Complex analysis, lecture 2: examples of standards

Successful

Successful, or S, work completely and correctly solves the given problem, and is clearly and legibly explained. There may be at most one or two very minor errors such as typos or misspellings, but nothing that significantly affects the mathematical content.

For example, suppose the problem is the following (say we are learning how to solve quadratic equations):

Problem. Find all positive solutions to $2x^2+x=3$.

An example of a successful solution is as follows:

Solution. Moving the $3$ to the other side of the equation, this is $2x^2+x-3=0$. Now we can factor the left-hand side: $(x-1)(2x+3)=0$, so we have either $x-1=0$ or $2x+3=0$, i.e. (adding $1$ to both sides) $x=1$ or (subtracting $3$ and dividing by $2$) $x=-\frac{2}{3}$. Since the problem asks for positive solutions, only $x=1$ works, and so the answer is just $x=1$.

This answer is almost completely correct, and it is (reasonably) easy to follow what is being done. However it contains a small error: the solution to $2x+3=0$ is actually $-\frac{3}{2}$, not $-\frac{2}{3}$. Since both are negative, this doesn’t affect the final result; more importantly, it is an arithmetic mistake, not an algebra one, and since it contradicts what the student says they’re doing may even be essentially a typo. Most importantly it doesn’t change the fact that the student clearly understands conceptually how to solve the problem.

Another example of a satisfactory solution is as follows.

Solution.

$$2x^2+x=32x^2+x-3=0$$

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}=\frac{-1\pm \sqrt{1^2-4\cdot 2\cdot (-3)}}{2\cdot 2}=\frac{-1\pm \sqrt{25}}{4}$$

which is $-\frac{3}{2}$ or $1$. Since $1$ is positive it is the only correct answer.

This solution gets the right answer, and the explanation is reasonably clear (leaving out the words is fine, though sometimes risky). The conclusion that $1$ is the only correct answer because it is positive is not, strictly speaking, correct; it is also necessary to check that $-\frac{3}{2}$ is negative. However, it’s clear that this student understands what is going on and how to solve the problem, and no error is really significant.

Partially successful

Work is graded Partially successful, or P, if it is has significant errors or displays an understanding of the underlying concepts which is less than completely correct, but shows a partial understanding and progress towards a correct solution.

Let’s use the same example problem as above. Here is an example of a solution which would receive the grade P.

Solution. This is the same thing as $2x^2+x-3=0$, which we can factor as $(x-1)(2x+3)$, so either $x=-1$ or $x=\frac{2}{3}$. Only the second one is positive, so the answer is just $x=\frac{2}{3}$.

This student has correctly set up the equation and factored, but they do not completely understand the factoring method: they know that the solution is somehow related to the factors, but not why and therefore have made a sign error on each term. This will take some studying and discussion to fully understand this concept, but they do have an understanding of the concept, if incomplete, and it is possible that the student has just made a straightforward sign error.

An example of a more serious mistake, but which would still receive the grade P, is as follows.

Not yet successful

Work is graded not yet successful, or N, if it demonstrates a general lack of understanding of the underlying concepts, is significantly incomplete, does not address the question, or is illegible to the grader. An example is as follows:

Solution. The left-hand side is $x(2x+1)$, so either $x=3$ or $2x+1=3$, i.e. $x=3$ or $x=1$. Both are positive, so the solution is both.

This exhibits a major misunderstanding of factoring, and the student could have easily checked that the resulting answer is not correct. However, the student has made a serious effort, and can expect helpful feedback to improve their understanding for next time.

In contrast, consider the following two example solutions:

Solution. $2x^2+x=3=\sqrt{3-x}$ none

It is very difficult to guess at the student’s train of thought here. If the grader is feeling especially kind they might do so and try to figure out what their error probably was and try to give helpful feedback. However, it is not reasonable to expect this at this level of work, and so answers like this should be expected to in general receive minimal feedback.

Solution. $x=1$.

Although strictly speaking the final answer is correct, there is nothing here to grade: any solution necessarily includes the method, and this has none.