[Numeracy 519] Re: Another perspective on numbers, operations, and negatives
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Mon Aug 30 01:33:47 EDT 2010
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I'll jump in here. One reason for using negative numbers is that they
help answer mathematical questions that can't be answered with positive
numbers. (The same is true of the rational numbers and real numbers.)
The positive integers suffice for doing addition and multiplication.
When we introduce subtraction, however, we also introduce a problem: not
all subtractions yield positive integers. There are two ways to address
this problem. One is simply to forbid subtracting larger from smaller
numbers. This is essentially the approach we use in elementary school
before we introduce negative numbers, and it appears to be the approach
that Michael finds appealing. The other approach is to extend the number
system to include negative numbers. Allowing the negative integers gives
closure to subtraction just as addition and multiplication are closed.
(Addition of two positive integers always yields another positive
integer. Likewise, multiplication of two positive integers also always
yields a positive integer. However, subtraction of two positive integers
may not yield a positive integer. By extending the number system, we now
have a way to evaluate expressions such as 15 - 25, and addition and
multiplication are still closed in the extended system.
Like subtraction, division is the other problem. When we first learn
about it, we start with "undoing" multiplication, so we are careful to
use only divisions that will yield whole numbers. Unfortunately, there
is no simple comparison that will tell us whether an expression will
evaluate to an integer. With a few exceptions, we have to actually carry
out the division in order to determine whether the result is a whole
number. Moreover, no matter how we try, we can't make 0 fit into our
division scheme. Division by 0 is simply forbidden. It causes too many
intractable problems. Once we realize that some division problems don't
yield whole number answers, we introduce the concept of remainder. The
remainder allows us to continue using whole numbers for a while.
However, there are plenty of situations where whole numbers are simply
not enough. We extend our number system again to include fractions: the
rational numbers. Ignoring 0 as a divisor, the rational numbers are
closed under division. Every division expression between two rational
numbers (excluding 0) yields a rational number as a result.
This is a mathematical explanation for why we need negative numbers and
fractions. It may not be very satisfactory to someone who wants to know
how they will benefit them in their daily lives. Perhaps someone else
has a more concrete explanation.
Chip Burkitt
On 8/28/2010 1:44 PM, Michael Gyori wrote:
> Hello Ladnor and and Mark,
> I understand more clearly now what both of you have shared. Thank
> you. All that said, I am still undecided about how a spatial rendering
> of negative integers (e.g. the Boston transit) would ultimately
> benefit ABE/GED students, except in the event of a need to perform
> operations on integers correctly on a test.
> What is the real-world applicability of integers for someone whose
> life does not revolve around the abstract in part - as yours, for
> example, as mathemeticians, and mine and as educational linguist?
> That is the question I still haven't answered to my own satisfaction.
> I appreciate your contributions very much!
> Michael
>
> Michael A. Gyori
>
> Maui International Language School
>
> www.mauilanguage.com <http://www.mauilanguage.com/>
>
>
>
> ------------------------------------------------------------------------
> *From:* Ladnor Geissinger <ladnor at email.unc.edu>
> *To:* The Math and Numeracy Discussion List <numeracy at nifl.gov>
> *Sent:* Fri, August 27, 2010 7:48:44 PM
> *Subject:* [Numeracy 515] Re: Another perspective on numbers,
> operations, and negatives
>
> Let me expand a bit on Mark's reference to The Mathematics of Trips,
> where positive and negative numbers are used as labels for the number
> of stops going outbound or inbound on the subway, and the arbitrary
> choice of the benchmark stop for dividing inbound from outbound.
>
> In many board games that kids play there is a path of squares that
> must be followed with your game piece to get to the final goal, and
> one can pick up cards from a shuffled deck that say move ahead 5 or
> move back 3. So when they get to school it wouldn't be a big hurdle
> if the teacher suggested that we give simple names to such moves, say
> M5 and M-3, etc. If we first do M5 and then do M-3 we could write
> that down simply as M-3 M5. And then it would be clear that this
> combination is equivalent in its effect on our game piece to M2, no
> matter where the piece was when we started these moves. So it might
> be convenient to write M-3 M5 = M2. Similarly M5 M-3 = M2. And more
> generally Mk Mj = M[k+j]. Working on this more we see that for any
> string of Moves that we can write down, there is a single move Ms that
> has the same effect and the label s is simply the usual addition sum
> of the integer labels on the string of moves. So addition of
> integers can be used to simplify our description of the result of a
> string of moves.
>
> Of course the same thing holds for motion along a path of any kind
> where there is a sequence of marked stops or positions, or moving
> along a railroad track where 1 step goes from one tie (or city) to the
> next, or moving up and down in an elevator where a single step gets
> you from one floor to the next above or below, etc. Our natural
> mathematical model for this is a motion along a geometric line, where
> usually we choose the positive direction to be toward the right or
> upward, and the opposite is labeled the negative direction. Often we
> choose a unit distance u along the line and M5 then might mean move 5
> unit lengths in the positive direction, while M-3 means move 3 unit
> lengths in the negative direction. Now suppose we also choose an
> arbitrary base point and label it 0. If we start at 0 and make the
> move M5 we can label the resulting point 5, and similarly for any
> positive or negative integer. That is how we get our basic Cartesian
> number line. Now we can ask where we get to if we start at position 7
> and make the move M5, well of course, we get to position 12 = [7+5].
> So now addition is giving us the final position 2 = [5+(-3)] if we
> start at -3 and move M5 or if we start at 5 and move M-3. And that
> leads to questions like, if we start at position k and we want to get
> to position n, what motion do we need to make? Ah, M[n--k], so now
> subtraction tells us something useful.
>
> Exactly the same thing works for motions in the plane or 3-space. A
> vector v is determined by a distance and a direction (an arrow), and
> we can let Mv be the motion which carries any point P into a point Q
> where the arrow from P to Q has the same direction and length as v.
> Then the composite of two such motions Mv and Mw is the motion with
> label v+w, the vector sum. etc. And we can choose an origin 0 and a
> Cartesian coordinate system, etc.
>
> In other settings we can label simple operations with integers and
> there integer multiplication will play a useful role in simplifying
> things.
>
> Ladnor Geissinger
>
> On 8/27/2010 6:55 PM, Mark.Trushkowsky at mail.cuny.edu wrote:
>>
>> Michael,
>>
>> I would highly recommend reading "The Mathematics of Trips", which is
>> the Appendix of Robert Moses' book,"Radical Equations: Civil Rights
>> from Mississippi to the Algebra Project". His work was with
>> children, but I think his practice speaks a lot to our work with
>> adult ABE and GED students.
>>
>> Moses starts off the appendix writing about a student named Ari,
>> whose method of working with signed number was to ignore the signs.
>> If Ari was asked 5 + -7, he would convert it to 5+7 and give the
>> answer 12. It occured to Moses that this was because Ari only had
>> one question on his mind about numbers: "How much?" or "How many?".
>> Moses realized he needed to find another question to help out his
>> student. Eventually he came to the other kind of question he wanted
>> to put in Ari's mind: "Which way?". Moses says that "Ari, like
>> everyone else, already has this kind of question, but Ari had not put
>> it together with the "how many" question about his concept of number.
>> The next question that Moses had to answer was how to develop that
>> question in Ari's understanding - his inspiration came from the mass
>> transit system serving the Boston metropolitan area where he was
>> working. If you are unfamiliar with the Boston T system, it
>> addresses the question of "Which way?" because it has trains that are
>> "inbound" or "outbound".
>>
>> The rest of the appendix talks about how he uses a simple
>> conversation about the heights of two students to change his class'
>> idea about subtraction - not just a tool to talk about the difference
>> in people's heights, it can be used to keep track of the position of
>> two measurements relative to each other. He then moves to describe
>> his wonderful work to help students synthesize their verbal
>> description and written diagrams of their trip on the Boston T into a
>> deep understanding of the subtraction of signed numbers and a
>> relationship with symbolic representations of numbers coming after
>> they were already comfortable with the concepts (student understood
>> the "Which way?" aspect of number concepts through their experiences
>> on the inbound and outbound trains). This leads his students to
>> understand the need for negative numbers because they need them to
>> answer "In what direction and how many stops is Park Street from
>> Central Square?", and "In which direction and how many stops is
>> Harvard Square from Park Street?" One of this things that is
>> interesting is that students can choose from between 5 train stops to
>> be the benchmark dividing inbound from outbound. But of course,
>> whichever they choose, the answer to the question is the same. That
>> made me think back to the idea that someone raised about temperature
>> not working because the benchmarks dividing positive and negative
>> were arbitrary because freezing could be zero degrees or 32 degrees.
>> But consider 5 degrees above 32 (37) and 6 degrees below 32 (26
>> degrees). Isn't 26-37 is the same as -6-5? What seems important is
>> that student recognize the benchmark and then recognize that they
>> need to find how many and which way.
>>
>> Mark
>>
>>
>>
>>
>>
>>
>>
>> *Michael Gyori <michael_gyori at yahoo.com>*
>> Sent by: numeracy-bounces at nifl.gov
>>
>> 08/27/2010 04:37 PM
>> Please respond to
>> The Math and Numeracy Discussion List <numeracy at nifl.gov>
>>
>>
>>
>> To
>> chip.burkitt at orderingchaos.com, The Math and Numeracy Discussion
>> List <numeracy at nifl.gov>
>> cc
>>
>> Subject
>> [Numeracy 513] Re: Another perspective on numbers, operations,
>> and negatives
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> Hi Chip and Ladnor,
>>
>> I think the overarching question might be whom we're teaching math to
>> and why. My inclination, for the purposes of a list devoted to
>> numeracy (mathematical literacy), is to consider a more abstract
>> consideration of negative integers as exceeding the needs of the
>> population of learners and their teachers this list is perhaps
>> intended for.
>>
>> That said, Ladnor: if you had learners well below college level,
>> would you be able to frame your thoughts below in a way (perhaps by
>> way of concrete examples) that would inform an actionable syllabus
>> for ABE teachers and learners alike?
>>
>> Thanks,
>>
>> Michael
>>
>> Michael A. Gyori
>>
>> Maui International Language School
>>
>> _www.mauilanguage.com_ <http://www.mauilanguage.com/>
>>
>>
>>
>>
>> ------------------------------------------------------------------------
>> *From:* Chip Burkitt <chip.burkitt at orderingchaos.com>*
>> To:* numeracy at nifl.gov*
>> Sent:* Sat, August 21, 2010 3:35:26 AM*
>> Subject:* [Numeracy 508] Re: Another perspective on numbers,
>> operations, and negatives
>>
>> I'm not familiar with Mathematics as the Study of Patterns, but it
>> makes sense. What attracted me to mathematics in the first place was
>> the very powerful abstractions which could be manipulated
>> independently of any connection to concrete reality. However, I think
>> most adult learners who have struggled with math in the past do not
>> find that aspect of math attractive. They find it daunting. For them,
>> the more we as teachers can connect math concepts to known, concrete
>> reality, the more our students will understand. I have students who
>> struggle with problems such as this: "A box holds 24 cans of soda. If
>> you pack 729 cans of soda into boxes, how many cans will be left
>> over?" Some struggle because they don't know how to do division. Some
>> struggle because they don't know that division is required. Some
>> don't understand why there should be any cans of soda left over. For
>> these students, I would like to be able to give them boxes and soda
>> cans and have them solve the problem mechanically, but that usually
>> proves impractical. Nevertheless, I try to always make explicit what
>> abstractions I'm making and why.
>>
>> Chip Burkitt
>>
>> On 8/20/2010 3:44 PM, Ladnor Geissinger wrote:
>> > I think that some of the comments about equality, negative numbers,
>> > and operations such as those in Numeracy 505 and predecessors in that
>> > thread are a bit skewed and in some ways ask unreasonable things of
>> math
>> > tools. My intent here is to give a slightly different perspective to
>> > the discussion.
>> >
>> > Most of us have seen in print the brief description of Mathematics as
>> > the Study of Patterns. That is, math tools are distillations of
>> > patterns that people have drawn out of (i.e., abstracted from) the
>> > study of a great variety of physical settings. They are mindtools
>> > constructed to aid in analysis of phenomena and prediction of outcomes.
>> > When we come upon some situation where our previous math tools don't
>> > seem to apply directly or do so but only quite laboriously, then we
>> > either generalize the old methods to a new class of settings, or if
>> even
>> > that doesn't give us efficient analytical methods then we invent
>> > something new. This may lead to hubris, to thinking that the new tool
>> > should apply everywhere: invent a hammer and everything looks like a
>> > nail. Negative numbers were invented for some specific purpose, and
>> > then it turned out they are very convenient to use for many other
>> > purposes, but we shouldn't expect them to be useful in all settings
>> > involving measurement or finance.
>> >
>> > The idea of natural numbers gradually condensed over a very long
>> time as
>> > a way to record how many things there are in a collection of objects,
>> > especially large collections where simple tallies are not efficient.
>> > Numbers don't have to be imbued with any fancy metaphysical existence
>> > to make them a useful tool. All we need is a simple way to
>> generate and
>> > write down numbers that we have all agreed we will use to label sets of
>> > items at the end of a standard "counting process". That is, we start
>> > with the ordered list of digits and we learn how the "next number
>> > algorithm" works and so we get the strictly ordered sequence of
>> numbers:
>> > 1,2,3,4,5,6,7,8,9,10,11,12,13,... .
>> > Now every other set S of objects is compared to the numbers by the
>> > counting process -- arrange the items in S in order one after the other
>> > at the same time pairing each new item with the next number. When you
>> > run out of items, label the set S with the last paired number and call
>> > it "size of S" or the "number of items in S". Great, so now we can
>> > count and record our results. But actually we can do more. It is easy
>> > to decide which of two numbers s and t is the smaller, that is, comes
>> > earlier in the number sequence. So when we count two sets S and T and
>> > get sizes s and t respectively, we can decide which of the sets has the
>> > smaller number of items in it. If I need to keep records of some
>> > standard inventories of different types of objects, I will also find it
>> > convenient to use 0 to indicate having none of some type -- so I've
>> > effectively added 0 to the counting numbers.
>> >
>> > But suppose I am the king's accountant and two people bring in big bags
>> > S and T of coins which they have independently counted, and they
>> tell me
>> > the bags have s and t coins. I need to be able to record the total
>> > number of coins. Can I do no better than to "count on", essentially
>> > count the coins in T but beginning the pairing with the "number after
>> > s", and so effectively count the whole combined collection of coins.
>> > Now is the time to invent "addition" of numbers, which we indicate
>> > briefly by using + and call the result the sum. From basic counting
>> > principles we can prove the elementary properties of addition, and then
>> > we can begin to work out practical algorithms for computing sums.
>> So in
>> > effect addition is fast counting -- it allows us to replace actual
>> > counting of the combination of two sets by the operation of addition of
>> > numbers, which we expect will be more efficient.
>> >
>> > One could give a similar description of multiplication as another form
>> > of fast counting when there is regularity ( What is the total count if
>> > we have n bags each having k items? n*k).
>> > Then maybe for measuring quantities more precisely we may find it
>> useful
>> > to invent fractions and fill in spaces between counting numbers. And
>> > very soon people will wonder if they can generalize addition and then
>> > multiplication to these "rational numbers", and will the results be
>> > useful in some settings?
>> >
>> > I have already described in an earlier email note [Numeracy 478?] a
>> > geometric situation that could lead to introducing negative numbers.
>> > Then various uses of this idea would lead to investigating how to
>> > generalize addition and multiplication to the negative rational
>> > numbers. In the history of development of mathematics we are now at
>> > about the time of Vieta and Descartes (1637) and the promulgation
>> of the
>> > idea that geometry could be done very efficiently using real
>> numbers and
>> > coordinate systems [analytic geometry]. Soon it became clear that
>> > efficiency and ease of use required the usual rule of signs
>> (-1)*(-1) = 1.
>> >
>> > I'll end there and hope others will comment on this and further the
>> > discussion.
>> >
>> > Ladnor Geissinger
>> >
>> > ----------------------------------------------------
>> > National Institute for Literacy
>> > Math& Numeracy discussion list
>> > _Numeracy at nifl.gov_ <mailto:Numeracy at nifl.gov>
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>> http://www.nifl.gov/mailman/listinfo/numeracy
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>> <mailto:chip.burkitt at orderingchaos.com>
>> >
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>
> --
> Ladnor Geissinger, Emer. Prof. Mathematics
> Univ. of North Carolina, Chapel Hill NC 27599 USA
>
>
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