Discussion Question 1
Marty
in school, involve only universal quantifiers. For example, the
Pythagorean theorem is a theorem about sides of any right triangle,
and trigonometric identities usually hold for all real numbers.
Give some examples of results in basic (high-school or below)
mathematics that require existential quantifiers. You can use the
California content standards (at http://www.cde.ca.gov/be/st/ss/mthmain.asp)
for a guideline to school mathematics.
Do you think that students assume that "solutions always exist" to
equations? Do you think that students assume that all operations can
be performed on all real numbers? At what level do you think it is
appropriate to introduce exponents involving an arbitrary rational
exponent? an arbitrary real exponent? Why?
Patrick
> Give some examples of results in basic (high-school or below)
> mathematics that require existential quantifiers.
In algebra they might see some theorems about the existence of
solutions. For example, that any cubic polynomial has a root.
Or if they do any Euclidean geometry (I did a little), you have axioms
like "for any two points, there exists a line connecting them" or "for
any line, and any point not on the line, there exists a line through
the point which doesn't intersect the given line".
Or also in algebra, they may see that for any integer, there exists a
unique prime decomposition.
Martin Weissman
(1) Please write in a more formal, less conversational, academic style with full paragraphs.
(2) Give some specific evidence. If you claim that "they might see some theorems about the existence of solutions [of cubic polynomials]", back up that claim with evidence. Find an adopted state mathematics standard or a widely used textbook that contains such a theorem. How does such a source talk about "existence of solutions"? Does it mention existence, or does it simply give a formula? Do not simply rely on personal recollections.
(3) At what stage do students learn about existence and uniqueness of prime factorization? They often learn how to perform such a factorization early -- long before such a theorem might be mentioned. If you think that such a theorem is mentioned, find a source. How is "existence of factorization" expressed in this source?
(4) In Euclid's treatment of geometry, the existence and uniqueness of a line through two given points is stated as an axiom. Is this done in textbook treatments of geometry, or is it just assumed?
- MW
Jen Mogel
equations?
The way students are introduced to problem solving in the seventh and
eighth grade leads many to live under the impression that there is a
solution to every equation, at least at first.
According to the California State Board of Education, students are
first introduced to solving equations of one variable in the seventh
grade math curriculum. There, except in one instance, there is a
solution to every equation. Even in Algebra 1, again according to the
State Board guidelines for the course, the students begin their
equation solving with linear equations. Before they learn about
solving quadratic, cubic, or other higher order polynomial equations,
students are first taught to factor. Then from there, use their
factoring to solve polynomial equations. Only later, when students
are taught to use the quadratic formula in Algebra 1 or logarithms in
Algebra 2, will they come across the possibility that there could be
no solution.
Even the way solving polynomial equations are taught leads students to
believe "there is always a solution." The word "solve" comes plays a
large part in this. Algebra 1 guideline 4.0 states: "Students
simplify expressions before solving linear equations and inequalities
in one variable, such as 3(2x-5) + 4(x-2) = 12." Or 14.0 states:
"Students solve a quadratic equation by factoring or completing the
square." This leads students to think that polynomials will always
have a solution, even if they can't find it.
At what level do you think it is appropriate to introduce exponents
involving an arbitrary rational exponent? An arbitrary real
exponent? Why?
After students learn whole number exponents, they are taught the
square roots, and other roots, (Number Sense sections 1.2 and 2.4,
Algebra and Functions section 2.2) using the inverse method. In other
words, teaching the students to find the square root of 4 by asking
themselves what whole number's square is four. I think that this
would be an appropriate time to introduce using rational number
exponent notation and how the two notations are interchangeable. If
students get used to this notion early, teaching them how to simplify
roots will be more natural, since they learn exponential operations
when they learn exponents. Rational powers are currently covered in
Algebra 1.
Exponents are first taught as an extension of multiplication, two
squared is two times two, so trying to teach a student to
conceptualize what 2 to the power of 2.1 or negative 2 is can be
challenging. It is possible to introduce arbitrary real exponents
when you work with exponents in the beginning, working with the
exponential rules in Algebra 1 and 2. This way, students don't get
stuck in the idea that exponents are always whole numbers. Students
will learn to use the exponential rules for any type of number in the
exponent.
Ref: California State Board of Education, Academic content standards
for kindergarten through grade twelve: http://www.cde.ca.gov/be/st/ss/mthmain.asp