Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics

Students’ images of two-variable functions and their graphs

“…It might appear that the research on visualization (Krutetskii, 1969(Krutetskii, , 1976Presmeg, 1986Presmeg, , 1992Presmeg, , 2006) is related to Lana and Jesse's ways of thinking about graphs. For instance, one could consider Jesse's thinking about graphs to be related to Krutetskii's visual reasoning construct, and Lana's thinking about graphs to be related to Krutetskii's analytical reasoning construct.…”

“…For instance, one could consider Jesse's thinking about graphs to be related to Krutetskii's visual reasoning construct, and Lana's thinking about graphs to be related to Krutetskii's analytical reasoning construct. One could also view Lana's thinking as related to Presmeg's construct of prototypes (Presmeg, 1986(Presmeg, , 1992 because Lana used shapes to help her associate a face (the graph) with a name (the equation). On the other hand, the connection becomes less clear when we note that Jesse expressed his visual reasoning in symbolic notation, and Lana's interpretations of her symbolic formulations were in terms of associated shapes.…”

Mathematical visualization process of junior high school students in solving a contextual problem based on cognitive style

“…Some research which discussed the benefit of visualization related to solving mathematical problem [1], [11], [12]. Besides, New Brunswick Mathematics Grade 8 Curriculum Guide (2011) stated that visualization was one of essential components which became the mathematics education goal.…”

“…It was the process in understanding the question of the algebra and visual shape by using the students' ability in observing the pictures by using their eyes and then connected it to their brains to solve the mathematical problem. The mathematical visualization was a mental ability or a mental activity where individual improved the connection between an internal construct and something which accessed through senses [1]. Mathematical visualization was one of abilities which was developed in a formal operational stage (about 12 years old until adult), which meant they were first grade students in the senior high school [2].…”

A Study on mathematical imaginations shown in children's mathematical narratives

“…그러나 은유 없 이는 수학도 발달할 수 없다고 가정하고, 수학에 깔려 있는 은유적인 토대인 '수학적 아이디어 분석 (Mathematical Idea Analysis; MIA)' 하였다 (Núñez, 2000a(Núñez, , 2000bLakoff & Núñez, 1997, 2000. 수학교육 분야에서도 수학학습이나 수학교실문화를 밝히고자 은 유 분석이 국내외에서 실행되었다(예를 들면, van Dormolen, 1991;Chui, 1994Chui, , 1996Sfard, 1994Sfard, , 1997Sfard, , 1998Sfard, , 2000Chapman, 1997Chapman, , 2008Lim, 1999;Presmeg, 1992Presmeg, , 1997김상미, 2005;김상미 외, 1997;김진호 외, 2014).…”