>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.
- -------------------------------------------------- From: "Joe Niederberger" <email@example.com> Sent: Saturday, December 01, 2012 9:25 AM To: <firstname.lastname@example.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