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Hexadecimal numbers in the Primary Grades
Posted:
Dec 31, 2009 5:00 PM



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 ASCIItoUnicode 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.
00toFF is your range of options (permutations) for one byte, or eight bits.
00001111 is room for 16 single hex digits, 0F.
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 09 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 highlighted 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 decimaltohex 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 numbers and adding vertically, talking about place notation and carrying, 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 additional topics into view.[3]
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 importance of the hexadecimal type is already a given.
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



