I had a student recently argue the merits of his algorithm. He was multiplying a binomial by a trinomial. To do this he was taking each term of the binomial and multiplying it by each term of the trinomial. This was resulting in the answer in one step. I at the time was trying to explain the concept of the distributive property and how it applies to multiplication of polynomials. At first I did not see the immediate proof for his algorithm working. After a few seconds of thought I saw that he was doing the distributive property in one step by memorizing how the terms would go together in the last line. This seems to work for any polynomial polynomial multiplications. This is all great except he was calling it the distributive property and it is not. The distributive property is specific. It is when you multiply in the form of a(b + c) = ab+ac . Now you can use the distributive property to prove that his algorithm is correct. The issue is the student does not know that. He only knows the algorithm and that it works.
This brings me to pose a fundamental question in Math education. Why do we not just teach algorithms? (in fact a lot of us just do) Students learn to get the answers with these algorithms. Isn’t that the goal of math education?( I say that sarcastically.) As you can tell I disagree. I believe it is the journey to the answer that is important not the answer. That is the way I grade and that is the way I teach. If the goal was just the answer then why not just have a computer do all math. There are many many computer programs that can do just that. Why teach math at all. Since we still teach math and don’t have students just use computer programs to do problems, we should not be teaching algorithms at all. Getting the answer in one step is not the journey we are interested in. We want students to understand how the math concepts fit together so they can reconstruct those concepts into new concepts. They cannot do that if they are only doing math as a series of formulas and procedures (that is how a computer looks at Math). So math like life is not about getting to the answer it’s the journey to get there. I leave you with a few people who agree with me:
Focus on the journey, not the destination. Joy is found not in finishing an activity but in doing it.
Standing still is the fastest way of moving backwards in a rapidly changing world.
No one else can speak the words on your lips. Drench yourself in words unspoken. Live your life with arms wide open. Today is where your book begins. The rest is still unwritten.
It may be when we no longer know what to do, we have come to our real work, and that when we no longer know which way to go, we have begun our real journey.
It has never been, and never will be easy work! But the road that is built in hope is more pleasant to the traveler than the road built in despair, even though they both lead to the same destination.
Marion Zimmer Bradley
I believe that life is a journey, often difficult and sometimes incredibly cruel, but we are well equipped for it if only we tap into our talents and gifts and allow them to blossom.
I believe that life is a journey towards God, and that no one has the right to insist that you go a certain road.
It is better to travel well than to arrive.
There are two mistakes one can make along the road to truth…not going all the way, and not starting.
Does this path have a heart? If it does, the path is good; if it doesn’t, it is of no use.
You should live every day like it’s your last day because one day you’re gonna be right.
Man improves himself as he follows his path; if he stands still, waiting to improve before he makes a decision, he’ll never move.
We are not human beings on a spiritual journey. We are spiritual beings on a human journey.
For a long time it had seemed to me that life was about to begin-real life. But there was always some obstacle in the way, something to be gotten through first, some unfinished business, time still to be served, a debt to be paid. Then life would begin. At last it dawned on me that these obstacles were my life.
A person often meets his destiny on the road he took to avoid it.
Jean de la Fontaine
The journey between what you once were and who you are now becoming is where the dance of life really takes place.
How we spend our days is, of course, how we spend our lives.
Be willing to be a beginner every single morning.
Do not go where the path may lead, go instead where there is no path and leave a trail.
Ralph Waldo Emerson
Whatever you do, you need courage. Whatever course you decide upon, there is always someone to tell you that you are wrong. There are always difficulties arising that tempt you to believe your critics are right. To map out a course of action and follow it to an end requires some of the same courage that a soldier needs. Peace has its victories, but it takes brave men and women to win them.
Ralph Waldo Emerson
You don’t create your mission in life – you detect it.
Let your mind start a journey thru a strange new world. Leave all thoughts of the world you knew before. Let your soul take you where you long to be…Close your eyes, let your spirit start to soar, and you’ll live as you’ve never lived before.
Two roads diverged in a wood, and I – I took the one less traveled by, and that has made all the difference.
How often I found where I should be going only by setting out for somewhere else.
We may run, walk, stumble, drive, or fly, but let us never lose sight of the reason for the journey or miss a chance to see a rainbow on the way.
Say not, I have found the truth, but rather, I have found a truth. Say not, I have found the path of the soul, Say rather, I have met the soul walking upon my path. For the soul walks upon all paths. The soul walks not upon a line, neither does it grow like a reed. The soul unfolds itself, like a lotus of countless petals.
It‘s not where we stand but in what direction we are moving.
Johann Wolfgang von Goethe
The way is not in the sky, the way is in the heart. For the traveler who knows his direction, there is always a favorable wind.
Stuart Avery Gold
Trust in what you love, continue to do it, and it will take you where you need to go.
There is no enlightenment outside of daily life.
Thich Nhat Hanh
Sooner or later we must realize there is no station, no one place to arrive at once and for all. The true joy of life is the trip.
Robert J. Hastings
The future remains uncertain and so it should, for it is the canvas upon which we paint our desires. Thus always the human condition faces a beautifully empty canvas. We possess only this moment in which to dedicate ourselves continuously to the sacred presence which we share and create.
You cannot have a happy ending to a miserable journey.
Tell me what you yearn for and I shall tell you who you are. We are what we reach for, the idealized image that drives our wandering.
Seek out that particular mental attribute which makes you feel most deeply and vitally alive, along with which comes the inner voice which says, ‘This is the real me,’ and when you have found that attitude, follow it.
It’s pretty hard to get what you want when you’re busy doing and being what you don’t want.
Most of us do not ‘sculpt’ our lives. We accept what comes our way, then we gripe about it.
We cannot escape fear. We can only transform it into a companion that accompanies us on all our exciting adventures…
Listen to me. We’re here to make a dent in the universe. Otherwise why even be here?
As one gets older one sees many more paths that could be taken. Artists sense within their own work that kind of swelling of possibilities, which may seem a freedom or a confusion.
In the degree that we remember and retell our stories and create new ones we become the authors, the authorities, of our own lives.
Often the search proves more profitable than the goal.
E. L. Konigsburg
To be born is to start the journey towards death.
A journey of a thousand miles must begin with a single step.
It is good to have an end to journey towards, but it is the journey that matters in the end.
The great lesson…is that the sacred in the ordinary, that it is to be found in one’s daily life, in one’s neighbors, friends, and family, in one’s back yard.
Do not look for rest in any pleasure, because you were not created for pleasure: you were created for joy. And if you do not know the difference between pleasure and joy you have not yet begun to live.
Every man has his own destiny: the only imperative is to follow it, to accept it, no matter where it leads him.
One’s destination is never a place but rather a new way of looking at things.
Life is what you make of it. Always has been, always will be.
By losing your goal – you have lost your way, too!
The more intensely we feel about an idea or a goal, the more assuredly the idea, buried deep in our subconscious, will direct us along the path to its fulfillment.
True religion is the life we lead, not the creed we profess.
I would love to live like a river flows, carried by the surprise of its own unfolding.
Rowing harder doesn’t help if the boat is headed in the wrong direction.
Heroes take journeys, confront dragons, and discover the treasure of their true selves.
The great awareness comes slowly, piece by piece. The path of spiritual growth is a path of lifelong learning. The experience of spiritual power is basically a joyful one.
M. Scott Peck
We don’t receive wisdom; we must discover it for ourselves after a journey that no one can take for us or spare us.
Some stories don’t have a clear beginning, middle and end. Life is about not knowing, having to change, taking the moment and making the best of it, without knowing what’s going to happen next. Delicious ambiguity.
Do not let your fire go out, spark by irreplaceable spark, in the hopeless swamps of the approximate, the not-quite, the not-yet, the not-at-all. Do not let the hero in your soul perish, in lonely frustration for the life you deserved, but have never been able to reach. Check your road and the nature of your battle. The world you desired can be won. It exists, it is real, it is possible, it is yours.
Each one of us has a fire in our heart for something. It’s our goal in life to find it and keep it lit.
Mary Lou Retton
All the arts we practice are apprenticeship. The big art is our life.
Be patient toward all that is unsolved in your heart and try to love the questions themselves like locked rooms and like books that are written in a very foreign tongue. Do not now seek the answers, which cannot be given you because you would not be able to live them. And the point is, to live everything. Live the questions now. Perhaps you will find them gradually, without noticing it, and live along some distant day into the answer.
Rainer Maria Rilke
The only journey is the journey within.
Rainer Maria Rilke
The good life is a process, not a state of being. It is a direction, not a destination.
Grace, not willpower, is what ultimately empowers us to live loving lives. Creativity, both in what spawns within the artist and the artifact, can be a vital source of that grace.
To be alive, to able to see, to walk…it’s all a miracle. I have adapted the technique of living life from miracle to miracle.
And you? When will you begin your long journey into yourself?
Journey from the self to the Self and find the mine of gold. Leave behind what is sour and bitter and move toward the sweet.
You have brains in your head. You have feet in your shoes. You can steer yourself any direction you choose. You’re on your own, and you know what you know. And you are the only one who’ll decide where you’ll go.
Sometimes in your life you will go on a journey. It will be the longest journey you have ever taken. It is the journey to find yourself.
Success is a journey, not a destination.
There was never yet an uninteresting life. Such a thing is an impossibility. Inside of the dullest exterior there is a drama, a comedy, and a tragedy.
Your journey never ends. Life has a way of changing things in incredible ways.
As I ramble through life, whatever be my goal, I will unfortunately always keep my eye upon the doughnut and not upon the whole.
The road of life twists and turns and no two directions are ever the same. Yet our lessons come from the journey, not the destination.
Don Williams, Jr.
The spiritual path – is simply the journey of living our lives. Everyone is on a spiritual path; most people just don’t know it.
Understand that the right to choose your own path is a sacred privilege. Use it. Dwell in possibility.