A talk I gave at "Mathematics Education for the Mathematical and Physical Sciences" (http://mathsed.mst-online.org/default.aspx)

1. Mathematics Education for the Mathematical and Physical Sciences
Nov. 29, 2007
Digital technology and doing
or learning mathematics
Yishay Mor, London Knowledge
Lab

2. (right to left): (first line) 1 sheep, 1 unit of textile, 1 unit of
honey, 1 unit of oil, (second line) 1 unit of textile, ?, 1
ingot of metal.
Susa, Iran, ca. 3300 B.C.
Denise Schmandt-Besserat
http://www.utexas.edu/cola/centers/lrc/numerals/dsb/dsb.html
Courtesy Musée du Louvre, Départment des Antiquités Orientales, Paris.

3. Prior to 1940 the Oksapmin in Papua New-Guinea used a 27-body part count
system. Anything beyond 27 was “Fu!”, literally translated as ‘a bunch of many
things’.
Between 1940 and 2005 the meaning of “Fu” changed to 20 and then to “double X”
http://gse.berkeley.edu/faculty/gsaxe/Research.html

4. Quick exercise
CXL x
IV
_____
???

6. Some observations
• Mud and paper have different affordances.
• The technology by which we do maths
shapes the maths we do.
Communication
Cognition
Culture
Body
Environment

7. America, pre 1800
The colonial schools had no blackboards, slates, or maps […]
Almost all of the school supplies for pupils were homemade […]
In fact, a teacher was sometimes hired more for his ability to cut and
mend quills than for his ability to teach. […]
1961 U.S. Department of Health, Education, and Welfare, Office of
Education publication called quot;Technology in American education,
1650-1900.quot; in http://www.pballew.net/mathbooks.html

8. The one
technology
that
transformed
mathematics
education?

10. Some new slates
• MoPIX
• NetLogo
• WebLabs

11. http://remath.cti.gr/

12. NetLogo:
It’s a complex
world...
ccl.northwestern.edu/
netlogo/

13. 13

14. 14

15. ToonTalk

16. And back

17. 17

18. 18

19. 19

20. 20

21. number sequence activities
quot; Add-1 ► Add-a-num ► Add-up
quot; Guess my robot
quot; Convergence & divergence

22. activity I: basic secuences
• Construct the natural numbers
► any arithmetic sequence
► iterative sequences
• partial sums (function on sequence domain)

23. construct
1, 3, 6, 10…
1, 2, 3, 4…
3 4 5
2
6
1

24. explore and report

25. discuss

26. findings: process, structure, parameters
Just go into it, I have to change it but I won't
have to erase it, I could add 1 for that 1
there [points to add this hole], I could just
make it add 2. I could just change its
function to have to add 2.

27. findings: a new language?
In your Explore section you say that the triangle
numbers cannot be generated by your ADD robot since
quot;add thisquot; keeps changing. Isn't this true for the
sequence:
8, 16, 32, 64,...
which you claim could be generated by it.?
quot;add thisquot; is times 2 (*2)

28. activity II: guess my robot
quot; Develop a shared mathematical language.
quot; Gain proficiency in manipulating
mathematical tools to generate and
analyze sequences.
quot; Experience fundamental issues of
mathematical argumentation
conjecture, hypothesis testing, proof,
equivalence.

29. WebReport Emmy (Sofia)
Ana (Lisbon)

30. games kids like
I don't receive any comments to my sequence,
because is too easy...

31. tit for tat

32. old language

33. activity III: convergence & divergence
quot; Can a sequence get smaller and smaller and
never go below 0?
quot; What happens when you sum the terms of such
a sequence?
quot; How can you describe the differences between
one sequence and another?
quot; How do these differences affect the
convergence of the corresponding sum series?

34. Predict test reflect
This is the real graph that was
produced by the cumulate total of the
halving-a-number robot.
it looks like the top of my graph but i
made the fatal mistake of thinking it
started at zero.
I also said it wouldn't go over 100,
which was very wrong.

35. make your own

36. what did you like?
um, like the debates were great, about um, is
there a limit and can we prove it and um, also
I ve learnt a lot more on how to prove things
in Algebra, with the ak for the terms and
everything about which I didn t know before.
You have to build the robot yourself and so you
know exactly what it s doing.

37. and learn?
You gotta have a proper method instead of just,
like, um, try and fail
Its about logically thinking things through,
rather than just um, like you being told that
this is the equation you have to do and say oh,
yeah .
We sort of have to discover for ourselves, and
you then have to think through things more
logically. Think more in depth into things.

38. DrunkBot
http://www.weblabs.org.uk/wlplone/Members/michele/my_reports/Report.2003-11-25.5513

40. Stephen Wolfram (2002)
A New Kind of Science
http://www.wolframscience.com/nksonline/

41. Wolfram's 2,3 Turing Machine Is Universal!
October 24, 2007--Wolfram Research and Stephen Wolfram
today announced that 20-year-old Alex Smith of Birmingham,
UK has won the US $25,000 Wolfram 2,3 Turing Machine
Research Prize.
In his 2002 book A New Kind of Science, Stephen Wolfram hypothesized that a
particular abstract Turing machine might be the simplest system of its type
capable of acting as a universal computer.
In May 2007, the Wolfram 2,3 Turing Machine Research Prize was established
to be awarded to the first person or group to prove either that Wolfram's Turing
machine is universal, or that it is not.
Alex Smith was able to demonstrate--with a 40-page proof--that Wolfram's
Turing machine is in fact universal.
This result ends a half-century quest to find the simplest universal Turing
machine.
It demonstrates that a remarkably simple system can perform any computation
that can be done by any computer.

42. Take-away bullet points
• The tools we use shape the things we do, the
thoughts we think.
• Digital technologies allow us to confront old
ideas in new ways, and to construct new ideas
But!
• The medium is not the message.
What is your message?

43. References
• Saxe and Esmonde (2005).
Studying Cognition in Flux: A Historical Treatment of Fu in the Shifting Structure of
Oksapmin Mathematics. Mind, Culture, and Activity, (12)3/4:171-225, Lawrence
Earlbaum.
• Schmandt-Besserat (1992). Before writing: from counting to cuneiform. University of
Texas Press, Austin.
• Cajori (1929) A history of mathematical notation. Vol. 1.: Notations in elementary
mathematics. The Open Court Publishing Co.
• Peggy Kidwell. Slates, Slide Rules and Software: Teaching Math in America. 2002.
• Mor, Noss, Hoyles, Kahn and Simpson (2006).
Designing to see and share structure in number sequences. the International Journal
for Technology in Mathematics Education, (13)2:65-78.
• Cerulli, Chioccariello and Lemut (2007). A micoworld to implant a germ of probability.
5th CERME conference - congress of European Society for Research in Mathematics
Education.

44. Links
• http://gse.berkeley.edu/faculty/gsaxe/Research.html
• http://www-history.mcs.st-andrews.ac.uk/
• http://en.wikipedia.org/wiki/Egyptian_numerals
• http://www.utexas.edu/cola/centers/lrc/numerals/numerals.html
• http://americanhistory.si.edu/teachingmath/
• http://remath.cti.gr/
• http://ccl.northwestern.edu/netlogo/
• http://www.lkl.ac.uk/kscope/weblabs/
• http://www.wolframscience.com/nksonline/