[Numeracy 523] Re: Another perspective on numbers, operations, and negatives
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Mon Aug 30 19:36:07 EDT 2010
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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
> >
> > ----------------------------------------------------
> > National Institute for Literacy
> > Math& Numeracy discussion list
> > *Numeracy at nifl.gov* <Numeracy at nifl.gov>
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> http://www.nifl.gov/mailman/listinfo/numeracy
> > Email delivered to *chip.burkitt at orderingchaos.com*<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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