Ratio and Proportion

Penalaran prorporsional dikenal sebagai dasar untuk keberhasilan siswa dalam menyelesaikan masalah di berbagai topik mata pelajaran Matematika. Pada kenyatannya, masih banyak siswa kelas VI-VIII yang mengalami kesulitan materi perbandingan, khususnya ketidakpahaman masalah perbandingan tertentu atau mengapa strategi yang digunakan berhasil dalam menyelesaikan masalah. Penelitian ini menganalisis proses dan kesalahan 115 siswa kelas VII dalam menyelesaikan masalah perbandingan dan proporsi setelah mempelajari materi Rasio dan Skala. Dengan menggunakan metodologi kualitatif, proses berpikir siswa diselidiki berdasarkan lima tingkatan dalam menyelesaikan masalah rasio dan proporsi. Hasil yang diperoleh menunjukkan bahwa lebih dari 75% siswa tidak mampu membedakan masalah yang termasuk proporsi dan yang bukan. Kesalahan ini dikarenakan kurangnya latihan dan pengenalan siswa tentang masalah perbandingan dan proporsi. Siswa cenderung menentukan satuan per unit dan menggunakan strategi perkalian silang untuk penyelesaian masalah nilai yang dicari, namun mereka belum memahami mengapa mereka menggunakan strategi teresebut.

Razones, proporciones y proporcionalidad constituyen un campo ampliamente investigado en los últimos cincuenta años. Evaluaciones recientes muestran que estos objetos de conocimiento siguen siendo difíciles de aprender para la mayoría de los estudiantes, lo que constituye un certero indicador de la necesidad de hacer mayor investigación didáctica que permita nuevas comprensiones de dicha problemática y, por esa vía, lograr mayores impactos en el sistema educativo. En este artículo se revisan y comentan algunas investigaciones recientes sobre razón, proporción y proporcionalidad. De acuerdo con la perspectiva de análisis, se agrupan en tres momentos: cognitivo, epistémico y semióticoantropológico. Finalmente, se plantean algunos problemas de investigación a manera de conclusión.

In the transition years, Grades 7 to 9, the shift from natural numbers to rational numbers and the associated multiplicative concepts prove challenging for many learners. The new concepts, operations and notation must be mastered if the student is to thereafter rise to meet the challenges of algebra and more advanced and powerful mathematics. The multiplicative conceptual field (MCF) groups together such concepts as fraction, ratio, rate, percentage and proportion all of which are related yet subtly distinct from one another, each with its own challenges. Rasch analysis allows us to compare the difficulty of mathematical problems located within the MCF while, on the same scale, locating the degree to which individual learners have mastered the necessary skill set. Such location of problems and learners on the same unidimensional scale allows for fine-grained analysis of which aspects of the problems being analysed make one more difficult than another. Simultaneously it gives the teacher clear evidence of which students have mastered which concepts and skills, thereby allowing more targeted assistance to the class and individual learners. This paper illustrates the process involved in such analysis by reporting on results located within a larger study. It is suggested that implementing  Rasch analysis within the school classroom on appropriately designed assessment instruments would provide clarity for the teacher on the location of difficulty within the problems used in the assessment and the relative degree to which individual learners are achieving success at mastering the targeted concepts. 

Keywords: Rasch measurement; multiplicative conceptual field; ratio and proportion; assessment

We believe that the problem of teacher preparation is one of development of “pedagogical content knowledge” rather than “subject knowledge” per se. It has been found that even experienced teachers may not be aware of the misconceptions that learners tend to exhibit, or at what stage of development and in what areas of the curriculum these are likely to be manifested. This pedagogical knowledge is, we believe, important to teachers’ mental models of their learners, and hence their teaching effectiveness. In this study, we aim to contribute to teachers’ awareness of their pupils’ strategies and misconceptions in the field of “ratio”: a topic that is difficult to teach and learn in the middle school years. Towards this aim, we constructed a diagnostic instrument which reveals children's proportional thinking. Our instrument contains two versions, one with “models” thought to be of service to children’s proportional reasoning and one without. It is also designed to function as a que...