And Left Me In Reputeless Banishment

The importance of definitions to our discussions cannot be overstated.

But in the oh-so-often-so-called “trenches”
of the mathwars… the classrooms and
one-and-one tutoring sessions that
have made up my working life for
a quarter-century…
definitions are made not to matter.
“Textbook exercises” like those of
Tuesday’s ramble will have
appeared sufficient for our student’s work
so far and they’ll deeply resent and resist
attempts to persuade them that
technical terms used correctly
{\it by themselves} are necessary
before we’re doing Mathematics at all
(as opposed to “scribal” work).

Anything that gets students to produce mathematical code, with their own hands, that they’re prepared to actually {\it talk} about, is a win in my book. And there’ll be much to talk {\it about}.

Handwriting again, for example. Since the upper-case “A” and lower-case “a”
have different meanings in our context, a tutor will have opportunities to urge
students to be {\it clear} in their use of handwriting {\it where it matters}:
one should ideally have several handwritten typefaces.
I myself have upper-and-lower case “print” and “script” faces
for example and have used ’em all since about 5th grade.
They ain’t pretty and I’ve never had much reason to care.
What they are is distinguishable and legible.

Many students
under-rate such skills to the point of
actual hostility to the whole idea.
Generally they’ll do anything they can
to get out of actually moving their pencils at
this stage and this looks like just one
more silly doggone obstacle thrown
in the way. There’ll have been
some trusted source in their background
telling them that upper-and-lower-case
{\it doesn’t} matter. And who are they
gonna believe: a loved one or
a math teacher?

Exercises with “A” and “a” having different meanings
provide “teachable moments”. Anybody can easily
see that it matters {\it for now}. If the student appears
actually to {\it care}, one can open it up into
an important generalization by considering,
say, a triangle having capital (roman) letters
for points and lower-case (italic) letters
for sides. (These are widespread conventions,
by the way, and should be introduced
as soon as possible and
maintained wherever convenience
allows.) What about the {\it angles}?
Well, conventionally one has
{\it Greek} letters… and it’s perfectly
easy for a willing mind to see
that this is {\it good idea} since,
for example, betas look like bees.

Page layout.
A display along the lines

would be harder to read but cheaper (by taking up less space on the page).
The “tall” presentation much more clearly reveals the “structure”
of this string of symbols (or so it appears to me)…
specifically, the pairings for the “set braces” are
more obvious.

Set-theory problems are rife with page-layout issues.
Lots of really good math arises if one or more of
the teacher and student refer {\it reasons} along with
all the this-is-how-you-do-it.

The {\it student’s} reasons are the ones that matter.
Administrators cannot believe this or they’d vanish
like some matter-antimatter annihilation.
Hence, “constructivism”: an elaborate system
of speaking of this fact-that-dare-not-speak-its-name.

University maths for highschool students: cut out the middlemen.