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Re: Some important demonstrations on negative numbers > a MACS syllabus
Posted:
Dec 8, 2012 1:28 PM
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>From far, far, below ?
"So, Clyde, can you briefly sketch your approach to introducing negative
numbers and their rules, esp. the infamous one? Joe N "
Joe:
As requested, a sketch of a Mathematics-As-Common-Sense^TM development of
signed numbers is easily provided, below. But it has not been so easy to
fulfill your request for brevity. Over-brevity would be a call for all kinds
of questions and challenges that would call for my having to make many
direct responses. So I have selfishly included (hopefully) enough detail to
preclude most of the needs for subsequent clarifications, as best I could
anticipate them.
Although the mathematics, itself, is quite simple, the development requires
step-wise achievement of under-standing fluencies ? spread over grade-levels
2-5 ? but typically achievable with adults in about an hour ? depending on
what else they know. However, in the interest of at least some brevity, I
have occasionally resorted to formal looking expressions ? for purposes of
communicating with our readers ? so using formulae which I do not use with
mathematical infants.
Source:
For purposes of discerning how humans can most common-sensibly learn
commonly troublesome topics from curricular mathematics ... by reasoning
their own way through those topics (as shepherded by their mentors) ?
Clinical R&D uses scientific methods [called "syllabus methods" ... an
adaptation of the critical-path methods widely used in the
operations-research arena of the managerial sciences]
http://arapaho.nsuok.edu/~okar-maa/news/okarproceedings/OKAR-2005/clgreeno-part2.html .
In the interest of personal mathematical health, clinical mathematics
instruction is maximally eductive. The learners are only minimally "told"
whatever mathematics is clinically targeted for them to learn. Instead, the
clinical instructor guides the learners to progressively develop
functional, personal mathematical theories which are instructor-targeted for
the learners to own. One outcome is the clinical identification of
mathematical syllabi that are maximally commonsensible to the majority of
children and adults.
The learning-portal into operations with signed numbers is no exception.
Below is sketched
the MALEI Clinic's mathematical syllabus for guiding children (and adults)
to create "the laws of signs" as personally concluded theorems within their
functional, personal mathematical theories. This syllabus has yet to be
broadcast to the field, but eventually could guide a re-formation of the
curriculum through signed numbers. Other mathematical syllabi also might so
serve ... conceivably, even better so ... but the one, below, typically
suffices.
It was developed years ago, and since has been used with invariable success
... meaning that after the learners are so led to personally create the
"sign laws" theorems, they then can provide others with common-sensible
explanations for what the laws are, and why they work (far better than can
most authors of scholastic textbooks!).
Your request for "briefly" is my excuse for not airing any of the possible
curricula (of the kind that R.H. might prefer) that implement this
particular mathematical syllabus. Even at that, disclosure of this
Mathematics-As-Common-Sense^TM (MACS) syllabus cannot be as "brief" as some
would prefer. When you have the time, read on ... but as for "...Any key
points?", I leave such highlighting as "an exercise for the reader."
Entry (level-2):
The safest development of the arithmetic of signed numbers is to work upward
from (grade) level-2 ... even with adult students. The students are presumed
to own an adequate interval of Arabic numerals ... and the up/down (+/-)
translations along that interval. At that level, children still rely on the
mathematics of *quantities*.
[Herein, a "level-w learner" means one who has reached at least level-w.
Thus, "level-2" includes PhDs. Those who might challenge the levels for
childhood achievement within this development may choose to think of
MALEI-levels, rather than of "grade" levels.]
The laws of signs eventually are achieved through ("a child's") construction
of systems of integers ? as quantities. In the "3 apples" quantity, the
noun, "apples" is that quantity'sand denomination ( a
nomen/name for all objects of the particular *kind* (class) of things to
which that label refers). The 3 is that quantity's *numerator* ... and the
"scalar multiplier" or "coefficient" that Crabtree attends as an adjective.
In this construction, the integers evolve as quantities ? eventually,
"negative 3" is seen as "3 negatives" and "positive 4" as "4 positives" ?
using whole numerators with "positive" things, with "negative" things, and
with "neutral" things ? as with dollar-quantities in a checks-record book.
Building on level-2 additions, level-3 typically introduces whole-numbers
multiplication as repeatedly adding-on ? as with 3*5 meaning 0+5+5+5. That
kind of multiplication suffices as the mathematical footing for a childlike
construction of "multiplications" and "divisions" of integers.
However, for mathematical infants, the notion of "negative" numbers has no
meaning. It requires later achievement of the notion of "negation." Far
better to leave "negativity" out of the picture until it actually has a
meaning ... which happens *after* they have (level-5) derived the "sign
laws." Instead, those laws are hereby derived from (lower-level)
consideration ... of quantities which much later evolve into "negatives" and
"positives."
The first step upward is to achieve some kind of quantities that can
eventually become integers. The best route is to begin with a context that
is very meaningful even to the youngest. The best choice found, so far, is
[level-2] temperature ... as in "too hot/cold for comfort."
By separating the span between "freezing" and "boiling" into 100 levels,
obtain an *un-numbered* scale for comparing "how much too hot/cold." [A
"scale" is a line consisting of successive "marks" that alternate between
successive "spaces.]
Use a Celsius thermometer and a water-basin to find a personal "just right"
mark on that scale. Thereby, set "0" as meaning any "desired level" ... as
with inventories, rather than in its cardinal meaning. Set up the personal
temperature scales ? as bi-directional lines of whole-numerator/scalar
quantities ... 0(rigins), their A(bove-0) elements and their B(elow-0)
elements. As with verbal weather news, those quantities are said as in "
five above or below." The quantity-labels are soon abbreviated, as in 3A
(=3 Above) and 4B (= 4 Below).
The (vertical) personal "thermometers" are thus labeled ? 2B, 1B, 0, 1A,
2A, ?. [While not essential, it is enriching to surface that Anders
Celsius's "0" ? the freezing point of tap water ? is above Daniel
Fahrenheit's "0" ? as the freezing point of salt water ? both of which are
well below the personally comfortable "0" within the shower.]
[In the more rigorous context, vector algebra, the A-unit is (0,1) ... and
the B-unit is (1,0). Thus, 3A = 3*(0,1) and 4B = 4*(1,0). The linear
ordering of the BOA-line does not destroy the (B,A) plane ... in which
occurs the familiar "ordered-pairs construction of the integers" ... and in
which also occurs the familiar "ratio-pairs construction of the fractions."
The latter's ratio-lines (the proportions) are what eventually resolve "...
the infamous
one ..." into personal common sense (well below.)]
[Each such B0A line (of wholes-scalars quantities) is a discrete *scale* of
quantities ... but does not yet qualify it as being a "number line", until
the qualifying additions and
multiplications among the quantities are imposed. Without those operations,
the B0A thermometer-points are neither "positive" nor "negative."
Scalar operations (level-3):
The level-2 construction of BOA "thermometers" under-stands and supports the
level-3 generalization of BOA lines to other contexts ? altitude, longitude,
latitude, etc. Level-3 also is where begins the construction of operations
along the line.
Because the level-3 learner soon owns (repeated addition) multiplication of
whole-numbers, it is NOT mathematically necessary to achieve
integer-additions before achieving integer multiplications. However, the
safer route for creating multiplication of signed numbers builds on ratios
and proportions ? which identify with level-4. So, the level-3 emphasis
neglects integer multiplication while focusing on the *scalar* operations
along the BOA scales.
Every non-degenerate discrete scale [e.g. the alphabetic list of U.S. states
( http://state.1keydata.com/ )], admits to two kinds of non-trivial
translations ... the "ups" and the "downs" ... characterized by counting
the scale-spaces in between the scale's "marks." Building on the Grade-1
"plus/minus" operations, "+w" still means "up-w" and "-w" still means
"down-w." Along one such scale, Alaska+5 = Connecticut, and
Connecticut-5=Alaska.
That is where negation first arises: the -5 translation and the +5
translation negate each other (when successively composed ... a la the
ordered-pair construction of the integers ? as translations). But such
negation among *line-translations* does not make any of the line-points
(or states) "negative."
Along all lengthy B0A lines, the up/down translations yield 4A+2= 6A, and
4A-7= 3B, etc. Invoking the up/down, level-1, translation meanings of "-",
"+", "plus" and "minus" as meaning "up" and "down", makes palatable, early
in level-3, the weather-news use of "-4" to
mean "4 below 0".
Without first achieving the +/- translations of scales, it is non-sensible
to speak of
4-below as being "minus 4" ... which means "0-4" along some BOA line.
Otherwise, the "0 minus 4" phrase directly contradicts the students'
entrenched cardinal meanings of subtraction. Worse, it is irresponsible to
speak of 4-below as being "negative 4" ... because mathematical infants
presently have no meaning for "negativity" among numbers. It is even far
more
irresponsible to speak of 4-below as being "the additive inverse of 4" ...
before the learner already owns the "additions" of signed *numbers.*
Along each line of whole-numbers numerated quantities (e.g. 3 apples), the
nature of scalar additions and subtractions (na (+/-) ma) are at level-1,
and the nature of scalar multiplication is at level-3 ? as with 3*(5
apples) -> (3*5)(apples). Likewise, along the BOA lines ? 3*(5 aboves) ->
(3*5)(aboves). Of course, such scalar multiplication of quantities by
numbers does not define "multiplications" among the quantities, as such ?
no "colds*colds = hots", etc..
It is quite another thing to use BOA quantities as "balances" for tug-of-war
competitions (temperature, money, motions, etc.). Here, the translations
among shower temperatures are driven by increasing/decreasing the supplies
of hot/cold water ? adding-in hots, taking-away hots [Hey! Who started the
laundry washer while I was taking a cool
shower!#?] ? adding-in colds, taking-away colds [Hey! Who flushed the
toilet while I was taking a hot shower!#?] Just *try* to forget that
temperature model while taking showers in a busy household!
For acquiring that "balances" theory, the level-2 learners must know how to
control the
water temperature in a dual-handle water supply. With that kind of physical
(perhaps sometimes physically painful) under-standing knowledge about
adding-in & taking-away hots & colds, young children are easily led to
create their personal theories of how adding-in things of two kinds can
oppose each other ? and how taking-off such things can do likewise.
[Achieving fluency with BOA additions/subtractions with less bodily stuff
... e.g. travels, cartography, finances, inventories, etc. ... identifies
with passage into level-4.]
Level-3 learners get those "balance" rules even without using numbers. But
for the numbers, meaningful tabletop games can be created from (2-color)
playing-cards and (2-color) game-pieces (e.g. checkers, backgammon, go,
poker).
In such (level-3) "competition" systems, each B0A translation can be caused
by either of two balance-operations. 4A+2 can happen from 4A+2A ... or from
4A-2B. So are derived the "sign laws" for additions/subtractions of the BOA
quantities. From those, and from the level-3 use of repeated up/down
translations, naturally follow the construction of *scalar* multiplications
and divisions of the BOA quantities ? as with 3*(5A) -> (3*5)A ? and 18B
divBy3 -> 6B.
In the process of those scalar constructions, the conditions that 0-w
identify with wB, and that 0+w identify with wA, make it now reasonable to
use the (grade-1) "+" and "-" symbols ... and the words "plus" and "minus" ?
also as meaning "above" and "below" ? 0. Thus "-4" and "minus 4" now mean
4-below, as well as a down-4 translation. Likewise, "+3" and "plus 3" now
mean 3-above, as well as an up-3 translation.
[Using the plus/minus labels of the scale-translations, also as labels for
BOA quantities, manifests the order-isomorphism between the B0A scale and
its family of translations. It is neither a mathematical conflict, nor does
it bother students ... as long as their instructors clearly reveal what
context is being used. [As yet, there still is no sense to "positivity" and
"negativity" for the BOA quantities. There is nothing intrinsically
"negative" about cold water, or about being below sea level.]
Hm! Even when entering level-4, the B0A quantities still are not yet
"numbers."
Although we now have the additions/subtractions of quantities, we do not
yet have
the needed "multiplication" of quantities by quantities. Although we now own
"negation" among the translations, we cannot yet access "negativity" or
"positivity" of those quantities ... because those notions are married with
*multiplications* among the BOA quantities (A*A, A*B, B*A, B*B)
Ratio-nal "multiplication" of quantities (level-4):
Students' difficulties with "multiplications" and "divisions" of signed
numbers result largely from educators being shackled to the level-3
perception of "multiplication" ? as being "repeated addition." That
interpretation suffices for multiplication among whole numbers ... but not
for "multiplication" among fractions, decimal-point numerals, or signed
numbers. As seen throughout this mathteach thread, all efforts to resolve "
... the infamous one ..." on the basis of repeated addition flounder in
ambiguities and presumptions which are not at all natural for children.
The key is that repeated additions entail a "secret ingredient" which; (1)
is widely overlooked in elementary-level education; (2) is a unifying
mathematical basis for all multiplications/divisions in all number systems
attended in the core-curriculum ... including complexes; (3) is commonly
seen in the middle grades and above; (4) but is not duly recognized as being
the essence of multiplications, and so, (5) routinely is messily and
ineffectively taught in introductory algebra.
In particular, "multiplication" of integers is an extension not merely of
repeated addition of whole numbers, but even more so of its "hidden
ingredient". The common-sensibility of the sign laws for multiplication and
division is discerned by exploring that "hidden" aspect of repeated
addition.
In level-3, learners are led to perceive "multiplication" as being
"repeated addition-translations" ... as in 0+5+5+5. Their perceptions of
"division" (if any, at all) likewise begin with "repeated
subtraction-translations" ... as in 0-6-6-6-6. So, we build on those notions
as standpoints.
The phrase, "3-times-5", literally means "5, used 3 times" ... and "3 of 5"
(as in "3*5") likewise means "5, used 3 times." In both cases, the 5 serves
as a *rate* ... as in "3 units @ 5 ea." Even "3 by 5" ... the meaning of
"3-cross-5" ... can be interpreted as "take 3 @ 5 ea." Hidden under the
repeated addition-translations are their *rate-multiplications* ? from which
are derived the sign laws for "multiplying" integers.
The "n@r_ea." interpretation of repeated addition is built into the
coinage/currency system, clocks, household measuring systems, and family
purchasing activities ... plenty of "space" for level-3 learners to become
fluent with the concept. [The calculator can greatly help develop the
"n@r_ea." imagery.]
For a real-life development, consider the (levels 3-4) inventories-values
graph for "cans@ 5$ each". The ("inventory") H-axis is the scale of
cans-quantities; the ("values") V-axis is the scale of $-quantities.
The graph of "cans@ 5$ each" is a non-vertical, straight, discrete line that
begins at (0,0). All of its points are of the form, (n,5n).That line also is
the "times-5" table from the repeated application of the +5 translation.
That same line also is a *binary proportion* and its points are
*proportional ratios" ... 5-per-1, 10-per-2, etc.
Therein lies the (level-4) reason for regarding the "n@r_ea." operations as
being "rational" (as in "ratio-nal") multiplications. In all number systems
attended by the core-curriculum, their "multiplications" are of that
ratio-nal kind : @ r-per-1 ... as with e@pi-per-1 ? even where "repeated
additions" make little or no common sense.
As a preliminary hint about sign laws for "multiplications", the
"cans@5$ea." line presents that "going forward" (with the inventory of cans)
results in an "upward climb" in value ? while
"going backward" along the same line results in a "downward climb" in the
values. Hold onto that thought ... @ (the rate of) 5$ per 1 can, forward
flow forces the values upward; backward flow forces the values downward.
[Even without attending a B0A scale, a "law of signs" surfaces directly from
*applying* repeated addition, in a 2-denomination space. See where its
headed?]
Obviously, going with [ 3_cans @ 5$_per_1_can ] results in 15$. But it works
the same, regardless of the kinds of quantities. 3 apples @ 5
bananas_per_apple equates with 3*5 bananas (via "repeated addition").
Regardless of the denominations ... but by using "/" to mean "per", rather
than meaning division ... 3@(5/1) -> 15. But more revealingly,
3q@(5p/1q) -> (3*5) p.
The "secret" is that within 3*5 = 0+5+5+5 is the "hidden" process of
proportionalizing: 3@(5/1). Just as "the ugly duckling'' was revealed to be
a swan, "the infamous one" can be pleasingly re-viewed in the light of
proportionality.
That times-5 proportion passes through (1,5) as the 5-per-1 ratio ... the
5:1 or 5/1 [not division], *per unit ratio* ... whose height is the (@)
*rate* of increasing values. All other values are whole-number multiples of
5.
On the same grid, also graph the other inventories->values lines for
whole-numerator quantities. Each passes through its own (1,r) point ... and
is the "times r" table of whole multiples of r. Those (1,r) points
constitute a vertical ("tangent -to-the-unit-circle")
line through (1,0) ... so serving as per-unit-rate measures of the angles
of inclination their respective proportions. It also could be called "the @
line."
[By failing to so develop the concept of "slope" from the multiplication
tables
for the whole numbers, the makers of the algebra curriculum have made a very
simple construct to be largely incomprehensible to most children. In
"reverse Polish" format, the rational multiplication formulas for those
proportions are written "xr" ? variable, x, parameter, r, ? meaning "go
x@r-per-1". In more traditional. "Polish" format, the same functions are
written "rx" ? meaning, "@r-per-1, go x". In introductory algebra, the same
formulas are written "mx" ? where the m (meaning @m-per-1) is called the
"slope number" for that line. The same formulas also are written "kx" ?
where the rate-slope-k (meaning, @k-per-1) is also called "the constant of
proportionality."
The mx ? kx ? rx ? xr?xm ? xk ? lines are the "multiplication table"
proportions ? and also are the UNARY *rational multiplications* used
throughout the familiar systems of numbers. That aspect of
number-multiplications has been hidden for decades by educators' focus on
the less natural and more cumbersome *binary* operations. Of course, those
unary proportion-functions are used throughout the curriculum ? but rarely
for actually defining "multiplications."
The level 4 achievement of rational multiplication of quantities ... as in
"taking nQ @ (the rate of) rP-per-1Q ->( n*r)P" ... does not
require surfacing such an abstract formula with students. But the coordinate
graphing of rational multiplications (mx or xr ) by using whole-numbers
numerators sets the
stage for later grasping the common-sensibility of *ratio-nal*
"multiplication" of signed numbers.]
======= esp. the infamous one? ===esp. the infamous one? ====
Multiplications & divisions of integers (levels 4-5):
Given a level-4 fluency with coordinate graphing of families of (unary)
rational
@-multiplications ... mx or xr ... the BOA quantities lead directly to the
4-quadrant, longitude-altitude planes ... where the "multiplication" and
"division" of signed numbers becomes visually sensible.
Since we are still using only whole-number numerators for our BOA
quantities, that plane is of the "pegboard" kind. its four directions are
up(U), down (D), forward (F), and backward (B). Along the vertical, (0,BOA)
axis, the (0,A) points are upward from (0,0), and along the horizontal,
(BOA,0) axis, the (A,0) points are forward from (0,0). As usual, it suffices
to label all points by using only the numerators of the quantities.
On such a plane, each non-vertical, straight line of planar points
accommodates both forward and backward motion ... and also accommodates
either level, or upward and downward motion. [Each such line may be taken to
be the graph of some linear relationship, or to be a longitude-altitude
diagram of some straight pathway.]
An "upward" line is one along which forward motion increases the altitude
... and backward motion decreases the altitude. A "downward" line is one in
which forward motion decreases the altitude, and backward motion increases
the altitude. Thus, the "laws of signs" are common-sensibly grasped even
without reference to numbers ...(forward on uphills goes higher; forward on
downhills goes lower; backward on uphills goes lower; backward on downhills
goes higher).
What remains is to ratio-nalize those laws in terms of the BOA-quantities'
numerators. Each non-vertical, 4-quadrant proportion passes through some
point on the vertical (1A, mQ) @-line? where Qs are either A or B, and m is
a whole number, perhaps even 0. The
Q determines whether the line climbs uphill or downhill ? and the m
determines how fast. When Q is A ? as with (1A,5A) ?the proportion is
uphill. Thereon, going forward, (to nA) results in an uphill climb to (nA,
nmA) ?and going backward (to nB) results in a downhill climb, to (nB, nmB).
When Q is B, the proportion is downhill and the directions of climb are
reversed.
As for directions (signs), going any A@A/1 yields an A-quantity ?
+3@+5/(+1) = +15 ?and going any B@A/1 yields a B-quantity ... -3@+5/(+1)
= -15. Likewise (and just as common-sensibly), going any A@B/1 yields a
B-quantity ? +3@-5/(+1) = -15 ? and going any B@B/1 yields an A-quantity
...-3@-5/(+1) = +15. Its all a matter of snowmobiling back/forth and up/down
on "the slopes."
Because the Q@rP/1A operations meet the "number systems" criteria for
"multiplications", the B0A line thus becomes a line of *numbers.* Its sign
laws for such rational multiplications are A*A->A; A*B->B; B*A->B; B*B->A.
Note that using a B (as a multiplier or as a B/1 rate) for ratio-nally
"multiplying" reverses the direction of its partner ? thereby *negating* its
partner's direction. There is nothing "negative" about the Below-0 numbers,
themselves. Rather, their "negativity" lies in how rationally *multiplying*
of/by below-0 numbers reverses the directions of its partners. The below-0
numbers can be negatively used to negate the directions other numbers.
[That's a bit like how image-complexes are used for rotating the directions
of complexes.] What now are commonly called "negative" numbers are more
accurately called "negator numbers." But it might be tough to convince
traditional educators that below-0 is not "negative."
The 4-quadrant graphs of the x@(r/1) [ or (m/1)x ] proportions clearly
present those sign laws for unary *multiplications* within all linear
systems of numbers ? especially when the vertical "tangent protractor"
through (1,0) is used to @-display the (per-unit-rate) slope-numbers. [With
complex-systems, the graphics are a bit more complicated.]
Those same graphs also present the meanings of the associated "divisions" of
signed numbers ? and their sign laws. The divisions are simply to find the
proportions' per-unit rates (their slope-numbers) from the coordinates of
other ratios ? as with: for the ratio of 12B-per-4B, the @-rate of climb is
3A-per-1A.
So ends this effort, as requested, to "? briefly sketch [my] approach to
introducing negative
numbers and their rules, esp. the infamous one? [With M]Any key points?"
Hopefully, my gross violations of brevity serve for the greater clarity and
usefulness to the readers.
Cordially,
Clyde
- --------------------------------------------------
From: "Joe Niederberger" <niederberger@comcast.net>
Sent: Saturday, December 01, 2012 9:25 AM
To: <math-teach@mathforum.org>
Subject: Re: Some important demonstrations on negative numbers
> Clyde says:
>>Bottom line: student's common-sensible mathematical encounters with
>>mathematical quantities are their only (conceptual understanding) means of
>>personally deriving arithmetic and functional "numeracy." Without
> quantitative derivations, they are forced to play the (risky) game of
> tying to play scholastic "conventions."
>
> So, Clyde, can you briefly sketch your approach to introducing negative
> numbers and their rules, esp. the infamous one? Any key points?
>
> Joe N
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