計測自動制御学会東北支部 第 312 回研究集会 (2017.11.24)
資料番号 312-11

iOS 上における離散数学の論理的な学習用アプリの開発に関する研究
Logical Learning System of Discrete Mathematics on iOS○信濃健介 ∗ ，森和好 ∗○ Kensuke Shinano∗ , Kazuyoshi Mori∗*会津大学*University of Aizu
キーワード : 学習 (Learning), 論理性 (Logicality), 離散数学 (Discrete Mathematics)，集合 (Sets)，iOS (iOS).
連絡先 :〒 965-8580

福島県会津若松市一箕町鶴賀 システム解析学講座

信濃健介，Tel.: 0242-37-2615，Fax.: 0242-37-2747，E-mail: m5201124@u-aizu.ac.jp

2. iOS

1. Social Background
MEXT (Ministry of Education, Culture, Sports,

Apple smartphones of 6.5 million in the 1st quar-

Science and Technology) of Japan announced thatter of 2016 and one of 8.5 million in the 1st quarter

textbooks in elementary school and junior highof 2017 were sold in Japan [17] [18]. These break-school are changed to electoric books in 2020 [1].

downs are Figs. 1-2. Apple smartphone is the sec-

Therefore all paper textbooks are abolished andond most popular in the world(see Figs. 3-4) [19].

students will take the lessons using electoric ter-

From these figures, we saw that smartphone of Ap-

minal. There are many various learning appli-

ple is one of the most popular smartphones.

cations in the world, for example, e-Text [4] [5],iTex viewer [6] and digital text in Japan [7] [8].
Furthermore thre are many theses of learning sys-

3. Sets of Discrete Mathematics

tem, for example, electrical engineering [9], elec-

We can learn discrete mathematics, especiallytromagnetics [10], fusion of education, entertain-

sets, using the Logical Learning Application. In

ment and training [11], computer science [12],mathematics, a set is a collection of distinct ob-distributed learning system [13], digital logic de-jects, considered as an object in its own right. For

sign [14], multilingual user interface [15] and on-example, the numbers 2, 4, and 6 are distinct ob-line learning [16]. But these applications lack log-jects when considered separately, but when theyicality. Thus, we implemented an logical learning

are considered collectively they form a single set

application.

of size three, written {2, 4, 6}. The concept of aset is one of the most fundamental in mathematics.

–1–Developed at the end of the 19th century, set theory

4.3

is now a ubiquitous part of mathematics, and can beused as a foundation from which nearly all of mathematics can be derived. In mathematics education,
elementary topics such as Venn diagrams are taught
at a young age, while more advanced concepts are

What is proof

The proof of Section 4.1 cannot prove the casesof Figs. 6-7. Therefore this proof is not perfect.
Everybody will agree to a logical proof which includes all cases. This time, we implemented theapplication that we can learn a logical proof.

taught as part of a university degree [20].

5.

4. Logicality
4.1

Important Points of the Implemented Application

Venn Diagram
The application has five important points thatAt first, we consider to proof by venn diagram,for example
A ∪ (B ∪ C) = (A ∪ B) ∪ C.

we consider to be important. These are “What weneed to show”, “Symmetric”, “Detective”, “Step(1)From Fig. 5, because the shadow of A ∪ (B ∪ C)

by step” and “Interactive”.

5.1

What we need to show

is same as one of (A ∪ B) ∪ C, Equation(1) is true.
When we do not know procedure of proof, weThis proof is venn diagram.

cannot solve. Thus, we need to know what we need

4.2

to show. We can know how to solve a problem

GeoGebraby making clear what is goal of a problem (seeGeoGebra [21] is dynamic mathematics software

Fig. 10).

that made the geometry, the algebra, the spreadsheet, the chart, the statistics and the analysis one

5.2

Symmetric

package which is easy to use. GeoGebra is a comDiscription is sometimes parallel. We can com-munity which consists of millions of users of allover the world. GeoGebra is supporting innovationof STEM (scientific: Science, technological: Technology, engineering: Engineering and mathematics: Mathematics) education, learning and guidance all over the world. GeoGebra can operate
easily by a GUI, can draw a chart of a numericalformula, can change the parameter, and can judge amovement by a slider. GeoGebra has the numericalpare diﬀerence of proofs by making it symmetricvisually. Fig. 11 shows the whole proof of Equation(1). This consists of two columns. One sideshows “A ∪ (B ∪ C) ⊃ (A ∪ B) ∪ C”. Other sideshows “A ∪ (B ∪ C) ⊂ (A ∪ B) ∪ C”. They arelocated symmetrically. As a result, learners can understand the parallelness of the description of theproof.

formula process functions of the factorization andthe diﬀerential and integral calculus.

5.3

Detective

Detective is to expect and argue unknown things
with known things. It is possible to make learners–2–

consider what continue after by not giving all sen-

Siri [22]. Xcode [23] [24] has a interactive sys-tence of proof (see Figs. 12-15). For example, totem concerning a speech framework. Third future

stop only “(ii) x ∈ B ∪ C のときさらに” in Fig. 14

work is to be desired that the learning application

then additional, “(ii-1) x ∈ C または (ii-2) x ∈ B

includes items, for example, contraposition, proofとなる。” in Fig. 14, they will think continuationby contradiction and clarity of definition. We willof proof. So, to stop makes to think.

include the other fields of education in the application. Final future work is to get feedback of students
that use the application.

5.4 Step by step
We can obtain a solution to deduce graduallywhen we solve a problem. To give learners hints

参考文献

little by little, they can learn certainly stepwise(seeFig. 12).

1) 文部科学省: 「デジタル教科書」の位置付けに関する検討会議 中間まとめ (平成 28年 6 月)

5.5 Interactive2) 文部科学省: 「デジタル教科書」の位置付

There are various methods to show solution ofproof: For example, to show all of sentences, toshow oneline sentence at a time, to show sentences
parallely, etc. Thus, students can select a method
as they can learn. Students can choose from several
items that they want to learn (see Figs. 8-9). Students can choose a solution method they like to tap
the buttons in Figs. 11-15. To tap the button “表示”in Figs. 11-15 every time, oneline sentence appears.
When students want to know proof early they can
see to tap the button “一括表示” in Figs. 11-15. Ifstudents still do not understand, they will be able

けに関する検討会議 最終まとめ (平成 28年 12 月)

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6. Result and Future Works
We have implemented the application about the8) 「デジタル教科書」の位置付けに関する検討

notion of sets for discrete mathematics in iOS in-会議, http://www.mext.go.jp/b_menu/

cluding “What we need to show”, “Symmetric”,

shingi/shotou/110/“Detective”, “Step by step” and “Interactive”. Firstfuture work is to add animation and venn diagram toimplemented application. Second future work, tomake the application more interactivity, we would
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Fig. 1
Share in Japan 1stquarter of 2017

Fig. 2
Share in Japan 1stquarter of 2016

–5–

Fig. 3
Share in the world
2014-2017

Fig. 4
Share in the world
2016-2017–6–

Fig. 5Venn diagram 1–7–

Fig. 6

Venn diagram 2

Fig. 7

Venn diagram 3–8–

Fig. 8 Screen shot1 ofApplication

Fig. 9 Screen shot2 ofApplication–9–

Fig. 10
Screen shot3of Application

Fig. 11
Screen shot4of Application– 10 –

Fig. 12
Screen shot5of Application

Fig. 13
Screen shot6of Application– 11 –

Fig. 14
Screen shot7of Application

Fig. 15
Screen shot8of Application– 12 –