[Numeracy 538] Re: Another perspective on numbers, operations, and negatives
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Tue Sep 7 17:43:27 EDT 2010
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I help my students see the correlation between numbers and words, each with
their own pieces or parts. I write "word" on the board, and a 3 or 4 digit
number. I ask, what are words made up of? Letters of the alphabet are the
pieces or building blocks that words are built out of. So then I show them
that digits are the "alphabet" of numbers. This leads into a discussion
about place value and our base-ten number system. This introduction helps
them see what digits are, and why their place value is important.
Leslie
On Sat, Sep 4, 2010 at 6:43 PM, ladnor <ladnor at email.unc.edu> wrote:
>
> I don't see what is confusing or difficult about the relation between
> "number" and "digit". The digits are of course the first ten whole numbers
> (starting with 0), and then the rest of the numbers are constructed out of
> the the digits by the simple process of forming strings of digits. So that
> after 9 there are the 2-digit numbers 10, and then 11, and then 12, ...
> and after 99 come the 3-digit numbers -- all built up successively in the
> way we all understand [I call it the next-number algorithm, but you can
> call it what you wish]. Can anyone tell me what a student might find
> simpler as a statement of what a number is, than a string [list] of digits
> (and of course the string shouldn't start with 0)?
>
> As far as I'm concerned, the point is not to concentrate on a specific name
> for a concept, or the identification of a keyword, but to concentrate on a
> simple way to describe a concept using ordinary language with the least
> amount of math lingo. And to keep the number of such concepts as small as
> possible.
>
> Ladnor Geissinger
> ========================================================
> On Thu, 02 Sep 2010 15:00:54 -0500, Chip Burkitt
> <chip.burkitt at orderingchaos.com> wrote:
> > Susan,
> >
> > Having taught both GED and college remedial math courses, I couldn't
> > agree more. I have found that it is essential to introduce new
> > vocabulary slowly, relate it to known concepts, and then use it
> > consistently and avoid sloppiness myself in order to have a hope that
> > some students will get it. The distinction between "number" and "digit"
> > is by no means self-evident, and many students will need to see it and
> > use it themselves repeatedly before they grasp it.
> >
> > Chip Burkitt
> >
> > On 9/2/2010 11:40 AM, Susan Jones wrote:
> >> Yes, the word "set" is more exotic.
> >>
> >> The word "number" is actually *used* by the students; digits much less
> >> so.
> >>
> >> We are fluent in this language and vocabulary; our students are not. I
> >> have not yet figured out a way to educate (some) math teachers about the
> >> "math language fluency" problem. It's so simple to them, so it *must*
> >> be simple to the students... except it isn't. Believing that it should
> be
> >> doesn't make it so.
> >>
> >> The real way to completely lose them is the "what about this is hard,
> >> anyway???" So... it's easy, it's fun, it's joyful... and if it
> >> isn't? If you tell me something is easy and it doesn't happen to be...
> >> then I must have something wrong with me.
> >>
> >> It's a wonderful, exhilarating thing when something finally "clicks" for
> >> a student. Students accustomed to success do expect confusion to come
> >> first sometimes, but if its' explained in Greek from day one, they are
> >> just trying to survive. It doesn't matter whether or not I *tell* them
> >> that they need this class for something... usually, that's exactly why
> >> they are there. It's a necessary evil in their lives; a rite of
> passage,
> >> a "gatekeeper" course to sift out those not willing to endure. There
> >> *is* some value in arduous rites of passage, but only if the rites are
> >> completed successfully, and an awful lot of our students don't do that.
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >>
> >> Susan Jones
> >> Academic Development Specialist
> >> Center for Academic Success
> >> Parkland College
> >> Champaign, IL 61821
> >> 217-353-2056
> >> sujones at parkland.edu
> >> Webmastress,
> >> http://www.resourceroom.net
> >> http://www.bicycleuc.wordpress.com
> >>
> >>
> >>
> >>>>> Ladnor Geissinger<ladnor at email.unc.edu> 8/31/2010 6:22 PM>>>
> >> Ms. Steinke,
> >> I have a few questions about your note below.
> >> 1. You seem to be saying that using the term "whole numbers" is ok with
> >> your students but it's not ok to use "natural numbers". Why so, I
> >> thought these terms were both used commonly and meant the same thing.
> >> 2. If your students wouldn't like my definition of a number as a list
> >> of digits not beginning with 0, then what meaning do they or you when
> >> you teach them attach to the word number?
> >> 3. Is the word "set" somehow more exotic or off-putting than
> >> "collection" of buttons, or "sack" of lollipops, or "box" of empty soda
> >> cans?
> >> 4. I don't think your students will be troubled by a few ordinary words
> >> used to describe basic mathematical objects. Are they running away from
> >> math, or have they not been helped to see that mathematical ideas were
> >> created by people who wanted to know more about their environment and
> >> developed efficient and effective tools to help them analyze phenomena
> >> and then try to predict outcomes -- to understand their world and try to
> >> get a more secure sense of what is happening around them? Of course
> >> saying they have to take this class or pass this test in order to get on
> >> with things is not great motivation to be interested in their classes.
> >> 5. I bet they all know how to write down the next natural number after
> >> any one that you give them -- so they know the "next number algorithm"
> >> but might not recognize my name for it.
> >> 6. What about exponents are you trying to teach them that seems so hard
> >> for them? And why should they have to know anything about exponents
> >> beyond maybe the convenient shorthand of T^2 instead of T*T ,and T^3
> >> instead of T*T*T, and T^4 instead of T*T*T*T, ..., especially if T is an
> >> expression like (278 +19* U+5*V)?
> >> That is enough for now. I will suggest a very nice geometric model for
> >> multiplication later.
> >>
> >> Also, I will try to look at some books of the kind you mentioned to get
> >> a better idea of what list of topics some textbook authors think ought
> >> to be taught. But I will still want to know why the textbook authors
> >> think everyone should know what they have in their books -- what's the
> >> rationale?
> >>
> >> Ladnor Geissinger
> >>
> >> On 8/30/2010 8:48 PM, steinkedb at q.com wrote:
> >>> Ladnor -
> >>> You asked "who the teachers on this list are teaching math to."
> >>> I am one of those teachers. I teach GED math and first-level
> >>> developmental math (whole numbers, decimals, fractions) at a community
> >>> college. Except for these two of your definitions,
> >>> D. addition is fast counting
> >>> E. multiplication is more efficient fast counting -- when we have
> >>> regularity
> >>> the language of what you are saying (sets, natural numbers, etc.) will
> >>> send my students running as far away from math as they can get. The
> >>> majority of them seem to have no aspiration in math other than "get
> >>> past this class" or "get past the GED test."
> >>> They struggle with the concept of exponents. They can do the "rules."
> >>> They do not always get a physical feel for how much bigger the next
> >>> power of a number is.
> >>> When I introduce multiplication and division by decimals and
> >>> fractions, they struggle mightily to understand why multiplication by
> >>> a number less than 1 yields an answer that is LESS than what you
> >>> started with (and division, of course, the opposite). For them,
> >>> multiplication is "always" supposed to give you a bigger answer. I'm
> >>> asking them to change their world view.
> >>> As for references, look at the major publishers' texts for "basic
> >>> mathematics" or "pre-algebra" at the college level. That will give you
> >>> some hints as to what my students are asked to learn.
> >>> Dorothea Steinke
> >>> Front Range Community College, Westminster, Colorado
> >>>
> >>> -----Original Message-----
> >>> *From:* numeracy-bounces at nifl.gov
> >>> [mailto:numeracy-bounces at nifl.gov]*On Behalf Of *Ladnor
> Geissinger
> >>> *Sent:* Saturday, August 28, 2010 10:25 PM
> >>> *To:* numeracy at nifl.gov
> >>> *Subject:* [Numeracy 518] Re: Another perspective on numbers,
> >>> operations, and negatives
> >>>
> >>> I'd like to respond to Michael's questions and comments below.
> >>> 1. I don't have an answer to the question of who the teachers on
> >>> this list are teaching math to, exactly what math they are
> >>> teaching, and why. I agree that it is an important question and I
> >>> wish I knew some answers. I hope that several people respond with
> >>> their thoughts about that and suggestions for references where I
> >>> can learn some answers to that question.
> >>>
> >>> 2. Negative numbers are clearly very useful, as Mark's latest
> >>> note points out, so we certainly don't want to eliminate negative
> >>> numbers from whatever is taught. But then I don't understand
> >>> Michael's concern about "a more abstract consideration of
> >>> negative numbers as exceeding the needs of the population of
> >>> learners". What "more abstract consideration" is he talking
> >>> about? Does he mean my description of negative numbers introduced
> >>> simply as labels on a number line in a direction opposite from the
> >>> points labeled by positive numbers? What definition of negative
> >>> numbers is simpler than that?
> >>>
> >>> 3. Finally Michael mentions the way I frame my suggestions (see
> >>> below) for what I claim is a very simple way to think and talk
> >>> about elementary math, a way that may be not quite the standard
> >>> presentation of schoolmath. The numeracy teachers on this list
> >>> are facing students for whom the standard version of mathtalk has
> >>> failed to make sense -- so why not at least consider a slightly
> >>> different way.
> >>>
> >>> My intent is to: make the definitions of elementary math as
> >>> simple and concrete as possible, keep the number of basic ideas
> >>> small, and state the ideas and methods in a way that allows for
> >>> easy generalization and further development as our range of
> >>> problems to be worked on grows.
> >>> Here is a list of very brief versions of my suggestions for
> >>> definitions (which appeared with more details before).
> >>> A. a natural number is a string of digits not beginning with 0
> >>> B. the natural numbers are ordered by the "next number algorithm"
> >>> C. to count a set S, order its elements one after the other and
> >>> pair them with the next number
> >>> D. addition is fast counting
> >>> E. multiplication is more efficient fast counting -- when we have
> >>> regularity
> >>> F. on a line choose a point 0, mark points with natural numbers at
> >>> multiples of a unit length in a positive direction, then start at
> >>> 0 and do the same in the opposite direction using the new labels
> >>> -1,-2,-3... Now we have all the integers to use in modeling
> >>> problems.
> >>> G. now extend addition to all integers as suggested by models of
> >>> concrete settings
> >>>
> >>> Of course each time we extend something beyond previous uses, we
> >>> have to carefully check out what works in the proposed new
> >>> environment. We have to give proofs so that we can be absolutely
> >>> sure -- many others in society are depending on the correctness of
> >>> our math principles.
> >>>
> >>> I hope that some others on this list will jump in with their
> >>> thoughts so we can discuss these ideas.
> >>>
> >>> Ladnor Geissinger
> >>> =================================================================
> >>> On 8/27/2010 12:23 PM, Michael Gyori wrote:
> >>>> 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
> >>> --
> >>> Ladnor Geissinger, Emer. Prof. Mathematics
> >>> Univ. of North Carolina, Chapel Hill NC 27599 USA
> >>>
> >>>
> >>> ----------------------------------------------------
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> >>> Numeracy at nifl.gov
> >>> To unsubscribe or change your subscription settings, please go to
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> >>> Email delivered to ladnor at email.unc.edu
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