[Numeracy 514] Re: Another perspective on numbers, operations, and negatives

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Mark.Trushkowsky at mail.cuny.edu Mark.Trushkowsky at mail.cuny.edu
Fri Aug 27 18:55:19 EDT 2010


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

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.


Michael Gyori <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>

chip.burkitt at orderingchaos.com, The Math and Numeracy Discussion List
<numeracy at nifl.gov>

[Numeracy 513] Re: Another perspective on numbers, operations, and

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,
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) =



> I'll end there and hope others will comment on this and further the

> discussion.


> Ladnor Geissinger


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