TitleLearning descriptive statistics with the use of dynamic contentgenerated by CindyJS (Study of Mathematical Software and ItsEffective Use for Mathematics Education)

Author(s) Kaneko, Masataka; Noda, Takeo

Citation 数理解析研究所講究録 = RIMS Kokyuroku (2019), 2105:137-145

Issue Date 2019-02

URL http://hdl.handle.net/2433/251883

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

137

Learning descriptive statistics with the use ofdynamic content generated by CindyJS

Masataka Kaneko

Faculty of Pharmaceutical SciencesToho University

Takeo Noda

Faculty of ScienceToho University

1 Introduction

The importance of learners’ manipulations on physical models has been repeatedly emphasizedin 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 isnot to furnish a simple copy of it: It is to act on it so as to transform it and graspwithin these transformations the mechanism by which they are produced. To know,therefore, is to produce or reproduce the object dynamically; but to reproduce itis 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 balancethe inverted pendulum [2].

WIRE TOCOM PUTER

TO BALANCE1 TEST ANGLE \rangle 102 IFTRUE FORNARD 83 TEST ANGLE \langle-104 IFTRUE BACK 8

TURTLE KEEPS ROD FROM FALLING 5 WAIT 1BY MOVING FORWARD AND BACK 6 BALANCEPOTENTIOMETER IN HINGE PROVIDESINFORMATION FOR FEEDBACK. END

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

137

138

Cinderella (https://www.cinderella.de) have created an environment where a wide rangeof mathematical objects including geometric shapes and function graphs can be handled bydragging 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 HTMLformat 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 CindyJSsystem 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 barriersare 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 lineregression, the authors prepared a content related to a more advanced topic, orthogonal lineregression. Firstly, we will examine its theoretical background.

While y‐x regression or x‐y regression is usually used in descriptive statistics, orthogonalregression 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 quitenatural 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 anorthogonal regression line, the targeting function is the square sum of the distance betweeneach 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)=0is the same as that of the usual line regression which implies that the resulting line should

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

hold.

b=\overline{y}-a\overline{x}

138

139

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})=0we 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})=0It is a quadratic equation of the form

Ka^{2}+La-K=0

in which K is the covariance s_{xy} and L is the difference v_{x}-v_{y} in the variances of x and y.

Therefore there are two real solutions to this equation which yield candidates for the extremumpositions 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 datapoints 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 abovementioned 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 forphysical 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 remainingthree data points are moved automatically following the symmetry rule. These movements arecontrolled via the quite simple scripting below.

139

140

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 theywere 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.

140

141

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. Theactivities of some groups were recoded via VTR. Some hints were given to the groups whoseoperating 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 discontinuousphenomenon 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_{\llcorner B\ovalbox{\tt\small REJECT}}o* $ \iota\partial \tau T3 :..\not\geq\not\in\phi\eta\cdot\llcorner 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 \mapsto\S k\hat{\prime} \vartheta tk

Group I Group II

M\mathcal{B}1\ \backslash Mt_{-}filt’ thb15^{t}xH*oxe1V-//svFg\Re 0(XJ-7/X lU−7N/$t e- Rg_{1}R\yen 1n1_{-}\approx*h^{t}bbtS rxmmoxp j-// sv Fe } R0(XJ-V/XJ-7-\#\#e/ kWb\prime e\#\dot{b}*-A\phi S/*n\tau) . \epsilon n_{b} &&oTR *&#b \langle\# l\beta f6 ( Zlll 1T\circ\circ_{-}0- *\aleph br_{t\vartheta^{l_{b}}*-A\Re} *n6\ \infty \mathcal{D}\tau \mathfrak{R}**\#b\langle $ \Ret6 (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

a\gamma P1yL4 Q\rangle\underline{(}*1 \subset\ell t\epsilon 6\ovalbox{\tt\smallREJECT} 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\{ A(1V \underline{\gamma_{a'}}tkb ct\bigcap_{\vee}t\acute{\dot{\varpi}}^{r}\backslash 1 r_{\perp 3}Pit g\hat{q}\Theta\hslash Eb^{f}(\iota\llcorner\tau\hslash \not\in l-L\prime 3C\llcorner\overline{\phi}\prime m_{-} 4L 6s\llcorner f.\wedge-1ii_{\overline{\llcorner}} (\sim 5 o\backslash \cdot-\ovalbox{\tt\small REJECT} \nwarrow Z \lambda p\kappa(T hSuFj\hslash^{t}) (\Supset\geq(- \underline{\delta}\sqrt{}\xi_{\phi} oe \hat{Q} s. A q $b ( 4arrow\chi)\neq r\underline{g}/\hslash\bullet\vartheta\overline{g}\ovalbox{\tt\small REJECT} ePt \mathfrak{k}

f ( C \yen. arrow\partial\nearrow< fae \bigwedge_{c}\vartheta-( ,č t.. r\overline{.}

\otimes-\prime\backslash (. \backslash arrow b\sqrt{}*3 T^{\iota}(f \llcorner B\tau AE / \frac{\not\supset}{f}E\not\in\# i\backslash A \kappa r

Group III Group IV

& n^{1u*\infty\}*\vartheta^{\{}b\mathfrak{h}*\dot{2}'I**J//^{\backslash }\ni/ト**\mathfrak{h}(\lambda^{\ovalbox{\tt\small REJECT}})-/\alpha ae\#\vartheta\prime}の \# 1\square \aleph\tau\Delta K\infty(\rfloor\dot{2}\}.\Re{\}t6).ffffibk\infty\hslash\Leftrightarrow\ * \tau n^{1\tau}v_{-\mathring{\ }}^{-} しs\mathfrak{s}\overline{\tau}\frac{\approx}{\Delta}A\#/*?tt).e^{x}n\mathfrak{s}\ _{\mathcal{D}^{\backslash }} \circ\phi^{1}\ \tau a**Rb\langle R\Re f6(\lceil r\varepsilon\circ f\prime\lambda^{\ovalbox{\tt\small REJECT}})-7, uaeu\neq rt_{-}R\hslash\vartheta^{i}b\mathfrak{y}*,\prime xR*oxparrow J//\ni/bk\hslash \mathfrak{h},(\lambda^{1})-7/\lambda^{1}J-n\prime\neq f\vartheta\prime\emptyset\vartheta\emptyset\sim 2\tau\Delta\Delta 1_{\vee}^{\vee}*(bb_{-\cdots J\emptyset A\dot{o}\}_{-\#\Re\tau \mathfrak{z})\alpha na^{\mathfrak{p}}\}i4\ 3\iota\tau ro_{\sim}^{r}\mathring{\succeq}}'}\ n^{1}\prime,\cdot b_{-}H’

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,. \Phi 13c11\#\backslash 1 . \iota.*jE\cdot t^{A}\underline{w}

\prime \mathfrak{r}t'\prime 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 groupsIII and V described their thinking as seen in Figure 6, groups I, II, and IV left almost nocomments.

141

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) \gamma' \uparrow r\delta\delta\#\prime_{4}t^{-} \{ lP 1 * î t \mathfrak{p}\uparrow\# og_{I}' r_{d} ∼@/ f\eta l_{\epsilon\dot{c}06}k

\zeta l1) (?\varsigma) (\uparrow\triangleleft) tL11\{b *\# 1ii\# f \eta ’ , t

\eta $ A t(\eta 1 ’ 1 *f, f_{j-}\cdot Pt\dagger\backslash [1\triangleleft) b \xi 13\dot{\eta};]^{\wedge}\prime.{\}* \cap R\llcorner t, \eta\{ . \vdash\tau 1

\varpi fr1)\eta\}--\lambda 1 t_{A}, t a 1 tt_{S^{N}n}* b.

n\ell. x \wedge ae\ovalbox{\tt\small REJECT} f*\llcorner\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}^{-}, \triangleright\cdot o A k. t ‐ tt 1, c . \neq 1, \lambda*tX*tn8 \eta p;*r_{\theta}t

zn;\rho_{\vee}*p \{_{1}:: ” \partial v *\{, t bf_{W}*3.

Group 111 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 IVwas linked to the mathematical condition

Ka^{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}\Rightarrowx\tau*Ji5.

*\lambda_{f1}\Re r'\backslash \ltimes i\iota \mathfrak{x}fflc\backslash (‐. i 7‐. . \theta*\prime x\nwarrow_{a\iota ID^{t}}NB\# g_{\triangleleft}\varphi l_{\backslash }31\vartheta

ka_{I}^{a}{}_{t}Le_{1}-1<. ‐ o \pi\frac{-x^{-}-}{-}

\theta\prime-\frac{-\llcorner 3\overline{L^{i}.+tk^{a}}}{zK} a.=\overline{\prime\chi^{1_{-\overline{\gamma}^{a}}'}}

7^{\backslash }.\delta)' ’ a_{Z}n^{j\uparrow g_{\iota 7k^{\backslash }3}7^{=xa5}}...

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

a_{1}\ddot{\cdot}.\Re_{[}/\Re_{r}.;X33. 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 \backslash

k ‐‐ \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.

R ff2\downarrow\emptyset a*\}_{-}^{\vee}0\iota\backslash \tau.ur\mathfrak{g}t^{i}\tau_{7\delta I^{\alpha}}\not\equiv 6,.*z\tau*A5*x\dot{s}Af\lambda\delta i^{3*j\cdot\gamma 1?\delta*R_{r}r\wedge 3l}ffl\iota l77Rai{\}^{L_{4a}\prime}-\kappa.z_{l}\frac{\backslash _{a}@}{\overline{},\alpha^{\overline{\gamma}^{\dot{\wedge}}}\neg r)e}a_{\eta}\frac{-La\sqrt{\llcorner+\backslash \kappa}0r17;_{nh}A\epsilon\aleph\yen\# 9\prime x}{zK}.\phi\sim Y*,3. 5 a2Aの\hslash*1_{-}^{\vee}\mathcal{D}t\backslash \tau,,\Re\Re t^{;}\tau\neq 6\vartheta\cdot*x\tau*A\dot{o}_{J\cdot\backslash \Phi}*\cdot.' 3ka_{4a.\frac{-\pi\copyright J}{7}}-\kappa z_{\overline{l}}\mapsto a_{;}\frac{-L\llcorner 0\prime s\dagger*r\ltimes A\epsilon K\neq l9\prime x}{zK}a_{in}\emptyset g_{7}m_{T}*,5\gamma.\rho g\iota R_{r}\#h3l a:'\Re_{\hslash l/\gg\S\tau Ri3}\mapsto L.\cdot*t>\llcorner ba\backslash . arrow 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 solutionwas 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, theycould 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.

142

143

5 Discussions and Conclusions

Whereas many groups found the discontinuous phenomenon, almost no group could infer itsmathematical background. This result seems to give some insights for the effective use ofdynamic content.

Together with the above six samples, the descriptions made by all groups did not includeany inequalities though some groups treated the geometric situations which could be specifiedby some “range” of parameter. In the case of this task, many subjects were able to obtain the

solution 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 tobe caused by the learners’ lack in experience of inferring with the use of inequalities.

Moreover, the comparison between two complicated quantities a_{1}= \frac{-L\pm\sqrt{L^{2}+4K^{2}}}{2K} and

a_{2}= \frac{K}{v_{x}} which appear in Figure 7 and 8 is also problematic. In fact, in the curriculums of

Japanese high school mathematics and university calculus courses, almost all of the descriptionsrelated to some locus or region are given by inequalities with simple variables like x, y , or \theta

as seen in the samples of Figure 9.

- \frac{\pi}{2}\leqq\theta\leqq\frac{\pi}{2} , 0\leqq r\leqq a\cos\theta 0\leqqr\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 asthe subjects worked out the solution.

In conclusion, the results of this case study indicate the following should be taken intoaccount 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 couldfind it clearly illustrates the effectiveness of using a dynamic geometry system.

143

144

2. Care should be taken that learners’ interactive operations on the mathematical modelcan 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 elementsand the necessary “commands”’ through some script editor to handle those models. Now, a

function called “Cindyeditor” which acts as a script editor is now being developed. Figure 10shows 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 oftheir 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 theymove the content collaboratively. A serious problem is that it is not easy to grasp whichmember 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 asseen in Figure 11.

Figure 11. Wearable recording device

144

145

Correlations between learners’ physical operations and communications shoud help us to clarifythe 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 CindyJSon tablets, Lecture Notes in Computer Science 10931, pp.379‐388, 2018

145