Conventionally, a grade of a C means "average". But have you ever thought about what "average" means? In order to have an average, you must have some population over which you compute this average. What is the average for any given class? Is it the classroom itself? I would argue not, because that implies that even for a classroom full of students who fully understand the material, a certain percentage must receive a D or F. Is it average over the entire population? No, that's not it either, for then almost every student in math will be a C or better student. Is it the historical average knowledge of students in the past who have taken this class? One might make that a theoretical argument, but in practice I have no idea how students did last year.
In my syllabi for this semester, I have given the following description of the letter grades:
A = Highly proficient
B = Proficient
C = competent
D = Minimally competent
F = Not competent
This may seem like a minor distinction to some, but I think it provides an appropriate framework for a lot of pedagogical claims. For example, "Everybody can pass this class." Under the "average" system, there must necessarily be a certain subset of the population that cannot pass. This, of course, does not say that everyone *WILL* pass.
Thursday, August 23, 2007
Friday, August 10, 2007
Students as Clients
Link to the article
I just finished reading an extremely insightful article that was referenced in "Enhancing Scholarly Work on Teaching and Learning." The article is titled "Students as Clients in a Professional/Client Relationship" by Jeffrey J. Bailey.
I'm quite pleased that this article exists, as it reflects some of my own perspectives on teaching that I already have and enhancees it by increasing the depth of those views. What is even more exciting is that this article exists in a journal for management education, not a journal for mathematics education. One of the main points of "Enhancing Scholarly Work" is that cross-disciplinary reading in education is both valuable and necessary because there are education issues that extend beyond the boundaries of a particular subject.
There were a few poignant quotes:
This statement expresses something that I've already felt and even codified in my "Standard Syllabus" in the contract that I have at the end. I feel less odd about the contractual nature of that document now.
This helps to put the classroom experience in the right perspective from the students' side. If you go to the gym but don't exercise, do you expect to derive any benefit from the experience?
I wish I had this article five years ago, when I started as a TA. It would have given me a much clearer explanation to students as to why it is unproductive to argue for a better grade when one it is clearly not deserved.
I just finished reading an extremely insightful article that was referenced in "Enhancing Scholarly Work on Teaching and Learning." The article is titled "Students as Clients in a Professional/Client Relationship" by Jeffrey J. Bailey.
I'm quite pleased that this article exists, as it reflects some of my own perspectives on teaching that I already have and enhancees it by increasing the depth of those views. What is even more exciting is that this article exists in a journal for management education, not a journal for mathematics education. One of the main points of "Enhancing Scholarly Work" is that cross-disciplinary reading in education is both valuable and necessary because there are education issues that extend beyond the boundaries of a particular subject.
There were a few poignant quotes:
The enhanced role of the professional and client in the client metaphor (compared to a sales clerk/customer metaphor) embodies additional rights, responsibilities, and expectations for both professors and students. The client rightfully has expectations that the professional operates within accepted standards and ethical guidelines and will fulfill responsibilities associated with being a member of the profession.
This statement expresses something that I've already felt and even codified in my "Standard Syllabus" in the contract that I have at the end. I feel less odd about the contractual nature of that document now.
As Franz notes, the attainment of physical fitness cannot be given to the client but must be accomplished by the client. A trainer can, however, show a client what to do, encourage him or her, and provide accountability. Similarly, students need to realize the importance of their active involvement in learning. The students must work at learning just as the trainee must exercise (work) to obtain physical fitness.
This helps to put the classroom experience in the right perspective from the students' side. If you go to the gym but don't exercise, do you expect to derive any benefit from the experience?
The professor/student relationship has dimensions to it that parallel the accounting firm/client relationship. If a student is not satisfied with a grade, he or she does not get it changed simply because of the dissatisfaction. Just as the audited client cannot say, “I’m not satisfied with your audit so change some numbers here to make me satisfied,” so too the student is generally bound by the grade the professor has assigned.
I wish I had this article five years ago, when I started as a TA. It would have given me a much clearer explanation to students as to why it is unproductive to argue for a better grade when one it is clearly not deserved.
Thursday, August 9, 2007
New Reading Material
I can't quite figure out where I came across this title, but I've been reading "Enhancing Scholarly Work on Teaching and Learning" by Maryellen Weimer and I must say that I'm pretty impressed with what I've seen so far. The primary discussion at hand is how to view the growing body of work that addresses teaching from a scholarly perspective. Without going into a history that will be full of errors, let me simply say that this is an area that is still struggling to find its identity and figure out what standards it should set for itself.
But as I've been reading through this book, I've also been getting my hands on the exemplary articles that she suggests, such as "Confidence in the classroom: Ten maxims for new teachers" and "Helping students understand grades," both of which are interesting reads, and I might have more to say about them later (after I have had some more time to think about them).
But as I've been reading through this book, I've also been getting my hands on the exemplary articles that she suggests, such as "Confidence in the classroom: Ten maxims for new teachers" and "Helping students understand grades," both of which are interesting reads, and I might have more to say about them later (after I have had some more time to think about them).
Sunday, August 5, 2007
"Opposites" of numbers
I'm taking a little break from writing lecture notes to complain about things that textbooks do that are really stupid. I'm going through the Math 097 textbook, and I have reached the third or fourth time where I just shake my head at the textbook and hope that somehow either the textbook or the author will feel my frustration and it will be changed in the next edition. I suppose I could write a letter to the publisher or the authors, but I'm not going to do that now.
Why does it matter? I have an expectation that the students will read the textbook. As a consequence of this expectation, I am reading the textbook as well to make sure what they read actually makes sense.
As an aside, in what classes do students pay $100+ for texts that they have no expectation of reading? As far as I know, it's only the math classes.
Anyway, something that really bothered me was the discussion of negative numbers. They talk about 1 and -1 being "opposites". This random terminology that math people would never use reminds me of some terminology I ran into in high school with the CPM program. Apparently, somewhere along the line someone felt that a "common denominator" should be called a "fraction buster". If you don't believe me, just Google "fraction buster." Anyway, I suppose the idea is that when solving an equation involving fractions , you can "bust" the fraction by multiplying by the right number... or something like that. Besides the absurdity of nonstandard terminology, it also means that when you add fractions with different denominators that you would have to introduce the "common denominator" as a separate concept, even though it's the exact same thing.
But back to "opposites". The standard terminology for this is "additive inverse" for the simple reason that the relationship between them is additive, namely that if you add them together and you get zero (4 + (-4) = 0), and zero is the special number that doesn't change the result if you add it to something else (10 + 0 = 10). Then there's a link between this idea and the "multiplicative inverse" of a number. The relationship between a number and it's multiplicative inverse is that when you multiply them together you get one (4 * (1/4) = 1), and one is the special number that doesn't change the result if you multiply it by something else (10 * 1 = 10).
It just got worse as I pushed onward into the following section. Here is a quote from the text:
Furthermore,
If you talk to anyone who does math (this includes chemists, engineers, physicists, and many others), and pointed to -x and told them to read it, they would all say "negative x." So while it would make sense to a student brought up in this way, to the rest of the world, it's nonsense.
But it gets worse. Consider the following sentence "Multiply x by the opposite of x." If given that sentence outside of the context of this subject, I would consider "opposite" to be related to "multiplication" and end up with x * (1/x) instead of x * (-x). The book will likely introduce a different word, "reciprocal" for 1/x. Unfortunately, all this does is introduce more words for students to memorize, and those words are not particularly informative to the nature of the relationship. Why is "opposite" additive and "reciprocal" multiplicative? It just is. Memorize it. (At least "reciprocal" is a common terminology to people who use math...)
I think that's enough of a rant. I need to get back to writing up lecture notes. I won't get into the "Rules for Addition of Real Numbers" table right now...
Why does it matter? I have an expectation that the students will read the textbook. As a consequence of this expectation, I am reading the textbook as well to make sure what they read actually makes sense.
As an aside, in what classes do students pay $100+ for texts that they have no expectation of reading? As far as I know, it's only the math classes.
Anyway, something that really bothered me was the discussion of negative numbers. They talk about 1 and -1 being "opposites". This random terminology that math people would never use reminds me of some terminology I ran into in high school with the CPM program. Apparently, somewhere along the line someone felt that a "common denominator" should be called a "fraction buster". If you don't believe me, just Google "fraction buster." Anyway, I suppose the idea is that when solving an equation involving fractions , you can "bust" the fraction by multiplying by the right number... or something like that. Besides the absurdity of nonstandard terminology, it also means that when you add fractions with different denominators that you would have to introduce the "common denominator" as a separate concept, even though it's the exact same thing.
But back to "opposites". The standard terminology for this is "additive inverse" for the simple reason that the relationship between them is additive, namely that if you add them together and you get zero (4 + (-4) = 0), and zero is the special number that doesn't change the result if you add it to something else (10 + 0 = 10). Then there's a link between this idea and the "multiplicative inverse" of a number. The relationship between a number and it's multiplicative inverse is that when you multiply them together you get one (4 * (1/4) = 1), and one is the special number that doesn't change the result if you multiply it by something else (10 * 1 = 10).
It just got worse as I pushed onward into the following section. Here is a quote from the text:
A symbol like -x, which has a variable, should be read "the opposite of x" or "the additive inverse of x" and not negative x," since to do so suggests that -x represents a negative number.
Furthermore,
As you read mathematics, it is important to verbalize correctly the words and symbols to yourself. Consistently reading the expression -x as "the opposite of x" is a good step in this direction.
If you talk to anyone who does math (this includes chemists, engineers, physicists, and many others), and pointed to -x and told them to read it, they would all say "negative x." So while it would make sense to a student brought up in this way, to the rest of the world, it's nonsense.
But it gets worse. Consider the following sentence "Multiply x by the opposite of x." If given that sentence outside of the context of this subject, I would consider "opposite" to be related to "multiplication" and end up with x * (1/x) instead of x * (-x). The book will likely introduce a different word, "reciprocal" for 1/x. Unfortunately, all this does is introduce more words for students to memorize, and those words are not particularly informative to the nature of the relationship. Why is "opposite" additive and "reciprocal" multiplicative? It just is. Memorize it. (At least "reciprocal" is a common terminology to people who use math...)
I think that's enough of a rant. I need to get back to writing up lecture notes. I won't get into the "Rules for Addition of Real Numbers" table right now...
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