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I teach math with about two years of experience now. In general I have found that all the cliches about graduate school teaching you "how to research but not how to teach" are true. But I have also found many great resources (at my institution, online, in print, etc.) that are helping me, over time, work through my teaching deficits.

...in all things but grading.

I struggle with many grading decisions: from small things, like grading individual homework problems, all the way up to life-changing things like evaluating master's degree defenses. What I lack is a coherent philosophy of grading that might motivate my various grading policies/strategies/choices.

Interestingly, I have not found good resources for this. Yes, my institution provides a tiny bit of guidance, but it is very broad. This forum contains a hundred or so questions tagged with "grading", which is a good start, but I wonder if there are resources that provide a more cohesive treatment of the subject.

I want to hear about theories of grading. I want principles which flow naturally from the theories. I want applications and strategies which build on the principles. I want to hear different viewpoints on the issues so I can evaluate their relative strengths and weaknesses as I come to understand my own thoughts better. In short, I want it all.

Do any such resources exist? Can anyone point me in the right direction? If these don't exist, why not?

  • Are you asking about specifically the act of assigning a grade to a piece of work, deciding on the relative quality of work, or providing feedback? – StrongBad Mar 2 '17 at 19:51
  • To the first option, yes. To the second, no, except insofar as addressing the first option spills over into it. To the third, no, but that would make for another good discussion. – sdg238 Mar 2 '17 at 20:00
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    Practice! I think it varies widely and is very subjective. I heard one teacher say "Before coffee: F's! After coffee: A's." On the other side you have teachers who grade like machines. My practice has always been how much progress did the student make compared to how much progress the student could have made, but that plays to my pretentious nature of feeling I understand humanity. TLDR: There is no guide. Do what feels right and communicate it clearly so your students understand. – Dave Kanter Mar 2 '17 at 20:02
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    Does your university offer tuition waivers for faculty? One option would be to take/audit an assessment theory class. At my university, we also have a teaching resource center designed to help with specific things like assessment. – guifa Mar 2 '17 at 22:43
  • Your question title does not match your actual questions. – einpoklum Mar 3 '17 at 21:50

11 Answers 11

31

One principle is that the instructor should aspire to grade in a way that results in the grade being most closely correlated with those aspects of the knowledge and skills being assessed that the instructor considers important.

As an example, it is a common belief among instructors of university level mathematics courses that exam questions should test conceptual knowledge rather than facility with arithmetic and other low-level calculations. For this reason, most instructors would take off only a very small number of points, or not take off any points, for a small error of arithmetic in an otherwise conceptually correct answer. However, some instructors might consider accuracy to be an important skill to test in and of itself, and may impose a heavy penalty of points for even small arithmetic errors. So you see, the grading function chosen by the instructor reflects that instructor's view of what's important. Decide in advance what's important to you, and grade accordingly.

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    I'd add that it's then important to clearly communicate your philosophy of what's important to you, perhaps on your syllabus and/or during the first class session. Since different instructors consider different elements to be more or less important, it's only fair to make sure students know where you stand, so nobody is surprised later. – Zach Lipton Mar 3 '17 at 4:59
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    I would change the word instructors view to course objective. This is because math dept. teaches the courses from Basic Math 1 for Fresh Biz Studs to the Information Theory for Math PhD. The same lecturer, but two totally different objectives. – user3644640 Mar 3 '17 at 9:00
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    I am not sure whether this answer answers the question "How do teachers learn to grade?", or rather the question "How should teachers grade?" – O. R. Mapper Mar 3 '17 at 10:03
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    @O.R.Mapper OP said "I want it all". My answer is a subset of "all". – Dan Romik Mar 3 '17 at 10:17
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    This answer doesn't seem to answer the question at all. The question is "How do teachers learn to grade?" and then "Do any such resources exist? Can anyone point me in the right direction [implicit: to these resources]? If these don't exist, why not?". This is about the process of learning how to grade, not about the process of grading in itself. I'm just commenting this here since it is the highest-voted answer at the moment, not because it is particularly bad or something like that. – AnoE Mar 3 '17 at 14:26
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There are a couple of restrictions:

  1. Everyone must be graded equally.
  2. A better answer should get a more points.

These two do leave one with quite many degrees of freedom, but satisfying even these two is not trivial.

My methodology is as follows:

  1. Solve the problem I am about to grade.
  2. Start reading the answers. Try to understand them.
  3. After I feel that I understand most of the answers (there are no more surprising answers), start using the red pen. Mark errors, problems and omissions; maybe write comments to the extent warranted by the context. Give the obvious zero points or full points as well.
  4. Come up with criteria for giving points.
  5. After having marked several answers with errors etc., and maybe only after going through all the answers with a red pen, I start grading them. I postpone any difficult answers and put them to the bottom of the pile. I keep track of all ad hoc scoring criteria I come up with (known as a rubric).
  6. Done.

Grade one problem at a time for all students; then tackle the next one. This helps with consistency.

This algorithm tries to make the scores as consistent as possible.

For the scoring criteria, I try to divide the question into separate and distinct components, so that it is possible for a student to fail any of them while succeeding at the others. This is not always possible.

Depending on the answers, I give out criteria which are worth a point (6 points for a complete answer are common hereabouts) and then simply calculate how many the student got. Sometimes, this takes the form of assuming full score and giving negative points for missed portions of proof or for misunderstood concepts.

I usually do not give out penalties for writing complete nonsense in the answer, but sometimes I do give a maximum; say, a maximum of 5 out of 6 points if the response equals a false claim about continuity.

This all is mostly based on folklore (discussion with more experienced lecturers) and on own thinking.

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    My approach is similar to yours. I'd like to add one step: Sometimes, at the end, I'll grab a subset at random, and sort them according to the number of points received. I then check to make sure I think the quality of the answers align with the number of points given. It's just a sanity check to make sure I'm being consistent, particularly when I'm grading a relatively large number of students. – J.R. Mar 3 '17 at 22:49
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    > I keep track of all ad hoc scoring criteria I come up with.. These are called Rubrics. – Joel Coehoorn Mar 4 '17 at 0:24
  • @JoelCoehoorn Edited the terminology into the answer. – Tommi Brander Mar 6 '17 at 16:01
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Having taught in both the US and the UK there are different philosophies associated with assigning grades. In the UK work is compared to a fixed set of marking criteria. This means that everyone could get A's or no one could get A's. In the US most grading is done on a curve such that a predetermined distribution is achieved.

In the UK some portion of the work is double (and in some cases tripled, quadrupled, ...) marked. This means that someone else independently marks the work and the two markers meet to discuss any discrepancies. This provides direct feedback on your marking and the learning is rapid. That said, the process is time consuming since it doubles the work load (plus the time to resolve discrepancies).

In the US, assigning grades is really a two step process. The first is dividing the work into groups of roughly equal quality. To me this is the hardest aspect and requires the grader to understand what is important in terms of learning outcomes. Once the groups are made one then needs to decide how much variation from the desired distribution is acceptable. Sometimes there are clear break points and a bimodal (or trimodal) distribution more than a continuous normal distribution.

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    "In the US most grading is done on a curve" [citation needed]. I can't recall any of my classes being norm-referenced, they all had criterion-referenced assessments. I had heard of strict curves being used in engineering and MIS courses, but I have to admit, I'd be pretty peeved if a professor tried to give me a lower grade than someone in a different section/semester even though my quality of work was better just because I was with a different cohort of students. – guifa Mar 3 '17 at 1:05
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    @guifa: As a U.S. person, I am likewise amazed/appalled that anyone grades to a curve (and never had that in my schooling). But I do get lots of signals that's the custom. Krantz, How to Teach Mathematics (American Mathematical Society), 3rd ed., sec 2.15 (on training TA's): "Likewise, the TAs can be allowed to set the curve for grading (under supervision) and to perform the other ordinary functions of the instructor." – Daniel R. Collins Mar 3 '17 at 3:16
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    Most of my college classes were graded on a curve, but usually not in the sense that grades were adjusted after the fact to fit a predetermined distribution. Instead, it was more like the difficulty of work and the grading standards were calibrated to the skill level of typical students such that, over the long term, the grade distribution should turn out to fit certain constraints (e.g. average C, <50% A-range, or whatever). I mention this because I suspect there's some disagreement about what is meant by a "curve". – David Z Mar 3 '17 at 5:48
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    The variation in the difficulty of a test is likely to be as large or larger than the variation in the ability of a class, and this probability increases as the size of the class increases. Therefore grading to a curve is likely fairer than grading to a pre-determined script. – Jack Aidley Mar 3 '17 at 9:23
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    @O.R.Mapper: When I say variation in difficulty of the test I mean between cohorts or locations. This variation cannot be by design. – Jack Aidley Mar 3 '17 at 10:25
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Rubrics

When developing an assignment, I always want a rubric to go with it. A rubric tells how you to grade the different results you would expect to see from a student.

Rubrics also help keep you on-topic for grading, to ensure that appropriate weights are applied to each of the areas. For example, a paper assigned in a literature class should demonstrate, with good grammar, that a student has read and understands the work, and is able to provide an analysis of the work at the appropriate level. A rubric here might assign point values to each of those areas: grammar, writing level, demonstration of content in the work, and literary analysis. An algebra assignment might use a rubric to allow an instructor to weight between using the right formula or procedure vs getting the arithmetic right. An assignment for a programming 101 course might weigh between stylistic concerns like comments and naming choices, correctness (does the program do what it's supposed to), and higher-level concepts targeted by the assignment (did they use an array with 10 elements like you told them, or did they declare 10 separate variables?).1

Rubrics keep you objective. They help you defend against charges that you prefer some students over others. A good rubric will allow you to objectively demonstrate why a particular assignment received the grade it received.

When I first started building my own rubrics for assignments, I quickly figured out that I'm kind of bad at it. A rubric tries to anticipate the kinds of things a students will do, to assign the appropriate results, and a new instructor just can't anticipate everything. I eventually learned NOT to build the full rubric in advance. You should know enough to have the category areas for the rubric done ahead, but how things break down within those categories isn't always clear at the outset, especially the first time through an assignment for a new instructor. Instead, I will rough grade a few or several assignments, and then use my findings from those first few attempts to flesh out my rubric. At this point, I always go back and re-check those assignments using the rubric, once it's written.

Sometimes I'll find something unexpected late in the grading such that I need to revise a rubric. This means going back and re-checking papers with the updated rubric, because, again, objectivity is the goal. In spite of this, I view building the rubrics as a time-saving measure. It lets me grade by looking for more specific things in the assignment, rather than subjectively evaluate quality. The specificity really helps with being fast.

Eventually, you want to be confident enough in your rubrics that you can publish them to students (though not everyone agrees this is a good idea), so that students can also have a clearer idea of what you expect. In some subject areas, I've seen rubrics given to students to use as peer-review guides. Good students, with good rubrics, can produce the same grade that would be assigned by the instructor in a peer-review situation.


1 As an aside, I remember being told in my own first programming class that a full half of the grade was dedicated to stylistic concerns, rather than correctness or assignment objective. Today I better understand some of the reasons for this: force students to develop good habits early, have some easier "soft" points to buffer the grades (see this Erno de Weerd's answer for why that's a good thing), force students to spend time in design instead of just jumping into code, and because naming and style really is that important to programmers.

  • @sdg238 You should be writing the rubric when you create a test. How can you grade a piece of assessment if you do not know what it has been specifically designed to assess? And yes you should give the students the rubric so they know how best to succeed and will question your marking far far less. It saves everyone so much time in the long run. – Ian Miller Mar 6 '17 at 14:46
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My experience as a grader (TA): Let's say there are four problems in the homework set. The professor has established how many points each problem is worth. First, I work the problems myself, consulting the solution key if necessary. Next, I start looking at some student work for Problem 1. When I encounter a solution that is completely off base, that's easy -- the score is 0. I try to find a couple of representative solutions of "not quite right" and "deserves some partial credit." I use those as reference points and come back to them as needed, to make sure my calibration is remaining constant -- especially if I get interrupted or if I get tired. It's easy to get tired if you're grading 50 to 100 submissions.

To check my calibration, I would go back and check the first 5 to 10 that I did, to make sure I hadn't slid in one direction or another as I went along.

Once in a while someone would use a completely different approach than what I used, and then I would have to put on my thinking cap to make sure it held water.

Similarly with exams.

In conclusion, if the problem is worth 3 points, it's helpful to hold onto a sample one-pointer and a sample two-pointer to use as reference points.

In terms of feedback written on the papers... I often wished I had some rubber stamps, such as "Does not follow," and "How is this germane?"

If there was an extensive calculation such as matrix manipulations, I would circle the first value that was in error. Then I would assume it was correct and check if the subsequent calculations, and logical conclusions, followed. If so, I would take off a bit but not a whole lot.

If there were multiple errors in one problem, I would look at the paper from the opposite point of view. Instead of looking for meaningful errors for which to subtract points, I would ask myself, is there anything of value in this paper? If so, partial credit.

If the student was struggling in general, I would make a point of looking for something on his paper that I could fairly praise. It is so discouraging to get back an assignment with nothing but negative feedback on it.

I learned most of this by grading alongside more experienced TAs. The rest, by doing.

When I was grading technical writing papers, I would provide feedback on drafts and ask the student to make some improvements and hand it in again. Then in the final version I would mainly pick out some aspects to provide positive feedback on. I figured, this would help the student have a clear idea of what I was looking for, in preparation for the next assignment.

I was looking for growth in writing, ability to write a good outline (which I required that they hand in), ability to self-edit, ability to respond to developmental guidance. I didn't see it as my job to separate the A's from the B's or the C's.

Perhaps this is not as theory-based as you were requesting. But it worked for me.


My mother wrote up a true story about a professor or hers who looked for succinct essay answers. He said he graded by tossing the projects (each of which was neatly stapled) down a staircase. In other words, he gave the best grades to the shortest papers (assuming they all addressed the question to a reasonable extent, which they did, since it was a graduate level course).

  • I like the anecdote at the end. Gives new meaning to "grading on a curve". – Dan Romik Mar 3 '17 at 10:02
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When I was teaching maths in secondary school, I ran into an interesting issue: I was too precise in calculating grades. (our grades range from 1 to 10)

This might have been be due to the fact that I was a math teacher. I would have a very precise weighing of the individual part of the tests and the end grade would be rounded from an indefinite number of decimals to the nearest first decimal.

Until the moment I sat in a meeting to discuss the grades per student by all teachers. A PE teacher would adjust his grade by a full point on the fly to allow a student to have a nicer list... The contrast with my precision made me rethink my grading.

I ended up with giving whole grades and not going below 4. My reasoning:

  1. Any weak student receiving less than a 4 would hardly be able to compensate a 1, 2 or 3 and a 4 would already fail them. A strong student would be able to compensate a 4 (a 7 or 8 would be sufficient).
  2. I graded the exercises separately, assigning points to each exercise in ratio to the other exercises. Calculated the sum of all the points and made grade 'buckets'. E.g.: 0-20 points => 4, 20-30 points => 5, 30-40 points => 6 ... 80-90 point => 10

The effects

  1. Students never felt locked into a bad year grade due to one or two low (<4) grades
  2. Less 'living on the edge' by students: "If I can get a 5.75 or higher, my year's average will be 5.5 so I will pass"
  3. A 10 was reachable even if you made a couple of mistakes
  4. No fake precision; I didn't feel comfortable with comparing a student's skills in geometry to the student's skills in algebra and expressing the difference in one decimal.
  5. Less discussions with students about their grades, in most cases getting a few more points for an answer would not change their grade.
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I have some experience in this. My view is that grading should be chiefly about fairness and consistency.

By fairness, I mean the exam should have enough questions to be marked out of 100. It should also be constructed so that the results will form a bell curve around 60, for arguments sake, with the first standard deviation at 12.5

In this way, students can be compared against each other. The actual marks can then yield a "z score". This might look like 58 ± 10, meaning the test was actually harder than normal, but quite bunched. On the other hand It might look like 70 ± 15 which would indicate too easy a test, on average, but too many found it too easy.

In both cases, this z-score can be used to "standardise" the results.

so in the hard case (58 ± 10), someone who got 63 (above average), would get:

(60) + (63 - 58)*12.5/10 = 66

in the easy test, someone who got 63 would get:

60 + (63 - 70)*12.5/15 = 53.

This also helps fairly align groups of students from one school compared with another, provided they all sit one common exam - once.

The last point is about consistency. Was last years test harder than this years ? This is important to keep standards consistent across years, so that dilution or excessive rigour is also protected against. This is more political, but interestingly tends to be the case in more communist countries.

In cases where i've marked lab work for 20 students at university, I will get the books and spend considerable time making a rough ordered list from best to worst. I will start with the best work, and assign that the top mark, then work my way down, using the best one as the benchmark. The mark will be pencilled in, just in case I need to move things around as I work through. Then I'll make a final run through to assign final scores.

One final thought - I always give full marks for the correct answer (in maths), regardless of showing the "working". However it is in the students interest to show working, because marks can be awarded in this case, depending on how far through mistakes were made.

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This answer is about resources for understanding grading philosophies and how those can be practically enacted. This was my interpretion of the OPs main question.

Do any such resources exist? Can anyone point me in the right direction? If these don't exist, why not?

One book that came highly recommended to me was McKeachie's Teaching Tips (http://www.cengage.com/c/mckeachie-s-teaching-tips-14e-mckeachie ) It includes three chapters on assessments and grading: "Assessing, Testing, and Evaluating: Grading is Not the Most Important Function", "Good Designs for Written Feedback for Students", and "The ABCs of Assigning Grades".

The book stresses that the choice of assessment and non-grade feedback are just as important as the grading strategy. The philosophy and practice of these items should be complementary.

Other potential resources (which I have not read personally) are: the chapter "Grading Practices" in Tools for Teaching by Barbara Davis and the book Effective Grading: A Tool for Learning and Assessment by Walvoord and Anderson.

In addition, you may want to google "grading strategy for teaching and learning", since many major universities have centers for teaching and leaning which include grading ideas and resources for faculty.

1

Multiple interesting comments (including content in answers) on this one. e.g., I liked this part from Dave Kaye's comment:

There is no guide. Do what feels right and communicate it clearly so your students understand.

The "question" section asked these questions:

Do any such resources exist? Can anyone point me in the right direction? If these don't exist, why not?

Those are excellent questions to which I am not directly answering. However, I do address the topic of the title question, "How do teachers learn to grade?"

Growing up, I heard that before a person becomes a teacher, they go through a process of being a "student teacher" where they learn things like this.

Well, let me tell you my experience. (I imagine this will raise some eyebrows.)

A college reached out to me and asked me to become an instructor. I agreed. I don't recall ever being taught how to assign grades. So I did things by trial and error.

It turns out that the college department I joined was failing in multiple ways. The drop-out rate was high. For those that didn't drop out, the department chair taught courses and gave students solid As, but they obtained degrees without sufficient skill/knowledge. I decided to do something about this, and I gave lower grades when I determined they were deserved.

Sometimes, grades would end up being too low. A student once said that I took the Gandalf approach to making tests. ("You shall not pass!") I recall the time in my second set of classes where the top grades on the midterm were something like an 82, a 76, and a 58. Clearly I didn't want to flunk the vast majority of the class (which might have triggered numerous more drop-outs). My tests were not too hard: The intent of this department was to train people well enough to take industry certifications, and those were slightly harder than what I gave. So I had to be stern and have them rise to the challenge.

At the end of every quarter, after final exams were collected and graded, I looked at the grade of every student. Then I would make adjustments to the value of assignments and tests, as needed, so that the final grades made sense to me. I had some students who were clearly trying the hardest, succeeding at tests the best, and learning the most. I figured they ought to get an A, and any deviation from that would be a failure of the assessment system, not a failure of the student. So I made sure the people who deserved top marks got them, and I looked at the grades of every other student and thought about what their grade looked like and how well they were learning the material. I made sure that nobody got a lower grade than I thought they deserved, and the vast majority of the students got exactly what grade I thought they deserved. If that wasn't true, then I adjusted some assignment/test weights equally across the board.

In the end, I was usually satisfied with every student's grade, but sometimes a couple of students would end up with a grade that was a notch (or maybe two) higher than I intended, and I couldn't adjust their grade without making someone else look wrong (giving someone else a grade lower than deserved). In those cases, I just figured that the student, whom I thought was performing a bit low, must've actually done a bit better than I thought, and so I finalized things with them getting a slightly higher grade than I would have thought. (I figured that even if a student got answers right due to luck, I had no grounds to penalize them.)

There was some grumbling about people not getting solid A's as easily, and the increased workload I may have been expecting, and tests being much harder. Nobody ever actually challenged a grade of mine, or asked me why they got the grade that they did.

I view the results of my efforts as entirely positive. Grades were doing a better job of reflecting reality. I feel like I improved things as much as possible without causing so much student frustration as to skyrocket the dropout rate. Actually, as my involvement increased (particularly as I led the department, since the department chair left within 90 days of my start), drop-outs happened less. More accuracy, stronger resulting students, and better critical numbers all seemed like good things.

I do recall the college president speaking to the instructors, showing that too many As were being given out. In the entire college, over half of all grades were As, and most of the rest were Bs. He told us to make such good grades harder to get. (I happily thought about how I was doing exactly that.) Besides that one point of instruction, I don't recall other teaching about how to assign the final grades. There were some guidelines that had to be followed (relative weights documented on the syllabus, rubrics for some specific college-wide assignments), but we still had some flexibility (which I definitely utilized) and ultimately it was up to the instructor's authority to determine what made sense.

So, in my case, the answer was largely trial-and-error (but fixing the errors before the finalized grades got posted). I'm sure that at another university I've been a part of, graders had specific rubrics about everything and they had to follow those very closely, and be able to justify how every grade fit within the rubric they were assigned to, with much less flexibility.

Like another answer of mine, the ultimate conclusion is that specifics vary A LOT between different organizations (including different institutions, and possibly between different departments. Different instructors are very likely to have significantly different experiences. Although I'm sure people have written books on the subject (as well as all kinds of other subjects on the planet), that doesn't mean that instructors actually go through any sort of centralized/standardized training in practice. I know that the state I live in has some standardized test for teachers of Kindergarten through 12th grade, but ironically, there can be a lot more variance in the higher education.

0

I had some students who were clearly trying the hardest, succeeding at tests the best, and learning the most. I figured they ought to get an A

This part of another answer deserves to be highlighted. While I'm strongly for objective grading, we must realize that most educational institutes and societal expectations compare grades between courses, and some teachers close more eyes to serious conceptual errors than others, so I think fairness requires us to ultimately assign grades that convey to others the level of mastery of the course content that we feel the students have.

This has two implications. I personally feel it is necessary to give raw scores for assignments and tests that reflect objectively how much the students grasp the course content. However, I equally gladly adjust the final grades (maintaining the relative order) that would end up in their academic records so as to give others the right impression. For example, if we teach a course that involves much more difficult concepts than the average course, then it makes no sense to give a B to some student who normally achieves A+ in ordinary courses and has put in the same hard effort into this course, just because the raw score says so, since this would only give other people the incorrect impression due to not knowing the relative difficulty of such a course.

Ultimately, I think that one should give a grade that one can defend as being exactly what the student deserves to get on his/her academic record, but at the same time give detailed feedback totally apart from the grade, so that the students know precisely what they know and what they do not know. That pedagogical component of grading is far too often lacking, especially in mathematics, because a lot of graders faced with an incoherent answer try their best to figure out what the student means and even attempt to patch a fundamentally flawed answer, giving more credit than is usually due. This results in the students rarely aware of their mistakes, and worse still some of them realize (and tell others) that they can get away with handwaving.

  • Incidentally, and not surprisingly, good students like this approach to grading. – user21820 Mar 6 '17 at 12:30
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There are some great answers here so i'll try and be more direct about the question "how do teachers learn to grade?".

We learn to grade from observation, implementation, and feedback. It's crucial to see how others have marked in your subject area to get an idea of what is top level, mid level, etc. From there you can mark work and ask another teacher to take a look at some items and give you feedback which you can use when you mark the rest.

If this isn't an option and you're really flying by wire, then grouping work into piles based on a quick look "Good", "Average", "Improvement required" and then from there you can make each group into 3 piles of "top level, mid level, low level" and assign grades A*- Ungraded (or however it is you do it at your end!). SHOW THE STUDENTS EXEMPLAR WORK. I cannot stress this enough! If they don't know what amazing work looks like, how will they know where they are? If you show them "This is a grade A item from a similar question last year!" then they can understand your expectations.

protected by eykanal Mar 6 '17 at 14:30

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