[Numeracy 508] Re: Another perspective on numbers, operations, and negatives

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Chip Burkitt chip.burkitt at orderingchaos.com
Sat Aug 21 09:35:26 EDT 2010

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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