Digital technology and doing or learning mathematics

Yishay Mor, London Knowledge Lab

Mathematics Education for the Mathematical and Physical SciencesNov. 29, 2007

(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-Besserathttp://www.utexas.edu/cola/centers/lrc/numerals/dsb/dsb.html Courtesy Musée du Louvre, Départment des Antiquités Orientales, Paris.

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

Quick exercise

CXL x IV _____ ???

Some observations

• Mud and paper have different affordances.• The technology by which we do maths

shapes the maths we do.

Environment Body

Culture

CommunicationCognition

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 "Technology in American education, 1650-1900." in http://www.pballew.net/mathbooks.html

The one technology

that transformed mathematics education?

Some new slates

• MoPIX• NetLogo• WebLabs

http://remath.cti.gr/

NetLogo:

It’s a complexworld...

ccl.northwestern.edu/netlogo/

13

14

ToonTalk

And back

17

18

19

20

number sequence activities

" Add-1 ► Add-a-num ► Add-up

" Guess my robot

" Convergence & divergence

activity I: basic secuences

• Construct the natural numbers

► any arithmetic sequence

► iterative sequences

• partial sums (function on sequence domain)

p

construct

1

23 4 5

6

1, 2, 3, 4…

1, 3, 6, 10…

explore and report

discuss

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.

findings: a new language?

"add this" is times 2 (*2)

"

In your Explore section you say that the triangle numbers cannot be generated by your ADD robot since "add this" keeps changing. Isn't this true for the sequence:

8, 16, 32, 64,...

which you claim could be generated by it.?

activity II: guess my robot

" Develop a shared mathematical language." Gain proficiency in manipulating

mathematical tools to generate and analyze sequences.

" Experience fundamental issues of mathematical argumentation� conjecture, hypothesis testing, proof,

equivalence.

Ana (Lisbon)

A

WebReport Emmy (Sofia)

games kids like

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

tit for tat

� old� language

activity III: convergence & divergence

" Can a sequence get smaller and smaller and never go below 0?

" What happens when you sum the terms of such a sequence?

" How can you describe the differences between one sequence and another?

" How do these differences affect the convergence of the corresponding sum series?

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.

make your own

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.

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.

DrunkBot

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

http://video.google.com/videoplay?docid=726806631713175370

Stephen Wolfram (2002)A New Kind of Science

http://www.wolframscience.com/nksonline/

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.

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?

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.

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/