Grading and the 'extra credit' device

The results of the semester’s third test have just been returned to the students. There are only three more lectures, a short break, and then the final examination to go. Student Alan is on track for a B-plus letter grade, and is out of range for an A-minus grade. After the lecture, Alan speaks to the lecturer.

Alan: “Sir, I see that I am out of range for an A-minus grade. Can I do something for extra credit?”

Lecturer: “Yes, you are out of range for an A-minus grade. You can still aim for a B-plus grade.”

“I am worried about my GPA and graduate school. I need extra credit.”

“How many marks would you like the extra credit to be worth?”

“Fifteen marks will be enough to get me into the A bracket.”

“Yes, I can set an extra test worth 15 marks. It will cover all topics to date. We conduct the extra test during consultation time immediately following the second lesson from today.”

“Sir, that’s not fair. It should cover only the topics in Test Three.”

“You’ve been losing marks since Test One. That’s how you became out of range of an A-minus grade. If the extra test covers only Test Three topics, then you’re essentially re-taking Test Three. The new test would be only a supplementary Test Three, and the score will replace (not add to) your Test Three score.”

“That will not give me enough marks to get an A grade.”

“Well, you do your own calculations. Another thing: I must offer the same extra test to all the other students – in fairness to them. Of course, they can opt to stay with the marks they have, and not take the extra test.”

“That’s for them to decide. I’m interested only in my extra credit.”

“There’s one more thing. With the extra test, your letter grade will be computed by taking your total marks (including from the final examination) as a percentage of 115 marks, instead of as a percentage of 100 marks.”

“What? No! That’s not fair! I will likely be out of range of A again! You add marks only to the numerator, not the denominator.”

“Alan, your misunderstanding is quite common. Grades are awarded on the basis of the percentage of offered marks that you earn, not on the basis of the number of marks you earn. This common misunderstanding is caused by the common practice of using a 100-mark scale in assessment, when percentages are calculated also on a 100-point scale. People (students, lecturers, administrators) conflate the two. If grade computation uses only the numerator, then a student who earns 100 out of 1,000 offered marks must be awarded an A-plus letter grade, and be considered “High Distinction”. But this is clearly absurd, since the student has mastered only ten percent of the course material. Consequently, students who take the extra test will have their letter grades awarded on the basis of their percentage of 115 marks, whilst students who do not take the extra test will have their letter grades awarded on the basis of their percentage of 100 marks, in the usual way.”

“None of our other lecturers does extra credit this way. I will be speaking to the student union about this, and about you.”

“So do you want the extra test or not?”

“I want the extra test. And I want the denominator to remain at 100 marks.”

END

To not bell curve or to bell curve grades

To not bell curve or to bell curve grades

In May 2019, Singapore Management University decided to review the letter grades of all 169 students in a class – because the lecturer had awarded every student an A grade, despite the university’s rule to limit the number of As in any one class to a third of the class. On 17 June 2019, The Straits Times ran a feature article on the non-use and use of bell curve grading in Singapore universities.

In this article, I would like apply some critical thought to grading on a bell curve.

We must first establish the purpose of grading. It is to inform some third party (not the lecturer or student) how competent the student has become in the subject at the end of the course. The most likely third party is the student’s future prospective employer. I exclude the lecturer because he or she already knows the answer from class interaction and formative assessments (class exercises, homework, and other non-credit work). The student also knows the answer from the above and from self-reflection.

Hence, the bottom line in discussing grading method is what information it provides to that third party, most likely the student’s future prospective employer. We return to this at the end of this article.

Typically, a student’s competence in the subject is measured through various modes such as quizzes, tests, presentations, projects, essays and examinations. Various weights are assigned to each mode, with the total amounting to 100 marks. That is to say, students are measured on a unique 100-point scale.

This is unlike such scales as for temperature (Celcius, Fahrenheit, Kelvin), length (imperial, metric), electrical (volt, ampere, ohm). A measure of 300 degrees Centigrade means exactly the same thing to a physicist, meteorologist, doctor, or indeed anyone. In contrast, one mark in any given course means one mark in only that course. Each academic measuring scale is unique to the course in which it is used.

At the end of each module, each student would have earned some number of marks for that module. Let’s call this the numerical mark.

The institution, faculty, department or lecturer (the decision-maker varies) sets a minimum number of marks as the pass mark. This establishes the minimum point beyond which the student is said to be competent in the subject. Typically, this minimum varies between 35 and 60 marks. Beyond this, we have a 65-40 point scale of competence.

This is the point where grading kicks in.

Grading is typically not reported on such a multi-point scale to the third party. The typical scale used is the letter grade scale, ranging from A-plus to D-minus – a twelve point scale. The pass mark range is typically equally divided into twelve parts, each corresponding to a letter grade. Each student is awarded the letter grade into which his or her numerical mark falls.

We need to collapse a 65-40 mark pass range into a 12-point letter grade range. Dividing equally, each letter grade covers anything between 5.4167 (65 ÷ 12) and 3.3333 (40 ÷ 12) marks.

Here is a simple example. We set the pass mark at 40 marks. That yields a 60-mark pass range. Dividing this into 12 parts gives us five marks per letter grade. A student scoring 88 marks would get the letter grade “A-“.

This is grading without using a bell curve.

The term “bell curve” is we commonly call the “normal distribution” (which when graphically expressed exhibits a bell shape). This is the distribution of any given feature (such as height, weight, intelligence) in an unbiased population. The caveat is that the measuring instrument must be sensitive enough to distinguish small differences in the occurrence of the given feature. For example, it is good enough to time marathons to the nearest second, but 100-metre races must be timed to the hundredths of a second.

In the same way, a well designed academic measuring scale will uncover student competence in a normal distribution or bell curve shape. This result may be taken as the criterion of good test design. (I once achieved a perfect bell curve with a class of 17 students).

As mentioned above, Singapore Management University set a rule for no more than one third of a class being awarded an A grade. This is a maximum. What happens if more than a third of the class achieves 85 marks or more (to continue our simple example)? What happens if no student achieves at least 85 marks?

This is where the bell curve method of grading would kick in – if mandated.

Percentages (not necessarily equal) are assigned to each letter grade or grade group (B-, B and B+ grades etc.) and students are awarded a letter grade as they progressively fulfil the percentage in each letter grade or grade group.

Bell curve grading is essentially a method to award grades according to a student’s rank (colloquially known as position) in the class.

We can now answer the question: What information does a letter grade provide to the third party, most likely the student’s future prospective employer?

If the letter grade was awarded based on the student’s numerical mark (that is, no bell curve was applied), the letter grade represents the student’s achieved competence relative to the academic measuring instrument / scale used in that course or module.

If the letter grade was awarded based on the student’s rank in class (that is, a bell curve was applied), the letter grade represents the student’s achieved competence relative to the other students in that course or module.

But the basis of awarding the letter grade is invisible to the third party.

Hence, a student showing an “A” for a given course in his or her academic transcript must be understood to be a student who was EITHER a student who achieved top competence relative to the academic measuring instrument / scale used in that course or module OR a student who achieved top competence relative to the other students in that course or module.

This is as good as it gets. The prospective employer must also use other criteria to assess whether it should hire that job applicant.

A final comment: It would be much more informative to the third party to report BOTH a student’s numerical mark (eg. 76 marks) AND the student’s class position (eg. 11 / 17).

Cheers.

It's how we frame the issue of euthanasia

As usual, and as expected, my letter to The Straits Times Forum page was not published. Fortunately, Blogger exists, so here it is.

I refer to the letter “Consider legalising euthanasia” by Seah Yam Meng (The Straits Times, 1 Aug), and wish to contribute just one point to the discussion.

Some objectors will base their objection on the claim that putting someone to death is murder – the worst and most immoral thing one person can do to another. I wish to question this claim.

There is a fate worse than death. It goes by the name “torture”.

Imagine a patient suffering a painful terminal illness, while undergoing costly and hopeless medical procedures, and who knows he is burdening his family with financial and emotional distress, while being personally unable to contribute anything meaningful or useful to anyone or anything. He cannot see any justification for his continued existence.

When we deny this patient the facility to be put to sleep (let’s use the humane phrase here, the one we use when we do the same thing for suffering animals), we can reasonably be described as forcing this patient to suffer torture – until he dies of bodily damage or exhaustion. And then we heave a sigh of relief – glad that he is now “at rest”.

I believe that when the issue as framed as “put to sleep” versus “torture”, we will react differently to the suggestion of legalising euthanasia.

END

Misunderstanding the Michelin Guide

The latest Michelin Guide for Singapore has added five new one-star restaurants to the list. Columnist Wong Ah Yoke ("Questions remain over selection process", The Straits Times, 27 July 2018), and other critics, have bemoaned the exclusion of other deserving restaurants.

This complaint arises from the interpretation of "not in the list" to mean "not worthy to be in the list".

This inference is illogical -- because there is no suggestion that all restaurants in Singapore were visited, and only five were found to deserve a Michelin listing. All that a listing in the Michelin Guide means is: Michelin visited these restaurants, and found that they deserve to be listed. It says nothing about any other restaurant in Singapore.

This illogical leap (fallacy) has a name: Argumentum ad Ignorantiam (Appeal to ignorance). Its argument form goes this way: "If there is no proof that X is true, then X is false". In this context, "Since there is no proof that Restaurant A is worthy to be listed in the Michelin Guide, then Restaurant A is not worthy to be listed in the Michelin Guide."

This fallacy has a mirror version: "If there is no proof that Y is false, then Y is true".

All we need to do when we encounter a logical fallacy is to expose the argument as a fallacy. We do not need to mount any kind of rebuttal, counterargument or alternative argument to the claim.

END

Discussing how to determine the optimal subsidy for public transport

The commentary piece “Determining the optimal subsidy for public transport” (The Straits Times, 20 July 2018) is a useful exercise for critical thinking.

First, a logical principle: “To determine optimal subsidies for a transport system, we need to identify the justifications for subsidising the system.” This statement is in line with the key principle in critical thinking: that every position must be supported by a good argument.

A “commonly invoked argument” is described: “Public transport subsidies benefit the poor since the poor are more likely than the rich to use public transport.” But transport is a “small portion” of the poor’s household expenditure. Hence, this argument is rejected.

The “main argument” for public transport subsidies is an “economy of scale” argument. Subsidies increases ridership, which reduces the average cost of providing public transport services (total operation cost divided by more users). More ridership also means more frequent services, which reduces commuters’ waiting time, which encourages more commuters, and so on. This is known as the Mohring Effect. Economists Ian Parry and Kenneth Small say this argument is used to justify subsidies in London, Washington and Los Angeles – especially during off-peak times.

Public transport subsidies encourage people to switch from cars to public transport. This reduces road congestion, pollution and accident costs. Professor Stef Proost and Dr Kurt Van Dender find this “congestion externality” consideration to be “quantitatively more important” than the “economy of scale” factors in justifying subsidies in Brussels and London.

On congestion costs: Singapore imposes a congestion tax via the Electronic Road Pricing system. Associate Professor Leonardo Basso and Assistant Professor Hugo Silva find that this lessens the congestion reduction benefit of public transport subsidy. The writer here adds: “Nevertheless, the benefit of subsidies on reduced congestion on Singapore’s roads is likely to be a non-negligible amount”. [No argument is presented to support the subsidiary claim of “non-negligible amount”.]

On pollution costs: Economists found that “the opening of a new subway network decreases particulate concentrations by about five percent in the 10km disk around the city centre”.

So subsidies provide the following benefits: lower average cost to service provider, lower waiting time for service, less road congestion, less pollution, less accident costs.

We are next introduced to some “arguments against public transport subsidies”.

Public transport subsidies increase ridership – which increases crowding, and hence increases discomfort and extends boarding and alighting times. These are the main costs of subsidies: discomfort, longer boarding and alighting times. Since boarding and alighting times are independent of distance travelled, subsidies should be a function of distance travelled less a fixed amount for boarding and alighting delays. This should result in “commuters traveling short distances with a smaller fraction of their fares subsidised”. This is a subsidiary point – relating to the quantum of individual subsidy, not the fact of general subsidy.

Money spent on public transport subsidies means money not spent on “other purposes, such as education, healthcare, or welfare.” This is the “opportunity cost” of public transport subsidies. Professor Proost and Dr Van Dender find that for Brussels, this opportunity cost “wipes out” the economy of scale benefits.

“Combined, these reasons determine the optimal subsidy for public transport operations.”

This is the key sentence. It tells us that the considerations relevant to public transport subsidy are: average cost to service provider, waiting time for service, road congestion, pollution, accident costs, passenger discomfort, boarding and alighting times, money available for other purposes. It is interesting that the actual money value of subsidy is not mentioned.

This list of considerations suggest that the justification argument is utilitarian in nature. However, this is not explicitly stated, so we must not presume so.

Dr Parry and Professor Small find optimal subsidies “to be very large” (mostly over 90 percent of operation costs) in Washington, London and Lost Angeles. Prof Proost and Dr Van Dender “find the optimal subsidy for Brussels during peak periods should be close to zero.” Prof Basso and Prof Silva find “optimal subsidies in Santiago to be about 55 percent of operation costs”.

The Brussels finding is clearly a recommendation, since it says “should be”. The statements regarding the other cities use the phrase “to be”, which precedes a fact (meaning that recommendations were implemented). The question that interests us is: How were these optimal subsidies determined?

The differences “stem from the various cities’ characteristics, including congestion levels, commuters’ preferences for different transport modes, and commuter valuation of travel time”. This is just a summary (as we have also done above) of the relevant considerations.

The essay concludes: “There is no hard and fast rule for optimal subsidies that Singapore can easily adopt. Instead, the Government will have to determine optimal levels by assessing the merits of each consideration in the Singapore context.”

Here is the remaining problem: The “considerations” may be as listed above, but we have not been told how to use them to determine the optimal (value of the) subsidy. So, referring to the opening logical principle: the “justifications for subsidising the system” have not been fully identified. Worse, what does the unequivocal declaration that “there is no hard and fast rule for optimal subsidies” portend for any further discussion of this matter?

END

How not to identify fake news

A recent news report ("Brazilian kids being schooled to fight fake news", The Straits Times, 14 July 2018) said schoolchildren in Brazil are being taught how to identify fake news.

Teacher Ms Lucilene Varandas said students are taught to "look at the articles, who wrote them, who could be interested in them and where they're published, which are all ways of questioning the information."

In logic, we recognise a fallacy called Argumentum ad Hominem, which says that a person's character or circumstance is irrelevant to the truth or falsity of what that person says or writes.

A learned professor can possibly get something wrong (perhaps by accident); a fool can possibly get something right (perhaps by sheer good fortune); a campaigner for some cause can possibly conceal or invent some data just to forward his or her cause.

The stated methods of identifying fake news are entirely fallacious, meaning they are errors in reasoning. It is a pity that these are the methods that are being taught to those schoolkids.

END

Let's think about fake news

What is news?

News is an account of what has happened (eg. Christchurch has been struck by an earthquake), been done (eg. The result of a British referendum was "leave the European Union), or been said (eg. Trump said he will build a southern wall as president).

In contrast, analysis and commentary is not news. These are opinions held by various persons. More generally, these are speculations -- because only the principal actors (such as politicians) know the true motives behind their actions and words. (By the way, only persons can hold opinions; organisations cannot.)

What is fake? Fake is the opposite of true.

So: True news is an account of what has in fact happened, been done, or been said. And fake news is an account of what has not in fact happened, been done, or been said.

But this is not enough. It is also important to avoid selective truth and embellished truth. Selective truth is when one reports something true, but omits other relevant truths. Embellished truth is when one reports something true, and adds some falsehoods. True news is captured by the old court requirement of the witness: To tell the truth, the whole truth, and nothing but the truth.

In this regard, there are two mainstream organisations that we need to watch out for: Advertisers and Public Relation agents. These people exist precisely to disseminate information just to serve the client's or organisation's agenda -- and the information may (but not always) be either selected or embellished. When these happen, we have fake news.

Fortunately, advertisements and public relation missives are usually flagged as such to their audience. This alerts the audience to be more discerning of the information provided.

The danger of fake news arises when no flag is provided. The audience is not warned.

Asking a third party to curate the alleged news is not a solution. It merely pushes the question back one step: Can the third party to be trusted to disseminate only true news? How can one be sure of this? Reputation is no real help -- because reputation is earned only by performance, which makes the matter rather circular.

The solution lies in teaching the audience to be more discerning and critical. Access news from different sources to seek consistency -- it is difficult for several sources to tell the same falsehoods, or make the same selections or embellishments. Ask if the given information conflicts with common sense (eg. causes occur before effects) and generally known facts (eg. the earth orbits the sun).

These strategies should take us a long way towards protecting ourselves from fake news.

Should schools teach philosophy?

On Sunday, 16 April 2017, I sent a letter to The Straits Times Forum page for hopeful publication. By Friday, 21 April 2017, the letter had still not been published. Here is the letter:

I refer to the recent discussion on teaching philosophy in schools (Philosophy focus can come in handy, April 2; Think carefully about philosophy in schools, April 13; Wrong to dismiss philosophy as ‘armchair reasoning’, April 15).

Everyone runs, sings and writes; but only some are runners, singers and writers. In the same way, everyone thinks – but training makes the thinking better. The only discipline that explicitly teaches the art and science of rational and rigorous thought is philosophy, specifically the branch called logic.

If we want our students (and adults) to develop the skill to think rationally and rigorously, one excellent way is to expose them to some logic. The default is that people learn to think by imitating everyone around them. They adopt the good and bad habits – with no inkling of the difference.

The first benefit of studying philosophy is learning to think rationally and rigorously.

Philosophy is unique in being characterised by dispute. Hardly any two philosophers completely agree in what they think to be true. Yet when one reads the classic philosophers, one finds their arguments utterly persuasive – though leading to utterly contradictory conclusions.

Several benefits arise from this experience. One becomes less dismissive of views unlike one’s accustomed or favoured view, more aware of subtleties and nuances in the issues discussed, and more aware that one could be wrong.

Many (perhaps most) controversial issues today are ethical issues. Should robots (when they gain rationality) be accorded rights? Should the autonomous car be programmed to crash into the lamp post or the jaywalkers? Should we allow people to openly carry guns?

Philosophy is the only discipline that explicitly discusses how to rationally and rigorously think about such questions – in the branch called ethics or moral philosophy. (The ethics modules in other disciplines tend to acquaint students with ethical decisions that have been made by either the law or the relevant regulatory body.)

There are thus many benefits in studying philosophy. The next question is: Is school time a good time to expose people to philosophy?

If students are not exposed to rational, rigorous and diverse thought about controversial topics, they will osmotically absorb the ideas they find around them – whatever the quality or truth. That cannot be a better alternative.

But here is a cautionary note.

Socrates, the father of western philosophy, was sentenced to death in 399 BC – for “corrupting the minds of the young”.

END

"Illegal immigrants" is self-contradictory, hence problematic

The phrase "illegal immigrants" keeps occurring in news reports about American President Donald Trump's thoughts and policies regarding people who slip into, and stay in, the USA without going through the proper channels.

The phrase "illegal immigrant" is a self-contradiction which, by virtue of its self-contradiction, evokes conflicting reactions from various people. We attempt in this essay to clarify the situation.

The word "immigrant" refers to someone who gains domestic nationality and residence by going through the proper channels and procedures. This way of entering, staying and working in a country is legal and legitimate -- and any call to arbitrarily expel, deport or in any way remove such a person from the domestic country rightly should be opposed.

Notice that an immigrant is someone who enters and stays in a country via legal means. This is why the phrase "illegal immigrant" is a self-contradictory.

Someone who enters and stays in a country illegally is not an immigrant. Such a person is referred to by the word "trespasser" -- and any call to remove him or her rightly should be supported. To not support such a call amounts to nullifying the rule of (immigration) law.

One reason why there is such strong objection to the idea of removing these trespassers is the objectors focus on the word "immigrant" and do not notice the word "illegal". The word "immigrant" connotes "legal, legitimate, approved" -- and hence such persons rightly should not be removed.

On the other hand, people who notice the word "illegal" -- and rightly view those who slip over the border as trespassers (not immigrants) -- feel little or no objection to preventing their entry and their removal. This is entirely separate from these trespassers competing for jobs or committing crimes while in the US. It is simply a matter of their illegal status -- and respect for law.

The problem, and appropriate action, will become much clearer once the self-contradictory phrase "illegal immigrant" is replaced by the correct term: "trespasser".

END

Facts, actual facts, and alternative facts

As almost everyone knows, the phrase “alternative fact” was recently coined – and widely derided as being an attempt to legitimise a falsehood or lie. This essay attempts to unpack the controversy.

We begin by asking: What is a fact? This is quite easily answered. A fact is a state of affairs in the world. For example, the cat sat on the mat.

Since a fact is a state of affairs in the world, the word “actual” in the phrase “actual fact” is redundant. A state of affairs in the world is by definition actual.

A fact is different from a statement expressing that fact. If the state of affairs in the world is that the cat sat on the mat, then the statement “the cat sat on the mat” is true. But if the state of affairs in the world is not that the cat sat on the mat, then the statement “the cat sat on the mat” is not true, hence false.

The words “true” and “false” can be attached only to statements, not to facts. There is no such thing as a false fact, precisely because facts are states of affairs in the world. All facts are by definition trivially true, but the adjective “true” is meaningless – because it cannot be contrasted with the adjective “false”.

It follows from this that it is not possible to say that the phrase “alternative fact” means a falsehood or lie. This is because facts are not capable of being called “true” or “false” (only statements can be called those).

Nor is it possible to say that facts have alternatives (except perhaps in other possible worlds). This is because a fact is the state of affairs in the world. What is (or is not), is (or is not); there can be no other.

So what can the phrase “alternative fact” mean?

Let’s consider the word “alternative”. It is used to indicate the sense “either A or B”. Which then brings us to the question: What were A and B? Specifically, were A and B statements or facts?

In the context of the coining of the phrase “alternative fact”, A was the statement “the crowd did not stretch from the building to the memorial”, and B was the statement “the crowd was larger than what you claim”. The alternatives are statements, not facts.

So how did facts and lies get into the story?

When the statement A was uttered, it was declared to be a description of a fact (which by definition is trivially true). Hence, anything contradicting A must be false and a lie.

But that was precisely the dispute: whether or not the statement A described the fact. The opposing contention was that the statement A did not describe the fact; that the fact was described by statement B.

So here is what the phrase “alternative fact” means: Here is a statement (B) that is an alternative statement to the earlier statement (A). Further, statement B describes the fact, whereas statement A does not describe the fact.

And here is the moral of the story: Merely uttering a statement does not make that statement true.

END