Thelonius Monk’s Shapeshifting Chord

monk-chord

One of my part-time hobbies is being a Monk interpreter. A Monk interpreter not only learns how to play Monk’s compositions, but also makes a point of getting into the head of this eccentric man. The reason to do this is that Monk was an improvisor – and he was driven by an inner vision. If you can tap that inner vision, then you can generate Monk-like music – and improvise on it…even while playing Beatles songs.

I wrote a piece in 2013 about Monk as a mathematician.

screen-shot-2017-01-15-at-2-43-39-pm

Math can be about patterns (visual or sonic). Math does not always have to be expressed in numbers. Monk once said,“All musicians are subconsciously mathematicians”.

A Symmetrical Chord

The chord I’m talking about has four notes. It is typically used as a dominant chord – which naturally resolves to the tonic. Unlike the classical dominant-seventh, this chord has a flatted fifth – which makes it slip into a symmetrical regime – as shown in the picture above – inscribed in the circle of fifths.

dominant_seventh_flat_five_chord_on_c

 

According to Wikipedia, this chord is called the “Dominant Seventh Flat-Five Chord“. The cool trick about this chord is that it can resolve to either of two different tonics – each being a tri-tone apart.

So for instance, a chord with these notes:     Eb   F   A   B      can resolve to either Bb or E as the home key.

This chord also happens to contain 4 of the 6 tones in a whole tone scale, which Monk famously used (often as a dominant arpeggio).

If you are not familiar with music theory, you may still appreciate the beauty of sonic geometry and how it can generate such variety. If you apply similar concepts to rhythm as to harmony then you have a wonderfully rich canvas for endless musical expression. I like the way Monk wove these geometries together in a way that makes the foot tap and the ear twinge – and the brain tweak.

Monk was of course not the only one to apply these ideas – but he did accomplish something remarkable: the application of embodied math. If you have spent as much time as I have learning his language, listening to him improvise can cause a smile – or the occasional giggle – to pop out. Like an inside joke.

There is plenty of material on the internet about Monk. Here’s one voice among the many who have acquired an appreciation for Monk:  How to Listen to Thelonius Monk – by George H. Jensen, Jr.

Enough with this Square Root of -1 Business!

Like so many other people, I was kept from appreciating the beauty and utility of mathematics because of the way it was taught to me.

The majority of introductions to complex numbers start with the elusive and mysterious square root of -1, denoted by i.

A number that has an i stuck on to it is called “imaginary” (a convenient differentiator to “real”). Being asked to learn something that is called “imaginary” is not very motivating to young learners who work best starting with concrete metaphors.

51tPEpSto+L._SX322_BO1,204,203,200_

The imaginary number is counterintuitive and confusing. And it’s not the coolest part. Sure, i was an important invention at a critical stage in the history of math when there was no good way to express z2 = -1. And yes, it makes a good ending to a long story (which happens to be true): math has advanced through several expansions of the concept of “number” … from the counting numbers to the wholes – to the negatives – to the fractions – to the irrationals – and finally to complex numbers – where i came along and saved the day.

But…does this mean that invoking i is the best way to explain complex numbers to novices – to everyday people? I join many others in saying that there is a better way to learn about the wonderful world of two-dimensional numbers. One voice among those is Kalid Azad.

kalidphoto-color-homeHe speaks in metaphors and freely engages the visual mind to help us grasp math concepts using our whole brain. In his explanation on complex numbers, Azad says this about i: “It doesn’t make sense yet, but hang in there. By the end we’ll hunt down i and put it in a headlock, instead of the reverse.”

…..

When you get an intuitive, aesthetic feeling for why certain mathematical ideas are being taught, you become more motivated to learn the notation. The corollary: learning math notation without understanding why is like learning musical notation before ever being allowed to listen to or play music.

Paul-Lockhart2Paul Lockhart, in A Mathematician’s Lament, compares the way math is taught to a nightmare scenario in which music is taught to students using sheet music notation only (no actual music is played or heard) – until the student is advanced enough to start “using” it.

 

What is a Two-Dimensional Number?

When I read that complex numbers are really no more “imaginary” than real numbers, I decided that I would start dismantling my old worldview. Why should I assume that numbers have to be one-dimensional? Over time, I became more accustomed to the notion that a number can occupy a plane (the complex plane) and not just a line (the number line). Learning how to make images of the Mandelbrot Set helped a lot.

Think of Multiplication as Rotation

Screen Shot 2015-12-18 at 9.49.07 AMInstead of trying to wrap your mind around i, and how it magically makes equations come out right, let’s start with geometry. Think of multiplication as rotation and expansion. In the blog Girls Angle, Ken Fan introduces complex number multiplication in a nice visual way… here.

Here’s a video explaining complex numbers in terms of physical metaphors, and eventually explaining why the square root of -1 becomes a necessary part of the notation.

screen-shot-2015-12-17-at-4-03-15-pm

Squaring

Consider the following diagram showing what happens when you square certain complex numbers that lie on the unit circle:

complex plane gravitation

The dot on the right represents the complex number (1+0i). When you square it, it stays the same (no surprise: 1×1=1). The number at the left is (-1+0i). When you square that, it becomes (1+0i). But when you square the number at top (0+1i) it “rotates” by 90 degrees to (-1+0i). Finally, at the bottom, the number (0-1i) rotates…but would it be correct to say that it rotates by 90 degrees clockwise to (-1,0i)? Depends on how you look at it. Rotating by 270 degrees counter-clockwise has the same result. This is the nature of rotation and angular reality: it is periodic – it cycles…it repeats.

What an awesome idea. Multiplication is like doing a whirling dervish jig.

Animated Squaring

Here’s an interactive tool I made that allows you to play with 200 dots (complex numbers) randomly scattered on the complex plane. You can experience what happens when complex numbers are squared. It also allows you to multiply the dots (using a complex number dot that you can drag along the screen).

http://ventrella.com/ComplexSquaring/

Screen Shot 2015-12-17 at 3.57.18 PM

This interactive tool might make you feel as if the dots on the screen are obeying some sort of gravitational law of physics. Well, in a way, yes, that’s what’s happening. When you add, multiply, or exponentiate numbers, you get a new number. In the complex plane, the space where that change takes place is two-dimensional. That’s cool! We like images.

Here’s another visual tool: when we multiply two complex numbers, such as (a+bi) and (c+di), we can visualize the operation in this way:

mult

In pseudocode:

realPart      = (a*c) - (b*d);
imaginaryPart = (a*d) + (b*c);

This explanation of multiplication does not require i.

To this day, I STILL do not feel very much music when I think about the square root of -1.

On the other hand, the more I play around with visualizing and animating complex numbers, the more intuitive they become, and the deeper my sense that these numbers are as real as any old one-dimensional number.

They are not imaginary at all.

Math Word Problems are Problematic

Mark Twain said: Never let school interfere with your education.

Here’s a math riddle:

“Peter has 21 fewer marbles than Nancy. If Peter has 43 marbles, how many marbles does Nancy have?”

The first sentence requires me to do some linguistic fiddling. There is an implication that both Peter and Nancy possess marbles – but it is not directly stated. The second sentence begins with “If”, which means the primary grammatical elements in the question are postponed until the end. Let’s re-phrase this riddle to say:

14_shutterstock_2584271“Peter and Nancy each have a bag of marbles. Peter has 43 marbles in his bag. Peter has 21 fewer marbles than Nancy has. How many marbles does Nancy have in her bag?”

….

This might make the riddle easier to solve. Or it might not. Either way, I can say for sure that all this wordy bullshit s irrelevant to the actual math.

Math, like Music, is a Universal Language

Now consider what it would be like if you were naturally talented in math, and you were faced with a math riddle expressed in English…but English were not your first language. You may have to spend more time on the question, and you may make some critical mistakes. The subtleties of one language may not translate to another language, causing you to trip up.

We are playing with words here. Now, playing with words is fine; it’s part of how we learn to speak, listen, read, and write. In fact, playing with words that have mathematical content is a good exercise. But this should not come into play for testing students on math skills. The problem (as always) is in the testing.

Here’s another one:

“Sue has two pencils. She spends one hour at the store and buys three more pencils. How many pencils does Sue have in all”.

bivdg4wrggssxq8akk2r

WTF does “spends one hour in the store” mean? Is this just narrative fluff, or is there some clever hint in there?

If I had been presented with this problem as a young student, I would have spent some time mulling over “spends one hour at the store”. However, this is irrelevant and unrelated to the answer.

How to Obfuscate Mathematical Thinking With Clever Language

For dyslexic students, students who learn through action (kinesthetic learners), students who are visual thinkers, and students who learn best by building things, this wordsmithing can be a recipe for failure.

In the real world of adults getting things done and making a living, math is rarely experienced in the form of clever riddles. Math – at its best – is manifested deep within the texture of our daily actions.

Here’s another one:

“You have 24 cookies and want to share them equally with 6 people. How many cookies would each person get?”

Let’s think about this. I “have” 24 cookies. (That’s a lot of cookies – why would I have so many cookies?) I “want” to share them with 6 people. Okay. I have a desire to share cookies. So far so good. I’m a generous guy! But then the second sentence appears unrelated: “How many cookies would each person get?” Wait a minute: am I about to give these cookies to these people? And what exactly does “equally” mean?

I know it may seem trivial for me to analyze these details. As an adult I know what this sentence means. But as a young student, I may not have had the full vocabulary or grammatical wherewithal to jump right to an answer. Also, as a “narrative learner”, I would have really wanted to make sure I understood the characters involved, their motivations, etc. I could imagine getting easily get swept up by the storyline (simple as it is).

In short, by working out the characters of this story and their motivations, I may not actually be doing math: I might be engaging in language craft and storytelling. Which is great! But this should not interfere with my being tested on my innate math skills.

Here’s another:

“Kennedy had 10 apples. She gave some to John. Now she has 2 apples left. How many apples did she give to John?”

The tense of this little story jumps back and forth between past and present. At age 55, I am now quite facile with language, but when I was 10, I would have had to put in some effort into parsing these shifts in tense.

tumblr_ma5ndonxYD1rgqecyo1_500

In fact, my language skills were quite poor when I was 10, and this had an impact on all my school subjects (not just math). Later in life, after I had escaped school and actually started to gain some relevant skills, MIT offered me an opportunity to earn a Master’s degree. They did not ask me any math riddles. MIT knows better than that.

Language

One might argue that language skills are fundamental and important for learning most anything. That’s accurate. Reading, writing, speaking, and listening are fundamentally useful, and the better you are at language, the better you are likely to become at most other skills.

If this is the case, we might conclude that mixing grammatical sentence structure with mathematical logic is a valuable skill.

Indeed.

But school curriculum designers should not confuse the ability to parse a cleverly-crafted sentence with one’s innate mathematical abilities.

The problem, as always, is with TESTING.

I’ll close with this:

An Open Letter to the Education System: Please Stop Destroying Students

Why Nick Bostrom is Wrong About the Dangers of Artificial Intelligence

emvideo-youtube-VmtrvkGXBn0.jpg.pagespeed.ce.PHMYbBBuGwNick Bostrom is a philosopher who is known for his work on the dangers of AI in the future. Many other notable people, including Stephen Hawking, Elon Musk, and Bill Gates, have commented on the existential threats posed by a future AI. This is an important subject to discuss, but I believe that there are many careless assumptions being made as far as what AI actually is, and what it will become.

Yea yea, there’s Terminator, Her, Ex Machinima, and so many other science fiction films that touch upon deep and relevant themes about our relationship with autonomous technology. Good stuff to think about (and entertaining). But AI is much more boring than what we see in the movies. AI can be found distributed in little bits and pieces in cars, mobile phones, social media sites, hospitals…just about anywhere that software can run and where people need some help making decisions or getting new ideas.

John McCarthy, who coined the term “Artificial Intelligence” in 1956, said something that is totally relevant today: “as soon as it works, no one calls it AI anymore.” Given how poorly-defined AI is – how the definition of it seems to morph so easily, it is curious how excited some people get about its existential dangers. Perhaps these people are afraid of AI precisely because they do not know what it is.

Screen Shot 2015-09-02 at 10.51.56 AMElon Musk, who warns us of the dangers of AI, was asked the following question by Walter Isaacson: “Do you think you maybe read too much science fiction?” To which Musk replied:

“Yes, that’s possible”….“Probably.”

Should We Be Terrified?

In an article with the very subtle title, “You Should Be Terrified of Superintelligent Machines“, Bostrom says this:

An AI whose sole final goal is to count the grains of sand on Boracay would care instrumentally about its own survival in order to accomplish this.”

godzilla-610x439Point taken. If we built an intelligent machine to do that, we might get what we asked for. Fifty years later we might be telling it, “we were just kidding! It was a joke. Hahahah. Please stop now. Please?” It will push us out of the way and keep counting…and it just might kill us if we try to stop it.

Part of Bostrom’s argument is that if we build machines to achieve goals in the future, then these machines will “want” to survive in order to achieve those goals.

“Want?”

Bostrom warns against anthropomorphizing AI. Amen! In a TED Talk, he even shows a picture of the typical scary AI robot – like so many that have been polluting the air waves of late. He discounts this as anthropomorphizing AI.

Screen Shot 2015-08-31 at 9.51.57 PM

And yet Bostrom frequently refers to what an AI “wants” to do, the AI’s “preferences”, “goals”, even “values”. How can anyone be certain that an AI can have what we call “values” in any way that we can recognize as such? In other words, are we able to talk about “values” in any other context than a human one?

Screen Shot 2015-09-01 at 3.49.13 PMFrom my experience in developing AI-related code for the past 20 years, I can say this with some confidence: it is senseless to talk about software having anything like “values”. By the time something vaguely resembling “value” emerges in AI-driven technology, humans will be so intertwingled with it that they will not be able to separate themselves from it.

It will not be easy – or possible – to distinguish our values from “its” values. In fact, it is quite possible that we won’t refer to it at “it”. “It” will be “us”.

Bostrom’s fear sounds like fear of the Other.

That Disembodied Thing Again

Let’s step out of the ivory tower for a moment. I want to know how that AI machine on Boracay is going to actually go about counting grains of sand.

Many people who talk about AI refer to many amazing physical feats that an AI would supposedly be able to accomplish. But they often leave out the part about “how” this is done. We cannot separate the AI (running software) from the physical machinery that has an effect on the world – any more than we can talk about what a brain can do that has been taken out one’s head and placed on a table.

Screen Shot 2015-08-31 at 9.56.50 PM

It can jiggle. That’s about it.

Once again, the Cartesian separation of mind and body rears its ugly head – as it were – and deludes people into thinking that they can talk about intelligence in the absence of a physical body. Intelligence doesn’t exist outside of its physical manifestation. Can’t happen. Never has happened. Never will happen.

Ray Kurzweil predicted that by 2023 a $1,000 laptop would have the computing power and storage capacity of a human brain. When put in these terms, it sounds quite plausible. But if you were to extrapolate that to make the assumption that a laptop in 2023 will be “intelligent” you would be making a mistake.

Many people who talk about AI make reference to computational speed and bandwidth. Kurzweil helped to popularize a trend for plotting computer performance along with with human intelligence, which perpetuates computationalism. Your brain doesn’t just run on electricity: synapse behavior is electrochemical. Your brain is soaking in chemicals provided by this thing called the bloodstream – and these chemicals have a lot to do with desire and value. And… surprise! Your body is soaking in these same chemicals.

Intelligence resides in the bodymind. Always has, always will.

So, when there’s lot of talk about AI and hardly any mention of the physical technology that actually does something, you should be skeptical.

Bostrom asks: when will we have achieved human-level machine intelligence? And he defines this as the ability “to perform almost any job at least as well as a human”.

I wonder if his list of jobs includes this:

Screen Shot 2015-09-02 at 12.45.02 AM

Intelligence is Multi-Multi-Multi-Dimensional

Bostrom plots a one-dimensional line which includes a mouse, a chimp, a stupid human, and a smart human. And he considers how AI is traveling along this line, and how it will fly past humans.

Screen Shot 2015-08-31 at 9.51.34 PM

Intelligence is not one dimensional. It’s already a bit of a simplification to plot mice and chimps on the same line – as if there were some single number that you could extract from each and compute which is greater.

Charles Darwin once said: “It is not the strongest of the species that survives, nor the most intelligent that survives. It is the one that is most adaptable to change.”

Is a bat smarter than a mouse? Bats are blind (dumber?) but their sense of echolocation is miraculous (smarter?)

parrot

Is an autistic savant who can compose complicated algorithms but can’t hold a conversation smarter than a charismatic but dyslexic soccer coach who inspires kids to be their best? Intelligence is not one-dimensional, and this is ESPECIALLY true when comparing AI to humans. Plotting them both on a single one-dimensional line is not just an oversimplification. By plotting AI on the same line as human intelligence, Bostrom is committing anthropomorphism.

AI cannot be compared apples-to-apples to human intelligence because it emerges from human intelligence. Emergent phenomena by their nature operate on a different plane than what they emerge from.

WE HAVE ONLY OURSELVES TO FEAR BECAUSE WE ARE INSEPARABLE FROM OUR AI

We and our AI grow together, side by side. AI evolves with us, for us, in us. It will change us as much as we change it. This is the posthuman condition. You probably have a smart phone (you might even be reading this article on it). Can you imagine what life was like before the internet? For half of my life, there was no internet, and yet I can’t imagine not having the internet as a part of my brain. And I mean that literally. If you think this is far-reaching, just wait another 5 years. Our reliance on the internet, self-driving cars, automated this, automated that, will increase beyond our imaginations.

Posthumanism is pulling us into the future. That train has left the station.

african cell phoneBut…all these technologies that are so folded-in to our daily lives are primarily about enhancing our own abilities. They are not about becoming conscious or having “values”. For the most part, the AI that is growing around us is highly-distributed, and highly-integrated with our activities – OUR values.

I predict that Siri will not turn into a conscious being with morals, emotions, and selfish ambitions…although others are not quite so sure. Okay – I take it back; Siri might have a bit of a bias towards Apple, Inc. Ya think?

Giant Killer Robots

armyrobotThere is one important caveat to my argument. Even though I believe that the future of AI will not be characterized by a frightening army of robots with agendas, we could potentially face a real threat: if military robots that are ordered to kill and destroy – and use AI and sophisticated sensor fusion to outsmart their foes – were to get out of hand, then things could get ugly.

But with the exception of weapon-based AI that is housed in autonomous mobile robots, the future of AI will be mostly custodial, highly distributed, and integrated with our own lives; our clothes, houses, cars, and communications. We will not be able to separate it from ourselves – increasingly over time. We won’t see it as “other” – we might just see ourselves as having more abilities than we did before.

Those abilities could include a better capacity to kill each other, but also a better capacity to compose music, build sustainable cities, educate kids, and nurture the environment.

If my interpretation is correct, then Bolstrom’s alarm bells might be better aimed at ourselves. And in that case, what’s new? We have always had the capacity to create love and beauty … and death and destruction.

To quote David Byrne: “Same as it ever was”.

Maybe Our AI Will Evolve to Protect Us And the Planet

Here’s a more positive future to contemplate:

AI will not become more human-like – which is analogous to how the body of an animal does not look like the cells that it is made of.

tree-of-lifeBillions of years ago, single cells decided to come together in order to make bodies, so they could do more using teamwork. Some of these cells were probably worried about the bodies “taking over”. And oh did they! But, these bodies also did their little cells a favor: they kept them alive and provided them with nutrition. Win-win baby!

To conclude, I disagree with Bostrom: we should not be terrified.

Terror is counter-productive to human progress.

The Evolution of Mathematics on Planet Earth

tumblr_ncxcqixe9k1qzy92fo1_1280

math-heartMany people couldn’t imagine Math and Biology going out on a date. Flirting with each other from time to time…maybe. But a date? Never! Math is precise, abstract, cool, and distant. Biology is messy, unpredictable, prone to mood swings, and chemically dependent…as it were.

But this may be changing.

“The conversion of biology into a more quantifiable science will continue to the extent that it might even become the main driving force behind innovation and development in mathematics”

Philip Hunter

Let me explain why I think Math and Biology are ultimately compatible, and in fact, part of a Single Reality.

tumblr_my165xRv5f1swaxzgo1_250

I have written a few articles on the subject of math, and raised questions as to the universality, truth-status, and God-givenness of Math. Here is something to consider about Math and Biology:

Math Evolved in the Biosphere

Let’s start with numbers. Imagine a mother crow busily feeding her three chicks. She would become worried if she came back to her nest to suddenly find two chicks instead of three.

House_Crow_feeding_chicks

She would know there something is wrong with this picture…because crows can count (they can subitize small numbers, like about 2 or 3).

How did it come about that some animals, like crows and humans, can count? First of all, in order for intelligent beings to be able to count, they have to live in an environment where countable objects are found, and where counting has some evolutionary benefit. Consider a gaseous planet where fluids intermix and there is no way to detect a “thing” or “event” and to compare that with another “thing” or “event”. In this kind of world, there is nothing to count.

seahorseFor that matter, it is unlikely that an intelligent entity that can count could ever evolve on such a planet in the first place, because structure and differentiation at some physical level are required for living things to bootstrap themselves into existence.

Theories of autopoiesis, negentropy, and the emergence of mind from matter rely on the existence of a prior structure to the universe where it is possible for self-regulation, and self-creation to arise. One might say that the origins of life had a head start long before those first molecules started dancing together and accidentally reproducing. Maybe it wasn’t such an accident after all.

imgres

…which brings me to a core concept: since Earth’s biosphere gave rise to animals that can count, as well as those things that can be counted – at the same time, we must understand ourselves as in and of the biosphere – we and it all evolved together: one did not come before the other.

70212-1024x603Which came first: the chicken or the egg? Neither. They have both been in a continual state of becoming since egg-like things and chicken-like things have existed. And if you go back in time far enough, these things look less and less like chickens and eggs.

We animals have evolved to understand containment, and that is partly because hierarchy evolved within the fabric of physical biology. We know what it means for something to be “inside” or “outside” of something else. We clumpify, categorize, differentiate, compare, and identify. All animals need some degree of this compartmentalization of nature in order to operate within it.

We cannot separate our math from the environment from which it evolved. The very foundations of math evolved within the bodies and minds of animals as a part of evolution. At least this is what several recent scientists and philosophers are suggesting. (Mathematicians are more likely to claim that math is universal, constant, and unchanged by biology.)

OctoMath

In a previous article I consider what kind of math would have emerged if octopuses has evolved to become the complex and dominant species on earth, instead of humans. This is not so hard to imagine, considering how intelligent they are.

Screen Shot 2015-08-10 at 12.19.31 AM

Would an advanced octopus race have stumbled upon complex numbers? Would they have become as obsessed with the Cartesian coordinate system as we are? Since they have no skeletons, would they have formulated a geometry based on angles and lengths? Of course we can’t know, but it is likely that they would have created some math concepts that we may never achieve. And that would be because the long history of math that we have built and that we rely on to create new math has taken our brains and societies too far away from the place where an octopus-like math would naturally arise.

mouroborobius2Now consider aliens from a completely different kind of planet than Earth. What kind of math would originate in that world? Many people would argue that math is math and it doesn’t matter who or what discovers or articulates it. And there may be some truth to this. But we can only hope and imagine that this is the case.

Until we meet aliens from another planet and ask them if they understand and appreciate the fibonacci sequence, I have to assume that their math is different than ours.

What do you think?

(I would have consulted one of my octopus friends on the subject…but I don’t speak their language).

………….. The Beauty of Gray Code …………..

30386a802345a35b20138b350d9ca50a34df7016

http://www.mathworks.com/matlabcentral/fileexchange/40928-generate-gray-code-disk

 ———————–

graycodebinaryclock_wallclock_render

http://anthony.liekens.net/index.php/Misc/TrueBinaryTime

———————–

greyXall-1

http://vision.middlebury.edu/~schar/papers/structlight/p1.html

———————–

opticalEncoder-italsensordotcom

 http://www.jeffreythompson.org/blog/tag/gray-code/

 ———————–

singleturn

http://www.fachlexika.de/technik/mechatronik/sensor.html

 ———————–

Gray Code is an alternative binary representation, cleverly devised so that, between any two adjacent numbers, only one bit changes at a time. If there is an error reading any bit that has changed then, at worse, the read value will never be out by more than one unit.

This has tremendous value in the real world. Computers might be digital, but we live in an analog world. Interfaces between these need to be carefully considered.

encoder2

http://www.qsl.net/oe5jfl/encoder.htm

Our Colorful Mathematics Revolution

math

Education bureaucrats are trying to gently and safely tweak a broken system so that fewer students fail math.

bbb

Meanwhile, a colorful revolution is taking shape outside the walls of a crumbling institution. A populist movement in creative math is empowering an unlikely crowd.

Authors of Wikipedia math pages aren’t contributing to this populist movement. They are intent on impressing each other; competing to see who can reduce a mathematical concept to its most accurate, most precise (and least comprehensible) definition.

Screen Shot 2015-01-31 at 11.22.53 AM

A debate rages on a “new way” to do subtraction. Oh does it rage. But step back from that debate and consider that these tricks, algorithms, processes, hacks, become less relevant as new tools take their place. When calculators entered into the classroom, something started to change. That change is still underway.

Do students no longer need to learn to do math by hand? No. But calculators (and computers) have changed the landscape.

Rogue amateur mathematicians, computer artists, DIY makers, and generative music composers are creating beautiful works of mathematical expression at a high rate – and sharing them at an even higher rate. This is a characteristic trait of the “new power“.

math art

Technology

(1) Computers are better at number-crunching than we are. If used appropriately, they can allow us to apply our wonderfully-creative human minds to significant pattern-finding and problems that we are well-suited to solve.

(2) Computer animation, generative music, data visualization, and other digitally-enhanced tools of creativity and analysis are becoming more accessible and powerful – they are helping people create mathematically-oriented experiences that not only delight the senses, but express deep mathematical concepts. And they also help us do work.

(3) The internet is enabling a new generation of talented people (amateurs and professionals) to exchange mathematical ideas, discoveries, and explanations at a rate that could never be achieved via the ponderous machinations of university funding, publishing, and teaching. There will never be another Euler. Mathematical ideas now spread through thousands of minds and percolate within hours. It is becoming increasingly difficult to trace the origins of an idea. Is this good or bad? I don’t know. It’s the new reality.

Five things You Need to Know About the Future of Math

According to Jordan Shapiro:

1. Math education is stuck in the 19th Century.
2. Yesterday’s math class won’t prepare you for tomorrow’s jobs.
3. Numbers and variables are NOT the foundation of math.
4. We can cross the Symbol Barrier.
5. We need to know math’s limitations.

math-fail2

We can (and will – and should) debate how math should be taught. Whether the “symbol barrier” is a actually a barrier, and whether memorizing the multiplication tables is necessary, no one can ignore the seismic changes that are rumbling underfoot.

110511_4lo


-Jeffrey