In the course of debating SteveH over the last few days, I had my own Aha! moment. I'm beginning to realize that the investigations I've been writing for several decades for my students were rudimentary examples of a somewhat new approach to finding mathematical 'truth' known as Experimental Mathematics. The Wikipedia article gives an excellent account of this from which I will excerpt the following:
Experimental mathematics is an approach to mathematics in which numerical computation is used to investigate mathematical objects and identify properties and patterns.[1] It has been defined as "that branch of mathematics that concerns itself ultimately with the codification and transmission of insights within the mathematical community through the use of experimental (in either the Galilean, Baconian, Aristotelian or Kantian sense) exploration of conjectures and more informal beliefs and a careful analysis of the data acquired in this pursuit."[2]
[edit] History
In one sense, mathematicians have always practised experimental mathematics. Existing records of early mathematics, such as Babylonian mathematics, typically consist of lists of numerical examples illustrating algebraic identities. However, modern mathematics, beginning in the 17th century, developed a tradition of publishing results in a final, formal and abstract presentation. The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten.
Experimental mathematics as a separate area of study re-emerged in the twentieth century, when the invention of the electronic computer vastly increased the range of feasible calculations, with a speed and precision far greater than anything available to previous generations of mathematicians. A significant milestone and achievement of experimental mathematics was the discovery in 1995 of the Bailey-Borwein-Plouffe formula for the binary digits of π. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous proof found.[3]
[edit] Objectives and uses
The objectives of experimental mathematics are "to generate understanding and insight; to generate and confirm or confront conjectures; and generally to make mathematics more tangible, lively and fun for both the professional researcher and the novice".[4]
The uses of experimental mathematics have been defined as follows:[5]
- Gaining insight and intuition.
- Discovering new patterns and relationships.
- Using graphical displays to suggest underlying mathematical principles.
- Testing and especially falsifying conjectures.
- Exploring a possible result to see if it is worth formal proof.
- Suggesting approaches for formal proof.
- Replacing lengthy hand derivations with computer-based derivations.
- Confirming analytically derived results.
High-speed computers allow for a different approach to 'proof', an idea that I used to find inconsistent with the formal structure of mathematical logic and truth. However, I'm beginning to open my mind to the possibilities afforded by technology and numerical analysis. In fact, I often wondered why, as a student, I was only shown the finished product, the formal abstract theorem, not the slightest hint of the trial and error and conjectures that were an integral part of the derivation. When I became a teacher, I continually reminded my students of this important facet of mathematical truth - the research aspect. Even though part of me continues to resist the idea of a computer-assisted proof by consideration of a large but finite number of cases, I cannot ignore the most remarkable result of this approach:
"The discovery in 1995 of the Bailey-Borwein-Plouffe formula for the binary digits of π. This formula was discovered not by formal reasoning, but instead by numerical searches on a computer; only afterwards was a rigorous proof found."More to follow...
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8 comments:
Dave, I've been reading your
previous posts about teaching
mathematics.
Your point makes perfect sense
if one teaches bright students
who remain curious over the
lessons. Those would really
benefit from your suggested
style of teaching. But those
who are a bit slower and not
so smart might find it even
harder to learn because they
won't be able to seperate the
"learning experience" from the
learning goal (=whatever the
system says they need to know).
If the experience of discovery
and experimentation with math
is the goal, it will be hard
to define which things a student
has to know.
So how would you suggest to bring
these two things together?
Thank you for that astute comment, Florian. It is much more difficult to apply this approach with less motivated students or with students who lack basics. However, I have done these with precisely these groups! The activity is more structured, each step clearly defined and illustrated. I spend more time reviewing prerequisite skills, but eventually, they get into it. I try to choose interesting scenarios as with the Unsummable Numbers activity you may have seen. That was by far the most successful with these youngsters.
Some students have a negative mindset to start with and will simply shut down if left alone. Pairing them, giving them a gentle nudge in the right direction and any other motivational device I can think of may help. These youngsters are convinced they cannot do anything that requires original thinking because they are simply not used to being challenged like this until they came into my class. Sometimes we would all get frustrated, but that too passes. I usually assume that if the activity did not go well, it was MY fault, not theirs. I learn from this and revise on the fly or for the next day or the future. Teaching is an experimental science too, Florian!
Again, I did not do these investigations frequently but when I did them with the lower-achieving groups I did not rush this process. Shockingly, some admitted they enjoyed it!
What is your opinion of the Haken-Appel proof of the Four Color Theorem then? That required a lot of computer manipulation; some people till don't trust it.
"The numerical examples that may have led a mathematician to originally formulate a general theorem were not published, and were generally forgotten."
Back when I taught computer science I decided that I didn't like the way textbooks presented all problems as cleaned-up, final versions of a lot of starts and stops and mistakes. I wanted students to see the process I go through when I try to figure out a problem.
I would occasionally (not always, because it took a long time) do a routine like a binary tree insertion or deletion problem from scratch on the board. I did not figure out what I was going to do beforehand. I would take a problem and talk it out at the board. I would tell the students what I was thinking. They would see my mistakes and corrections.
It didn't work. It went right over their heads. They did not have mastery of the basic skills to keep up with how far ahead I could think. This wasn't a course on programming technique, it was a course on data structures - new material.
I decided that class is for teaching and homework is for discovery.
Steve--
The way you described your lesson with the binary tree is how I taught many of my lessons in CS and other more advanced classes. I spent considerable time 'planning my spontaneity' and 'figuring out where to go' at the board. Sometimes it failed, sometimes it worked. When it worked, I experienced elation, when it failed I tried to figure out why for the next lesson. I distinctly remember teaching why the harmonic series diverged, developing the argument logically but with considerable input from the class. They couldn't tell at first where it was going. Then all of a sudden a few began to catch on. There was electricity in the air. In the end I applauded them for their 'discovery'. I believe the feeling was mutual!
Steve, no one is ignoring the problems we have in our schools. My new-fangled ideas are not so new. Teaching what Jonathan calls 'Enriched Basics' has been going on forever in some classes. If we reduce the number of basic topics, there is more opportunity for all to do this. If you don't believe this is possible, visit Jonathan's classroom or the classrooms of many others who have already made it happen.
"If you don't believe this is possible..."
I never said that this wasn't possible, but there is a price in time to be paid, and you cannot possibly cover all of the things you have to cover using discovery in class.
As you say, this is nothing new. Teachers have used leading questions and lessons for ages. But you're lucky if you get the light bulb to go off for just a few students. This technique is not enough. There is still a lot of hard work and discovery that has to be done individually by each student with homework.
This technique might help with motivation, but it could also frustrate a student whose light didn't go on. Just because some students are excited doesn't mean that they all are.
All of this is moot if the class only gets through two-thirds of the material.
Eric the computer proof that
prooved the the 4 color theorem
by brute force is a great example
of ugly math. Many mathematicians
were let down because the proof
didn't reveal why the 4 color theorem
was true. Only that it was true.
No new insights or understanding
was won.
I think experimental math in the
classroom allows students to gain
new insights only if it is done
without programmable computers.
Ok, let's stop talking about those hot words like discovery, exploration, experimentation...
I would be ecstatic if I walked into a 4th grade class who had been working on their times tables up to 10's and the teacher wrote the following addition problem (but in vertical format):
12+12+12+12+12+12+12+12+12+12.
She states, "The first 3 who write the correct answer win a prize...". Some logistics there in collecting the papers but this is one of those 'leading' questions, Steve, that develop conceptual understanding of multiplication and multiplication by 10 (leading up to powers of 10 later on).
Teachers need:
(a) To Believe that such experiences are critical for the child's development
(b) To Recognize that this type of question does not replace the need to expect mastery of basic facts, but it helps to reinforce basic (as would any applied problem)
(c) To have resources that include these kinds of concept-based questions at their disposal
(d) To Believe that this does not take much time but has great benefit
(e) To reinforce these ideas with outside work
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