After mathematical modelling processes along “the Grammar of Science” by Karl Pearson and its extensions as PDCA cycle and PPDAC cycle for general problem solution are explained to clarify the significance and role of mathematical sciences, some illustrative examples of the modelling process are introduced on mathematical and statistical models for classification, systematic and/or cyclic trends and general associations, which should be utilized in elementary and secondary education in Japan.

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We have developed a new concept representation method named Manga Card Sort to achieve learning process model in Manga Case Method. We have experimented to confirm the effectiveness of Manga Card Sort with Manga Cases dealing with business and with two types of subjects who have different degrees of achievement in business. The result shows that Manga Card Sort makes learners examine their own concepts and modify them (learning process I). This result is not depenent on degree of achievement. It also shows that Manga Card Sort makes learners find contrarieties and improvements in others’ concepts and point them out (learning process II), but only if learners’ degree of achievement is low, or the number of discussion group members is two. We couldn’t confirm the effectiveness to achieve learning process II when learners’ degree of achievement is high or the number of discussion group members is over three.

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Previous studies on mathematical modeling have reconsidered the value of school word problems by capturing traditional word problems relatively from a mathematical modeling perspective. However, it is not sufficiently clear how pupils in elementary school interpret a problem situation in a more authentic word problem. The purpose of this paper is to explore how pupils interpret the problem situation, and clarify the peculiarity of more authentic word problems. To achieve this purpose, the current study developed a set of problems, including one with division story problems involving remainder. This paper analysed pupils’ responses to them. The results reveals that pupils are able to interpret the situation based on sympathy, generality, adjustment, equality and pleasure. These interpretations become the fundamentals of better interpretations, and therefore it is suggested that dealing with more authentic word problems can become the fundamentals of mathematical modeling.

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In this study, a lesson model related to the phenomenon of the rusting of iron was developed to confirm the effect of modeling by drawing on the understanding of chemical reactions. The model was applied in the non-science course “Science study (chemistry)” classes comprising third-year undergraduate students in Tokyo Gakugei University. Students constructed images of the rusting of iron, in addition to attending lectures with frequent discussions, without any prior written communication to students, as well as taking notes, and the students completed the subsequent process of drawing as well as its explanation. Students drew pictures with their own images and explained their drawings appropriately as per their own understanding of chemical concepts and the phenomenon at hand. By the evaluation of the lesson model via the lesson of an “activated complex” as a representative of the model by the process of drawing and self-explanation, as well as an intelligibility test, students could understand chemical concepts and images of the rusting of iron by actually drawing, i.e., the task of drawing could strengthen the student’s explanatory ability and intelligibility about the rusting of iron; in particular, it was effective for intelligibility even for students who obtained a low score of 0 points, besides the limitation of the effect on reinforcement of explanatory capability of the phenomenon for the 0 points score students, suggesting that learning with imaginative thinking and behavior is imperative for students to understand chemical reactions by evaluation.

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Aiming to foster competencies expected in chemistry education, experimental lessons with model-based learning for chemistry were implemented. The study verified how competencies expected in the discipline were fostered with respect to the practices of instructions, practice, and evaluation, through implementing lessons with and without focus on model-based learning. The mutual evaluation table (Goto, 2013) was used for evaluating the practice, mainly looking at the description written in the discussion part of students’ reports. Verification of the practice was based on the curriculum management model for science lessons (Goto and Matsubara, 2015). The experimental lessons demonstrated revealed what kind of impacts and challenges exist for practicing model-based learning for future implementation.

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The present study developed a new scheme of model-based learning in chemistry education and extracted units of a round of activities of experiments in chemistry, aiming to foster competencies expected in the discipline. These units consist essentially of modeling, experiment plan, verification, and report writing. Applying the idea of TEIKEIBUN (fixedly-formed sentence), elements of the modeling were fixedly-formed and the relationships between the elements of modeling, and the experiment plan and the items used in the report writing were shown. As a challenging issue, the study has revealed that a change in perspectives is necessary for individual topics and a theme, which is the underlying main idea among similar topics.

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Mathematical modeling is one of the activities through which various real-life situations can be considered from a mathematical perspective. It also provides an opportunity for the practical application of the knowledge and skills learned in the textbook. However, in reality, most of the students in Japan have never experienced mathematical modeling. In this report, we will present a summary of the “Modeling Challenge Program” which was held in Kyoto twice, and also the students’ activities of mathematical modeling we observed in this program. At the same time, we will describe the two modeling teaching materials that were used in the program. The first one is related to the “vote-value disparity and political districting problem” and the second one is related to an “expanding sphere problem.” In this program several high school students attempted to solve the above problems and experienced the mathematical modeling.

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This study aims to illustrate the effectiveness of model-making and simulations for solving social problems in high school information study classes. The instructional material seeks to foster scientific decision-making capability through modeling by simulation of random events.
The methods of the instructional design are as follows: First, we divided the scientific decision-making capability into five capability components: “modeling”, “representation”, “method”, “interpretation, evaluation, and improvement”, and “communication”. Second, we developed an instructional procedure which consists of three steps, “Modeling aimed to solve unstructured problems”, “Implementing the simulations by random events”, and “Decision-making by using a result of the simulation”. Third, we designed a lesson plan including formative assessment and team cooperative problem solving through interaction, and put it into practice.
As a result of analyzing the performance of students from different aspects using their practical records, within “modeling”, “representation”, and “method”, we found that some students begun to think from diverse points of view and continued onto the phase of sociality.
The result of this study illustrates that interaction and formative assessment are effective means that shift decision-making capability to the phase of sociality, revealing that model-designing and simulation activities with interactions aimed to solve unstructured problems help foster the students’ scientific decision-making capability, thus proposing the implementation of these activities in schools.

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The purpose of this study was to investigate the development of observational skills in an undergraduate student’s lessons.
In this study, we developed a training method for observing lessons and studying lessons to improve observation skills in lessons. The following method was used: In the lesson, both the undergraduate student and an expert teacher recorded the lesson by using their iPad. In the lesson study, ① the undergraduate student explained his observations on the basis of the video he took with his iPad; ② the expert teacher explained his observations on the basis of the video he took with his iPad; and finally, ③ they reflected on the lesson study together. We repeated this procedure four times. We analyzed the lesson study by looking at the following three factors: a) the number of sentences produced by the undergraduate student and the teacher and the contents of their sentences; b) mutual understanding between the two individuals through their interaction; and c) the interview of the undergraduate student.
As a result, the undergraduate student was able to enhance his lesson observational skills by continuously having these lesson studies.

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This paper examines whether elementary school students’ understanding of parallel and serial connections of dry cells was improved by teaching the concept of electric voltage.
Students in the experimental group were taught the concept of electric voltage for 20 minutes using analogy of water flow, such as, the dry cell is like a water pump to uplift water and the voltage stands for the height of uplift.
The experimental group, who was taught electric voltage, showed a significantly higher achievement score than the control group in the post test. Moreover, the way of thinking about the brightness of the miniature bulb differed between groups on the circuit in which three dry cells were connected parallel.
On the other hand, students in the experimental group showed more scientific answers to a descriptive question about electricity compared to the control group. The concrete understanding about phenomena regarding electricity may promote a scientific way of thinking.
The results of this study imply that we can introduce the concept of electric voltage to elementary school science class if we exercise ingenious ways of teaching, even though the concept is not being taught under the present elementary school curriculum in Japan.

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By focusing on the unit “function of saliva” in the second grade of a lower secondary school, this study clarified the concrete effects of hypothesis formulation, including aspects such as sharing and sophistication of the hypotheses, on data interpretation in order to obtain fundamental knowledge on the development of data interpretation skills. The following three findings were obtained:
(1) Sharing and sophistication of the hypotheses formulated by the students increased in the proportion of scientifically correct descriptions of data interpretation; for example: “Starch was broken and changed to glucose by saliva” and “Saliva reacted (mixed) with starch to form glucose.”
(2) Sharing and sophistication of the hypotheses formulated by the students increased in the proportion of students who connected the hypotheses with a scientifically correct interpretation of the data.
(3) Sharing and sophistication of the hypotheses formulated by the students increased in the proportion of students who made scientifically correct data interpretations based on evidence.

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We developed teaching materials about the flow-direction and the characteristic of the cationic ray in vacuum. In general, Crookes tube uses plate-shaped electrodes for observation of the flow-direction of the electron ray. By the use of needle-shaped electrodes instead of plate-shaped ones, the purplish-red color discharge was generated in the low vacuum and the impeller was rotated toward the cathode from the anode opposite to the case of Crookes tube. This phenomenon is considered to indicate that the cationic particles collided with the surrounding neutral ones and those gained larger momenta, as a result of continuous collision. Thus, it was observed that the purplish-red color discharge was deflected to the direction of the Lorentz force by the use of magnet. It is expected that our teaching materials are effective in promoting the observation and understanding the relationship between the current flow and the magnetic field.

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