Do The Math

Apparently math comes in handy sometimes and our cultural predisposition to be dismissive of innumeracy is coming home to roost. We've fallen behind and we're not catching up. We're not even trying. Our K-12 system, largely public schools, are to blame. The pandemic shutdown may have exacerbated the problem, certainly drew attention but wasn't the cause. This has been going on for a long time as public schools have pushed conceptual understanding over indepth comprehension. They favor labels and are putting out a product that doesn't live up to the credential. Colleges and universities across America are finding incoming students who've passed high school calculus who simply cannot handle algebra as demonstrated by placement tests. So what happens? Instead of starting ahead of the curve, these students are taking remedial courses in high school algebra, necessary to have any chance in a college level math course.

Maybe some teachers are trying, shrugging off the mantra that learning has to be fun (and games) and going old school. Learning is knowing things and acquiring skills. One teacher literally ditched the games and stated (out loud) that:

"You have to explicitly teach the content.”

It isn't clear if she's been fired [yet] or if the union knows about this departure from doctrine. Research indicates that students actually learn math when they are explicitly told the rules of the road rather than relying on serendipity and intuition. This debunks the trendy notion that Inquiry Based Learning is a silver bullet, a belief held even though research strongly suggests that IBL works best in graduate level courses where students have sufficient base knowledge to support curiosity. They know enough to know what they don't know. Your eight grader doesn't. 

There is a chance this whole "teaching and learning" thing will catch on as the path (ed: really?) is already paved with a return to phonics. For our children's sake let's hope it does.

Olde School

New School Algebra

by

George A. Wentworth

Copyright 1898

From the preface, a total of two pages from a small-format text:

The first chapter of this book prepares the way for quite a full treatment of simple integral equations with one unknown number. In the first two chapters only positive numbers are involved, and the beginner is led to see the practical advantages of Algebra before he encounters the difficulties of negative numbers.

The definitions and explanations contained in these chapters should be carefully read at first; after the learner has become familiar with algebraic operations, special attention should be given to the principal definitions.

Where to begin?

First, let's ignore the dated and politically incorrect use of the male pronoun as this was published well before women had the vote in the US or Eleanor Roosevelt published "It's Up To The Women." Then, the text is four hundred seven pages (not counting a whopping six pages of front matter) published in a four and a half by seven inch format starting with definitions and simple equations and covering multivariate equations, imaginary numbers, quadratics and simultaneous equations, properties of series and the binomial theorem. On average the book is more than fifty percent exercises (you know, where the learner does the work of learning by actually solving problems) with Chapter XII comprising fourteen pages with less than a page and a half of text and the remaining space dedicated to 66 exercises. Today's student might wonder what happened to all the pictures, the visual pop-outs of definitions and "further study," and the multi-cultural-correct photos of children having fun doing something that must be related to math if for no other reason than it is in a math book.

Today's pedagogical theorists would dismiss this as antiquated "drill and kill." Obsolete. Ineffective. Out of step with today's students. Nothing could be further from the truth. This is actually "skill and thrill." By progressively working thru exercises that demonstrate and elucidate the principles explained in the text the student learns--masters--techniques and concepts necessary for a deep understanding of and proficiency in more advanced mathematics.

This one book can take a learner from colours and counting to pre-calc. Yet you'll not see this in any modern school today. If you have a child you wish to see as a tiger-cub you don't need to subscribe to modernity in mathematics. Instead you should go to a used book sale and buy a text no less than fifty years old. Have your child learn alongside the minds that put a man on the moon. 

Math Sucks

Long gone are the days when slide rules, used aboard the Apollo missions putting boot prints on the moon were usable here on earth. Not because they fall short when solving some seriously sophisticated problems but because slide rules don't do the simplest arithmetic operations: addition and subtraction. In the age of slide rules it was a reasonable assumption that a slide rule user could add two or more three or more digit numbers.

We are now confronted with a reality (and a future) where very few if any of the people born in the United States can add or subtract. We're talking the simplest of arithmetic, not math[1]. This has come to the fore in local mass media with a debate on how we teach addition to children, specifically the various journeys between $$ 19 + 27 = ? $$ and $$ 46 $$ Old farts write down 19 then write 27 below aligning along implied decimal points and draw a bar below 27. Then they add 7 to 9 and come up with 16 writing the 6 below the bar and the 7 and then they write the 1 above the 1 in 16. This is called "carry the one"[2] and this is then added to the 1 from 16 and the 2 from 27 with the resulting 4 written to the left of the 6 below the bar. Voila! 46! This approach is well understood by folks who can count points and keep score in bridge. Explains the dearth of Bridge Clubs in Georgia Public Schools, eh?

There are some serious takeaways from the Great Addition Debate. First, we've lowered the bar to the point that we'll celebrate if the average student graduates FROM high school with the ability to add two numbers. Not three nor any more than three. We'll throw a party if more than five out of ten graduates arrive at "46" for the above problem even if they have to copy the answer or have it given them by a teacher. The second issue is the addition algorithm schools are using: $$ 19 + 27 =  $$ $$ (10 + 9) + (20 + 7) = $$ $$ (10 + 20) + (7 + 9) =  $$ $$ 30 + 16 = $$ $$ 30 + (10 + 6)  = $$ $$ (30 + 10) + 6 = $$
and finally $$ 40 + 6 = 46 $$
Execution of this algorithm in a classroom environment pretty much precludes adding numbers with more than two digits due to limitations on time and "Smart Board" space. Should Georgia Public Schools be successful we will pump out graduates who can successfully add any two two digit numbers. Given enough time and paper.

While this situation may have devolved from a more proficient past it is not entirely unintentional and is currently embraced. Asking today's teachers to do a Mad Minute[3] would produce appallingly disappointing results so don't ask them to teach and demand this of tomorrow's leaders (and teachers). Other rationalizations include the bruised and broken self esteems brought on by the Drill and Kill approaches that are pretty much required by the Old School techniques and the fact that the New Math supports "conceptual learning." Or at least the learning of concepts. This is said to include powers of 10[4], as they decompose "16" to "one in the ten's column and six in the one's column" with the illuminating explanation that the "ones column" is "ten to the zeroth power" and the "ten's column" is "ten to the oneth power" at which point we've completely abandoned addition in favour of a hallucinagenic trip down a rabbit hole.

Suppose we demand the basketball couch embrace a concept-centric pedagogy[5]. He would be forced to draw out (with 'x's and 'o's? that almost looks like algebra. or does he need manipulatives? animated power point?) a simple pick and roll play, making sure the team groks the concept. These students also grok dribbling, layups, free throws and general ball handling, even the bounce pass, it's origins, benefits over other forms of passing and the cultural implications of its behind the back use. Now these kids have little or no actual practice and an outstanding performance would be dribbling five times before bouncing the ball off your own foot sending it out of bounds or directly into the hands of an opposing player. Of course the coach is not expected to field a winning team. No, he is going to field a championship team, as good as any in the state, probably the country. All the parents, the principal and even the superintendent of the school system will declare this team "straight A ball players." But honestly, if they ever took the court against a pick-up team of 3rd graders who actually play for 3-4 hours a day who do YOU think would win?

But focusing on concepts over capabilities  moves us from the objective world of "the answer is 46 and all other answers are wrong" to the more comfortable subjective world of "critical thinking" and "let's understand the concepts" where flexibility in grading supports the demand for A's. And we also hide behind the calculator excuse (everybody has one, who needs to do their own arithmetic?) to explain our exclusive presentation of the abstract and subjective rather than the concrete and tangible. To be fair the critical thinking part, were it really there, might actually trespass on mathematics, asking and hopefully answering the question: "are these the two right numbers to add and how do we know?"

Similar but perhaps more disturbing is the "nobody really uses math" excuse for cultivated math ignorance. It's almost like Godwin's law, but you cannot have a tweet fest on this topic without someone dragging out "I took calculus at Tech twenty years ago and haven't used it since--it was just a clean out course." That cannot be allowed to slide by without comment. If calculus were your biggest hurdle, you should not have gotten in to Tech. Even twenty years ago. And even if you had Dr. "Death" Wray, calculus doesn't rise to the level of "electromagic", "heat transport" or "p-chem". Those are clean out courses.

But suppose we entertain the notion of  this "don't teach it if they ain't be usin' it" model of "education" for just a bit. How many times do you find yourself in a sales presentation only to realize this is the perfect time for that Shakespeare sonnet you had to memorize in your Junior year? Not so much, eh? Then why teach Shakespeare? Or Chaucer. Or Mark Twain. Or Steinbeck. Really want to touch a PC nerve? Toss out Maya Angelou. What about history? Just because one person way back when said "those who do not learn from history are doomed to repeat it" does not mean we need to study history. First we keep hearing about the good old days and second what makes everyone think only the bad things will be repeated? Isn't it equally likely that only the good things will repeat? Ditch history. Then there is "Art" and whatever that is it isn't something you do unless forced. Ditch it too. Then there is English or Language Arts or whatever PC fluff name used these days. Here is where the calculator excuse applies. No one needs to know grammar or how to spell--your word processor does that for you. Don't know what a word means? Look it up online. Ditch Language Arts too. If we only teach what kids will need to know in the real world we can start handing out high school diplomas after kindergarten.

Want to really piss folks off and get a clear understanding of what is really going on at the same time? Ask how many high school football players make it into the NFL. That low? Then ditch all football programs in public school. Now you've hit a hard stop. Folks, mostly parents, will come out of the woodwork touting the benefits, mostly indirect (like building character) offered by sports competition. Perhaps so. It certainly fills a man-made void created when classroom academic competition was declared verboten. So if we're going to justify sports and all the fuzzies based on their secondary carry-over benefits then just what makes math so special that it can be excluded as "unused in the real world?"

The excuses presented sound a lot like mass societal self-inflicted stupidity. Until we recognize how we really do use math on a daily basis[6] and stop accepting "I'm just not good at math" as an excuse to not only fail a math course but to avoid taking them altogether we as a society will continue to Suck At Math.

[1] We'll not here go down the rat hole of "arithmetic IS math" but instead will deal with the corrosive impact of "Term and Topic Inflation" in education in a separate diatribe.

[2] We leave it as an exercise to the reader to PROVE that when adding only two numbers with an arbitrary number of digits the most that will be carried from one column to another is 1. Extra credit if you can prove that the maximum carry-over when adding "n" numbers is "n-1".

[3] A test with approximately 20 arithmetic problems (addition, subtraction, multiplication and division) to be completed in one or two minutes. Often used as a mental warmup/stretching exercise at the beginning of a math class.

[4] Set aside for the moment that exponentiation is special form of multiplication and if multiplication is reduced to iterative addition that is still higher order arithmetic than the problem being solved. Just. Don't. Go. There.

[5] Can you apply the term "pedagogy" to coaching? What the hell, if educrats can call "arithmetic" "math" let's abuse one of their words.

[6] If you want an excellent introduction to how mathematics is involved in every thing we do every day Jordan Ellenberg's "How Not To Be Wrong: The Power Of Mathematical Thinking," is highly recommended.

To Infinity...

...and beyond!

Do you spend much time thinking about infinity?

Probably not since you are wasting perfectly good infinity-thinking time reading this tripe. And ponder this: you don't have an infinite amount of time to ponder infinity. Due to your lack of infinity-thinking you probably just think of infinity as a lazy eight all laid over on it's side. Might make you think $$ \infty < 8 $$ or at least that infinity is somehow inferior. Perhaps inebriated.

If you think much about infinity then like most folks you may think of infinity as the largest number there is which is not a bad definition at times. Suppose you think of a large number, larger than you ever thunk before and you can give it a name. That number still isn't infinity because infinity is the largest there is and is therefore larger than yours. In fact, it is larger than yours plus one. Or wrap your noggin' around this: infinity is larger than your largest-ever number PLUS INFINITY! In fact infinity is bigger than infinity plus infinity. All because infinity is the largest number there is and like the national debt it just keeps getting bigger and just when you think it is getting big as fast as it possibly can it starts getting bigger even faster.

Now we just bounced around some quizzical ponderings about adding a couple of numbers and comparing to infinity to get you out of that self-absorbed all-about-Dunwoody mind-trap you've been stuck in and get a few of those remaining synapses fired up. Now it's time to get a bit more formal.

Now let's be clear before this goes any further. We're not talking arithmetic here--this will tread dangerously close to math. You know. That shit you forgot. And this is not modern math--the kind that seems to always involve toothpicks or pizza--this is that classic stuff with the Greek letters and what not.

And this is not the kind of question that you're kid is gonna see on the CRCT. No siree. Nothing at like:

If Johnny has 5 reefers and gives 3 to Suzie for a peek down her sweater, what can we say about Johnny?

1. Johnny's gonna score with Suzie.
2. Johnny is hanging with the wrong crowd.
3. Johnny has 2 spliffs left.
4. Johnny IS the wrong crowd. 
5. All of the above.

Nope. This problem has fewer and much simpler answers. In fact, it only has one answer. And the question is "What is the sum of all integers greater than or equal to one?" Or, to get your geek on:
$$\sum\limits_{i=1}^\infty i = ? $$
While all the round-eyes are headed down to Kroger to buy more toothpicks before going all Rain Man on us let's get all the Asian kids to put down their violins and talk this thing thru.

First off we know the answer just has to be big. After all we start at 1 and add every other number on top of it so how can it not? We've already learned it can't be bigger than infinity--or can it?

This really makes your head hurt.

We're adding up an infinite number of numbers all of which are greater than one and the answer cannot be greater than infinity because infinity is the biggest number there is or ever will be but that is also the number of numbers we're adding up. Can it also be that the infinite sum of numbers greater than one cannot grow faster than infinity itself? Is this the mathematical equivalent of the physics conundrum of two objects approaching one another, each traveling at three quarters the speed of light and yet each sees the oncoming speed as less than the speed of light except that in this case we're talking about infinity and infinity is actually accelerating? Can we at least agree that infinity has no mass?

So if the sum of an infinite number of numbers greater than one cannot be infinity then surely it must be something else. If it is indeed less than infinity what other number could it be less than? If it is less than one other number can it be less than others as well? Could an infinite sum of numbers itself be less than an infinite number of numbers? But since we're adding integers then surely the sum of integers must also be an integer otherwise our entire understanding of the universe would collapse--dogs would meow, cats would bathe, pigs would fly and Democrats would pay taxes. The world as we know it would be over.

And so it is.

The answer to this infinite sum of integers is indeed less than a well known though relatively recently discovered number--zero. And just for fun it is a fraction as well.

Spoiler alert!

The answer is:
$$\sum\limits_{i=1}^\infty i = -1/12 $$
Don't believe it? Why, because half the shit you read here is made up? Fine. Check this out:



Of those who actually watched this video some probably have the same feeling you get when you see David Copperfield make the Eiffel Tower disappear--it looks believable but deep down you just know it isn't true. The rest are mumbling to themselves that one sixth really is bigger than one fourth and they knew it all along.

And the math world is all a-flutter as well but not for the reason you might think. Turns out the answer is correct but there are those who disagree with the method.

But given this answer perhaps the best way to look at this is if you had a dozen eggs and Johnny steals one this number represents the egg Johnny took.