137
Learning descriptive statistics with the use of
dynamic content generated by CindyJS
Masataka Kaneko
Faculty of Pharmaceutical Sciences
Toho University

Takeo Noda
Faculty of Science
Toho University

1

Introduction

The importance of learners’ manipulations on physical models has been repeatedly emphasized

in mathematics education. For instance, J. Piaget [1] pointed out that
The figurative aspects of the cognitive functions, though obviously useful and nec‐
essary to the knowledge of states, are incapable of accounting for thought, becauseby themselves they cannot succeed in assimilating the transformations of reality.
The reason is that knowledge is not a static copy of reality. To know an object is
not to furnish a simple copy of it: It is to act on it so as to transform it and grasp
within these transformations the mechanism by which they are produced. To know,
therefore, is to produce or reproduce the object dynamically; but to reproduce it
is necessary to know how to produce, and this is why knowledge derives from theentire action, not merely from its figurative aspects.

In line with this perspective, S. Papert proposed many “models” and “ingredients” whichlearners can use to experience mathematizing for themselves. Figure 1 shows his proposal fora physical model with which children write the program (as seen in the right panel) to balance
the inverted pendulum [2].

WIRE TOCOM PUTERTURTLE KEEPS ROD FROM FALLING

TO BALANCE
1 TEST ANGLE \rangle 10
2 IFTRUE FORNARD 8
3 TEST ANGLE \langle-10
4 IFTRUE BACK 8
5 WAIT 1

INFORMATION FOR FEEDBACK.

6 BALANCEEND

BY MOVING FORWARD AND BACKPOTENTIOMETER IN HINGE PROVIDES

Figure 1.

A“formal physical” model of the stick balancing situation

Some recently developed mathematical software systems have enabled learners to manip‐ulate a large variety of mathematical models. Among them, dynamic geometry systems like

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Cinderella (https://www.cinderella.de) have created an environment where a wide rangeof mathematical objects including geometric shapes and function graphs can be handled by
dragging a mouse. However, in order for learners to handle mathematical models via these tech‐
nologies, they must install those systems into their PCs and master their usage. Fortunately,
Cinderella has been equipped with a function which converts graphical outputs into HTML
format at the push of a button. The generated materials can be moved via the CindyJS system(http: //cindyjs. org) on ordinary web browsers [3] [4] [5]. Some case studies using CindyJS
system have been made in the authors’ previous works [6][7].
In this paper, the methods and results of the other case study are described. The theme ofthis study is the fundamental concepts in descriptive statistics including correlation coefficientsand line regressions. The learners’ worksheets are analyzed in order to clarify what the barriers
are to their reasoning with the aid of physical operations on models.

2

Theoretical Background

Besides learning materials related to elementary topics like correlation coefficients or line
regression, the authors prepared a content related to a more advanced topic, orthogonal line
regression. Firstly, we will examine its theoretical background.
While y‐x regression or x‐y regression is usually used in descriptive statistics, orthogonal
regression is also important in some fields of statistics including principal component analysis.
Regarding the meaning of “regression” as the best approximation of the data set, it is quite
natural that the least squares method is applied to not only the errors in the values of x or
y but also to the distance between each data and the resulting line. Thus, in the case of an
orthogonal regression line, the targeting function is the square sum of the distance between
each data (x_{i}, y_{i})(i=1 , m) and the line y=ax+b given as follows.

f(a, b)= \frac{1}{1+a^{2} \sum_{i=1}^{m}(ax_{i}-y_{i}+b)^{2}
Hence, at the extremal point of the function f(a, b) , the following conditions should be satisfied.

\frac{\partial f}{\partial b}=\frac{2}{a^{2}+1}\sum_{i=1}^{m}(ax_{i}-y_{i}+b)= 0\frac{\partial f}{\partial a}=-\frac{2a}{(a^{2}+1)^{2} \sum_{i=1}^{m}(ax_{i}- y_{i}+b)^{2}+\frac{2}{a^{2}+1}\sum_{i=1}^{m}x_{i}(ax_{i}-y_{i}+b)=0

The abbreviated form of the first condition\sum_{\dot{i}=1}^{m}(ax_{i}-y_{i}+b)=0
is the same as that of the usual line regression which implies that the resulting line shouldpass through the point
hold.

(\overline{x},\overline{y})=(\frac{1}{m}\sum_{i=1}^{m}x_{i},\frac{1}{m}\sum_ {i=1}^{m}y_{i})b=\overline{y}-a\overline{x}and that the following equation should

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By substituting \overline{y}-a\overline{x} for b in the abbreviated form of the second condition\sum_{i=1}^{m}(ax_{i}-y_{i}+b)(ay_{i}-ab+x_{i})=0
we obtain(\overline{xy}-\overline{x}\cdot\overline{y})a^{2}+(\overline{x^{2} - \overline{x}^{2}-\overline{y^{2} +\overline{y}^{2})a-(\overline{xy}-\overline{x} \cdot\overline{y})=0
It is a quadratic equation of the formKa^{2}+La-K=0
in which K is the covariance s_{xy} and L is the difference v_{x}-v_{y} in the variances ofxand y.

Therefore there are two real solutions to this equation which yield candidates for the extremum
positions of f(a, b) . Further computations lead to the conclusion that only one of these twocandidates which has the same signature as s_{xy} is minimal and the other is not any extremum.
As the equation degenerates when K=s_{xy}=0 , a discontinuous phenomenon may occur if wevary the data so that the value of s_{xy} passes through 0 and its sign changes.
In order to simplify the reasoning process, the authors chose the condition that the data
points are situated symmetrically regarding line y=x . In this case, \overline{x} is the same as \overline{y} and L
should be 0 because of its definition. Thus, in this case, the solutions are a=\pm 1 . Moreover, if
a=1, b should be 0 . These results imply that there can be some discontinuous phenomenonof the extremal position of f(a, b) when we move the data points while keeping the above
mentioned symmetry. Though direct computations would show which of \pm 1 gives the minimalvalue of f(a, b) , the authors made a lesson in which students tried to find the correct choice andthe conditions controlling the choice while moving dynamic content generated by CindyJS.

3

Methods

To enable students to find the discontinuous phenomena and the conditions associated withit, the authors prepared the dynamic content shown below by using CindyJS. The screen for
physical operation is shown in Figure 2.

Figure 2. Screenshot of the main content
In this content, students can move three data points A, B , and C freely. The remaining
three data points are moved automatically following the symmetry rule. These movements arecontrolled via the quite simple scripting below.

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D. xy= [ A.y , A. x ]; E. xy= [ B.y , B. x ] ;\Gamma.xy= [ C.y , C. x ];xl=A.x;x2=B.x;x3=C.x;yl=A.y;y2=B.y;y3=C.y ;m=(xl+x2+x3+yl+y2+y3)/6 ;draw( [[m, -13], [m, 13]] , dashtype‐ >2 ) ; draw( [[-13,m], [20,m]] , dashtype‐ >2 ) ;sx2 =x1^{-}2+x2^{-}2+x3^{-}2+y1^{\wedge}2+y2^{\wedge}2+y3^{\wedge}2;vx=sx2/6-m^{-}2 ;
cv=((xl-m)*(yl-m)+(x2-m)*(y2-m)+(x3-m)*(y3-m))/3 ;plot ( [m+vx*t,m+cv*t] , start‐ >-50 , stop‐ >50 );Lengthl =(xl-yl)^{-}2+(x2-y2)^{\wedge}2+(x3-y3)^{-}2 ;Length2 =(xl+yl-2*m)^{arrow}2+(x2+y2-2*m)^{\sim}2+(x3+y3-2*m)^{arrow}2 ;if (Lengthl <Length2,
plot ( [m+t,m+t] , start‐ >-50 , stop‐ >50 , color‐ >[0,1,0] );plot ( [m+t , m‐t], start‐ >-50 , stop‐ >50 , color‐ >[1,0,0] );) ;As a result, the usual regression line is drawn in blue. To specify the orthogonal regressionline, green or red is used in accordance with the choice of \pm 1 as a . Together with some othercontents, this content was implemented on iPads.

The subjects were the first grade university students (19 years old) who were non mathe‐
matics majors. After fundamental notions of orthogonal line regression and basic usage of thecontent were presented as shown in Figure 3, iPads were delivered to the subjects and they
were asked to move the content with a partner.

Figure 3. Scenes of experimental class

Figure 4. Students’ physical operations
Students had a choice of lessons which were presented to them on iPads. Their tasks were thefollowing.

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1. Find some phenomena which they think are mathematically important and report themwith the use of screenshots.

2. Explain the mathematical background of those phenomena.
Their worksheets were collected and the descriptions in them were analyzed qualitatively. The
activities of some groups were recoded via VTR. Some hints were given to the groups whose
operating process seemed to have gotten stuck.

4

Results

Among the 21 groups in the class, 6 groups used the content cited in the last section. As seenin Figure 5 which shows their worksheets for task 1, almost all groups found the discontinuous
phenomenon together with its relation to the slope of the usual regression line.
Ra_{1}oeM1=*\aleph\vartheta^{i}b9f\dot{o}'eB*\emptyset xp1V-//S^{\backslash } \backslash /\vdash k1W\mathfrak{d}(\lambda^{r})-f/\lambda^{\mathfrak{l}}J-Jl,*\# \vartheta/
ligl aet91_{-}\mathfrak{T}*\vartheta^{i}b0t5\prime x\mathfrak{R} \wedge\emptyset\lambda P^{1/}-//\exists^{\backslash }r ト& \alpha \mathfrak{y}(\lambda^{\wedge}J-7/\lambda)-SIbf\vartheta/*\aleph u\prime e*5t-A\# e-enr)\{n6e\alpha_{0}zm*e\#u \langle StOPt 6 ( I\aleph 1TO\circ r_{\vee}\cdot ot_{-}*\hslash b\prime g\vartheta^{t}b*-A* /&ll/t) . *n_{b}g\propto oTRl*\#V\langle R\mathfrak{M}t6(\lceil Ee1T\circ\circ t_{-3-} \vee\omega\vartheta^{t} zn2T\Delta\Delta i_{-}^{\vee}X\{bb_{-}\cdots J\emptyset\lambda\tilde{2}1^{\vee}lt\Re t6) \mathbb{R}\hslash bf_{-}\mathfrak{N} St\ *AtRO_{\vee}^{\vee}\geq \emptyset{\}^{*} \Phi l2T\Delta\Delta\}_{-}^{\vee} K(bb_{-}\cdots\rfloor\emptyset\lambda\dot{?}\}=\Re\Re t6) \ \hslash bk\#*f\ \ kT\otimes v_{\sim}^{-}\geq.

k

e

bX

1v1

F1H

L^{I},

)

b

B

(* 1

1i^{b\wedge}.

\underline{\varpi}1\eta r_{\l corner B\ovalbox{\t \smal REJECT} o* $ \iota\partial\tau T3:. \not\geq\not\in\phi\eta\cdot\l corner t*\overline{B}^{\ulcorner 0\circ\varphi_{\eta}\star 0 \wedge J_{L^{t}} Lt \{tP^{s}r 1\phi \rangle\backslash 1 1'\otimes t7;_ {i}\acute{\not\in}b \star,h\vartheta\mapsto\Sk\hat{\prime}tk
Group IGroup II

Rg_{1}R\yen 1n1_{-}\approx*h^{t} bbtS rxmmoxp j-// sv Fe } R0(XJ-V/XJ-7-\#\#e/*\aleph br_{t\vartheta^{l_{b}}*-A\Re}*n6\ \infty \mathcal{D}\tau \mathfrak{R}**\#b\langle $ \Re t6 (1\wedge 1TOOf_{-}\cdot\circ t\simeq\omega\# \Sigma\{l2T\Delta\Delta\}.K(bbk\cdots J\emptyset A5\}_{\vee}^{\vee} a\Re T6) (tnbf_{-}gtt\ g\hslash YBV_{arrow}'\mathfrak{x} \emptyset h^{C} = \{\backslash 2Y\Delta\Delta 1_{-}'*(bb\gamma_{-}\ldots\rfloor\emptyset X5\}_{-}' \ovalbox{\tt\small REJECT}\Re T6) \propto\hslash bt_{-}\Re t14*>\iota\tau\emptyset tarrow k

M\mathcal{B}1\ \backslash Mt_{-} filt’ thb15^{t}xH*oxe1V-//svFg\Re 0(XJ-7/X lU−7N/$t ekWb\prime e\#\dot{b}*-A\phi S/*n\tau) . \epsilon n_{b} &&oTR * &#b \langle\# l\beta f6 ( Zlll 1T\circ\circ_{-}0-a\gamma P1yL4 Q\rangle\underline{(}*1 \subset\ell t\epsilon 6\ovalbox{\tt\small REJECT} 1f, E_{c}\underline{\varepsilon}\nearrow\wedge g)_{\acute{V}}\underline{ \overline{q}}fib \underline{b}\parallel\hslash 2 Uf.g B Q/\hat{\not\cong}ae\{
4L 6s\llcorner f.\wedge-1i _{\overline{\l corner} (\sim 5 o\backslash \cdot- \ovalbox{\t \small REJECT} \nwarrow Z \lambda p\kappa(TA(1V \underline{\gamma_{a'} tkb ct\bigcap_{\ve }t\acute{\dot{\varpi} ^{r} \backslash 1 r_{\perp 3}Pit g\hat{q}\Theta\hslash Eb^{f} (\iota\l corner\tau\hslash \not\in l-L\prime 3C\llcorner\overline{\phi}\prime m_ {-}(\Supset\geq(\underline{\delta}\sqrt{}\xi_{\phi} oe \hat{Q} s. A q $b
\mathfrak{ }( 4ar ow\chi)\neq r\underline{g}/\hslash\bullet\vartheta\overline{g} \ovalbox{\t \smal REJECT} ePt

hSuFj\hslash^{t})f(\otimes-\prime\backslash (.

の&arrow\partial\nearrow< fae \bigwedge_{c}\vartheta-( ,č t.. r\overline{.}C \yen.
\backslasharrow b\sqrt{}*3T^{\iota}(f\llcorner B\tau AE / \frac{\not\supset}{f}E\not\in\# i\backslash

A \kappa rGroup III

Group IVn^{1u*\infty\}*\vartheta^{\{}b\mathfrak{h}*\dot{2}'I* J//^{\backslash }\ni/ト** \mathfrak{h}(\lambda^{\ovalbox{\t \small REJECT}})-/\alpha ae\#\vartheta\prime}

uaeu\neqrt_{-}R\hsla h\varthea^{i}b\mathfrak{y}*,\primexR*oxparowJ/ \ni/bk \hsla h\mathfrak{h},(\lambda^{1})-7/\lambda^{1}J-n\prime\neq f\varthea\prime\ mptyset\varthea\emptyset\sim2\tau\Delta\Delta1_{\ve } ^{\ve }*(b _{-\cdotsJ\emptysetA\dot{ }\_{-\# Re\tau\mathfrak{z})\alpha na^{\mathfrak{p}\i4\ 3\iota\ uro_{\sim}^{r}\mathring{\suc eq}'\ n^{1} \prime,\cdotb_{-}H\#1\square\aleph\tau\DeltaK\infty(\rflo r\dot{2}\. Re{\}t6). f ibk\infty\hsla h\Leftrightarow\*\taun^{1\tau}v_{-\mathring{\}^{-} しs\mathfrak{s}\overline{\tau}\frac{\ap rox}{\Delta}A\#/*?t).e^{x}n\mathfrak{s} \_{\mathcal{D}^{\backsla h} \cir \phi^{1}\ tau *Rb\langleR\ ef6(\lceil r\varepsilon\cir f\prime\lambda^{\ovalbox{\t smal REJ CT})-7,

01\dagger^{t}\vee\vartheta t,\hat{\eta}_{\triangleright}ffit\{l_{J}'\prime ff\xi\mu\psi\gamma1)’ p1\dot{\varphi}\eta_{\vee}^{11e_{rf_{*\prime}r\prime q}}\prime\vee t,(i) \#_{\vee}'.r,.
\prime \mathfrak{r}t'\prime’\Phi 13c11\#\backslash 1.

\iota.*jE\cdot t^{A}\underline{w}ul\{k1_{i}st\{k_{nt^{*}}

x\{t^{l}.\vee h\cdot-1_{Y}1k{\}^{\backslash }k ttlb.

Group v

Group VI

Figure 5. Worksheets for task 1
Contrarily, most groups seemed to have faced great difficulty in doing task 2. While groups
III and V described their thinking as seen in Figure 6, groups I, II, and IV left almost nocomments.

142
R82\neq\emptyset flfl1_{-}^{\vee}a\backslash \backslash \tau. \aleph\geq r\prime xi\aleph\Re\vartheta:T\approx 6\vartheta\cdot g\lambda Tb\lambda\dot{o} \Re 82\neq\emptyset\Re*t_{-}^{\vee}ov(. a*n\prime_{JM\Re h<cg_{6\vartheta 1*XTbZ5}}\perp\cap t.

C

(1)’

1?(;1)A(Tf)$

A t(\eta 1’

1

tL11\{b*f, f_{j-}\cdot Pt\dagger\backslashlP\{*\# 1ii\# fat_{A}, tx \wedge ae\ovalbox{\t \small REJECT} f*\l corner\oint sf_{1}x\underline{i}_{i} \not\simeq\llcorner\not\in r_{xZ}(1F\mathcal{O}Z^{\forall}b-a\overline{x}t^{u} 1t^{r}-\prime;kf\zeta t\{_{x3}^{-},

Group 111

\triangleright\cdot o

A

k.

1

*
, t

tt_{S^{N}n}*t ‐ ttzn;\rho_{\vee}*p \{_{1}::”î t \mathfrak{p}\uparrow\# og_{I}'1\eta ’\cap R\llcorner t, \eta\{ . \vdash\tau[1\triangleleft) b \xi 13\dot{\eta};]^{\wedge}\prime.{\}*\varpi fr1)\eta\}--\lambda 1
n\ell.

\uparrow r\delta\delta\#\prime_{4}t^{-}\gam a'\zeta l1) (?\varsigma) (\uparrow\triangleleft)

\eta

*\{,

∼@/ f\eta l_{\epsilon\dot{c}06}k

1

b.

1, c
\partial v

r_{d}\neq 1,.

t\lambda*tX*tn8\eta p;*r_{\theta}tbf_{W}*3.

Group v

Figure 6. Worksheets for task 2

As is easily seen, these descriptions remain surface level “observations” which are not connectedto the essential points given in section 2. As seen in Figure 7, only the discussion in group IV
was linked to the mathematical conditionKa^{2}+La-K=0
given in section 2 though they needed some advice from the teachers.
RZ2\downarrow\emptyset H*1_{-}^{\vee}Ot\backslash T, \Re\Leftrightarrow 8nr_{9} \iota\ovalbox{\tt\small REJECT}\beta E\gamma_{j};-c\not\equiv 6fj^{1}\Rightarrow x\tau*Ji5.

*\lambda_{f1}\Re r'\backslash \ltimes i\iota \mathfrak{x}fflc\backslash (‐.

ka_{I}^{a}{}_{t}Le_{1}-1<.

‐

i 7‐. .

o\theta\prime-\frac{-\l corner 3\overline{L^{i}.+tk^{a} {zK}7^{\backslash }.\delta)' ’\theta*\prime x\nwarrow_{a\iota ID^{t} NB\# g_{\triangleleft}\varphi l_{\backslash }31\vartheta\pi\frac{-x^{-}-}{-}
a.=\overline{\prime\chi^{1_{-\overline{\gamma}^{a} '}a_{Z}n^{j\uparrow g_{\iota 7k^{\backslash }3}7^{=xa5}}..

\sqrt{\llcorner^{2}+fK}\cdot>\llcorner e_{4?^{-}}

a_{1}\d ot{\cdot}.\Re_{[}/\Re_{r}.;X3 . g_{A\otimes^{\iota}1_{7}^{3}E\hslash*}^ {\grave{\overline{\prime} t\sim\backslash .-\backslash 1h^{\backslash } \backslash \theta_{\eta\lambda^{A_{\neg}\otimes\hslash bfS_{q}\backslash \backslashk‐‐\overline{If}-\overline{7}\tilde{e}n_{1n}\eta^{J}\doteqdot 1J-5k15..

Figure 7. Worksheet of group VI
The two points where they needed advice are specified with red rectangles in Figure 8.

Rf2\downar \emptysaa*:\'}\_{-R^ev_e{}\0\hiostalabschksll/h\\gtau.\rSm\tthafauk{gR}it3^}i\\mau_a{7pdselttoaI^L{\.\pchad}onto*t\>eq\uivl6c,.*oz\rtanueA5r*xb\daot\{sb}Aafclamkbsdla\sehlta.i^{3*j\cdotgam 1?\delta*R_{r}\wedg3l}f\iota7Ri{\}^L_4a\prime}-\kap.z_{l}\fracbkslah_{}@\overlin{},\aph^{overlin{\gam }^{\dot wedg}\ner)}a_{\et frac{-L\sqrtlconer+\backslh\ap}0r17;_{nh}A\epsilonaeph\yn#9\primex}{zK.\phismY*,3.

5a2Aの\hsla *1_{-}^\ve}mathcl{D}t\backsl h\tau,Re\ t^{;}\auneq 6\varthe\cdot*x\auAdot{}_J\cdotbacksl h\Pi}*cdot.'3ka_{4.\frac{- \picoyrightJ}{7-\kap z_{\overlin{}\mapsto_{;}\frac{-L\lorne 0\primes\dager*\ltimesA\pilonK\eql9\primex}{zKa_in}\emptyseg_{7}m {T}*,5\gam .\rhogitaR_{r}\#h3larrow fkH^{\cdot}r_{\dot{7} ^{b^{\backslash } s-t*\cdot\theta n7\wedge\cdot sh\delta\#.

Figure 8. Points where advice was needed

The members could solve the quadratic equation and assume that the signature of the solution
was related to the slope of the orthogonal regression line. However, they could not decide thesign based on the form of the solution in the first place. Similarly, in the second place, they
could not infer that the slope of the usual regression line was decided by the sign of thecovariance s_{xy} because the variance v_{x} is always positive.

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5Discussions and Conclusions

Whereas many groups found the discontinuous phenomenon, almost no group could infer its
mathematical background. This result seems to give some insights for the effective use of
dynamic content.
Together with the above six samples, the descriptions made by all groups did not include
any inequalities though some groups treated the geometric situations which could be specified
by some “range” of parameter. In the case of this task, many subjects were able to obtain thesolution form

\frac{-L\pm\sqrt{L^{2}+4K^{2} }{2K} and understood the necessity to consider its sign. However,their reasoning did not lead to the comparison between \sqrt{L^{2}+4K^{2}} and L . This gap seems to
be caused by the learners’ lack in experience of inferring with the use of inequalities.
Moreover, the comparison between two complicated quantitiesa_{1}= \frac{-L\pm\sqrt{L^{2}+4K^{2} }{2K}

andappear in Figure 7 and 8 is also problematic. In fact, in the curriculums ofaJapanese_{2}= \frac{K}{v_{x} whichhigh school mathematics and university calculus courses, almost all of the descriptions
related to some locus or region are given by inequalities with simple variables likeas seen in the samples of Figure 9.

x,y,or \theta- \frac{\pi}{2}\leqq\theta\leqq\frac{\pi}{2} , 0\leqq r\leqq a\cos\theta 0\leqq r\leqq a, 0\leqq\theta\leqq\pi, 0\leqq\varphi\leqq 2\piFigure 9. Inequalities appearing in a university text

(M. Sugiura: Introduction to Calculus I)
In the case of this task, both K and L are computed from the data set in a complicated mannerand their relation to the position of data points is indirect. Thus it can be assumed that thelink between mathematical expressions and their geometric images was lost along the way as
the subjects worked out the solution.
In conclusion, the results of this case study indicate the following should be taken into
account when dynamic contents are used in the mathematics classroom.

1. Discontinuous phenomenon in this case study could not be found without learners’ in‐teractive operations on mathematical models. The result that almost all groups could
find it clearly illustrates the effectiveness of using a dynamic geometry system.

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2. Care should be taken that learners’ interactive operations on the mathematical model
can be seamlessly linked to the mathematical expressions of that model. When thoseexpressions are complicated, some preliminary training is needed.

6

Future Work

With respect to the current state of education, it is not easy for average university students toconstruct these mathematical models using programming language by themselves. However,
especially for talented students, it will be quite useful if they can input the geometric elements
and the necessary “commands”’ through some script editor to handle those models. Now, afunction called “Cindyeditor” which acts as a script editor is now being developed. Figure 10
shows a screenshot of the Cindyeditor associated with the content in this case study.

Figure 10. Screenshot of Cindyeditor

A web based editorfor CindyJS widgets is currently being developed by Jürgen Richter‐Gebertand Aaron Montag as part of the CindyJS‐project. An online demo of the experimental stateis available at https: //cindyjs. org/ editor. The users can observe how changing parts of
their code would affect the widget in real‐time. The authors think that the verification of theeffect of using Cindyeditor for STEM education is an urgent need.
Moreover, investigating the real states in which dynamic geometry systems are used isneeded. One of the most reliable ways is to track communications between learners as they
move the content collaboratively. A serious problem is that it is not easy to grasp which
member made which utterance. To overcome this difficulty, the authors are now planning touse a system which analyses communications with the aid of wearable recording devices as
seen in Figure 11.

Figure 11. Wearable recording device

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Correlations between learners’ physical operations and communications shoud help us to clarify
the above mentioned states.

References[1] J. Piaget, The Role of Action in the Development of Thinking, In: Overton W.F., Gal‐lagher J.M. (eds), Knowledge and Development, pp.17‐43, 1977
[2] S. Papert, Teaching children to be mathematicians versus teaching about mathematics,
International Journal of Mathematics Education in Science and Technology 3, pp.249‐262,
1972[3] M. von Gagern, U. Kortenkamp, J. Richter‐Gebert, M. Strobel: CindyJS— mathematicalvisualization on modern devices, Lecture Notes in Computer Science 9725, pp.319‐326,
2016[4] M. von Gagern, J. Richter‐Gebert: CindyJS plugins— extending the mathematical visu‐
alization framework, Lecture Notes in Computer Science 9725, pp.327‐334, 2016

[5] A. Montag, J. Richter‐Gebert: CindyGL— authoring GPU‐based interactive mathemati‐
cal content, Lecture Notes in Computer Science 9725, pp.359‐366, 2016[6] M. Kaneko: Using tangible contents generated by CindyJS and its influence on mathe‐
matical cognition, Lecture Notes in Computer Science 10407, pp.199‐215, 2017[7] T. Noda, M. Kaneko: Collaborative use of mathematical content generated by CindyJS
on tablets, Lecture Notes in Computer Science 10931, pp.379‐388, 2018