Keywords: Number Theory, Algebra,
Ref: Josh1
Author(s): Crossfield, Don
Date: 1997
Title: (Naturally) Numbers Are fun
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 90, Number 2, pp. 92-95
Reviewer: Josh
Date of Review: 22 March 1999
The author begins by describing banners he put up in his room a few years ago: each brightly colored, and each including a sequence of natural numbers (primes, powers of 2, cubes, and squares). Beginning with basic questions like "What is the next number in the sequence?", he then used these banners to develop a stronger sense of numberical patterns in his students. The colors of each banner, rather than just a possibly meaningless name, provided coherence for each set of numbers. For example, can you add two numbers of one color to get a number of the same colors? (Here Crossfield mentions the possibility of exploring the Pythagorean Theorem in the squares and Fermat's Last Theorem in the cubes.)
The patterns of these sequences and the relationships among them are essential tools which students must either possess or develop at many levels of mathematics. Crossfield notes that the presence of these banners improved the number sense of his students, provided many interesting questions to ask and answer, and gave concrete examples for future topics, including approximating irrational roots, partial sums of series, and modular arithmetic.
I liked this idea because I have always thought classroom walls can provide some beautiful illustrations of mathematics for students. This suggestion goes further by incorporating wall posters into the classroom activity. Students get a feel for the experimentation present in math, the occasional unexpected result, and the power of a proof in number theory.
Keywords: Issues, ,
Ref: Josh2
Author(s): Wu, Hung-Hsi
Date: 1999
Title: Professional Development of Mathmatics Teachers
Journal or Publisher: Notices of the American Mathematical Society
Volume, Issue, Pages: 46(5), pp. 535-542
Reviewer: Josh
Date of Review: 14 April 1999
Amid all our discussion of standards and progressive curricula, I have had a question that I haven't asked because it doesn't much apply to the people in our class. This article addresses that question: what difference will all the changes in what we're teaching make if the teachers don't understand it themselves? I suspect this may be one reason for negative reactions to the integrated curricula: they don't present math in the way the teachers themselves understand it, and they may expect learning at a level beyond what the teachers know. Near the beginning of the article, Wu asserts what seems to be so obvious yet is ignored in the discussions of how to improve math education: "the only way to achieve better mathematics education is to have better mathematics teachers" (p. 535).
The main material of the article presents and assesses some sessions Wu attended for "inservice professional development," specifically the remedial development of math teachers' understanding of mathematics. (The other kind of inservice professional development mentioned by Wu is enrichment, which advances the understanding of teachers already confident in math through the high school level.) Wu lists five important considerations in remedial teacher training, with mathematical, pedagogical, and practical aspects. I was initially surprised that one must teach to teachers in the same way one should teach any other group, since I would expect the teachers to be more attentive and active in the material, but I guess just because they know how they want students to behave in their classrooms doesn't mean they'll follow those expectations on their own when learning.
Then Wu describes the activities of three sessions, with various topics, strengths, and weaknesses. He claims these "give a fair representation of the state of professional development from one segment of California" (p. 537). The first two sessions were on discrete mathematics and connections in graph theory; two of Wu's criticisms are that neither session really taught concepts key to K-12 mathematics, and that some related ideas that came up and are essential for teachers to understand were not addressed fully. The third session was on technology (appropriate, given our current concerns in class). Wu make a comment following this section that states the technology problem very clearly, I think: "The benefit of technology should not just be to save labor. It can enhance mathematics education if it is put to creative uses such as instant and easy experimentations with new ideas or providing test cases for conjectures. But a prerequisite for using technology this way is an a! dequate understanding of the relevant mathematics, and this is why one must always include the pertinent mathematics in any technological presentation" (p. 541).
This article does not deal with giving teachers ideas on how to be more effective in the classroom; it proposes to the mathematical community some necessary steps to improving all math education by better educating the teachers. I believe this is the greatest and surest way to help students succeed in mathematics.
The references at the end of the article include the author's website and articles of his that can be found there, http://math.berkeley.edu/~wu/.