[Numeracy 524] Re: Another perspective on numbers, operations, and negatives
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Mon Aug 30 20:48:29 EDT 2010
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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
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.
Front Range Community College, Westminster, Colorado
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,
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
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
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.
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?
Michael A. Gyori
Maui International Language School
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,
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.
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
> 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
> 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
> 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
> 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
> write down numbers that we have all agreed we will use to label sets
> 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
> 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
> 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
> 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
> 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
> 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
> S and T of coins which they have independently counted, and they tell
> 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
> we can begin to work out practical algorithms for computing sums. So
> effect addition is fast counting -- it allows us to replace actual
> counting of the combination of two sets by the operation of addition
> 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
> 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
> idea that geometry could be done very efficiently using real numbers
> coordinate systems [analytic geometry]. Soon it became clear that
> efficiency and ease of use required the usual rule of signs (-1)*(-1)
> I'll end there and hope others will comment on this and further the
> Ladnor Geissinger
Ladnor Geissinger, Emer. Prof. Mathematics
Univ. of North Carolina, Chapel Hill NC 27599 USA
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