[Numeracy 527] Re: Another perspective on numbers, operations, and negatives
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Tue Aug 31 19:22:59 EDT 2010
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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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> Math& Numeracy discussion list
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> 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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