Sunday, 16 December 2018

Framing Mathematics

We had a numeracy PD day a few weeks ago.  This filled me with trepidation having barely survived high school mathematics.  It began with a warning about how we frame the subject:

Fair enough.  Evidently I'm not the only one who treats mathematics with caution, but I can see the point about how negatively framing maths with students can cause problems.  If you don't think maths are a useful tool that can help you solve real world problems then you've been living under a rock.  Everyone should develop basic numeracy.  I'll try and do better with how I'm framing it, but that doesn't mean maths gets a free pass on how it's delivered.

We then did a maths based online escape room exercise with Edtechteam.  This was an engaging process, but it cast a bright light on what was for me one of the problems with trying to learn maths: parsing poorly written word problems.

When one of our group (a published playwright with a Masters in English) suggested that the questions were vague to the point of being misleading the math teacher in our group said, "yeah, but any language based question is going to be somewhat unclear."  The English teacher looked at her quizzically and said, "no it isn't."

Therein lies the problem.  If a teacher who has never focused on developing strong language skills gets lost in creating nuanced word problems to get at complex mathematics, you can see where this might go wrong for everyone.

From the point of view of someone who doesn't pick up maths easily, confusing language doesn't engage me, it does the opposite.  I'd rather (and I speak as an English major) have the maths served straight up without any confusing or misleading language in the mix, but maths teachers seem determined to lean on language skills they don't have in order to confuse the numeracy they do have.

This problem appeared again when we got out to an exercise where we (again, in groups) were supposed to find factors in an array of numbers, but rather than simply explaining the logic involved, the activity was dressed up in a tax avoidance theme that made no sense to me or the science and history teachers I was working on it with.  So far this morning both maths activities had demanded that we embrace confusing and contradictory language in order to get at the logic below.

In this activity, if you selected a number to get paid the 'tax man' got all the factors of that choice.  So if you picked twelve, the tax man got 1  2, 3, 4 and 6 dollars.  When I asked how I was being taxed $16 on the $12 I made I was told that the taxes don't actually come out of the money I was making, which isn't helpful.  When I suggested that people should pay taxes in order to support all the  benefits of society they enjoy and shouldn't be trying to dodge paying them, I was told that I was putting too much thought into this.  At least someone is.  This has always been the way with me and mathematics, especially when it dresses itself up in confusing language in a desperate attempt to appear more interesting.

I think I'm a pretty sharp fellow.  I've been able to calculate binary subnets in order to build networks and I've never had trouble doing the maths needed to be a mechanic or a technician.  When the maths are immediate and real I'm able to get a handle on it, but the bubble gum world of high school mathematics has always alienated and confused me.  It seems arbitrary and nonsensical because it often is.

Maybe the best way we can frame mathematics is to stop trying to make it into something it isn't.  If we treated it like the tool it can be instead of trying to turn it into some kind of spy based action adventure or libertarian tax dodging daydream, we wouldn't have so many people feeling alienated by it.

Of course, the solution is obvious but how we solve it is prevented by how we organize education into departments.  If we collaborated on word problems with the English department, we'd remove a lot of that confusion.  If we applied our mathematics through science, business and technology we wouldn't get lost in the confusion of maths for maths' sake.  We could be applying mathematics in the statistics we use in social sciences or  the ratios we use in art, but we separate numeracy off in high school and let it atrophy in a maths classroom that struggles to connect to the real. 

Ironically, our PD followed these two engaging but ultimately confusing activities up with two teachers telling us about their experimental manufacturing technology-mathematics combined course which encourages applied maths students to work through manufacturing technology in solving real-world problems.  No imaginary tax schemes.  No escape rooms.  Just applying maths to real world problems in an unobstructed and meaningful way that leads to outcomes that are transparent and obvious.

This would mean combining mathematics with other courses and then working to integrate numeracy into those subjects in a constructive and transparent way.  There could still be an academic/abstracted mathematics stream for the tiny percentage of students who would need it, but for those of us who aren't aiming to be theoretical physicists or academic mathematicians, we need our math served up without the garnishes.  Knowing what we're doing it and why we're doing it would go a long way to alleviating the maths anxiety so many of us have.

Saturday, 8 December 2018

The Neverending Story of Rational Reductionism

Remember the first time you went away from home without your family?  I'd done scout weekends and that sort of thing, but the first extended time away was when I was heading to Air Cadet Basic Training in Trenton for two weeks in the summer of 1984.  Just before I left I saw The Neverending Story.  As a creative kid who was neck deep in Dungeons & Dragons and art, and whose dad kept telling him to stop wasting his time and take real courses that led somewhere, it resonated.

It's been thirty-five years since fifteen year old me saw that film and an awful lot has happened in the meantime.  Having just watched it again, I'm stunned by how strange a film it is.  What I took as a high fantasy romp when I was a teen is actually a bizarrely meta (physical) narrative that would make a suicidally depressed Hamlet snort with amusement.  The film was directed by famed German director Wolfgang Petersen, and boy does das kopfkino it produces lay on the schadenfreude thick.

The film's message, that your imagination can save you from the banality of existence, suggests that you need something more than rationality to justify your reason for being.  Or, back to Hamlet again, "There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy."  I find a great deal of comfort in recognizing the complexity of existence, though many people seem terrified of it and go to great lengths to simplify it.

The film's thesis is that imagination allows us to withstand the pointlessness of existence and offers hope.  If you turn yourself off from the impossible it prevents you from holding despair at bay.  The scene in the film where Atreyu's horse gives up hope and sinks into the mud of a swamp (of Sadness no less) is one of the most powerful in the film.

The quest that drives the story forward is the destruction of Fantasia, an alternate reality that exists as an expression of human creativity and imagination.  It's being destroyed because people are losing their hopes and dreams, the very things that cause Fantasia to exist.


Viewing this film produced one of those strange lateral connections for me that science minded people put down to coincidence but artists thrive on.  I've just finished reading Michael Crichton's Travels, an autobiographical book by the popular author where he reflects on his travels, both physical and spiritual.  As a hardening atheist (thanks to reading Dawkins' The God Delusion) I found myself suppressing eye rolls as Crichton attempts new-age spiritualism again and again in search of something tangible beyond the science he started with as a Harvard trained medical doctor.  But Crichton's canny speech at the end of the book offers an approach to the unknowable that I couldn't help but agree with.

It's worth reading Travels just go get to to the closing speech that he never gave.  It deconstructs a number of scientific prejudices that hard rationalists cling to even though they aren't particularly logical, such as surgeries carried out to prevent a possibility of illness with no clear scientific benefit, or the long history of fake experimental results that are accepted because they support a current world view rather than the truth of things.  Hard rationalism is as susceptible to fantastic thinking as any other human endeavour.  Crichton's final lines highlight the space he has made for human understanding beyond the limitations of rational inquiry:

"...we need the insights of the mystic every bit as much as we need the insights of the scientist. Mankind is diminished when either is missing. Carl Jung said: The nature of the psyche reaches into obscurities far beyond the scope of our understanding."

Our rational understanding of things allows us to do many relatively mundane things in the real world, but our existence reaches deeper than that, and we ignore what we are capable of if we limit ourselves to the realms of what our remarkable but limited intellects can comprehend.  Put another way, there is understanding to be found in our being as well as in our thinking.

Lao Tzu's Tao Te Ching had this covered 2500 years ago.  We've
forgotten a lot of that wisdom in our information age.
In addition to critiquing science's hypocrisy, Crichton also bounces back 2500 years to Lao Tzu (who I have a weakness for) and describes how the founder of Taoism understood how our rational minds and our irrational existence must work together to bring us into a fuller understanding of our place in the universe.  It's powerful stuff, and a reminder that there is no simple (ie: only mind-based) answers to the big questions.  It takes all that we are to even begin to attempt answer them.  In embracing our existential intelligence we also come to a more balanced understanding of our place in the world.


With Crichton's angle on how we frame the impossible in my mind, I was slapped in the face by The Neverending Story's strident attack on reductive, 'feet on the ground' rationality in the face of the threat of non-existence.  The brief scene between Bastion and his father is stark and cruel, but I think it points to something obvious.  It's never mentioned how Bastion's mother dies, but the father's unwillingness to acknowledge it in any way suggests a shameful death, and we all know which kind of death is the most shameful and must not be spoken of.

"When a visibly sad Bastian tells his father that he's had yet another dream about his mom, he responds that he understands, but quickly adds that they have to move on, emphasizing that they can't let her passing stop them from getting things done. And just when you think he'll soften up and help Bastian process his pain, Bastian's father lays into his son for doodling in his notebook during math class."

Considering the metaphysical message of this film and that strange dialogue between father and son, I was left hanging on the edge of tears.  My Mum was upstairs the last time I saw this film.  She's been dead six years this time around, but that sense of loss is always surprisingly quick to surface.  Her life as an artist was frequently derailed and undervalued, and her end was, I suspect, similar to Bastion's mom's.  The Neverending Story suddenly took on a resonance that it didn't have before.

The evil that is destroying the world in The Neverending Story is The Nothing.  It is quite literally non-existence.  Bastion's father's brusque 'move on and keep your feet on the ground' advice suggests (quite obviously I think) that his mother commited suicide.  The entire narrative in Neverending Story is based around Bastion trying to summon his imagination to battle this existential disaster, something that Lao Tzu and Michael Crichton would both agree can't be done with reason alone.  The film's only weakness is it's reductive imagination is the answer philosophy.  Imagination is vital in bringing you to a place beyond the rational, but populating it with make believe isn't the goal once you get there.  Imagination is what allows us to see beyond the world around us and plumb those existential mysteries.


From Kermit the Frog pondering Rainbow Connections to Alice looking down rabbit holes, there is a lot of art that seeks to explore the limitations of rational inquiry and how it fails to answer the big questions.  Creativity is hard enough without tying your hands up with rational absolutism, so I can appreciate why many artists lean more heavily on the hidden intelligence found in existentialism for their inspiration; there is power in our being that cannot be easily explained.  

Our ability to reach down into our selves and gain inspiration and insight makes us powerful in a way that thinking never can.  For the Bastions of Neverending Story, travelling Crichtons and other artists out there, it's something we should never let the hard rationalists of science ever try and trivialize away as flights of fancy.  There are truths in our being that can't be found through rational inquiry.

Imagination by itself is a fine thing, but when it's used as a means of opening the door to existential comprehension it really comes into its own.  Crichton describes how measurement always misses the quiddity of a thing, it's inherently reductive to say anything can be completely understood through its measurements.  A wholistic, existential understanding, along with specific, rational comprehension, is the most complete way a human being can relate and understand the world.  Crichton's closing lines encouraging us not to ignore and belittle the irrational - something that The Neverending Story also argues, though it gets lost in imagination for imagaination's sake.

Valuing both rational and irrational human comprehension offers us a more balanced and effective way forward, and gets us into the vicinity of answering the big questions.  The trick is not to get carried away with imagination or rationalism and end up treating either one as the answer to everything.  As in all things, balance offers more insight.

Other notes:

Atreyu: If you don't tell me, and the Nothing keeps coming, you will die too, both of you! 

Morla, the Ancient One: Die? Now that, at least, would be *something*.

Urgl: I like that, the patient telling the doctor it's all right. It has to hurt if it's to heal.

I'm not the only one picking up on the weird vibe this film is giving:

From a 2018/the sky is falling/we're-all-illiterate-because-of-technology point of view, the book keeper's scorn when talking to Bastion, the pre-teen main character way back in 1983 (over two decades before smartphones) is interesting:

Koreander: The video arcade is down the street. Here we just sell small rectangular objects. They're called books. They require a little effort on your part, and make no bee-bee-bee-bee-beeps. On your way please.

... and reminds me of the Socrates quote and that we're most prejudiced with our own children.  It's also a timely reminder that the tech of our time doesn't define us any more than video arcades did in the '80s.  I grew up in them and it didn't make me illiterate.

The Way: