Dan and I have been co-teaching a couple classes at the Robinson Center this quarter: *Zeno: Fractions within fractions* for 4th and 5th graders, and *Turtles All The Way Down *for 8th through 12th graders.

One of the areas of emphasis in our teaching is the art of asking questions. Without good questions, it is very hard to have good answers. Interestingly, the practice of asking questions can be very difficult for students to develop. The older the students are, the harder it gets, and this inverse proportionality illustrates a basic principle about question-asking: it is largely a matter of *ownership of the material*. Students who feel control over their relationship with a subject will, in general, ask good questions. Students who feel someone else controls the subject will not. But all students, of any age, need to practice asking questions to get a feel for finding those really exciting ones. To illustrate this, consider this list of questions our 4th and 5th graders asked on the second day of class in *Zeno*.

Student Questions from the opening circle:

Can you add infinity?

Why are ten millimeters the same as one centimeter?

What approximation of π should be used in calculations?

What is the largest fraction between 0 and 1?

How do you convert a repeating decimal into a fraction?

How do you convert any decimal into a fraction?

Why are fractions used in math?

Are there different sizes of infinities?

Which came first, decimals or fractions?

Why is it easy to get the fraction form of 0.0303… while 0.3939… is hard?

Can all fractions be turned into a decimal and vice versa? Are decimals a division problem like fractions are? Can a decimal be a part of a fraction?

Why is it called ‘math’?

What fraction represents π?

Can 0.9…. be a fraction?

What equation is used to find π?

Why is the top number called the numerator and the bottom the denominator?

Why do you start counting with the number 1?

None of these are bad questions, and some are quite good. However, for contrast, look at this list of questions from yesterday, our seventh day of class.

If there is more than one kind of infinity, what are their names?

If Hilbert charged $n per night for Room n, how much would he make in a night with the hotel full? Does he make more money if the hotel is full, but then he rents out more rooms? How can Hilbert make as much money as possible?

Is π-e irrational? What about πe, π+e, and other combinations?

Does π show up in the decimal expansion of e?

Does φ show up in π?

How can there be different infinities?

Would an infinite spiral of finite diameter be infinitely long if you unrolled it?

Is the amount of numbers between 0 and 1 infinity squared?

What is the significance of e? How is it defined?

Are there infinitely many different infinities?

Does infinity squared equal infinity?

Does infinity to the infinitieth power equal infinity?

What is π! ? (Read about factorials here).

What number occurs most often in the decimal expansion of π?

Are there any patterns in the decimal expansion of π?

How do you prove that π goes on forever without repeating?

Wow! I was thrilled to hear the questions the kids were coming up with on this day. To understand how deep some of these questions on the second list are, you have to know a fair amount of mathematics. Some of these questions remain unsolved and are active areas of research in math. The kids are an unstoppable force with their questions now, like a giant snowball flying off the side of a mountain. Questions fly like sparks, bouncing from one child to another, and the excitement and enthusiasm for the task of constructing a monument to everything we’d like to know is deeply felt by all.

It takes a little patience and effort to develop good question skills. Since one of the major objectives for us in this course was to hone the students’ ability and appreciation of question-asking, we allotted time at both the beginning and end of each class: the first 10 to 20 minutes of each class, and a few minutes at the end. In the beginning of class, as we go through the room taking questions, we write them on the board so that we have a running record. Those questions with quicker answers we handle on the spot. Otherwise, we may comment, but many questions are left unanswered. We encourage the kids to write down any questions that interest them so that they can return to them later. The kids have learned how to generalize and extend each others’ questions, which is one of the most useful skills a working mathematician can have; this sharing and re-possessing of questions brings mathematics into the heart of a community. It also raises the stakes on the questions, since generalizes often lead to questions with deeper and more difficult answers. The students are hungry for answers, and we’ve told them we won’t be able to answer everything in course–there simply isn’t time. However, we use their questions to determine the rest of the class, so the kids get to experience an education in which *their *interests dictate the flow. The traditional ownership of math gets turned on its head.

*Zeno* has been an exciting success, with only one more session to go. We’ll be sad to end the course, but the good news is we are offering it again, and some other courses. If you know any 2nd-12th grader in the Seattle area who is interested in some extreme math, registration for our classes at the Robinson Center start Monday at 8am… that’s tomorrow!

## Comments 1

Good evening mr. , my name is Emanuel and i’m a math student in Romania .

I have a problem , and if it doesn’t bother you too much i would very much appreciate your opinion on the matter.

Open problem 1 :

f is a continuous functions for which the limit at plus infinity does not exist but whose liminf and limsup exist and are finite. Then there exist two sequences that converge to infinity x_n and y_n such that x_n1 and f(x_n) , f(a_n x_n) tends to different limits .

If these propositions are false do you know some restrictions to f that makes them true ? is periodicity the single one ?

Thank you, best regards from Romania