Think of one of your favorite classes. What's come to mind? For some, learning certain content is memorable. For some, the classroom environment can be remembered in detail - working in groups, the nature of discussion, experiencing student-led learning. For others, a class that connects to future employment is important. And maybe, for some or possibly even many, it's a combination of these and other parts of a class.

But, what about learning to fail? That could seem like more of a nightmare than a pleasant and important remembrance of a class. I've been teaching at Davidson College for over a decade and, increasingly, I hope my students learn, yes, to fail. I hope they pass my class but also that they learn to fail and grow by doing so.

Do I facilitate uncomfortable, frustrating courses? In fact, many students comment on how they enjoy the courses and approach. How? It's important to understand what I mean by failing. Let me summarize this concept with the following image, drawn by a studio artist and math student at Davidson College.

The artist's name is Tife Odumosu. I see his drawing as a visual representation of one method to effective learning. Tife didn't sit down and immediately draw the perfect cartoon. He started drawing and iterated into his more finalized sketch. Some strokes of his pencil improved his work and others did not.

Is this failure? I don't think so. But, taken in another context, it could be seen as so. I teach math, where the odd problems often have answers in the back of the book. Life rarely comes, outside such a class, with answers in the back of the book. You check your answers in the future with outcomes that result from your choice.

Learning math shouldn't by marked by those occasions when someone gets a correct answer. What about those times you must redo your work? As you iterate in your work, what might you learn about your tendencies? How might you improve for the next time, even if you do it incorrectly once again? In fact, in some areas of mathematics there is no right answer.

I'd like more people to trust their own ideas and learn to patiently make imperfect attempts as they observe their thoughts evolving. To encourage this type of thinking, I'm leading a Massive Open Online Course, often called a MOOC, through DavidsonX, which is a partnership between Davidson College and edX. The course comes in two parts with part one launching in late February. The topic is applications of linear algebra generally in the context of computer graphics or data mining.

Let's see an application in graphics. A pixel in a color image can be stored with three values representing the red, blue, and green intensities of the corresponding pixel. Each value ranges from 0 to 255, where 0 means no red and 255 means the full intensity of red in the corresponding pixel in the image.

Simple enough. Now, let's use equations for lines to alter an image. Remember, y = mx + b, where m is the slope and b is the y-intercept of a line? Here, x will be the red, blue, or green intensity in a pixel in an image. Then, y will equal the new intensity. Choosing m and b and applying this to color intensities allows us to recolor an image with linear equations. For example, we might take m = -1 and b = 255 forming y = -x + 255. This would turn a value of 255 (full intensity of red) to 0 (no red) in the corresponding intensity value of a pixel.

Let's give the Mona Lisa a mathematical make-over. We simply need equations for the red, green and blue intensities in the image. Then, we'll apply this to every pixel in the image and have your new image. I'll take

(new red pixel) = (old red pixel) - 80

(new green pixel) = -0.75(old green pixel) + 150

(new blue pixel) = -0.5(old blue pixel) + 200

Here is what results.

Is that what you expected? I didn't know exactly what to expect. I had a certain idea but also simply had to run the program to see. Like Tife drawing his cartoon, I simply tried and saw what I got. Then, I adjusted my attempts, learning from what I'd done and iterating to what I wanted. At times, my iterations were based on mathematical calculations. Other times, I mathematically doodled, in a sense, with numbers.

So, soon, my MOOC will launch and we'll see what participants from around the world do with this and other similar exercises. For those ready to explore the ideas offline and outside a MOOC, there is also my book, *When Life is Linear: From Computer Graphics to Bracketology*. It covers related topics.

To close, I'll share what I tell my students with increasing frequency each year:

Look not only for where you succeed but where you believe that you didn't. It is here that you can learn how you learn, how you achieve, and what challenges you. You can learn to trust your thoughts.

For this, I share a paraphrasing of Albert Einstein's article *Education for Independent Thought*; this quote appears in the one-man play *Walking Lightly: A Portrait of Albert Einstein* by Len Barron.

The purpose of education is to nurture thoughtfulness. The lesser function of thinking is to solve problems and puzzles. The essential purpose is to decide for oneself what is of genuine value in life. And then to find the courage to take your own thoughts seriously.

In our education, we should learn to stand in the unknown and wait - for what? You don't know. Will the answer come? Often. But not always. Will the image match your expectations? If not, try again and see what you learn. When you simply don't know, stand there, for it is when you stand - not knowing, that the answer may come. Here I share words by William S. Burroughs:

Your mind will answer most questions if you learn to relax and wait for the answer.

And yes, I want students to try and make attempts and learn from them. For this, I share a quote by Thomas J. Watson, former chairman and CEO of IBM:

If you want to increase your success rate, double your failure rate.

I want students to feel affirmed in my classes. I strive to craft moments that they can remember when asked about the course. I want them to grow in their willingness to make attempts and iterate and improve in their insight about underlying concepts. Experiencing this process is something they can remember and benefit from for years to come. I want them to learn to make attempts, successful or not, and see the benefit either way.