You can do it! You must tell yourself that. Don't think because you haven't done a problem before, that you can't do it, or that someone else must show you how to do it first (a myth some people want to use to keep others in ignorance). You can do it! Don't be afraid. You've learned how to do the hardest things you'll ever do, walking and talking--mostly by yourself. This stuff is much easier! Say loud and clear, often, "I can do it !!"

Don't worry about making mistakes; we all do. I make a lot of mistakes. My brighter students usually make more mistakes because they try an answer, it doesn't work, they change it until they get a solution. See On Making Mistakes at bottom of this page, 

Stay with it, don't give up, and have fun! This is going to take time. Keep a journal or diary of your work. Write down what you are doing, what you're thinking about, what's easy, what you don't understand, questions you have. Make up your own problems, your own tests.

Talk to someone else about what you are doing; teach it to a friend.

Don't get discouraged if you are confused or don't understand it; most of us are confused about most things, but we don't want others to think we are.

Look for patterns; look at differences and ratios. Don't write one number for an answer, but write it so you can tell where it came from and how to get the next answer and all answers. Try to predict what will happen next.

Don't just memorize, try to understand; think about what you are doing. Ask yourself often..What am I doing? Where am I going? What am I trying to do? Be able to get the answer later on; come back to it again.

Ask a lot of questions; you might get a lot of questions answered by others. Ask questions like: What happens if I change this number? What happens if I change this graph? Can I predict what happens next? Is there a pattern here?

Guess. Try a number..see if it works. Test it out. This will get you going. Don't wait.

Write a program on a calculator or computer to help solve the problem. Use your calculator to do the "dirty work".

Graph everything. Look for patterns!

Try to find 2 or 3 ways to do everything. Be able to check your work, that builds confidence. Do problems more than once, but in a different way.

If a problem is difficult, work hard on it, take a break by working on a different problem, but then come back to the original problem. Change it to one like it, but easier, solve that one, than go back to the harder one.

All the problems were done by the author, his students, or what the author understood of the mathematician's works at this time.

This learning will never be finished..there will always be new questions and new problems..enjoy!

Sir Ernest Rutherford, President of the Royal Academy, and recipient of the Nobel Prize in Physics, related the following story:

Some time ago I received a call from a colleague. He was about to give a student a zero for his answer to a physics question, while the student claimed a perfect score. The instructor and the student agreed to an impartial arbiter, and I was selected. I read the examination question:

"Show how it is possible to determine the height of a tall building with the aid of a barometer."

The student had answered: "Take the barometer to the top of the building, attach a long rope to it, lower it to the street, and then bring it up, measuring the length of the rope. The length of the rope is the height of the building."

The student really had a strong case for full credit since he had really answered the question completely and correctly! On the other hand, if full credit were given, it could well contribute to a high grade in his physics course and certify competence in physics, but the answer did not confirm this.

I suggested that the student have another try. I gave the student Six minutes to answer the question with the warning that the answer should show some knowledge of physics. At the end of five minutes, he hadn't written anything. I asked if he wished to give up, but he said he had many answers to this problem; he was just thinking of the best one.

I excused myself for interrupting him and asked him to please go on. In the next minute, he dashed off his answer, which read, "Take the barometer to the top of the building and lean over the edge of the roof. Drop the barometer, timing its fall with a stopwatch. Then, using the formula x=0.5*a*t^2, calculate the height of the building."

At this point, I asked my colleague if he would give up. He conceded, and gave the student almost full credit. While leaving my colleague's office, I recalled that the student had said that he had other answers to the problem, so I asked him what they were.

"Well," said the student, "there are many ways of getting the height of a tall building with the aid of a barometer. For example, you could take the barometer out on a sunny day and measure the height of the barometer, the length of its shadow, and the length of the shadow of the building, and by the use of simple proportion, determine the height of the building."

"Fine," I said, "and others?"

"Yes," said the student, "there is a very basic measurement method you will like. In this method, you take the barometer and begin to walk up the stairs. As you climb the stairs, you mark off the length of the barometer along the wall. You then count the number of marks, and this will give you the height of the building in barometer units. A very direct method."

"Of course. If you want a more sophisticated method, you can tie the barometer to the end of a string, swing it as a pendulum, and determine the value of g [gravity] at the street level and at the top of the building. From the difference between the two values of g, the height of the building, in principle, can be calculated."

"On this same tack, you could take the barometer to the top of the building, attach a long rope to it, lower it to just above the street, and then swing it as a pendulum. You could then calculate the height of the building by the period of the precession".

"Finally," he concluded, "there are many other ways of solving the problem. Probably the best," he said, "is to take the barometer to the basement and knock on the superintendent's door. When the superintendent answers, you speak to him as follows: 'Mr. Superintendent, here is a fine barometer. If you will tell me the height of the building, I will give you this barometer.'"

At this point, I asked the student if he really did not know the conventional answer to this question. He admitted that he did, but said that he was fed up with high school and college instructors trying to teach him how to think.

The student was Niels Bohr (1885-1962), Danish Physicist, Nobel-Prize winner in 1922, best known for proposing the first 'model' of the atom with protons & neutrons, and various energy states of the surrounding electrons--the familiar icon of the small nucleus circled by three elliptical orbits ... but more significantly, an innovator in Quantum Theory.

On making Mistakes

  From Don's email to his student Grace in Chicago: Thank you for working with me Grace, I really enjoy seeing you do these great things. You know Grace, what we're doing is normally done in 11th grade?! That's why when you make a mistake now, that's ok, because you don't have to think you need to know everything right away, like in school, like for a test on Friday! You are getting a fine start, on important mathematics, that even the best mathematicians struggled with and made lots of mistake with. Here is the quote from Kline's book about this:

"In his first paper on the Calculus (1669), Newton proudly introduced the use of infinite series to expedite the processes of the calculus...

As Newton, Leibnitz, the several Bernoullis, Euler, d'Alembert, Lagrange, and other 18th-century men struggled with the strange problem of infinite series and employed them in analysis, they perpetuated all sorts of blunders, made false proofs, and drew incorrect conclusions; they even gave arguments that now with hindsight we are obliged to call ludicrous."

   -from 'MATHEMATICS THE LOSS OF CERTAINTY' by Morris Kline