Mono- and Multi-Representational Learning of the Covariational Aspect of Functional Thinking

Using multiple external representations is advocated for learning in STEM education. This learning approach assumes that multiple external representations promote richer mental representations and a deeper understanding of the concept. In mathematics, the concept of function is a prototypical content area in which multiple representations are used. However, there are hardly any experimental studies investigating the effect of learning functional thinking with multiple representations compared to learning with only one form of representation. Therefore, this article reports on a quasi-experimental intervention study with students from Grade 7, using three measurement time points. The study compared the multi-representational learning of functional thinking with both tables and graphs with mono-representational learning with either tables or graphs. The results show that multi-representational learning led to advantages in learning qualitative functional thinking. However, in quantitative functional thinking, learning with both graphs and tables did not result in higher learning gains than learning exclusively with graphs. Furthermore, students were better able to transfer their knowledge from graphs to tables than vice versa. The results also indicate that multi-representational learning requires more time than mono-representational learning but can lead to higher learning gains. In sum, the results show that the effect of learning with representations is a complex interaction process between learning content and the forms of representation.

EXPLORING PROSPECTIVE ELEMENTARY MATHEMATICS TEACHERS’ KNOWLEDGE: A FOCUS ON FUNCTIONAL THINKING

The importance of students being acquainted with algebraic ideas before secondary education has been revealed in the research literature. It is therefore essential that prospective elementary teachers (PTs) be prepared to instill an early algebra perspective in their teaching. However, PTs often show difficulties in algebra content knowledge, which need to be diagnosed aiming to assist them in developing the required knowledge to teach according to that perspective. This study aims to understand what aspects of functional thinking Spanish and Portuguese elementary PTs exhibit at the beginning of their teacher education program. The findings show that although PTs from both countries use different strategies to generalize functional relationships, the occurrence of successful strategies is low. Also, most participants provide local approaches in their interpretation of relationships between variables and reveal difficulties in understanding and connecting different representations of functions. These difficulties show that PTs lack important knowledge about functional thinking. By providing a framework concerning the functional thinking required for PTs to teach within an early algebra perspective, we shed light on a necessary step for teacher education programs to diagnose PTs’ functional thinking and to assist them in developing the needed mathematical knowledge to teach accordingly.

Young Students’ Functional Thinking Modes: The Relation Between Recursive Patterning, Covariational Thinking, and Correspondence Relations

A theoretical model describing young students’ (Grades 1–3) functional-thinking modes was formulated and validated empirically (n = 345), hypothesizing that young students’ functional-thinking modes consist of recursive patterning, covariational thinking, correspondence-particular, and correspondence-general factors. Data analysis suggested that functional-thinking tasks can be categorized on the basis of the proposed model. Analysis traced three categories of students that represent different functional-thinking profiles. Category 1 students exhibited a recursive-thinking profile. Category 2 students utilized a combination of recursive and contextual strategies and exhibited an emergent covariational and correspondence-particular thinking. Category 3 students approached functional-thinking situations flexibly, using a combination of covariational and correspondence strategies. A structural model showed two parallel paths from recursive patterning to correspondence-general through correspondence-particular or covariational.

Designing digital technology tasks for the development of functional thinking

In this paper we introduce a theoretical framework concerned with fostering functional thinking in Grade 8 students by utilizing digital technologies. This framework is meant to be used to guide the systematic variation of tasks for implementation in the classroom while using digital technologies. Examples of problems and tasks illustrate this process. Additionally, results of an empirical investigation with Grade 8 students, which focusses on the students’ skills with digital technologies, how they utilize these tools when engaging with the developed tasks, and how they influence their functional thinking, are presented. The research aim is to investigate in which way tasks designed according to the theoretical framework could promote functional thinking while using digital technologies in the sense of the operative principle. The results show that the developed framework—Function-Operation-Matrix—is a sound basis for initiating students’ actions in the sense of the operative principle, to foster the development of functional thinking in its three aspects, namely, assignment, co-variation and object, and that digital technologies can support this process in a meaningful way.

STAGES IN PARTIAL FUNCTIONAL THINKING IN THE FORM OF LINEAR FUNCTIONS: APOS THEORY

Purpose of the study: The purpose of this study is to describe students' partial functional thinking processes in solving mathematical problems based on APOS Theory. The problem in this study was formulated into the question, what are the stages of students' partial functional thinking in solving mathematical problems based on APOS Theory?. Methodology: This study was conducted by with 44 students from the Department of Mathematics Education. The subjects of this study were asked to solve mathematical problems developed from (Wilkie, 2014). Then some of them were interviewed to learn their functional thinking processes. The subjects’ partial functional thinking processes were analyzed using APOS theory. Main Findings: The results showed that, based on APOS theory, the students’ partial functional thinking consisted of several stages: 1) identifying the problem, 2) organizing the data, 3) determining the recursive patterns, 4) determining the covariational relationships, 5) generalizing the relationships between variations in quantities (correspondence), and 6) re-checking the generalization results. In this case, the students generalized the relationships between variations in the form of functions done partially using the arithmetic formula . Applications of this study: The findings of this study can help teachers understand the stages in students' thinking processes in solving problems about functions and the difficulty faced by the students in understanding the functions. Novelty/Originality of this study: The researchers identified stages in students' partial functional thinking in solving mathematical problems in the form of functions based on APOS Theory.

Pendidikan Kewargaan Negara: Pengkajian Hak dan Kewajiban Warganegara dari Aspek Ilmu Politik dan Ilmu Perilaku

The study on the State Citizenship Education has experienced the development of discourse by experts, especially in political science and education science. In the democratic system, one's personality is still valued as something unique or specific. That is why there is a statement that mentions differences of opinion are still valued as diversity. In the State Citizenship Education, this discussion on personality provides an understanding of the main importance of person in the democratic system and everyone tries to play a role in it as a citizen who has the rights and obligations as appropriate as possible. This paper underlines that, behavioral and political science that use the basis of functional thinking and systems approach can be utilized by the State Citizenship Education to understand the use of citizens' rights and obligations in the society.