The Splitting Group

2012 ◽  
Vol 43 (5) ◽  
pp. 557-583 ◽  
Author(s):  
Anderson Norton ◽  
Jesse L. M. Wilkins
Keyword(s):  
Quantitative Analysis ◽  
Student Learning ◽  
Algebraic Groups ◽  
Rational Numbers ◽  
Learning Trajectory ◽  
Mathematical Structures ◽  
Hypothetical Learning Trajectory ◽  
Eighth Grade Students ◽  
Piagetian Theory ◽  
Mental Operations

Piagetian theory describes mathematical development as the construction and organization of mental operations within psychological structures. Research on student learning has identified the vital roles of two particular operations–splitting and units coordination–play in students' development of advanced fractions knowledge. Whereas Steffe and colleagues (e.g., Steffe, 2001; Steffe & Olive, 2010) describe these knowledge structures in terms of fractions schemes, Piaget introduced the possibility of modeling students' psychological structures with formal mathematical structures, such as algebraic groups. This paper demonstrates the utility of modeling students' development with a structure that is isomorphic to the positive rational numbers under multiplication–the splitting group. We use a quantitative analysis of written assessments from 58 eighth grade students to test hypotheses related to this development. Results affirm and refine an existing hypothetical learning trajectory for students' constructions of advanced fractions schemes by demonstrating that splitting is a necessary precursor to students' constructions of 3 levels of units coordination.

scholarly journals Kajian Hypothetical Learning Trajectories dalam Pembelajaran Matematika di Tingkat SMP

2020 ◽  
Vol 1 (1) ◽  
pp. 1-11
Author(s):  
Agnes Ivana Hendrik ◽  
Christine K Ekowati ◽  
Damianus D Samo
Keyword(s):  
Student Learning ◽  
Mathematics Learning ◽  
Student Thinking ◽  
Learning Activities ◽  
Learning Trajectory ◽  
Learning Trajectories ◽  
Hypothetical Learning Trajectory ◽  
Main Components ◽  
Learning Flow ◽  
Improve Student Learning

Mathematics learning in this era requires teachers to be able to develop learning models that pay attention to the characteristics of students, so it is important for teachers to know learning trajectory. This study aims to describe the conception of learning trajectory and hypothetical learning trajectory, as well as presenting the design of hypothetical learning trajectories in mathematics learning. Based on the theories studied, it is concluded that hypothetical learning trajectory is a learning design in the form of allegations on student learning activities based on initial understanding and characteristics of students to achieve higher understanding. HLT has three main components namely learning objectives, learning activities, and hypotheses of the learning process. HLT is used by teachers as a guide to predict and prepare learning flow designs that are appropriate to the stages of student thinking and can improve student learning outcomes. The findings of previous studies show that to design HLT in mathematics learning, educators need to consider student obstacle learning, a hierarchy of material, and learning support capacity, so that learning activities to be hypothesized are built based on these things. It is recommended that future researchers can use the existing HLT, and also develop HLT in other materials that have not been tested by further studying HLT and whatever affects it, research with different differences can be used.

scholarly journals Study of Hypothetical Learning Trajectories in Mathematics Learning

2020 ◽  
Vol 9 (3) ◽  
Author(s):  
Agnes Ivana Hendrik ◽  
Yuanda Oktrin Lay ◽  
Yohanes Sanca Nele Amuntoda
Keyword(s):  
Student Learning ◽  
Mathematics Learning ◽  
Student Thinking ◽  
Learning Activities ◽  
Learning Trajectory ◽  
Learning Trajectories ◽  
Hypothetical Learning Trajectory ◽  
Main Components ◽  
Learning Flow ◽  
Improve Student Learning

Mathematics learning in this era requires teachers to be able to develop learning models that pay attention to the characteristics of students, so it is important for teachers to know the learning trajectory. This study aims to describe the conception of learning trajectory and hypothetical learning trajectory, as well as presenting the design of hypothetical learning trajectories in mathematics learning. Based on the theories studied, it is concluded that a hypothetical learning trajectory is a learning design in the form of allegations on student learning activities based on initial understanding and characteristics of students to achieve higher understanding. HLT has three main components namely learning objectives, learning activities, and hypotheses of the learning process. HLT is used by teachers as a guide to predict and prepare learning flow designs that are appropriate to the stages of student thinking and can improve student learning outcomes. The findings of previous studies show that to design HLT in mathematics learning, educators need to consider student obstacle learning, a hierarchy of material, and learning support capacity, so that learning activities to be hypothesized are built based on these things.

scholarly journals THE CONTEXT OF ARCHIPELAGO TRADITIONAL CAKE TO EXPLORE STUDENTS’ UNDERSTANDING IN INTEGERS DIVISION CLASS VII

2019 ◽  
Vol 14 (1) ◽  
pp. 91-100 ◽  
Author(s):  
Elis Muslimah Nuraida ◽  
Ratu Ilma Indra Putri
Keyword(s):  
Junior High School ◽  
Design Research ◽  
Mathematics Learning ◽  
Learning Trajectory ◽  
Mathematics Problems ◽  
Learning Path ◽  
Starting Point ◽  
Hypothetical Learning Trajectory ◽  
Integer Division ◽  
Division Operation

This study aims to explore the students’ mathematical understanding in integer division operation through the context of archipelago traditional cakes in class VII. This research is related to the Indonesian Realistic Mathematics Approach (PMRI) as a learning approach used. The methodology used in this study is Design Research consisting of three stages: preliminary design, experimental design, and retrospective analysis. The study was conducted on VII grade students of Palembang 1 Junior High School. The learning path (Hypothetical Learning Trajectory) in design research plays an important role as a research design and instrument. The Hypothetical Learning Trajectory (HLT) was developed together with a series of activities using the context of archipelago traditional cakes such as: omelette roll, bakpia, milk pie, etc. The medium used in this study was the Students’ Activity Sheet. The results of this study indicate that exploration using the context of traditional archipelago cakes can help students understanding in multiplication and division of integers. The conclusion of this study is the use of archipelago traditional cakes as starting point in mathematics learning in integer division operation material helps the students to explore their understanding in solving mathematics problems.

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