Tag Archives: thinking

A Mathematician’s Lament: on education

Posted on by

Paul Lockhart wrote an accessible read on what is wrong with math education and the popular perception of math that is reinforced in culture that has been shared on the Web in quite a few corners. It deserves a wider read: “A Mathematician’s Lament”:

The art of proof has been replaced by a rigid step-by step pattern of uninspired formal deductions. The textbook presents a set of definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students copy them into their notebooks. They are then asked to mimic them in the exercises. Those that catch on to the pattern quickly are the “good” students.

The result is that the student becomes a passive participant in the creative act. Students are making statements to fit a preexisting proof-pattern, not because they mean them. They are being trained to ape arguments, not to intend them. So not only do they have no idea what their teacher is saying, they have no idea what they themselves are saying.

Even the traditional way in which definitions are presented is a lie. In an effort to create an illusion of “clarity” before embarking on the typical cascade of propositions and theorems, a set of definitions are provided so that statements and their proofs can be made as succinct as possible. On the surface this seems fairly innocuous; why not make some abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem-generated. To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it.

The point is you don’t start with definitions, you start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.

Related:

Kevin Devlin: “Lockhart’s Lament – The Sequel”

Slashdot: “A Mathematician’s Lament — an Indictment of US Math Education”

G.H. Hardy:

A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

Four Links on Ideas (and Perspective)

Posted on by

My friend Arpit Mathur posted a great piece today on ideas and implementation: “On ideas, implementation and iteration”. I can relate to this – including the self admonition to hunker down and execute.

The following three links try to make an additional point on what an idea needs – partners. People who have different perspectives from which ideas can clash, evolve, and grow. There is an opportunity in sought out diversity of background and opinion to make something that much stronger.

Or to put it another way, if your project has its Lennon, does it also have its McCartney?

YouTube: “Steven Johnson: Where Good Ideas Come From” (via Boing Boing):

Slate: Joshua Wolf Shenk: Two Is the Magic Number: A new science of creativity.: “a new body of research has begun to show how growth and achievement emerge from relationships”.

And while this last link is not centered on ideas and creativity, it applies directly if you take the above to heart I think:

The Most Powerful Word In The Human Vocabulary: Perspective:

Choosing a career path is tough, and the education system doesn’t make it much easier. They try to guide you towards one of these worlds that exists on our planet. The problem is that most of the time, schools don’t understand the unique perspective of their students, and they don’t adapt their needs to the situation. This is why finding your passion is so important. Passion is not the be-all-end-all, but rather the guiding force that allows us to make the right decisions as we travel through life. Finding your passion, connecting with your tribe and achieving your goals are the steps that you must take in order to find a career that you will find rewarding. But having a sense of perspective for the world around you allows you to understand that your opportunities are endless.

Never let someone with little perspective guide you away from your passion. Next time you have a conversation with someone who doesn’t understand why you are doing what you are doing, try to understand their perspective. How were they raised? What was their path in life? This will often let you understand why they are making the statements they are. By getting this, you can understand where they are coming from. People that are not in your world will very often not have the same perspective to you. Remember, that’s what makes the world so unique and wonderful.

This post is pretty much me talking to myself as I have a few coals in the fire and some additional perspective will go a long way, but I think the links are inspiring and hope you enjoy.

Update: Arpit shares a link that in some ways reinforces the idea above: WSJ: “The Origins of Good Ideas” – “the adjacent possible.”.

http://dailylocal.com/articles/2010/09/24/opinion/srv0000009463806.txt

Douglas R. Hofstadter “analogy as the lifeblood, so to speak, of human thinking”

Posted on by

Douglas R. Hofstadter: Analogy as the Core of Cognition:

My point is simple: we are prepared to see, and we see easily, things for which our language and culture hand us ready-made labels. When those labels are lacking, even though the phenomena may be all around us, we may quite easily fail to see them at all. The perceptual attractors that we each possess (some coming from without, some coming from within, some on the scale of mere words, some on a much grander scale) are the filters through which we scan and sort reality, and thereby they determine what we perceive on high and low levels.

“Programming is an exercise in overcoming how wrong you’ve been in the past.”

Posted on by

Kickingbear: “Blog Archive » Don’t Be A Dick: Compiled Flash and You.”:

Programming is an exercise in overcoming how wrong you’ve been in the past. At first you’ll overcome the syntax errors, then you’ll overcome the structural errors, and then you’ll come to align your code with the standards of a greater community and you’ll feel safe and like you’ve made it. You haven’t – you’re still wrong because you’re always wrong. You are playing a game you cannot win. And let’s face it – if it was a game you could win you’d not be playing at all.

via Arpit’s Web Quotes tumblr

“Fear drives out reason”

Posted on by

Reverend Dr. Martin Luther King: Beyond Vietnam — A Time to Break Silence:

The truth of these words is beyond doubt, but the mission to which they call us is a most difficult one. Even when pressed by the demands of inner truth, men do not easily assume the task of opposing their government’s policy, especially in time of war. Nor does the human spirit move without great difficulty against all the apathy of conformist thought within one’s own bosom and in the surrounding world. Moreover, when the issues at hand seem as perplexed as they often do in the case of this dreadful conflict, we are always on the verge of being mesmerized by uncertainty; but we must move on.

And some of us who have already begun to break the silence of the night have found that the calling to speak is often a vocation of agony, but we must speak. We must speak with all the humility that is appropriate to our limited vision, but we must speak.

via stilicho, via Anil:

…Dr. King shows the best possible way to testify, the highest calling of declaring one’s faith publicly. Most public declarations of faith are unseemly, full of preening and judgement. I grew up in an area where it seemed most Christians acted anything but, so it was a revelation to me for a public figure to have championed his religion so humbly, honestly and respectfully. Before I encountered Dr. King’s speeches, I didn’t understand that true manifestations of faith could cause someone to embrace those who were different or those with whom we disagree. It’s obvious why so many, regardless of their own faith or lack thereof, found common cause and a comfort in Dr. King’s values.

Al Gore: MLK Day 2006: Speech at Constiution Hall:

Fear drives out reason. Fear suppresses the politics of discourse and opens the door to the politics of destruction. Justice Brandeis once wrote: “Men feared witches and burnt women.”

The founders of our country faced dire threats. If they failed in their endeavors, they would have been hung as traitors. The very existence of our country was at risk.

Yet, in the teeth of those dangers, they insisted on establishing the Bill of Rights.

Is our Congress today in more danger than were their predecessors when the British army was marching on the Capitol? Is the world more dangerous than when we faced an ideological enemy with tens of thousands of missiles poised to be launched against us and annihilate our country at a moment’s notice? Is America in more danger now than when we faced worldwide fascism on the march-when our fathers fought and won two World Wars simultaneously?

It is simply an insult to those who came before us and sacrificed so much on our behalf to imply that we have more to be fearful of than they. Yet they faithfully protected our freedoms and now it is up to us to do the same.

(via Atrios)