No abstract

Usually, multiplication is introduced to students to represent quantities that come in group. However there is also rectangular array model which is also related to multiplication. Barmby et al (2009) has shown that the rectangular model such as array representations encourage students to develop their thinking about multiplication as a binary operation with row and column representing two inputs. Considering that finding, this study focusses on a design research that was conducted in Indonesia in which I investigate second grade students' (between 7 and 8 years old) in Madrasah Ibtidaiyah Negeri (MIN) 2 Palembang, Indonesia, ability to structuring the situation and their ability to represent rectangular model into multiplication sentence. The results shows us that students activity to structuring the situation, looking the number of objects in row or in column, lead them to repeated addition and transform it into multiplication sentence.

Integral is one of major topics in Calculus course. Many university students fail in calculus course because their understanding on the concept of Integral especially in understanding definition of integral is low. This situation encourages us to design a calculus lesson to build students understanding on Integral especially integral definition by using Realistic Mathematics Education as an approach. For that, we try to design a series of instructional activities starting from introducing sigma to definition of Integral in order to know students difficulties in understanding this topic. Some first-year university students in Universitas Negeri Padang are chosen to test our Hypothetical Learning Trajectory (HLT) in our preliminary research. The result is students had difficulties in connecting the sigma, area and definition of integral. This happens because the students did not see that the activities that have been designed as a whole that cannot be separated. Therefore, it is recommended to design a series of instructional activities which is focus on connecting sigma, area and definition of integral.

This research aimed to develop local instruction theory that is valid, practical, and effective to help elementary school students developing their mathematical problem-solving skills. Therefore a sequential activityis design on dailybasis to encourage students to develop their ability to solve mathematical problems, especially on the topic division. To achieve the goal, realistic mathematics approach was implemented to grade three elementary students in the learning process. The designed activities were validated by experts on the aspects of mathematical contents, language, didactical process based on realistic mathematical approach. Data were analyzed with descriptive statistics and parametric statistics. The validation results show that the local instruction theory was valid, and the implementation shows that the local instruction theory is practical and effective in improving students' mathematical problem-solving skills.

The low learning outcomes and student’s difficulties in understanding calculus courses, espe-cially integral topic, encourage us as researcher to design a calculus lesson for first year students by using realistic mathematical approach (Realistic Mathematics Education). For that we try to design a series of instructional activities starting from understanding integral as an anti derivative. The in-structional activity is designed and developed based on the learning process that occurs in the class-room by involving 30 first year students in FMIPA UNP. The classroom learning compared with our Hypothetical Learning Trajectory (HLT). The results of teaching experiment show that students' un-derstanding of derivative plays important role in understanding the integrals, in general students have difficulty in communicating their ideas in determining the anti-derivative of a function. Through class-room discussions students can get ideas and discuss it in determining the anti-derivative of a function. Based on these results it is recommended to conduct classroom discussions to build students' under-standing in studying calculus especially integral as an anti derived.