Keywords: Trigonometry, ,
Ref: Zdenek1
Author(s): Katz,Victor
Date: April 1995
Title: Sines & Cosines of the Times
Journal or Publisher: Math Horizons
Volume, Issue, Pages: April,1995,pg.5
Reviewer: Zdenek
Date of Review: 4/10/99
This article described the history of certain concepts of calculus. Of course this can help in developing an intuitive understanding of the basic ideas of certain subjects, and consequently lead to the understanding necessary for us to apply certain techniques in problem solving. Katz attempts to describe the graphical representations of the derivatives of sin,cos,and tan, using the unit circle. As it turns out, the tangent line at any particular point on the circle can be presumed to be the hypotnuse of a right triangle whose base is dx, and whose height is dy, the infinitesimal change in the sine.
I very much appreciate any way to represent tricky material. I
found this out in my first micro teaching lesson. In particular,
I had trouble defining sin, and got stuck in a loop not knowing
whether I should try to explain some of sin's complications, or
assume that the class allready knew what was required of them. I
see this article as a way to perhaps a sub-lesson plan to further
ground students with trigonometric knowledge.
Keywords: Curriculum, ,
Ref: Zdenek2
Author(s): Hungerford, Thomas
Date: 1994
Title: Future Elementary Teachers: The Neglected
Constituency
Journal or Publisher: Mathematical Association of America
Volume, Issue, Pages: Vol.101,No.1, Jan., 1994, 15-21
Reviewer: Zdenek
Date of Review: 4/17
In this article, Hungerford supports the claim that our current elementary school mathematics teachers are not as good as they could and should be. Hungerford claims that the mathematical preparation elementary school teachers have is perhaps the weakest link in our nation's entire system of mathematics education. Hungerford sites a growing body of evidence suggesting that students' attitudes toward mathematics and science are well established by the time they enter high school. He calls for a curriculum change not only at the elementary level, but also in educating elementary teachers at college. Apparently, of the 54 states and territories that certify teachers, more than one-require no mathematics at all for elementary certification. At most 10 percent require more than 6 semester hours of mathematics for elementary certification. To alleviate the situation, Hungerford thematics education in the setting of a mass nonelite audience, and more respect for the intelligence and precious talent of educators who manage to achieve success in that setting.
For the most part, I agree with what Hungerford is assessing. He praises
the curriculum changes established by the NCTM, and encourages elementary
educators to use these and other resources that are out there in terms of
updated curriculum. For the instructor of the instructor he suggests that
one should encourage article reviewing and attendance at some mathematical
education gathering. Looks like you're doing all right by his standards.
Keywords: Tests, ,
Ref: Zdenek3
Author(s): Hilton, Peter
Date: 1993
Title: The Tyrrany of Tests
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol. 100, No.4, Apr., 365-369
Reviewer: Zdenek
Date of Review: 4/17
In this article, Peter Hilton testifies of some of the problems in our current test taking policy. Most of the article is about undergraduate education and testing, but it definitely applies to the secondary and elementary levels of math teaching and math testing. One of the main points Hilton stresses is that in our current milieu of testing, the student who thinks quickly is favored over the student who thinks deeply. This is a consequence of examinations where time is a constraint. Hilton claims that the constraints imposed upon students through tests place them in an environment under which no mathematician would ever contemplate doing mathematics. Hilton also warns about the problems of teaching towards a test. With the genuine focus of students in our society being strictly obtaining the degree, it is easy for our standards to fall into a pattern that promotes teachin ented course and we cannot claim that we are teaching for a genuine understanding and imparting knowledge which the students will then be able to use and apply.
For me, test taking has become quite a bother. I used to be quite the
test taker. I had no problem with the setting or the questions or
performing welluntil physics sophomore year. I blanked and all I could
think of was the pencil scratchings I was hearing from the 50 or so other
students crammed into that small room. As a result of this experience,
I've had great difficulty performing in the traditional test setting. So
I'm likely to read and support change. I agree with everything Hilton has
to say. I especially liked the suggested multiple choice grading scale
which allows for partial credit. Hilton says, and I contend, that the
only test that can be trusted is one provided by continuous evaluation by
the instructor, reinforced by occasional oral examinations conducted by
experts in oral examination.
Keywords: Calculus, ,
Ref: Zdenek4
Author(s): Bressoud, David
Date: 1992
Title: Why Do We Teach Calculus
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol. 99, No. 4, Aug-Sep, 615-617
Reviewer: Zdenek
Date of Review: 4/17
In this article, Bressoud is questioning why calculus is required in most college curriculum. To this question he poses two answers. One is the usual reply you could get from asking any math major stressing the natural connections and spatial relations that can be modeled and analyzed using calculus. The other answer is one of historical significance. The significance can be equally passed down to high-school, and is, because Bressoud wrote the article as a result of some advanced placement high school calculus he had been teaching. Bressoud claims that the usefulness of calculus is not a sufficient answer to why do we teach calculus. He goes on to say that we teach calculus because it is important for an understanding of who we are in a society and we do a tremendous disservice to our students in the first year of calculus if we do not convey this excitement. Historically ans of finding local extrema. Bressoud is convinced that current calculus needs to be taught because it is central to our human scientific society now, and has evolved to become so. He feels historical pedagogy could be used to better teach calculus, and I agree. Bressoud included an interesting quote from Henri Pioncare: The task of the educator is to make the child's spirit travel again where his father's have passed, crossing certain stages rapidly but suppressing none of them. In this regard, the history of science must be our guide.
The use of history in terms of teaching math is a subject that has yet to
come up in class. I find it quite appealing, but much of that history is
foreign to me. As to where the history applies in terms of non-calculus
math, I am both unsure and unaware. In retrospect, the ideas I used and
learned in calculus were never presented in a historical format, but I
feel if they had, perhaps my interest would have been strengthened.
Keywords: Assessment, Communication, Curriculum
Ref: Zdenek5
Author(s): Schoenfeld, Alan
Date: 1993
Title: Partnerships
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol. 100, No. 10, Dec., 926-929.
Reviewer: Zdenek
Date of Review: 4/19
Alan H. Schoenfeld makes a fair assessment of our current state of mathematics education. He does so by looking at where we are now, and comparing it to where we were 20-25 years ago. In all of the social sciences through the 60's and 70's, the methods employed tended to be rigorous and scientific, with the focus predominantly on experimental studies and statistical analyses. Many experiments took place in the lab and those studies which took place in the classroom tended to downplay the complexity of classroom interactions. The line between mathematics and math-ed communities used to be well defined, but thanks to people like Polya, we no longer have such separation. A validation representing of how far we have come can be seen in the existence of educational organizations including: The Mathematical Sciences Education Board, Mathematics and Educational Reform, American Mat athematics, Professional Standards for Teaching Mathematics, NCTM and their annual meetings, and the general community of mathematics educators. The classroom, once seen by most as too complex for careful studies of mathematical thinking and learning, is now seen by many as the natural place for such studies.
I found Alan H. Schoenfeld's article to be quite exciting. He stresses
the importance of where math ed has come from, where it is, and where we
can make it go. He also recommends that we not get complacent about our
mathematical advances, because there is much work to be done, however
noting how far we have come can be seen as quite a reward. Now is a very
exciting time in mathematics education.
Keywords: Communication, ,
Ref: Zdenek6
Author(s): Bullock, James
Date: Oct. 1994
Title: Literacy in the language of Mathematics
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol. 101, No. 8. pp. 735-743.
Reviewer: Zdenek
Date of Review: 4/24
"Mathematics differs from other languages not because of its inclusion of the abstraction, but because of its complete detachment from the complications of what we experience by direct observation." P. 736. James Bullock attempts to put mathematics on an equal plane with other natural sciences in a forum that is essentially philosophical, from the perspective of education. Bullock claims that all of human understanding is based on metaphors, and that equating the earth with a sphere is no different than stating, for instance, that I am the oppressor of generations of minorities. Physically, I have not oppressed anyone (at least in any past generation), but because I am white, I can be modeled to a perspective of what is white. Perhaps this is a bad example, but getting back to the mathematics of it all. Saying that the earth is a sphere is a metaphor that is explicitly mathematical in nature. Bullock stresses that we should seek to improve our language so as to incite more rich and colorful metaphors as a way of interpreting our world and our existence. This is mathematics. Bullock claims that these types of illustrations and interrelations are what needs to be incorporated in the curriculum. Bullock also insists that we discard the strict drill and answer forum that math students currently exist in. This forum will never be seen by students after the test. "The goal of the scholar is not technical dexterity, but insight." P.740.
I found all of what Bullock said to be accurate. I agree with what he
said about training vs. Education, and their respective differences. I
enjoyed the way Bullock analyzed education and the learning process, and
then deduced that mathematics is an essential foundation element of
education. His assertions are right on pertaining to the need to expand
our vocabulary and master the contexts in which this vocabulary is to be
administrated.
Keywords: Communication, Curriculum, Standards
Ref: Zdenek7
Author(s): Halmos, Paul
Date: 1994
Title: What Is Teaching?
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol. 101, No. 9, pp. 848-854.
Reviewer: Zdenek
Date of Review: 4/25
Hamlos claims that if problems presented to the students that are just barely within their reach, their interest will be sparked. Often times students may marvel at the amount or cardinal of information we as teachers, or prospective teachers, have on our subjects. When they wonder how we remember all of that, our response should be that we don't remember it, we understand it. Hamlos emphasizes problem solving, and ideally thinks that problem solving should be the foundation of all math courses. He comments further on his capacity to teach courses, and the amount of thought that is required in order to succeed. Hamlos finishes his article by requesting educators to tell the who, where, how much, & such, "but when it comes to the why, stay out of their way so they may proceed full steam ahead."
I thought Paul Hamlos' article was fairly accurate, but not too complete.
Hamlos didn't specifically say how to implement problem solving for all
math courses. Of course this wouldn't fit into an article. I agree with
him on the issue of letting the student figure out why certain things are
the way they are. 3.2
Keywords: Calculus, Curriculum,
Ref: Zdenek8
Author(s): Ferrini-Mundy, Joan; Graham, Karen
Date: 1991
Title: An Overview of the Calculus Curriculum Reform
Effort
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol. 98, No. 7, Aug.-Sep., pp.627-635
Reviewer: Zdenek
Date of Review: 4/26
The Authors of this particular article claim from many sources that Calculus is the most important course. The Reform Calculus curriculum effort is attempting to discover where we lack and how we can improve teaching calculus. Most of the problems that calc students solve are strictly computational. The higher level of calculus problems are ignored in most classrooms. These authors give several suggestions for enhancement. Vary order of topics. This leads to a greater conceptual understanding of the material. Never omit the application. These authors also stress that curriculum development research must continue, in particular research on student learning. The article talks specifically about pros and cons of the derivative and integral, in relation to what students know, and what they don't know. The authors call for curriculum that 1.provide awareness of errors and mi
I like all the articles that I've read so far concerning curriculum
change. This one, although is was calculus specific, was in essence no
different. Perhaps the fact that these authors desire error awareness in
the curriculum is a bit different, but error awareness will be applied in
any course simply because of the grading done for that course. I like
errors, and I'd like to take advantage of error teaching when the
opportunity arises. When an error is made, it's not as if the student
wasn't thinking. Those trains of thought that fall under the error guise,
can also be learned from.
Keywords: Curriculum, Activities,
Ref: Zdenek9
Author(s): Weissglass
Date: 1993
Title: Small-Group Learning
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol.100, No.7,pp.662-668
Reviewer: Zdenek
Date of Review: 4/26
Julian Weissglass pays tribute to the benefits of Small-Group learning. She, like so many others, sat through lecture after lecture in high school, and then took notes through lecture after lecture in college. Seeing how disengaging the structure of courses were, she decided to incorporate small-group learning in her classroom. She loved the results. This article contains valuable student responses to the new method. Most of which were pleased. She offeres an example to try. Take a square grid of squares and count how many there are. Share and discuss how you came to that conclusion. Take the upper left hand square out of the grid, and now have the students count. Of course discuss how the answers came about. Julian goes on to warn of the possible problems that may arise considering evaluation.
Having experienced all the microteaching lessons, and having the guest
speaker come to class, I fell it's obvious which method is more
beneficial. Of course the forum has to be suitable for small group
learning. Elementary and junior high seem to be formidable settings for
such modes of instruction.
Keywords: Issues, ,
Ref: Zdenek10
Author(s): Renz, Peter
Date: 1993
Title: Thoughts on Innumeracy: Mathematics Versus the
World?
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol.100, No. 8, pp. 732-740.
Reviewer: Zdenek
Date of Review: 4/27
It's about time I feel validated. I really was annoyed with Paulos'
book. I just hated the way he threw that word out and accused the world
of falling victim to the perils of innumeracy, while he, the untouchable
author, reigned supreme with his unexplained conjectures, denouncing all
to be essentially unfit people. This book made me feel like an idiot, but
I was reluctant to express that. After reading Peter Renzs' article,
Paulos seems to be nothing more than wrong about many assertions. Renz'
comments about the arrogant tone of the book, and denounces Paulos'
blatant unsubstantiated torts. Renz gets technical and computes the real
probability of the Caesar's breath example, citing all of Paulos errors,
not only calculational, but also incorrect assumptions. Renz critiques
another of Paulos' examples. He realizes that Paulos thinks that Mt.
Fuji is as long in its base as it is in its
height:
a cone. Mt. Fuji is of course
not and the margin of error is approximately 595,000 years. Contemplating
all
Paulos has said about
people not understanding large numbers, this assertion should turn his
face
pink. Renz also picks apart
the AIDS example, criticizing Paulos for downplaying the devastating
importance
of this disease. Renz
concludes that innumeracy is a problem, but Paulos' book is not part of
the
solution. I concur.
Keywords: Communication, ,
Ref: Zdenek11
Author(s): Cowen, Carl
Date: 1991
Title: Teaching and testing Mathematics Reading
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol. 98, No.1, 50-53
Reviewer: Zdenek
Date of Review: 5/17/00
Carl Cowen is stressing the importance of teaching students to read and understand mathematics. Since most students drop what they learn after the test, Cowen feels that the teaching should be done in a manner that promotes understanding, not simply memorization. Cowen is stressing self initiated out of class reading for the student. He feels that mathematics instructors should communicate to the students that they think it is reasonable and that students should read mathematics and understand mathematics so as to come to expect this to be everyday behavior.
Cowens plea is not anything new. Personally, I feel he is asking quite
alot from his students. To be able to explain material never before
touched on in class is almost absurd. Yes, I do agree with promoting a
self sustainable mathematical mind, but demanding that students simply
read more math texts is not necessarily the only way to go.
Keywords: Issues, ,
Ref: Zdenek12
Author(s): Allen, John
Date: 1990
Title: Review of Innumeracy
Journal or Publisher: American Mathematical Monthly
Volume, Issue, Pages: Vol. 97, No.1,88-91
Reviewer: Zdenek
Date of Review: 5/17/00
John Allen further attests to the arrogant demeanor of Paulos' Innumeracy. The book's existance and name do nothing more than contradict. Paulos has nomed a problem, and attempted to address it. Fine, but does addressing the problem mean that everyone should take joy and pleasure in computing meaningless numbers? Allen argues the Paulos is preaching to the already converted, ie, those who enjoy problem solving. Allen feels, and I concur, that any person who would fall into paulos' innumerate chategory would shun this book. The problem of innumeracy is addressed, and then Paulos does a mathematical song and dance to wow the innumerate reader. This does nothing more than address the problem, and mock the victims. This is not problem solving.