Figure - available from: International Journal of Research in Undergraduate Mathematics Education
This content is subject to copyright. Terms and conditions apply.
A typical function diagram like those commonly used in Mendelson’s (2012) “Introduction to Topology” textbook
Source publication
Visual representations, such as diagrams, are known to be valuable tools in problem solving and proof construction. However, previous studies have shown that simply having access to a diagram is not sufficient to improve students’ performance on mathematical tasks. Rather, students must actively extract information about the problem scenario from t...
Citations
... As stated above, the coordination between different representations is a central cognitive mechanism in mathematics education (Arcavi, 2003) taking place in different areas of school and university learning-for example, fractions (Rau et al., 2009), trigonometry (Cooper et al., 2018), probabilities (Zahner & Corter, 2010), proof construction (Gallagher & Infante, 2022), and, particularly, (multivariable) functions (e.g., De Bock et al., 2015;Kabael, 2011;Makonye, 2014;Martínez-Planell & Gaisman, 2012;Yerushalmy, 1997). Based on the central functions of multiple representations to foster learning (Ainsworth, 1999), numerous researchers report a strong connection to and a positive effect of multiple representations on knowledge acquisition and problem-solving skills (e.g., Even, 1998;Gagatsis & Shiakalli, 2004;Rau et al., 2009;Rosengrant et al., 2007;Souto Rubio & Gómez-Chacón, 2011;Trigueros & Martínez-Planell, 2010;Villegas et al., 2009). ...
In mathematics education, students are repeatedly confronted with the tasks of interpreting and relating different representations. In particular, switching between equations and diagrams plays a major role in learning mathematical procedures and solving mathematical problems. In this article, we investigate a rather unexplored topic with precisely such requirements—that is, vector fields. In our study, we first presented a series of multiple-choice tasks to 147 introductory university students at the beginning of their studies and recorded students’ eye movements while they matched vector field diagrams and equations. Thereafter, students had to solve a similar coordination task on paper and justify their reasoning. Two cluster analyses were performed including (i) transition and fixation data on diagrams and options (Model 1), and (ii) additionally the number of horizontal and vertical saccades on the diagram (Model 2). In both models, two clusters emerge—with Model 1 distinguishing behaviors related to representational mapping and Model 2 additionally differentiating students according to representation-specific demands. Model 2 leads to a better distinction between the groups in terms of different performance indicators (test score, response confidence, and spatial ability) which also transfers to another task format. We conclude that vertical and horizontal saccades reflect executive actions of perception when approaching vector field coordination tasks. Thus, we recommend targeted interventions for mathematics lessons; these lessons must focus on a visual handling of the vector field diagram. Further, we infer that students’ difficulties can be attributed to covariational reasoning, thereby indicating the need for further investigations. From a methodological perspective, we reflect on the triangulation of eye-tracking and verbal data in (multiple-choice) assessment scenarios.
... Within this organizational frame, students' proofs progress through three phases: from collaborative construction in small groups, through whole-class presentation at the board by one of the constructors, to a posteriori reflection. As part of a larger project, we study proof progressions in a course in topology -a content area that has been rarely explored in university mathematics education (for exceptions, see Gallagher &Engelke Infante, 2021 andStewart et al., 2017). Our third goal is to illustrate opportunities for mathematics learning and teaching that proof progressions entail. ...
... Within this organizational frame, students' proofs progress through three phases: from collaborative construction in small groups, through whole-class presentation at the board by one of the constructors, to a posteriori reflection. As part of a larger project, we study proof progressions in a course in topology -a content area that has been rarely explored in university mathematics education (for exceptions, see Gallagher &Engelke Infante, 2021 andStewart et al., 2017). Our third goal is to illustrate opportunities for mathematics learning and teaching that proof progressions entail. ...
Coming from a social perspective, we introduce a classroom organizational frame, where students’ proofs progress from collaborative construction in small groups, through whole-class presentation at the board by one of the constructors, to a posteriori reflection. This design is informed by a view on proofs as successive social processes in the mathematics community. To illustrate opportunities for mathematics learning of proof progressions, we present a commognitive analysis of a single proof from a small course in topology. The analysis illuminates the processes through which students’ proof was restructured, developed previously unarticulated elements, and became more formal and elaborate. Within this progression, the provers developed their mathematical discourses and the course teacher seized valuable teachable moments. The findings are discussed in relation to key themes within the social perspective on proof.
... Regarding research methodology, our review showed that visualization was still predominantly investigated with qualitative research approaches, using, amongst others, task-based interviews and observations of small samples (see Presmeg, 2006). As these research methods allow visualization processes to be captured, they were often aimed at analyzing students' or teachers' interactions with visualizations during learning or teaching, respectively (e.g., Gallagher & Infante, 2021;Hollebrands & Okumuş, 2018). In addition to qualitative research approaches that examine visualization in depth, quantitative and mixed methods approaches can help validate theoretical considerations, e.g., theory of self-generated drawing (Van Meter & Garner, 2005), or contribute Numbers and operaƟons Analysis Algebra Geometry Probability and data analysis Logic and set theory n/a to collecting evidence on the effectiveness of visualization trainings on visualization use and learning outcomes (e.g., . ...
External visualization (i.e., physically embodied visualization) is central to the teaching and learning of mathematics. As external visualization is an important part of mathematics at all levels of education, it is diverse, and research on external visualization has become a wide and complex field. The aim of this scoping review is to characterize external visualizations in recent mathematics education research in order to develop a common ground and guide future research. A qualitative content analysis of the full texts of 130 studies published between 2018 and 2022 applied a deductive-inductive coding procedure to assess four dimensions: visualization product or process, type of visualization, media, and purpose. The analysis revealed different types of external visualizations including visualizations with physical resemblance ranging from pictorial to abstract visualizations as well as three types of visualizations with structural resemblance: length, area, and relational visualizations. Future research should include measures of visualization products or processes to help explain the demands and affordances that different types of visualizations present to learners and teachers.
... Prospective mathematics teachers face mathematical proofs in many courses during their education. According to Gallagher and Infante (2021), courses such as calculus, linear algebra, abstract algebra, and real analysis are the requirements for the completion of undergraduate level mathematics in general. Abstract algebra is full of definitions and theorems that all require proof, and students need to understand every definition and theorem they learn and be able to organize the concepts needed to prove theorems (Agustyaningrum et al., 2020). ...
... According to Selden and Selden, (2008), teachers should not teach a complex process such as proofs only through lectures and by trying to convey existing proofs directly to the student; they should also take into consideration the mutual interactions in the form of teacher-student and student-student. Giving students helpful tools that will facilitate their intuitive understanding of proofs can make complex ideas more comprehensible by helping students with reasoning and may increase students' participation in the proving process (Gallagher & Infante, 2021). Key points and ideas positively affect the performance of students in the proof process (Karaoglu, 2010). ...
The aim of this research was to examine prospective mathematics teachers’ completion processes of mathematical proofs supported by key ideas in the field of abstract algebra. Prospective teachers were asked to complete the parts of a proof left incomplete using those key ideas. It was decided together with three academics who are experts in the field of algebra which theorems to use and which proof sections to leave missing in the proof completion form. The proof completion form consisted of 5 proofs and was applied to 5 participants. The participants' processes of completing the proofs were evaluated in semi-structured interviews supported by the think-aloud method. Interviews were recorded by video and transcribed, and descriptive analysis was performed. The findings obtained were analyzed with think-aloud protocols. According to the results, key ideas can be said to play an active role in the proof process and the teaching of proof. In addition, developing activities consisting of proofs supported by key ideas can contribute to prospective teachers’ abilities to prove without memorization and internalize proof processes.
... (1) To interpret the tasks, and further lead the students to identify the theme, structures and how to express their opinions logically and creatively [9] . (2) How to raise students' cultural confidence of China and awareness of the depth, beauty and diversity of Chinese traditional culture and traditional virtues. ...
Reading should not only be focused on the learning of grammar and vocabulary, but more importantly, to learn new things. Therefore, high school English teachers should optimize their teaching process in terms of quality and quantity to promote high school English learning ability. For example, in “Focus on the Cultural Consciousness of High School English Grading Group Reading Study: In Jintang County,” the author taught a lesson on “When Accidents Happen.” This paper takes this course as an example, to discuss about the practice and thinking of high school English graded group reading during the teaching process.
