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It is well known that primary school children may face difficulties in acquiring mathematical competence, possibly because teaching is generally based on formal lessons with little opportunity to exploit more multisensory-based activities within the classroom. To overcome such difficulties, we report here the exemplary design of a novel multisensory learning environment for teaching mathematical concepts based on meaningful inputs from elementary school teachers. First, we developed and administered a questionnaire to 101 teachers asking them to rate based on their experience the learning difficulty for specific arithmetical and geometrical concepts encountered by elementary school children. Additionally, the questionnaire investigated the feasibility to use multisensory information to teach mathematical concepts. Results show that challenging concepts differ depending on children school level, thus providing a guidance to improve teaching strategies and the design of new and emerging learning technologies accordingly. Second, we obtained specific and practical design inputs with workshops involving elementary school teachers and children. Altogether, these findings are used to inform the design of emerging multimodal technological applications, that take advantage not only of vision but also of other sensory modalities. In the present work, we describe in detail one exemplary multisensory environment design based on the questionnaire results and design ideas from the workshops: the Space Shapes game, which exploits visual and haptic/proprioceptive sensory information to support mental rotation, 2D–3D transformation and percentages. Corroborating research evidence in neuroscience and pedagogy, our work presents a functional approach to develop novel multimodal user interfaces to improve education in the classroom.

For countries concerned about equity and access to higher education, providing support services for academically underprepared students is critical to increasing their probabilities of success. Corequisite math courses provide opportunities for underprepared students to develop their basic math skills through simultaneous enrollment in college-level and developmental courses. However, adding the additional developmental course also increases students’ academic workload, potentially impeding their progress early in their college journey. Our study explores the effects of being assigned to this additional developmental math class at one of the largest technical colleges in Chile. Few remedial studies have examined the effects of corequisite courses in the technical college context, either in the United States or abroad. Using a regression discontinuity design, we find that for students attending a technical college and scoring close to the cutoff on a standardized mathematics placement exam, being assigned to a corequisite math course increases the grades in their college-level math course and marginally decreases their likelihood of withdrawing during the first semester, compared to students assigned to a single college-level math course. The effects on credits earned and first semester grades are driven by those pursuing less advanced technical degrees.

As content-specific educational coaches, elementary mathematics specialists (EMSs) have emerged as school-based professionals who are needs-driven and work closely with school stakeholders in regard to mathematics teaching and learning. While leading mathematics education organizations have identified the specialized knowledge and skills required for EMS positions, how to best prepare these individuals is knowledge that the field is still exploring. This paper first presents a theoretical model for EMS preparation that aligns an emerging coaching tool, the Decision-Making Protocol for Mathematics Coaching (Baker & Knapp, 2019, [DMPMC]) with the Professional Development Design Framework (Loucks-Horsley et al. in Designing professional development for teachers of science and mathematics, Corwin Press, 2010). The paper then presents a descriptive case study that examines the application of this model in an EMS preparation course. The findings indicate that assessing the coaching situation fostered administrative partnerships, revisiting goals increased specificity of anticipated outcomes, and applying research-informed practices increased EMS self-efficacy and advanced coaching agendas. Taken together, these findings suggest that integrating the DMPMC into an EMS preparation course led to positive changes in EMS candidate learning of professional development design. Notably, this is one of the first studies that documents the influence of a coaching education tool on EMS candidates’ professional development design.

This article reports on a second national survey of higher education institutions in the USA to answer the question “Who teaches mathematics content courses for prospective elementary teachers, and what are these instructors’ academic and teaching backgrounds?” and addresses valuable information not collected with the first survey conducted in 2010. We surveyed 1740 institutions and a faculty member from each of 413 institutions (23.7%) participated in the survey. The survey results demonstrate that the majority of these institutions are not meeting the recommendations of the Conference Board of Mathematical Sciences (2012) and the Association of Mathematics Teacher Educators (2017) for prospective elementary teachers to take at least 12 semester-hour credits of mathematics content designed specifically for them. The data do indicate that there is movement toward more activity-based approaches for these courses as compared to the 2010 survey. Additionally, there is an increase in these courses having instructors with doctorates in mathematics education as well as an increase in instructors having grades 7–12 teaching experience. Most instructors for these courses do not have elementary teaching experience and have likely not had opportunities to think deeply about the important ideas in elementary mathematics. While most institutions still do not provide training and/or support for these instructors, formalized support and training appear to be increasing since the 2010 survey.

The study of pivotal teaching moments (PTMs) offers significant insights into mathematics classroom interactions. PTMs are student-generated instances within a lesson that provide opportunities for teachers to modify planned instruction (Stockero & Van Zoest, 2013). Given the importance of interactions and discourse in students’ mathematical growth and understanding, we examine them in the context of PTM episode triples. PTM episode triples consist of three main elements: (1) a PTM, (2) a teacher response to the PTM, and (3) the immediately ensuing student utterance. Analysis of data from nine elementary mathematics lessons shows that teacher responses that explicitly “pursued students’ thinking” elicited significantly higher cognitive levels of student discourse than teachers who "ignored or dismissed" PTMs or simply "acknowledged [PTMs] but continued as planned." While many researchers have focused on PTM episode “doubles” (combinations of student-generated interruptions and the teacher’s response), no studies to our knowledge have addressed PTM episode “triples” as we have. Implications of these findings prove helpful for identifying concrete ways that educators can increase the cognitive level of student discourse within their classrooms. Further, our integration and modification of existing theory and frameworks for analyzing PTM episode triples—incorporating the relationship between PTMs and teachers’ responses to them, on the one hand, and the cognitive level of the ensuing student utterance, on the other hand—are a unique contribution to the field and provide a method by which researchers and educators can study classroom practice.

In 1990, Dr. Jean Schmittau (1937–2015) traveled to Russia to study the Davydov elementary mathematics program. Later she worked closely with Davydov to translate the program into English and oversaw an implementation of the program at a local private school. This tribute overviews her tremendous work in making the program known to so many scholars.

This multi-case study examines how three elementary mathematics methods instructors, in the same teacher education program, provide their prospective teachers with learning opportunities. Qualitative data were collected through interviews, classroom observations, and artifacts. The findings suggest that the instructors’ beliefs associated with teaching philosophies influence both the content that prospective teachers have the opportunity to learn (what) and the nature of the prospective teachers’ opportunities to learn (how). Through analytic induction, three assertions were developed to understand and explicate: similarities in opportunities to learn, differences in opportunities to learn, and perceptions about the purpose of the methods courses across the three cases. Specifically, the first assertion examines how all three methods instructors focused on developing conceptual understanding and combating mathematical misconceptions for which prospective teachers most often experience opportunities to learn through representations and approximations. The second and third assertions place more emphasis on differences across the cases based upon observed instructor actions and their beliefs. This study is significant because it helps us gain a deeper understanding about prospective teachers’ opportunities to learn within one teacher education program, and therefore, may point toward what can be done in the future to better prepare teachers in elementary mathematics education and the development of ambitious instruction. Additionally, this study unpacks how prospective teachers in the same teacher education program may have varying experiences and thus varied access to opportunities to learn.

Recent scholarship around teaching elementary mathematics supports the learning of early algebra with 5- to 12-year olds. However, in spite of the recognition of the affordances of early algebra, issues about how to introduce it remain open. Within this context, Davydov’s work is often cited as a source of impressive demonstration of young learners’ capacity for algebraic thinking. This work requires further exploration in order to yield a clearer picture of a very particular teaching approach, which focuses on early abstractions and symbolic language. We argue that in order to fully understand how Davydov’s work contributes to current conversations and what Davydov was trying to do, we need to shed light on the context- and time-specific discourse of the 1960 Soviet educational reforms that made it possible for Davydov to develop his vision about algebraic thinking and to set in motion appropriate teaching approaches for young learners. In this paper, we look back to the Soviet debates that unfolded in Russia on the integration of early algebra in elementary school word-problem solving. Drawing on these debates and the results of Davydov’s school experiments, we lay out the developmental axes of capacity building. This can be done by emphasizing ascent from the abstract to the concrete using a variety of representational modeling tools to support the emergence of algebraic thinking while targeting particular habits of mind within carefully designed learning activities. We conclude with some insights about current arithmetic-algebra debates, and how these could be enriched and deepened by Davydov’s work, which yet remains open to future discussion and reflection.

Professional noticing of children’s mathematical thinking (Jacobs et al. in J Res Math Educ 41(2):169–202, 2010) is a construct that is used frequently to understand the ways in which teachers attend to, interpret, and respond to children’s mathematical thinking. In this paper, we present our analysis of prospective elementary mathematics teachers’ (PTs’) responses to a pedagogical activity designed to engage them in an approximation (Grossman et al. in Teach Coll Rec 111(9): 2055–2100, 2009) of professional noticing of children’s mathematical thinking in the moment of teaching, with a focus on developing responses which extend the student’s thinking. PTs were provided two different examples of student work and a series of questions designed to elicit information about their ability to professionally notice student work. We present data and analysis of 73 PTs’ abilities to engage in the individual skills associated with noticing as well as their abilities to use the skills cohesively in concert. Our results demonstrate the type of representation used determined PTs’ success in all three facets of noticing, with a particular effect on interpreting. Further, our focus on a specific type of response move, along with the introduction of a scaffold for that response move, made a difference in PTs’ ability to successfully respond to student work when compared to a prior study with the same PTs (Tyminski et al. 2014). We unpack these results along with suggestions for pedagogical practice as well as directions for future research.

This paper explores peer interactions in an elementary mathematics classroom (ages 9–10) where the teacher intentionally shared authority with her students and supported them in learning to share authority with one another. Authors examine how students shifted between shared, concentrated, and contested social and intellectual authority relations in partner and small group work during a three-week unit on place value. Findings show that (a) students were able to share both social and intellectual authority, and did so often; (b) the distribution of social authority was more dynamic than that of intellectual authority; and (c) when groups shifted into shared intellectual authority, shifts were usually preceded by a student making some aspect of the collaborative task public. We connect these findings to research on authority in mathematics classrooms that serve racially and linguistically minoritized students and offer directions for future work.

This case study explored Prospective Elementary Mathematics Specialists’ (PEMSs) pedagogical practices during a K-5 Mathematics Endorsement (K-5 ME) program, specifically focusing on the authentic residency with mentor sessions. Participants included nine PEMSs, with qualitative and quantitative data collected via individual interviews, professional portfolios, and observations of classroom practices. Upon completion of the K-5 ME program, all participants were implementing instructional practices learned about in the program in substantial ways. Participants were honing their skills for orchestrating effective discourse, and the application of those skills became more meaningful as they saw the impact it was having on student learning. This implementation and documentation program coursework was rigorous as PEMSs worked to develop themselves as mathematics teachers and teacher leaders. The collaborative support during the authentic residency and mentor sessions seem to have been significant for PEMSs in supporting implementation of newly learned instructional practices. Ultimately, PEMSs completed the program feeling more prepared as elementary mathematics teachers and teacher leaders.

This study compares web usage data with interviews from 41 participants, who are members of an online professional development site called the Everyday Mathematics Virtual Learning Community (VLC), to explore how elementary school teachers learn from classroom video. Web usage data reveal that the commentary surrounding video posted to the VLC is sparse and surface level, possibly indicating a lack of serious attention to the videos. Interview data, however, indicate that participants felt they learned from this resource. Participants reported that the videos provided them with the opportunity to view and reflect on model lessons, plan curricula, and consider student thinking, among other learning outcomes. Participants also identified key factors that prevented them from posting comments to the site to convey their learning. These results can be used to understand not only how teachers perceive their own learning from classroom video, but also to redesign online professional development experiences to promote expression of that learning.

In many discussions of the ways in which abstraction is applied in computer science (CS), researchers and advocates of CS education argue that CS students should be taught to consciously and explicitly move among levels of abstraction (Armoni Journal of Computers in Mathematics and Science Teaching, 32(3), 265–284, 2013; Kramer Communications of the ACM, 50(4), 37–42, 2007; Wing Communications of the ACM, 49(3), 33–35, 2006). In this paper, we describe one way that attention to levels of abstraction could also support learning in mathematics. Specifically, we propose a framework for using abstraction in elementary mathematics based on Armoni’s (2013) framework for teaching computational abstraction. We propose that such a framework could address an enduring challenge in mathematics for helping elementary students solve word problems with attention to context. In a discussion of implications, we propose that future research using the framework for instruction and teacher education could also explore ways that attention to levels of abstraction in elementary school mathematics may support later learning of mathematics and computer science.

Incorporating computational thinking (CT) ideas into core subjects, such as mathematics and science, is one way of bringing early computer science (CS) education into elementary school. Minimal research has explored how teachers can translate their knowledge of CT into practice to create opportunities for their students to engage in CT during their math and science lessons. Such information can support the creation of quality professional development experiences for teachers. We analyzed how eight elementary teachers created opportunities for their students to engage in four CT practices (abstraction, decomposition, debugging, and patterns) during unplugged mathematics and science activities. We identified three strategies used by these teachers to create CT opportunities for their students: framing, prompting, and inviting reflection. Further, we grouped teachers into four profiles of implementation according to how they used these three strategies. We call the four profiles (1) presenting CT as general problem-solving strategies, (2) using CT to structure lessons, (3) highlighting CT through prompting, and (4) using CT to guide teacher planning. We discuss the implications of these results for professional development and student experiences.

The inclusion of alternative and multimodal methods for online interaction and knowledge construction in mathematics teacher education is still an incipient field. In this paper, we present a multiple case study of three blended courses in an elementary mathematics teacher education program at Western University. In these blended courses, the online component included the construction of collaborative mind maps. Through constructivist grounded theory methods, we analyzed teacher candidates’ mind maps as (a) final products, and (b) interaction processes. The resulting theory describes how pre-service mathematics teachers interact and construct knowledge while they engage in online collaborative mind mapping. The study provides insights into the ways that collaborative and multimodal technologies affect mathematics teacher education, and in turn, suggests how its implementation may be improved.

As block-based programming grows in popularity within education, the integration of its application within teacher education programs needs to be investigated. This article examines block-based programming as a viable tool to teach elementary mathematics conceptually drawing on the connections pre-service teachers made between computational thinking and mathematics. This cross-comparative case study that includes a convergent, mixed-methods design examines how block-based coding and computational thinking for conceptual mathematics (B2C3Math) facilitated ten pre-service teachers’ connections made at a large southeastern US university. Connections between computational thinking and mathematics concepts were investigated while the pre-service teachers’ attitudes towards block-based programming as a conceptual mathematics teaching tool were queried. A focus is not placed on interactive games or simulations made using block-based programming to teach mathematics, but rather on pre-service teachers’ ability to design a lesson teaching mathematics conceptually through the programming process. Descriptive analysis reveals changes in pre-service teachers’ attitudes of the proposed teaching strategy following the implementation of B2C3Math. Pre-service teachers’ reflections and lesson designs provide insight into the quality of the connections made between computational thinking and mathematics for the purpose of conceptual mathematics teaching. Three critical cases are identified to illustrate the spectrum of all ten participants in relation to the major findings. Implications for research and practices using block-based programming for teacher education are discussed.

Popularly used in marketing and business, gamification has been gaining interest in educational contexts for its potential to invigorate otherwise mundane or difficult processes. A gamified environment transfers motivational elements of games to learning activities thereby engaging learners in the learning task thus transforming dull classroom environments to smart ones. This paper presents the design process of a gamification intervention in geometry at elementary level, based upon Huang and Soman (Gamification of education. Research report series: behavioural economics in action, 29. Rothman School of Management, University of Toronto, Toronto, 2013) model. We describe how insights from various sources helped us to refine an intervention previously used in one school. The design focuses on gamifying the tangram, an unplugged resource, through incorporating game-based elements of leader boards, points/stars and challenge levels to motivate young learners individually and in teams. Cognitive and motivational scaffolding undergird five challenge levels to bring affordances to self and social elements for learner participation in increasingly complex geometry tasks. There are limited theoretical models to guide educational researchers, especially ones that do not require digital resources. This paper presents our insights and recommendations to support scaffolded learning in student-centred gamified learning environments.

Existing research suggests that young children can develop a partial understanding of the equal sign as an operator rather than as a relational symbol of equivalence. This partial understanding can be the result of overemphasis on canonical equation syntaxes of the type a + b = c in elementary school mathematics. This paper presents an examination of context and syntax nuances of relevant sections from the grade 1 Greek series of textbooks and workbooks. Using a conceptual framework of context variation, the analysis shows qualitative differences between equations of similar syntax and provides a nuanced determination of contextual and structural aspects of ‘variation’ in how the equal sign is presented in elementary mathematics. The paper proposes that since equations have context-specific meanings, context variations should constitute a separate element of analysis when investigating how the equal sign is presented. The implication for practice and future research is that nuanced considerations of equation syntax within varied contexts are needed for elaborating analyses of the equal sign presentation that move beyond dichotomized categorizations of canonical/non-canonical syntaxes.

Teachers’ decision-making and pedagogical reasoning often become visible in their lesson planning sessions. This study explores how the professional learning opportunities presented by teacher-facilitator interactions in planning sessions. By tracing shifts in teachers’ discourse in planning sessions, we examine the development of their pedagogical reasoning. The findings of this study illustrate that these interactions create four kinds of professional learning opportunities: (1) reflecting on students’ learning in past lessons; (2) attending to the details of the complexity of teaching; (3) negotiating and adjusting lesson goals and activities by making direct and informed decisions for teaching; and (4) anticipating specific features of instruction. This study also illustrates how refining and elaborating teaching complexity supports teachers in re-conceptualizing and expanding their teaching knowledge.

This article focuses on conversations about gender and mathematics among other intersecting identities such as age, education level, and professional expertise. I used positioning theory to unpack a conversation at a science museum (between museum volunteer John, prospective teacher Ann, and myself) to share a story about gender and mathematics. I argue that the impact of this conversation from our perspectives as young women who do work in elementary mathematics education should be made visible to recognize and address inequities for women and girls within mathematics and mathematics education spaces. Implications for elementary teacher education include providing opportunities for prospective teachers to share their stories and gendered experiences with attention to intersections of age, gender, and connected disciplinary narratives (in this case, mathematics).

Professional learning experiences (PLEs) provide teachers with opportunities to improve their understanding of mathematics content and teaching practices. However, PLEs are often conducted in person and in small groups—hence costly and localized. The purpose of the current study was to explore different ways for teachers to engage in PLEs and how these approaches might enable the field to scale up these efforts in a sustainable manner. We compared the impact of three PLE formats on the early algebra knowledge and teaching practices of elementary mathematics teachers: (1) a facilitated summer workshop, (2) a multimedia course completed on teachers’ own time, and (3) learning resources provided in the algebra curriculum unit that teachers used individually. Our findings suggest that all three formats can be mapped against a set of principles for quality professional learning. Analysis of pre- and post-treatment measures indicate that participating teachers’ knowledge of algebra content and best practices significantly increased, regardless of the PLE format with which they engaged. Interviews with a subset of the teachers from the three groups point to the key features of each of the formats that can be capitalized on by designers of PLEs.

Translating a mathematics lesson from curriculum materials into instruction is not straightforward, and launching, or beginning, a lesson such that students will become productively engaged in the target mathematics is nontrivial. The purpose of the study was to investigate preservice teachers’ curricular noticing as they draw upon curriculum materials to design instruction, with a focus on how to launch that lesson. We engaged preservice elementary licensure students at two universities in a four-part process of analyzing mathematics curriculum, planning a lesson, demonstrating their visualization of enactment through animating their lesson launch, and reflecting on the process. Findings indicate that the focal case study pair modified the curricular materials to model mathematical aspects of fractions, adapted the introduction of key academic vocabulary, and introduced materials not mentioned in the curriculum to draw children into the lesson. We discuss implications for preservice teachers’ planning of the lesson launch and their curricular noticing.

Background There is anecdotal evidence that many elementary teachers integrate mathematics lessons and art activities by having students first make colorful, rich material that is subsequently used in an instructional activity. However, it is unclear whether such activities effectively promote learning and transfer of mathematical concepts. The goal of the present research was to examine the use and effectiveness of such “math-and-art” activities on children’s ability to acquire basic fraction knowledge. We report the results of a survey of practicing elementary school teachers in the United States, their use of activities involving physical material, and the resources they use for ideas to supplement the standard curriculum. Two experiments examined first-grade students’ learning, transfer, and recognition of fraction knowledge from rich, contextualized material versus simple, generic material. Results The survey results confirm that many U.S. teachers use math-and-art activities and are often inspired by informal sources, such as Pinterest and YouTube. Experiment 1 examined the effectiveness of colorful, contextualized student-constructed material (paper pizzas) versus simple, pre-made material (monochromatic paper circles) in an instructional activity on fractions. Students who used the pre-made circles scored higher than those who used the student-made pizzas on pre-instruction tests of basic fraction knowledge, immediate tests of learning, and delayed tests of transfer. Experiment 2 tested students’ ability to spontaneously write fractions to describe proportions of pizzas and circles. Students who answered generic circle questions first were markedly more accurate than those who answered pizza questions first. Conclusions These findings suggest that rich, contextualized representations, including those made by the student, can hinder students’ learning and transfer of mathematical concepts. We are not suggesting that teachers never integrate mathematics and colorful, contextualized material, and activities. We do suggest that elementary students’ mathematics learning can benefit when initial instruction involves simple, generic, pre-made material and opportunities for students to make and use colorful, contextualized representations come later.

The purpose of this study is to investigate the impact of teacher education by examining instructional visions of graduates from a teacher education program, 2 or 3 years post-graduation. An instructional vision is a teacher’s idealized image of teaching practice, encompassing aspirations and hopes of what could occur in her classroom. We conducted an inductive analysis of participants’ self-reports of: (1) their instructional visions for teaching elementary mathematics, (2) whether their visions changed over time, (3) whether teacher education influenced changes in their visions, and (4) what their teacher education program promoted as an ideal instructional vision. We administered an open-ended questionnaire to 81 graduates from two cohorts of the same teacher education program. Using latent class analysis, we identified three profiles. Two profiles (89% of the participants) appeared to align with teacher education, suggesting more than one manifestation of the impact of teacher education. Our results indicated alignment between instructional visions and teacher education experiences with respect to teaching mathematics for conceptual understanding and engaging students in productive struggle. A contribution of this study is an awareness that teacher education programs could impact the instructional visions of their graduates in more than one way, with different subsets of intended visions remaining as residue. We conjectured that experiencing instruction that aligned with these instructional visions as learners while in teacher education influenced the development of these graduates’ visions.

This paper explores how problem solving is approached in two 6th grade mathematics textbooks and national elementary mathematics curriculum in Turkey. The curriculum and textbooks’ problem solving approaches are analysed and compared in the light of three approaches of ‘teaching for, about and through problem solving’. Four learning objectives from different content areas (e.g. algebra, data processing) from the curriculum are determined and their treatments in the textbooks are specifically analysed. The analysis showed that the curriculum did not have a prescriptive stance on any one of problem solving approaches. Similarly, the textbooks also did not follow any approach faithfully. ‘Teaching for problem solving’ approach was not employed while ‘teaching through problem solving’ approach was used for only one learning objective in the textbooks. ‘Teaching about problem solving’ approach was encountered in terms of only one problem solving heuristic (table drawing). We discuss the implications of these findings and note some further research areas.

Whole number arithmetic is the foundation of higher mathematics and a core part of elementary mathematics. Awareness of pattern and underlying problem structure promote the learning of whole number arithmetic. A growing consensus has emerged on the necessity to provide students with the opportunity to engage in algebraic reasoning earlier in their education. In fact, the U.S. National Council of Teachers of Mathematics curriculum standards and the nationwide mathematics reform call for algebra readiness in elementary mathematical learning. The purpose of this study was to evaluate the effect of Conceptual Model-based Problem Solving (Xin, Conceptual model-based problem solving: Teach students with learning difficulties to solve math problems. The Netherlands: Sense Publishers, 2012) intervention program, with the Singapore bar model method (Kaur, The model method—A tool for representing and visualising relationships, In: Conference proceedings of ICMI study 23: Primary mathematics study on whole numbers, pp 448–455, 2015) serving as the bridge towards symbolic mathematical model equations, on enhancing additive word problem-solving performance of elementary students who are struggling in mathematics. Findings from this study indicate the promise of this program. Through representing a range of additive word problem situations in one cohesive mathematical model equation, students are prepared for generalized problem-solving skills.

With louder and more widespread calls to include computer programming as a core element of school curriculum, global efforts to define innovative and distinct coding curricula are underway. We take a different tack in this paper, one oriented by an investigation of the common ground between learning to program and learning mathematics. We observed 9- and 10-year-olds as they learned to build and program Lego Mindstorms EV3 robots over 4 days, attending in particular to the ways that programming robots to move might support the development and integration of powerful instantiations of number, arithmetic and multiplication. Our findings suggest that children’s understanding of number, and their transitions from additive to multiplicative thinking, can be powerfully supported by engaging in practical tasks rather than practice exercises.

This study investigates the link between basic math skills, remediation, and the educational opportunity and outcomes of community college students. Capitalizing on a unique placement policy in one community college that assigns students to remedial coursework based on multiple math skill cutoffs, I first identify the skills that most commonly inhibit student access to higher-level math courses; these are procedural fluency with fractions and the ability to solve word problems. I then estimate the impact of “just missing” these skill cutoffs using multiple rating-score regression discontinuity design. Missing just one fractions question on the placement diagnostic, and therefore starting college in a lower-level math course, had negative effects on college persistence and attainment. Missing other skill cutoffs did not have the same impacts. The findings suggest the need to reconsider the specific math expectations that regulate access to college math coursework.

A critical role of mathematics teacher education is to equip teachers with understandings and ambitious practices that support effective mathematics teaching for students from diverse backgrounds, specifically connecting to children’s mathematical thinking (CMT) and children’s linguistic, cultural, and family funds of knowledge (CFoK). Drawing on data from a larger research project, TEACH Math, this longitudinal case study uses the lens of authority to examine one teacher’s, Sena’s, understandings and practices related to CMT and CFoK over a 4-year period, i.e., through mathematics methods, student teaching, and early career teaching. Findings identify changes made in understandings and practices related to CMT and CFoK, with a focus on the development of recognition and realization rules. Emphasis is placed on how authority was shared in the classroom and how this may have impacted one teacher’s development of two very different learning-to-teach trajectories for connecting to CMT and CFoK. Implications for teacher educators focused on supporting novice teachers in developing ambitious mathematics teaching practices are discussed.

Writing is recognized as a critical skill in science, technology, engineering, and mathematical fields, as it provides opportunities for individuals to think critically about learned information, integrate knowledge, communicate understanding, and contribute new information to the field. Requirements for applying mathematical knowledge to writing require a seamless blend of content-specific knowledge, domain-specific vocabulary, and an understanding of written expression. Relatively, little is known about how mathematics writing is currently being assessed in K-12 classrooms to promote the acquisition and growth of students’ writing. This exploratory survey research sought to evaluate how undergraduate students across various career majors, at a large university in the USA, perceive the writing quality of elementary mathematics writing samples. Research determined that respondents valued accuracy of the mathematics, logical explanation, effort of the response, as well as organization/mechanics of the writing. The extremes of mathematical writing had the most reliable scores, bringing attention to the need to develop quality assessments (e.g., rubrics) that address subjectivity of writing quality indicators and explicitly communicate expectations of mathematics writing. Implications for practice and future research are presented.

Reports show that only 40% of 4th-grade students are proficient in math and American students are failing to demonstrate proficiency in key areas of mathematics (National Center for Educational Statistics 2011). Improving students’ fluency of basic math multiplication facts has been recommended as a way to increase math proficiency (Psychology in the Schools, 47:342–353, 2010). Incremental rehearsal is one promising strategy for improving multiplication skills. In this study, three 5th-grade students with disabilities who had Individual Education Plan math goals received one-to-one multiplication instruction using an incremental rehearsal strategy twice per week for 7 weeks. Data indicated immediate and large effects with intervention and growth rates for each student that exceeded expectations. Implications are discussed including the importance of considering both statistically significant and instructionally meaningful results in interpretation.

The UTeach Observation Protocol (UTOP) was designed to inform STEM teacher education. The instrument has been used in prior studies examining inter-rater reliability and relationships to teacher value-added scores. However, prior work has not shown examples of how rating with the UTOP works in practice nor has it discussed the instrument’s strengths and limitations. Here, we describe how the UTOP draws upon theories and practices heavily emphasized in teacher preparation—including deep student engagement, classroom management, STEM content fluency, lesson structuring, and innovative instructional models. We then present the ratings of three sample elementary mathematics lessons on the UTOP. We show how the UTOP reveals important aspects of teachers’ instruction, and discuss key strengths and weaknesses of the instrument. We find that the UTOP provides a broad view of instructional practice useful for informing systemic professional development, while also addressing content-specific teaching behaviors critical to STEM teaching. However, it may be cumbersome to consider so many teaching indicators simultaneously, and less emphasis is given to theory-driven indicators of the development of mathematical reasoning. This article provides a novel theoretical, empirical, and practical base of knowledge for using or making decisions about whether to use the UTOP for math classroom observations.

As teacher education shifts to focus on teaching beginners to do the work of teaching, assessments need to shift to focus on assessing practice. We focus on one teaching practice, eliciting student thinking, in the context of elementary mathematics. We describe assessments in two contexts (field and simulation). For each assessment, we describe the eliciting of three prospective teachers what could be seen about the skills of group of prospective teachers (N = 44). We report on how three prospective teachers had differing opportunities to demonstrate their skills in the context of the field assessment, but similar opportunities in the context of the simulation assessment. Although both contexts make important contributions, in each case, contributions are counterbalanced are by significant challenges. Although the authors do not argue for one assessment context over the other, they offer insights into the affordances and challenges of each so that teacher educators can make responsible decisions.

The study presents the results of a project in which a series of digital games were used for teaching Mathematics to first, fourth, and sixth-grade primary school students (ages 6–7, 8–9, and 11–12). Mathematics was selected as the teaching subject because of the difficulties students face in understanding basic math concepts. Although digital games are used quite extensively for educational purposes, they are scarcely used for teaching Mathematics. The games were developed by the classes' teachers using Microsoft's Kodu Game Lab. The learning outcomes were compared to two other groups of students. The first was taught using the model proposed by Driver and Oldham while the second was taught conventionally. Data was collected using questionnaires and evaluation sheets. A total of 201 students participated in the study coming from three schools in Athens, Greece. Results indicated that students in the games group outperformed, in most cases, students in the other groups. Students' views for the games were highly positive. The implications for software engineers and education administrators are also discussed.

The difficulties children with ADHD experience solving applied math problems are well documented; however, the independent and/or interactive contributions of cognitive processes underlying these difficulties are not fully understood and warrant scrutiny. The current study examines two primary cognitive processes integral to children's ability to solve applied math problems: working memory (WM) and math calculation skills (i.e., the ability to utilize specific facts, skills, or processes related to basic math operations stored in long-term memory). Thirty-six boys with ADHD-combined presentation and 33 typically developing (TD) boys aged 8-12 years old were administered multiple counterbalanced tasks to assess upper (central executive [CE]) and lower level (phonological [PH STM] and visuospatial [VS STM] short-term memory) WM processes, and standardized measures of mathematical abilities. Bias-corrected, bootstrapped mediation analyses revealed that CE ability fully mediated between-group differences in applied problem solving whereas math calculation ability partially mediated the relation. Neither PH STM nor VS STM was a significant mediator. When modeled together via serial mediation analysis, CE in tandem with math calculation ability fully mediated the relation, explained 79% of the variance, and provided a more parsimonious explication of applied mathematical problem solving differences among children with ADHD. Results suggest that interventions designed to address applied math difficulties in children with ADHD will likely benefit from targeting basic knowledge of math facts and skills while simultaneously promoting the active interplay of these skills with CE processes.

There is a critical need to identify primary level students experiencing difficulties in mathematics to provide immediate and targeted instruction that remediates their deficits. However, most early math screening instruments focus only on the concept of number, resulting in inadequate and incomplete information for teachers to design intervention efforts. We propose a mathematics assessment that screens and provides diagnostic information in six domains that are important to building a strong foundation in mathematics. This article describes the conceptual framework and psychometric qualities of a web-based assessment tool, the Primary Math Assessment (PMA). The PMA includes a screener to identify students at risk for poor math outcomes and a diagnostic tool to provide a more in-depth profile of children's specific strengths and weaknesses in mathematics. The PMA allows teachers and school personnel to make better instructional decisions by providing more targeted analyses.

This paper proposes a framework for identifying the mathematical knowledge teachers activate when using curriculum resources. We use the term knowledge of curriculum embedded mathematics (KCEM) to refer to the mathematics knowledge activated by teachers when reading and interpreting mathematical tasks, instructional designs, and representations in mathematics curriculum materials. The KCEM framework is situated within existing research on content-specific teacher knowledge. Through analysis of elementary mathematics teacher’s guides from the USA, we identified elements of curriculum resources teachers interact with when using them to plan instruction. These findings were complemented by interviews of teachers using curriculum guides to plan lessons, uncovering how they interacted with different elements of their guides. From this analysis, we propose four overlapping dimensions of KCEM: foundational mathematical ideas, representations and connections among these ideas, problem complexity, and mathematical learning pathways. Using an excerpt from one elementary curriculum guide and one of the teacher interviews on using this guide to plan a lesson, we illustrate how these dimensions might be activated as teachers read and use their guides.

The purpose of this study was to investigate the relationship between prospective teachers’ (PTs) instructional practises and their efficacy beliefs in classroom management and mathematics teaching. A sequential, explanatory mixed-methods design was employed. Results from efficacy surveys, implemented with 54 PTs were linked to a sample of teachers’ instructional practises during the qualitative phase. In this phase, video-recorded lessons were analysed based on tasks, representations, discourse, and classroom management. Findings indicate that PTs with higher levels of mathematics teaching efficacy taught lessons characterised by tasks of higher cognitive demand, extended student explanations, student-to-student discourse, and explicit connections between representations. Classroom management efficacy seems to bear influence on the utilised grouping structures. These findings support explicit attention to PTs’ mathematics teaching and classroom management efficacy throughout teacher preparation and a need for formative feedback to inform development of beliefs about teaching practises.

Understanding place value is a critical and foundational competency for elementary mathematics. Classroom teachers who endeavor to promote place-value development adopt a variety of established practices to varying degrees of effectiveness. In parallel, researchers have validated models of how young children acquire place-value understanding. Faced with competing demands, teachers may resist abrupt, disruptive, or time-intensive changes to existing practices. In this study, we explore ways of introducing changes in pedagogical supports that minimally disrupt existing classroom routines that nonetheless offer potential gains in children’s understanding of base-ten number concepts. Specifically, we examine the impact of modifying how teachers demonstrate and discuss representations of the ten-for-one structure during the familiar “Days in School” counting routine on young children’s place-value understanding. We report on a 2-year longitudinal study of a cohort of students beginning in grade one, conducted in nine classrooms across a control school and a treatment school. We describe the modifications in the counting routine and report the results of children’s development of base-ten number concepts. The percentage of children who identified base-ten number concepts was three times higher for the children in the treatment group than for the children in the control group. Moreover, nearly twice as many children in the treatment group as compared with the control group advanced beyond a concatenated single-digit conception by the end of second grade when counting two-digit quantities. We conclude with a discussion of the implications of our findings for early elementary mathematics education more generally.

Conventional teacher-led instruction remains dominant in most elementary mathematics classrooms in Taiwan. Under such instruction, the teacher can rarely take care of all students. Many students may then continue to fall behind the standard of mathematics achievement and lose their interest in mathematics; they eventually give up on learning mathematics. In fact, students in Taiwan generally have lower interest in learning mathematics compared to many other regions/countries. Thus, how to enhance students’ mathematics achievement and interest are two major problems, especially for those low-achieving students. This paper describes how we designed a game-based learning environment, called Math-Island, by incorporating the mechanisms of a construction management game into the knowledge map of the elementary mathematics curriculum. We also report an experiment conducted with 215 elementary students for 2 years, from grade 2 to grade 3. In this experiment, in addition to teacher-led instruction in the classroom, students were directed to learn with Math-Island by using their own tablets at school and at home. As a result of this experiment, we found that there is an increase in students’ mathematics achievement, especially in the calculation and word problems. Moreover, the achievements of low-achieving students in the experimental school outperformed the low-achieving students in the control school (a control group in another school) in word problems. Moreover, both the low-achieving students and the high-achieving students in the experimental school maintained a rather high level of interest in mathematics and in the system.

The purpose of the study is to explore how elementary preservice teachers’ mathematics–science integrated teaching strategies changed as a result of participating in exemplary interdisciplinary activities with multiple themes across school curricula. The participating elementary preservice teachers (n = 28) were recruited for this study from the College of Education students enrolled at a medium-sized southwestern research university in the United States. A qualitative methodology with pre-and-post data collection from open-ended surveys was used in the current study to explore the development of preservice teachers’ mathematics teaching strategies with connections to science themed activities before and after an 8-week intervention. In general, the results from the pre-and-post surveys revealed that the preservice teachers’ interdisciplinary knowledge of using science-themed activities as instructional approaches for teaching mathematics had remarkable changes across all four science content areas including physics, chemistry, biology, and environmental and space science. This study provided additional empirical evidence on how contextualized mathematics educational activities, in the current case using the association between science and mathematics, can be used as effective teacher education resources for developing teachers’ capacity for designing mathematics lessons.

This study examines the impact of the Primarily Math Elementary Mathematics Specialist program on K-3 teachers’ mathematical content knowledge for teaching, attitudes toward learning mathematics, and beliefs about mathematics teaching and learning. Three cohorts of teachers participating in the program were compared to a similar group of non-participating teachers. Teacher outcomes were measured longitudinally across 5 years. Participating teachers showed changes in their knowledge, attitudes, and beliefs in line with program goals immediately after completion of coursework. Moreover, these changes were sustained in subsequent years, following program completion. Relative to the comparison group, participants demonstrated greater gains in knowledge as well as greater improvements in attitudes and beliefs. Implications of these results for professional development design, implementation, and evaluation are discussed.

Many in the field of mathematics education call for elementary schools to have elementary mathematics specialists (EMSs) who provide needed mathematical expertise and support for children and teachers. EMSs serve as a reasonable, immediate alternative to the challenges generated by elementary teachers needing improved mathematical knowledge for teaching in the classroom. However, limited inquiry has explored how to best prepare EMSs and how program features and learning activities influence their development. This mixed-method study identifies some of the interrelated benefits from a K-5 Mathematics Endorsement Program designed to prepare EMSs through examining changes in mathematical beliefs, specialized content knowledge (SCK), and classroom teaching practices during the program. Data (n = 32) were collected over the 2-semester program via belief surveys, a content knowledge assessment, observations of teaching practices, and individual interviews from elementary teachers participating in the program. The findings show some changes in beliefs can be made relatively quickly, other shifts in beliefs take more time and continued support, and changes in SCK and adoption of various aspects of standard-based pedagogy require considerably greater opportunities to learn. The described program features and learning experiences provided a context for these changes and offer considerations for EMS preparation programs.

This paper reports on 2 studies that examine how mathematical problem posing is integrated in Chinese and US elementary mathematics textbooks. Study 1 involved a historical analysis of the problem-posing (PP) tasks in 3 editions of the most widely used elementary mathematics textbook series published by People’s Education Press in China over 3 decades. Study 2 compared the PP tasks in Chinese and US elementary mathematics textbooks. This allows for the examination of PP tasks from an international comparative perspective, which provides one point of view about the kinds of learning opportunities that are available to students in China and the USA. We found evidence that the inclusion of PP tasks in the Chinese textbook series reflected, to some degree, changes in the curricular frameworks in China. However, the distribution of PP tasks across grade levels and content areas, as well as the variety of types of PP tasks included, suggest a need for greater intentionality in the design and placement of PP tasks in both the Chinese and US textbook series. Findings from the 2 studies reported in this paper not only contribute to our understanding about the inclusion of PP tasks in curriculum both historically and internationally, but also suggest a great need to systematically integrate PP activities into curriculum and instruction. The fact that both Chinese and US curriculum standards have heavily emphasized PP in school mathematics, despite there being only a small proportion of PP activities in both Chinese and US elementary mathematics curricula, suggests the existence of challenges that are delaying the implementation of reform ideas such as problem posing in school mathematics.

Preservice teachers in six sections (n = 87) of a sequenced, methodological and process-integrated elementary mathematics/science methods course were able to reconcile an issue centered on a similar area of epistemology. Preservice teachers participated in a science inquiry lesson on biological classification and a mathematics problem-solving lesson on polyhedra. Preservice teachers were asked why it was possible to derive a mathematical formula for polyhedra and not organisms and what this implies about the similarities and differences between the epistemologies of mathematics and science. An area of epistemology known as simplicity was triangulated as “simple versus complex” emerged from the constant comparative method (28 %) and “justification and simplicity” (96 %) was coded with a content analysis. Implications include how methods course instructors can create other appropriate activities which incorporate and reconcile other areas of epistemology.

Since Felix Kleins’ influential statement on “Elementary Mathematics from an Higher Standpoint” many scholars have discussed the role of mathematical knowledge in teacher education and generated diverse arguments for selecting and delivering university subject matter courses. This article contributes to the discussion by considering the special case of Galois Theory. It conducts a subject matter analysis of Galois Theory by drawing on several frameworks from general didactics, subject matter didactics and teacher competence, and by presenting a concrete university course that emphasizes the idea of symmetry and connections to school mathematics (also available as a Klein-Vignette, Leuders 2016b). Thereby it proposes an example for a “Subject Matter Analysis with a Perspective on Education and Teachers”.

In unequal societies the effectiveness of formal education depends on the socioeconomic status (SES) of students. Studies have shown that poverty affects the development of the brain in ways that might compromise future learning, thus increasing the differences between groups with different SES. Interest is growing in the development of tools that might change this state of affairs. This article presents a tablet-based study aimed at determining the cognitive abilities related to primary school children’s math learning. The study followed the students’ changes during a short intervention, the purpose of which was to improve students’ performance of some of the core components of mathematical cognition; in particular, of the approximate number system (ANS), a system that supports one’s ability to estimate quantities and to compare time intervals. The article presents the study’s characteristics and shows how the variables that were evaluated—ANS precision, time discrimination accuracy, digit span, and mathematical achievement—depend on SES. We employ multiple regressions to show that the variance in mathematics performance attributed to SES can be explained by differences in underlying cognitive factors. The study also indicates that those students of low-SES schools who participated in more tablet activities increased their performance more than students who did fewer activities. Although the intervention’s initial objective was to influence mathematical development and the study is not a randomized double-blind study, we argue that training the ANS can have positive effects in mathematics learning, and that this might benefit children living in low-SES contexts more than those in the general population, perhaps because of the former’s initially low levels of performance in school mathematics.