A Mathematician’s Lament: on education

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The art of proof has been replaced by a rigid step-by step pattern of uninspired formal deductions.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.


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.

Screaming Architecture: on systems

In “Screaming Architecture” Uncle Bob lays out one of the biggest wins by designing to the problem domain, instead of your weapon (ahem.. framework) of choice:

“If you system architecture is all about the use cases, and if you have kept your frameworks at arms-length. Then you should be able to unit-test all those use cases without any of the frameworks in place. You shouldn’t need the web server running in order to run your tests. You shouldn’t need the database connected in order to run your tests. Your business objects should be plain old objects that have no dependencies on frameworks or databases or other complications. Your use case objects should coordinate your business objects. And all of them together should be testable in-situ, without any of the complications of frameworks.

Anemic Domain Model: on systems

Martin Fowler wrote a piece in 2003 that addresses a subtle anti-pattern – developing your domain model code devoid of behavior. It’s a short, interesting read, that is related to the development of fat controllers in MVCish applications: “AnemicDomainModel”:

“In general, the more behavior you find in the services, the more likely you are to be robbing yourself of the benefits of a domain model. If all your logic is in services, you’ve robbed yourself blind.”

Dear Emma

Dear Emma,

Hi Tinker. You’ve started Kindergarten, it is a big deal, even if you won’t remember it as such down the road, and I wanted to write you a letter, a message in a bottle. I hope you find this one day on an Internet like mine, one that empowered me to reach for my dreams.

In 1988, before I met Mommy, my younger brother and I were in trouble. It’s a long story, but importantly, both of us are blessed with a fantastic family now. We had to overcome much to get to where we are.

One of the tools to achieve that, for me, was a book of short stories that contained nuggets of wisdom. Wisdom that I saw few around me followed, but those that did seemed… happier… more at peace with themselves and the nature of the world. This didn’t result in complacency, but in an openness that enabled them to see the big picture, act upon it, and be true to themselves in their journey.

I learned about the book from an unlikely source, from a Rolling Stone article in 1990, on Steven Tyler, and Aerosmith, and its fight back from destruction.

It’s called, “All I Need to Know I Learned in Kindergarten”, by Robert Fulghum. Emma, my hope is you’re going to experience these lessons growing in your heart and mind (hopefully Mommy, me, all your grandparents, aunts and uncles, have demonstrated these things for you – I think we have).

The ideas in the book seem simple, but in practice, they aren’t. For example, ‘play fair’, when you will find, and it breaks my heart that this is so, that the world isn’t fair, and that some actually consider the idea of playing fair… weak. It’s not. Or take not hitting people. Always try and uphold it, however, there will be times you’ll be faced with a choice to defend yourself and others. And it will test your integrity and what you believe.

It is confusing and messy. But know this:

If you need to ask anything, ever, your entire family will try and answer it. There is no such thing as a dumb question. And if someone implies otherwise, have them see me (or your Grandpops, or uncles, or Grandmoms, or Mommy). We’ll set them straight.

Experience is a hard teacher, because she gives the test first, the lesson after. Ask!

Most important, we all have an internal compass pointing at our true North. Listen to it. If you are feeling something is wrong – trust yourself – and think of the credo outlined at the beginning of this book.

It is a credo your Mommy and me believe in (she embodies this so much – she leads and teaches by example), and we hope you will too:

Each spring, for many years, I have set myself the task of writing a personal statement of belief: a Credo. When I was younger, the statement ran for many pages, trying to cover every base, with no loose ends. It sounded like a Supreme Court brief, as if words could resolve all conflicts about the meaning of existence.

The Credo has grown shorter in recent years – sometimes cynical, sometimes comical, and sometimes bland – but I keep working at it. Recently I set out to get the statement of personal belief down to one page in simple terms, fully understanding the naïve idealism that implied.

The inspiration for brevity came to me at a gasoline station. I managed to fill my old car’s tank with super deluxe high-octane go-juice. My old hoopy couldn’t handle it and got the willies – kept sputtering out at intersections and belching going downhill. I understood. My mind and my spirit get like that from time to time. Too much high-content information, and I get the existential willies. I keep sputtering out at intersections where life choices must be made and I either know too much or not enough. The examined life is no picnic.

I realized then that I already know most of what’s necessary to live a meaningful life – that it isn’t all that complicated. I know it. And have known it for a long, long time. Living it – well that’s another matter, yes? Here’s my Credo:

All I really need to know about how to live and what to do and how to be I learned in kindergarten. Wisdom was not at the top of the graduate-school mountain, but there in the sandpile at Sunday School. These are the things I learned:

Share everything.

Play fair.

Don’t hit people.

Put things back where you found them.

Clean up your own mess.

Don’t take things that aren’t yours.

Say you’re sorry when you hurt somebody.

Wash your hands before you eat.


Warm cookies and cold milk are good for you.

Live a balanced life – learn some and think some and draw some and paint and sing and dance and play and work every day some.

Take a nap every afternoon.

When you go out into the world, watch out for traffic, hold hands, and stick together.

Be aware of wonder.

Remember the little seed in the Styrofoam cup: The roots go down and the plant goes up and nobody really knows how or why, but we are all like that.

Goldfish and hamsters and white mice and even the little seed in the Styrofoam cup – they all die. So do we.

And then remember the Dick-and-Jane books and the first word you learned – the biggest word of all – LOOK.

Everything you need to know is in there somewhere. The Golden Rule and love and basic sanitation. Ecology and politics and equality and sane living.

Take any one of those items and extrapolate it into sophisticated adult terms and apply it to your family life or your work or your government or your world and it holds true and clear and firm. Think what a better world it would be if we all – the whole world – had cookies and milk about three o’clock every afternoon and then lay down with our blankies for a nap. Or if all governments had as a basic policy to always put things back where they found them and to clean up their own mess.

And it is still true, no matter how old you are – when you go out into the world, it is best to hold hands and stick together.

We love you sweetheart,

Mommy and Daddy

Congratulations to Paul Ford and family

Paul Ford just welcomed twins into the world yesterday. A few months back, in a piece at the Morning News he shared personal details to get here, titled “The Age of Mechanical Reproduction”.

Paul Ford’s blog has been a source of many thought provoking pieces for me over the years and if he’s reading this, well I just wanted to say thank you and congratulations.

Understanding the Maker Movement

YouTube: Maker Faire: “The Long Slow Make: Understanding the Maker Movement”:

“Anil Dash shares his observations and insights into the development of the Maker movement He sees it as a kind of political movement that is apolitical in nature but also radical and inclusive. This conversation with Anil and Dale Dougherty, founder of MAKE magazine and Maker Faire, touches on the social context of making, and what it means for individuals, families and communities. How will a “long, slow make” transform our society?”

Check out the post at Boing Boing: “Understanding Makers, a conversation with Anil Dash & Dale Dougherty” for more.