[Numeracy 523] Re: Another perspective on numbers, operations, and negatives

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Kathleen Meilink meilinkk at gmail.com
Mon Aug 30 19:36:07 EDT 2010


After attending a 3-day conference on teaching math this summer at our T-TC,
I am going to using Algeblocks this year. This hands-on approach will
start, as all math should, with concrete move to representational before
moving to the abstract. Although I was always a very good math student, I
never understood WHY a negative times a negative was a positive until doing
some of the lessons from this program!! I think hands on is the way to go!!

Kathleen

On Sat, Aug 28, 2010 at 2:44 PM, Michael Gyori <michael_gyori at yahoo.com>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

>

>

>

>

>

>

> ------------------------------

> *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> <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><numeracy at nifl.gov>

>

> To

> chip.burkitt at orderingchaos.com, The Math and Numeracy Discussion List

> <numeracy at nifl.gov> <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><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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> >

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>

> --

> Ladnor Geissinger, Emer. Prof. Mathematics

> Univ. of North Carolina, Chapel Hill NC 27599 USA

>

>

>

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