**2. Background**

that is the number percentage of sugar solution and sedimentation time of "ball" on each of

**Figure 3** represents various concentrations of solutions. Practically, we used more details of sugar solutions by inserting 15, 25, 35, and 45% of sugar solution. Instructional design of mathematics based on the science using didactical phenomenology approaches was pre‐ sented to the students, so that the students have a sense of the relationship between two quan‐ tities. Students have the ability to represent it in a variety of mathematical representations. Furthermore, numbers obtained are processed and then packaged in various forms of math‐ ematical representations. Using this reflective thinking, students can think mathematically and scientifically to instill their awareness to be able to live in a healthy life and harmony.

The chapter presents the results of this study synergically involving scientific activities. Didactical phenomena of instructional design of mathematics were designed by the team in the laboratory activities, recorded using video camera, transferred to the power point presen‐ tation. The student tasks in the classroom were to make mathematical model or mathematical graph related to the data as the result of observation was recorded in video camera and power point presentation. These data were presented in the form of table. The pair numbers in the table as coordinates were then be plotted in the Cartesian coordinates. By graphing this phe‐

As a design proposed by Verschaffel, Greer, and De Corte (2002, cited by Turmudi et al. [2], **Figure 4**), the process of learning mathematics that involves the modeling process includes the observation of phenomena (reading), understand the situation, modeling, analyzing the

The teaching materials of mathematics in this study used the didactical phenomenon in the form of a natural situation or similar situations that conditioned the models created in the science laboratory. The students in the classroom are faced with these instructional materials

> Mathemacal Model

Generang from model Mathemacal Analysis

model, formulate the results, interpretation, and then make a communication:

and the materials were manipulated in the form of power point presentation.

Situaon Model

Evaluaon

Unde Modeling rstanding

Interpretaon

Interpretaon Communicaon

nomenon, the students were asked to interpret the phenomena.

Reports Results of

**Figure 4.** Diagram of mathematical modeling process.

these solutions.

10 Science Education - Research and New Technologies

Observed phenomena After doing some research in a clump of innovative teaching, research teams are interested in trying out the learning of mathematics with science‐based didactic phenomena within rea‐ sonable time. Research studies on realistic mathematics have been conducted with the result that is very promising and could attract the attention of students [3, 4]; encourage teachers in the Bandung city area to realize that they already know their innovative ideas but do not have the ability to present learning with innovative ideas because of the absence of exemplary prototypes [5, 6], the teachers realized that the training through lesson‐study has opened their insight to continue working and improving the learning for teaching of mathematics in the classroom [7] but the teachers still require exemplary prototype of mathematics learning using scientific approach that can be implemented in the classroom, so that openly they can watch in open‐lesson setting, and in turn the teachers be able to implement it in their respec‐ tive classes.

Results of research on mathematical modeling by [2, 8] show that the students involved in the study felt there was something new in mathematics instruction. For example in view of the variables that were not visible, but after attending the workshop the participants were able to see the variables in the phenomena, so they were able to make the association among vari‐ ables that exist to make a mathematical model. Consider the following figures:

**Figure 5** is the pattern model that originally taken from the floor of JICA‐FPMIPA building of UPI on the second floor (**Figure 6**, personal collection of photograph), the tile patterns can be seen in the image (**Figure 7**, personal collection of photograph).

At first time, the images were just as the pictures without meaning, the students are not too concerned with tile patterns like that, but with a small call "Let us see and we noticed a pat‐ tern (in **Figure 7**), as well as how to process it so we could have an interesting mathemati‐ cal concept." The invitation make a number of students feel surprised by the mathematical patterns that exist in the JICA Building, in Bandung. When the students were able to see the pattern of the picture and are able to associate with the image number and the area of the geometry shapes, then they obtained a mathematical model that previously did not figure

**Figure 5.** Patterns of floor in JICA‐FPMIPA building.

**Figure 6.** JICA‐FPMIPA building in UPI.

out. This association when we continued to the 4th, 5th , 6th terms and so on until the n‐th term, it will get the functional formula by following form of F(n) = n2 + 2n with n is the image num‐ ber (term) and F(n) is the area of geometrical shapes. By knowing how to build and determine the formula of model, the students can reach through the process of horizontal and vertical mathematization [9] and the process of progressive mathematization [10]). Teacher's capa‐ bilities to view such situations nowadays are needed urgently, so that they can try out these competences at the microlevel of the classroom. Such capabilities are seemed urgent, because the implementation of the 2013‐curriculum [11], which was characterized scientifically, in fact, was not absorbed completely by the teachers during the 2013‐curriculum upgrading. Therefore, the presence of such learning is a positive contribution toward building the current curriculum innovation in Indonesia.

Provisioning capabilities of mathematical concepts, and the pedagogical content knowledge (PCK) of mathematics for teachers would give effects to a teacher as an actor of mathematics learning in action in front of the class. The ability to see the phenomena should be the part of the teachers' as well as the students' competences, so that they always think continuously

**Figure 7.** Floor in JICA‐FPMIPA building.

over time and will always able to find connections between the existing phenomena by taking into account the specific quantities or the shapes in the flat geometry (2D) or space geom‐ etry (3D), therefore it can intertwined the functional relationship between the quantities that appeared in Ref. [54].
