### 5/8 of MEdZ 0.3

“algebra 1” students in enormous numbers
are confronted every semester (or quarter)
with a survey-of-number-systems.

${\Bbb N}, {\Bbb Z}, {\Bbb Q}$, and ${\Bbb R}$

…(to give them their standard symbols…
as the textbooks, more or less of course,
do *not* do [effectively; sometimes
they *gesture* at these symbols])…

denote the sets of
Natural numbers, integer numberZ,
rational (Quotient) numbers,
& (so-called) Real numbers.

the texts then go on to *ignore* their own “survey”.

& students get worse-than-nothing out of it.
in many cases, they’ll have seen this treatment
*many* times (and become *worse* prepared to
think about the rest of the course material
[about “graphing” and “factoring” and so on]
*every* time).

then there’re these exercises. wherein one is
required to say, about *each* of a (given) collection
of numbers, which of our “number-sets” include it
(and which don’t)… for instance,
“pi” is real but not natural, integer, or rational
“-11” is integer, rational, and real, but not natural
etcetera. right on the front page of the final…
the same damn *problem*, verbatim, for years,
in at least one instance known to me… and yet
substantial subsets of *every* class of students
miss these incredibly-easy-if-the-mere-vocab-
ulary-is-understood “exercises”.

because the one thing they know for sure
after all this time is “i will never understand
what any of these words mean” (so “just show me
how to ‘do’ the problems”).

whereas.
rational number arithmetic is routinely taught
to children in functional schools (and families).
with a certain amount of effort, of course.
but it’s “easy” enough if it’s done *clearly*.

so here’s something that *ought* to be in the text.

not necessarily *instead* of
$\{ {n\over d} : d \not= 0, d$ and $n$ are integers $\}$“,
but certainly somewhere *nearby*
(if the deliberately-obfuscatory
high-theory “set-builder”
thingum
*must* somehow be included in our treatment).

namely, a table (showing possible numerators
and denominators & their Quotients), such that
${\Bbb Q}$ is “all the numbers in the table
(and their opposites)”.

(the little “$\pm$” icon at upper-left
is meant to indicate that, e.g. “-3/2”
(the “opposite” [or “additive inverse”] of “3/2”)
should be considered to be “in the table”…
this is a nuance, improvised for the zine
[and if i were giving this lecture today
at the blackboard, i’d very likely draw
out another part of the table and include
some negative rational numbers explicitly].)

there it is. we’re looking right at it.
*now* we can talk about it.
(and, say, how various “decimals”
do or don’t “give us” a
“number on the table”.)

(secretly, of course, “we” know this means
“a number that can be represented as [an]
integer-over-rational-number”… but
it does no good to keep on *saying so*
once we know darn well we’re being
“tuned out”.)

now. about those dotted-line circles.
(… get the student to talk…)
hey, “lowest terms”. wow, cool.

same idea, different picture.
each “lowest terms” ratio appears
as the *nearest point to zero*
on a “line” having the “slope”
represented by that number.

(by “nearest” [point to zero],
one here means “nearest on the
‘integer lattice’ ${\Bbb Z} \times {\Bbb Z}$“…
but of course one will not
[necessarily] *say so* explicitly;
this is… or was… a “lecture
without words” more-or-less precisely
*because* the real work is getting
the *student* to do a lot of the
talking [and pencil-moving];
“nearest *on the diagram* is sufficient
for our purpose [a clear view
of the set-of-rational-numbers,
in case you’ve forgotten].)

you’ve gotta give ’em hope.

#### 1 Comment

1. don’t point that thing at me
unless you intend to use it.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)