Preservice Teachers’ Mathematical Knowledge about Repeating Patterns and their Ability to Notice Preschoolers Algebraic Thinking

Background: Several studies have shown that many preservice teachers (PTs) who teach in the early years have a superficial knowledge about repeating patterns (RPs), which affects their knowledge about children’s algebraic thinking. Objective: This article aims to understand PTs’ algebraic thinking and their ability to notice preschoolers algebraic thinking and how these two domains articulate within a teacher education experiment. Design: The study follows a qualitative methodology based on participant observation, complemented by document collection. Setting and participants: The study stems from a teaching experiment carried out in a school module focused on patterns and algebra of a degree in basic education, with two pairs of PTs as participants. Data collection and analysis: The data come from the written productions and discussions between the elements of each pair of PTs within the scope of the tasks proposed in the teacher education course, adopting an original analysis framework. Results: The results reveal that the PTs successfully identify the structure of the RPs and the general position of each term; however, one of the pairs still find difficulties in fully understanding that mathematical object. The pairs attend to relevant aspects of children’s algebraic thinking, although sometimes with limited interpretation. Conclusions: This study highlights the importance of creating opportunities in initial teacher education for PTs to develop their algebraic thinking from an early algebra perspective and to analyse, in this context, the preschoolers’ work.

Urban ethnic enclaves and migration industries: The urban choices of mobile people

When migrants come in large numbers, they tend to segregate in enclaves where they lead a familiar lifestyle alongside people who can provide a support system. But how do these enclaves come about? This paper engages with migration industries literature, saying that it is ‘the labour involved in managing, facilitating and controlling migration’ that makes it an industry. Relying on the case of privileged Jewish migration to Israel, I argue that while the state remains central in facilitating and controlling migration, migration industries and migrants’ social networks dictate in which urban areas privileged migrants settle, creating unequal urban geographies. To illustrate this, I rely on qualitative data gathered in two research projects I completed in Israel over the past decade, in various Israeli cities relating to migrants and second-home owners from Western countries. I look at how and why people decide where to settle upon migrating and the role of various migration industries actors in their choices. I argue that what seems like individual decision-making is in fact a ‘structured agency’, repeating patterns of the imagined urban geographies produced by agents of migration and various urban stakeholders. The result is unequal patterns of location and consumption, where privileged migrants locate in urban enclaves, distancing themselves from other groups and causing gentrification.

Wallpaper group kirigami

Kirigami, the art of paper cutting, has become a paradigm for mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns that derive inspiration from geometric art forms and simple planar tilings. Here, we complement these approaches by directly linking kirigami patterns to the symmetry associated with the set of 17 repeating patterns that fully characterize the space of periodic tilings of the plane. We start by showing how to construct deployable kirigami patterns using any of the wallpaper groups, and then design symmetry-preserving cut patterns to achieve arbitrary size changes via deployment. We further prove that different symmetry changes can be achieved by controlling the shape and connectivity of the tiles and connect these results to the underlying kirigami-based lattice structures. All together, our work provides a systematic approach for creating a broad range of kirigami-based deployable structures with any prescribed size and symmetry properties.

Effects of figural and numerical presentation formats on growing pattern performance

Prior work exploring preschool-aged children’s reasoning with repeating patterns has shown that patterning ability is an important predictor of math achievement; however, there is limited research exploring older children’s growing pattern task performance. The current study tested whether presentation format impacts performance on growing pattern problems, and whether the effects of presentation format extend to transfer word problems for which no patterns are provided. Sixth grade students were randomly assigned to complete several growing pattern tasks in one of three presentation formats (figures, sequences of values, or tables of values), and later completed transfer story problems with no figures, sequences, or tables provided. Findings suggest that presenting growing patterns as figures can benefit performance, although these benefits may depend on both pattern type and task. No differences were observed in performance on transfer problems, likely because students rarely spontaneously generated figures. Additional exploratory analyses suggest that performance on growing pattern problems may be related to both standardized math ability and fraction task performance, whereas inhibitory control may only be related to performance for specific patterning tasks. These findings have implications for educators because describing/expressing patterns is critical to algebra and higher-level mathematics.

No Association Between the Home Math Environment and Numerical and Patterning Skills in a Large and Diverse Sample of 5- to 6-year-olds

Selecting a large and diverse sample of 5–6-year-old preschool children (179 boys and 174 girls; Mage = 70.03 months, SDage = 3.43), we aimed to extend previous findings on variability in children’s home math environment (i.e., home math activities, parental expectations, and attitudes) and its association with children’s mathematical skills. We operationalized mathematics in a broader way than in previous studies, by considering not only children’s numerical skills but also their patterning skills as integral components of early mathematical development. We investigated the effects of children’s gender and socioeconomic status (SES) on their home math environment, examined the associations between children’s home math environment and their mathematical skills, and verified whether these associations were moderated by children’s gender and/or SES. Parents of 353 children completed a home math environment questionnaire and all children completed measures of their numerical (e.g., object counting) and patterning skills (e.g., extending repeating patterns). Results indicated no effect of children’s gender on their home math environment. There was no effect of SES on the performed home math activities, but small SES differences existed in parents’ math-related expectations and their attitudes. We found no evidence for associations between children’s home math environment and their mathematical skills. Furthermore, there were no moderating effects of gender or SES on these associations. One explanation for these findings might relate to the characteristics of the general preschool system in the country of the present study (Belgium). Future studies should consider the effect of the preschool learning environment because it might explain differences between studies and countries with regard to the home math environment and its association with mathematical skills.

Follow a pattern

Seize the moment to support children to explore ideas to do with repeating patterns as they follow their own play and exploration. Snakes can provide the inspiration – by threading beads on to a string or using loose parts outdoors.