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Before ItŐs
Too Late: Glenn Commission Report |
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"Michigan State University" |
Michigan State University | |
University of Missouri | |
Western Michigan University | |
University of Chicago |
To advance the research base and leadership capacity supporting K-12 mathematics curriculum design, analysis, implementation, and evaluation. |
Doctoral Program |
Goals: | |
Increase the number and diversity of professionals prepared in doctoral programs in mathematics education at CSMC institutions. | |
Strengthen the quality of doctoral programs through collaboration and program enhancement. | |
Prepare doctoral graduates to assume leadership roles in scholarly work related to mathematics curriculum. |
Conference Announcement |
DOCTORAL PROGRAMS IN MATHEMATICS EDUCATION: A DECADE OF PROGRESS | |
Marriott Country Club Plaza | |
Kansas City, Missouri | |
September 23-26, 2007 | |
Register at: mathcurriculumcenter.org | |
Deadline: March 31, 2007 |
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"Course lecture responsibilities shared between..." |
Course lecture responsibilities shared between doctoral students and instructor (some students worked in pairs and others went solo) | ||
Each class period involved presentations, discussions, questions, conjectures and proofs | ||
Besides daily lecture responsibilities, students prepared three papers: | ||
1) Curriculum analysis of current high school textbook treatments of polynomials | ||
2) A detailed lesson plan and presentation of higher degree polynomial applications for secondary teachers | ||
3) A detailed paper on a polynomial topic not covered in this class but important for doctoral students in mathematics education |
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"The polynomial x^{2}+5x+3..." |
The polynomial x^{2}+5x+3 in R[x] induces the function f:R¨R defined | |
by the rule: f(a) = a^{2}+5a+3 for each a in R. This assignment defines an surjective ring homomorphism R[x] ¨R[a] = ring of polynomials in the variable a. | |
Remark: There is a distinction between polynomials and polynomial functions (these rings are not isomorphic in general). | |
Example: Consider the unequal polynomials x^{4}+x+1 and x^{3}+x^{2}+1 in | |
Z_{3}[x] and the induced functions | |
f: Z_{3}¨Z_{3} , f(a)= a^{4}+a+1 for each a in Z_{3} | |
g: Z_{3}¨Z_{3}, g(a)=a^{3}+a^{2}+1_{ }for each a in Z_{3} | |
f(0)=1=g(0); f(1)=3=0, g(1)=3=0; f(2)=19=1, g(2)=13=1 | |
Therefore, f=g, but the polynomials that induced these functions are not equal, (i.e., the above homomorphism is not always injective). | |
Note: The existence of such an example is clear, since the polynomial ring Z_{3}[x] is infinite, but there are only a finite number of functions f: Z_{3}¨Z_{3}. | |
Question: Is there a similar example if the coefficient field is infinite? | |
Answer: NO!!! R[x] isomorphic to R[a] iff R is infinite. |
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