"Inventing" number rules
Murray Bourne
Regards,
Murray Bourne
===================================
Interactive Mathematics: http://www.intmath.com/
squareCircleZ: http://www.squarecirclez.com/blog/
Daily Math Tweet: http://twitter.com/intmath
Maria Droujkova
Cheers,
Maria Droujkova
Make math your own, to make your own math.
Edward Cherlin
> Hi all
> I think I first heard of the Brizuela book Mathematical Development in Young
> Children through the Mathfuture group.
Mathematical Development in Young Children: Exploring Notations (Ways
of Knowing in Science and Mathematics)
I was disappointed that the focus here is on notation. I'm looking for
studies of math ideas in pre-literate children to expand my
Kindergarten Calculus project. There are a number of branches of
mathematics that have basic ideas with a visual or tactile model,
where we can introduce the idea early, and leave notation,
calculation, and proof until much later. Symmetries and permutations,
for example, which are the essence of group theory.
> I suspect Maria has something to add to the discussion about European number
> systems!
> http://www.squarecirclez.com/blog/is-there-a-place-for-invention-in-math/6053
Number Words and Number Symbols: A Cultural History of Numbers, by
Karl Menninger
The Universal History of Numbers: From Prehistory to the Invention of
the Computer, by Georges Ifrah and David Bello
> I hope you find it interesting.
>
> Regards,
> Murray Bourne
> ===================================
> Interactive Mathematics: http://www.intmath.com/
> squareCircleZ: http://www.squarecirclez.com/blog/
> Daily Math Tweet: http://twitter.com/intmath
>
> --
> You received this message because you are subscribed to the Google Groups
> "MathFuture" group.
> To post to this group, send email to mathf...@googlegroups.com.
> 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.
>
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
milo gardner
Best wishes on your paper. My nephew sent along these two links that fit into this discussion by:
1. I thought you might enjoy this. http://worrydream.com/KillMath, the "Mathematician's Lament" thread
2. and the Interactive Exploration of a Dynamical System (http://vimeo.com/23839605), an way to visualize dynamic systems and paths to higher math
Milo Gardner
From: Maria Droujkova <drou...@gmail.com>
To: mathf...@googlegroups.com; Paul Libbrecht <pa...@hoplahup.net>
Sent: Wed, May 18, 2011 6:56:08 PM
Subject: Re: [Math 2.0] "Inventing" number rules
Maria Droujkova
Ihor Charischak
I recently watched "Waiting for Superman", a documentary about the failure of public schools. It was a terrible movie, at least for me, because it took the content of education (the curriculum) for granted, and focused only on the mechanism of education (the schools and teaching). It blindly assumed that learning twelve years of math (as determined by scoring well on a set of tests) equates to "success". It never questioned whether the "math" itself is actually beneficial in any way -- it only matters that the kids do the math. (There was even a cartoon scene of a teacher pouring a can of numbers into a student's head -- as if that represents learning!)
(And I believe a subject can be "beneficial" without necessarily being directly usable or even relevant -- emotional inspiration, intellectual exercise, inspiring intellectual connections, and simple curiosity are all acceptable reasons to learn something. But school math is useless, kills inspiration and curiosity, is mind-numbingly tedious, makes no connections to anything, and is forgotten immediately after the test. It's all negative.)
(Of course, that's no different than other mandated subjects. School science is a mockery of real science; it's more faith-based than scientific, involves no actual thinking or exploring, etc.)
One part of the movie asserted that in a year, a good teacher could cover 150% as much material as a standard teacher, whereas a bad teacher could only cover 50%. (This was helpfully illustrated with a bar graph with two bars.) But 150% of zero and 50% of zero are the same! If it's all worthless anyway, it doesn't matter what rate you go at.
It's so obvious to me that the reason high test scores correlate with success is entirely social, not intellectual. Better performing students are expected to succeed by authority figures and their peers. That expectation puts them in a better emotional position to succeed, and also grants them access to the higher castes of society (via college etc.) which puts them in a better societal position to succeed. It has nothing to do with their supposedly better understanding of fractions. It's no different than if the kids were being tested on memorizing bible verses.
The movie completely ignores this. It doesn't care what the kids are taught, or why. Many of the software engineers that I've worked with have only the most rudimentary mathematical skill. I've seen some struggle with basic algebra, literally eighth-grade stuff, and apologize that they "used to know this". My point is, if these professional engineers (whose careers would representstratospheric success for the inner-city kids in the movie) don't even need to know basic algebra to get by, who the fuck decided that every eighth grader in the country needs to know it? It's so painfully, obviously irrelevant.
Linda Fahlberg-Stojanovska
I have not been following this thread very closely, but I do think algebra very important. It should be teaching thinking, logic, abstraction and connecting skills. I mostly agree with Maria about content. With the exception of basics, it really doesn’t matter what you teach – as long as you enjoy what you teach and the kiddies learn to think and use math usefully. I like to think that sometimes we actually do this and that this is what makes “success in algebra” a consistent indicator of “higher earnings”. Not what we “pour” in.
However – what to do about number rules?
1. Yesterday, I was working on common-core standards and I still get the discouraging image “of a teacher pouring a can of numbers into a student's head”. Randomly picking a number rule: “Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.” Fun-yes, required-no. (And not a single mention of parameters or parametric functions – what is STE(M) without them?)
2. Yesterday, I had a college student of IT not understand why this equation is not correct: √x =x² . I said “substitute x=4”. Huh? I say what is √4? What is 4²? He immediately gets 2 and 16. Are they equal? No. Can I write √x =x²? He does not see the connection. Then he could not calculate: ¼-1. So I say: you give me 25¢, but you owe me 1 dollar. How much do you still owe me? He immediately says 75¢. If I tell you that this kid lives in Macedonia, has never been to the US, does not speak English, probably has never seen US coins … I say, “What does this mean for ¼-1=?” He does not see the connection.
What do I do? Do I fail him? For not knowing formally rational exponents? For not knowing formally common denominators? He has a high school diploma “certifying” his knowledge of these number rules.
Question – I am thinking to read Ashlock: Error Patterns in Computation: http://www.pearsonhighered.com/assets/hip/us/hip_us_pearsonhighered/samplechapter/0135009103.pdf
Does anyone know this book? Any comments?
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On Behalf Of Ihor Charischak
Sent: Friday, May 20, 2011 1:29 AM
To: mathf...@googlegroups.com
Cc: Droujkova Maria; br...@worrydream.com
Subject: Re: [Math 2.0] "Inventing" number rules
Milo,
Murray
Ihor: I share your pain. It's not only about how we write math, it's
fundamentally about which bits we should teach - and leave out - and
why.
Linda: Your student seems to know the mechanics of indices for numbers
at least, so I'm not sure I would fail him outright. It's probably
more useful to know sqrt(4) and 4^2 than it is to know whether
sqrt(x)=x^2, no? (The equation is correct for x = 1. Could that have
been his block?)
Regards
Murray
Alexei
multiplication, if I remember correctly. I liked the parts, where you
have to deduce what rules imaginary students use to obtain answers to
problems. Sometimes it's a challenge to think in particular erroneous
ways. I don't remember the book going into comparative frequency of
particular mistakes (maybe saying that adding the numerators and
denominators is common when adding fractions, but one does not need a
book for this piece of knowledge). After the errors are revealed, he
offers some suggestions on how to remedy them. The suggestions are OK,
but I don't remember them being really mind-opening. I would say, it's
a solid book, but not a revelation. I can see that it may be very
useful to show prospective teachers how difficult it is sometimes to
figure out students' thinking when they assign their own meaning to
math procedures.
On May 20, 3:47 am, "Linda Fahlberg-Stojanovska" <lfahlb...@gmail.com>
wrote:
> Question – I am thinking to read Ashlock: Error Patterns in Computation:http://www.pearsonhighered.com/assets/hip/us/hip_us_pearsonhighered/s...
Edward Cherlin
<lfah...@gmail.com> wrote:
> I have not been following this thread very closely, but I do think algebra
> very important.
True, if it is real algebra. Algebra has four parts.
1. Solving polynomial equations and sets of equations exactly,
including Diophantine equations over the rational numbers.
2. Solving equations inexactly, for example by numerical methods for
close approximation, and least squares for overdetermined systems.
3. Optimization, finding the best solution to a set of equations and
inequalities according to some measure. This includes Linear
Programming.
4. Understanding the structure of equations, beginning with the group
of roots of an equation and then continuing on independently of
equations to symmetry and permutation groups, rings, fields,
groupoids, monoids, knot theory, vector spaces, matrices, algebras,
and categories.
These other parts have split off from equation-solving to become
subjects in their own right. Inexact solutions are the foundation of
statistical analysis. Optimization is one of the most important
industrial applications of math. "Higher algebra" is one of the great
open frontiers of mathematics, with deep connections to quantum
physics, crystallography, cryptography, analytic geometry,
non-Euclidean and projective geometries, differential geometry,
General Relativity, topology, cosmology, and much more.
We teach the solving of linear and quadratic equations, and very small
sets of linear equations, manually, in an age when every desktop
computer is more than equivalent to a Cray-1, and even pocket
calculators and telephones are bigger and faster than mainframes of
old, with far better graphics than the first $100,000 graphics
workstations. We do not teach the solution of equations as in fact
they are solved today, and we do not teach any of the uses of
equation-solving. Nor do we teach the high intellectual adventure of
any of this.
> It should be teaching thinking, logic, abstraction and
> connecting skills.
These are the conventional math reform topics. They are grossly
inadequate to the real problem.
> I mostly agree with Maria about content. With the
> exception of basics,
Nobody in education has any idea of what actually is basic in math.
Among the alternatives that have been proposed are set theory,
first-order logic, combinatory logic, and recently topose theory. The
basic ideas of each can be taught to preschoolers, though of course
not the formal definitions, notations, terminology, calculations, and
proofs. We have no idea at what stages of development these can most
effectively be introduced, with computer assistance.
The essence of counting is the one-to-one mapping between an initial
segment of the ordinal numbers and a set of objects. The language of
math thus begins with the memorization of a number system of some
kind. All the rest can be derived from this starting point of
mappings--negative numbers, rationals (fractions), algebraic numbers,
real numbers, complex numbers, geometry, topology, set theory, logic,
and on and on.
> it really doesn’t matter what you teach – as long as
> you enjoy what you teach and the kiddies learn to think and use math
> usefully. I like to think that sometimes we actually do this and that this
> is what makes “success in algebra” a consistent indicator of “higher
> earnings”. Not what we “pour” in.
I learned in college that the topic of a course was almost irrelevant,
except for economics, which is almost all false. Well, some topics in
philosophy are also almost all false, such as Plato's Republic, but it
can be taught well, acknowledging the problems, which does not happen
in economics. The quality of the professor was essential. I was lucky.
I had only one dud math professor (an excellent mathematician who
couldn't teach), and two duds in philosophy. IMNSHO.
> However – what to do about number rules?
>
> 1. Yesterday, I was working on common-core standards and I still get the
> discouraging image “of a teacher pouring a can of numbers into a student's
> head”. Randomly picking a number rule: “Prove polynomial identities and use
> them to describe numerical relationships. For example, the polynomial
> identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate
> Pythagorean triples.” Fun-yes, required-no. (And not a single mention of
> parameters or parametric functions – what is STE(M) without them?)
and no mention of discovering polynomial identities, with appropriate hints.
> 2. Yesterday, I had a college student of IT not understand why this equation
> is not correct: √x =x² .
x = x⁴
1 = x³, unless x = 0.
Alternatively,
x⁴-x = 0
x(x-1)(x-r)(x-r²) = 0
where r is either of the complex third roots of 1, specifically cos
(4π/3) + i sin (4π/3) (because 2π/3 and 8π/3 are the same angle), and
cos (2π/3) + i sin (2π/3), since(4π/3) is twice 2π/3 and half of 8π/3.
The full set of solutions consists of the cube roots of 1, and 0.
> I said “substitute x=4”. Huh? I say what is √4?
> What is 4²? He immediately gets 2 and 16. Are they equal? No. Can I write
> √x =x²? He does not see the connection. Then he could not calculate: ¼-1.
> So I say: you give me 25¢, but you owe me 1 dollar. How much do you still
> owe me? He immediately says 75¢. If I tell you that this kid lives in
> Macedonia, has never been to the US, does not speak English, probably has
> never seen US coins … I say, “What does this mean for ¼-1=?” He does not
> see the connection.
>
> What do I do? Do I fail him? For not knowing formally rational exponents?
> For not knowing formally common denominators? He has a high school diploma
> “certifying” his knowledge of these number rules.
Does your college have remedial math courses?
> Question – I am thinking to read Ashlock: Error Patterns in Computation:
> http://www.pearsonhighered.com/assets/hip/us/hip_us_pearsonhighered/samplechapter/0135009103.pdf
>
> Does anyone know this book? Any comments?
It is an extremely important topic, and I thank you for bringing this
book to my attention. it seems from the title that it might take the
point of view that it is the student who is in error, and not the
teaching materials or methods. I hope I'm wrong about that.
--
Linda Fahlberg-Stojanovska
First, what a wonderfully concise and thorough book report - another thing
we need to emphasize in education.
Second, thanks for telling me that it only goes to fraction multiplication -
I really would like to know how kiddies (and adults) think (erroneously)
about algebra* (but am interested in any imaginary rules that people make up
- that is the value of planning your classroom questions/problems well and
color coding.)
Thank-you! Linda
*And by algebra, I rather mean taking concrete problems with specific
numbers and moving them into abstract concepts, i.e. into functions with
variables that have properties that one can explore and discuss and prove
and use to solve other problems.
-----Original Message-----
From: mathf...@googlegroups.com [mailto:mathf...@googlegroups.com] On
--
Alexei
I’m glad my summary was of use. And I agree that written assignments
of this kind should be more common.
Regarding errors and such, I find this book more interesting.
Children’s Mathematics 4-15: Learning from Errors and Misconceptions
Previews are available from Amazon and Google Books. It goes into
algebra, though, obviously, covers a lot of elementary material. I
like that many errors come with evidence about their prevalence.
Many sources of errors are pretty clear. Transferring from previously
encountered situation is, probably, the most frequent mechanism. On
the other hand, to look into what particular erroneous procedure is
employed is not necessarily productive. Too often it is just
“magical” manipulations with numbers/symbols that may never be
replicated. There is a school of thought claiming that rather than
untangling the origins of misconceptions, it is easier to reteach from
“scratch”. An idea not without some merit.
For more detailed analysis of particular errors/misconceptions you may
try Google Scholar search. Many articles have full texts available
from authors’ pages or elsewhere. If you access to a library with good
subscriptions, then everything is even easier. Unfortunately, finding
a useful article is more difficult than one would expect.
One suggestion on how to deal with errors. Asking students to
articulate their processes in words, not formulas or symbols will
force some self-reflection. To save time, this can be a part of
students working in pairs/groups. This arrangement is likely to
require getting used to as well as simple and concise guidance
facilitating inter-student communication. In my experience, hoping for
big revelations among students who are new to mathematical
conversations is a sure way to disappointment.
There is one more thing about asking for explanations, that I find
easier said than done. It’s important to ask for explanations for
correct procedures from time to time too. Otherwise, the first
reaction to an explanation request will be “What’s wrong?”
On May 22, 1:27 pm, "Linda Fahlberg-Stojanovska" <lfahlb...@gmail.com>
wrote:
kirby urner
> Milo,
> Wow! Huge Kudos for your nephew. With that Vimeo video he's taking what
> Salman Kahn is doing to the next level. Also I really liked his commentary
> on the movie Waiting for Superman. It is the first truly intelligent
> critique that I've read. Instead of the usual knee jerk reaction to the
> movie being an anti-public school manifesto (which I don't think was
> Guggenheim's intent) he focuses on what the movie leaves out. I've taken the
> liberty of copying that portion of his blog entry below. (Taken
> from http://worrydream.com/KillMath/)
> -Ihor
>
I was a little distressed that the prey / predator sine waves (periodicity)
http://vimeo.com/23839605 was presented as a topic in differential
calculus. Not that we shouldn't bridge, but the notation is rather
intimidating right out of the gate, and not the only applicable set of
concepts.
Creatures eating each other and having populations is essentially a
discrete math problem. Games like SimCity and SimEarth show
flow diagrams, things affecting other things, without pretending to
have closed form solutions ala "diffy Q" ("diff eq").
Systems science is indeed deeply invested in these feedback
loops and the oscillations, chaotic interludes, these systems
contain.
However, this is more a domain for computer explorations, genetic
algorithms, artificial life and all that. Differential calculus has not
been "paving the way" in these respects. Look at Wolfram's NKS:
hardly a lick of calculus.
I have nothing against calculus per se, but I despise how it's used
as a filter, a "make or break" test of academic ability, pre-college
especially. What a shameful development, a wretched state of
affairs. In this is coming form a former high school calculus teacher,
with students who did well on the AP etc.
I do agree with the Bret's analysis of 'Waiting For Superman'
which documentary does waaaay to little to question the curriculum.
Lots was tragic in that film, like the little kid getting taken away
from his dear grandma to go to some boarding school that looked
ever so much like a prison in lock down.
If students were to boycott schools en masse until some real
reforms were instituted, including at the curriculum level, I'd be
sympathetic most likely (hearkens back to my 1990s "Math
Makeover" campaign).
The stuff they could do while on break, to self organize replacement
learning, would likely be enough better to make it impossible to
go back to "the bad old days" (i.e. now).
Kirby
Edward Cherlin
> On Thu, May 19, 2011 at 4:28 PM, Ihor Charischak <ih...@clime.org> wrote:
>> Milo,
>> Wow! Huge Kudos for your nephew. With that Vimeo video he's taking what
>> Salman Kahn is doing to the next level. Also I really liked his commentary
>> on the movie Waiting for Superman. It is the first truly intelligent
>> critique that I've read. Instead of the usual knee jerk reaction to the
>> movie being an anti-public school manifesto (which I don't think was
>> Guggenheim's intent) he focuses on what the movie leaves out. I've taken the
>> liberty of copying that portion of his blog entry below. (Taken
>> from http://worrydream.com/KillMath/)
>> -Ihor
>>
>
> I was a little distressed that the prey / predator sine waves (periodicity)
> http://vimeo.com/23839605 was presented as a topic in differential
> calculus. Not that we shouldn't bridge, but the notation is rather
> intimidating right out of the gate, and not the only applicable set of
> concepts.
We should present ideas like this in as many ways as we can.
Trig functions arise from triangles
Trig functions are easily animated using a revolving circle and a moving x axis
trig waves solve y´´ = -y, which goes with exponential y´ = y and
hyperbolic y´´ = y
The calculus falls easily out of various sums of x^n/n! with same
sign, alternating sign, even terms only, odd terms only
Exp(x) is a sum of hyperbolic functions. Exp(ix) is a sum of trig
functions in the complex plane.
Exp(ix) = cos(x) + i sin(x)
e^(iπ) + 1 = 0
Trig functions approximate the motion of a pendulum, or a mass on a spring.
The Sugar Measure activity does a Fourier analysis of any acoustic input.
I can go on.
> Creatures eating each other and having populations is essentially a
> discrete math problem. Games like SimCity and SimEarth show
> flow diagrams, things affecting other things, without pretending to
> have closed form solutions ala "diffy Q" ("diff eq").
>
> Systems science is indeed deeply invested in these feedback
> loops and the oscillations, chaotic interludes, these systems
> contain.
>
> However, this is more a domain for computer explorations, genetic
> algorithms, artificial life and all that. Differential calculus has not
> been "paving the way" in these respects. Look at Wolfram's NKS:
> hardly a lick of calculus.
>
> I have nothing against calculus per se, but I despise how it's used
> as a filter, a "make or break" test of academic ability, pre-college
> especially. What a shameful development, a wretched state of
> affairs. In this is coming form a former high school calculus teacher,
> with students who did well on the AP etc.
>
> I do agree with the Bret's analysis of 'Waiting For Superman'
> which documentary does waaaay to little to question the curriculum.
Indeed.
> Lots was tragic in that film, like the little kid getting taken away
> from his dear grandma to go to some boarding school that looked
> ever so much like a prison in lock down.
>
> If students were to boycott schools en masse until some real
> reforms were instituted, including at the curriculum level, I'd be
> sympathetic most likely (hearkens back to my 1990s "Math
> Makeover" campaign).
Children do not have human rights, however.
> The stuff they could do while on break, to self organize replacement
> learning, would likely be enough better to make it impossible to
> go back to "the bad old days" (i.e. now).
>
> Kirby
>
> --
> You received this message because you are subscribed to the Google Groups "MathFuture" group.
> To post to this group, send email to mathf...@googlegroups.com.
> 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.
>
>
--
Edward Mokurai (默雷/धर्ममेघशब्दगर्ज/دھرممیگھشبدگر ج) Cherlin
Silent Thunder is my name, and Children are my nation.
The Cosmos is my dwelling place, the Truth my destination.
kirby urner
> functions in the complex plane.
>
> Exp(ix) = cos(x) + i sin(x)
>
> e^(iπ) + 1 = 0
>
> Trig functions approximate the motion of a pendulum, or a mass on a spring.
>
> The Sugar Measure activity does a Fourier analysis of any acoustic input.
>
> I can go on.
>
I'm not discouraging you from doing so. 'Who Is Fourier?' remains high
on my list of favorite calculus teaching (LEX Institute).
Compare with O'Reilly Media's 'Head First' series, in terms of pedagogy
(andragogy).
'Concrete Mathematics' (Knuth et al) also an influence.
<< ... >>
>
> Children do not have human rights, however.
>
Many humans do not. Nor would they all pass the Turing test.
I'm for a Global University "floor" in terms of minimum acceptable
living standards. You need glasses? You get glasses.
Etc.
Whom do we contact? Food Services?
Curriculum matters.
Kirby
milo gardner
Intuition can be used to grasp hunter and prey diffy q. data. Nature's differential equations are not easily balanced in real life by man, as you conclude.
That is why I have chosen Egyptian food sources that were grown and used as money by 2050 BCE within a finite unit fraction system. Every commodity grown for wage payments was given given a scaled valued in equivalent grain hekats. Egyptians exactly measured the amount of grain used in a loaf of bread, a glass of beer and a quail, dove and geese fed over 30 days by an inverse proportion ... named pesu ... inventories were fairly allocated in decentralized estate and centralized estate granary systems as wages. During flood years, when commodity inventories fell, arithmetic proportions were developed that scaled back wages payments to every sector of the working class proportionally that approximated a modern differential equation analysis.
Note that 4,000 year old number rules worked very well for over 3,500 years. Unit weights and measures developed in Egypt were adopted by Greece, Coptics, Arabs and medieval Europeans before the 1454 AD break with Islam took place. After 1585 AD Europeans developed base 10 decimals and modified metric definitions that differed greatly from the unit fraction units use prior to 1454 AD. Sadly, modern institutional memory losses have needlessly clouded the Western Tradition's primal number system rules, and historical record, based on rational number finite arithmetic. The older arithmetic continues in use today, re-written into base 10 decimal terms, as anyone is free to independently discover.
My view is that children do not need to invent their own number systems. Yet, I did in small ways as a 10 year old. Box score information was double checked to study baseball teams and to value players. As a youth, Pacific Coast League statistics came out on Sunday. Sacramento Solon team statistics were updated after every game. Later ranking systems of players from all the teams were developed, some of which appear on Major League baseball sites today. Once numerate, gained from classrooms or personal hobbies and interests, students should be free to read the history of number before and after the development of base 10 decimals, and draw their own conclusions.
Each cultural math era offers important lessons to children and college students. My study of Egyptian math began reading a college history of math text book in 1963. Howard Eves did not fairly report the history of zero ( zero only appeared in Germany by 1200 AD per Eves). As an adult after 1987,a broader history of zero reported more than an exponent in the positional base 10 decimal system. Mesoamerican mathematicians used zero in a base 4, base 5 positional number system 2,000 years earlier (than Euro-centric historians oddly report as fingers and toes 'Piaget' base 20). Over 3,500 years years earlier in Egypt every rational number between the interval 0 to 1 was analyzed and structured to create the first money system. Zero was used as a critical limit that defined the first unit in its number system.
Breathing life into math history, absent its Euro-centric barriers, motivated students need assistance from math educators. Thank you one an all for participating in the intellectual war to educate youth on the real foundations of mathematics, empowered by number system rules from many cultures.
Best Regards,
Milo Gardner
From: kirby urner <kirby...@gmail.com>
To: mathf...@googlegroups.com
Sent: Tue, May 24, 2011 8:57:11 PM
Subject: Re: [Math 2.0] "Inventing" number rules
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To post to this group, send email to mathf...@googlegroups.com.
kirby urner
Planet Earth, Milo, at least stemming from the Egyptian branch,
which is huge.
With Bucky it was always back to Phoenicia, rhymed with Venetia
and the Veekings, and about those landlubbers getting trapped in
orthonormal XYZ thinking, a kind of Matrix (mind prison).
I've adopted some of his themes for Martian Math, one of a four
part Digital Mathematics curriculum sourced on Wikieducator.
Egyptian Math would map through Supermarket Math for me. We
could mix it with Python and its fractional number type.
I've been advocating for a school Baseball Statistics database
to write curriculum around. SQL (structure query language) is an
important part of what's to learn, and sports stats, world records,
athletic stuff in general, is a useful way to learn tabulation,
relational database concepts.
Yes, one may use predator / prey or Egyptian grain game arithmetic
to fathom diff eq stuff. No question.
However, whereas I used to teach calculus, I've since jumped track
and tend to a discrete math curriculum which fairly rigorously excludes
perfectly smooth continua (or perfectly solid solids) as a matter of
metaphysics.
We work in a quantized digital realm, and would be quick to point
out that those grains of wheat the Egyptians used were discrete
-- and concluding we need not surrender our right to study predator / prey
relationships on computer just because we're proudly eschewing
any analog math.
Kirby