Assessment in Mathematics
Marty
education.
Feel free to answer one or more of the following questions:
(1) What are the purposes of grades and numerical assessment, in
practice, and under ideal circumstances? Are grades simply ways for a
teacher to communicate pleasure or displeasure of a student's work?
Are grades simply ways for employers to determine qualification? Are
grades simply meant to modify the behavior of students by a system of
punishment and reward?
(2) How should assessments be used? How should you change your
method of assessment based on the ways the assessment is used by
others?
(3) What makes a grading system "fair"? How does your personal sense
of "justice" affect your grading system?
Please try to read and reference at least one outside source in your
response. Please respond by the evening of Tuesday, October 30.
Jen Mogel
A grade in a class, A through F, with N and U and others, is a code to
indicate a standard evaluation of a student by the teacher. This
evaluation covers mainly the level mastery of course material attained
by the student but also the degree of understanding as well as
originality or creativity as appropriate to the course. The grade also
indicates how well the student works independently and with what
effectiveness and how often they take the initiative in seeking new
knowledge outside the formal confines of the course. With these
questions in mind, a teacher evaluates the student's performance and
assigns a grade indicating outstanding, good, average, less than
average, and very very bad! (Reference the web address for VPAA
grades and assessments below.)
Different institutions use these evaluations in different ways. The
school uses it to evaluate the student's overall performance in the
school, again, assigning a code number, the GPA, to give the student
an idea of their average performance over the year and over many
years. The student should use the grade as an independent, expert
assessment of their competency of a certain subject for that term.
Students often say that they "know" a certain subject well, when, in
fact, they do not. The grade tells the student if they really know
the subject well. Employers use the grades to tell them two things,
their possible level of competency over certain subject matter and how
well the student works effectively at learning new things. For
example, by the description of an A on the VPAA website, a student
with many A's will be faster to train to effective or efficient
working competency in the workplace. This person will seek further
information outside training and will have unusually high initiative.
The A student will, perhaps, be a leader in the workplace. On the
other hand, the F student is probably lazy and incompetent.
As with any independent evaluation given a person, a grade will
motivate a student to improve themselves only if the student wants
to. It's exactly like a (very forgiving) job. You have a couple of
bad evaluations and you don't improve yourself, you get fired! It is
possible that students who do poorly in regards to their grades don't
know how to improve them, but that is part of life. In a job, the
employee must figure out how to improve their performance primarily on
their own. This is the same for a student's grades.
http://www.vpaa.villanova.edu/handbook/policies/gradesandassessments.htm
Marty
> A grade in a class, A through F, with N and U and others, is a code to
> indicate a standard evaluation of a student by the teacher.
Are you referring to the grades that you give? In your classes? By
everyone who gives grades? Is the ideal of what a grade "should be"?
According to whom?
>This
> evaluation covers mainly the level mastery of course material attained
> by the student but also the degree of understanding as well as
> originality or creativity as appropriate to the course. The grade also
> indicates how well the student works independently and with what
> effectiveness and how often they take the initiative in seeking new
> knowledge outside the formal confines of the course. With these
> questions in mind, a teacher evaluates the student's performance and
> assigns a grade indicating outstanding, good, average, less than
> average, and very very bad! (Reference the web address for VPAA
> grades and assessments below.)
>
Again, who is "a teacher"? Are you talking about ideals or practice?
Friends or yourself? Do you claim that you use metrics for
independence and initiative? How do you know that a grade has such
indicative power?
> Different institutions use these evaluations in different ways. The
> school uses it to evaluate the student's overall performance in the
> school, again, assigning a code number, the GPA, to give the student
> an idea of their average performance over the year and over many
> years.
Do institutions primarily use grades as a means of communication to
the student? Or do they use grades as a means of communication to
employers and other schools? Any evidence?
>The student should use the grade as an independent, expert
> assessment of their competency of a certain subject for that term.
What do you mean by "should"? Why should they? Is this your personal
recommendation to students? Is this an issue of ethics within some
system?
> Students often say that they "know" a certain subject well, when, in
> fact, they do not. The grade tells the student if they really know
> the subject well. Employers use the grades to tell them two things,
> their possible level of competency over certain subject matter and how
> well the student works effectively at learning new things. For
> example, by the description of an A on the VPAA website, a student
> with many A's will be faster to train to effective or efficient
> working competency in the workplace. This person will seek further
> information outside training and will have unusually high initiative.
> The A student will, perhaps, be a leader in the workplace. On the
> other hand, the F student is probably lazy and incompetent.
>
"perhaps"? "probably"? Any evidence for these claims? Are you
talking about grades in reality, or in the vision of an institution?
> As with any independent evaluation given a person, a grade will
> motivate a student to improve themselves only if the student wants
> to. It's exactly like a (very forgiving) job. You have a couple of
> bad evaluations and you don't improve yourself, you get fired! It is
> possible that students who do poorly in regards to their grades don't
> know how to improve them, but that is part of life. In a job, the
> employee must figure out how to improve their performance primarily on
> their own. This is the same for a student's grades.
>
> http://www.vpaa.villanova.edu/handbook/policies/gradesandassessments.htm
A general guideline to posting about these issues: Talking about
"what a grade means" is kind of like talking about "what Odysseus
feels". You can talk about what Homer is trying to suggest about
Odysseus's psyche -- the author has intent, while a fictional
character does not. A "grade" cannot talk. The giver of a grade
might try to convey a message. The message (grade) might be
interpreted differently by different audiences.
When giving "prescriptions", like saying that a teacher *should* do
something, or a student *should* do something, justify the
prescription.
When saying how things are used by various parties, try to give some
evidence. This is an opportunity to challenge your assumptions.
megan
grading systems change based on the teacher, the situation of the
student, and the level of the class. For me, if a student is taking a
basic course that will not have a huge effect in their future career,
then I am more willing to bump a grade up to a passing grade, as long
as they made a good effort in the class. But I would be highly
unlikely to do this for, say, an upper division math course, where the
majority of the people will need a firm grasp of the subject to do
well in future classes, or their career. On the other hand, if a
student has an extreme comprehension of the subject but doesn't do
enough of the material to pass, how can you not pass them? (I find
this doesn't come up too often in college, since most class grades are
dependent mainly on the midterms and finals)
This kind of relaxed system of grading does cause problems though. I
found it difficult this summer to find a good cut off grade. I had a
student that worked exceptionally hard and showed fluency in the
subject to me in office hours, but did not perform to their abilities
on the tests. So of course I wanted to give them a passing grade, but
I felt that if I passed them, then I couldn't fail people with a
higher percentage. In the end I chose to lower a C grade 2% in order
to pass them, and ended up passing another student that shouldn't have
passed.
I don't think there is a set grading system that is always "fair". I
believe it is fairer at times, to have a lenient grading system. The
tricky part is to figure out the best place to draw the line.
Jen Mogel
The grade "meaning" refers to the standard definition of the letter
grade code given by a set of guidelines like the one on the referenced
website. This was the only set of guidelines I could find. This
definition covers all types of classes and the teacher is ideally
supposed to look at these guidelines to determine how to assign grades
to their students. I am not saying that all of the questions,
initiative and the like, are actually taken into account by the
teacher for every class or even every student.
When I say a teacher "should" take these questions into account, it
means that the teacher is supposed to use the standard grade
definition used by the school as a basis for assigning grades. The
grades we give students in Math for example are not supposed to be
based on the students' scores because the scores are there. The
scores the students earn are how Math teachers tell what level of
competency a student has reached with the material and how much work
the student has put into the class. Suppose a student does no work
prior to the final and gets a legitimate 100% on the final. This is a
demonstration of the student's mastery of the material. According to
the definition of an A given by the referenced website, the teacher
for this course could give this student an A.
It has been my experience, observing teachers I have worked for, some
friends who are teachers, and myself, that we adjust the grades of a
student upwards when that student is observed putting more of an
effort into the class than the bulk of the other students.
As for the section:
>The student should use the grade as an independent, expert
> assessment of their competency of a certain subject for that term.
I really should (he he) have used the word can or could instead of
should.
As far as what Employers use a person's school grades for, I do have a
source beyond my own experience. Sorry I didn't mention it. Bonnie
Diefendorf was the owner and manager of Mogel Engineering Inc, a Civil
Engineering and Surveying firm in Sonoma County. She used applicant's
grades as part of her screening process and I was paraphrasing her
reply from when I asked her how. (She's my Mom, I hate to use
relatives as references and did so unconsciously, oh well.)
Hope this makes things a little better.
Jen Mogel
Our current grading system, both in its ideal form and in practice, is
an inadequate assessment of a student's accomplishments in a class for
only students and graduate schools.
The ideal form of our grading system is where all grades for
equivalent classes take the same type of information into account,
grade all on the same scale, within the school at least, and truly
reflect on how much a student learned in the class for that term. In
actuality, different teachers may grade harder or easier on their
students then their colleagues or not be able to truly focus on each
student well enough to determine how well they learned the subject due
to overly large class sizes.
First, let's take a look at the students' needs. The current standard
grading system only gives them an overall idea on how well they
performed in the class. All I have ever used grades for, as a
student, was for a motivation to work hard and to do better. I was
often more concerned with getting that good grade than I was in
learning. I know, bad me.
What about graduate schools? Again, current grades don't say enough
about the student. It does not take into account many factors
involved in the student's grades, like whether the student was ill for
a number of crucial weeks, or if the student started out with a couple
of bad tests or papers but pulled of a brilliant comeback in the end.
As the article by H. M. Teaf, Jr. in the Journal of Higher Education
pointed out, the student could have been brilliant but lazy or hard
working but not quite so smart, the graduate application committees
won't know the difference.
What we really need is a grading system that includes more
information. I remember in elementary school we received report cards
that gave us two grades for each subject twice each term. One grade
for effort put into the subject and the other for the resulting
knowledge gained. (Don't ask me how they determined that, I was a
little kid then.) There was also a little comment box where my
teacher could write down extra comments or recommendations. This
system would be difficult to do in a college like this one since there
are classes with up to 250 students and it is impossible for any
teacher to know enough about them to assign a grade with this much
detail. What we could adopt is a modified version. For non-major and
prerequisite classes, the current system would do just fine. I never
cared much what my own grades were outside of Math classes as long as
I was passing. For classes pertaining to our major, which would be
smaller (hopefully), we could adopt this version or something like it,
a grade for how much the student learned and a grade for the effort,
creativity, initiative, etc, plus a comment or two. This will give
the student more perspective into their performance in the class and
will tell a graduate school more about the student and how well they
might do in their graduate program.
There are other institutions that use grades. Medium to large
employers use grades to weed out their applicant pool. (Ref 2) They
are not interested in the nuances of the student's performance at
every stage of their classes; it would take too much time to read
through. They want the overall result because it's faster.
Ref 1 Teaf, H.M., What Price Grades, Journal of Higher Education
Ref 2 Diefendorf, Bonnie, Civil Engineer, (707) 542-6465 (work) (I
dare you to call her. Everybody should call her, it would be fun.
She'll disown me.)
Jen Mogel
Just thought I'd put that out there. Oh, I also hate A minuses. I
got one once I don't think I deserved. Down With A minuses and D's!
Jacob West
the above questions. Consider it a work in progress. I post it now
for the sake of further discussion...
---
Under ideal circumstances, grades would not be necessary. It is the
practical considerations of Universities that seek to award degrees
and employers who seek to correlate those degrees with some measure of
potential performance as an employee that necessitates something like
the current grading system. By practical considerations, I mean of
course economics.
Before I delve into the so-called "practical considerations", let me
first justify my comment on the "ideal circumstances". The idealized
goal of any educational system is the passing of knowledge from
educator to student. Note here that I make no claim as to the
*method* used to achieve this so-called passing of knowledge. That is
to say, and to avoid unnecessary argument, I make no claim that the
"passing of knowledge" is achieved via some traditional lecture
format, or any other preconceived preferred method of teaching for
that matter. Rather, "passing of knowledge" is considered here in the
more abstract sense as whatever mechanism by which a student manages
to gain knowledge within an educational system. In other words, in
the ideal setting, an educational system is intended to educate.
Perhaps that seems like an empty tautology, but I think it is often
forgotten by both students and educators alike. Many students blame
inadequacies in their knowledge on "bad teachers", while many teachers
complain about the inadequacies of their students detracting from the
course material they'd like to cover. This sort of transfer of
responsibility of ones education and subsequent denial of
responsibility for that education is a failure of student and teacher
alike; and, as I will argue later, a direct consequence of the current
"practical" grading system. Ideally, students would study a subject
under the guidance of an educator until a predefined level of mastery
was achieved, however long that might take. Individual students then
advancing at individual rates. In an ideal educational system, every
student would earn "A" grades, only some more quickly than others,
perhaps. Hence, the grade itself would become meaningless, with the
emphasis and "goal" of the educational system rightly placed on each
students educational progress.
Of course, this sort of idealization is not generally possible due to
many "practical considerations". In particular, one of the primary
goals of modern Universities is to award degrees. I suppose that the
reader does not find this a provocative statement, so much as a matter
of fact. The administration and economic maintenance of modern
Universities are no doubt complicated matters, but we can agree that a
regular influx of new students is an important aspect of the continued
life of any educational institution. Consequently, to maintain this
influx, the University must also produce an outflux (or grow to fill
the demand). The most common mechanism used to generate an outflux is
to impose some fixed time frame (usually 4-5 years) within which a
given student must either meet the requirements for graduation or
leave. These requirements are most often specified in terms of
classes satisfactorily completed, which in turn are given fixed time
constraints for completion to ensure that a student is "on track" to
graduate within the desired time frame. Eventually, these time
constraints translate into something like a quarter system, where
individual courses must cover a prespecified set of topics within some
number of weeks. It is at this stage of "practical constraints"
arising from "practical considerations" that a grading system must be
introduced to address what is meant by "satisfactory completion."
Even in the idealized educational system, the notion of what
"satisfactory completion" actually means within a given subject had to
be addressed, but in that case I tacitly assumed that such an
assessment would be self-evident and thus ignored any details
therein. The presumption being that in the ideal settings, an
educator would have intimate knowledge of their student's
understanding and educational progress through direct and regular
interaction. However, in the practical educational systems of today,
this level of attention is not generally possible. In fact, most
Universities today have an incentive to grow their student body size
(while maintaining time constraints) and no corresponding incentive to
grow their faculty size; so that as the student body size grows,
student to faculty ratios soar, and the scarcity of educator's time
per student becomes a primary difficulty in meeting the individual
educational needs of any given student. Therefore, we're left to
address what "satisfactory completion" means now in this settings,
where the answer may be not at all obvious. It is precisely this
confluence of practical constraints that gives rise to grading systems
designed to be time efficient, yet "sufficiently accurate" to define
satisfactory completion. Now, the definition of "sufficiently
accurate" is perhaps highly circumstantial, and in large part what
underlies the original questions above which prompted this response.
I will stop at this point and save further discussion for future
posts.
Jen Mogel
in two ways, by creating a strong and reasonable curriculum for all
levels of math education and by insisting on a rigorous level of
research before accepting the validity of various factions' claims.
The first task may not be as easy as it seems. The Mathematician in
me believes we can teach algebra as early as fifth or sixth grade in
all US schools and we should have statistics and calculus taught to
all High school students in addition to pre calculus. Just look at
what the "Stand and Deliver" guy, Jaime Escalante, did. If taught
well, I think we can teach students almost anything! This would be
included in a strong curriculum but not necessarily a reasonable one.
Finding the proper balance would take working together with math
education researchers, cognitive psychologists, and math teachers from
different levels. This group should be like Switzerland, neutral; it
should rise above the "party lines" or "war factions," and ignore the
unsubstantiated shouting of the hard liners on any side of the
conflict. The group takes all their ideas, the research, and the
experience and works together to build a reasonable curriculum from
the strong one. This is analogous to negotiators from different
countries working together to form a treaty, but in this case, each
part of the group would provide research to use both in the building
of the curriculum and in promotion of it when they try to sell the
idea to various states and to the country at large. I think that this
was attempted in a looser sense, as described in Klein's article, when
a group of Stanford Mathematicians rewrote California's Academic
Standards in 1997.
The second task is something that everybody should be doing, not just
Mathematicians. One needs to use unbiased information to support or
refute different sides of an argument. However, it is difficult to do
this for two main reasons. First, people misunderstand some of the
results from research. As stated in 'Applications and Misapplications
of Cognitive Psychology to Mathematics Education' math ed. people have
misunderstood the research the cognitive psychologists have done. The
way to fix this problem is to have competent "translators" to tell the
math ed. people what the cognitive psychologists are saying, to tell
the Mathematicians what everybody else is saying, and so on and so
forth. Science communication, I am told, is a growing field. There
is so much information out there on so many different things that it
becomes confusing. Chemists can't be expected to understand a paper
in math and vice versa. Science communicators can bridge the gap in
understanding and translate the findings in one field so that the
layman understands, not all the details, but the gist of the
information. These "translators" will reduce the accidental misuse of
information that is out there so our math war "battles" can be had
with well-maintained weapons like the X-wing the rebels used in Star
Wars and not junk like products from the ACME Company. The other
difficulty is harder to overcome. There is a tendency exhibited in
many national or worldwide debates where people will insist that their
side is right while blinding themselves to any opposing argument.
Their debates are more like shouting matches, who can drown out the
other through volume and insults, than discussions. It happens in
politics, it happened just last year, I think, when a protest group
ran military recruiters off campus by threatening violence. Though, I
am sure we don't have any thugs from NCTM throwing rocks through David
Klein's window, (yet?) we do have people calling others racists and
sexist or ignorant liberals just because they disagree with their
views. The only way I can think of to stop this sort of behavior is
to refrain from it myself and not tolerate it in others. So I'm going
to stop listening to you, you hippy liberals.
Thank you for you time!
Anderson, J.R., Reder, L.M., Simon, H.A. (2000 Summer). Applications
and Misapplications of Cognitive Psychology to Mathematics Education.
Texas Educational Review.
Klein, D. "A quarter century of US 'math wars' and political
partisanship", BSHM Bulletin, Vol. 22, 2007, Pg 22-33
megan
it is realistic. I think that for a select few of the student
population this would work, but for the most part students need more
motivation. Grades are a way to propel competition between a student
and themselves or other students.
Americans have become pretty lazy in recent decades, perhaps because
of video games, too much TV, lack of outdoor activity, etc. Whatever
the cause, I think for most k-12 students; if their education was left
up to them we would be worse off than we already are, especially in
math. And with a time and money driven society, we need students to
get through school as fast as possible for their own good. It is
pricey to live in this country, and since many students are expected
to support themselves by the age of 18, a high school diploma is key
for getting jobs, and for an increase in pay, a college education. I
found an article in the San Francisco Chronicle from 2004 (http://
www.sfgate.com/cgi-bin/article.cgi?f=/c/a/2004/06/04/BAGJ370QUK1.DTL)
that said that 70% of high school students graduate in four years.
Also in this same article they said that 23% of ninth graders go on to
graduate with a "C" or better. In another article, from ACTIVITY
2004, (http://act.org/activity/spring2004/ready.html), they said that
26% of high school students enrolled in a 2-yr college, 45% enrolled
in a 4-yr college, 4% other postsecondary, which is a total of 75%.
Clearly from the results in the Chronicle's article these students are
not prepared if 76% of high school students don't graduate with above
a "C" average. Personally I don't understand with these statistics
how 45% are accepted to a 4-yr program. In the ACTIVITY article they
do address how well students are prepared for college, "McCabe cites
studies that show 20 percent of entering students are underprepared in
reading, 25 percent in writing, and 34 percent in math." I'm sure
that because of students expectations of what colleges are going to be
like, and lack of preparation is why only 27.7% of the US population
(25 and older) actually receive a Bachelor's degree (Wikipedia).
These are horribly depressing statistics, which from my experiences is
mainly caused by students not doing homework, not studying, not
attending class, etc. I don't think that if we had an alternate
grading system that these problems would be fixed. I think what needs
to change is how we teach the subjects and our expectations of how
much a high school student can learn. In class, Jake was saying that
he would use humiliation in order to drive people to do their work. I
think that grades already do this. It made me embarrassed to tell my
parents and my friends that I got a bad grade in a class, not to
mention I was dissapointed in myself for letting down my teacher. I
think that this embarassement drives many students to work harder to
do as well as others or better. I do think there are a very select few
self-motivated students, and if you were to ask most students why they
work hard, it would be for the grade.