Thursday, September 27, 2007

The Seven Liberal Arts, Part I

The classic seven liberal arts were the trivium (grammar, rhetoric, and dialectic [or logic]), and the quadrivium (arithmetic, geometry, music, and astronomy). These were preparatory studies for the more advanced fields of philosophy and theology.

Are our students well-grounded in these fields? Should they be?

Many of my students claim not to have studied grammar at all. Some students do study grammar at the college level: by studying a foreign language. In classical times, much of grammar was also taught through studying other languages: Greek and Latin. Why is grammar so devalued now? Is there value to understanding that language has structure, and that different languages can be structured very differently?

It seems to me that grammar has been devalued because one can develop a good grasp of grammar without actually acquiring a workable fluency in speaking another language, or without acquiring good writing skills in one's own language. And people raised speaking several languages can speak those languages fluently and can write quite eloquently without consciously knowing grammar. But to use these observations as a critique of the study of grammar is to make a mistake about the purpose of studying grammar. The value of understanding the structure of language has little to do with whether it makes you a fluent speaker or a poetic writer. The latter skills are valuable in their own right, but are not the reasons for studying grammar.

The reason for studying grammar is to learn the different ways that language is structured. It is only when one closely studies the structures of different languages that one begins to understand the difference between language and thought (so closely related that most modestly-educated people think that there is no difference at all).

Tuesday, September 25, 2007

The Ideal of Liberal Education

Today is the day of the Crimmel Colloquium at St. Lawrence University. In honor of this event, I would like to quote from Professor Crimmel's book, sharing his view of the ideal of liberal education:

The ideal of liberal education is: "the development of wise people--that is, people who possess the capacity and inclination to act on the basis of knowledge of reality and ideality" (Crimmel, Henry H., The Liberal Arts College and the Ideal of Liberal Education: The Case for Radical Reform, University Press of America, 1993, p. 125).

Here is a later expansion: "The wise person [is] one who possesses the capacity and inclination for rational action. To act rationally is to act on the basis of knowledge of what is and what ought to be, and with prudence, and with the aid of moral virtues" (Crimmel, p. 217).

One more statement I would like to quote: "The wise person acts to transform reality into ideality" (Crimmel, p. 222).

And here are other statements of ideals that he does not think are as worthy:

Ideals Giving Priority to Theory:

To provide a religious faith
To provide specialized knowledge
To provide general knowledge
To provide both specialized and general knowledge
To understand the Great Books
To develop cultural literacy
To provide an understanding of Western culture
To provide an understanding of world cultures
To provide an initiation into the forms of knowledge
To develop the critical thinker

Ideals Giving Priority to Practice:

To provide vocational training
To prepare students for graduate school
To prepare for a mature, effective, adult life
To provide political liberation
To develop solidarity
To actualize human potential
To cope effectively with change
To develop the citizens of a free society
To develop "the democratic personality"
To develop a person

Ideals Giving Priority to Interests

To satisfy student interests
To satisfy a plurality of interests

It is not that all of these other statements of ideals are unworthy. He argues that his statement of the ideal is superior to all of these other statements. (His arguments can be found on pp. 127-163.)

As I look over his list of other statements of ideals, I see that the ones that give priority to theory all neglect the question "to what purpose?" The ones that give priority to practice point to goals but do not fully articulate them or justify them. Of each of those implicit goals, the further question "why?" still can be asked. This is not necessarily a problem. It just means that a person must choose such a goal outside of the educational system that supports that goal. But those who have set such goals for themselves might find such educational systems quite meaningful. I am inclined to agree with Professor Crimmel, however, that these would not count as institutions providing liberal education. I also agree that orienting education around interests is problematic.

I am struck most of all by the emphasis Professor Crimmel places on the study of ideality. I agree with him here (and have written previously about a similar theme here in the SLU Philosophy Blog). We seem to put more emphasis on the study of various dimensions of reality. And we often delude ourselves into thinking that "ideality" is not really real. Setting ideals and choosing values is just a matter of personal preference, and it is a bit rude (even an infringement of "academic freedom") to question each other about our values and our moral choices. And yet these relativistic attitudes about ideality, about what ought to be, miss the point completely. "Ought" gains much (most, all?) of its meaning from the reality of our essential interconnectedness with each other. It is crucial for us to be able to examine this, study it, question each other about it.

Our lives are permeated by the force fields of many "oughts" that compete for our attention. Our lives are so much more than aimless wanderings through a dispassionate world of what is. We always regard that world through lenses of "ought." All of our actions are oriented towards transforming the "is" we find ourselves in to the "ought" we want it to be.

It does seem to me that the wise person gives some serious consideration to the study of ideality, as well as the study of reality. To ignore paying explicit attention to understanding ideality is to unthinkingly follow the force-fields of "ought" that others have set up and that you happen to wander into unawares. I agree with Professor Crimmel that such a person is not very wise. Our "oughts" are not always good ones. Nor do we always succeed in effecting the transformations we hope for. This is why it is good to study ideality, and also good to study "practical wisdom," or, how to be effective in transforming reality to ideality.

Saturday, September 15, 2007

More on Grading: Living with What We Have

In yesterday's posting, I explained why I am unhappy with our system of numerical grading. But I do have to live with this system. So I thought I would pose a few questions about how to use it as responsibly as possible.

1. I have noticed that different faculty convert between a 100-point (or percentage) grading scale to our 4.0 scale in different ways. Is this a problem? Students get upset: to receive a 93% in one class may earn them a 3.25; in another class a 3.5; in another class a 3.75; in another class perhaps even a 4.0. The defense I heard from a math professor is that it does not matter that different professors align their scales differently, because some professors make their exams so hard that a 93% does indicate a remarkable achievement warranting the top grade of our grading scale (4.0), while others align their expectations a bit differently.

2. My own solution is to avoid all conversions altogether. I grade everything on the 4.0 scale, and then just average these grades (using weighted averages as appropriate). Here's how to round to .25 intervals: you take the raw averaged grade, multiply by 4, round this number to the nearest whole number, and divide by 4. It's easy to make a spreadsheet that does all of this for you.

3. My system of grading yields this question: Which is more appropriate for grading within the course, before the final averaging and rounding: (a) use only the .25-interval grades, (b) use even finer gradations (e.g., .125-interval grades), or (c) use coarser intervals (.5-interval grades, or even just the whole number grades, since these are the only ones whose meanings are defined: excellent, good, satisfactory, low-pass, and fail)? Or does it not matter? (Mathematically, do the averages mean something different on these three different scenarios?)

I hope someone can answer question 3 with a convincing and mathematically well-grounded rationale.

Friday, September 14, 2007

The Meaninglessness of Numerical Grading

The following is adapted from my web page on grading.

In 2005, St. Lawrence changed its grading system, from the grades of 4.0, 3.5, 3.0, 2.5, etc. (0.5-interval grading) to grades of 4.0, 3.75, 3.5, 3.25, 3.0, 2.75, etc. (0.25-interval grading).

It was the students who wanted the finer gradations. They said that they wanted the grades to more accurately reflect their performance in their courses. The faculty passed this proposal (but not without debate, and not without some faculty arguing in a very different direction). The new grading system went into effect in the 2005-2006 academic year.

It is interesting to note that this change is not really just a refinement of an existing system. The two systems are in fact different enough that it is inappropriate to think that the Grade Point Averages (GPAs) computed in each system can be directly compared.

The following chart shows how GPAs do in fact change if you round grades in different ways. This table shows the kinds of rounding historically used at St. Lawrence University. Other schools often use +/- systems, which numerically convert to grades such as 3.3, 3.7, 4.0. What my little table shows is that it is dubious to compare GPAs on 4.0 grading scales if the systems of rounding are different.


.25 Rnd

.5 Rnd

.0 Rnd





















Note that not all sets of grades would necessarily always round down on .25 intervals and up on .5 intervals. What is interesting is just that the GPAs are different. Two students with the same grades would have different GPAs depending on whether they started before or after the change in grading intervals – and yet those students who came in the midst of the change have both kinds of grades averaged together, as if averaging these incommensurable scales is legitimate!

Here is a table showing what happens when you average together the grades of students graded under both systems. Imagine these five hypothetical students who happen to get exactly the same raw grades in their courses every year (the grades from the above table) -- but the grading system changes for all except Student 1 sometime during their time here. This table shows the differences in their final GPAs at the end of their four years (the yearly GPAs are taken from the table above):

Yr 1 GPA

Yr 2 GPA

Yr 3 GPA

Yr 4 GPA

Final GPA

Student 1






Student 2






Student 3






Student 4






Student 5






In this case, the student lucky enough to have arrived before the change has the highest GPA. The student unlucky enough to have spent all four years under the new grading system has the lowest. Again, it is not the case that this change results in lower GPAs for all students -- the point is that the very same raw grades average out to different GPAs depending on the grading system. Worse, these GPAs are then compared to those of students from schools that may use the .3/.7 intervals (plus/minus grading) -- an altogether different system, but because it is also a 4.0 scale we think it is essentially the same!

We place a lot of faith in these numbers that we are not in fact even computed in a mathematically responsible way. Can we really say that the GPA has a stable and unambiguous meaning?

I wish we could switch to a system of grading that does not convert grades to numbers. My favorite option is to use a high-pass, pass, fail system. The "high pass" would be a grade specifically to indicate that the student did so well that you regard that student as having graduate school potential. After all, the main distinctions we wish to make are whether the student should pass, and whether the student has worked with the material so well that you would recommend them for more advanced study in that field.

But I would also be content with a return to A, B, C, D, F grading: no pluses or minuses, and no attempt to convert these grades to numbers. The meanings here are excellent, good, satisfactory, low-pass, and fail.

When I criticize the grading system, people often leap to the assumption that I want to replace it with narrative evaluations of all students. But that is not true. I see the value of our having a shorthand way of representing the quality of students' work. What I object to is converting grades to numbers, averaging these numbers, and tying so much to this average (sometimes taken out to the thousandth decimal place) when this use of numbers is not warranted and thus highly misleading.