# Resources on how to mark a maths exam

I am looking for some resources which tells someone how to mark a (1st year undergraduate level) maths exam paper. Ideally it would cover the basic stuff, like making an error at the top doesn't affect the marks at the bottom (so long as the question hasn't been overly simplified, etc.)

Google-ing has turned up nothing. I could try and write something myself, but time....so little time....... (Also, it would be nice to read something someone else wrote, just to see if how I was taught was in any way standard - such as crossed-ticks means "correct according to an earlier error").

I don't think there's a "universal" rubric or way to grade. But the people at CollegeBoard (a.k.a AP) think a lot about fair grading rubrics and have developed descent guidelines on how to grade the open response questions on the AP exam. Many of their philosophies translate to regular course-work grading.

Here's a few of their resources:

"Commentary on Instructions" (pdf)

Specific scoring guidelines for past problems can be found here.

When I've served as a reader for the exams, they also had general guidelines on how to handle decimal errors, etc. However, this information appears not to be public (or at least not easily found online) and frequently had to be tweaked to suite a given problem.

This answer is not a published resource, etc. but below are techniques that me and many of my colleagues use and it seems to work pretty well. My suggestion is to first go through the test yourself, question by question and determine a "rubric" for each question, by determining what are likely to be the "most popular" methods of solving each problem and assigning point values to each correct step. For example, if a question was

Find the roots of the following function: $f(x) = x^2 - 9$

and it was worth 6 points your rubric might look something like:

• Method A
• 2 pts: Correctly factoring (1 pt for each binomial)
• 2 pts: Setting factored binomials equal to zero (1 pt for each binomial)
• 2 pts: Finding the roots by solving each equation (1 pt for each root)
• Method B
• 2 pts: Setting equation equal to zero
• 1 pts: Isolating the $x^2$ term
• 3 pts: Correctly taking the square root of both sides and having both +/- 3

It will not always work out perfectly depending on the problem/methods used to solve the problem but it gets you started on being objective as possible. What I also recommend is to have a list of "sloppy/low-risk" mistakes and point deductions for each. By "sloppy/low-risk", I mean if the student makes this mistake they will receive a deduction but as long as they continued the process correctly, they should not be further penalized. Common mistakes I apply this rationale to are:

• dropping a negative sign (-1 pt)
• error in rewriting between steps (-1 pt)
• transposing digits/terms (-1 pt)
• getting the correct answer somewhere along the way but reporting the wrong answer (-1 pt)
• not fully simplifying (-1 pt)

This will depend on the test material and the level of expectations you have for your students, but i feel like these are good guidelines. Another suggestion is to grade problem by problem or page by page, i.e. grade the first problem/page of each students exam, then grade the second problem/page of each students exam, etc. This helps to keep it fresh in your mind exactly where you took off points for each question because not everything will fit nicely into the rubric you make, making it easier to be more objective. Once you finish a page, tally up the points gained/lost and write them in the bottom corner of each page, so it is easy to add them all up once the exams are marked.

On your question about a standard way to mark papers, to my knowledge, there is no standard way established on how to mark papers. I have seen "subtractive" grading where only deductions are marked and anything not marked is assumed to be full credit. I personally use "additive" grading where the student starts with a 0 and every point they earn is explicitly marked. I have also seen teachers just mark right or wrong and put the percentage of credit earned next to any marked wrong.

hope this helps!

• Although this is interesting, it is not really what I am after. You seem to be describing how to write a marking scheme. Rather, I am after something I can give to someone who is using a marking scheme. – user1729 Dec 5 '14 at 11:57
• Incidentally, despite subtractive and additive grading being equivalent to a rational entity, I don't like the former because it tends to engender a psychological 'ceiling effect' and the perception that the default is to ace the exam and that you are one of the few to do badly. Maybe this view is connected to pessimism or to the preponderance of upwards counterfactuals, but I (and likely most) prefer to think in terms of a lower bound (or worst case) plus possibly something rather than a perfect score minus something. I'd bet there's some interesting psychology in here. – Vandermonde Nov 17 '15 at 23:06