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Replies: 6   Last Post: Jan 2, 2010 1:26 AM

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 kirby urner Posts: 3,690 Registered: 11/29/05
Posted: Dec 31, 2009 5:00 PM
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BACKGROUND: REAL NUMBERS

We've been discussing whether to do dwell on the real number or fraction
type first, with Bill Marsh suggesting a "reals first" approach based on
intervals, a way of using binary sequences to progressively narrow the
location of a specific real number.

In executable logic (computer languages) we also have the float type,
decimal type, other species of number, none of which re precisely the real
type, per a mathematicians understanding. Reals include irrationals and
transcendentals (a subspecies of irrational) and so can't be expected to
conform to the needs of our finite state machines.

RELATED QUERY

This brings me to a related question for the digital math teachers: when,
how and where would you introduce hexadecimal numbers?

STORY PROBLEM

Here's an application, a story, some lore, just to set the stage.[0] Just
imagine the textbook if you will (like some PDF going around on an OLPC/XO2
someplace).

"""
Lindsey Walker, a popular musician in Portland, Oregon, takes an Apple
laptop to each venue where she plays and records four audio tracks for later
editing and converting to mp3s in Cubase. She then uploads the mp3s to her
web site.

The program hangs during the recording and the files on the hard drive do
not open in Cubase. Upon consulting the Internet, she realizes solving a
little math puzzle may fix the problem. She just needs to pull up the
improperly formatted file in a hexadecimal editor and figure out which
fields in the header might need to be filled in or corrected.

She has several uncorrupted files to use as examples. She has the following
information off the Internet...

<< snip >>

"""

RATIONALE

We live in a world thoroughly steeped in hexadecimal numbers, little endian
and big endian, and even middle endian (exotic species).

The story of ASCII-to-Unicode won't make as much sense minus the hexadecimal
piece, and that's a story we really want to tell, along with some stuff
about SQL (keeping tabs, Hollerith Machine, IBM...).

The jump to thinking in other bases besides 10, a core feature of New Math
in the 1960s, was in fact an important jump.[1]

I'm being abbreviated here as these themes are in no way new with this
author.

00-to-FF is your range of options (permutations) for one byte, or eight
bits.

0000-1111 is room for 16 single hex digits, 0-F.

0000 - 0
0001 - 1
0010 - 2
0011 - 3
0100 - 4
0101 - 5
0110 - 6
0111 - 7
1000 - 8
1001 - 9
1010 - A
1011 - B
1100 - C
1101 - D
1110 - E
1111 - F

That's 0-9 then A,B,C,D,E,F where A=10, B=11... F=15. That's 16 values in
all, counting 0 as one of them.

When Lindsey sees the total file size in decimal (a high-lighted number),
she knows she will need to convert it to hex before entering it into the
appropriate slot in the header. The example files cue the student to these
facts of life.

The exercises take it slowly, step by step, providing hints along the way.
Our storybook problem solver is free to use a decimal-to-hex converter, a
decimal calculator.[2]

So my question for math teachers here is: given we need to cover
hexadecimals without making students take calculus first, or even a lot of
algebra, where should we put them in terms of state standards (or university
standards if you're more like an MIT)?

My answer: right when we're first doing addition in base 10, stacking two
relating to the abacus. This is what New Math did right by investing
considerable time on positional notation, with an aim of then introducing
other bases besides 10.

Per Saxon and spiralling: just because we visit a topic does not mean we
never go back to it. On the contrary, every important topic gets repeated
attention, as the curriculum spirals forward through time, bringing

However, I would not spend equal time on all bases, nor develop our lesson
plans as if we must give equal time to bases 29 or base 91.

The hexadecimal numbers have special importance in engineering and getting
more practice with them is a curriculum objective. I'm speaking of our
variously branded Silicon Forest digital math curricula, where the

Kirby

[0] An interesting aspect of this story problem is it's all true.

[1] I'm quite aware that Tom Lehrer satirized the New Math in his song by
that title. Let's not forget Lehrer was a math teaching professional (his
day job) and that his lyrics were mathematically correct. He was
entertaining the adults while giving the nod to kids like me that he knew
and appreciated what we were learning.

[2] The former might be on a web page, the latter a part of the standard GUI
calculator that comes with MacOS X.

[3] note that I cite Saxon specifically, in connection with spiralling, in
my Python for Teachers handout for usa.pycon 2009 (venue: International
Room, Hyatt Regency near O'Hare, Chicago).
http://www.4dsolutions.net/presentations/p4t_notes.pdf

--
>>> from mars import math
http://www.wikieducator.org/Digital_Math

Date Subject Author
12/31/09 kirby urner
12/31/09 Michael Paul Goldenberg
12/31/09 kirby urner
1/1/10 Bishop, Wayne
1/2/10 Michael Paul Goldenberg
1/2/10 kirby urner
1/1/10 Robert Hansen