I plan on using this question for a math assessment. It is, in my view, an example of an excellent challenge of one’s maths reasoning and knowledge. There is no reason most of the students in a social structure that values the skills necessary to solve this problem couldn’t do so. We, with no reasonable or unreasonable doubt, are not in such a place.
Many of us are in a place to make it true for then and there. Despite the inane ability for the American education system to be obstinately confounding, we possess the wonderous gift of purveyors of learning. Life is the expression of curiosity and our biggest challenge is often directing the amplified abundance of it in the classroom. The more we can allow that energy to be on the growth of learning and challenging ourselves while nurturing the positive, the less we yell at kids, drink whiskey, and pour over faulted data for meetings.
I have yet to meet a person who does not feel accomplishment for solving a challenging problem.
How does this idea of addition/subtraction solve this statement.
I often introduce Zeno’s paradox in my math classes as a way to discuss asymptotic behavior and limits. I am not the only one to do this for many teachers find it a nice way to consider these things. The general arguments against Zeno involve limits but I was wondering if there was another way around it for even philosophers don’t feel that such mathematical disprove ends the philosophic argument. In my musings and considerations, I stumbled upon both a simpler mathematical approach and perhaps a different way to answer it in terms of metaphysics. At the center of Zeno’s argument is that you can not divide something and have nothing. This is true. Even if the limit definition says the same the thing for the limit is not about the point of dividing by infinity but what that point would look like if we could do so. Calculus added an extra definition for continuity at that point in case we care about its existence. If one wants to argue against the paradox using the mathematics of limits one must also include that the lim f(x)= f(a) and that f(a) exists.
Zeno, like all good Greek thinkers, loved proportions. He argued that the arrow would never make its target because it would always be 1/2 the distance closer. This simple statement is where he skipped a stepped of mathematics, logic, and philosophy. A proportion is a form of division. Division is the inverse of multiplication and multiplication is repeated addition. We do perceive, measure, and understand the passage of the arrow or Achilles by addition. (Recall that subtraction is just a different form of addition.) While we can never divide something to be nothing or multiply nothing to be something, we can easily add something to nothing or take away to be nothing. This element is true in the scenario but it was not expressed in the paradox. In all logical or mathematical arguments, we need to express these things in the beginning for the argument may lose sight of simple definitions. We turn the passage into a problem of division, which does not allow for the zero space but our original conditions do so.
The illusion of motion is turning the addition of intervals into a zero sized one. It is viewing flight as photograph and stating that because it does not travel in zero time so it never travels. Consider this photograph if we were in it, for that is how Zeno presented it. If time is frozen, we will not perceive any motion or much of anything for we will always be puppets of the infinitesimal. This moment is one of those semi-pointless exercises in philosophy for we know the arrow to travel and we know time to pass. Theoretical physics can change the speed of time, and even pause it for light, but not for us.
As for that original question, of which, I need to refind the source. Consider the whole path the arrow travels as one. It goes 1/2 the distance in 1/2 the time, and therefore makes 2 intervals. We can keep dividing the distance and increase the number of intervals. It will go 100 intervals of 1/100th the time. In this manner, we can find an infinite amount of tasks in a finite amount of time. The challenge really highlights our analytic need to make things discrete and countable and the simple continuity of a line.
Some posts are written down on paper first, this ain’t one of them. It’s really just disguised laziness. I like the challenge of problem-solving. I like teaching math and I liked the part about math where you work real hard to figure something out and then figure it out. It is a special accomplishment and why people always view math as hard and a measure of intelligence. It is often a challenge but it is not any more of a measure than any other study, we just measure it on a different part of the scale than we do other things.
I work to try and help people build their schema of math understanding and develop that part of reasoning and symbolic logic. Unfortunately, I am not the most organized individual so I haven’t refined how to best reach my goal. My goal also involves changing others’ viewpoints and that just don’t always work so well. People are a stubborn species and teenagers are a stubborn subset of people. Getting them to buy into a different view of the world, particularly one based in the unexciting system of algebra, feels like a pointless fight at times. My only advantage is that the students do believe that something about the subject is good for them, even though they have to ask “When will I use this?” every class.
That supremely frustrating question is why I am shifting to a class more focused on problem-solving rather than mechanical-computational skills. I am also moving in that direction because modern students have astoundingly little space or need to recall facts and skills. They are absorbing information at peak levels. They know fifty times more bands that I did at the same age. So much of the working memory I need access to for traditional learning is used up on things far more interesting to teenagers. So….how do I teach a subject that requires a significant amount of this cognitive process? I have faith, perhaps too much, in my ability to present the information to people in way that makes sense but the same people often don’t store it. I am trying to circumvent this issue by using a problem-solving approach or a method in which I guide students to construct their understanding of mathematical systems. Both methods require thinking and both require effort from the students. Effort is a challenge in the subject. Student often ask for help as soon as they read a problem. He or she may work on it for a minute or two but rarely long enough to reason through it.
I have worked on a problem-solving approach for a few weeks in one of my classes and I am seeing signs of change, though. I hear fewer more questions and students seem to be following my guidelines to find some manner of solution. Next up, setting up the problems so that individuals develop an understanding of the rules of higher math. Until now, we were working mostly with old computational skills. I am adopting my system from Exeter’s Harkness math but I am confined to less time and with a broader range of students. I do know they are more engaged and that students who could care less for a lecture on solving quadratics enjoy they can find different ways to solve a problem. Watching people use different methods has taught me a great deal about math over the past few years. Ehh, I just got tired and stuff so I will continue this later…
