On a scale from 0 to 10, you're at a -5: Reasonings behind the Danish grading system

As a physics education researcher, I often find myself trying to explain the Danish educational system to international colleagues. This is usually difficult, because the system does not seem logical. This article will show that the Danish grading system is the product of a series of apparently reasonable assumptions and transformations. Whether the result is reasonable, is a reflection left to the reader.

The basics

Ok, so let's get started. First of all, the Danish grading system consists of seven grades: -3, 0, 2, 4, 7, 10, and 12. The system has been designed to match the seven-letter international ECTS scale (Richardson et. al 2004), which was desired at the time (European Commission 2009, pp. 41-43). Here is the official English translation: https://ufm.dk/en/education/the-danish-education-system/grading-system, where the Danish grades are mapped to the letter scale: Fx, F, E, D, C, B, A. Both scales have five passing grades (A-E, 12-2) and two failing grades (Fx and F, 0 and -3).

A bit of the historical context

In the early 2000s the Danish government perceived a need for a new scale. They installed a commission to investigate the need for a new scale and at the same time come up with a suggestion for a new scale. Two main arguments were in play: Grade Slipping and Convertibility. Grade slipping the propensity for a grade scale to allow grades to become increasingly higher over time. Convertibility refers to the problem of converting grades between educational systems. The grade scale in use at the time was called the 13-scale, with 7 passing grades (6, 7, 8, 9, 10, 11, and 13) and three failing grades (00, 03, and 5). This scale was, it was argued, impossible to convert to other systems (see Richardson et. al 2004) - particularly the ECTS-grading system that had been meant as part of the Bologna-process (Karan 2004, 2005, European Commission 2009).

The commission set up five criteria for the new scale. The scale must allow for (see Richardson et al 2004, pp. 29-30):

1. International applicability. Danish grades were to be more easily converted to international grades and the ECTS-grade system was chosen as the benchmark. This meant a requirement of five passing grade (corresponding to A, B, C, D, and E) and two failing grades (corresponding to Fx and F). It also meant that the new scale needed to categorise students in terms the expectations from the ECTS scale: For a sufficiently long period of time, 10% of students, who pass, should receive an A, 25% a B, 30% a C, 25% a D, and 10% an E.
2. Consistent use across the educational system and using the scale to assess the degree of achievement of learning goals. It was hypothesised that basing the grade solely on the degree to which the student met the (precisely defined) learning goals would secure a consistent use across the educational system.
3. Using the same scale in all parts of the educational system. This was to secure horizontal as well as vertical mobility between educational subsystems. Presumably, the transition from compulsory school to upper secondary programs can be seen as "vertical mobility" while shifting from e.g. a school teacher education programme to a bachelor in mathematics is horizontal mobility. Furthermore, having the same grade system throughout educational systems was hypothesised to promote citizens' knowledge and accept of the scale.
4. Clear grade differentiation. The number of grades and the numerical difference between grades should allow for certainty in the choice between a grade and the neighbouring grades. Furthermore, the scale should be fine grained enough to be usable for formative assessment purposes. It is unclear precisely what this means; perhaps that the jumps between grades are not so large as to make it seem impossible to bridge the gap between them.
5. Calculating means. The Danish educational system makes use of mean grades to distribute students on different educational tracks. Thus, the grading system needed to readily support this. This was the argument for a system of numbers rather than letters. This appears to be the sole criterion for calculating means, although other criteria could have been chosen to ensure that means calculation be meaningful (see, for example, Kolen, Brennan, & Kolen 2004).

It was believed that a scale using these criteria would prevent grade slipping and ensure convertibility. Thus, using these five criteria, the commission argued that the existing scales were insufficient. The most prevalent scale used was the so-called 13-scale, with grades: 00, 03, 5, 6, 7, 8, 9, 10, 11, and 13 - with 00, 03, and 5 as failing grades. Instead, they suggested the 7-point scale. Let's see how that came about.

The reasoning behind the Danish 7-point grade scale

First of all, we base the grade scale on the idea that the quality of student achievement of learning goals can be measured and converted to a single number. This number can be made to fall into the range 0-100 (the red bar on Figure 1. We then say that for a certain number P>0, the student has passed (the blue bar on Figure1).

Remember now that the distribution of passing grades on the ECTS scale is: 10% get E, 25% get C, 30% get D, 25% get B, and 10% get A. This scheme is now converted to intervals on the blue bar in Figure 2.

It is not entirely clear, how this conversion is made, but manual measurements on the original figure indicated that the blue bar is 10 cm long and that the 10%, 35%, 65%, and 90% lines fall at 1 cm, 3.5 cm, 6.5 cm, and 9 cm, respectively. Since the blue bare represents scores between P and 100, the length is 100-P. This means that the upper limit of the interval that results in what would be an E, is 0.1*(100-P). Thus, the conversion seems to be that 10% of the people who pass will get a score between P and P+0.1*(100-P). 25% of the people who pass will get a score between P+0.1*(100-P) and P+0.35*(100-P), and so on.

Next, the interval P to 100 is rescaled to a new 100-point scale. See the green bar in Figure 3. The grade systems is now further developed based on that scale.

The minimal passing grade is 0 and the maximum passing grade i 100, thus reflecting the "population percentage mapping" constructed in Figure 2. Thus in the green bar, the lowest passing grade is 0 and the highest passing grade is 100. 50 is chosen as the middle grade because it is in the midpoint of the interval (35-65) that captures the 30% from the ECTS scale, who should get a C on the ECTS letter scale. Likewise, 22.5 is chosen as the midpoint of the interval capturing the 25% who should should get a D, and 77.5 captures the midpoint between the interval capturing the 25% who should get a B.

The reader will now notice that we have five passing grades: 0, 22.5, 50, 77.5 and 100. For convenience, these grades are divided by 10 and rounded off to nearest integer. See Figure 4.

The ECTS system does not contain any set numbers for how many should fail or how many should achieve each of the two failing grades Fx and F. Therefore, it is not possible to use a scheme like the one above to construct numbers for the failing grades in the new system. Instead, the REF organises the failing grades symmetrically around the passing grade, which is 0. This results in the two failing grades: -2 and -5.

And there you have it! If you deliver a performance in Denmark which is "unacceptable in all respects" (see https://ufm.dk/en/education/the-danish-education-system/grading-system), you are a -5 on a scale from 0-10.

Richardson et al. (2004) argues that the distribution of grades will most likely map closely to the distribution of grades given in the ECTS grade system. Furthermore, they expect that the mean grade will also allow for at-a-glance assessment of the relative placement of a student. For instance, they argue that it should be expected that 80% of students should achieve a grade of 8,0 or less.

They also question their own assumption that the degree of achievement of learning goals can be mapped to a scale between 0 and 100, which was the starting point. They argue that this is an ideal situation that will not always be feasible. However, they argue that one can get close to this ideal situation by defining precise learning goals.

Wait a minute! Where's -3, 4, 7, and 12?

Good catch! Richardson et al. (2004) argues that the scale shown in Figure 5 (top) is the "natural choice". However, they recognise that having 0 as the lowest passing grade and having negative failing grades would send out wrong signals and stand in the way of appropriate use of the scale. This is why the number 2 was added to the natural scale, yielding the final scale:

• 12 (ECTS: A), "for an excellent performance displaying a high level of command ofall aspects of the relevant material, with no or only a few minor weaknesses."
• 10 (ECTS: B), "for a very good performance displaying a high level of command ofmost aspects of the relevant material, with only minor weaknesses."
• 7 (ECTS: C), "for a good performance displaying good command of the relevant materialbut also some weaknesses."
• 4 (ECTS: D), "for a fair performance displaying some command of the relevant materialbut also some major weaknesses."
• 2 (ECTS: E), "for a performance meeting only the minimum requirements for acceptance."
• 00 (ECTS: Fx), "for a performance which does not meet the minimum requirementsfor acceptance."
• -3 (ECTS: F), "for a performance which is unacceptable in all respects."

Did the scale live up to the expectations?

In short, no. The Danish Evaluation Institute has made thorough analyses and evaluations of the use of the scale (EVA 2019, accessed 9/7/2024). However, documenting the discussions, critique, defences, and evaluations of the new grade scale is beyond what I wanted to do here. The interested reader can follow the references given. They are in Danish, but Google Translate captures Danish well enough that the main points should be attainable.

My personal take is that it is not surprising that the scale has not solved the problems it was meant to solve. The fundamental problem - as I see it - is that reducing the complexity of a person's knowledge, skills, and competencies to a single, static number is probably not meaningful. In that sense, I would not expect this scale to perform neither better not worse that any other scale.

References

European Commission, Directorate-General for Education, Youth, Sport and Culture. (2009). ECTS users' guide, Publications Office. https://data.europa.eu/doi/10.2766/88064

European Commission, Directorate-General for Education, Youth, Sport and Culture. (2015). ECTS users' guide 2015, Publications Office of the European Union. https://data.europa.eu/doi/10.2766/87192

EVA. (2019, January 01). Evaluering af 7-trinsskalaen. Danmarks Evalueringsinstitut. https://eva.dk/udgivelser/2019/jan/evalueringen-af-7-trins-skalaen

Karran, T. (2004). Achieving Bologna Convergence: Is ECTS failing to make the grade?. Higher education in Europe, 29(3), 411-421.

Karran, T. (2005). Pan‐European Grading Scales: Lessons from National Systems and the ECTS. Higher Education in Europe, 30(1), 5-22.

Kolen, M. J., Brennan, R. L., & Kolen, M. J. (2004). Test equating, scaling, and linking: Methods and practices.

Richardson, K., Br?ndum, S., Haastrup, K., Hansen, I., Hellesh?j, H., Padde, S., Schmict, E.M., Shapiro, H. (2004). Bet?nkning om indf?relse af en ny karakterskala til erstatning af 13-skalaen: Bet?nkning nr. 1453. K?benhavn: Undervisningsministeriets Forlag. https://static.uvm.dk/publikationer/2004/karakterer/karakterer.pdf