[Numeracy 516] Re: Another perspective on numbers, operations, and negatives
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Sat Aug 28 14:44:05 EDT 2010
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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
________________________________
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
>
>
>
>
>
________________________________
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
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>>
>----------------------------------------------------
>National Institute for Literacy
>Math & Numeracy discussion list
>Numeracy at nifl.gov
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>
>
> ---------------------------------------------------- National Institute for
>Literacy Math & Numeracy discussion list Numeracy at nifl.gov To unsubscribe or
>change your subscription settings, please go to
>http://www.nifl.gov/mailman/listinfo/numeracy Email delivered to
>ladnor at email.unc.edu
>
-- Ladnor Geissinger, Emer. Prof. Mathematics Univ. of North Carolina, Chapel
Hill NC 27599 USA
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