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In learning mathematics in the 21st century and after the COVID-19 pandemic, a multimodal role in the process of drawing conclusions involving mathematical symbols or signs is very much needed. This process is called multimodal semiotic reasoning. This research aims to study the literature on multimodal semiotic reasoning research articles. Until April 2023, there were 32 articles on the keyword multimodal semiotic reasoning with the subject of mathematics in the Scopus, Science Direct, and ERIC databases. Furthermore, it is identified based on a) suitability of criteria; b) the similarity of the articles in the three databases. Finally obtained 24 articles that meet the requirements for further research. A literature review on multimodal semiotic reasoning is explored based on a) kinds of mathematical reasoning involving symbols and signs using multimodal, b) multimodal used in research, c) the most researched mathematical concepts and the mathematization process that occurs, d) the role of multimodal semiotic reasoning in mathematics education. The results showed that a) the most dominant type of mathematical reasoning involving objects, symbols, and signs using multimodal was Algorithmic Reasoning (AR), b) the most multimodal source was verbal answers, c) the concept of geometry through a process of vertical-horizontal mathematization was the most researched, and 4) the biggest role of multimodal semiotic reasoning for the benefit of knowledge of mathematics education. In conclusion, multimodal semiotic reasoning has had a major impact on mathematics education and research on this multimodal semiotic reasoning will continue to thrive in the future. [Note: The publication year (2023) shown in the citation on the PDF is incorrect. The correct publication year is 2024.]

Reversibility thinking carried out mentally in mathematical operations has an important role in the process of understanding concepts as it involves developing a thinking process from beginning to end and from end to beginning. This qualitative research aims to describe students' reversible thinking processes in solving algebra problems, specifically in climbertype adversity quotient. A total of 2 students who have the climber type were selected as participants from 36 potential subjects involved in the research. Data collection was carried out by providing Adversity Response Profile questionnaires, Test of Thinking Reversible, and semi-structured interviews. Our participants solved problems according to polya stages and through forward and reverse processes. The results of data analysis show that in the forward process, there are 2 aspects of reversible thinking, namely negation and reciprocity, while in the reverse process, they involve 2 aspects of reversible thinking, namely the capability to return to initial data after obtaining the result and negation. The results suggest that student's ability to perform mental reversals indicates a strong reversible thought process, leading to more precise cognitive thinking.

This article presents a study investigating the use of games in undergraduate mathematics instruction, with a specific focus on students' perspectives regarding post-instructional quiz games. The study explores students' opinions on the implementation of a linear algebra quiz game as a supplementary learning tool. A survey was administered to 78 students to examine their perceptions of the game's usefulness in preparing for their linear algebra examinations. Students reported high levels of attention and interest while engaging with the game and described it as fun, dynamic, exciting, and beneficial for reviewing linear algebra concepts. The findings suggest that quiz games can be a valuable addition to undergraduate mathematics education, fostering student engagement and providing an enjoyable and effective means for reinforcing course material.

This article aims to analyze the process of the progressive mathematization of the concept of the function by secondary school students through the resolution of a free fall situation. A free-fall mathematical situation was designed and experimented using the Tracker software to obtain movement data. Worksheets and conversations were collected between students and the teacher, and experiments and class discussions were filmed. The analysis consisted of identifying the development from informal strategies associated with the context of the problem to formal procedures. The results show that the process brings meanings to mathematical ideas such as variation, variables, and the relationship between variables, allows coordination and transition between representations and contributes to the learning of functions as a representation of the variation of physical phenomena.

High school math projects center on solving real-world examples in a wide variety of situations that require a creative skill set. Creativity in high school mathematics takes place in a subtle form. In contrast, creativity presents itself conspicuously in art. Therefore, this paper will explore the utilization of polynomial equations in drawing various figures as part of algebra class projects. Additionally, these projects incorporate integrated digital tools in mathematics. The paper will analyze three algebraic projects: the first project involves drawing a bird, the second project focuses on drawing a butterfly, and the third project entails drawing a bat using polynomial equations with the technological tool Desmos. All three projects are products of student work that encompass figures requiring up to forty polynomial equations. The students' impressive work demonstrates the power of mathematics in pushing the boundaries of other subjects. As digital technology becomes increasingly integral, mathematical tools are positioned as universal tools for the future and are poised to rival artists in producing visual art. This raises the intriguing question: can a mathematician create better art with mathematical equations through digital technology than an artist?

Mathematical creative thinking skill often becomes the orientation of mathematics learning, aiming to enhance students' creativity in mathematics. Recognizing that creativity encompasses the capacity for thinking creatively and creativity disposition is essential. Building on this conceptual foundation, the primary objective of this study is to develop a comprehensive model illustrating the relationship between students' aptitude for mathematical creative thinking and their creative disposition. The research methodology employed in this study aligned with the framework of cause-and-effect analysis. The study cohort consisted of 36 students, carefully selected by a cluster random sampling technique. The research instruments included a mathematical creative thinking ability assessment and a creative disposition scale. The data was analyzed using the Non-Recursive Structural Equation Modeling. The results showed the reciprocal cause-and-effect dynamic between mathematical creative thinking ability and creative disposition, exhibiting a mutually influential relationship with determination coefficients of 21.83% and 21.05%. This shows that mathematical creative thinking ability is better at explaining mathematical creative disposition than mathematical creative disposition explaining mathematical creative thinking ability, with a relatively small difference (0.78%). This study also concluded that an optimal approach to mathematics pedagogy entails a balanced and simultaneous focus on nurturing mathematical creative thinking ability and disposition.

Teaching "out-of-field" is a present obstacle in mathematics education in many countries, and developing professional learning programs aimed at upskilling non-specialist teachers is urgent. A teacher study group was established wherein two non-specialist teachers of mathematics (Years 7-10; aged 12-16) engaged with algebra to develop a deeper understanding of the subject and its teaching. The study group lasted one school year, during which the teachers actively participated in the learning. Multiple data sources were collected, including reflection forms, open-ended questionnaires, and a storytelling form. Analysis of the teachers' solutions to a sample of algebra problems and self-reports suggests that the study group enabled teachers to acquire new knowledge/skills in mathematics pedagogy, gain a new understanding of how student thinking and understanding develop, adapt new classroom instruction strategies, develop ability/confidence in problem-solving, and develop new knowledge/ability in algebra. The study findings contribute to our understanding of supporting non-specialist mathematics teachers' professional development.

An important approach for developing children's algebraic thinking involves introducing them to generalized arithmetic at the time they are learning arithmetic. Our aim in this study was to investigate children's attention to and expression of generality with the subtraction-compensation property, as evidence of a type of algebraic thinking known as relational thinking. The tasks involved subtraction modelled as difference and comparing the heights of towers of blocks. In an exploratory qualitative study, 22 middle primary (9-11-year-old) students from two schools participated in individual videoed interviews. The tasks were designed using theoretical perspectives on embodied visualization and concreteness fading to provide multiple opportunities for the students to make sense of subtraction as difference and to advance their relational thinking. Twelve out of 22 students evidenced conceptual understanding of the comparison model of subtraction (subtraction as difference) and expression of the compensation property of equality. Four of these students repeatedly evidenced relational thinking for true/false tasks and open equivalence tasks. A proposed framework for levels of attention to/expression of generality with the subtraction-compensation property is shared and suggestions for further research are presented.

In this investigation, the extent to which differences were present in the mathematics achievement by the ethnicity/race of Grade 3 students in Texas were analyzed. Data obtained from the Texas Education Agency Public Education Information Management System for all Texas Grade 3 students who took the State of Texas Assessment of Academic Readiness Mathematics exam were examined for the 2016-2017, 2017-2018, and 2018-2019 school years. In all three years analyzed, statistically significant differences were revealed in overall mathematics achievement and in all four Mathematics Reporting categories. Inferential statistical analyses revealed a clear stair-step effect. Asian students were the highest performing ethnic/racial group in all four Mathematics Reporting categories and all three performance level standards, followed by White, Hispanic, and Black students. Suggestions for future research and implications for policy and practice were provided.

The study aimed to investigate the effect of the flipped classroom strategy based on ADDIE Model on Prince Faisal College students' development of algebraic thinking skills. To achieve the goal of the study, the researchers developed a measure of development of algebraic thinking skills. The study used the quasi-experimental approach with a pre-post design, and the study was applied to (53) students, who were divided into two groups: an experimental group of (27) students, and a control group of (26) students. The results showed there is an effect of the strategy on the dimensional measurement of the test of developing all algebraic thinking skill. Flipped Classroom helped develop algebraic thinking skills. The study recommended using the stages of ADDIE model in applying the flipped classroom strategy, and flipped classroom based on ADDIE model enhances student learning, because the strategy integrates student learning at home and in the classroom.

In the field of physics education research, numerous studies have been dedicated to investigating the relationship between gender identity and physics learning. However, these studies have predominantly employed binary gender measurement methods, which may limit the range of research questions that can be explored and impede the discovery of crucial insights. In this study, we adapted gradational measures from prior research to investigate students' self-identified femininity, masculinity, and androgyny, as well as their reflected appraisal of femininity, masculinity, and androgyny (i.e., perceptions of how others perceive them) in both algebra-based and calculus-based introductory physics courses. The use of gradational measures revealed significant variation in students' self-identified femininity, masculinity, and androgyny within the binary categories of women and men, providing new insights into gender dynamics in physics. We found that self-identified women in the calculus-based courses, where they are underrepresented, tend to perceive themselves as more masculine and less feminine than how they believe others perceive them. Similarly, students of color are also more likely than White students to perceive themselves as more masculine than they believe others perceive them. Using structural equation modeling, we found that students' gender stigma consciousness plays an important role in mediating the effects of identifying as women and students of color on the observed discrepancies. Additionally, we found that women also exhibit a tendency to perceive themselves as more androgynous than they believe others perceive them in both algebra-based and calculus-based physics courses, and this phenomenon is also related to gender stigma consciousness. Moreover, our analyses revealed that students in the calculus-based courses tend to have a higher level of gender stigma consciousness even after controlling for gender and race. Our findings underscore the potential of gradational gender measurements in deepening our understanding of gender-related issues in physics education, shedding light on the complex interplay between students' gender identity, perceptions from others, and their educational experiences in the field.

Quadratics provide a foundational context for making sense of many important algebraic concepts, such as variables and parameters, nonlinear rates of change, and views of function. Yet researchers have highlighted students' difficulties in connecting such concepts. This in-depth qualitative study with two pairs of Year 10 (15 or 16-year-old) students investigated the potential of figural pattern generalisation--a context not traditionally used for teaching quadratics--to stimulate students' coordination of visual and algebraic reasoning and attention to quadratic function concepts. Theorisations of embodied visualisation, algebraic thinking, and student noticing were drawn on to analyse the pairs responding to 19 quadratic figural pattern generalisation tasks interspersed throughout their class topic on quadratic equations. It was found that students became adept at connecting the generality of different types of structural aspects of figures (square, rectangular, linear, constant/invariant) to their symbolic expression in quadratic equations. Students' construction of numeric instantiations of figural aspects was found to support pairs in moving towards symbolic generalisation. Task prompts to find different (but equivalent) algebraic equations for the same pattern evidenced pairs beginning to distinguish among general, factorised and standard forms of quadratic equations. One pair's attention to first and second differences (between total quantities of figures in a sequence) highlighted both the difficulty of and potential for connecting quadratic rate-of-change concepts and parameters visually. Implications for including figural pattern generalisation when teaching quadratics and suggestions for further research are shared.

An important topic of study in secondary mathematics is non-linear functions, including quadratic equations. In this study, findings from 25 Year 11 students indicated that difficulties with critical prerequisite concepts such as algebraic conventions impeded students' success in understanding and working with quadratics. Analysis of student errors identified misconceptions associated with the null factor law, and the nature of quadratic equations. This paper proposes that these findings are a result of limited timeframes nominated for learning quadratic topics outlined in the enacted curriculum. The implication of this is that the enactment of the Australian Curriculum: Mathematics F-10 did not support the development of conceptual understanding or procedural fluency with key mathematical concepts for these students. Without purposeful attention to prerequisite knowledge, and suitable time allocated to develop understanding and fluency, students' proficiency with topics such as quadratics is negatively influenced. A mastery approach to the hierarchically organised curriculum is supported by findings of this study.

Recognizing, building, describing, extending, and analyzing patterns are key components of algebraic reasoning. Oftentimes pattern-based instruction is used to support the understanding of functions and variables. In this study, elementary preservice teachers (n = 23) were asked to explore, extend, explain, create equations for, and analyze linear, geometric, and quadratic integer patterns using color tiles. Then they created, extended, described, and analyzed their own patterns while working in groups (n = 8). The research question was: What algebraic errors do PSTs demonstrate in the process of constructing, explaining, and analyzing their own linear, geometric, and quadratic patterns using color tiles? The analysis, which focused on errors and misunderstandings preservice teachers demonstrated in their final unit, revealed three specific areas of difficulties: figural errors, explanation challenges, and improper use of algebraic language. Participants failed to connect their algebraic images in a coherent manner, there was confusion in articulating their algebraic thought processes, and algebraic terminology was improperly used. Implications for teaching and further research are described.

We describe how an instructor integrated embodiment to teach the Fundamental Homomorphism Theorem (FHT) and preliminary concepts in an undergraduate abstract algebra course. The instructor's use of embodiment reduced levels of abstraction for formal definitions, theorems, and proofs. The instructor's simultaneous use of various forms of embodiment primed students for the formalism and symbolism, highlighted and disambiguated students' referents, amplified students' contributions to develop conceptual fluency, and linked students' body form catchments to motivate the FHT. Our results offer practical implications for teaching by illustrating examples of how embodiment can assist in making abstract concepts concrete.

This paper discusses several linear algebra activities designed to help enhance students' skills in collaborating, exploring mathematics, and linking together abstract and visual ways of approaching mathematics. Most of these activities are short, accessible, engaging, and easy to incorporate into any classroom. In addition, we discuss some questions instructors can ask themselves to design novel and engaging activities when constrained to teaching from a particular curriculum.

Though posters have been used as a pedagogical tool in a variety of fields, the literature suggests that the use of posters as an educational assessment tool by the mathematics community has been limited. This is unfortunate given the numerous potential benefits of a poster, including the development of oral communication skills and the ability to examine a mathematical topic at a deeper level. We describe the use of a poster project in an introductory abstract algebra course with specific implementation guidance including a schedule of assignments and assessment criteria. We also include reflections on the success of this project over several semesters as well as advice for adapting the project to a remote course. Samples of student posters are available upon request to the authors.

In this article, the authors share their favorite "Construct It!" activity, which focuses on rate of change and functions. The initial approach to instruction was procedural in nature and focused on making use of formulas. Specifically, after modeling how to find the slope of the line given two points and use it to solve for the y-intercept, the authors gave students time to practice. They recognized that their approach lacked an opportunity for students to engage in reasoning and sense making. According to Lithner (2008), imitative reasoning (i.e., a focus on memorizing) is ubiquitous in mathematics classrooms in the United States, whereas creative reasoning (i.e., reasoning that results in knowledge creation) is rare. To help students generalize the process of identifying a linear relationship that holds for two ordered pairs, explain the connection to the slope formula, and engage in creative reasoning, the authors designed the activity they call Turn A Into B. This activity focuses on reasoning about the relationship between two quantities and reflecting on how this is related to rate of change and functions.

In this note we demonstrate two instances where matrix multiplication can be easily verified. In the first setting, the matrix product appears as matrix element concatenation, and in the second, the product coincides with matrix addition. General proofs for some results are provided with a more complete description for 2×2 matrices. Suggested for consolidation of the algorithm rather than as an introduction, we believe the example base that can be created, combined with an analysis for understanding the context, acts as a positive motivator to secure mastery as it needs to be practised.

The study advances the instrumental approach to mathematics education (Drijvers et al., 2013; Trouche, 2003), aiming to elucidate the interplay between students' reasoning competency, conceptual knowledge and tool utilisation in dynamic digital geometry and algebra environments. The dynamic properties of these environments pose a nuanced predicament, as the outsourcing of translation between visual and algebraic representations raises concerns regarding students' conceptual development and reasoning competency. To mitigate this issue, a prediction task is proposed, focusing on the dynamic behaviour of variable points in GeoGebra. I introduce a comprehensive framework adapting Toulmin's argumentation model into the instrumental approach, emphasising processes of justification. This is complemented by the application of components of Vergnaud's (1998) scheme concerning generative and epistemic ways to approach how students' conceptual knowledge has played a part in these processes. Through a case study of a student pair solving a prediction task, I explore the links between instrumented justification, students' mathematical reasoning competency and conceptual understanding, and how students' use of GeoGebra tools is intertwined with their justification processes. The analysis reveals the intricate interplay between data production and interpretation, and it is grounded in inference drawn regarding students' implied theorems about concepts, dynamic behaviour and progression in terms of techniques. The results indicate that the progression of technique is driven by the experience of the inefficiency of techniques and artefacts related to the goal of justification. Essentially, the framework links students' reasoning competency to their use of tools and conceptual knowledge, as well as demonstrates that predicting dynamic behaviour can enhance knowledge-based justification.

This study was performed to determine preservice elementary mathematics teachers' solution performance of systems of linear equations using geometric representation, algebraic representation and transition from geometric representation to the algebraic representation and the predictive role of preservice elementary mathematics teachers' performance of different representation usage on their systems of linear equations self-efficacy perceptions. The research follows the prediction model, which is one of the relational screening models. The data were obtained by using the 'Systems of Linear Equations Self-Efficacy Test' developed by Kardes [Kardes, D. (2010). Matematik ögretmen adaylarinin lineer denklem sistemleri çözüm süreçlerinin öz-yeterlik algisi ve çoklu temsil baglaminda incelenmesi [An investigation of the processes of pre-services mathematics teachers? Solving systems of linear equations within the context of self-efficacy and multiple representations] [Master's thesis]. Marmara University. Turkish Higher Education Council thesis repository (#264144). https://tez.yok.gov.tr/UlusalTezMerkezi/] and the 'Systems of Linear Equations Performance Test' developed by the researchers. Because of the research, it was found out that preservice teachers' algebraic representation performances and their self-efficacy perceptions were at the maximum level when the number of equations and unknown variables was equal (m = n). Additionally, due to the multiple regression analysis, preservice teachers' performances on geometric representation, the transition from geometric representation to algebraic representation and systems of linear equations in R[superscript 3] were determined to be significant predictors of their self-efficacy perception. [This study was presented as a verbal announcement at the International Symposium on Active Learning-ISAL 2019 symposium.]

Using a mixed methods approach, we explore a relationship between students' graph reasoning and graph selection via a fully online assessment. Our population includes 673 students enrolled in college algebra, an introductory undergraduate mathematics course, across four U.S. postsecondary institutions. The assessment is accessible on computers, tablets, and mobile phones. There are six items; for each, students are to view a video animation of a dynamic situation (e.g., a toy car moving along a square track), declare their understanding of the situation, select a Cartesian graph to represent a relationship between given attributes in the situation, and enter text to explain their graph choice. To theorize students' graph reasoning, we draw on Thompson's theory of quantitative reasoning, which explains students' conceptions of attributes as being possible to measure. To code students' written responses, we appeal to Johnson and colleagues' graph reasoning framework, which distinguishes students' quantitative reasoning about one or more attributes capable of varying (Covariation, Variation) from students' reasoning about observable elements in a situation (Motion, Iconic). Quantitizing those qualitative codes, we examine connections between the latent variables of students' graph reasoning and graph selection. Using structural equation modeling, we report a significant finding: Students' graph reasoning explains 40% of the variance in their graph selection (standardized regression weight is 0.64, p < 0.001). Furthermore, our results demonstrate that students' quantitative forms of graph reasoning (i.e., variational and covariational reasoning) influence the accuracy of their graph selection.

Inverse and injective functions are topics in most college algebra courses. Yet, current materials and course structures may not afford students' conceptual understanding of these important ideas. We describe how students' work with digital activities, "techtivities," linking two different looking graphs that represent relationships between variables in the same dynamic situation, can help to address this problem. To bolster our argument, we provide empirical data from a broader study, drawing on student work and instructor interviews across three institutions. We conclude with illustrations of how techtivities can serve as instructional anchors for injective and inverse functions.

This paper describes Alkaline, a size-reduced version of Kyber, which has recently been announced as a prototype NIST standard for post-quantum public-key cryptography. While not as simple as RSA, I believe that Alkaline can be used in an undergraduate classroom to effectively teach the techniques and principles behind Kyber and post-quantum cryptography in general. Classroom experiences with individual concepts used in Alkaline support this belief. In addition to cryptography, linear algebra and abstract algebra classes would be good candidates for the use of Alkaline. A few exercises suitable for use in these classes are included.

The Miller-Rabin test is a useful probabilistic method for finding large primes. In this paper, we explain the method in detail and give three variations on this test. These variations were originally developed as student projects to supplement a course in error correcting codes and cryptography.

Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K-1 students' thinking about parity arguments for sums of evens and odds, as well as underlying concepts of pair and parity of a number. Data are drawn from classroom lessons of a Grades K-1 early algebra instructional sequence, as well as task-based interviews conducted at four timepoints during the implementation of the sequence. While most students at the beginning of the study (Grade K) did not know the concepts of even or odd and could not make any viable arguments regarding parity, by the end of Grade 1 students were largely constructing representation-based arguments to justify number parity and claims about sums of evens and odds. Results of this study align with other research that shows young learners can develop viable arguments to justify mathematical generalizations. Results also provide preliminary evidence that the instructional sequence used here can foster students' practice of argumentation from the start of formal schooling.

Backward transfer is defined as the influence that new learning has on individuals' prior ways of reasoning. In this article, we report on an exploratory study that examined the influences that quadratic functions instruction in real classrooms had on students' prior ways of reasoning about linear functions. Two algebra classes and their teachers at two comprehensive high schools served as the participants. Both schools drew from low-socioeconomic urban populations. The study involved paper-and-pencil assessments about linear functions that were administered before and after a four- to five-week instructional unit on quadratic functions. The teachers were instructed to teach the quadratic functions unit using their regular approach. Qualitative analysis revealed three kinds of backward transfer influences and each influence was related to a shift in how the students reasoned about functions in terms of an action or process view of functions. Additionally, features of the instruction in each class provided plausible explanations for the similarities and differences in backward transfer effects across the two classrooms. These results offer insights into backward transfer, the relationship between prior knowledge and new learning, aspects of reasoning about linear functions, and instructional approaches to teaching functions.

Machine learning (ML) methods for causal inference have gained popularity due to their flexibility to predict the outcome model and the propensity score. In this article, we provide a within-group approach for ML-based causal inference methods in order to robustly estimate average treatment effects in multilevel studies when there is cluster-level unmeasured confounding. We focus on one particular ML-based causal inference method based on the targeted maximum likelihood estimation (TMLE) with an ensemble learner called SuperLearner. Through our simulation studies, we observe that training TMLE within groups of similar clusters helps remove bias from cluster-level unmeasured confounders. Also, using within-group propensity scores estimated from fixed effects logistic regression increases the robustness of the proposed within-group TMLE method. Even if the propensity scores are partially misspecified, the within-group TMLE still produces robust ATE estimates due to double robustness with flexible modeling, unlike parametric-based inverse propensity weighting methods. We demonstrate our proposed methods and conduct sensitivity analyses against the number of groups and individual-level unmeasured confounding to evaluate the effect of taking an eighth-grade algebra course on math achievement in the Early Childhood Longitudinal Study.

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.

After nationwide school closures due to COVID-19, virtual learning environments (VLE) have seen tremendous increase in usage. The current study identified teacher activities for orchestration using an Algebra VLE during school closures, and whether these activities were related to student achievement. In May 2020, we collected survey data on how 213 teachers were using a VLE for Algebra with 10,590 students, along with system logs and student achievement data. Results indicated that teachers made several changes to teacher strategies due to school closures, including allowing students more time to complete assignments. Multilevel modeling showed that teacher orchestration activities, particularly those related to regulation/management and awareness/assessment, were positively related to student achievement. We discussed the results and provided implications for practice (Q&A setting, assignment flexibility).

The Building Painted Cubes Task is a groupworthy algebraic task. Students build cubes using linking unit cubes, search for algebraic patterns, and report findings on posters. This task can create spaces for students to see themselves as doers of mathematics.

Providing students with opportunities to build, and test and revise ideas, can help them develop critical thinking and problem-solving skills. In this article, the authors describe how core measurement and geometry concepts were embedded within a "Model It!" task. They replicate an industry-based, interdisciplinary problem, but only focus on the mathematical ideas here. Twenty-six Algebra 1 students worked on organizing gingerbread houses within a community and designing the path and operation of a mini-trolley tour. Starting work on these tasks required that students keep the task specifications in mind. With guidance from their teacher, the students sub-divided their modeling work into creating three designs: (1) trolley path design, (2) gingerbread community design, and (3) a two-dimensional (2D) model of the gingerbread community to fit the resort ballroom.

In this paper, we examine the relationships between teachers' subject matter preparation and experience in teaching and their performance on an instrument measuring mathematical knowledge for teaching Algebra 1. We administered the same instrument to two different samples of teachers-high school practicing teachers and community college faculty--who teach the same algebra content in different levels of institutions, and we compared the performance of the two different samples and the relationships between the measured knowledge and their educational and teaching background across the samples. The comparison suggested that the community college faculty possess a higher level of mathematical knowledge for teaching Algebra 1 than high school teachers. The subsequent analyses using the Multiple Indicator Multiple Causes (MIMIC) models based on our hypothesis on the factors contributing to the differences in the knowledge between the two teacher samples suggest that experience teaching advanced algebra courses has positive effects on the mathematical knowledge for teaching Algebra 1 in both groups. Highlighting the positive effect of algebra-based teaching experience on test performance, we discuss the implications of the impact of subject specific experience in teaching on teachers' mathematical content knowledge for teaching.

Generalizing is a critical aspect of mathematics learning, with researchers and policy documents highlighting generalizing as a core mathematical practice. It can also be challenging to foster in class settings, and teachers need access to better resources to teach generalizing, including an understanding of effective forms of instruction. This article proposes Classroom Supports for Generalizing (CSGs), investigating how multiple elements--such as tasks, teacher moves, student interactions, and representations--interact to meaningfully foster student generalizing. Drawing on class video data from a middle school teacher and two high school teachers, we present the CSG Framework, which identifies three categories of supports: Interactions for Generalizing, Structures for Generalizing, and Routines for Generalizing.

On 24 June 1994 at Fairchild Air Force Base, during practice for an air show, a low-flying B-52H aircraft banked its wings vertically and crashed. Emphasizing the activity of modeling drag and gravity, these notes examine the possibility of recovery with several models. First, with algebra, historical data lead to a model where in a free fall near Earth's surface, the distance fallen is proportional to time squared. Calculus then gives an ordinary differential equation to model free fall near Earth's surface. Second, with calculus, historical data lead to various models of drag on objects moving through air. Third, combining models of drag and gravity with Newton's Laws of Motion leads to ordinary differential equations that model free fall in air. Fourth, predictions from such models with ordinary differential equations are consistent with the aircraft crash. Further models examine the possibility of increasing engine thrust to regain the vertical component of lift. All models fit in a first course in differential equations, without requiring any computational machinery. However, numerical experiments show how uninformed use of professional software can produce rounding errors to cause the modelled aircraft to plunge to the bottom of the deepest oceans and shoot up into space.

When justifying a mathematical statement, textbooks often interweave several instances of justification, creating, what we term, "paths of justification." This study aims to characterize the paths of justification of mathematical statements that are offered in eight 7th grade Israeli mathematics textbooks. The study attended to context and compared paths of justification in different mathematical domains (algebra and geometry) and in textbooks intended for students with different achievement levels. Analysis focused on the number of instances of justification in a path, the types of justification of those instances, and their order. Comparing paths of justification between algebra and geometry, and between textbooks intended for students with different levels of achievements, revealed several similarities and differences. The findings portray a complex picture of textbook justifications, underlining the need to be sensitive to context when studying textbook justifications. The study results are discussed in light of relevant literature, highlighting potential factors involved in the way a textbook path of justification of a specific mathematical statement is structured.

Recent research has highlighted the role of functional relationships in introducing elementary school students to algebraic thinking. This functional approach is here considered to study essential components of algebraic thinking such as generalization and its representation, as well as the strategies used by students and their connection with generalization. This paper jointly describes the strategies and representations of generalization used by a group of 33 sixth-year elementary school students, with no former algebraic training, in two generalization tasks involving a functional relationship. The strategies applied by the students differed depending on whether they were working on specific or general cases. To answer questions on near specific cases they resorted to counting or additive operational strategies. As higher values or indeterminate quantities were considered, the strategies diversified. The correspondence strategy was the most used and the common approach when students generalized. Students were able to generalize verbally as well as symbolically and varied their strategies flexibly when changing from specific to general cases, showing a clear preference for a functional approach in the latter.

We present a model for roleplaying (RP) for university students that is aimed at fostering the interactions of each learner with the teacher and coursemates, self- and peer-assessment, as well as an awareness and meaningful learning of mathematics. Specific features of our model are its focus on metacognition and individualized learning, based on a careful distribution of the tasks according to the individual student's profile. Within our model, the students play, alternately, the role of the teacher and their own role: firstly they assign problems to their peers, then they solve problems formulated by their peers, and, finally, they assess their peers' problem-solving processes. We discuss an implementation of the RP within a teaching/learning path regarding mathematical induction, aimed at promoting robust learning and use. We analyse the outcomes of a pilot study carried out with engineering students and identify their common praxeologies in solving the tasks. Moreover, the effects on students' attitudes and learning are highlighted at the linguistic, cognitive, metacognitive and affective levels.

This paper reports on a novel corequisite design and implementation for College Algebra at Florida International University. The corequisite course uses online, just-in-time, prerequisite assignments delivered on an open-educational platform. Students get help from near-peer learning assistants inside a math emporium environment. The course remains three credits after the redesign and now supports all students who would otherwise be taking Intermediate Algebra as a prerequisite course.

Analogy has played an important role in developing modern mathematics. However, it is unclear to what extent students are granted opportunities to productively reason by analogy. This article proposes a set of lessons for introducing topics in ring theory that allow students to engage with the process of reasoning by analogy while exploring new (to the students) mathematics. In this way, students come to creatively establish new concepts that they may take ownership of. I provide insights from previous implementations and conclude by reflecting on what has (and has not) worked well in my experience with implementing the lesson.

Procedural flexibility is an important skill for algebra. Although prior work has focused on measuring students' procedural flexibility using arithmetic problems, word problems may also capture students' flexibility because of their open-ended nature. To date, no published study has examined the use of word problems as another measure of procedural flexibility. The present study aims to establish flexibility on fraction word problems as a predictor of algebra learning and to determine whether flexibility demonstrated on fraction arithmetic versus fraction word problems differentially predicts two types of algebra learning (e.g., algebraic feature knowledge and algebra equation-solving). Middle-school students (N = 350) completed fraction arithmetic problems, fraction word problems, and algebra measures at the start and end of the school year. We coded fraction arithmetic and word problems for demonstrated procedural flexibility. Path models showed that overall flexibility significantly predicted both algebraic feature knowledge and equation-solving. When examining flexibility by type, procedural flexibility on arithmetic problems significantly predicted end-of-year algebraic feature knowledge and algebra equation-solving, whereas procedural flexibility on word problems significantly predicted algebraic feature knowledge. Procedural flexibility on arithmetic problems was still predictive of performance when accounting for fraction magnitude skill. These findings build on prior literature that establishes procedural flexibility as a crucial predictor of algebra learning and extends measures of procedural flexibility to include word problems.

This study examined opportunities provided for preservice secondary mathematics teachers (PSMTs) to learn reasoning and proof in algebra from the perspective of college instructors. We analyzed interview transcripts of 15 course instructors recruited from three teacher education programs in the United States. We examined the reported opportunities provided for PSMTs to engage in proving-related activities, including making conjectures, investigating conjectures, developing arguments, evaluating arguments, and disproving by using counterexamples. We also analyzed instructional strategies reported by the instructors. We found the inconsistency between instructors' perceptions of the importance of reasoning and proof in algebra and instructor-reported opportunities to learn. Findings also indicated that developing arguments was reported the most frequently. In addition, instructors reported more pedagogy-focused general teaching strategies than proof-specific teaching strategies.