

2007-05-23T03:25:00.000-07:00

Someone asked how New Math integrates logic and math. And as it happens Right Wing Prof just wrote a blog entry on just this very issue. I'll banter about mundane details and then get to the less familiar.

As you know, I am slave-driving my eleven year old through this very same book. We've had two minor hurdles so far. The first was graphing inequalties.

I have no idea why initially this was so difficult but 200 problems later he got over it. Sometimes you just can't neatly cover a topic per day as planned.

(For those not using this book, graphing inequalities on a simple number line can be found on pg. 584 of Foerster's, no exercises on this topic in Singapore NEM or in Jacob's. I have no idea where this is covered in Teaching Textbooks)

Not only did he have to graph them but he had to negate them and graph that as well which meant he had to learn DeMorgan's Laws...only they didn't call it that. It's not called anything, it's just used. It's interesting how not everything is taught directly in this book, sometimes you learn it because it's a result that you derive in one of the problems in the exercises. For example today, we came across how to write an equivalent statement to an implication. Using a truth table he "discovered" (does this make it discovery math???) that " ~p or q" is an equivalent statement to p ---> q.

The second hurdle was learning to deal with the form of assertions rather than their actual meaning. The symbols of Ps and Qs are one thing, but natural language was another and he really tripped on the fact that he was required to negate something that was really true or that he should state a negation that was false. In other words, the objective was to ignore the actual content of what was being stated and just toy with it.

"If two lines intersect, then they are parallel."
Contradiction: Two lines intersect and they are not parallel.

The statement "Some squares are rectangles"is false.
Contradiction: The statement some squares are rectangles" is not false.

Where is this all leading to and what is the monster at the end of the chapter? A soon upcoming topic in this chapter is going to be indirect proof.

"Sometimes in order to prove that a statement is true we resort to the strategy of proving that its contradiction is false. This maneuver is known as proving the statement indirectly."

But you can't do that if you don't know what a contradiction is, hence, the chapter on logic which also covers the converse, inverse, and the contrapositive. The problems in the end look like this:

-Prove that that the null set is a subset of every set. (Not technically left as an exercise for the reader.)

How about this one, I have to write it in prose since I don't know if they symbols will show up properly,

A, B, C are sets. Suppose that A is a subset of B and B intersects C equals the null set. Prove the statement A intersects C equals the null set by the indirect method.

Sounds like something that was befuddling to me in Hernstein. I'd be thrilled to see my son even attempt it and I'll be sure to go back and attempt the exercises on set theory again in Herstein if only to confirm that working through an, ahem, ninth grade algebra text wasn't what I needed.

By the way, the Dolciani Algebra II text has incredible sections of "oral exercises" which amount to mental math. Dolciani expected more mentally than Jacob's does written (I know, I know, apples to oranges, I don't have Doliciani's book one though) and she very explicitly states that the student needs to be very good with the standard algorithms and offers plenty of practice.

"...my daughters have been studying (Chemistry) at the University for several semesters already and think that they have learned the differential and integral calculus in College; and yet they still don't know why,
x * y = y * x.
Edmund Landau

For more information on the history of rigor in mathematics see this article written for the lay-person. Because of the connection it makes with the Greeks, drat them, this may prove of interest to those involved in classical education.

Math motivation: now and then

2007-05-16T18:09:00.000-07:00

Got a 1963 Dolciani (Algebra II) in the mail today.

If any of you are still looking for a proofy algebra book this is another one. A few people have asked about the Allen texts, I can't find a trace of them online any more. I still prefer Allen's text because of the gigantic chapter on logic that we're mired in, Dolciani lacks that. There doesn't seem to be a "development" of algebra via the proofs, but there is clearly a lot of time spent on proofs, not just filling in the justifications but also in learning about the different varieties. They look like fun, "Prove 1 is greater than 0." The set theory is de minimus.

Photo above is from a side bar on statistics. "This mathematician is constructing a statistical model for analyzing production data."

Here's another one:

" Increased technical knowledge has produced new aids for education. The professor above uses an analog computer to solve a problem in chemical kinetics involving exponential equations. The solution appears as a curve on the oscilloscope."

I ought to tell you something more substantial about this book but it's the photos that make the biggest impact. Buzz cuts standing with chalkboards full of equations and gadgetry. I wonder how much that computer cost back in the day? What do all those buttons do? Oooooo, I want to touch them!

Compare and contrast Dolciani to an arbitrary modern college algebra text that I picked up at a used book store:

The introductory chapter:
1963: Face of the moon, caption reads, "The mathematical knowledge which aids modern astronomers in their study of the moon, seen above through the 100-inch telescope at the Mount Wilson Observatory, was not available to the 17th-century astronomer (at right) with their "celestial machine."

2003: Photo of bees on honeycomb. No buttons, no people, just.....bees. Caption promises there will be thought provoking explorations available through internet resources. No buttons. And did I say no buttons?

Chapter on Quadratic Equation
1963: A scientist working on the nose cone of a missile. Another illustration, a woodcut, of Faraday in a highly geeked out lab. First portion of caption reads, "A knowledge of quadratic relations is basic to an understanding of the mathematics of low-temperature chemistry..."
2003: An affectionate couple is walking along a wooden pier lined with docked sailboats. The caption contains a reference to this activity, "you will be asked to verify the hypothesis that income rises with age, reaches a peak, and then declines..."
Chapter on Functions:
1963:

2003: Traffic. Sunset.

Chapter on Progressions and Binomial Expansions:

1963: Buzz cut with black glasses looking intently at manual typewriter which can both be use to type musical notation while simultaneously making a punched paper ribbon copy of it. Again, it's the "lots of button" theme.
2003: A biker laying down his bike, can't see a human in the helmet. In fact, in no picture can you ever see a human. No buttons either. Caption killed whatever inspiration I had...

Adventures in Space Surplus

Numbers and numerals and names, oh my!

2007-05-09T16:08:00.000-07:00

My own personal newsflash: Doing math is easy as long as you don't have to define anything.

Follow the progression and tell me where I got lost in this book.

Fractions are not formally defined but you can use them.

Pg. 149 Division is defined as the inverse operation of multiplication. The fraction bar is the symbol denoting the operator

pg. 163 The multiplicative inverse is defined and the same notation is used to denote it as the fraction and division.
Proof on page 163 shows that multiplying by the multiplicative inverse "produces
the same effect" as dividing by a number.

In other words it is to be shown that a/b = (1/b) * a.

The problem is that if you let a = 1, then you are supposed to show that 1/b = 1/b. That is, that 1 divided by b = b inverse. The fraction bar on the left side is supposed to denote division, but the fraction bar on the right side is part of a symbol that denotes the inverse of an element.

Now, to make things more confusing, they really are equivalent, which is what we are supposed to be proving here, but up to this point they are supposed to be distinct things. Maybe it would have been less confusing had a different "name" such as b' for the inverse of b instead of 1/b. Then what you would have to show is that a/b = b' * a. Toss in a typo and things get really fun.

It's a way of talking about a single idea three different ways using the same notation each time. Not unlike Carlos Mencia's "dee followed by dee followed by dee."

Of course, I brought up this craziness of proving tautologies to Tater, and he said that if I wanted to prove things that weren't always true then maybe I'd be interested in physics.

P.S. I was glad for my initial confusion with this problem because it brought a post on the Text Savvy blog to mind. It discusses a similar problem that children have in seeing an expression such as 2 divided by 3 (2/3) and the answer, the number two thirds: "More On Troublesome Fractions."

Another Interlude

2007-05-04T19:23:00.000-07:00

First thing's first since this is what most folks want to know about: Adrian has another screed on Calculus and New Math.

And now me:

I thought I could pop into Amazon and order the Well-Tempered Clavier in a few clicks, but it turns out that there are 236,476,209 recordings of it and I got bogged down in reading reviews for hours until I decided on the "best" recording. I got this one by Bernard Roberts.

I am fascinated by Prelude 3 and so spent a few hours this afternoon trying to play it. I have no idea how the fingering goes so it took a lot of experimentation on my part. Fortunately, I found a good video of it on YouTube to help me out, but don't for one second think that I can play nearly as quickly as this guy. I can play about the first nine seconds of what that guy does and at my best I can only do it at a third of the speed. If I only use my right hand I could keep up with him, but when the Mr. Left Hand joins in everything sloooooows doooooown. It might also help if I could sight read music and not constantly have to say "every good boy does fine" to figure out the notes.

I also found some fun videos of kids (not mine) playing the piano and for some time I've been looking for an excuse to blog about them so here they are.

Girl #1 - a montage of her progression over 18 months. Incredible

Boy #2 at piano

Ricky Skaggs at age 7 singing Ruby, my favorite bluegrass song.

The quality this next one is very poor but at the end when the kid looks into the camera you can tell that he's not more than about nine or ten years old.

Another Sample of New Math

2007-05-02T17:07:00.000-07:00

Right Wing Prof, who commented below on "Blog A Log" entry, has a post up on a lesson from Frank Allen's Algebra I book with sample pages. Take a look. Also, the table of contents is here

I will be updating my own blog on progress my son is making in that particular book. There will be more for me to say in September than right now, we haven't even begun to hit any proofs yet. We're still in the unit on logic and it's come to pass that my son "gets it" before I do. Very sad day indeed for Myrtle who is being shown up by a fifth grader. I'd like my university to refund me my tuition.

Worrying about State Standards

2007-04-25T16:07:00.000-07:00

The above, if built today, would meet my state's (building code) standards.

But this wouldn't...

I was going to say something about state math standards too, but I forgot what it was.

Bond-Aids

2007-04-17T10:16:00.000-07:00

I am not entirely convinced that a young child that doesn't understand that subtraction is the inverse of addition needs to back up all the way to the use of manipulatives. I think there is another way of looking at the relationships of the terms involved that might give the insight needed in order to formally manipulate a problem such as 40 + ? = 55 and rewrite it as 55 - 40 = ?

Each addition or subtraction problem can be rewritten as a relationship between numbers represented by a big circle (sum or minuend) and two smaller circles. By P2 the children are already familiar with this triangular representation of the relationship between numbers in addition and subtraction problems. I gave my son the blank number bonds and he filled in the circles according to the equation that was given to him on the left. He even included the question mark.

I think he needed somewhere between 15 - 20 of these before he caught on, however, he was already familiar with number bonds. The second step was to have him translate number bonds into subtraction or addition problems, again this would be a review for a child in Singapore and if your child isn't familiar with this then you might want to start with this step. I provided the question mark, he provided the operator. Here is what that format looks like
This is fairly easy, if the big number at the top is missing, it's an addition problem, if a small number is missing it's a subtraction problem. I didn't explain it to him that way, whenever he'd write down something incorrect such as 3 + 7 = 4 I'd respond to the equation written instead, "So you are saying here that if you have a small group of 3 and a small group of 7 that together they will make a big group of 4?" In other words, I asked him to justify his assertion.

The third step was to begin with the problem, translate it into a number bond, and then translate that into a subtraction or addition problem:

Once this is mastered, the next step is to remove the number "bond-aid" so that the child can take an addition or subtraction problem on the left and go straight to the step on the right.

Singapore Bar Models: Primary 2

2007-04-13T20:03:00.000-07:00

I am going to demonstrate how I taught word problems on the topic of weight to my six-year old. This topic is introduced at the end of the first grade (along with length) and at the beginning of the second grade. It is difficult because it is harder for weight to be "abstracted" to a bar model than it is for length. I think there are a couple of reason for this.

-Long skinny rulers look like long skinny bar models.

-The manipulatives used for teaching measurement of weight are lumpy piles of objects in bucket balances.
-Other activities on the topic of weight in Singapore in the early grades are really more about reading scales than about weighing or word problems. This is great for learning how to read a scale, but it is not so good for learning word problems.

A typical one-step weight word problem in P1 is phrased something like this,

"Kelly weighs 11 kg. Susan weighs 3 kg less. How much does Susan weigh?"

The same topic in P2 simply uses larger numbers,

"A papaya weighs 850 grams. A mango is 170 grams lighter than the papaya. What is the weight of the mango?"

Using hex weights, which are stackable, I modeled the weights of the two items in word problems like this (see pic below). To be prevent confusion I also rewrote the word problems so that the weights involved could be exactly represented using the hex weights we had, I did not always use the original numbers in the word problems. The resemblance of weight to the bar model next to it is clear....to adults.

In fact, when I stacked up the weights my son did not "get it". I had to back up, place the hex weights on paper, and trace around them. When I lifted them off the paper he could then see the connection to the rectangular bar model.

The outline of the bar model above can be used in all of the addition and subtraction problems in the early grades. With an exacto knife and a piece of card stock I cut out a template with which my child can trace the same bar model over and over again. What remains is the process of labeling each bar, writing the given information in the correct place, and identifying the "?"

Not related to modeling itself was the fact that my son would just as likely write the corresponding "number sentence" as 3 + ? = 11 as he would write 11 - 3 = ? I suppose that is another issue for me to discuss in a different post. Within the next couple of weeks I will be teaching two-step word problems on addition and subtraction and I will post an update on how to do this with the basic two column bar diagram from above. I've already done this once with my older son so this should be a bit easier than it was the first time.

You can order hex weights from Rainbow Resources.

Classic Adrian on the role of calculus

2007-04-11T20:07:00.000-07:00

I wanted to share with you the typical of the kind of discussion I hear in real life so I've lifted this from elsewhere online (beause it's a fair representation) and reposted it. After you read it you can imagine what sorts of issues that someone without a background in math, such as myself, has to reconcile when confronted with these kinds of statements.

The original question posed was "Should everyone take Calculus? and "Is Calculus the best course of study for all students?" The following is Adrian's response, not mine:

The running joke in our house is "it's a promotional topic". That's what Andre Toom said about teaching fractals to high school students. It's just selling points -- saying "we teach X!" The truth is that calculus is a highly advanced subject that should require at least a year of abstract algebra followed by point-set topology as its pre-requisites. Then, you can step into your topology class with a background proving theorems directly before you try to tackle the much more subtle indirect arguments encountered there. You will also have the knowledge of the axiomatic development of the rational numbers and motivation for developing the real numbers and the complex numbers by way of irrational and complex roots and transcendental numbers. Topology will come in and give you a geometric foundation and context for doing calculus. Your main goal, though, would be to really master the contrapositive argument and epsilon-style arguments. Then, finally in your calculus class, you would be developing the real numbers rigorously using Dedekind cuts or equivalence classes of Cauchy sequences of rationals or some such thing, reviewing your algebra and then reviewing the topology and then moving on to the extraordinarily powerful theory you can develop from there. You would never have to come up with heuristic arguments for some theorem which is nothing more than a spurious and fallacious way of justifying formal a priori assertions like math theorems. Instead, you could just prove it the right way the first time and not be trained to think fallaciously. Of course, the class I am describing is usually called "Analysis" and it can often make or break a math major in their junior or senior year largely because they tend to take it concurrently with Abstract Algebra and with barely any experience proving theorems. So, then, they just get blitzed with everything all at once and it all becomes a big IQ test or more likely a test of one's determination and resolve.

It is bad enough that we try to teach this to incoming freashman who will mostly comprise the next generation of civil engineers and who couldn't care less about topics that are 2 or three times removed from anything they will ever have to think about when they're done. Now in the eternal escalation of bragging rights we are trying to teach it to high school students. Pretty soon calculus will be included right along with coloring in kindergarten. Of course, it doesn't matter if everyone in the world starts doing that or if they out number mathematicians 1,000,000 to one. Just because everyone starts trying to turn calculus into a child's toy, Analysis will not just suddenly cease to exist. And, then more than ever a good education will become increasingly impossible and completely unkown -- hidden away -- for rank-and-file members of society. Only a select few people in the in-crowd of folks that were lucky enough to happen upon it will even know about it.

I just don't think people, including the very mathematicians that ought to know better -- that write all these calculus textbooks, in the first place -- appreciate that it is getting worse and it will just keep getting worse. All that our efforts to educate the masses has accomplished is to make the real content of an education all the further removed from the masses. And, the few people that really want it won't even be able to find it let alone acquire it. We are selling out the people we should care most about and the real future of the field when we water down our material for commercial use. And, all just so that one person can look down their nose at another and say "*I* took Calculus," when neither of the two could possibly care less about math or the actual content of the very shibboleth in contention.

Spoon-Fed Math

2007-04-11T10:49:00.000-07:00

I must have overdone it with the whining about mathematical induction at my house. Some people just don't appreciate having a six week long philosophical discussion about something that ought to be mastered in five minutes.

The original problem, the one that I skipped, is this:

Prove by mathematical induction from Axiom II tht if a₁,a₂,...,a_n are positive, then the sum a₁ + a₂ + ... + a_n and the product a₁a₂...a_n are positive.

Axiom II: If a and b are members of the set P of positive numbers, then the sum a + b and the product ab are members of the set P

And my idea was that I could set it up as an inequality greater than zero since after all that's what it means for a sum to be postive. But no, I was told that ">" hadn't been defined so I couldn't use it. I had no response to that. Glazed eyes and a string of drool don't translate well to math. Subsequently there was a great confusion involving dominoes and bumper cars, suppositions, hypotheses, and implications. And short of implanting a feeding tube I'm not sure it can get any more broken down and spelled out for me than this.

Oh, the humiliation.

But thank you. I know that took a lot of time to do.

Down the Rabbit Hole

2007-04-08T19:38:00.000-07:00

It makes me cross when I can't figure something out. To wit, my success rate:

In the first problem set from this book, which I thought would be some sort of glorified algebra review. You know, "when you multiply both sides by a negative you flip the signs." Just read the reviews and see how wrong I was. I had to skip the problem on induction. That alone is very, very sad. I didn't even understand the solution in the back of the book.

Second problem set:
1. I thought I proved the first one, but didn't, and it took a good deal of explaining.

2. I reworked the problem four times trying to come up with a different outcome. As it turns out my outcome was correct, but I didn't recognize it as such. I had to go back and reread several sections as well as get my own personal Blue's Clues spoon-fed lecture.

3. Got it right but figured it didn't "count" since it was too similar to # 2.

4. Another problem involving induction: Prove the general ">" transitivity rule for inequalities.
I had to ask for help. And that's Nick Burns, Your Company's Math Guy, helping me out.

I think I'm going to have to start a notebook in which I will write every proof by induction I can find. Lagnappe for bearing with me...

Watch Video

An Inconvenient Tooth

2007-04-06T18:36:00.000-07:00

- Let me pull it for you.
- No.

Has the AP revolution changed high school math?

2007-04-01T18:32:00.000-07:00

Roy Smith speaks about high school math, and AP courses, especially AP calculus. I've chosen to reprint the bulk of his post. There is a fascinating and very long discussion that ensues at the above link in which the objective of high school math is debated. The participants seem to be research mathematicians, graduate students, and high school students. For those of you who found the Singaporean thread critiquing NEM interesting, this one is even better. Roy discusses SMSG materials later in the thread and I subsequently emailed him asking if he could give me specific titles. The emphasis within the text in mine.

"Some remarks on high school preparation for a college education:

Some time ago I argued that XXX could not be considered a particularly "hard" school in comparison with many others, because it had so few AP courses. Now that this is changing, I have begun to have some reservations. I had hoped AP courses would strengthen the program by upgrading weaker course offerings, rather than competing against the excellent courses already in place. I believe that in the country as a whole, this may have been a principal result of the proliferation of AP courses, tending to a sort of standardization of advanced instruction, bringing a reduction in quality of education at good high schools, rather than an overall upgrading of the level of the average course offering. I also did not realize that graduates of "Advanced Placement" courses would take the term too literally and try to place out of substantive courses in college which they should have taken. In the case of those AP students who repeat beginning college courses I have also found the problem of trying to teach in depth a college subject to people who think they have already learned it.

I am most familiar with mathematics, which I teach at the University of Georgia, so I use that example. The AP designation in calculus refers to a specific list of "topics" on which one must be prepared to work problems. A year of this AP material coincides with the content of one or two quarters of non honors college calculus at Georgia, but a full year college course, and especially an honors course, not only covers more ground, but treats the material at greater depth. It is ironic that AP calculus courses, which are taken by honors high school students, are comparable at best to non honors college courses, which the best such students would not elect.

As a result many entering AP college students place either into advanced, but less stimulating, non honors courses, or into intermediate honors college mathematics courses for which they are not prepared. Before the AP revolution, students prepared by getting a better grounding in algebra and geometry (and sometimes logic) than is found in high schools today, then took a first year college calculus course which included theory. Introductory college calculus courses for gifted mathematics students which teach theory as well as computation are hard to find today because so many students exempt this course with AP credit. The disappearance of the most outstanding introductory college calculus courses is thus a direct result of the proliferation of significantly inferior AP courses.

In view of its unsuitability, it is ironic that AP credit has begun to be designated as the "prerequisite" to some advanced courses, even though the true prerequisite for advanced work is often just the ability to think in a certain way. This may be the case even when the college catalog says otherwise. At Stanford for example, the prerequisite listed recently for honors intermediate calculus is a certain score on the AP calculus exam, but when asked, the departmental advisor said "Of course that's not the real prerequisite" (his emphasis). The real prerequisite? "To be able to handle proofs, with no apology". The book used in that course is volume 2 of Apostol, an outstanding text treating calculus with theory. Presumably the right preparation is to learn beginning calculus from volume 1 of Apostol, but where can the interested student find such a course? Stanford does not offer it (that's the course that was replaced by the AP courses), and it certainly is not available in most high school AP classes; (books used in the XXX course are ordinarily one or even two levels of sophistication below Apostol). The result of this at Stanford is roughly a 70% attrition rate (after the first week!) in the honors intermediate calculus class, among those students who have the required score on the AP test. Surely many of those students who must drop out are disappointed that they are not in fact prepared for the course, and possibly the career, they had wanted.

Unfortunately AP calculus courses and the standardized testing mentality have helped to eliminate, not just from the college Freshman mathematics curriculum but also from high schools, classes in which theory and proof (i.e. systematic logical reasoning) are taught, since "proofs" are seldom included on AP tests. For example the 1982 and 1987 AP BC calculus tests in my practice book have less than 3% proof questions, whereas the exams in the Stanford course above are said to be 100% proofs. This phenomenon has accompanied years of decline, and the current near extinction, of adequate teaching of geometry in high school, which worsens the problem of learning either calculus or deductive reasoning.

I conjecture that these negative effects are not so great in some subjects where AP exemption is less common. For instance my impression is that in the recent past students from XXX's non AP honors English courses have been superbly prepared for beginning college courses in that subject. Presumably the reason is that in these classes, students learn to read, write, and discuss their ideas. I hope these courses are never replaced by ones designed to prepare people to answer multiple choice questions on the correct author of some obscure poem.

Interestingly, although our data at the University of Georgia shows no correlation (and even some negative correlation!), between scores on the quantitative SAT test and performance in our precalculus and basic non honors calculus courses, there does apparently exist a positive correlation (almost a direct one) with scores on the verbal SAT test. My own theory is that the verbal test, as bad as it is, at least measures vocabulary (mathematics is a language), and the ability to comprehend what one has read. Consequently the demise not only of instruction in reasoning in mathematics, but the decline in the ability of the average student to read and write, has steadily tracked the drop in performance also in basic mathematics courses for non honors students. (One might even argue from this that the claim that Saxon's books raise SATQ test scores, also suggests that they may lower average performance in college mathematics courses.)

What is my conclusion? I suggest the school seriously reconsider the practice of creating AP courses in subjects which are already represented by excellent honors courses, since this may well lead to the demise of the superior course, and a decrease in the quality of student preparation. In subjects where AP courses already compete with non AP courses, I strongly urge students to select the course which involves the most writing, and the deepest analysis, without regard to which one boasts a syllabus sanctioned by the Educational Testing Service. In cases where an AP course has already driven a superior course out of existence, I feel there is a strong argument for creating, or recreating, a non AP honors alternative.

I do not oppose taking AP classes in principle, but since (in my experience) they do not play an appropriate role in advanced college placement, I do believe they must justify themselves based simply on their educational merits. I also strongly suggest that a graduate of an AP class consider taking an introductory college honors course in the same subject rather than skipping the introductory course altogether.

The only case in which I see a reason to consider creating an AP class is in a subject where the existing course work is currently on an inappropriately low level. Even in such cases I think it likely that a non AP honors course designed by the teacher may be an even better option. In my opinion such an opportunity exists at XXX in the physics program, which I understand does not ordinarily offer a calculus based course. One possible way to make good use of the existing AP calculus course would be to offer a subsequent or concurrent calculus based physics course, or even a course that combined the two subjects. Since Newton invented calculus precisely to do physics, this is one of the best possible ways to learn both physics and calculus.From my own perspective I believe there is also a real need for new substantive mathematics courses which are not just oriented towards performance on standardized tests. When I tell my colleagues at the University of Georgia that XXX does not offer a year long course in geometry for example, they do not readily believe me. I would also like to see innovative, faculty - sponsored, courses on other subjects of current or abiding importance in mathematics and related areas, such as linear algebra (an easier and more fundamental subject than calculus), finite mathematics and probabilty, computer programming, algorithms, numerical analysis, or computer aided design.

In general, I believe those of us who are "consumers" of XXX educations, parents and students, should have faith in the knowledge and scholarship of the teachers; these outstanding individuals should be considered at least as qualified to select the content of their courses as the faceless people who write standardized tests. Some of these teachers value and use an AP syllabus in their own courses, which is a recommendation to me of the positive aspects of some AP programs. Others prefer to design their own curricula. Such distinctive courses offer opportunities unique to XXX, and I believe they play a large role in the school's impressively successful identity. Some teachers may even be holding back exciting proposals thinking we want only standardized education from them. I hope such individually conceived courses will continue to be encouraged, and valued for the rare gems that they are.

Axiomatics

2007-03-25T19:00:00.000-07:00

1. I had a six hour conversation today with my husband on proofs vs. calculations. How can I be so sure that this proof approach is the right thing to do? While I personally think it's entertaining, how do I know that the kid will do okay on the SAT or won't be counting on his fingers later on?

2. I learned that "Birkhoff's postulates" are sometimes pronounced in the same breath as "Hilbert's axioms" but that no one has ever heard of "Jacob's axioms" except Solomonich who describes Jacob's Geometry as “Irish soup” of various approaches (Birkhoff, Euclid, Hilbert, - all in one pile, – the more the merrier) complemented with a few superfluous postulates proposed apparently by the author himself in spite of already having too many of them." No doubt the devastating rejoinder that completely deflects Solomonivich's criticism is "whatever works for your family."

3. I've exhausted my interest in the topic of Fuzzy Math and I'm now interested in Junk Geometry.

4. Birkhoff's Geometry, the answer key, and the teacher's manual will be joining the ranks of the other geometry books we have. Someone *cough cough* is turning into a math curriculum junkie. More info on Birkhoff in Wiki.

3. Singapore NEM doesn't get around to the quadratic formula until NEM 3. That's right, the ninth grade which is how we do it in the United States. My fantasy about NEM covering all of algebra I and II before the tenth grade has been thoroughly disabused.

4. I've discovered that Frank Allen teaches algebra "axiomatically" more or less. He defines a "well-ordered number field" and then uses those properties as axioms in all the following proofs. I spent all day yesterday flipping page by page through the book typing up the theorems. I even had a fantasy about putting it in HTML online. If I do I'll post an update here.

5. Patrick Suppes, a logician, makes the argument for why high school math should be developed axiomatically. This might be interesting for those interested in the role of logic in a classical education and math.

6. Apostol Calculus arrived in the mail. I can only give hear-say testimony about this book that seems to have a cult following. Let's see, it was used at Cal-Tech for several decades. It is used as honors calculus at UW Madison, (home of Rudin), SUNY and MIT. The introductory chapter has problem sets that look suspiciously like what was done in Frank Allen's book but bears no resemblence at all to anything in Singapore math ...or Thomas & Finney. And this brings me to my last observation...

7. EPGY uses this book for Analysis in their gifted youth program. I would have never thought of EPGY as the type to offer a "pathetically dumbed down calculus course" but here is one reviewer's take on the kind of math program that leads to the necessity of using this book:

I don't understand people that constantly knock this book.The vicious barrage of critisms levied against this text is usually by arrogant math majors at top level schools.Thier attitude is basically that,"If Rudin is too hard for you,you are too dumb to learn this,get over it." You know,the first edition of Rudin was written over 4 decades ago, when calculus was usually first exposed to high school students on a regular basis and eplison-delta proofs were not uncommon in a college level calculus course.Therefore,after a meaty,theoretical calculus course that taught limits,derivatives and integrals carefully in addition to related rates,differential equations and the applications that today's watered-down calculus courses laughingly consider mathematics,those students of past generations were READY for something brutally terse like Rudin.The sad truth is that in today's pathetically dumbed down mathematics eduation system in the US-where high schools are happy if they can get students to use thier CALCULATORS to add and subtract correctly-Rudin or Apostol are simply way past the preparation level of any but the best students after calculus.The need for a "bridge" course that gave students the minimum exposure to a hard core approach to calculus was realized in the early 1980's-and Ross' book is still,to me,the best of the lot.

Hmmmmmmmm.
Maybe my husband isn't crazy about using an axiomatic approach. Maybe he is right.

Blog a Log

2007-03-17T22:41:00.000-07:00

Frank Allen's lesson on logarithms. This is the very first introduction to the topic in the book. From the stamp on the inside cover this text was used for high school Algebra II in the state of Florida at some point in the 1960's.

The copy I have appears to have never been used at all. It's been sitting on someone's shelf for 50 years. Very exciting. If you can't contain your curiosity to see how the lesson proceeds here is the next page:

And for the record, since this information doesn't seem to be recorded accurately any place online:

Modern Algebra: A Logical Approach Including Trigonometry Book Two
by Frank B Allen and Hellen R Pearson
copyright 1966, by Ginn and Company
Library of Congress Catalog Card No: 64-4164

Here is the entire table of contents for book two

Sometime in the future someone will do a keyword search and this lowly blog entry will be the only hit. Dear future reader looking for this book, there are only four copies of this book in university or public libraries in the United States, get thee to your librarian and get an inter library loan before the last copy is removed from the shelf and tossed in the trash bin.

It is full, full, full of proofs. Not only does it present each lesson in definition (a rather discursive definition) theorem, proof fashion but following each lesson are two sets of problems. The first set are the typical exercises involving rune manipulations. The second, or set "B" problems, are the proofs. Not all lessons have set "B" exercises but about half do. Unusual topics include systems of quadratic equations, and the gigantic fat chapters on logic and set theory. The authors say that this book is intended for use in a three semester course in high school.

There are still some things that I need to do. I have to harmonize the problems in Gelfand's monographs with this series. I also have to go get NEM 2 & 3 and match up their word problems with Allen's Algebra. We are waiting for Apostol's Calculus to arrive. If this turns out to be too advanced I guess we'll end up trying to get copies of the original SMSG Calculus. It's supposed to be in the public domain but how difficult it will be to convince someone to allow us to photocopy their particular copy remains to be seen. I also would love to get my hands on an older Dolciani and see how it compares to Allen.

Chessboard problem

2007-03-13T14:03:00.000-07:00

The original problem was for a board that was 4 x 4 squares. We extended the problem to a regular 8 x 8 board and saw an interesting pattern.

The problem: Move a token from one corner of the board to the opposite corner. Each move may be "up" or "to the right." How many different ways can you arrive in the opposite corner?

Step 1: On my first move I only have two choices.

That means that there was only one way in which I could arrive at those adjacent squares. How many ways to the diagonal square?

Only two ways to get there:
And there are three different ways of arriving at this one:

Rather than illustrating each path what I need is a rule on how this works. And the rule happens to be: Add the square below it and the square to the left. My son wanted to multiply and even my husband ( for about .0001 of a second) thought it should be multiplied. In case you aren't convinced that adding is the key insight I'll illustrate using pathways that there are only six ways of arriving at the indicated square. (What's the HTML tag for hand waving?) You can see that there aren't nine pathways which is what you would get if you multiplied.

So, then the way to proceed seems to be to assign each square a number which indicates the number of pathways to get to that square. Each is obtained by adding the one below and to the left. The whole chessboard ends up looking like this and a neat pattern emerges:

Flip the chessboard around and you'll see Pascal's Triangle...

Next time you are here, you'll have something to think about:

Singapore Math: Deficits

2007-03-09T08:58:00.000-08:00

It's Singaporeans talking about us talking about them and their math. The original post isn't interesting; it's all the comments that are worth reading:

HERE

I wish I had found that thread before making that last post. It's useful to hear from people who have gone through the program. Read the original thread to get the context, and of course there is plenty discussion of the role of proof:

"I don’t think we have any idea how to create formulas in the first place."

"a lot of people trying out Singapore Math want to implement it wholesale for K-12. Then they get upset at how Maths Syllabus ‘D’ (E Maths) is missing all sorts of things…

"I don’t think the Sg math syllabus lets us understand how the principles work at all. It’s all a matter of telling us the formulas and teaching us how to use them;"

"I think our curriculum is awesome for the elementary school. After that it kind of loses steam as it gears up toward pre-specified ‘O’ and ‘A’ level syllabuses which may or may not reflect the skills needed in modern jobs."

"I also agree about the benefits of proof-based mathematics for logical reasoning. I grade the homeworks for a lower-level calculus course at my university and the number of elementary logical fallacies students commit is astounding."

By the way, I did finally find a proofy Algebra II text. The librarian said there were only five remaining copies of it (Frank Allen's second book) in the United States and I offered to pay a $20 fee to get it through interlibrary loan. Then the very next day I got an email from a used book seller saying that he too had a copy and sold it to me for $8. Neither has arrived.

I also recently found a neat chess board problem that I guess I'll blog about next time...either that or Greek. I haven't talked about how Greek is going in a while and today is the release of "300."

What qualifies math as higher order thinking?

2007-03-05T05:23:00.000-08:00

Stage One: A couple of years ago I was under the impression that the objective of math education was to get the kid to solve a "problem" as fast as possible. I thought that memorization and speed were the keys to math success, whatever that means, mostly because I had to projectile spew trig identities to pass tests in college, which I did a good job of. So then my experience proved what math was all about. The 340,978,312,945 liberal arts majors out there with the same experience confirmed it and so now we can sit around online and discuss how to duplicate this experience for the next generation because that's all there is to math. And if you need further confirmation that math is all about memorizing and regurgitating ask the engineer husbands of the liberal arts majors.

Application of formulae isn't just a part of math. It is math. The more formulae you have memorized, the more mathy you are.

Stage Two: I became a homeschooling parent reciting various mantras about the importance of math. This was garden variety sloganeering on my part. (The same thing happens when discussing Latin. "It trains the mind.") Lace knitting and learning to diagnose engine trouble trains the mind as well. It troubled me that no one could distinguish what made math special and different from lace knitting or car mechanics. It really should have troubled me that I didn't know the difference but I figured that it was someone else's responsibility to figure this out for me.

The conceptual approach used in some K-6 programs can be, after all, just a gimmick to get the kid to spew answers out real quick. And besides does it really matter anyway? Parents promoting classical education will, from time to time say things like, "I never learned that much math and I am OKAY." (A member of my extended family never made it past the third grade and was basically illiterate. He was OKAY too. He raised and supported a large family and had some sort of an intellectual life by participating in debates about religion.)

Then came along little bits of information here and there that put tiny cracks into that view and by that I mean I couldn't reconcile what I was hearing with my beliefs. Something had to give. I could become more extreme in my view and think the people saying these things were nuts or I could decide that my beliefs about math needed a makeover.

Below are two examples of discussions that chip at me. And what makes these significant is that they are not talking about the causes of failure of remedial students. They are talking about kids who did Tony-the-Tiger greeeeeat in high school, "it worked for them", and just couldn't hack it in college. As someone explained it to me, "The problem isn't that arithmetic is taught as arithmetic, the problem is that math is taught as arithmetic." I think that is supposed to mean that a child needs some basic level of facts and facility with arithmetic, but that then it's time to become "mathematically mature." But what does that mean?

Wait, wait...I almost forgot. Before I get an angry email from someone I don't know who is offended that I might possibly be referring to something that they are doing, I have to add in the disclaimer: Of course no one is talking about your pet program. Your pet program is just greeaat. And you really do know math concepts, because like me, you can now do mixture problems and know where a cannonball will land.

Here is Ralph Raimi explaining:

"It must not be thought that all high school students who are "accelerated" in mathematics, even those who become engineers or scientists, have actually thereby received a benefit. Every college mathematics teacher experiences every year the plague of high school calculus suffered by students who have been told they know something they do not, and who are puzzled and ultimately angry when their acceleration proves to have been nothing at all. The ritual of mathematical rote is not confined to fourth grade exercises; even calculus can be turned into a catechism, and alas, in most cases it is. In fact, most high school students who show early promise, and are therefore moved along ahead of the rest, end up with as stultifying an experience under the honorific title of "advanced placement" mathematics as they would have received under any older rubric, "trigonometry," perhaps, or "logarithms."

It is as if they had learned music by being taught, note for note, over a two year period, how to play Für Elise, without learning anything at all about scales, arpeggios, rhythms or modulations. Painless Beethoven, Advanced Placement Beethoven; but what comes next? The student now wants to play Opus 110, but insists on being
told, note by note, where to place his fingers. It is his right; why are his college teachers now "making it hard," talking about chord sequences instead? Why do they refuse to teach him the Beethoven he needs? He soon gives up, and so do most of our eighteen year olds who are crippled by high school calculus. Yes, most of them. Others, a minority of the elite (and an infinitesimal minority of the general population), survive. "

And this ties in with a comment I saw in a comment box at Kitchen Table Math. Here the anonymous author is responding to a quote.

“The whole point of math is to avoid the need to pull the magical answer rabbit out of the hat. If you know math, then you don't have to think higher. If you can do the math in an instant, is that higher ordered thinking?”

But at some point, math problems become higher order problems.

Warning: Purely anecdotal evidence ahead. :-) I was always terrible at math. In elementary school, when we had to do hundreds of addition/multiplication problems, I’d usually have the lowest score in the class. In middle school and beyond, I noticed that the people who scored at the top of the Math Olympiad, Putnam Exam etc. were the people who were could pull the magical answer out of a hat, instead of using some complicated technique they’d learned, so spent most of the time I would have spent doing homework doing random problems out of interesting problem collections instead.

In high school, my grades weren’t so great. Since I didn’t spend much time doing basic problems, I’d make basic arithmetic errors, leave off a +C when doing an integration, or what have you. HS teachers were pretty strict about that sort of thing, and would usually give me 0 when I did that. My grades got a lot better in college. I didn’t get any better at avoiding dumb mistakes, but Professors would give me 80-90% of the credit for the problems if I obviously knew how to do them but just forgot a step. In grad school, I aced almost everything; I still made the same mistakes I always did, but Professors would just make a note that I missed something silly, and not deduct any points.

Most people’s scores had the opposite progression. They spent years learning how to do problems where you can tell what technique to use by looking at the section header, so they could ace everything in high school. I’d get 80s on most exams and the average grade in the class would be about 90. In college, there were more novel problems, and Professors didn’t care as much about the details, so the scores were reversed: I’d get 90s and the class average would be in the 80s. In grad school, where most of the problems required coming with a novel, “magic”, solution, the average grade on most exams was somewhere between 20 and 60, and almost everyone would gripe about how none of the material on the exams was taught in class. I was used to being forced to come up with magical answers, so I mostly got 100s.

Whether or not you want to spend time learning how to do magic depends on the sort of problems you want to be able to solve. I never did come close to winning a Math Olympiad or becoming a Putnam fellow, and I still make dumb arithmetic errors all the time, but I don’t have problems doing graduate level math. Personally, I like that tradeoff."

My question: Must it be a trade off?

I don't want math methods for the sciences...or computational techniques needed for a particular vocation, but good old fasioned liberal artsy math that reveals truth about the world and trains the mind.

Who is Viete and what does he have to do with quadratics? Don't know. No time for that, it's off to trig you go.

And one last chipping anecdote,

"Stieljes was an unlikely victor. As a student he had failed his university examinations three times, much to the despair of his father, a member of the Dutch parliament and an eminent engineer responsible for building the docks of Rotterdam. But Thomas's failure was not due to laziness. He had simply been too distracted by the pleasure of reading real mathematics in Leiden's library rather than mugging up on the technical exercises required for his examinations." SOURCE

New Math

2007-03-04T16:27:00.000-08:00

For anyone interested in our latest math book saga. We got a new book:

Modern Algebra: A Logical Approach
by Hellen R Pearson and Frank B Allen. 1962.

I've been reading everything I can find online about Frank B Allen. He was at the forefront of New Math in the 1960's and also a contributor at the mathematicallycorrect.com website. He raps too. "None of your new ideas are good and none of your good ideas are new." It's all coming out on a new CD called the Sigma Enigma featuring songs such as, "When da bound don't fit." Okay, that last sentence isn't true, but reading Allen's opinions about math is much more interesting than reading through the National Math Panel's transcript.

Ralph Raimi, another new math guy, had a long article about Frank Allen's Modern Algebra: A Logical Approach for high school students and said that Allen's algebra book was unnecessarily rigorous. This prompts husband to say that being too rigorous in math is like having too much money or being too thin. Knowing that this was considered unnecessarily rigorous made it instant math nip.

It's a "proofy" book but not in the creative artsy way that Gelfand is. Allen's kind of proofiness is more derivational, more mechanical. Ugly. And the word problems seem too easy. What I'm going to have to do is supplement from Singapore's NEM series to make this work. I know my kid can do this since he's been doing exercises out of an old college algebra text.

Here's part of the table of contents.

And here's the first page of a cumulative review:

Well, what do you think? Wierd? Easy?

Math Session with Dad

2007-03-02T16:36:00.000-08:00

"If you can prove that then you're done." Dad left the room and said son (10 years old) came up with this entirely on his own. Does anyone recognize this?

"All I did was write down everything I knew about negative one and x."

Fast Reckoning

2007-03-02T15:43:00.000-08:00

I gave my husband a number fact drill on the Flashmaster. I told him to go neither fast nor slow, just do it casually. I think he momentarily tripped up on 11 x 12, but ended up with 16 problems answered in 60 seconds. That's about what I expect the kids to do. If they are down around 10 or 12 per minute I feel like there might be a little more practice that can be done.

Barring an extreme situation where the child is still counting on fingers or taking 30 seconds to figure out something like 7 x 8, I just don't see the benefit of being speedy with basic facts. What concept is a first grader going to have trouble with because he isn't able to write down the answer to 21 addition facts manually within 60 seconds? Jump in and comment if you know. Maybe this really is just me not getting it. There seems to be a lot of emphasis on speed drills, not because it's a kind of repetitive exposure that helps in mastering basic facts but speed has become an objective unto itself. Does this happen in English also? I wouldn't send my kid to an auctioneer to develop skill in persuasive writing.

Auctioneer

Wow! That guy is really good at English!

Minding my P's and Q's

2007-02-25T19:48:00.000-08:00

A few days ago I found a fun little problem that is supposed to be for 8th graders. ...ahem. I tried to prove it. With that sort of an opening, if you know me, you'll guess that there is only one way that this blog entry can end: With a really bad pun.

Nonetheless, I do solemnly affirm that all the facts herein are true.

To get into the mood of what is about to unfold sing Tom Lehrer's New Math song because this problem deals with numbers in different bases, to wit, "In binary we count upon our fists rather than our fingers." So then, how would one prove that any number, in the land of ten fingers as well as the land of two fists, that ends with an even number, (a digit really), is even? What if it is in base 3? What if you had to come up with a way of predicting if a number is odd or even but you didn't know the base in advance? That was my problem but it was phrased more mundanely like this:

"State and prove a condition (involving the representation of a number) which allows us to determine whether the number is odd or even in base n."

And fortunately the book supplied me the general form for a base n number. Deciphering it was my first obstacle.

Not too bad. That's just scientific notation using letters instead of numbers. I earn 10 life points for surviving that Chinese ~~water~~ ellipses torture. Drop, drop, drop, dot, dot, dot. Obstacle #2 was notational difficulties. The trauma that sigma notation has inflicted upon me is worthy of yet another blog entry as well as several years therapy. I considered myself lucky when I found this online video lesson to remediate my deficiencies during which I lost all my life points from watching the woman sloooooowly add up fractions.

Progress continued as revealed below in the following rough draft (written in black pen) until impeded by Mr. Math and his comments (in pencil). I didn't even get a chance to make it to "case 2" in which I consider bases that are odd. Then there was an attached note, not shown below, containing an exposition of p's and q's kindly informing me of necessary logical form that assertions should have, "Stylistically speaking you should have blah blah contrapostive blah blah bidirectional equivalence blah blah. " I thought everything was going well and now all I have is this sea of symbolic vomit.

Looking over the proposed corrections this thought came to mind:

But the real punchline is that the answer in the back of the book had nothing to do with symbols at all and amounted to just a few normal English sentences. I don't know what I want to do now. I could redo the whole thing correctly because I do have the second half scrawled out some place around here or I could just move on.

Your Brain on Bar Diagrams

2007-02-24T22:28:00.000-08:00

Somebody really did do a study on this. MRI of brain on bar diagram vs. traditional algebra.

"In Singapore schools, algebraic word problems are taught using two methods: formal algebra and model. The latter depicts relevant quantitative relationships between unknowns in a pictorial format. In this study, we used fMRI to investigate whether the two methods are subserved by different cognitive processes. "

It would have been more interesting to see this study done in children rather than adults. I'm guessing it would have turned out more like this:

Mathy programs for mathy kids

2007-02-23T12:23:00.000-08:00

Okay, I must....resist....urge to respond to this.....

"If you have a mathy kid you should use XYZ curriculum, otherwise, use this other curriculum."
How do you think the "mathy" kid got mathy to begin with? Maybe they are just born that way. High achieving countries have genetically mathy kids. And the best course of action is to isolate the nonmathy kid from a rigorous math program. That's the answer.

Reminds me of my grandma complaining about physical therapy, "I don't want to go because it hurts!" "It hurts because you are weak. When they finish with you it won't hurt anymore, granny, and you'll be able to walk."

Now I've said it and I feel better now.

Invert and Multiply, Redux

2007-02-22T08:50:00.000-08:00

I have always known the rule "invert and multiply." That's never been the problem for me. I applied it flawlessly in arithmetic and I had no problem in college algebra. But when pressed by my son to explain why this works, I had no answer. When he questioned the fraction overlay that I was using I was screwed.

"I see THAT it works when I look at the fraction overlay but I don't understand why it works."

I spent the rest of the evening trying to figure out how to manipulate fractions so that I could divide them without actually inverting and multiplying. An analogous situation in English is defining a word without using that word in the definition, but in math this is hard. How do you divide two fractions without inverting and multiplying? By golly, like this!

And if you use letters instead of numbers it turns into a proof.

Theorem: a/b ÷ x/y = a/b x y/x

Proof:

I guess I am supposed to add the lawyer clause of "where zero isn't in the denominator."

TextSavvy is now running a series on dividing fractions.

Instructions for Climbing the Stairs

2007-02-20T09:49:00.000-08:00

Julio Cortazar is an extremely well known Argentine writer (well-known to those who know him well). If the goal of your Spanish teacher was to have you enjoy chats with the carpet-installer rather than expose you to Spanish literature you may not have heard of him and have my sympathy.

For the rest of you anglo-parlantes I tried to find a translation of this in English online but I am not sure that it exists; therefore you are stuck with my hastily translated version which wasn't extremely easy to translate but I'm not sure that every clause in the original was intended to be crystal clear. Having attempted to sort through overly formal definitions and instructions for skills and concepts, especially in math, by those who think that "good teaching" is all about breaking a skill into yet smaller simpler steps with the objective of making sure their student doesn't suffer from a moment of confusion, I hope this be an antidote to that line of thought. Sometimes it's best to just "do it" and not be instructed to death.

There is no one that hasn’t observed that frequently the floor will fold in such a manner that one part of it rises at a right angle with the plane of the floor, and later the following part is placed in a parallel manner to this plane, giving way to a new perpendicular, and that this conduct repeats as a spiral or broken line up to an extremely arbitrary height.

Each one of these footholds, formed along the way by two elements, is situated equally higher and more forward than the former, a principle that gives meaning to the staircase since whatever other combination will produce a form perhaps more beautiful or picturesque, but incapable of translating the lower floor to the upper floor.

Stairs are climbed from the front, since climbing them from behind or the side will result particularly uncomfortable. The natural attitude consists in maintaining oneself on one’s feet, the arms loosely hanging at the side, the head erect but not so much so that the eyes stop seeing those elevated footholds immediate to the one that is being tread upon, and breathing slowly and regularly.

In order to climb a stair case one commences by raising that portion of the body situated at the below right, covered almost always in leather or rubber, and almost without exception fits exactly upon the foothold.

Said part being placed upon the first foothold, and in order to abbreviate we will hitherto call it “foot”, the equivalent part on the left is then removed (also called foot, but one must not confuse it with the aforementioned foot.) and raising it to the height of the foot, it is to be made to follow until it is placed upon the second foothold, upon which the foot may now may rest, and upon the first the other foot may rest.

(The first footholds are always the most difficult, until acquiring the necessary coordination. The coincidence of the name between foot and foot make the explanation difficult. Be especially careful to not raise the foot and the foot at the same time.)

Arriving in this manner to the second foothold, it is enough to alternate the movements until one finds oneself at the end of the stairs. One can easily leave them with a light blow of the heel that fixes it in its place, from which it shall not until the moment of descent.

Julio Cortazar in English: Cronopios and Famas

The Russian Math Conspiracy

2007-02-18T16:16:00.000-08:00

I just watched a video of a conference of mathematicians interested in working with children where one of the participants says that in the 1960s Russian public math education wasn't all that it needed to be. Here's the website And here is a direct link the video in QuickTime It's not nearly as exciting and professional as the MJ McDermott "Math Education: An Inconvenient Truth." In the conference video the sound is bad in a few places and there are distracting pocket protectors and microphones gone wild.

Said by Dmitri Fomin about 28 minutes into the video:

"In St. Petersburg it began some some point in the 1960's when a lot of local enthusiasts professors, teachers of math, graduate students, they've just realized that they need to educate school kids differently outside regular elementary high school, middle school curriculum because mathematics needed to be done differently. (Unintelligible) They needed to teach them some stuff different from the same old stuff that they had been taught very repetitively."

Am I reading too much between the lines or is that not just a diplomatic way of saying that math education was broke beyond the point of repair? The mathematicians decided that it would be easier to have a direct pipeline to the students rather than try to reform the curriculum.

And they made it happen.

They skipped right over the school system and worked directly with kids in what is called "extracurricular math education." They met after school. This extracurricular math education comes in the form of "Math Circles" which, if you listen to the entire video, is supposed to be a hook to get kids eventually involved in real math...as in research. The last speaker in the video, Alexander Soifer who starts speaking at about 48 min., speaks of his experience in the math circles as a child and went on to become one of the only three graduate students to have Erdos as an advisor. And I just love the reference to his mean mama as the reason that he did so well in math as a kid. He says that it was less work to make an A in his math class than to explain to his mother why he made a non-A. It's not all about "happy math" and instant gratification.

Other people who speak about this direct pipeline go out of their way to affirm current educational practices. That article protesteth too much. Can't you just say that you work with kids without reaffirming that everyone else is doing a Tony-the-Tiger-greeeeeat job?

The conspiracy is to get a direct pipeline in the United States as well. There is information in the videos with contact information about how to bring this to your school. Steve Olson, a journalist, (here's his book he wants you to buy), discusses his role as a parent in the school system. He shows up in a couple of clips on that site.

Off topic, but the conference begins with a discussion (five minutes into the video) about how the absolute numbers of math majors has declined in the United States since 1966. It's not the percentage of math majors that has declines, it's the absolute numbers that's gone down. I think the report that is referred to at the beginning of the video is this one: CBMS 2000 So maybe if U.S. mathematicians want more and better undergrads they are going to have to get involved like the Russians did and train them in high school themselves.

Here's some more Fomin lagniappe. It's almost an entire chapter of problems out of his Mathematical Circles online. Bear in mind that this is for 7th and 8th graders. It is neither fuzzy nor is it "traditional" computational word problems. So what is it?

Harmonic Series: The AfterMath

2007-02-17T14:25:00.000-08:00

Okay, no more posts about the harmonic series, just this last one. Bear with me. This time from the point of view of my husband. He posted this someplace else first so it has other elements in it other than my issue:

Discussing the harmonic series with my wife this morning, I realized that this is precisely the experience I want my kids to have. I want them to have as many of these moments that my wife had this morning as possible in their math education. So, how do we bring this experience (as I'll elaborate on below) to the masses? Since it doesn't exist much in public or private schools, I'm homeschooling -- what are the good textbooks for this? At any rate, let me tell you the story so you know what "this" is....As many of you know the harmonic series diverges. Perhaps the "standard" proof of this is done by observing that the harmonic series is greater than the series that goes 1 + 1/2 + 1/4 + 1/4 + 1/8 + 1/8 + 1/8 + 1/8 + ... = 1 + 1/2 + 1/2 + 1/2 + .... The trick which is not obvious at first is that you will get 2^(n-1) terms of 1/2^n so that the moving parts kind of cancel each other out and you are just left with a constant term 1/2. My wife, on the other hand, tried to use the fact that any unit fraction can be expressed as the sum of unit fractions to figure out just exactly what the sum of the harmonic series was. For instance, 1/k can always be expressed as 1/(k+1) + 1/[k(k+1)]. I thought this approach was interesting because it didn't just bound the harmonic series below by another divergent series. It actually rearranges the terms to get an exact sum of so many terms of the harmonic series. So, we spent some time talking about how to go about really using this to prove that the harmonic series diverged. I had started to see what I could write down "real quick" to do it but never could quite get a proof out before being interrupted by some other inevitable long discussion about how what we had so far wasn't enough and so on. Well, sure enough, it eventually occurred to me that I need something equivalent to the fact that the series diverges to really do this kind of proof. (In other words, this probably won't work.) To appreciate why, I'll have to disclose a little more of the program of the proof. The idea is to rearrange the series into a sum of infinitely many 1/2's just like with the "standard" proof. So, for instance, you get 1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ... =1 +1/2 +1/3 + 1/6 +1/4 + 1/12 + 1/7 + 1/42 + ... Each row sums to a half and so you just get the series from the "standard" proof of mostly just 1/2 added to itself. You write down the first two terms 1 and 1/2 and then to go from one row to the next you just break each term of the form 1/k up into 1/(k+1) + 1/[k(k+1)]. The problem is that on the sixth row you start repeating yourself by breaking 1/5 up into 1/6 + 1/30 when you have already used 1/6 up in the third row when you broke up 1/2. It doesn't do us any good to show that the series diverges when you repeat infinitely many of its terms. My wife says "but you can just keep breaking that up if you get repeats, so doesn't that prove it?" Both me and my father (who is a retired mathematician) say "while it does seem like that is true -- that you can always go out further if need be -- just saying that doesn't really *prove* it -- no."So, this problem of hers comes up on and off for a while until last night (Myrtle's note: Yeah, he woke me up at two in the morning to tell me this ) I was thinking about how you get this explosion of terms as you start getting repeats -- a single term like 1/6 will turn into the sum of a whole litany of terms to avoid repeating any from the previous rows. And then, when you get on over to other terms in the row they explode out into a bunch of terms for the chunk that derives from the 1/6 being broken out. It really explodes pretty fast. (There was some discussion about that between me, my father and my wife.) And, it occurred to me that if it "exploded fast enough" it would converge. What you need to show is that given any natural number M, you can write 1/2 as the sum of unit fractions all with denominators greater than M. That way I really can make sure and go out far enough whenever I have to so that there are no overlaps. But, this is basically the same thing as saying that the series diverges in the first place -- I'm saying that no matter how far out in the tail I go, I can always get at least 1/2 out of it. So, then I tell my wife that this may not really work out for her to prove it this way, and she just looks at me in disbelief. After explaining it to her, she is still unconvinced and though she knows that me and my dad "must be right" she still doesn't really know why her proof didn't prove it. So, finally, I give her an example of the geometric series (which we all, including her, know converges). Take 1 + 4/5 + (4/5)^2 + (4/5)^3 + .... This series converges to 5. It turns out that if you take the 3rd and 4th terms they add to (4/5)*36/25 > 4/5. In fact, I can then start at (4/5)^4 and add up the next three terms to add up to greater than 4/5. So, "Can't I just do the same thing with this series," I ask her, "and write it as 1+ 4/5 + 4/5 + 4/5 + ...? Can't I just 'always go out a little further' if I need to and get some more terms to get my next 4/5?" And, that's when she realized that since the series converged, obviously you can't just keep going out and gather some more terms. Something about how you need more and more terms to do it foils your plans on this series in a way that it doesn't on the harmonic series. But, at any rate, I think she "feels" like her proof doesn't prove it now as opposed to before when I was just giving her technical counterexamples to specific things she might have said. (Of course, the gist of what she is saying is true, though, or otherwise the harmonic series would converge.)Now the thing here wasn't that she just needs to explore the wonders of infinitude or something -- perhaps take a good calculus class. It was the fact that she was trying to prove something in the first place and really thought she more or less had it. She probably learned more in the realization that her proof of a result that she already knew of did *not* work than she has *ever* acquired in memorizing and applying theorems or heuristically figuring out a new result.

Hand Waving Boosts Mathematical Learning

2007-02-16T11:34:00.000-08:00

I can't get Paul Simon out of my head. "The problem is all inside your head, he said to me, the answer is easy if you take it logically." I did have my meeting and it didn't go the way I planned, Stan.

--So how did you prove it? Oh, I see, hand waving. Well I know how I would prove it.
--Really? Tell me.
--No.

Where is the love? The hand holding?

--I'm sure it can be done, but your running with scissors approach is dangerous.

I haven't figured out the answer to my problem, but I have learned the classic proof, what a zeta function is, how sigma notation is used, where this shows up in Euclid, 500 years of proofs of this series, and that Gauss probably did all this when he was ~~four years old~~ a sperm cell.

Valentine's Day Edition

2007-02-14T20:11:00.000-08:00

I forgot what I wanted to say.

Quick Trick Brick Stack

2007-02-11T15:41:00.000-08:00

And here's a new trick, Mr. Knox. You can pile up bricks and blocks.

The top brick is two inches long. It overhangs the brick beneath it by 1 inch The second brick overhangs the third brick by 1/2 inch. The third brick overhangs the fourth brick by 1/3 inch The k brick overhangs the k+1 brick by 1/k.

If we wish for the bricks to extend even more over Mr. Knox's face we will add the next brick to the bottom. In this way we can "lean" the tower over as we please without the center of gravity shifting.

To find out how far the end of the first block extends beyond the end of the k block,

add the displacements (in inches) 1 + 1/2 + 1/3+ 1/4+ ...+ 1/k.

(There is someone leaning over my shoulder saying, "You haven't shown that the series diverges." I guess that is the next thing I have to figure out.)

Update: I did it. I knew those Egyptian unit fractions would come in handy one day. So my idea was that any fraction can be expressed as the sum of "unique" unit fractions (and I can show this with induction, that came in handy too, thanks guys!!!!!!! ) and that an integer is just the sum of fractions, say the sum of halves. So for any whole number you choose "n" , n can be expressed as the sum of 2n halves. Now each of those halves can be expressed as a sum of two or more unit fractions no one of which is ever used more than once in the sum. In other words, let the fractions in the series above be a set of fractions from which you can choose addends to "make halves." Good lord, this is difficult to express in prose but it all makes sense when I look at it on paper. And now my next assignment is to express this formally.

Mathematical Circles

2007-02-08T16:46:00.000-08:00

Induction keeps rearing it's head in all the math books I look at Mathematical Circles being no exception. (Here are much better reviews of this book: Andre Toom's review , Amazon reviews, American Mathematical Society. ) I'll try to say something that hasn't been said before. From the day this book arrived it's been nothing but a big distraction. Like all truly good children's toys, the grown ups want to play with it first which means endless hours of entertainment arguing with each other. Characteristics of note:

-It's for children 12 - 14 years old.

-It contains problems which are unsolvable. What a wonderfully wicked thing to inflict on a child! The fun is in figuring out why they can't be solved. The book explains it this way.
Many problems in this section deal with proofs that certain situations are impossible. Indeed, when a question asks whether some situation is possible, the answer in this section is invariably "no". This poses some difficulty for mathematically naive students. Their first reaction is either frustration that they cannot find the "correct" situation (fulfilling the impossible conditions) or a declaration that the situation is impossible, without a clear conception of what it might take to prove this.

-This book is meant to be used as a supplement over the course of two years. I didn't add up the second year problems but there are about 340 first year problems.

I am having difficulty understanding the wording of some the problems. It's not clear to me if this is due to the translation or my mental acuity. For example,

"Distribute 127 one dollar bills among 7 wallets so that any integer sum from through 127 dollars can be paid without opening the wallets."

Huh? Of course a wallet is opened up when you pay someone! Response: "Duh, you could hand over the entire wallet without opening it." I had to go look at the solution to understand what the question was asking. The wording might make more sense like this,

"Distribute 127 one dollar bills among 7 envelopes so that any integer sum from 1 through 127 dollars can be paid by handing over some combination of envelopes."

And here is one I actually tried out on my kid from the chapter on parity, but it involved a half hour discussion to clarify what the problem was getting at and rephrase it:

A closed path is made up of 11 line segments. Can one line, not containing a vertex of the path, intersect each of its segments?

Not only did I have to completely reword it, but also because I'm working with a ten year old, I thought it wiser to use fewer line segments.

There are four towns, four bridges, and one river. Arrange the bridges so that each one crosses the river. Each bridge may connect two and only two towns. Each town has two and only two bridges leading out of it. You may place the towns anywhere along the river. The bridges can criss-cross each other but they must be straight.

When he instantly drew a diagram I asked him to do it again with six bridges and six towns. He worked on this maybe a minute, gave up and said it wasn't possible. I'm sure Dewey would roll in his grave if he heard my verbal response to that one so I'll skip the gory details and say the kid went back to work and after some grunting and whining, but not a peep of help from me, came up with this:

He was so pleased with himself that he vowed never to give up again. There is only one humane way to respond to such dedication from an innocent and trusting child: Give him an impossible task. "Connect seven bridges and seven towns in the same manner as before."

Math plus with pixie dust

2007-02-02T14:15:00.000-08:00

I thought I had this all figured out before. So the inductive step is why it works or the inductive step is what makes it work? Why do I get to assume that if n=k that it is divisible by 5 but I have to prove that it still works for k+1?

Here's the original explanation as stated:

What is "traditional" math?

2007-01-31T13:29:00.000-08:00

I got this from a college algebra text with a 1953 copyright. I hope that counts as an era in which we could agree that "traditional" math was taught. The 50's should be a safe decade. This particular text was used at a small tier four university with a population consisting of mostly rural southern boys. In other words, had Larry the Cable Guy gone to college he would have seen this book. In its very, very short preface I found this statement:

I looked up what those proofs might be. They were considered supplementary material but were present just as the author promised. Here is one such page:

I guess the solutions to those might look like regular ole algebraic manipulations. Not very exciting.

I tried my hand at a different problem from a theorem that was at the beginning of the book. (I didn't get any hints so I'm really happy to come up with what I did) What I'm not clear on is the form that all this should take because before this I had been trying to prove set identities in a completely different book and in that you have to prove that what is on the left side is contained on the set that is on the right side and then do it the other way. (A is contained in B, B is contained in A. That wasn't very exciting either because that sets in the problems were equal was obvious without "proving" it. I'm interested in those inasmuch as it will help me understand what's on the last page of the book which is a proof that PI is irrational.) So that was the only "model" that kept popping into my mind.

The problem: Prove that for every value of n, (a^n - b^n) has (a-b) as a factor.

That was fun, if not tedious.

And did you know that "degenerate" is a technical math term? As in, "You have included an irrelevant and degenerate section in your proof."

Beyond Bar Diagrams

2007-01-25T15:53:00.000-08:00

A comment on this blog post by mrc motivated me to change my approach to word problems. My emphasis is added:

"I know that if I were to pose the geometry question to my high school classes (mostly 10th grade) there will be 1 at most out of 80 students who will get it. Another 5 will make decent attempts. The rest will either glaze over immediately, or else scrawl nonsense."

Okay, Mr. C. Point taken. My son scrawls nonsense when he answers word problems. Coming up with the correct number hasn't been the problem, it's everything that precedes the answer which is scrawl and I have no one to blame but myself. So, I've ditched the bar diagrams, a Singaporean heuristic device for demonstrating relationships between numbers, and I'm opting for something a little more formal. Granted, he's leaving out the steps that involve solving for x, but I'll give him a break on that since he's ten and hasn't taken algebra yet.

I'm now requiring him to define his variable in a complete sentence. (Dad said to use the "implies that" arrow.) If any of my readers remember my blog entry, "Its a hard, hard language" in which I was accused of introducing coconuts and sparrows into a word problem, well, we've been working a lot with integers since then. The workbook page below represents about thirty minutes of work with most of that time dedicated to navel gazing and pencil dropping. He tells me the hardest part is defining x. No mistakes have been corrected from this page by me and he never did answer the question in "C"...

Thanks to Dave at Math Notations for a blog entirely devoted to word problems.

In which I don't understand the number "one" anymore.

2007-01-21T07:56:00.000-08:00

All I wanted to know was why different fractions can add up to be "one" if they are all different sizes. I don't mean that the denominators are different sizes. I mean that they are different sizes relative to "one." And I wouldn't have thought this thunk if not for a fifth grade math problem* which required converting a repeating decimal to a fraction. It was fun to do it with something like 0.242424 but 0.9 repeating is disturbing because it's the same as "one." Here we go, 0.9 repeating written out as a sum of fractions.

(9/10) + (9/10 of 9/10) + (9/10 of 9/10 of 9/10) ...

But maybe this is just a trick of the decimal system. If we switch to a different base maybe this won't work out. I can get closer to "1" and greater than .999 repeating in hexadecimal. But evidentally it does work out in a way because 0.fff repeating is one. But then that means that "nine" isn't really nine because .9 repeating in hexadecimal is not the same thing as .999 repeating in decimal. Maybe nine really means "the last digit" in whatever base you are working in. Or rather nine only means nine when its in the units place.

"I don't see much sense in that," said Rabbit. "No," said Pooh humbly, "there isn't. But there was going to be when I began it. It's just that something happened to it along the way."

And in base four the last digit is 3 so 0.999 repeating is really .333 repeating. Converting it to fractions and adding it all up would look like:

(3/4) + (3/4 of 3/4) +(3/4 of 3/4 of 3/4) +...

And in binary the last digit would be 1 so that .1111 repeating equals one so that summing the repeating "decimal" of 0.111 as a fraction would look like this:

(1/2) + (1/2 of 1/2) + (1/2 of 1/2 of 1/2) + ...

All these fractions which represent repeating "last digits" in different bases are different sizes but they all add up to the same thing. How disturbing is that!

Then I asked Owl. And he talked and talked and talked and wrote and wrote and wrote. See the uploaded memento above.

"But, Owl, I don't understand."
"You must concentrate!"

*Converting repeating decimals to fraction problems are in Primary Mathematics Intensive Practice 5B US edition, pg. 14. See also NEM 1, pg. 64.

Once I knew that the secret was in decomposing the decimal in terms of ninths rather than tenths I figured the rest out. The dealio with limits and the so-called "geometric series" is another matter.

Socrates and Math in Plato's Theaetetus

2007-01-12T20:09:00.000-08:00

Theaetetus is the adolescent student of Theodorus a geometrician. He gets into a conversation with Socrates and tells him about the discovery of an entire new class of numbers: the irrationals. Up to this point they knew there were issues with expressing the diagonals of some squares as a ratio with the length of the side, but it turns out, says Theaetetus, that there are a bunch of numbers like this and they should get special recognition as a class of numbers and a name. And what does Socrates do? Basically, he blows it off and changes the subject, "Yeah, great, whatever kid, yeah, you sure are smart, so...uh back to my original topic, what is knowledge?"

Oh, no, he di'in't!

And you really do not want Socrates as a math teacher either.

Check out this reasoning.

An object of knowledge consists of the sum of its parts. For example, you know a syllable because it consists of more elemental building blocks such as the sounds from which it is composed. The syllable cannot be more or less than the sum of the sounds that create it. So too a number is an object of knowledge. Part of what we know about a number, say "6", is that it has equivalent expressions such as 2 +4 and 2 x 3, etc. These expressions are the same as 6 itself. They are also a part of what 6 means; they are a part of six. But they can't be a part of six because six can't be greater than the sum of its parts. And as we all know a part of a number must be less than that number itself. Oh dear, a paradox, however shall it be resolved?

The Theaetetus isn't even about the math. These were just some things that stood out to me while I was reading. Theaetetus is about refuting subjective notions of truth. Why look! It's the same old arguments and refutations of subjectivism that you still see in online debates. Is it just as clever today as it was 2000 years ago?

"Protagorus, for his part, admitting as he does that everybody's opinion is true, must acknowledge the truth of his opponents' belief about this own belief, where they think he is wrong."

And Socrates later says, "Must we not suppose that Protagorus speaks this way to flatter the ears of the public?" Ayup. Tell 'em what they want to hear. No one is every wrong. We are all right in our own way.

Another glaring passage is the first reference to philosophy babble. Poor Theodorus (Theaetetus' teacher who had the way cool discovery math class) can't stand talking to the philosophers at Ephesus. All they speak, according to Theodorus, is philo-babble and when he asks what their phrases mean they just come back with more philo-babble.

When you put a question, they pluck from their quiver little oracular aphorisms to let fly at you, and if you try to obtain some account of their meaning, you will be instantly transfixed by another, barbed with some newly forged metaphor.

Theodorus gave up on ever having a normal discussion with these folks and headed to Athens.

How about the first recorded description of the social-skills-lacking Geek in history? It's the perfect Platonic Geek. Attic Geek. The Greek Geek laughs at inappropriate moments.

Then, if that is your wish, let us speak of the leaders in philosophy, for the weaker members may be neglected. From their youth up they have never known the way to market place or law court or Council Chamber of any other place of public assembly; they never hear a decree read out or look at the text of a law. To take any interest in the rivalries of political cliques, in meetings, dinners, and merry makings with flute girls, never occurs to them even in dreams...

and this goes on and on until,

And so, my friend, as I said at first, on a public occasion or in private company, in a law court or anywhere else, when he is forced to talk about what lies at his feet or is before hie eyes, the whole rabble will join the maidservants in laughing at him, as from inexperience he walks blindly and stumbles into every pitfall. his terrible clumsiness makes him seem so stupid. He cannot engage in an exchange of abuse, for, never having made a study of anyone's peculiar weaknesses, he has no personal scandals to bring up; so in his helplessness he looks a fool. When people vaunt their own or other men's merits, his unaffected laughter makes him conspicuous and they think he is frivolous."

The next book after Theaetetus is Parmenides and it doesn't look be nearly as entertaining. The editor says of it, "The argument runs on and on in words that appear to make sense and yet convey nothing to the mind." I've read a few online arguments like that too.

It's a hard, hard language

2007-01-11T13:27:00.000-08:00

A bottle can hold 250 ml of orange juice. How many such bottles are needed to hold 4 3/5 liters of orange juice?

Child's answer: 18 2/5

Mom: How can you have 2/5 of a bottle? You only have whole bottles.

Child: You cut the bottle.

Mom: Of what use is a cut bottle? (hand waving a vertical cut)

Child: Duuuuh, you cut the bottle horizontally.

Mom: 2/5 of a bottle doesn't mean 2/5 of a bottle, it means 2/5 of the volume of the bottle.

Child: You can still cut the bottle.

Mom: What if the question read "A baby can drink 250 ml of orange juice. How babies are needed to drink...

Dad: But if bottles means "volume of the the bottle" then he answered the question correctly. Because 18 and 2/5 means 18 and 2/5th of the volume of another bottle. You are the one bringing up the coconuts and sparrows.

Child: You can have 2/5 of a baby.

Nice idea while it lasted

2007-01-07T17:28:00.000-08:00

I was wrong... they aren't all math geniuses. (See my post below) Just 0.16% of the students. So we might be talking about the pedagogical miracle of turning a silk purse into a silk purse after all.

Joshua Zucker blogs about his trip to Russia to find out about how middle and high school students math circles work.

One of my frustrations today was that several members of our group, including me, tried to ask various forms of the question “What is math curriculum like for those kids that aren’t in a math circle or attending a special school like this?” (In StP there are about 60,000 sixth graders; about 100 of them attend this school; and I think only 25 of those are in this circle. So what do the other 59,900 do?) We never got an answer, except for the honest “Getting worse and worse, and soon it will be as bad as America.” True enough, but bad in which ways? Lots more lecture, lots less “seatwork” and “going over homework” in the bad sense in which you see it in America. But not the amazing problem-oriented approach like we use at Castilleja and like they use at school 239.

In the following I think the reference to Bourbaki is to the group of French mathematicians that started new math in that carried over to America 60's. And, yikes! Sossinsky doesn't sound like Mr. Happy Math:

The Russian way: problem-oriented, with the active participation of students.
E.g., “Ivanov, come to the board, and show us how to solve this problem.” [So far, sounds a lot like Castilleja, particularly the new Exeter curriculum.] If they fail to solve the problem at the board, give them a good long tirade, and perhaps make them cry. But “we can’t do that with American students.” [says Sossinsky. He’s talking about the college senior math majors who they get as visitors at the IUM.] It’s not the French formalized way; the Russian way is geometric, concrete, with lots of applications. [Also sounds a lot like Castilleja in that sentence!] “We don’t think of applied math as being third-rate” [in contrast to the French/Bourbaki style approach, so axiomatic.] “Mathematics is that branch of physics where experiments are cheap”. They are Platonists, view math almost as a natural science, investigating the reality of the mathematical objects with an experimental attitude much as a scientist. Which is not to say that they don’t also value rigor.

I'm not sure what to do now. This issue of proofy math is something I've been chewing on for ten months. I think it's time to conjugate some Greek verbs and chew on this some more. I'd like to know more about the role of the Gelfand correspondence school which Zucker mentions. He also might be in a position to compare Gelfand to EPGY.

Some Russian word problems from the math circles free online translated to English from the Teaching Math Circles website.

Word Problems and Abstract Thinking

2007-01-06T16:52:00.000-08:00

I can't remember exactly when I became disenchanted with the math wars. When my son was in kindergarten in public school I was sure that there had to be a way to "get 'r done" that wasn't being done. After we began homeschooling, we chose Singapore because it ~~has "gaps" and doesn't cover every freakin' topic under the sun~~ goes into a lot of depth on each topic. It's super heavy on word problems. Kids coming out of this program are capable of doing all sorts of word problem gymnastics. But there was still one thing lacking.

How does a kid make the transition between manipulating terms in an expression to proving theorems? I don't mean 2-column proofs in 10th grade geometry. I mean something more like Vieta's theorem or the Cauchy-Schwarz inequality.

Try to find a discussion of that in k12 math education! Leave the URL in the comment box below.

Americans simply don't believe that anyone under the age of, oh say, about 21 is capable of doing such a thing. Maybe it magically happens when you are a junior math major in college and run across mathematical induction for the first time after having completed your K-14 race to calculus. Or maybe we just stubbornly resist the idea that the mathematical ability to tackle real problems is a skill that can be nurtured at all and keep insisting that, "Annie's not a mathy person."

Meanwhile, in some other nations, explicitly Germany and Japan, 50%-90% of classroom time at the high school level is spent working on proofs. (Because I suppose they are the master math race or something)*. I also know that it's a big part of Russian math education as well so it's not that crazy to think there is a plan out there some where to achieve this.

Permit me to entertain some wild assertions, mine can't be any worse than some of the other pedagogical theories that have come out so far and I'll be summarizing and paraphrasing Toom a lot: Just as there is a critical age for learning a foreign language, there is a critical age at which a child develops the abstract reasoning required later on for proving theorems. And having recently read a couple of years worth of online arguments that he's made as well as all his articles at his website, he believes that children can be groomed to think in a "proofy" way starting in the first grade. This is not new math or set theory.

Andre Toom says that word problems are the key. He makes some important distinctions when he says all this. First of all, he doesn't claim that just any old world problem will do. Here are some of the things that he does NOT consider good word problems with respect to developing higher mathiness:

1. Word problems meant to prompt/trigger the child to respond in a particular way.

I think by this he means that the word problem is meant to be answered by using a particular algorithm/bar diagram etc.

2. Word problems which are applications in other areas such as science or pizzalogy sic. I don't think he means that application word problems have no value, they certainly do teach the topic at hand, they simply don't reach the level of abstraction or draw out the creativity that's needed.

Then there is the queen of all word problems: The mental manipulative. Here's an example, "What's the largest number of sections into which a rectangle can be separated when 4 line segments are drawn through the rectangle?"

A digression, Toom also says that problems should not be gimmicky to interest a child. Counting Bratz dolls instead of coins isn't the key to generating interest in math. Interesting problems are the key to generating interest.

"To me a mathematical problem is interesting and educationally useful because of its intrinsic mathematical structure. I strongly disagree with the idea to attract students to mathematics pretending that it helps to play baseball, organize marching bands or enjoy rock music. That is a false promise."

And happily he recommends to us word-problem-impoverished Americans things such as the Math Olympiads (this originated in Russia?) to work on these skills. There's no magic, it's just tackling problem after problem. Here's the other thing that he also seems to say, the topics of fun problems should be presented in a systematic way. So while it's possible to find puzzle problems online there seems to be no structure to the collection. I did find a single book which has hope of supplying me with "mental manipulatives" that I've haven't seen teh usual suspects reference before.

Mathematical Circles: Russian Experience (Mathematical World, Vol. 7) (Mathematical World, V. 7) Dmitri Fomin

Toom's review of the book is here.

"Problems are on very simple levels, but building to more complex and advanced work ... [contains] solutions to almost all problems; methodological notes for the teacher " Which is cool because something like Gelfand's Algebra does NOT, ahem, have an answer key...yet. Chapter zero is supposedly at the level of a ten or eleven year old.

Toom's article on word problems:
Between Childhood and Mathematics: Word Problems in Mathematical Education.

Depending on your background you may or may not find this article interesting:
Toom's How I Teach Word Problems

A sixth grade Russian math text in translation. But it cost money to buy and it's on my wish list. Too bad the entire series isn't out. It would have been interesting to see their K-6 program is better than Singapore or not.

And also for those of use who aren't particularly interested in entering our children's as competitors in the race to calculus here is another intriguing idea: Introduction to Number Theory. When I get the book I'll let you know what I think. What are the chances that there will ever be an AP Abstract Algebra or AP Number Theory exam?

On Wed, 8 Oct 1997, Tad Watanabe wrote:> And, to know mathematics one first needs to _____?

Good question. There are several things that can be written here.For example: to have brains, to study mathematics and in most cases to be taught mathematics well. Those who have been taught pizzalogy instead of mathematics will not know mathematics.

Andre Toom, University of Sao Paulo

*This is a nebulous fact that I'm recalling from either "The Teaching Gap" or "The Learning Gap"

Andrei Toom

2007-01-05T06:43:00.000-08:00

Toom, a Russian research mathematician, has noticed the "anti-theoretical bias" in American math education. By that, he means American K-12 math education is all about calculations and not about proofs. While there are excellent reasons why it ought to at least churn out proficient calculators there are some some good reasons why it ought to be proofy as well:

1. Mathematicians don't sit around doing calculations. This actually comes as a surprise to many people who just imagine that math majors are not much more than better cipherers than the engineering and physics majors. The job of a mathematician is to PROVE theorems. Children graduate high school with at least a vague notion of what a professional biologist might do, what geologists might do, etc, but they don't know what mathematicians do.

2. Proofs are justification for mathematical belief. For instance, almost everyone has a belief that the sum of angles in a triangle is 180 degrees. Almost everyone has some sort of justification for this belief. The question is whether or not that justification is a fallacious argument from authority: "Because my math teacher told me so, or if it is an actual deductive proof that can withstand scrutiny.

In Plato's Meno knowledge isn't just "true belief" but justified true belief. That in a nutshell seems to summarize what I believe an education to be about. It is ancient and timeless. It is a lot bigger than our transient technological interests. Like most parents, I very idealistically want my children to catch a glimpse of that. There will be plenty of time to pick a vocation. But, they will likely never get an education as busy adults off pursuing a career. And, that is why now is the time for these impractical ideals that lead to no particular vocation.

Toom was in charge of teacher training for the Gelfand correspondence school in Moscow and then came to the United States. (He's now in Brazil.) He refers to word problems as simply problems and what we call problems, numerical equations, he refers to as "exercises." I will talk about his view of the role of word problems in developing abstract reasoning in a child in my next entry. However, this article "A Russian Teacher in America" which is best read printed out since it is more than twenty pages long is about the culture shock that Toom had trying to teach math in America at the college level.

At one lecture I wrote a theorem on the blackboard and said to the students: "Look what a beautiful theorem it is!" Some laughed. I asked what was the matter.Then one explained: "Professor, it is nonsense, a theorem cannot be beautiful!" And I understood that these poor devils, who had always learned under the lash of grades, never from natural curiosity, really could not imagine that an abstraction might be beautiful."

"I understand that some students are the first in their families to get a higher education. Their parents did monotonous work all their lives, tried to make more money for less work and were right, of course. Now their offspring do monotonous exercises at universities. try to make more grades for less work, and nobody in the family sees anything wrong with it."

Toom is the only math-ed person I have ever seen to explain the transition between a child in arithmetic to a young adult proving theorems. He is NOT talking about teaching kids set theory or some other new math thing and he insists on calculation competency.

"I had to learn that every technical calculation, which I was used to ignoring, was a considerable obstacle for my students. It took a considerable amount of time for me to understand how poor they were in basic algebraic calculations."

Next up: Toom on Word Problems

Bad Phone Call

2006-12-30T15:34:00.000-08:00

Saturday evening.

Doc's office just called me to say that they found something wrong,"elevated", in my blood work and they are going to pawn me off to a rheumatologist. Just looked up what sorts of diseases that these folks treat in Wikipedia and it looks scary. Lupus, for example. And I know a test for that was ordered. Rheumatologists also treat scores of forms of "vasculitis", whatever that is.

My symptoms included almost constant tingling in my extremities, arms fall asleep at night, legs fall asleep all day long. Something lightly pushing against me, like the edge of a chair, the bed, a kid throwing their leg over me, becomes painful very quickly. The worst part of my day is laying down to go to sleep at night. I know it's going to hurt.

I can't believe the doctor's office can leave me hanging like this over the weekend without telling me exactly what they found. My family has minimized my problems from the beginning, "This is just part of aging." "You just need to exercise more" Meanwhile an acquaintance tells me, "A friend of mine has Lupus, she's lost a leg in the fight."

WHAT THE ...?! Oh, I feel so much better now. JUST the encouragement I need.

I have to figure out how to get it together by tomorrow. Tomorrow I pick up my son from the airport and he's going to be in very happy mood.

Other news, I am still reading Gulag Archipelago and I am glad that I am reading it. It's giving me a lot to think about. Solzhenitzyn was a math student before WWII cut his studies short and he become an officer. He has a whole lot to say about engineers, how much they were blamed for every little thing that went wrong and sent off to concentration camps. One bizarro reference he made was about Esperantists being rounded up and hauled off to the concentration camps. (Hitler had it out for the Esperantists also, he mentions them in Mein Kampf, and had them hauled off as well.) And sure enough, it turns out Esperanto was a big deal just about the turn of the century. It wasn't just a language fad but represented a very small ideological movement that, well basically they thought that languages with irregular verbs were elitist and excluded the little guy off in some third world country trying to learn them. A universal language was needed to level the political playing field and Esperanto would be it. As far as I can tell they were a relatively small but slightly nutty harmless pacifist group. There are some 25,00o original works of literature published in Esperanto and they annoyingly refuse to have any of them translated into any other language. So I guess the world will miss out on the definitive history of the Esperantist holocaust which is only written in Esperanto.

Two Steps Forward, One Step Back

2006-12-27T12:58:00.000-08:00

Cost of bilateral upper and lower venous Doppler and various other sonograms: $123,203,135,394

Cost of neurological testing: $204,278,678,304,305

My physician's comment before ordering these tests for me? Priceless: "Well my prediction is that that nothing will turn up."

Greeeeeeat.

I could deduct the entire cost of the Iraqi war out of my HSA and still not meet my freakin' deductible.

Time to regroup.

Christmas

2006-12-22T13:42:00.000-08:00

Someone gave me volumes two and three of Gulag Archipelago for Christmas. Guess I'll be reading the rest of that after all.

Finally I have a whole slew of medical tests lined up for some disturbing symptoms I've been having. It's the same ole neurological stuff that I complained about before. The doc warned ahead of time that they could very well turn up nothing, but having serious things ruled out would be a relief.

And in the dead tree department, here's what I've read since last

time:

Son of the Revolution: Depressing autobiography about being a child during the Cultural Revolution. Every last character is corrupt, even, or especially, the children. Thumbs up.

The Rape of Nanking

Summary: Lots of shock and outrage, predictable cliches. The first thing that turned me off was using "a proud people" to describe the Japanese. No insight into behavior. The Japanese are dismissed as psychopaths. The German Nazi that rescues the Chinese isn't a "real" Nazi. I suppose the lack of analysis necessitated the gory photos of corpses to keep the reader interested as if this were some kind of second rate true crime novel.

"One death is a tragedy; a million is a statistic. "

The Great Chinese Revolution

Some would consider this a dry historical academic book, but it "worked for me." This guy does a great job of connecting the dots between imperial China and communist China. He gives an elaborate description of the educational system in imperial China. Chinese children had to learn up to 200 new characters per day in order to develop the reading fluency it would take to be able to read classical Chinese literature. To top it off they only had a 1% chance of passing the prequalifying exam for taking the civil service exam! How about spending your entire life studying for that test and trying over and over again to be one of the lucky? Calculus is a piece of cake compared to Chinese civil service exams.

Life and Death in Shanghai

2006-12-16T15:35:00.000-08:00

I subsituted reading about the Russian Gulag for the Chinese equivalent.

I really don't have more to blog about, but Life and Death in Shanghai was a super book. It's history and moral philosophy all mixed up together. Cheng does not espouse the typical American view of "hate and violence" as the root of evil but rather attributes it weak-willed middle-class soft people and well intentioned ideology. (That hit a little too close to home.) I'm sure that isn't the point of her book though; it's what stood out to me the most. I usually take notes when I read but I didn't this time so I have no nifty quotes to share with you. I'll save this one for my son and if I'm still homeschooling a few years from now I'll assign it as history.

Intrigued by Chinese history, I went to the library this afternoon and checked out a stack of books about China. I've just cracked open "The Rape of Nanking." From what I can tell the hero of the book saves 300,000 people from the Japanese but he's a rabid Nazi. This book looks short so hopefully it will be just a few hours before I can figure this out.

One of these days I'll get back to math. I feel the call of the siren.

Procrastination

2006-12-13T14:08:00.000-08:00

I've been meaning to read this book for a while. Solzhenisyn won the Nobel Prize in literature in 1970 and incredibly the guy is still alive. Since I prefer non-fiction to fiction, as well preferring cold weather survival stories, I thought I couldn't go wrong with this author. Little did I know I would soon plunge into the world's longest political screed mixed in with eye-glazing details of Soviet jurisprudence. Having read 450 pages by this afternoon I thought I'd be disciplined and go ahead and push through the remaining 150 pages of my book until I got online later and realized that I have in my hands only the first of four volumes which means if I chose to finish this I'd have another 1400 pages. Oh yeah, you can even see the little "v. 1" on the library sticker at the bottom but since no other volumes were on the shelf next to it at the library I didn't give it a second thought when I checked it out. I am now.

On to the next topic.

How 'bout pig racing and whiskey? You gotta love Texas! A neighborhood in a town near me is upset over the construction of a mosque. As part of some sort of passive-aggressive revenge on the attendees of said mosque one of the upstanding and cultured citizens of this community, with property next to the mosque, has decided to hold pig races every Friday night. I think whiskey was mentioned as well. I bet the neighbors who were anxious over the potential noise and congestion are just thrilled with the idea of adding pig racing and whiskey to the mix.

For those of you using or considering using Athenaze for Greek this website has audio files of the vocabulary as well as the readings in chapters 1- 10. He also has supplementary grammar examples for you to look at. This is the task I am on right now. I'm supposed to be getting all those accent marks and vowels straight before moving on.

Control of Education by the State

2006-12-05T08:12:00.000-08:00

From the Education of Cyrus I have gathered that the Persians were into "social engineering" and the Greeks were not. In Persia it was the state that educated the child rather than the parents. This kind of intrusion was something that would strike a Greek as odd. (I wonder how the Spartans would have viewed the following passage from Xenophon) Evidently, it was odd enough to require a bit of explanation on the part of Xenophon.

Most states permit their citizens to bring up their own children at their own discretion, and allow the grown men to regulate their own lives at their own will, and then they lay down certain prohibitions, for example, not to pick and steal, not to break into another man's house, not to strike a man unjustly, not to commit adultery, not to disobey the magistrate, and so forth; and on the transgressor they impose a penalty. [3] But the Persian laws try, as it were, to steal a march on time, to make their citizens from the beginning incapable of setting their hearts on any wickedness or shameful conduct whatsoever. And this is how they set about their object..."

The reference to "to steal a march on time," I believe, means that the unusual thing that the Persians did that the Greeks didn't do was use the state to prevent crimes through education, and in fact later in this passage the goals of the Persian educational system amount to virtue, justice, and the martial arts rather than basic literacy or academics.

Did the Persians imagine that when the state does not control education that parents would bring up their children to be antisocial weirdos or did they feel that the state itself was somehow threatened by the lack of a universal program? In the particular case of Persia that might mean that state control insured a steady supply of obedient motivated soldiers for empire building rather than simple defense of the state. However, Xenophon explicitly names what we consider basic moral behavior as the goal.

But the Greeks cared about the proper upbringing of their children too even without the state telling him what to do and how to do it. Here is a passage from Euripides's Medea.

"...The childless, because they have never proved whether children grow up to be a blessing or curse to men are removed from all share in many troubles; whilst those who have a sweet race of children growing up in their houses do wear away, as I perceive, their whole life through; first with the thought of how they may train them up in virtue, next how they shall leave their sons the means to live..."

Where do the Romans fit in when it comes to insuring a virtuous citizenry? To what extent did they "steal a march on time" versus leaving it up to the parents and punishing the transgressors?

The Story of Oedipus Performed by Vegetables

2006-12-01T20:26:00.000-08:00

It's the pagan version of VeggieTales.
"A sword and salad epic, featuring a Potato, a Tomato, Broccoli, Garlic, and Billy Dee Williams as the Bartender."

Greek Update: I made it through the ninth chapter of Athenaze!

Ancient History Deniers

2006-11-28T14:03:00.000-08:00

I've heard folks that deny that AIDS is caused by HIV infection. I've heard of Holocaust deniers. I never knew there were people that denied the entirety of ancient history. Enter stage right Anatoly Fomenko, an otherwise respectable Russian mathematician.

Ancient Rome, Greece and Egypt were crafted during the Renaissance by humanists and clergy; Jesus Christ was born in 1053 and crucified in 1086.

According to Wiki it's not just Fomenko but a group of Russian mathematicians that have the theory of "New Chronology ." It's an impressive synchronization of tin foil hats. Here is one of his conclusions that support his idea that the Romans and Greeks basically never existed and it's all a conspiracy of medieval Benedictine monks. Classicists and linguists be prepared for some mighty bad news.

In his final analysis of an eclipse triad described by the ancient Greek Thucydides in History of the Peloponnesian War , Fomenko dates the eclipses to 1039 , 1046 and 1057 AD. Because of the layered structure of the manuscript, he concludes that Thucydides actually lived in medieval times and in describing the Peloponnesian War between the Spartans and Athenians he was actually describing the conflict between the medieval Navarrans and Catalans in Greece from 1374 to 1387 AD.

Below is a video promoting Fomenko's book at Amazon, which amazingly receives four and a half stars with a total of 35 comments. (Aside, these reviews should be "Exhibit A" in the case against end of semester student course evaluations, or why educational content should not be democratically determined.) There were a series of commercials made that were meant to be aired on the History Channel. The History Channel declined. I just can't imagine why. This video claims that modern astronomy has it all wrong, you know Thales's eclipse didn't happen. It's worth watching for the Carmina Burana playing dramatically in the background and for the one split second that you see the Russel Crowe psychomath proving it all. Here's a different one with eight minutes of jaw-dropping claims.

If you want to earn a little cash working at home in your spare time his publisher has offered a $10,000 reward to anyone who can disprove his claims. So there you go, ten Fomenko claims equal one Goldbach conjecture.

Also one of his fellow New Chronologists has a free ebook you can read about re-dating the birth of Christ. I briefly browsed it to get a flavor of what these guys sound like. While Fomenko has dismissed radio-carbon dating, metallurgy, biology, and modern astronomy his worst crime was not to explain how the Benedictines invented the concept of mathematical "proof," in other words, no word of Euclid.

I'm not quite sure what motivated me to blog about this. Perhaps there isn't enough of dysfunctionality in my family lately and I just needed to be exposed to an extra dose of it or maybe it's the deep-seated resentment over my inability to master the diacritical marks of Greek and I'm going to spoil it all for the classicists. So take this, you macron-lovers, what do you think of your field now?

"Be wary of mathematicians, particularly when they speak the truth."

Britney Spears reviews Rex Warner's Antigone

2006-11-27T17:31:00.000-08:00

And it's a bargain!

Link to Christie's Live
Location: New York, Rockefeller Plaza
Sale Date: Dec 04, 2006
Lot Number: 104
Sale Number: 1730
Lot Title: Britney Spears
Estimate 500 - 700 U.S. dollars
Lot Description:

Britney Spears A page taken from Britney Spears' junior high school notebook containing her handwritten review of Rex Warner's translation of Sophocles' story Antigone, written in black ballpoint pen on either side of the page, Britney's review annotated by her teacher with corrections to her spelling and comments including ...Nice cover Organized Watch your spelling... and Write more neatly and her grade: 88; and a corresponding piece of yellow card decorated with the book's title Antigone in black felt pen -- 12x9in. (30.5x20.8cm.) (2)

Lot Notes: Britney Spears donated 170 personal items to an on-line auction in aid of the Britney Spears Foundation on May 22nd, 2003.

hat tip: The Superficial

Hooked on φonicς

2006-11-27T15:38:00.000-08:00

A website on Greek Education in Ancient Egypt. I don't know why this was a news flash for me: The folks that brought us the alphabet used a phonics approach. I'd like to know if the students were native Greek speakers or Egyptians. This particular papyrus seems to be all about Bob:

hat tip: Philo of Alexandria

Most Obnoxious Toy Ever Made:

Bulls-Eye ball. I can't even begin to describe the obnoxious sounds that this makes. Child slams marbles against a pad which clang about while the machine makes "sound effects" at a much louder than necessary level, because evidentally clanging marbles alone just don't do the trick. Comes with a sports announcer voice that verbalizes the non-stop scoring. No volume control.

It's the perfect passive aggressive "revenge" gift this season.

Take the one at our house. No, really.

Sins of Ignorance

2006-11-27T14:25:00.000-08:00

Xenophon, who wrote Cyropaedia, was a student of Socrates and a peer of Plato. In this snippet Tigranes tells Cyrus about the fate of a mutual friend. The comment about corruption brought to my mind the accusation against Socrates and it's interesting to see more details about what such a charge might consist of, but more interesting is the parallel to Luke 23:24. However, to clarify, here the son is asked to forgive the father for sinning in ignorance rather than the son asking the father to forgive someone else.

"Do you not know," he said, "that my father put him to death?" "And why?" said Cyrus, "what fault did he find in him?" He thought he corrupted me," said the youth ; "and yet, I tell you, Cyrus, he was so gentle and brave, so beautiful in soul, that when he came to die, he called me to him and said, "Do not be angry with your father, Tigranes, for putting me to death. What he does is not done from malice, but from ignorance; and the sins of ignorance, I hold, are unintentional."

War on Diacritics

2006-11-24T23:43:00.000-08:00

Apologies to true Greekophiles, but I'm having a personality conflict with diacritical marks. This is mostly due to me not wanting to learn the rules on placing all those pesky accent marks. I'm lazy. I simply want to learn to read Greek. So here's my ~~pathetic justifications~~ argument about why I shouldn't be subjected to accent marks.

1. Plato didn't.

2. It's not really Attic Greek. Accent marks were invented by people who spoke Koine Greek. And by the way, if you are using Wheelock's or Henle Latin, they stress the importance of macrons in Latin, but they aren't used on the AP Latin tests and I don't think it's used on the SAT subject area test either. At my house the macrons got thrown out with the leftover turkey.

3. Not unlike the accent marks in Spanish, Greek diacritics are supposed to help us Barbarian non-native speakers. With a "cure" such as these complicated rules I'd rather have the disease. If you really want to be helpful to Barbarians use two different colors of ink to distinguish the roots of verbs and nouns from their suffixes.

I am reasonable and willing to compromise. We'll can use a Timothy Dexter* solution: The first five pages or so of any book originally written in Attic Greek will contain nothing but an uninterrupted string of graves and circumflexes. Then those who wish may "pepper and salt as they please" whatever unadulterated text that follows.

Now that I'm involuntarily using Linux/Firefox I've got to figure out a new way to easily type in Greek.

PS I'm on chapter 9 of Athenaze. Also I've got several reading projects going.

- I've dropped Thucydides.
- I picked up Hanson's "War Like No Other."
- I flipped through a copy of Newton's Principia.
- I'm half way through Xenophon's Education of Cyrus (Cyropaedia) To bad I printed it out in size 8 font or I'd be done now. It's really good.
- I picked up a free history of Spain text that someone was about to throw out and got interested in the Romans in Spain.

* Original quote "fouder mister printer the Nowing ones complane of my book the fust edition had no stops I put in A Nuf here and thay may peper and solt it as they plese"

Audio on the legacy of Lord Timothy Dexter

Dating in Thucydides

2006-11-18T14:24:00.000-08:00

I don't mean the amorous type of dating, I mean the dating of events. How long-winded is this?

"In the fifteenth year (after the conquest of Euboea) , the forty-eighth year of the priestess-ship of Chrysis at Argos, during the ephorate of Aenesias at Sparta and in the last month but two of the archoship of Pythodorus at Athens, six months after the battle of Potidaea and just at the beginning of spring a Theban force a little over three hundred strong..."

I don't know where you fall in the whole BC/BCE debate but either of those options is easier than establishing dates with pedigree and papers.

I finally made it to the description of the plague that hit Athens in book two of Thucydides and I was disappointed. I would really like to like this book more than I do but it's just too dry for me. I'm not saying that there is nothing that I don't like. The descriptions of seiges are interesting, for example.

Meanwhile, I finished chapter seven in Athenaze. I was up at four this morning hitting the books. Yes, it's voluntary. Just me and a pot of coffee. I like to test what I learn by flipping through Diogenes Laertius and seeing if I can read it in Greek. Well, I still can't. But I do know enough to know that the English translation that's on the right hand side is not exactly how it was said in Greek. I think the translator tries to dress it up a bit.

For now here is my list of books to read in 2007:

1. Thucydides
2. Xenophon's Hellenica
3. VDH's A War Like No Other
4. Plato's Republic (again)
5. Xenophon's Education of Cyrus & Symposium
6. Caesar's Conquest of Gaul
7. Aeneid (I've never read this before)
8. Virgil's Metamorphosis

Comparison of Three Greek Textbooks

2006-11-16T13:54:00.000-08:00

Which do you think is the best? I have my own opinion.

Greek an Intensive Course by Hansen and Quinn:

" An adjective which modifies a noun and is preceeded by an article agreeing with the noun is in the attributive position."

Now how hard was that?? But look at how Mastronarde's Introduction to Attic Greek handles the topic. I'll leave out the examples in Greek that follow it:

"The position of an adjective in relation to the definite article and the noun serves to mark it as an attribute: an attributive adjective is inside the article-noun group."

And here is the same topic explained in Athenaze. In Athenaze the examples in Greek are given first followed by the explanation. And in general Athenaze works like this: You get a giant reading passage thrown at you at the beginning of the chapter and then you'll read the grammar later. It's a token gesture at some sort of constructivist approach as languge teaching.

"The adjective is said to be in the attributive position in these examples, in which it is placed either between the article and the noun or after the repeated article."

At any rate, I'm now on chapter 7 in Athenaze and I'm more comfortable with the word "the" popping up in every nook and cranny of a sentence. I would have switched to Hansen and Quinn except that there is no answer key to that book.

Liberal Arts and Video Games

2006-11-14T07:37:00.000-08:00

The theory around my house is that I listen to NPR because of some sado-masochistic need. To support that theory I listened to this morning's segment about majoring in "Video Games."
(The audio from the link does not work on my computer.)

"This year, the University of Southern California enrolled its first class of undergraduate students who will major in video-game development. The school is not the first major university to have a program in video games. But the curriculum is not all about car races and shootouts. Laura Sydell reports."

Now video games really are part of the liberal arts education that was intended to make us civic-minded and philosophical. Oh, paideia!

Drat These Greeks!

2006-11-13T09:08:00.000-08:00

I am trying to learn Attic Greek on my own while reading the classics in translation. Occasionally I wander into math and philosophy. Greek seems unbelievably difficult to me but maybe I'm just getting too old.

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