Spatial thinking is ubiquitous in design. Design education across all age groups encompasses a range of spatially challenging activities, such as forming and modifying mental representations of ideas, and visualizing the scenarios of design prototypes being used. While extensive research has examined the cognitive processes of spatial thinking and their relationships to science, technology, engineering, and mathematics learning, there remains a knowledge gap regarding the specific spatial thinking processes needed for open-ended problems, which may differ from those assessed in close-ended, analytical spatial tasks. To address this gap, we used educational design-based research to develop a nature-inspired, design-by-analogy project and investigate the spatial thinking processes of young, novice designers. 16 children from an international school in the Netherlands participated in this five-week design project. Multimodal evidence from classroom recordings and children’s design works were triangulated to offer insight into the key spatial thinking processes involved in their creation of nature-inspired, analogy-based design prototypes. Our results revealed spatial thinking processes that might not align with those assessed in conventional spatial tests and may be unique to design or open-ended problem-solving. These processes include abstracting spatial features to infer form-function relationships, retrieving a range of relevant visual information from memory, developing multiple possible analogical matches based on spatial features and relationships, elaborating and iterating on the design concepts and representations to make creative and suitable solutions for the design challenge, as well as visualizing design prototypes in practical usage scenarios. By highlighting the nuanced differences between spatial thinking in open-ended, divergent thinking tasks and conventional spatial tasks that demand single correct solutions, our research contributes to a deeper understanding of how children utilize spatial thinking in design and open-ended problem-solving contexts. Furthermore, this case study offers practical implications for scaffolding children's analogical reasoning and nurturing their spatial thinking in design education.

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Extensive research has established that spatial ability is a crucial factor for achieving success in Science, Technology, Engineering, and Mathematics (STEM). However, challenges that educators encounter while teaching spatial skills remain uncertain. The purpose of this study is to develop a research framework that examines the interrelationships, barriers, and enablers amongst various educational components, including schools, teachers, students, classrooms, and training programs, that are encountered when teaching for spatial ability development. A thorough examination of international research, in combination with a detailed review of the primary Science and Mathematics curricula in Ireland, Latvia, Sweden, and the Netherlands, is undertaken to acquire a more concentrated comprehension of the incorporation of spatial components in the curriculum. The review seeks to establish the fundamental factors that enable or hinder teachers in terms of curriculum, pedagogy, pedagogical content knowledge, and spatialized classroom practices.

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Understanding and effectively using visual representations is important to learning science, technology, engineering, and mathematics (STEM). Various techniques to visualize information, such as two- and three-dimensional graphs, diagrams, and models, not only expand our capacity to work with different types of information but also actively recruit our visual–spatial thinking. Data physicalization is emerging as a beginner-friendly approach to construct information visualization. Mapping intangible data onto tangible artifacts that possess visual, spatial, and physical properties demands an interplay of spatial thinking and hands-on manipulation. Much existing literature has explored using formatted infographics to aid learning and spatial thinking development. However, there is limited insight into how children may leverage their spatial thinking to create information visualizations, particularly tangible ones. This case study documented the data physicalization activities organized in two design classrooms of an international school in Netherlands, with 37 children aged 11–12. Seven themes relevant to spatial thinking were identified from multimodal evidence gathered from the data physicalization artifacts, classroom videos and recordings of children’s making process, and semi-structured interviews with children. Our findings suggested that these children generated various ideas to create visual–spatial forms for data with the materials at hand, such as mapping quantities to tangible materials of different sizes, using spatial ordinal arrangement, and unitizing materials to set visual parameters. Meanwhile, they evaluated and adjusted the visual–spatial properties of these materials according to the numerical data they had, crafting feasibility, and others’ spatial perspectives. What was particularly interesting in our findings was children’s iteration on their visual–spatial understandings of the intangible numerical values and the tangible materials throughout the embodied making processes. Overall, this study illustrated the different types of spatial thinking children applied to create their data physicalizations and offered insights into how embodied experiences accompanying the open-ended visualization challenge allowed children to explore and construct spatial understandings.@en

Mathematics is of major importance in science subjects. Unfortunately, students struggle with applying mathematics in science subjects, especially physics. In this qualitative study we demonstrate that transfer of algebraic skills from mathematics in physics class can be improved by using pre-knowledge effectively. We designed shift-problems involving instructional models to carry out small interventions in textbook problems. Shift-problems are feasible for teachers to adopt in teaching practice. To gain insight in the extent to which students improved their application of algebraic skills including basic skills and symbol sense behaviour, we selected three grade-10 physics students. In round one, the students solved algebraic physics problems as they appear in physics textbooks. Two weeks later in round two, the same problems were presented as shift problems to them where we activated prior mathematical knowledge by providing systematic rule-based algebraic hints at the start of these tasks. Algebraic skills were presented in a similar way to how these were learned in mathematics textbooks. We observed that students' problem-solving abilities increased from 48.5 % in the first to 81.8 % in the second round, indicating the effectiveness of how we implemented shift-problems. Furthermore, we discussed the implications of our results for the international science audience.

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Learning mathematical thinking and reasoning is a main goal in mathematical education. Instructional tasks have an important role in fostering this learning. We introduce a learning sequence to approach the topic of integrals in secondary education to support students mathematical reasoning while participating in collaborative dialogue about the integral‐as‐accumulation‐function. This is based on the notion of accumulation in general and the notion of accumulative distance function in particular. Through a case‐study methodology we investigate how this approach elicits 11th grade students’ mathematical thinking and reasoning. The results show that the integral‐as‐accumulationfunction has potential, since the notions of accumulation and accumulative function can provide a strong intuition for mathematical reasoning and engage students in mathematical dialogue. Implications of these results for task design and further research are discussed.@en

Senior pre-university education (SPE) students experience difficulties applying mathematics to physics. This paper reports the outcome of an online explorative quantitative study of teachers' belief systems about improving transfer of algebraic skills from mathematics into physics, conducted among 503 mathematics and physics teachers working in SPE. We used a questionnaire with 16 beliefs about improving transfer, and asked teachers to select a top 5 and distribute 50 points among them. We used agglomerative hierarchical clustering to cluster qualified SPE teachers with more than 10 years of teaching experience. We found 3 large clusters, each containing naïve and desirable beliefs about transfer. These clusters turned out to be rather coherent sets of beliefs. Hence, these clusters can be interpreted as belief systems, to a certain extent justifying Ernest's [(1991). The philosophy of mathematics education. London: Falmer.] idea to cluster teachers based on their belief systems. We found relations between our groups and those of Ernest. Since naïve beliefs turn out to be weak in each cluster, science teacher educators can help science teachers to change their harmful naïve beliefs, into desirable transfer enhancing beliefs. Furthermore, we discuss some implications of our results for science teacher educators, curricula, teachers and textbooks.

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Students in senior pre-university education face difficulties in the application of mathematics in physics. This paper presents the results of a qualitative study on teachers’ core beliefs about improving the transfer of algebraic skills to physics. Teachers were interviewed about their beliefs regarding a transfer problem from mathematics to physics for which solution algebraic skills were needed. We obtained large amount of data which were reduced to sixteen core beliefs including constraints and affordances influencing students’ demonstration of coherent mathematics education (CME) and transfer of algebraic skills from mathematics into physics. These core beliefs were grouped into the five main categories ‘Collaboration’, ‘Curricula’, ‘Students’, ‘Teachers’ and ‘Textbooks’. We think that our approach to pattern coding is both elegant and generally applicable to reduce code trees including large amount of data. Four core beliefs were identified as naïve beliefs, which may impede transfer. We provided a powerful remedy against such unproductive beliefs: through professional development programs teachers with such beliefs should be made aware, reflect and reconcile their naïve beliefs with those required for transfer. These core beliefs contain data to extract teachers’ belief systems. Quantitative research could investigate to which extent this is the case and which beliefs these contain.@en

Students in upper secondary education encounter difficulties in applying mathematics in physics. To improve our understanding of these difficulties, we examined symbol sense behavior of six grade 10 physics students solving algebraic physic problems. Our data confirmed that students did indeed struggle to apply algebra to physics, mainly because they lacked both sufficient symbol sense behavior and basic algebraic skills. They used ad hoc strategies instead of correct, systematic rule-based procedures involving insight. These ad hoc strategies included the cross-multiplication, the numbering, and the permutation strategy. They worked only for basic formulas containing few variables. In problems with more variables, students got stuck. The latter two strategies substitute numbers for variables. The permutation strategy randomly checks several permutations to guess which one is correct. The numbering strategy substitutes numbers to check algebraic manipulations. Our results indicate insufficient focus on conceptual understanding of algebra in some mathematics textbooks, leading to reliance on poorly understood ad hoc strategies. Effective teaching of algebraic skills should not focus on either basic algebraic skills or on symbol sense behavior. Instead, both aspects should be taught in an integrated manner. Our operationalization of symbol sense behavior turned out to be very useful for analysis. In contrast to earlier qualitative studies, it provided us the opportunity to measure symbol sense behavior quantitatively. This operationalization should also be applicable to other science subjects. Furthermore, we discussed some implications of our results for curricula, teachers, science teacher educators, and textbook publishers aiming at successful application of mathematics in physics.

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Students in senior pre-university education encounter difficulties in the application of mathematics into physics. This paper presents the outcome of an explorative qualitative study of teachers’ beliefs about improving the transfer of algebraic skills from mathematics into physics. We interviewed 10 mathematics and 10 physics teachers using a semi-structured questionnaire that was based on an algebraic transfer problem. Almost all teachers acknowledged this transfer problem and considered it to be important. We found a continuum of teachers’ beliefs about aspects influencing transfer, including beliefs on improving this transfer. Together with identified improvement aspects about coherent mathematics education, these may help reduce physics teachers’ frustrations who spend extra time on re-teaching mathematics. Teachers think that transfer does not happen, because students see both subjects as separate disciplines. Contrary to most physics teachers, most mathematics teachers do not feel the need to collaborate with physics teachers. We found two extreme, opposite beliefs about the transfer of algebraic skills into physics. An intermediate group believes that only an integrated approach can solve the transfer problem. Some of the teachers’ beliefs could be organised into a beliefs system. Further research could investigate to which extent such beliefs systems exist and which beliefs these contain.

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