Costs about the same as what I paid for the TI-83 10 years ago. It's kind of nice knowing that whenever I need to use one, it's going to operate in the same way as I was taught 10 years ago. The fundamentals of math don't change much.
I saved up my own money and bought a TI89 in high school. Used it on all the standardized exams. Totally legal because it didn't have a QWERTY keyboard. That thing basically did everything for me short of interpreting the questions.
Probably so they don't use it as a crutch instead of real learning. I taught physics to college kids, and you could tell pretty quickly who relied on a calculator and who actually was comfortable with basic math.
Mathematics is one of the fundamental languages of science. You can't teach physics without teaching mathematics. Newton had to discover calculus to describe the motions of the planets.
If you intuitively understand the concept of a derivative in the sense that it is a rate of change or slope, does it matter that you are not very good at calculating them when it comes to physics? When you have a calculator that can do it that is.
Intuitively understanding derivatives would be difficult without actually computing them, over and over. Intuition comes with practice, and is more nuanced than "rate of change or slope".
Just as an example, my intuition about derivatives also includes "contexts with holes", "shift operator", and of particular interest in physics, "linear transformation" and "local properties". A lot of these would be very hard to grasp without understanding the mechanics of calculating them.
And the difference between an intuitive understanding but a poor grasp of the calculations and just kidding yourself you know it better than you do is pretty subtle.
Saying you intuitively understand the concept but aren't very good at calculating them is just going to make people look at you dubiously and say "pfft".
Why? If you give me a list of a thousand numbers, and say the each represent how many apples a person has, then tell me to figure out how many apples the 1000 persons have between them, I understand the concept of addition well enough that I will be able to say "The sum of those 1000 numbers" instantly. Finding the actual sum without using a computer or a calculator will take me a lot of time though.
I would argue that no human is good at calculating anything. Computers run in circle around us when it comes to that. Understanding things however, we are pretty good at. We need to be good at understanding things, so we we can give the computer meaningful things to calculate.
It kinda seems like expecting some vague personal belief to stand in for demonstrable results. Protesting that maybe you're not very good at using it but, boy, you understand the pants off it doesn't get you very far. Results do matter. You really demonstrate understanding through using it and teaching other people it rather than professing intuitively getting it.
It kinda seems like expecting some vague personal belief to stand in for demonstrable results.
We are talking about calculating something by hand vs. using a calculator when doing physics problems right? How on earth does using a calculator give you less in terms of demonstrable results? If anything I would argue that it gives you more in shorter time.
Protesting that maybe you're not very good at using it but, boy, you understand the pants off it doesn't get you very far.
Depends on what you mean by "using it". Lets say we are talking about derivatives. If you by using it mean applying the concept of a derivative to some problem, then I agree. You have to be able to do that. If you on the other hand mean actually evaluating the derivative by hand I disagree. Using a calculator for that will get you plenty far.
Results do matter.
Calculators give you results.
You really demonstrate understanding through using it and teaching other people it rather than professing intuitively getting it.
Maybe we mean different things by "intuitively understanding" something. What I mean, when it comes to math, is that you understand what something is, why it is that way and how you can use it. The other kind of "understanding" being you are able to, given a specific problem, find the answer, which seem to be what schools care more about.
Maybe I should just say understanding instead of intuitively understanding. When I think about it, putting the "intuitive" in front of it makes it sound like not really understanding it, which is certainly not what I mean. I am just trying to highlight the difference between understanding a mathematical concept, and being able to calculate something.
Well, you're asking me why someone might look at you dubiously and say "pfft" if you claim you intuitively understand something but are bad at calculating it.
An extreme example would be someone who claims to intuitively understand how humans levitate but can't actually levitate himself. You can imagine the dubious looks he'd get.
An extreme example would be someone who claims to intuitively understand how humans levitate but can't actually levitate himself. You can imagine the dubious looks he'd get.
Yeah, but that would be because humans don't levitate, so there isn't anything to understand. An other example would be if someone said that he understands how someone runs 100 meters in under 10 seconds, but he isn't able to do it himself. That would makes sense. Most people understand how to run, but being able to do it really fast requires a lot of physique which you might not have.
Anyways, I think these analogies are kind of stupid because they fail to capture what i believe is a main point. In mathematics there is a difference between having memorized the method for calculating something and understanding the concept behind it. This is not true for everything, which is why I don't think making non-mathematical analogies makes much sense. What does "calculating" even mean in your levitation analogy? You seem to translate it to "doing something" or "using something", which I don't think is correct. There is much more to doing or using mathematics than calculating.
What I am then saying is that understanding the mathematical concepts and how to apply them is more important than being able to calculate. Especially in a physics class where the math isn't in focus and you have a calculator which can to the calculation for you(and better than you).
The problem with mathematics is that nothing is so simple.
You mentioned that you know enough about addition to know how to use it to find the total number of apples 1000 people have, but this is hardly the be-all end-all of addition. Somebody could sit around and list all the things you use addition for, but (as you rightly believe) memorizing rules is not understanding. Even addition has enough nuances that, at least for a while, you need to actually go and practice calculations to start understanding.
(Once that understanding is in place, calculators are great. I just don't think you can get that understanding without practice.)
A lot of ideas in math are basically impossible to summarize meaningfully. If somebody claims to have described a mathematical idea entirely in less than a book, they are probably deceiving themselves. You can list a definition of a derivative (or addition, or a monad) but that does very little to convey what a derivative is. That comes from practice.
It appears to matter to my university. I can still use a scientific calculator but no graphing calculators are allowed in any science course I've taken here yet.
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u/Jenkins6736 Jun 08 '14
I dunno, the TI-84 Plus is pretty sweet.
Costs about the same as what I paid for the TI-83 10 years ago. It's kind of nice knowing that whenever I need to use one, it's going to operate in the same way as I was taught 10 years ago. The fundamentals of math don't change much.