Teaching Tolerance

[Algot Runeman]

This is a request twice removed from its origin on Twitter. It seemed
like a possible fit here.

"Colleague in my non-Twitter #PLN looking for examples of teaching
#tolerance in a math lesson. Anyone have ideas to share?"
by way of @ThalesDream on Twitter (a social studies teacher)

Love this Internet thing...

--Algot

--
-------------------------
Algot Runeman
algot....@verizon.net
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Twitter: http://twitter.com/algotruneman/
sip:algot....@ekiga.net
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[Christian Baune]

Teach to accept errors, error you do and error others do.

You can also find multiple way to solve a given problem and ask the student to choose the best one.

When they are fighting hard, you can explain that the "best strategy" depends of what you know, what you've to do and your means.

Kind regards,

Christian

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[Maria Droujkova]

One thing I do is tell everyone to write down (or draw, or show with manipulatives) answers to any questions. We have a quiet pause for that. Kids can talk to neighbors, but quietly. When everybody is done, only then we discuss answers. And everybody has a chance to share what they did. This works the best with open-ended questions, of course. And you can admire both simple and complex answers, for different reasons.

For example, when my 8yo group was making non-linear functions out of Lego blocks, one kid made the function that went 1-2-1-2-1-2 and another 1-2-4-8-16... At first, kids thought the first one was too simple, but we talked about periodic functions and it turned out very interesting! And several kids tried constant functions at first - even though they are linear, we had a good conversation going about it. Open questions with multiple answers, and a pause to admire every answer, is my example of math tolerance.

Cheers,
Maria Droujkova
919-388-1721

Make math your own, to make your own math

[Charischak Ihor]

Hi,

I hope you join us for this session on Wednesday, May 9th at 9pm EDT.

Thanks,

Ihor

Ihor Charischak

CLIME 

Council for Technology in Math Education

White Plains, NY

ih...@clime.org

http://climeconnections.blogspot.com

[mok...@earthtreasury.org]

Would a lesson on comparison tolerance help? Because binary numbers cannot
represent real numbers with more than a certain accuracy, numerical
analysts have developed the technique of fuzzy comparisons. Thus the
question whether two numbers, such as 1 and 3*%3 (3 times reciprocal of 3)
are equal, is not done with exact comparisons of the binary
representations. Instead, the question is whether the difference is less
than a certain multiple of the smaller number. The multiple is determined
in APL and J by a comparison tolerance system variable, by default 2^_44
(5.68434e_14).

Alternatively, how about this?

Peter Drucker (1909�2005)

The most serious mistakes are not made as a result of wrong answers.
The truly dangerous thing is asking the wrong questions.

Men, Ideas and Politics, Harvard Business Review Press, 2010

Quoted in Right Answer, Wrong Query, Statistics Roundtable, Quality
Progress, March 2010.
John Tukey (1915�2000)

Far better an approximate answer to the right question, which is often
vague, than an exact answer to the wrong question, which can always be
made precise.

"Sunset Salvo," The American Statistician, Vol. 40, No. 1, 1986, pp. 72-76.

Quoted in Right Answer, Wrong Query, Statistics Roundtable, Quality
Progress, March 2012.

In math, the sciences, and many other realms good new questions are better
and more important than good answers to old questions. Here is a question
the Greeks missed completely, even though they had all the geometry needed
to answer it: Why is the path of a water fountain, or water flowing from a
water clock, or any other low viscosity fluid in air, a parabola?

The question was only asked two millennia later, first by Leonardo da
Vinci, who got the answer but was not in a position to publish it, and
again more than a century after that by Galileo, who did publish his
solution. (Constant acceleration in a uniform gravity field, within the
limits of experimental error.) But then Kepler asked what figure the orbit
of Mars made, with accurate enough data (from Tycho Brahe, who asked a set
of critical questions with sufficient determination to make it possible)
to determine that it was an ellipse. And then Newton asked what could make
that happen, and came up with the inverse square law of gravity. And then
Leverrier asked what could explain the orbit of Mercury, and exhaustively
explored the possible Newtonian solutions, creating the question that
Einstein answered with General Relativity. We have a few questions about
that today, such as the nature of Dark Energy.

Similarly in math, the key question is, What is a number? We have not
finished answering that question yet. Some steps along the way were

Counting numbers

Rational numbers

Irrational numbers such as square root of 2

Negative numbers

Imaginary and complex numbers

Residues (remainders on division) in finite fields

Quaternions and octonions

Vectors, tensors, spinors, etc.

Cantor infinities, in a vast profusion of variations for every possible
set theory

Robinson non-standard arithmetic, with a different set of infinities and
infinitesimals

Conway surreal numbers and games, with different classes of infinities and
infinitesimals entirely

If you don't know all of these, don't worry. Nobody knows all about them,
and the others still being invented.

The Robinson and Conway number systems provide a new view on the perennial
chestnut, whether 0.9999999...is necessarily 1. Within the real numbers as
defined by Dedekind and others, the answer is yes, by definition, because
the limit of 0.9, 0.99, 0.999...is the Dedekind cut with all numbers less
than 1 on the left, and all numbers greater than or equal to 1 on the
right.

But within these other systems, this sequence defines other numbers. In
particular,

{(0.9, 0.99, 0.999...)|(1)} defines the number 1-omega, which is
infinitesimally less than 1.

There has been a great deal of intolerance expressed over these many kinds
of numbers over many centuries. Legend has it that the discoverer of
irrationals, a disciple of Pythagoras, was murdered for his heresy. Cantor
infinities were denounced as a disease within mathematics. If anybody is
interested, I can give many more examples.

On Sun, May 6, 2012 4:17 pm, Christian Baune wrote:
> Teach to accept errors, error you do and error others do.

Peter Drucker (1909�2005)

The most serious mistakes are not made as a result of wrong answers.
The truly dangerous thing is asking the wrong questions.

Men, Ideas and Politics, Harvard Business Review Press, 2010

Quoted in Right Answer, Wrong Query, Statistics Roundtable, Quality
Progress, March 2010.
John Tukey (1915�2000)

Far better an approximate answer to the right question, which is often
vague, than an exact answer to the wrong question, which can always be
made precise.

"Sunset Salvo," The American Statistician, Vol. 40, No. 1, 1986, pp. 72-76.

Quoted in Right Answer, Wrong Query, Statistics Roundtable, Quality
Progress, March 2012.

> You can also find multiple way to solve a given problem and ask the
> student to choose the best one.
>
> When they are fighting hard, you can explain that the "best strategy"
> depends of what you know, what you've to do and your means.
>
>
> Kind regards,
> Christian
>
>
> On 6 May 2012 22:06, Algot Runeman <algot....@verizon.net> wrote:
>
>> This is a request twice removed from its origin on Twitter. It seemed
>> like
>> a possible fit here.
>>
>> "Colleague in my non-Twitter #PLN looking for examples of teaching
>> #tolerance in a math lesson. Anyone have ideas to share?"
>> by way of @ThalesDream on Twitter (a social studies teacher)
>>
>> Love this Internet thing...
>>
>> --Algot
>>
>> --
>> -------------------------
>> Algot Runeman
>> algot....@verizon.net
>> Web Site: http://www.runeman.org
>> Twitter:

>> http://twitter.com/**algotruneman/<http://twitter.com/algotruneman/>

>> sip:algot....@ekiga.net
>> Open Source Blog: http://mosssig.wordpress.com
>> MOSS SIG Mailing List:

>> http://groups.google.com/**group/mosssig2<http://groups.google.com/group/mosssig2>

>>
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Edward Mokurai
(&#40664;&#38647;/&#2344;&#2367;&#2358;&#2348;&#2381;&#2342;&#2327;&#2352;&#2381;&#2332;/&#1606;&#1588;&#1576;&#1583;&#1711;&#1585;&#1580;)
Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
http://wiki.sugarlabs.org/go/Replacing_Textbooks

[Christian Baune]

That is because "number" has a to broad meaning.

[Linda Fahlberg-Stojanovska]

I was wondering whether these kind of questions might teach tolerance (maybe not though since I tend to have weird ideas in my head).

(a)    Writing “Why is a non-zero number divided by 0 not defined?” (perhaps for older students).

(b)   Using GeoGebra (et.al) to have students to draw their favorite enclosed figure with an area of 3 units?

(c)    Find different ways to describe a color to a computer.

 

P.S. We had an interesting discussion in my class about why we need both fractions and decimals.  Do you ever say “I want 0.5 of that piece of cake?”

[John Mason]

I have always assumed that by developing a conjecturing atmosphere in a
classroom, students would experience the positive effects of tolerance
and respect for other people's views. Working on tasks in which everyone
can get a different (looking) but equally correct answers, as in
expressing generality concerning geometrically presented repeating and
growth patterns) also demonstrates how people working together can
locate an essential agreement behind apparently different articulations.

But it is probably important not to belabour the point with students as
it then becomes preachy.

JohnM

[Patrick Vennebush]

> P.S. We had an interesting discussion in my class about why we need both fractions and decimals. 

> Do you ever say “I want 0.5 of that piece of cake?”

 

Interesting that you used the word “say.” I’d say “half,” but you’d never actually see the word written out. Truth is, I could write that as, “I want half of that piece,” or “I want 0.5 of that piece,” or “I want 1/2 of that piece,” or “I want 5/10 of that piece,” and in all cases I could just say “half” and you’d never know what written representation was being used.

 

If I were writing it, though, I would spell out the word “half,” and I wouldn’t use a numeric representation. (Maybe this is old-school, Elements of Style style that was inbred in me.)

 

Was your class convinced that we needed both? I think it might be interesting to run a class debate: “The politicos have decided that we can’t have both fractions and decimals. You can only have one of them. Which would you choose, and why?”

 

 

[Linda Fahlberg-Stojanovska]

 

1.       I am reminded of something my father once said to me:  “You cannot write the sentence ‘There are 3 words for 2 in the English language.’.”
Ha – I just wrote it, but he was thinking that you cannot properly choose between writing “to”, “two” or “too” (and of course the above writing 2 is not proper since numbers between one and ten must be written out (old-school indeed).
I always liked the sentence because it was mathy foolish fun statement.

2.       The classroom discussion was an oral discussion – but I believe we all got the point and had a good laugh (and hopefully they – like me at that very moment – realized that fractions are important).
As readers here may know, I fanatically prefer decimals to fractions – but as I was writing out a problem, it flashed on me that we don’t “speak” decimals.

3.       Also, today we were doing probability and the same problem occurred with percents (percentages?). What is the probability of a coin landing on heads? What normal person says “0.5” or even “one-half”? Everyone answers 50%.

 

P.S. I tell my IT students that “number/0” does not exist because the computer would go crazy being told to “allocate something to no place”. I would be interested in knowing other peoples (and their kiddies) responses.

--

[Donald Cohen]

Patrick,

I prefer saying if we share one cookie between 2 people, we each get one-twoth, with 3 people we each get one-threeth, and sharing between 4 people, we each get one-fourth, ...oh my, we get to the usual way on the fourth. I think it's important. Some of my students whose parents went to school in India, say one over 2, for one-twoth. I also say 1-ty 7 instead of seventeen which saves youngsters a lot of trouble because when they hear seventeen, they want to write 71. Haven't you seen these mistakes? Clearly the most indefensible writing on ditto sheets given to kids and still in textbooks comes with subtraction and negative numbers: I just saw yesterday (-2)-(-3)  instead of raising the negative signs ⁻2-⁻3 to make the difference between an operation and the negative number so the student will understand it better. This would take  care of a lot of confusion.

Don
   

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[kirby urner]

I appreciate the emphasis on nuance, including around "tolerance".

As Mokurai was noting, we have fuzzy digits, so-called "insignificant"
sometimes.

In some of the unit testing frameworks, used to "check your own work"
(and more), assertAlmostEqual is used.

I'm only sometimes a classroom teacher, holding forth, examples on Youtube.

Mostly I'm in touch with students through writing within a shared
interface, belonging to a school.

Either way, I sometimes drop into a mode and tone, backed up with
explication and behavior: that I'm teaching a language.

Language teachers use emphasis in specific ways, and have adopted the
attitude that others in the room are just beginning to develop various
associations. You might be explicit about allusion, etymology,
pronunciation. The mythographic implications of words might be in
order.

For example, in talking about "wisdom" I might deliberately conjure
the image of Athena, perhaps in some slides. The idea here is a
language is more than just a dictionary of words, it's also a web of
associations, shared circuits.

Mathematics teachers vary in their sounding out the highways and
byways. One reason people go to colleges and such places is to listen
for cues and clues regarding the current who's who. Who are we
talking about, who's big? The answers will vary by venue.

To make it more concrete, these days I'm hanging out with University
of Illinois alums and faculty, in a mix with some other folk. The U
of I has a long history of being rather deep into computers. That
heritage feeds into distance education and approaches to pedagogy that
make use of the Internet.

Even when not in a classroom, I'm emphasizing the "mathematics is a
language" approach and emphasizing doing, making, speaking,
performing. Our school's founding philosopher was much influenced by
his tour in Russia, where he saw the emphasis placed on oral
recitation, i.e. thinking on your feet, not just regurgitating content
or working through a set of memorized steps in a process.

This school is based in Sebastopol (California, not Russia).

Athena is oft depicted with Nike on one side, her Python on another.
Nike is her quick moving spy, her UAV (not offensive). The Python,
sometimes tucked behind a shield, represents the ancient intelligence
which informed her oracles at Delphi, a headquarters before the
Apollonians moved in.

That connection with Nike is reinforced in the "Just Use It" slogan
that Python (the language) is using. Watch the airport concourses.
The logo is like a playing card, with a reflected figurine, a kind of
yin-yang motif. Iceland is a dev center, kind of like Prague is for
FoxPro (another language with an animal brand).

Kirby

[mok...@earthtreasury.org]

On Tue, May 8, 2012 7:49 am, Patrick Vennebush wrote:
>> P.S. We had an interesting discussion in my class about why we need
> both fractions and decimals.

> Was your class convinced that we needed both? I think it might be
> interesting to run a class debate: "The politicos have decided that we
> can't have both fractions and decimals. You can only have one of them.
> Which would you choose, and why?"

0) New politicos.

1) My mother used to propose such questions as a teacher, except that she
did it in Civics class: It is obvious to everybody that aristocracy is the
best possible system of government, as long as they and their friends get
to be the aristocrats. Discuss.

2) I have participated in such a math discussion, at the APL89 Conference.
APL allows the user to specify that counting should start at an Index
Origin that could be set to either 0 or 1.

Thus, in Index Origin 1, where indices and initial segments of the
integers run 1 2 3...

(2 3)[1]
2

but in Index Origin 0, where they run 0 1 2...

(2 3)[1]
3

The plenary session was polled on the following questions:

Should we have only one Index Origin? Unanimous answer: Yes.

Who wants it to be 1? Roughly half the audience, mostly the business
programmers.

Who wants it to be 0? The other half of the audience, mostly the
mathematical, scientific, and engineering programmers.

Ken Iverson, inventor of APL, was a mathematician. He made 0 the only
index origin in his last version of APL, called J.

3) We could ask similar questions about angle measure: degrees, radians,
fractional circles, or other?

Celestial navigation: Earth-centered (with rotating sky) or Newtonian?

GPS: Special or General Relativity?

The last question actually occurred. The rocket scientist crowd was split
over the question of time corrections for orbiting atomic clocks, so both
solutions were programmed in the GPS satellites. They began with the SR
solution, considering only the speed of the satellites, but all of the
locations rapidly drifted away from where they were supposed to be, even
from the surface of the Earth. When the satellites were told to figure
their time according to GR, allowing for gravity at altitude, everything
worked correctly.

Some of these questions have real, unarguable answers, as in the case of
GPS, once it was in orbit. Others have real, unarguable lack of definitive
answers, where two or more formally equivalent systems are used for
convenience, and you will convince nobody that their convenience does not
matter.

[mok...@earthtreasury.org]

On Wed, May 9, 2012 3:03 pm, Donald Cohen wrote:
> Patrick,
>
> I prefer saying if we share one cookie between 2 people, we each get
> one-twoth, with 3 people we each get one-threeth, and sharing between 4
> people, we each get one-fourth,

In J, that would be 1r2 1r3 1r4, where the r notation signifies rationals.
So 1r1 is one oneth, and 0r1 is zero oneths.

> ...oh my, we get to the usual way on the
> fourth. I think it's important. Some of my students whose parents went to
> school in India, say one over 2, for one-twoth. I also say 1-ty 7 instead
> of seventeen which saves youngsters a lot of trouble because when they
> hear
> seventeen, they want to write 71. Haven't you seen these mistakes?

Excellent. I will add these to my collection. Can I interest you in
working on an Arithmetic book that I will be translating from APL to J?

> Clearly
> the most indefensible writing on ditto sheets given to kids and still in
> textbooks comes with subtraction and negative numbers: I just saw
> yesterday
> (-2)-(-3) instead of raising the negative signs ⁻2-⁻3

I'm sorry. My mailer broke your math: �2-�3.

Were you sending in 8859-1 or in Unicode?

> to make the
> difference between an operation and the negative number so the student
> will
> understand it better. This would take care of a lot of confusion.

That's what we did in APL. In J, which is written in ASCII, we use an
underbar instead to distinguish between the notation for negative numbers
and the negation function.

_2-_3
1

> Don
>
> On Tue, May 8, 2012 at 6:49 AM, Patrick Vennebush
> <pvenn...@nctm.org>wrote:
>
>> > P.S. We had an interesting discussion in my class about why we need

>> both fractions and decimals. ****
>>
>> > Do you ever say “I want 0.5 of that piece of cake?”****
>>
>> ** **

>>
>> Interesting that you used the word “say.” I’d say “half,” but
>> you’d never
>> actually see the word written out. Truth is, I could write that as, “I
>> want
>> half of that piece,” or “I want 0.5 of that piece,” or “I want
>> 1/2 of that
>> piece,” or “I want 5/10 of that piece,” and in all cases I could
>> just say
>> “half” and you’d never know what written representation was being

>> used.***
>> *
>>
>> ** **

>>
>> If I were writing it, though, I would spell out the word “half,” and
>> I
>> wouldn’t use a numeric representation. (Maybe this is old-school,

>> *Elements
>> of Style* style that was inbred in me.)****
>>
>> ** **

>>
>> Was your class convinced that we needed both? I think it might be
>> interesting to run a class debate: “The politicos have decided that we
>> can’t have both fractions and decimals. You can only have one of them.

>> Which would you choose, and why?”****
>>
>> ** **
>>
>> ** **

>>
>> --
>> You received this message because you are subscribed to the Google
>> Groups
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>> To unsubscribe from this group, send email to
>> mathfuture+...@googlegroups.com.
>> For more options, visit this group at
>> http://groups.google.com/group/mathfuture?hl=en.
>>
>
>
>
> --

> *"Learning, Living and Loving mathematics.."- the core of Don's teaching
> and books, **observed by Seth Nielson.*
> *The Math Program*
> *Don Cohen -The Mathman*

> 809 Stratford Dr.
> Champaign, IL 61821-4140
> Tel. 217-356-4555
> Fax: 1 217 356 4593
> Email: doncohe...@gmail.com
> Don's Mathman website URL: http://www.mathman.biz

> See Don's *new* *clickable* A Map to
> Calculus<http://www.mathman.biz/html/map.html>with student works and

> sample problems from Don's books at every node
> Our Shaklee website:
> http://marilynanddoncohen.myshaklee.com
>

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--

[Paul Libbrecht]

Linda,

are you in the Americas?

There's a whole lot more fractions in the Americas than in Europe.

It's almost as bad as "fractions for school only", and this is witnessed in a lot of the professional activity here in Europe.

A good way to impress europeans is to bring a box of drill-bits and tell them that joe-bo in America is able to tell you that a bit of 5/32 is smaller than 3/16. 

Or is it not more so?

paul

[David Chandler]

I think the primary reason fractions are important is they are needed to form rational expressions in algebra.  You can't use decimals for that.
--David Chandler

[Linda Fahlberg-Stojanovska]

@Patrick. Indeed I live in Europe (FYR Macedonia just north of Greece for 33 years), but I am US born and schooled (25 years). Luckily I was schooled in US before the old “new math” came in with set theory and blah, blah. (I am a theoretical mathematician and did my master’s in algebraic set theory and got on just fine without set theory in elementary school.)

 

With regards to the imperial system of 8ths, it is a mathematical pity that we were born with 10 fingers and not 8 :) . That would have solved a byteload of problems. (Feeble attempt at joke.)

 

We spent an entire year studying fractions (5^th grade) and I give a lot of credit for my success in math to this plus an incredible algebra 1 teacher who made us write everything down.

I lucked out with about 4 math teachers in a row that wanted the whole process from thinking through solution and were excellent teachers of the same.

They tolerated different thinking process (thank goodness), but did not tolerant sloppiness (again - thank goodness).

 

@David

>I think the primary reason fractions are important is they are needed to form rational expressions in algebra.  You can't use decimals for that.

I am sorry David, I disagree. Please feel totally free to explain why I might be wrong. (At age 58, I was amazed last week to have to change my mind about the degree of usefulness of fractions:) )

(a)    I think the “primary reason” we need fractions is that we use them in everyday speak. The questions here might be what denominators are REALLY necessary (CCSS is good here), what operations and how to explain them (i.e. not “Solve 3/(1/2) by invert and multiply.” but “How many 1/2 meter pieces can I cut from a 3 meter ribbon?” and how to use games for learning and understanding fraction operations (like the cool game Colleen King has for learning and understanding solving equations).

(b)   I don’t think your “average person” needs ‘rational expressions in algebra’. Certainly not with a variable in the denominator. And if there is no variable in the denominator, I think you could use decimals. (I particularly do not like fractional slopes, but I am sure to be in a minority here.)

Of course this leads to the question of the math needs of the “average person” (i.e. someone not entering a STEM profession)… and their “tolerances”.

Warm regards to all, Linda

 

 

 

 

[David Chandler]

@David

>I think the primary reason fractions are important is they are needed to form rational expressions in algebra.  You can't use decimals for that.

I am sorry David, I disagree. Please feel totally free to explain why I might be wrong. (At age 58, I was amazed last week to have to change my mind about the degree of usefulness of fractions:) )

I wasn't trying to minimize their usefulness for other purposes.  I was just pointing out one area where rules for fraction manipulation become necessary.  It is frequently the case that in solving equations you must divide both sides by some expression.  This puts you in the position of needing to work with fractions.  Ratios also give you fractions.  Decimals are limited to numerical work, but they are very convenient.

By the way, if you see decimals as just a way to standardize the denominators as powers of a base, giving you place value notation, there is no need to limit yourself to 10 as the base.  An ordinary English system ruler uses binary fractions: 1/2, 1/4, 1/8, etc.  You could use binary place value to read and compute with these fractions.  For instance, 5/8 could be represented by ;1010 since it is 1/2 + 0/4 + 1/8 + 0/16.  The distance from 3 5/8 to 5 7/16 could be found by subtracting
  5;0111 - 3;1010 = 1;1101 or 1 13/16. 
(I use the ; instead of a . to distinguish the base 10 whole numbers from the base 2 fractions.)  When I read an English style ruler I find it easier to identify the 13/16 mark as 1/2 +1/4+1/16, or 3/4+1/16.  If I want to know where to cut the board, 1;1101 works for me just fine.  I don't even do the conversion to 13/16.

By the way, again, the practice of using different bases for whole numbers and fractions is not unprecedented.  Angular measurements with degrees, minutes, and seconds is a base 60 place value system for the fractional parts of angles while the whole numbers are left in base 10.

(I haven't published this system and don't know if anyone else has.)

--David Chandler

[Donald Cohen]

I don't know either of these. Yours looks better than mine! Mine came out too short, yours a little long. I don't know about the underbar (I need to think about that). On my Mac, there is a way to raise the - symbol. At the top right of the screen there is a small square with an asterisk, click and choose "show character viewer", then pick which character you want to raise, under "Related Characters". When I wrote my books on an old PC, I had to raise the dash a certain # of steps. I thought it was a lot of trouble, before using superscripts, but that worked, and I'm glad I took the extra time to raise the dash for negatives.

--

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[Donald Cohen]

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David, this writing of binary fractions reminds me of infinite series in my books: take  1/2 + 0/4 + 1/8 + 0/16. What if you continue this 1/2 + 0/4 + 1/8 + 0/16 + 1/2 + 0/4 + 1/8 + 0/16 + ... What is the limit of this infinite series as a fraction? I never knew this before writing my books. Most people here know this, I guess.

Don

[David Chandler]

If you mean 1/2+0/4+1/8+0/16+1/32+0/64+... this is just a geometric series with common ration 1/4, so the limit is 2/3.  However, I'm not sure if that is what you mean.
--David Chandler

[Donald Cohen]

David,

How about another way to find it?

Don

[mok...@earthtreasury.org]

You can do what David suggests on the multi-base Abacus program in Sugar,
in bases from 2 to 20.

http://activities.sugarlabs.org/en-US/sugar/addon/4293

On Sat, May 12, 2012 3:16 pm, David Chandler wrote:
>>
>> @David****

>>
>> >I think the primary reason fractions are important is they are needed
>> to
>> form rational expressions in algebra. You can't use decimals for

>> that.***
>> *

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[mok...@earthtreasury.org]

On Mon, May 14, 2012 2:09 pm, Donald Cohen wrote:
> On Fri, May 11, 2012 at 12:14 PM, <mok...@earthtreasury.org> wrote:
>
>> On Wed, May 9, 2012 3:03 pm, Donald Cohen wrote:
>> > Patrick,

...

>> > Clearly
>> > the most indefensible writing on ditto sheets given to kids and still
>> in
>> > textbooks comes with subtraction and negative numbers: I just saw
>> > yesterday
>> > (-2)-(-3) instead of raising the negative signs ⁻2-⁻3
>>
>> I'm sorry. My mailer broke your math: ¯2-¯3.
>>
>> Were you sending in 8859-1 or in Unicode?

> I don't know either of these.

Ah, I see, below you explain that you were using a Macintosh, which has a
character set different from both of these.

Unicode is the universal character set that can be displayed correctly in
Windows, Mac OS, Linux, and any other operating system. 8859-1 is an older
8-bit character set much used on Linux. It would be useful for you to
learn about the mathematical character support in Unicode. For example:

���&#8730;&#8721;&#8734;&#8747;&#8743;&#8744;&#8800;&#8804;&#8805;&#8706;&#8714;

The above requires a Unicode math font to be installed for proper viewing.
Some are in the Macintosh Symbol font.

http://tlt.its.psu.edu/suggestions/international/bylanguage/math.html

I think you say below that you used this Character Palette utility.

http://tlt.its.psu.edu/suggestions/international/keyboards/charpalosx.html#math

>> > to make the
>> > difference between an operation and the negative number so the student
>> > will
>> > understand it better. This would take care of a lot of confusion.
>>
>> That's what we did in APL. In J, which is written in ASCII, we use an
>> underbar instead to distinguish between the notation for negative
>> numbers and the negation function.
>>
>> _2-_3
>> 1
>>
>> > Don

...

> Yours looks better than mine! Mine came out
> too short, yours a little long. I don't know about the underbar (I need to
> think about that). On my Mac, there is a way to raise the - symbol. At the
> top right of the screen there is a small square with an asterisk, click
> and
> choose "show character viewer", then pick which character you want to
> raise, under "Related Characters". When I wrote my books on an old PC, I
> had to raise the dash a certain # of steps. I thought it was a lot of
> trouble, before using superscripts, but that worked, and I'm glad I took
> the extra time to raise the dash for negatives.

J takes the rationalization of math notation much further. Here is the
early version of the explanation, math vs. APL.

http://www.jsoftware.com/papers/tot.htm
Notation as a Tool of Thought
Kenneth E. Iverson
Turing Award lecture

I am using J in this manner to create math texts in which every expression
can be evaluated, plotted, or dissected by the student. If you would be
willing, I would like to demonstrate how that would work with some of your
materials.