Developments in Interactive Visualization and Physics Simulation with Cinderella Study of Mathematical Software and Its Effective Use for Mathematics Education

1

Introduction

In this article I will describesomefuture directions ofthe Interactive

Geometry System

Cinderella,

summarizing

the

_{presentations}

at the RIMS

_workshops

in 2014 and 2016. Cinderella

_[10]

is a DGS that has been

published

in

Japan

[11]

in 2003. Instead of

describing

the

_system

infull

_depth

Irefer toa

presentation

given

at a RIMS

workshop

in the same _year. In

[1]

we describe the current state of

_Cinderella,

some advanced

learning

scenarios,

and we

_give

an outlook for the future of such interactive

_teaching

tools. In this article I will recallsomeof the

_aspects

presented

there and howthesehave

been

_implemented

and used —

or not —

during

the last decade. I will also describe

some of the current

_challenges

for

_DGS,

in

_particular

in the areas of 3\mathrm{D}

input

and

output

and how

_{technology development}

will enable ustoblend the

_physical

worldwith

abstract microworlds realized inaDGS.

Furthermore,

I will

point

outhow

_{technological}

development

made it

_possible

tostill

_present

Cinderella content onthe

web,

in

_spite

of

the death ofJavaonthe client side.

2

A decade of DG8

Since the introduction of DGS in the end of the 1980’s more than 30 years have

passed,

and some

teaching

scenarios that have been described in 1990’s have become

commonplace

in the classroom. In 2001 the DGS GeoGebra was created

Uy

Markus

Hohenwarter and since then it became

_popular,

in

_particular

because it haslater been

published

as

Open

Source

Software,

as

opposed

totheDGS Geometers’

_Sketchpad

and Cabri G?om?tre.

_Currently,

GeoGebra is the

_predominant

softwareavailable and_many

people

areconvinced that it is the

first,

the

only,

and the bestDGSthat has been and is

available. While thisisnot truein its

_generality,

itcanbe

acknowledged

that GeoGebra was abletomake DGS atool that is

being

used

regularly

_{in many classrooms.}

lTheresearchleadingtothese results has received_fundingfrom the_EuropeanUnionSeventhFrame‐

work_Programme

_{(\mathrm{F}\mathrm{P}7/2007-2013)}

under_{grant agreement}No. 610467—_project \mathrm{M}\mathrm{C}_Squared”. This publicationreflects_onlythe author’sviewsand Unionisnotliable for_anyusethat_maybe madeof the information contained therein. Theauthor has been_{supported by}the_{Japan Society}for the Promotion of Science underaShort TermFellowshipfor Researchin_Japan.

.. \prime ’

\mathrm{s}'\searrow.\cdot.

\mathrm{H}^{\backslash }1

: The

angular

bisector used

repeatedly.

On the

right

the

_point

\mathrm{B} has been _moved once around

point

\mathrm{A},

leading

to another instance in the

configuration

space of the

construction.

2.1

_Experimental

Mathematics

Thefirst

_learning

scenario described in

_[1]

is

_{doing Experimental Mathematics,}

where

aDGSis used to

explore

not

_only

one but many

examples

ofa

_geometric

construction. In

_particular,

_using

a

geometric

locus it is

possible

to

_explore

several

_{configurations}

at thesametimeandto

_change

the

_parameters

for_{many of them}

_{simultaneously.}

Thiscon‐

cept

hasbeen identified no

only

inInteractive

Geometry

but also as a

general technique

inInteractive Visualization

_by

Victor

_[13].

_However,

as describedin

_[3]

_{it is necessary} that the mathematical

_theory

_{behind any}

_system

usedtodomathematical

_experiments

mustbe

_highly

consistent anddeliver correctanswers eveninunforeseen

configurations.

The

_approach

ofmost

_geometry

software isto startwith the

requirements

for

_working

on basic

examples

_coming

from

secondary

school classrooms. But as soonas circles are

used inaconstruction thesoftware hastohandleboth the

_problem

_{of“vanishing”}

inter‐ sectionsand the

_problem

of

_multiple

intersections.

_Still,

Cinderellais the

_only

software that

_consistently

and

_correctly

handles such situations. It must be

_acknowledged

that

westill lack

examples

where thisindeed harms the

teaching

with

DGS,

but thiscould be

relatedtothe fact that DGS arenotused for freeand creative

_{experimentation}

thatcan

leadto

_deep

_insights,

but because

_they

areused in

teaching

scenariosthatarerestricted

toa_{very clear}

teaching goal.

This asks for

_empirical

studiesonclassroom

practice

with

respect

to DGS useand

_{experimentation.}

2.2

Mathematics

on

the Web

In 2003 we described how interactive Mathematics can be

represented

on the web.

The last years have shown that this is a_very

important

direction,

in

particular

with the

trendof

_Open

EducationalResources

_(OER),

that

_is,

freeto use,

modify

andredistribute material for

_teaching

and

_learning.

There are numerous websites that offer interactive content created with DGS. Iwould like tomentionthree ofthem.

>2

: The

Intergeo portal

(

\mathrm{i}2\mathrm{g}\mathrm{e}\mathrm{o}

.

net)

is

specialized

forcontent

using

interactive geom‐

etry

software.

1. The

_Intergeo

_project

_{[4, 6] (Fig. 2)}

createdaweb

platform

where interactive con‐

tent can be collected and

categorized

according

to

_{topics, competences}

and edu‐ cational levels.

Intergeo

solvedsome

problems

thatoccurinthecontextof

_searching

and

_finding

re‐

sources,in

particular

with

respect

to

_{multilinguality.}

Allresourcescanbe

equipped

with metadata that is

_{language‐independent}

and reflects

_topics

and

_competences

that are mentioned in curricula.

Also,

resources can be rated

_using

a

_question‐

nairethat builds ondidactical criteriaforinteractive media in

teaching.

Still,

the

platform

has not been

_{widely adopted,}

_probably

due to the additional overhead when

_adding

material and

_suboptimal

_usability

when

_searching

for content. 2. Thesetwoissuesaresolved withGeoGebranbe

(http:

//\mathrm{w}\mathrm{w}\mathrm{w}

.

geogebratube.

com),

which is

_{tightly integrated}

intothe GeoGebra Software. Youcan

publish

construc‐ tionsfrom GeoGebra

_directly

to

GeoGebraTube.2.

Onthe

_platform

_they

areavail‐

able without

_prior

installation of the

_software,

not

_only

forJava‐enaUled browsers but also for modern browsers

_using

HTML5. In

_particular,

theses constructions

2Actually,

this isnow the_{only supported} wayto _publishGeoGebra constructionsonthe web, as

the_originalHTMLexporthas been removed. This is a questionable_move, asthe terms ofuse and the_privacystatement of GeoGebraTube haveto be_acceptedifyouwant to _publishyourwork there

orembedcontentfrom therein yourownpages?whichisnot_compatiblewith theOpenEducational Resource idea.

can be used ontablets aswell

(see

alsoSec.

2.6).

The

_approach

behind GeoGebraTube is

_{community‐based:}

_Anybody

can

publish

his or her

work,

and

everybody

can comment on and rate resources. This allows

for a

huge

amount ofmaterial

_(about

140000 resources as of

January

2015),

but

the collection lacks structure and coherence. This shows the

_strong

connection to a

platform

like YouTube: You can find millions of videos

there,

but the

only

structureis

_given

_by

channels.

3. Mathe Vital

_{(Fig. 3)}

follows a different

approach

than GeoGebraTube. This col‐

lection has been created

_by

asmall team lead

_{by Jürgen Richter‐Gebert,}

andit is neither

_possible

to _{add your} own content nor to comment on resources. This

restrictionleads tothefactthat thecontent isof_very

_high

_quality

and

_consistent,

both in

_design

and inmathematical

_terminology.

MatheVital and GeoGebraTube show two extremes on a scale: A

large

community

can createthousands of_{resources, but the}

_diversity

ofthese resources makes itdifficult

for theusertousethecontent to_{learn without proper}

_guidance.

A smallteam can

only

create small

_collections,

but these

_might

be better suited for

_{unguided learning.}

There is noreason to

_prefer

one

approach

overthe other—for both

approaches

there are use

cases, and

depending

on the actual demands either

approach

can be better than the

other. It is

_interesting

to seethat these two

_approaches

seem to be a common theme

in

_todays

world: For all

types

of resources

(news

stories,

videos, music, recipes,

..

.)

there exist

_platforms

and

_portals

that either offer a

huge

amount of resources where

youcanselect from

_using

various

_criteria,

orcurated collectionsthat consist of

_carefully

pre‐selected

ormanufactured

resources.3

2,3

Interactive Exercises

Interactive Exercises have been a

major

feature of Cinderella since 1997.

However,

these have

hardly

been used and the feature was more or less removed in the latest

version. In our

view,

there aretwo reasons for the lack of

adoption

ofthis feature:

(1)

It was

always

difficult to use the built‐in Exercise Editor for

_creating

good exercises,

and

₍₂₎

in

_teaching

it is difficult to find situations where it makes sense to use such

self‐checking

exercises with automatic feedback.

During

thelast decade

_Learning

_Management

_Systems

_(LMS)

becamemoreandmore

popular.

A

_{good example}

is

Moodle4

that has beencreated in 1999‐2002 and isnow one

of themost

_popular

LMS worldwide. Such

_systems

allowfortheinclusion ofinteractive

content,

and we see this as a chance to revive the interactive exercises in Cinderella.

3You areinvitedtofindoutof what_typethe_{following portals}are: _{Twitter, Google News, Spotify,}

TV stations. Findmore_examples!

3

: The Mathe Vital

portal

(www.

mathe vital.

de)

offers a

huge

collection of

high‐

quality

interactivevisualizations. This is

_just

aoverviewoversomeofthem demonstrat‐

ing

the consistent

_design

of the various visualizations.

Here,

we have a

_system

that is

capable

of

recording

student’s _scores, soit makessense

to include such exercises in courses. First

approaches

for this

_scoring

are

currently

implement

inthe context of theMC

_Squared

_project.5

It should be noted that

_recently

_{(September 2014)}

aMoodle

plugin

has been

published

that allows for

_setting

_up

_questions

that canbe solved and checked

automatically using

GeoGebra

_(https:

//moodle.

_{org/plugins/view.}

_php?plugin

_qtype

_GeoGebra).

Un‐

fortunately,

there isnoadvanced

_support

for

checking;

the correctness ofa construction

just depends

on a

single

boolean value that must be created

_by

theauthor ofthe Geo‐ Gebra construction.

2.4

Interactive Whiteboards

Interactive Whiteboards are

widespread

in _{schools in many countries.} All manufac‐ turers deliver software for

_writing

on these boards and drivers to work with standard

desktop

software. There are even _many

applications

that are

specifically

designed

to

support

mathematics

_teaching.

Our

_experience

shows that thissoftwareis

_usually

not

up‐to‐par

with standard DGS

_systems.

_So,

most teachers who use mathematical soft‐

ware onwhiteboardsatusethemwith thesoftware

they

knowfromthe

Desktop

(usually

GeoGebra).

Advanced

_approaches

like \mathrm{e}‐Chalk have almost vanished.

A very

promising

alternative software is

_(or

_was, as it appears to be

discontinued)

Open‐Sankoré,

formerly

Uniboard.6

Thissoftwareisabletoinclude

_widgets

basedonthe

5http:

//\mathrm{m}\mathrm{c}2‐project,eu

\mathrm{W}3\mathrm{C}

standard7,

soit is

possible

touseHTML5‐based

_geometry

software,

evenwith_per‐

sistentdata

_{storage. Still,}

thereare no

capabilities

for

cross‐widget

communicationasin

the API

_{specification}

for

_widgets

inthe MC

_{Squared project.}

Basedonthe observations

abovewe claim that it is_{necessary to}

develop

a commonstandardfor the inclusion of

interactivecontent inLMS

_(coming

from SCORM and

_LTI)

andinwhiteboard software

(coming

from the \mathrm{W}3\mathrm{C}

_standard).

2.5

_Sketching

Themost

_promising

_approach

to

_Sketching

sofar ist

Sketchometry,

aweb‐based_geom‐

etry

software that uses

gesture

recognition

for

input

(http:

//\mathrm{d}\mathrm{e}.

sketchometry.

org,

Fig.

4).

This isdifferent from the

_approach

that has been

_implemented

in Cinderella and thatwas

presented

in

2003,

wherewe triedto

_{automatically}

_generateconstructions

from a

drawing.

In that

scenario,

users should draw

exactly

like

they

draw with a

pen. The necessaryrelations between

objects

ina

drawing

are

subject

to

_{sophisticated}

guessing

of the software: A line that is almost

parallel

to another line should become

a constructed

parallel,

aline that ismore or less round and _goes

through

three

_points

should become a circle defined

by

these three

points,

but if it _goes

through

one

_point

only

and there isa

_point

inthe center of

_gravity

of that line than it should be a circle

defined

_by

_midpoint

anda

_point

onthe

circle,

etc. Such

guessing

doesnotworkinsome

situations,

so we

implemented

additional

hinting by

pre‐selection

of elements. If _you

taptwo

_points

toselect them and then drawa

point

more orlessinthe centerof these

two

_points,

then the software will be more

_likely

to _{guess thatyou}meanthe

midpoint

of thetwo

_{pre‐selected}

_points.

_However,

we are

_missing

data that could show whether

this

_input

method issuitable for theuse inthe

classroom,

in

particular

the _necessary

deviceswere not

widely

availableten years ago.

Compare

thisto

_{Sketchometry:}

_Here,

a_gesturefor the

_midpoint

is

_used,

aline_segment

that containsasmall

loop

toindicate that instead of thesegment, the

_midpoint

of that

segment

shall be constructed. So far it is unclear whether this

_approach

is better or

not tothe

_{interpretation}

of

_drawings

with

_{pre‐selection,}

but the research team at the

University

of

_Bayreuth

is

_currently

_{investigating}

thisin aschool.

2.6

Mathematics

on

Mobile Devices

Mathematicsonmobile devices isa

rapidly

_growing

areaof interest. While therewere

first devices

_capable

of

_{high‐resolution}

color

_graphics

in the

_beginning

of the

_century,

we now have mobile

phones

and tablets that are

superior

to the

_laptop

_computers of that time.

_Actually,

we can assume that within the _{next ten years every} student in

developed

countries will have accesstoamobile device thatis

_powerful

_enough

todo all

EZ 4:

_Example

_gestures in

_{Sketchometry.}

Source: Documentation of

_Sketchometry

at

http:

//\mathrm{d}\mathrm{e}.

sketchometry.

\mathrm{o}\mathrm{r}\mathrm{g}/\mathrm{d}\mathrm{o}\mathrm{w}\mathrm{n}\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}/\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{e}-\mathrm{a}4

5: PhotoMath isan _app that solves maths

problems

from books

_{simply by taking}

a

picture

of them. Itnot

_only

shows the

_solution,

but it isalso

_possible

to_get

_{step‐uy‐step}

explanations.

Source: PhotoMath website

_https:

_//photomath.

_{net/en/presskit}

school mathematics. A

_stunning

example

for what

already

is

_possible

isthe PhotoMath

App

(Fig. 5).

A

_{technological}

drawbackisthat dueto

_political

reasonsthe universal Java

_platform

wasabandonedfor mobile devices.

Although

Javawas meant tobe “writeonce—

run

anywhere”’

and there were efficient Java Virtual Machines

_running

on

early

devices

(for

example,

on Windows CE or the

Sharp

Zaurus),

this hasnot been

_pursued.

ln‐

stead,

Google, Apple

and other

_large

_{companies pushed}

_JavaScript,

which is available

in Internet browsers bothon

Desktops

and mobile devices. The concentrationon this

technology

made it

_possible

tohave interactive content that is

_smoothly

_integrated

in

HTML_pagesandatthesametimemade

JavaScript

asfastasor evenfaster than

Java.9

8\mathrm{S}\mathrm{e}\mathrm{e}alsohttps://\mathrm{e}\mathrm{n}.wikipedia.\mathrm{o}\mathrm{r}\mathrm{g}/\mathrm{w}\mathrm{i}\mathrm{k}\mathrm{i}/Write once,‐run‐anywhere

9Asa_{starting point}Irecommendreading http://\mathrm{w}\mathrm{w}\mathrm{w}.royvanrijn.\mathrm{c}\mathrm{o}\mathrm{m}/\mathrm{b}\log/2012/07/java speed \mathrm{o}\mathrm{f}-\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}/, which_points outthateven in2012 _JavaScript could befasterthan Javawhen usedout‐of‐the‐Uox,

However,

mostinteractiveacademiceducational softwarehasbeen

_produced

in Javaso

far.

_Currently,

_manysoftware

_developers

switchto

_{JavaScript/HTML,}

either

_{by starting}

from scratchor

_using

_compensating

_technology

like the

_Google

Web Toolkit GWT. For

example,

GeoGebrais

_using

thesamecodebaseforboth its Java and

_JavaScript

_version,

while

_JSXGraph

has been written in

_JavaScript

as a

replacement

for GEONExT. For mobile

_devices,

the software that has been written

_natively

in

_JavaScript

is

_generally

faster and better

_adopted

to the device. The GeoGebra

_App,

for

_example,

_{is very slow} forreal‐world

_examples,

while

_{JSXGraph performs smoothly.}

Somemore

thoughts

with

regard

tomathematics education softwarecanbefoundin

[5],

for

example.

There isone

important

lessontolearn: It iseven more

important

to

_design

a

good

userinterface for

mathematicalsoftwareonmobile devicesthan it isfor

_{desktop applications}

—and it is

not

_helpful

at allto

_just

copy theuserinterface from the

desktop

tothe mobile device. For

_Cinderella,

we decided not to

just

translate the software fromone

_programming

language

to the

_other,

but to start from scratch in

_JavaScript

and

_provide

a soft‐ ware

library

named

CindyJS

that includes the

_major

technology

_{necessary for}

_running

Cinderella‐Uased

_examples

_[14].

This includes the

_{scripting language}

_CindyScript,

witha

full

_compiler,

the

_physics

simulation

_framework,

and the

_geometry

kernel

_including

com‐

plex tracing.

Allthisisavailableas

Open

Source Softwareat

_https:

//

_github.

com

_CindyJS.

Using

these

_components,

we can

provide

content that has been created in Cinderella

via an

_export

in interactive websites that work on_any

device,

in interactive electronic

books in

_Apple

_iBooks,

as standalone _appsfor

_smartphones

and

_tablets,

or in creative

books

₍

\mathrm{c}

‐books)

as

produced

inthe \mathrm{M}\mathrm{C}

Squared

_project

(http:

_{//\mathrm{m}\mathrm{c}2}

_‐project.

eu).

Fascinating enough,

it is

_possible

toaccess modern hardwarefrom

CindyJS

that has

not been available to Cinderella before. In

_particular,

it is

_possible

to access the 3\mathrm{D}

hardware that is

_present

in

_{todays’ devices, allowing}

for

_extremely

fast 2\mathrm{D} and 3\mathrm{D}

rendering.

Using

the

_{easy‐to‐learn}

_{CindyScript language}

it is

_possible

to execute code in

_parallel

on the

graphics processing

unit

(GPU) [8].

\mathrm{p},7

Collaboration

As for collaborative work the

_{disappointing}

_{message is that} this

_aspect

seems to be

neglected currently.

Although

technology

like multitouch interactive

_tables,

fastwireless

networking

with

_{Wi‐Fi, Bluetooth,}

and NFCinvitefor

_cooperation,

the classroom

_reality

is that either students work alone with their

_devices,

or the teacher is

using

a device

to

_present.

_Although

_many

_people

are

using

shareddocuments

(through

Google

Docs,

iCloud/iWork

or similar

infrastructure)

and work with them at the same

_time,

this

trend has not been followed

_by

DGS users. The

sharing

of constructions is

_usually

asynchronous

andnot

_{simultaneously.}

It seems that teachers and students

prefer

to (

(\mathrm{o}\mathrm{w}\mathrm{n}”

a construction— and there are

In

_[1]

we mentioned four directions of future research. Let us check what has been

achieved!

2.8.1 Visualization of

_Algorithms

In

_2003,

we

imagined

a

_system

based on Cinderellathat is able to run and visualize

algorithms. Meanwhile,

the

_Visage

_system1

has been

_presented.

With

_Visage

it is

possible

tovisualize

_{(not only)}

_Graph

_Algorithms

and

_manipulate

them

_visually.

In

_[2]

we describe how this can be used in

teaching.

2.8.2 A School PDA

As

_pointed

outinSec.

_2.6,

mobiledevices have

_{developed enormously. Moreover,}

_many teachers and even students

already

own these

capable devices,

like iPads or Android

tablets.

_Actually

this has rendered the vision of a School PDA

mostly

_unnecessary.

There is

_only

onereason left to have

specialiled

devices

only

for

mathematics,

which

is the use in examinations. A

problem

of tablets and mobile

phones

is that

_they

are

too

_powerful

and also have a connection tothe

_Internet,

so —

at least in

_Germany

—

theiruse in examinations is

prohibited.

The ban onsuch devices is amatter ofdebate for thenext_{years: In school} and

_particularly

in

_mathematics,

students should learnto

use _proper tools for

solving

their

problems,

and mobile

_computers

and the Internet are

proper toolsthat arealso used for

solving problems

in

university

mathematics. Instead

of

_fighting

these devices

_they

should Ue embraced and the curriculum should reflect their use,

leading

toeven

deeper insights

inmathematics.

2.8.3 Natural

_language

_input

The

_input

of construction via written

_language

wasanotherCinderellaresearch

project

in 2003. After the

prototype

implementations

this has not been

_pursued

_any

_further,

as it did not seem relevant for

day‐to‐day teaching.

With the

availability

of

_spoken

2\wedge _{ié\hat{\vee}}

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2pointsize(3);

3 _{$\kappa$ \mathrm{m}u\mathrm{n}\mathrm{d}((\mathrm{F}.\mathrm{y}+4\}*\mathrm{l}\mathrm{e}\mathrm{e})}; 4repeatCn, \mathrm{i},

5 \ovalbox{\tt\small REJECT}=\mathrm{s}\mathrm{q}\prime \mathrm{t}\mathrm{l}\mathrm{n}-\mathrm{i}\mathrm{I}\ovalbox{\tt\small REJECT}.2: 6 pointcolor(\mathrm{h}u\mathrm{e}\{\mathrm{i}/21*\otimes.7\}:

7 \mathrm{d}\mathrm{r}m(\mathrm{A}+oe[\mathrm{s}\mathrm{i}n(\mathrm{i}* $\nu$\}.\cos \mathrm{l}\mathrm{i}*\mathrm{W})\}\}:)

8 9

6

: Some

sample

code in

CindyScript

creating

asunflower

_pattern

language input,

for

_{example through}

Siri on \mathrm{i}\mathrm{O}\mathrm{S}

(or

the

Speechkit

library),

it

_might

become relevant

_again,

in

_particular

for

_handicapped

users whocannot use a mouse or

keyboard,

but stillwant towork with

_Dynamic

_Geometry

_{Systems. So,}

while there has been no _{progress in}the last

decade,

we think that

somebody

should start

_working

on

such aresearch

project again.

2.8.4

_Scripting

and Macros

Forexact formulationsof construction_sequences, for

_{fine‐grained}

controloverthe be‐

haviorof interactive

_figures,

orfor easier

manipulation

in

_{animations, just}

toname afew

examples, using

a

_scripting

language

bears_many

_advantages.

_Actually,

what has been discussed as a future

development

in 2003 has been realized

_through

the

_CindyScript

language

[9].

We refer tothe full manual

_[12]

_here,

which

_gives

details tothis

_language.

Also,

at

_https:

//\mathrm{d}\mathrm{o}\mathrm{c}.cinderella. de the full documentationisavailable

online,

inEn‐

glish

aswell as in

Japanese,

due to a translation

_provided

_by

$\lambda$'\Leftrightarrow^{\backslash }\mathbb{P}_{0}

. One

important

project

that has

_only

become

_possible

due to the available of the

_scripting

_language

CindyScript

KETCindy.11

As noted

_earlier,

the

_CindyScript

_language

can be used with the

JavaScript imple‐

mentation of Cinderella as well. It is even

possible

to write code in the

_CindyScript

language directly

embedded intoHTML.

ideas should not be ruled

_just

_by

technical

_feasibility,

but it cannot be detachedfrom it. Inorder to knowwhat and how toteach mathematics educatorsmust know about thetools thatare available. Thesecan

_suggest

new

approaches

andnew

_topics.

Onthe other

_hand,

mathematics itself should

_stay

_independent

ofthe tools that are available

—

the tools shouldfollow.theneedsof

_teaching,

notthe

_teaching

shouldfollow the tools. So this has tobe a

dialogue

between the tools and the

_teaching,

to be held

_by

maths education.

The other

_important

direction that

_might

have been

_neglected

forawhile is towork

onthe mathematicalfoundations of the softwaretoolswe are

using.

The

(micro‐)worlds

created

_by

softwareare

something

that is

experienced,

and this

_experience

can

only

help

learning

mathematics ifit matches the mathematical

_content,

and the mathematical

objects

inthese worlds should behave

_correctly.

\mathrm{g}_{,}\sim

き \mathrm{X}\mathrm{H}\not\simeq \mathrm{A}

[1]

Enno Brehm and Ulrich

_Kortenkamp,

Advanced

_Teaching

of

_Geometry

with In‐ teractive Tools. In

_{Ryosuke Nagaoka, Hideyuki Ishi,}

and Eckhard

_{Hitzer, editors,}

Proceedings

of

RIMS‐

_{Workshop ITMga 2003,}

number

_{1378, Kyoto,}

2004. RIMS.

[2]

Andreas Fest and Ulrich

_Kortenkamp.

_{Teaching graph algorithms}

with

_visage.

Teaching

Mathematics and

_{Computer Science,}

_7(1):35-50

, 2009.

[3]

Ulrich

_{Kortenkamp. Experimental}

mathematics and

_proofs‐

what issecure math‐

ematical

_knowledge?

Zentralblatt

_für

Didaktik der

_Mathematik,

_36(2):61-66

,

April

2004.

[4]

Ulrich

_{Kortenkamp. Interoperable}

Interactive

_Geometry

for

_Europe.

The Electronic Journal

_of

Mathematics and

_Technology,

_5(1),

2011.

[5]

Ulrich

_Kortenkamp.

Interaktives

_whiteboard,

iPad& co. —das Klassenzimmer der

[6]

Ulrich

_Kortenkamp,

Axel M.

_Blessing,

Christian

_Dohrmann,

Yves

_Kreis,

Paul Lib‐

brecht,

andChristianMercat.

_{Interoperable}

interactive

_geometry

for_europe‐first

technological

and educational results and future

_challenges

of the

_{intergeo project.}

In

_{Proceedings of}

theSixth

_{Congress of}

the

_European

_{Society for}

Research in Mathe‐ maticsEducation.

_January

28th‐

_February

1st

_2009,

_Lyon

_(France),

_Lyon

_(France),

January

2009.

_Proceedings

ofthe Sixth

_Congress

of the

_European

_Society

for Re‐ search inMathematics Education.

[7]

Ulrich

_Kortenkamp

and

_Jürgen

Richter‐Gebert. Blended

_{experimentation}

with

_dgs.

In

_{Proceedings of}

CADGME

2009,

2009.

[8]

Aaron

_Montag

and

_Jürgen

Richter‐Gebert.

_{Cindygl: Authoring}

_gpu‐based

in‐ teractive mathematical content. In Gert‐Martin

_Greuel,

T.

_Koch,

P.

_Paule,

and A.

_{Sommese, editors,}

Mathematical

_Software—

ICMS

2016,

volume 9725 of Lecture Notes in

_{Computer Science,}

_{pages 319‐326.}

_Springer,

2016.

[9]

Jürgen

Richter‐GebertandUlrich

_Kortenkamp.

Thepowerof

scripting:

DGSmeets

programming.

Acta Didactica

_Napocensia,

_3(2):67-78

, 2010.

[10]

Jürgen

Richter‐Gebert and UlrichH.

_Kortenkamp.

The Interactive

_{Geometry Soft‐}

ware Cinderella.

Springer‐Verlag,

Heidelberg,

1999.

[11]

Jürgen

Richter‐GeUert and UlrichH.

_Kortenkamp.

\grave{\backslash }J ン $\tau$レラ.

\displaystyle \ovalbox{\tt\small REJECT}\sqrt{\mathrm{p}}\frac{1\rightarrow\sim}{\lrcorner}

の $\gamma$-\leftarrow

めのグ

\overline{-7}フイ

_{クス.Springer‐Verlag, Tokyo,}

2003.

[12]

Jürgen

Richter‐Gebert and Ulrich H.

_Kortenkamp.

The Cinderella.2 Manual ‐

Working

with the Interactive

_{Geometry Software. Springer‐Verlag,}

2012.

[13]

Bret

Victor.

Up

and down the ladderofabstraction. a

_systemic

approach

tointer‐

active visualization.

_http:

_{//worrydream.}

_{com/LadderOfAbstraction/,}

2011.

[14]

Maftinvon

Gagern,

Ulrich

Kortenkamp, Jürgen Richter‐Gebert,

and Michael Stro‐

bel.

_Cindyjs.

mathematical visualizationonmoderndevices.InGert‐Martin

Greuel,

T.

_Koch,

P.

_Paule,

and A.

_{Sommese, editors,}

Mathematical

_Software—

ICMS

_2016,

volume 9725 of Lecture Notes in

_{Computer Science,}

_{pages 319‐326.}

_Springer,

2016.