[Numeracy 530] Re: negative numbers as a way of thinking
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Thu Sep 2 11:45:10 EDT 2010
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There is a autobiographical story by a man named Jesus Colon, whose
writings are collected in a book called "A Puerto Rican in New York." The
name of the story is "Kipling and I". It details Colon's early work
years, living in a squat and going from job to job, often not getting
hired because of his skin color or accent, and losing jobs because he
refused to be exploited. All thoughout this recollection, he talks about
the inspiration he drew from a framed copy of Rudyard Kipling's poem "If",
which he has hanging on the wall. He reads the poem everyday, commits it
to memory, even recites it to himself under his breath on the subway. And
then one particularly cold evening, he is freezing (dare I saw the
temperature may have been below zero) because there's no heat. He looks
around and can't find any paper or wood. He looks up at the poem for what
he describes as an eternity, and then he takes it down, breaks up the
wooden frame and burns it for heat.
In discussing the story with students, it often comes down to this
question of "day to day real life value" - to what end was the poem more
valuable in that sense - as a source of spiritual strength or as kindling?
For me, the answer is both and neither. The poem continues inside of him
- on one level, he had committed it to memory, on another level it had
already seen him through so many dark times, and on another level it's
message of how one should interact with the world had shaped the man he
had become. But on a cold night when his hands were so numb, such
abstractions were not his priority. One way (of many) I read the story is
that the value of the poem itself is the way it opened his mind to a
particular way of thinking.
In math class, I am interested in helping my students not only master
specific mathematical concepts and relationships, but also a sense of what
it means to think mathematically.
I could tell my students that in a circle, the circumference is equal to
pi times the diameter. But instead, I give students rulers, string,
circular objects (records, frisbees, wall clocks, cd, cans, quarters, etc)
and the words for referring to the distance around each circle and the
distance across each circle, passing through the center of each circle. I
then give students time to measure the circumference and diameters of all
their circles - part of which is them figuring out how to measure the
circumference (it is with the string). Then we share our findings and I
add students' data to a giant chart in front of the room. After the C and
d for each circular object is up, I ask them if they see any patterns in
the numbers. Students see that the circumference is always about three
times larger than the diameter (it happens, but is far less common for
them to say that the diameter is about 1/3 the circumference.
Will any of them ever have to use this knowledge of the relationship
between the diameter and circumference of a circle? I don't know. I
suspect few will ever have to use it to find the length of a bridge across
a circular lake, given the distance around the lake (a common test
question in this vein). But because the activity goes beyond a mere
depositing of information, and I am asking students to think and observe
and notice, I know that it is having some effect on the way that some of
my students think about a whole range of things.
Negative numbers came out of the same place as words - from a need to
express something about observed relationships between things in the
world. They are part of a way of thinking that is far more interesting
and applicable than the mere rules for their manipulation with operations.
Negative numbers are all around us, and our students, in the ways the
world is discussed.
- When we study history and talk about the year 500 BC, that year is an
answer to simultaneously considering "How many?" and "Which way?". It is,
on a historical timeline, essentially -500.
- I understand in Hawaii, your students might not have to deal with
temperatures below zero, but I think it is useful for them to be able to
look at a weather map of the US and understand what it means when another
region has negative temperatures. Besides, can anyone fully appreciate
the tropical bliss that is living in Hawaii without understanding just how
cold a midwestern winter can be?
- Last spring a student of mind brought in a graph he saw in the paper
which was dealing with population changes in different regions of the
world - the graph had actual numbers, but the shifts were also described
as percent changes - and some regions of the world have experienced a
negative percent change.
- An understanding of negative numbers as a way of thinking also came up
when we were looking at the effects of climate change on sea level on
coastal regions of the world which are below or close to sea level - the
whole idea of sea level is again connected to the idea of "How many?" and
"Which way?"
- my last example of the day - for your enjoyment, consider the way
negative numbers are tied into our moral code.
Consider the rules for multiplying (and dividing) with signed numbers:
a positive times a positive is a positive
a negative times a negative is a positive
a positive times a negative is a negative
a negative times a positive is a negative
Change "positive" to "good" and "negative" to "bad" and add the word
"people" and you get:
When good things happen to good people, it is a good thing
When bad things happen to bad people, it is a good thing
When good things happen to bad people, it is a bad thing
When bad things happen to good people, it is a bad thing.
Isn't that something we can all relate to in our day to day interactions?
Mark
Michael Gyori <michael_gyori at yahoo.com>
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08/31/2010 10:41 AM
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[Numeracy 524] Re: Another perspective on numbers, operations, and
negatives
Hello Mark,
You state, and rightfully so, that:
Without understanding the concept of negative numbers, how can students
answer, "In Chicago, Illinois, Monday’s temperature at 3 p.m. was 11
degrees. If it dropped to a low of -3 degrees that evening, how many
degrees did the temperature drop?"
Without understanding the concept of negative numbers, how could students
understand the distance between the top of Mt. Everest (at 29,028 feet
above sea level) and the Dead Sea shore (at 1,371 feet below sea level)?
Would you comment on why you believe such understandings might be useful,
let alone necessary? What is the day-to-day life value of such
understandings? I ask as someone who even spent time at the Dead Sea
shore... :)
I'm asking from a place of true curiosity and I hope that is not in doubt.
Thanks,
Michael
Michael A. Gyori
Maui International Language School
www.mauilanguage.com
From: "Mark.Trushkowsky at mail.cuny.edu" <Mark.Trushkowsky at mail.cuny.edu>
To: The Math and Numeracy Discussion List <numeracy at nifl.gov>
Sent: Mon, August 30, 2010 8:01:03 AM
Subject: [Numeracy 522] Re: Another perspective on numbers, operations,
and negatives
Michael,
The basic concepts of negative numbers are necessary when you need to know
how far you need to go to get back to zero. That happens in real life
situations all the time.
One fun way you can raise this concept with students is through playing a
game of Math Jeopardy.
There are a million ways to vary the game to fit your needs, but it
basically works like Jeopardy, with a set of given answers within a series
of categories. I use post-it notes or index cards taped to the wall.
Students choose a category and then they choose an answer based on its
monetary value and then they have to come up with a question that fits
that answer.
I have also done it, reversing the rules of Jeopardy, when the cards have
questions and students have to come up with the answers.
Usually when I do this, the concept of negative numbers is not the theme
of the categories. But they become necessary for groups to talk about
when say for example a group with 400 points gets a 1000 point question
incorrect - they need to understand that they are at -600, which is to say
they need 600 points to get back to zero. Sometimes I offer a team that
needs points some bonus questions like, "You are currently at -400 points.
I'll give you 300 points if you can tell me how many 200 point questions
you would need to have 600 points?"
You might also have students work on a project where they are figuring out
profits. Say you buy a boat for $35,000. Then you put an additional
$15,000 into it to raise its value. In terms of your profit, you are at
-$50,000. Maybe the boat market is slow and the best price you can get
would be $45,000. In terms of profit, you would then be a -$5,000.
Again, I go back to Moses' idea that students need to be able to ask not
only, "How many?" but also, "In what direction?"
Without understanding the concept of negative numbers, how can students
answer, "In Chicago, Illinois, Monday’s temperature at 3 p.m. was 11
degrees. If it dropped to a low of -3 degrees that evening, how many
degrees did the temperature drop?"
Without understanding the concept of negative numbers, how could students
understand the distance between the top of Mt. Everest (at 29,028 feet
above sea level) and the Dead Sea shore (at 1,371 feet below sea level)?
Mark
Michael Gyori <michael_gyori at yahoo.com>
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08/29/2010 11:49 PM
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[Numeracy 516] Re: Another perspective on numbers, operations, and
negatives
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>
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08/27/2010 04:37 PM
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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
>
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--
Ladnor Geissinger, Emer. Prof. Mathematics
Univ. of North Carolina, Chapel Hill NC 27599 USA
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