To be honest, I really don’t get why the SAT board allows using calculators anyway. They can certainly make the numbers in the math questions easier to calculate (make it divisible without any weird fractions/decimals). And to be honest if you are somewhere in the pre-calculus/calculus level you should have an intuition of how to roughly draw graphs by hand (without a graphing calculator).
It really seems like corruption of the College Board (and many other western educational institutions) at this point...
Trig functions are basically impossible to do that with unless you limit the angles drastically then you're still testing memorization instead of actual math skills.
There is a common issue with limiting the angles in trig, and only requiring division with an integer or simple fraction solution: It provides a lot of help to the test-taker, based on their test-taking skills rather than their math skills.
If the test-taker can discard mistaken answers because they don't look like the kind of answer the test designer would choose, the test diverges from testing for the desired skills.
Yeah I remember doing that too. Sometimes you could eliminate all the other answers because they weren't in the right area or the wrong sign. It was also interesting to see how answers would often have one that would be your result if you missed a step or had some other smallish error in your work.
You can teach the students trig identities to expand the choice of angles. Moreover, the purpose of the lesson should be to teach the method of solving the problem. Pushing the buttons on the calculator to get a final answer in decimal form (only an approximation) is extraneous to that goal.
Or they could go really old-school and provide a look up table...
sin(x) = y
for a variety of values of x.
Or, even nicer -- instead of having a correct numerical answer, they could write the answers as formulas. Then you wouldn't need to know the actual values at all. (Then again, this could also be a pain).
But some combination of the two should be possible without needing manual computation.
It's been a long time since I had to use the identities but looking over them quickly I don't see how they help you get the approximation of cos(70 deg) knowing the 5 basic values that I memorized back in school of (0,30,45,60,90).
Even if getting the answer needs a calculator the process to get to that point is still testing the algebraic/calculus/trig/geometry skills the test is supposed to test. You can often get rid of the need for a calculator by formulating the questions carefully so the numbers are simple to work with but sometimes you can't do that easily.
Even if getting the answer needs a calculator the process to get to that point is still testing the algebraic/calculus/trig/geometry skills the test is supposed to test. You can often get rid of the need for a calculator by formulating the questions carefully so the numbers are simple to work with but sometimes you can't do that easily.
You can also just accept cos(70°) as a final answer without asking students push buttons on a calculator.
For more advanced students you could teach them to calculate even weird values like 70° using a few terms of the Maclaurin series for cosine:
cos(x) = 1 - x^2/2 + x^4/24 - x^6/720 + O(x^8)
But that's all beside the point. Mathematics education should focus on understanding and applying the concepts, not using technology. The educators' rationale that "students need to learn how to use technology" doesn't really hold water when it comes to calculators. It's far easier to teach someone with a degree in mathematics how to operate a calculator, having never used one, than it is for someone totally dependent on a calculator for even basic calculations to obtain a math degree.
> For more advanced students you could teach them to calculate even weird values like 70° using a few terms of the Maclaurin series for cosine:
You /can/ but by the the time that is introduced, which is usually college since we're talking calculus, you don't want to test basic trigonometry. The tests at that level are about something else not about the particular values of the basic trig functions. Personally once I got to calculus very few of the questions were about the numerical result of a function.
You can get close-enough ballpark answers on the angles between (0, 30, 45, 60, 90). (If the questions are multiple choice, then you can design it in a way that you don't need a calculator) And this is actually quite an important handy skill you will have in real life, since later on when you're drawing graphics/diagrams either on software or hardware you really need that intuition of ball-parking angles.
> Especially since (at least in my experience) basically none of the math courses at University level involve a calculator.
A lot of non-math courses (e.g., in the sciences) at University level involve numeric computations that want some type of calculation device to avoid needless pain, and the math that is tested on the SAT feeds into those courses as well as math courses.
Yeah, but I really don't get that you need to be "educated" to use a calculator - it's just something that's just too banally simple. It's definitely not a skill that needs to be explicitly prepared for high schoolers getting into college.
And to be honest, you don't really need one of those fancy expensive TI graphing calculators to do all those physics/chemistry problem sets. You just need a good-enough scientific calculator (for example, a cheap Sharp one you can get for $20 bucks) and there would be almost no difference in the time it takes to finish your homework.
The first university I went to allowed calculators and required TI nspire for calc 2. When I transferred the second university did not allow graphing calculators and covered half the content in the same period of time. (calc 2 content was covered in calc 1 at first university)
Out of curiosity, which parts or topics of the 'calc 2' curriculum involved calculators? Where you using it for numerical analysis or for symbolic algebra.
While we're at it, maybe we should have students write their answers with ink and quill onto rag paper. The College Board is clearly in the pocket of Big Ballpoint.
If anything, a calculator is under-equipping students in an exam. Anyone using these skills professionally is going to be doing so with tools far more sophisticated than a graphing calculator. We need to be testing their ability to understand and apply principles, not pit hand-calculation skills (which they'll never use in real life) against a stopwatch.
