This is the third and final of my three part series on the principles of great assessment. In the first post I focused on the principles of assessment design, and in the second on principles relating to issues of fairness and equality. This final post attempts to get to grips with principles relating to issues of reliability and making assessments provide useful information about student attainment. I have been putting off this post because whilst I recognise how important reliability is in assessment, I know how hard it is to get to grips with, let alone explain to others. I have tried to do my best to synthesise the words and ideas of others. I hope it helps lead to the better use of assessment in schools.
Here are my principles of great assessment 11-16
11. Define standards through questions set
The choice of the questions set in an assessment are important as they ultimately define the standard of expectation, even in cases where the prose descriptors appear secure. Where there is variation between the rigour of the questions set by teachers, problems occur and inaccurate inferences are likely to be drawn. The following example from Dylan Wiliam, albeit extreme, illustrates this relationship between questions and standards.
Task: add punctuation to the following sentence to make it grammatically correct
John where Paul had had had had had had had had had had had a clearer meaning.
This question could feasibly be set to assess students’ understanding of grammar, in particular their knowledge of how commas and apostrophes are used to clarify meaning, which on the surface seems a relatively tight and definitive statement. Obviously, no right-minded teacher would ever set such an absurdly difficult example, which most of us, including English teachers, would struggle to answer correctly*. But what it highlights is the problems that can arise when teachers deploy their own understanding of the required standards independently.
A teacher setting the above question would clearly have sky-high expectations of their students’ grammatical understanding, or supreme confidence in their own teaching! More realistically, a question assessing for students’ grammatical ability would look more like the example below, which requires a far lower grammatical understanding.
Task: add punctuation to the following sentence to make it grammatically correct
John went to the beach with his towel his bucket his swimming trunks and his spade.
All this is yet more reason why summative assessments should be standardised. It simply cannot be that the questions some students face demand significantly greater knowledge and understanding than others who have been taught the same curriculum. The questions used in tests of this nature should be agreed upfront and aligned with the curriculum to remain stable each year. This is, of course, in practice really difficult: teachers may start teaching to the test, and thus invalidate the inferences from the assessment, or the question set one year is not of the same standard as the ones previously, thus making year on year comparisons difficult.
12. Define standards through exemplar pupil work
As well as defining standards through questions, standards can also be defined through student work. Using examples of work to exemplify standards is far better than defining those same expectations through the abstraction of rubrics. As we have seen, not only do rubrics tend to create artificial distinctions between levels of performance, but the descriptions of these performances are more often than not meaningless in isolation. One person’s notion of detailed and developed analysis, can easily be another’s highly sophisticated and insightful evaluation. As Hamlet says of Polonius’ speech, they are just ‘words, words, words’. They only mean something when they are applied to examples.
Whether we like it or not, we all carry mental models of what constitutes excellence in our subject. A history teacher knows when she sees a great piece of historical enquiry; she doesn’t need a set of performance descriptors to tell her it demonstrates sound understanding of the important causes and effects explained in a coherent way. She knows excellence because she has seen it before and it looked similar. Perversely, performance descriptors could actually lead her to lower the mark she awards, particularly if it is too formulaic and reductive, which seems to be the problem with KS2 mark schemes: the work includes all the prescribed functional elements, but the overall piece is not fluent, engaging or ambitious.
Likewise, the same history teacher knows when something has fallen short of what is required because it is not as good as the examples she has seen before that did, the ones that shape the mental model she carries of what is good. On their own rubrics really don’t tell us much, and though we may think they are objective, in reality we are still drawing upon our mental models whenever we make judgements. Even when the performance descriptors appear specific, they are never as specific as an actual question being asked, which ultimately always defines the standard.
If objective judgement using rubrics is a mirage, we are better off spending our time developing mental models of what constitutes the good, the bad and the ugly in terms of exemplar work rather than our misunderstanding abstract prose descriptors. We should also look to shift emphasis towards the kinds of assessment formats that acknowledge the nature of human judgement, namely that all judgements are comparisons of one thing with another (Laming, 2004). In short, we should probably include comparative judgement in our assessment portfolio to draw reliable judgements about student achievement and make the intangible tangible.
13. Share understanding of different standards of achievement
Standardisation has been a staple of subject meetings for years. In the days of National Curriculum Levels and the National Literacy Strategy English teachers would pore over numerous examples of levelled reading and writing responses. At GCSE and a Level in other subjects, I am sure many department meetings have been given over to discussing relative standards of bits of student work. From my experience, often these meetings are a complete waste of time. Not only do teachers rarely agree on why one piece of writing with poor syntax and grammar should gain a level 5, but we rarely alter our marking after the event anyway. Those that are generous remain generous, and those that are stingier continue to hold back from assigning the higher marks.
The main problem with these kinds of meeting is their reliance on rubrics and performance descriptors, which as we have seen fail to pin down a common understanding of achievement. The other problem is that they fail to acknowledge the fundamental nature of human judgement, namely that we are relativist rather than absolutist in our evaluation. Since we are probably never going to fully agree on standards of achievement, such as the quality of one essay over another, we are probably better off looking at lots of different examples of quality and comparing their relative strengths and weaknesses directly rather than diluting the process by recourse to nebulous mark schemes.
Out of these kinds of standardisation meetings, with teachers judging a cohort’s work together, can come authentic forms of exemplified student achievement – ones that have been formed by a collective comparative voice, rather than by a well-intentioned individual attempting to reduce the irreducible to a series of simplistic statements. Software like No More Marking is increasingly streamlining the whole process, and the nature of the approach itself lends itself much better to year on year standards being maintained with more accuracy. Comparative judgement is not fully formed just yet, but as today’s report into the recent KS2 trial, there is considerable promise for the future
14. Analyse effectiveness of assessment items
As we have established, a good assessment should distinguish between different levels of attainment across the construct continuum. This means that we would expect a marks for difficulty assessment to include questions that most students could answer, and others that only those with the deepest understanding could respond to correctly. Obviously, there will always be idiosyncrasies. Some weaker students sometimes know the answer to more challenging questions, and likewise some stronger students do not always know the answer to the simpler questions. This is the nature of assessing from a wide domain.
What we should be concerned about in terms of making our assessments as valid and reliable as possible, however, is whether, in the main, the items on the test truly discriminate across the construct continuum. A good assessment should contain harder questions that discriminate students with stronger knowledge and understanding. If that is not the case then something probably needs to change, either in the wording of the items or in realigning teacher understanding of what constitutes item difficulty.
How to calculate the difficulty of assessment items:
Step one: rank items in order of perceived difficulty (as best you can!)
Step two: work out the average mark per item by dividing the total marks awarded for each item by the number of students.
Step three: for items worth more than 1 mark, divide the average score per item by the number of marks available for it.
Step four: all item scores should now have a metric of between 0 and 1. High values indicate the item is relatively accessible whilst low values indicate the item is more difficult.
This is the formula in Excel to identify the average score of an individual item:
On an assessment with a large cohort of students we would expect to see a general trend of average scores going down as item difficulty increases i.e. a lower percentage of students are answering them correctly. Whilst it would be normal to expect some anomalies – after all, ranking items on perceived difficulty is not an exact science and is ultimately relative to what students know – any significant variations would probably be worth a closer look.
How to calculate item discrimination
There are different ways of measuring the extent to which an item distinguishes between more and less able students. Perhaps the easiest of these uses the discrimination index.
Step One: Select two groups of students from your assessment results – one with higher test scores and one with lower test scores. This can either be a split right down the middle, or sample at both extremes, so one group in the top third of total results, and one group in the bottom third.
Step Two: Divide the total of the sum of the range of the chosen high test score group minus the chosen low test score group by the number of students in the high score group multiplied by the marks available for the question
This is the formula to use in Excel:
The discrimination index is essentially the percentage of students in the high test score group who answer the item correctly minus the percentage of the students in the low test score who do not. It operates on a range between -1 and +1 with values close to +1 indicating the item does discriminate well between high and low ability students for the construct being assessed.
Values near zero suggest that the item does not discriminate between high and low ability students, whilst values near -1 suggest that the item is quite often answered correctly by students who do the worst on the assessment as a whole and conversely incorrectly by those who score the best results on the overall assessment. These are therefore probably not great items.
15. Increase assessment reliability (but not at the expense of validity)
Reliability in assessment is about consistency of measurement over time, place and context. The analogy often used is to a pair of weighing scales. When someone steps on a pair of scales, whether in the bathroom or the kitchen, they expect the measurement of their weight to be consistent from one reading to the next, particularly if their diet is constant. This is the same as reliability in assessment: the extent to which a test produces consistent outcomes each time it is sat. In the same way you wouldn’t want your scales to add or take away a few pounds every time you weigh in, you wouldn’t want a test to produce wildly different results every time you sat it, especially if nothing had changed in your weight or your intelligence.
The problem is that in assessment it is impossible to create a completely reliable assessment, particularly if we want to assess things that we value, like quality of extended written responses which we have already discussed can be very subjective, and we don’t want our students to sit hundreds of hour’s worth of tests. We can increase reliability but it often comes at a price, such as in terms of validity (assessing the things that we believe represent the construct), or in time, which is finite and can be used for others things, like teaching.
What is reliability?
There are two mays of looking at the reliability of an assessment – the reliability of the test itself, or the reliability of the judgements being made by the judges. Reliability can be calculated by comparing two sets of scores for a single assessment (such as rater scores with comparative judgement) or with two scores from two tests that assess the same construct. Once we get these two sets of scores, it is possible to work out how similar the results are by using a statistical term called the reliability coefficient.
The reliability coefficient is the numerical index used to talk about reliability. It ranges from 0 to 1. A number closer to 1 indicates a high degree of reliability, whereas a low number suggests some error in the assessment design, or more likely one of the factors identified from the Ofqual list below. Reliability is generally considered good or acceptable if the reliability coefficient is in or around .80, though as Rob Coe points out (see below), even national examinations, with all their statistical know how and manpower, only get as high as 0.93! And that was just the one GCSE subject.
How to identify the reliability of an assessment?
There are four main ways to identify the reliability of an assessment, each with their own advantages and disadvantages and each requiring different levels of confidence with statistics and spreadsheets. The four main methods uses are:
- Test–retest reliability
- Parallel forms reliability
- Split-half reliability
- Internal-consistency (Cronbach’s alpha)
This approach involves setting the same assessment with the same students at different points in time, such as at the beginning and end of a term. The correlation between the results that each student gets on each sitting of this same test should provide a reliability coefficient. There are two significant problems with this approach, however. Firstly, there is the problem of sensitivity of instruction. It is likely that students would have learnt something between the first and second administrations of the test, which might invalidate the inferences that can be drawn and threaten any attempt to work out a reliability score.
The other, arguably more, significant issue relates to levels of student motivation. I am guessing that most students would not really welcome sitting the same test on two separate occasions, particularly if the second assessment is soon after the first, which would need to happen in order to reduce threats to validity and reliability. Any changes to how students approach the second assessment will considerably affect the reliability score and probably make the exercise a complete waste of time.
Parallel forms reliability
One way round these problems is to design a parallel forms assessment. This is basically where one assessment is made up of two equal parts (parallel A and parallel B), with the second half (parallel B) performing the function of the second assessment in the test-retest approach outlined above. As with test-retest, correlations between student results from the parallel A and parallel B parts of the test can provide a reliability figure. The problem now is that, in reality, it is difficult to create two sections of an assessment of equal challenge. As we have considered, challenge lies in the choice of a question, and even the very best assessment designers don’t really know how difficult an item really is until real students have actually tried answering them.
Perhaps the best way to work out the reliability of a class assessment, and the one favoured by Dylan Wiliam, is the split-half reliability model. Rather than waste time attempting the almost impossible – and create two forms of the same assessment of equal difficulty – this approach skirts round the problem, by dividing a single assessment in half and treating each half as a separate test.
There are different ways the assessment can be divided in half, such as straight split down the middle or creating two parts by separating out the odd and even numbered items. Whatever method is used, the reliability coefficient is worked out the same way: by correlating the scores on the two parts and then taking account of the fact that this only relates to half the test by applying the Spearman-Brown formula**. This then provides a reasonable estimate of the reliability of an assessment, which is probably good enough for school-based assessment.
The formula for applying Spearman-Brown in Excel is a little beyond the scope of my understanding. Fortunately, there are a lot of tools available on the Internet that make it possible to work out reliability scores using Spearman-Brown’s formula. The process involves downloading a spreadsheet and then inputting your test scores into cells containing pre-programmed formulas. The best of these is, unsurprisingly, from Dylan Wiliam himself, which is available to download here. Rather handily, Dylan also includes some super clear instructions on how to use the tool. Whilst there are other spreadsheets available elsewhere that perform this and other functions, they are not as clean and intuitive as this one.
Internal-consistency reliability (Cronbach’s alpha)
At this point, I should point that I am fast approaching the limits of my understanding in relation to assessment, particularly with regards to the use of statistics. Nevertheless, I think I have managed to get my head around internal-consistency reliability enough to use some of the tools available to work out the reliability of an assessment using Cronbach’s alpha. In statistics Cronbach’s alpha is used as an estimate of the reliability of a psychometric test. It provides an estimate of internal consistency reliability and helps to show whether or not all the items in an assessment are assessing the same construct or not. Unlike the easier to use – and understand – split-half reliability, Cronbach’s alpha looks at the average value of all possible split- half estimates, rather than just the one that has been split in half.
It uses this formula:
If like most people, however, you find this formula intimidating and unfathomable, seek out one of the many online spreadsheets set up with Cronbach’s alpha and ready for you to enter your own assessment data into the cells. Probably the most straightforward of these can be found here. It is produced by Professor Glenn Fulcher and it allows you to enter assessment results for any items with a mark of up to 7. There are instructions that tell you what to do and are quite easy for the layman to follow.
Make sure everyone understands the limitations of assessment
Given that no school assessment which measures the things we value or involves any element of human judgement is ever likely to be completely reliable, the time has probably come to be more honest about this with the people most impacted by summative tests, namely the students and their parents. The problem is that in reality this is incredibly hard to do. As Rob Coe jokes, can anyone imagine a teacher telling a parent that their child’s progress, say an old NC level 5, is accurate to a degree of plus or minus one level? Most teachers probably haven’t even heard about standard measurement of error, let alone understand its impact on assessment practice enough to explain it to a bewildered parent.
The US education system seems rather more advanced than ours in relation to reporting issues of error and uncertainty in assessment to parents. This is a consequence of the Standards for Educational and Psychological Testing (1999). These lay out the extent to which measurement uncertainty must be reported to stakeholders, which US courts follow in their rulings and test administrators account for in their supplementary technical guides.
A 2010 report commissioned by Ofqual into the way assessment agencies in the US report uncertainty information when making public the results of their assessments showed an impressive degree of transparency in relation to sharing issues of test score reliability. Whilst the report notes that parents are not always directly given the information about assessment error and uncertainty, the information is always readable available to those who want it, providing of course they can understand it!
‘Whether in numbers, graphics, or words, and whether on score reports, in interpretive guidelines (sometimes, the concept is explained in an “interpretive guide for parents”), or in technical manuals, the concept of score imprecision is communicated. For tests with items scored subjectively, such as written answers, it is common, too, to report some measure of inter-rater reliability in a technical manual.’
To my knowledge we don’t really have anything like this level of transparency in our system, but I think there are a number of things we can probably learn from the US about how to be smarter with sharing with students and parents the complexity of assessment and the inferences that it can and cannot provide us with. I am not suggesting that the example below is realistic for an individual school to replicate, but I like the way that it at least signals the scope for grade variation by including confidence intervals in each of its assessment scores.
There is clearly much we need to do to educate ourselves about assessment, and then we may be better placed to educate those who are most affected by the tests that we set.
The work starts now.
* The answer to the questions is: John, where Paul had had ‘had’, had had ‘had had’. ‘Had had’ had had a clearer meaning
** The Spearman–Brown prediction formula, also known as the Spearman–Brown prophecy formula, is a formula relating psychometric reliability to test length and used by psychometricians to predict the reliability of a test after changing the test length.