To help with this list please comment to this post, including the errors you see AND the relevant math courses.
Disclaimer: These mistakes are not necessarily being made by current students. The people commenting below are not necessarily the current TAs or instructors for these courses, but may have taught, graded, or tutored students in the past. The ultimate goal of compiling such a list is to identify common problems so that they may be corrected.
22m55 - Some of my analysis students make logical errors such as proving Q implies P when asked to prove P implies Q.
ReplyDelete22m28, Calc III - Recently I have been seeing a mistake that goes something like this a lot: ((complicated expression)/6)/2) = (complicated expression)/3. In general, the better they are at manipulating complex algebraic expressions correctly and quickly, the fewer problems they will have with university-level math. Also, they all know that \sqrt{a^2+b^2} does not equal a+b, but when the expression inside the square root is much more complicated, they will frequently apply that "rule" anyway.
ReplyDeleteAll of the examples I can think of are instances of assuming (implicitly) that order doesn't matter in an instance where it really does. I am talking about Calc I, II, III here.
ReplyDelete* Equating f(x+y) with f(x) + f(y) (e.g. when f is a power function, particularly if the powers are 2, 1/2, and -1, or when f is a logarithm, exponential function, or trigonometric function.) The mistake is in assuming that it is immaterial whether one adds and then applies f, or vice versa.
* The related error f(x+y) = f(x) + y is another instance of this--- implicitly assuming that it doesn't matter whether you add y before or after applying f to x. [The most common example of this is when f(t) = -t; the error is then -(x+y) = -x + y.]
* Confusing a statement and its converse is the implicit assumption that "if P then Q" is insensitive to the order of P and Q.
* Integrating a product f(x) g(x) dx by integrating f and g separately and multiplying the results together assumes that the order of "product" and "integration" is immaterial. (I also see "reciprocal" and "square root" interchanged with integration a lot.)
* To compute f'(1) you need to compute f'(x) and then plug in 1; it won't do to evaluate at one first, and then take the derivative.
* In the formula for the length of the curve described by y = f(x) from x = a to x = b, you need to integrate the square root of 1 + [f'(x)]^2 from a to b. The order of _everything_ (taking the derivative, then squaring, then adding one, then taking the square root, then integrating) is important here. It simply cannot be done in any other way.
If I could get one message across in high school, it is that order matters. A lot of the complicated things we have to learn are complicated precisely because order matters. People often only pay attention to order when they see large numbers of parentheses. They should pay attention to order all the time.
Quite a few students try to manipulate trig. functions and logarithm functions as if they were quantities, not functions. So students think "cosine times x" not "cosine of x" or "ln times x" not "ln of x".
ReplyDelete