Science Forums 2

[CraigD]

Originally Posted by Don Blazys

Let all variables herein represent non-negative integers. Then, the recently discovered identity:

(T/T)a^x=T(a/T)^((xln(a)/(ln(T))-1)/(ln(a)/(ln(T))-1))

If the above expression assumes the usual precedence of operations (eg: exponentiation before multiplication), then the given equation is provably false.

As Jay notes, math is much more readable using hypography’s available LaTeX package, so I’ll translate Don’s text to that for this very short proof:

removing identities (ones),

expanding and combining common coefficients,

dividing both sides of equation by ,

we find

so,

except for special cases and .

Originally Posted by Don Blazys

… shows that it is algebraically impossible to "cross out" the cancelled T's. What does this mean to you?

Because the first given is false, no further comments are formally meaningful.

Originally Posted by Don Blazys

Should we stop teaching students to "cross out" cancelled factors and common factors? I say yes, and will present my reasons for doing so as this thread continues. What do you say?

I say yes, too, but with some qualification.

From several years of tutoring college Math, a combined couple of years of teaching remedial college and GED-track high school Math, and the common parental experience of helping several now adult children (my own and others) through grades K-12, I think there’s a great need to acquaint students with the idea of mathematical formalism – that is, the idea that, as someone put it, “algebra is a collection of exact rules for manipulating marks on paper” – at nearly the earliest age possible, 5 or 6 years.

Reliance on phrases like “cross out” and “cancel”, in my experience, usually show a lack of grasp of the fundamentals of formalism, a lack that can handicap students in Math and a large collection of similar disciplines for the rest of their lives. Unfortunately, in the present-day US, and, I suspect, many other states’, schools, formalism is usually introduced only in the last years of high school or later years of college (ages 16 to 21), and then only to students with specializing in Math-intense education paths.

IMHO, a strong grasp of formalism needs to be as essential as knowing ones alphabet, and taught as early and as well. My opinion is by no means original or untried, dating back at least to suggestions from turn-of-the-20th-century mathematicians and educators, and having made sporatic widespread appearances in curriculum under monikers such as “new math” for over 50 years, as well as being taught in isolated instances in many schools and by many teachers within and outside of traditional schools. My experience with attempting to promote this approach, however, has taught me that it’s not an easy task. It’s necessary to first win the hearts and minds of professional educators, who are often themselves math-adverse to the extent of near innumeracy. Learned early and well, formalism is, in my experience, an easy concept to understand and apply, but studied late in ones education, very difficult to really grasp and appreciate.

PS: Don, you can pick up the essentials of hypography’s implementation of the LaTeX math package and markup code tags by clicking “quote” on this post and inspecting the text. More in depth documentation on LaTeX can be found in many places, including a couple of my favorites, the wikimedia article “Help: Displaying a formula” and this easy-to-use tutorial. For documentation on hypography's supported markup tags like [math], see the BB Code List FAQ page.

In math, presentation – pretty handwriting, or its computerized equivalent – can be very effective, and is IMHO worth making a strong effort to learn well in whatever medium you find yourself.