**4. Theoretical framework**

out. This association when we continued to the 4th, 5th , 6th terms and so on until the n‐th term,

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

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

+ 2n with n is the image num‐

it will get the functional formula by following form of F(n) = n2

curriculum innovation in Indonesia.

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

12 Science Education - Research and New Technologies

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

Equipped teachers with a number of competencies [12] suggested teachers [mathematics, in addition to the author] to follow the development of professionalism in order to gain new knowledge and skills so as to improve their teaching in the classroom. Nevertheless, we do not deny the condition that the change turned out to be only on the surface, as stated in Ref. [13] "There were not a lot of professional development activities for teachers or other types of inno‐ vations implemented as a routine activity for the next stage but there is only the result of the professional development (PD) or innovation is communicated through questionnaires, inter‐ views, or a survey" (p. 77). Symptoms such as those indicating innovation through PD (semi‐ nars, training, workshops) face the problem of sustainability, so often teachers are still applying old habits, otherwise known as the "back to basic," even though they have attended a number of times the workshops, seminars, and others. But the situation now is different, although the general teachers feel less comfortable when seen and observed by other teaching [14, 55].

Now gradually the teachers' perception have changed, at least felt by the teachers who attended the lesson‐study in Bandung [7]. They have changed their habits according to an anecdote, quoted by [15], "Two jobs that do not like to see by other people. That are work as a teacher and work as a thief," and if this anecdote is true, then for teachers, they are now open to be observed by others either by the teacher (another)or by policy makers (supervisors, department heads, principals). Now, they are open to learn from each other in improving the quality of learning at the microlevel in the classroom. Openness like this makes the chances of a teacher to have the optimal ability to make the classroom productive and allow teachers to apply science‐based mathematics instruction, so that the students have an opportunity to be creative in learning mathematics and sciences.

Through the implementation of these learning materials, it was difficult for students to forget it, because it has a very deep impression and also encourages teachers to apply them in their own learning accomplishments.

In a study paper, Ref. [15] recommends to examine deeply whether the teachers' willingness to improve their professionalism in teaching tasks can improve their perform in teaching? Moreover, whether their better perform can improve students' achievement in mathematics? What kind of professionalism improvement could boost their strong willingness to innovate mathematics instruction? To answer the challenge of the recommendation, the author offers a study on the implementation of learning mathematics using science‐based of didactical phe‐ nomena [1, 54], and empirically tested the implementation of this learning in the classroom.

Mathematics classes with the types of "transmission" as described by Senk and Thompson [16], include the introduction of each topic by declaring a rule which is followed by an exam‐ ple of how to apply the rules (rules, the arguments, the law), and then given a number of exer‐ cises, have encouraged developers who are looking for alternatives. Now, the effort to reform the mathematics is to portray the students participation actively, to transform the learning characterized by the "transmission" and to the learning characterized by the "participation."

In studying mathematics and science, the role of the students is constructing knowledge with the teachers. The teacher reveals the problems, asking questions, listening to students' answers, pursuing with follow‐up questions (probing questions), and then wait for the responses of the students in the formation of knowledge or mathematical concepts expected. Teachers should be little patience to listen to the arguments, presentation, and reasoning expressed by the stu‐ dents, either in the form of oral or written communication.

Hearing the mathematical ideas of students is an important aspect in learning sound con‐ structivism, i.e., to shift from "telling and describing" to "listening and questioning" and "probing for understanding" [17]. With science‐based instruction of mathematics, students are directly retrieving data, processing the data, presenting the data in tables, and describing the data in the table into a chart and then it becomes possible to make a mathematical model of images.
