If you're teaching chemistry why the fuck should you care if the students use a calculator, you should be concerned that they know what calculations they need to do, not that they can calculate the logs and perform division.
My 3 calculus classes I had to take all allowed for the use of a calculator. Reason being? if you didn't know the math in the first place a calculator wouldn't help in the slightest.
That teacher (freshman chem) held us to the standard of having to know both, and I'm glad he did. When you take a class like Thermodynamics it is insanely helpful to understand how logs and exponential curves will look without having to go pull out your graphing calculator. Anyone who has to graph something to get a rough picture of what an increase in entropy/enthalpy/specific heat/ etc. will do to some system shouldn't be in the sciences.
With calculus (or most pure math) it doesn't matter as much because you are working with truly abstract mathematics. It doesn't matter that you can look at the integral of tangent and say, ohhh as x goes to blah then blah goes to blah. Because its abstract, there's no application to a real world system. You would still need that innate comfort with numbers/math for factoring equations and juggling terms to rearrange something into a form you can integrate.
That said, why would someone even need a calculator in a calculus class? I can think of a couple min/max problems that
might require one, but really the teacher should be writing questions that don't need them. Seriously though, someone help me. When would you need it? And don't say "to calculate the natural log of 5", because for this kind of math, "ln 5" is a much better answer than "1.609..."
On top of that, most graphing calculators will do derivatives for you these days, which kind of defeats the purpose of a calculus course.
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With regards to the notes, I have to say that the single best (and most difficult) class I ever took had tests that were open book, open note, bring whatever you wanted, and 4 questions long. It was my Physical Chem/Thermo class and my Quantum Mechanics class. Thing is, if you didn't know what you were doing before you walked in to take the test, you would fail. The professor was brilliant, every question had this incredible way of testing if you really
understood the science. That doesn't mean you know the equations, that means that you know what the equations mean. He tested true knowledge, and that's something that no amount of notes will teach you in an hour if you don't already know it. I've never taken another class like that since (funny story with that class: I had to ask the professor what the chemical symbol for gold was so I could answer some test question that required molecular weight or some such thing, I knew it was either Au or Ag, I always get it mixed up. He kind of stared at me like I was an idiot for a second and I shrugged my shoulders saying 'hey I get them mixed up'. So he procedded to interrupt the rest of the class that was taking the test and asked them if anyone could help me out. We all laughed. Good times, good times.)