Rich Mathematics Research Articles

Exploring ways to engage disaffected mathematics students through formative assessment processes with rich tasks

Researchers and educators continue to grapple with how to re-engage mathematics students who evidence disaffection and low achievement in class. In a fine-grained qualitative case study, observations and regular self-reports of six 16-17-year-old early school leavers were analyzed to investigate their responses to cycles of formative assessment activities in the context of rich mathematics tasks. The interplay of extrinsic and intrinsic motivational aspects was found in the students’ situational interest and value appraisal of the activities, dis/engagement related to experiencing activities unfamiliar to them, and the social dimension of engagement in peer-community-building leading to eventual engagement.

Exploring high school teacher’s design of rich algebra tasks

The phenomenon in schools today is that teachers rarely change their tasks. However, the teacher's activities to change their tasks, build students' arguments, support solutions, and maintain arguments without long debates are important and exciting things to learn. This study explores the ideas of high school teachers about task design and practice in teaching mathematics. The authors surveyed twelve high school teachers who teach mathematics in East Java Province. First, the authors conducted preliminary observations to observe the design of mathematics teacher tasks for six months in eight schools. Second, the authors state that teachers are engaging and consistent in designing rich algebra tasks. Third, the authors examine the teacher's ideas through direct observation and unstructured interviews. The results show how teachers' ideas about task design enhance students' creative thinking by reforming tasks from textbooks into rich mathematics tasks. The design of the task carried out by the teacher is to create to stimulate creative thinking. The teachers also use their knowledge and understanding of the material and curriculum to modify mathematics tasks in students' mathematics books. The task given by the teacher is to improve students' reasoning, not just memorize formulas or properties.

Ambitious instruction and student achievement: Evidence from early career teachers and the TRU math observation instrument

What constitutes effective instructional practices in improving student achievement continues to elude practitioners and researchers. Researchers have offered evidence on the association between different instructional practices and student achievement using various tools, but some aspects of instructional practices, such as formative assessment, were left out. Using TRU Math to capture rich elementary mathematics observation data coupled with student achievement data, we found that the extent to which teachers surface students’ ideas and build upon these ideas has significant associations with student achievement in mathematics. We also provide vivid descriptions of high quality instruction compared to lower quality instruction.

Abstract 1515: Shannon entropy of mutational signatures predict sensitivity of signature detection in targeted sequencing

Abstract Mutational processes can cause driver mutations which are a proximal cause of tumorigenesis. COSMIC identifies signatures of biological and environmental events that drive mutational processes and can be identified by examining patterns of somatic mutations. We describe the complexity of the COSMIC mutational signatures and their impact. We introduce Shannon entropy as a metric to measure complexity of trinucleotide distributions, thus capturing a previously ignored dimension of mutational signatures that may predict their identification. Shannon entropy derives from the rich predictive mathematics of information theory. We calculated entropy for all the COSMIC mutation signatures and found that SBS3, homologous recombination repair deficiency (HRD) signature, had highest entropy of 6.326, while SBS2 signature, activation of AID/APOBEC cytidine deaminases, had the lowest entropy of 1.758.We discovered that high entropy signatures (such as HRD SBS3) require more somatic mutations to be reliably detected than do low entropy signatures, important when planning signature detection using targeted panels that have smaller genomic footprint. We demonstrate this correlation with two approaches i) by simulation using MutSigSim, a tool we developed for simulating mutational profiles given the number of mutations, and ii) by using real datasets. Simulation was performed for high, medium, and low entropy signatures. We used cosine similarity of 0.7 as the threshold for detection. For signatures with entropy <2, a signature can be detected 90% of the time with 10 or more mutations. Similarly, signatures with entropies between 2-3, 3-5 and >5 can be detected 90% of the time with 30, 40, and 100 mutations, respectively. Mutation signatures of 1500 samples from a pan solid tumor cohort sequenced with Oncomine Comprehensive Assay Plus or Oncomine TMB were used to confirm these conclusions. We saw that signatures with higher entropy were difficult to detect at cosine similarity threshold of 0.7 in samples with a smaller number of mutations. In a cohort of samples with MSI-high, we detected mismatch repair (MMR) signatures with low to medium entropy with high specificity and good sensitivity. Furthermore, in a cohort of Ovarian cancer samples, it was difficult to detect the high entropy HRD signature at 0.7 threshold when the number of somatic mutations was not high. We show that ability to detect mutation signatures using targeted panels is directly correlated to the complexity of signature and number of mutations. For research use only. Not for use in diagnostic procedures. Citation Format: Ajithavalli Chellappan, Chintan Vora, Jagannath Patro, Shilpa Nair, Rushikesh S. Kanap, Fiona C. Hyland. Shannon entropy of mutational signatures predict sensitivity of signature detection in targeted sequencing [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2022; 2022 Apr 8-13. Philadelphia (PA): AACR; Cancer Res 2022;82(12_Suppl):Abstract nr 1515.

The Power of Mothers and Teachers Engaging in a Mathematics Bilingual Collaboration

Background: Making mathematics connections between home and school can have a positive impact on students’ mathematics learning. This is particularly important for students whose experiences and perspectives are underrepresented in school curricula. Although there is evidence for how crucial it is to make connections between mathematics instruction and children’s lived experiences, doing so is often a challenging task for teachers. Purpose: This study examines what teachers in two different settings—a dual language school and a Structured English Immersion school—learned from engaging in a mathematics collaboration with Latina mothers. We share the teachers’ perceptions of their students’ and families’ mathematics’ strengths and how they envisioned their future mathematics teaching. Research Design: This qualitative interpretive study explored how 15 teachers from two states made sense of their experiences working together in a two-year mathematics collaboration with mothers from their school community. We collected multiple teacher reflections and artifacts, and administered a small group interview with teachers and parents and an individual teacher interview. We used an iterative analysis by demarcating the words that pertained to the different data sources, as we looked for patterns of the teachers’ key ideas. We summarized the key ideas across the data sources that resulted in six major themes. Findings: The key findings in this study reveal that the teachers learned from the mothers more about their students’ lives outside of school, the mothers’ mathematics background, and ways to create multiple channels of communication with the mothers. The teachers also learned about the significant contributions that the mothers make to their children’s mathematics education and the importance of honoring the mothers’ ways of doing mathematics in their mathematics teaching. The teachers shared how the collaboration with the mothers helped to serve as a guide to leverage their students’ learning of mathematics in Spanish and English. Conclusions/Recommendations: This study suggests that mathematics collaborations between teachers and parents can be highly beneficial to student learning. It showcases the importance of providing teachers with opportunities to learn about the rich mathematics knowledge that parents from diverse backgrounds possess. This study also demonstrates that creating authentic relationships that enhance mutual understandings between teachers and parents can be key in supporting Latinx students’ mathematics learning. We recommend that these types of collaborations between teachers and parents be considered part of the preparation of preservice teachers as well as the professional learning of in-service teachers.

Teacher orchestration of student responses to rich mathematics tasks in the US and Japanese classrooms

Teacher orchestration of student responses to rich mathematics tasks in the US and Japanese classrooms

Exploration of Mathematics Culture and Stimulation of Ethnic Students’ Interest in Learning Mathematics

The interest, attitude, habit and ability of ethnic students in primary and secondary schools in Western China are not optimistic. In the mathematics classroom teaching of primary and secondary schools, appropriate infiltration of relevant mathematical cultural knowledge can enhance students’ self-confidence in learning mathematics, and stimulate students’ interest in mathematics learning, in order to promote students to understand and master mathematical knowledge and its ideological methods, and produce correct views on Mathematics and mathematics learning, which can promote the improvement of students’ mathematical achievements and the development of teachers themselves. The western region has many ancient cities, small border villages, classical culture, human geography, overview of natural scenery, unspoiled ecological environment, legends, wind washing dust, weddings and funerals, etc., all these are materials that can be used for mathematics classroom teaching in primary and secondary schools, which are of great educational significance to students’ mathematics development. According to the requirements of mathematics curriculum standards, primary and secondary school mathematics teachers should fully explore the rich and colorful traditional mathematics culture curriculum material resources in ethnic minority areas, reorganize the content of national general mathematics teaching materials, and design classroom teaching, as well as compile teaching plans for primary and secondary school mathematics classroom teaching according to the reality of students in the western region. It can improve the mathematics learning achievement of ethnic students in primary and secondary schools in the western region, so as to improve the quality of science teaching in the western region.

How Elementary Teaching Candidates’ Learning Opportunities Are Associated with Their Knowledge, Self-Efficacy, and Beliefs

Background/Context Many elementary teacher education programs seek to prepare candidates to enact ambitious mathematics instruction that supports students in engaging in rigorous, conceptually rich mathematics tasks. Extant literature suggests that preparedness to engage in ambitious elementary mathematics instruction is multifaceted and includes teaching candidates’ mathematical knowledge for teaching (MKT), self-efficacy with regard to teaching mathematics, and pedagogical beliefs about mathematics. Prior research has produced findings that provide discrete, and at times conflicting, information about teacher preparation. Purpose/Objective/Research Question/Focus of Study This study examined how elementary candidates’ learning opportunities in mathematics content courses, mathematics methods courses, and student teaching were moderated by their reports about the quality of their experiences in courses and field placements to seem to affect their MKT, self-efficacy, and beliefs. Population/Participants/Subjects The study participants were 220 elementary teaching candidates who were in their final year of teacher preparation at four universities in three states. Research Design We employed multivariate path analysis, an approach that is purposefully designed to probe heterogeneity in teaching candidates’ experiences in courses and clinical placements. Data Collection and Analysis We administered two surveys to each study participant: an elementary teaching candidate survey, which included measures of mathematics teaching self-efficacy and pedagogical beliefs about mathematics, and an MKT survey. Findings/Results The number of mathematics content courses that elementary candidates took was positively associated with their MKT and mathematics teaching self-efficacy only when they also reported having positive experiences learning mathematics. When candidates reported increased opportunities to engage with representations, decompositions, and approximations of mathematics teaching practices in mathematics methods courses, this was associated with higher MKT scores and pedagogical beliefs about mathematics. When candidates reported that their cooperating teacher was a high-quality mentor, increased opportunities to observe, attempt, and receive feedback on mathematics teaching practices during their field experience were associated with mathematics teaching self-efficacy and pedagogical beliefs about mathematics. Conclusions/Recommendations The findings from this multivariate path analysis, which account for both the reported quantity and the perceived quality of elementary teaching candidate experiences, may shed light on conflicting findings in prior literature. There is little agreement in extant literature about associations between facets of teacher preparation and candidate knowledge, self-efficacy, or beliefs. Explaining the positive associations in some samples and lack of associations in others may have more to do with the quality of teaching candidate experiences than with whether a candidate was exposed to a particular opportunity to learn.

Prospective and In-Service Mathematics Teachers’ Attention to a Rich Mathematics Task While Planning its Implementation in the Classroom

This study explores secondary mathematics teachers’ attention to the three components of Jaworski’s (1992) “teaching triad” (i.e., mathematical challenge, learning management, and sensitivity to students) as they planned a lesson revolving around a rich mathematics problem and assessed pre-designed student solutions for that problem. Written work was gathered from 17 cohorts. Qualitative analysis generated categories. Quantitative analysis revealed that some components of the teaching triad were attended to in both activities, some were not attended to in either activity, and some were attended to in one activity but not in the other. Findings are interpreted in light of theory.

Investigating peer-assessment strategies for mathematics pre-service teacher learning on formative assessment

Formative assessment practices for secondary mathematics have been advocated as valuable for students, but difficult for teachers to learn. There have been calls in the literature to increase the emphasis on formative assessment in mathematics teacher preparation courses. This study explored the use of peer-assessment strategies for helping pre-service secondary mathematics teachers (PSTs) cultivate formative assessment principles and practices for assessing school students. Twenty-seven PSTs participated in a peer-assessment cycle comprised of: sourcing a rich mathematics task; constructing an assessment rubric for it; and collecting and analysing a selection of secondary student responses to the task. Each PST then provided written and verbal feedback to a peer on his/her rubric and student solution assessments. We draw on theoretical conceptions of Teacher Assessment Literacy in Practice to characterize the PSTs’ perceptions of their experience of formative assessment processes for learning to assess school students, in terms of cognitive and affective dimensions of their conceptions of assessment. The cohort evidenced a wide range of levels of confidence with the various aspects of formative assessment practices but on average less confidence in assessing school student task responses themselves than in assessing peer work. In addition to highlighting specific changes to different types of assessment knowledge, the PSTs also evidenced an awareness of shifts in their attitudes, in coming to view student task responses with more appreciation and humility.

Connecting the Tangent Function to Cardinality: A Method for Introducing Set Theory to High School Students.

High school students often ask questions about the nature of infinity. When contemplating what the “largest number” is, or discussing the speed of light, students bring their own ideas about infinity and asymptotes into the conversation. These are popular ideas, but formal ideas about the nature of mathematical sets, or “set theory,” are generally unknown to high school students. The authors propose a method for introducing basic ideas in set theory to high school trigonometry students by connecting prior knowledge of the tangent function and the unit circle to Georg Cantor’s ideas about infinity. By doing so, high school teachers have an opportunity to inspire their students with rich mathematics.

A Math Field Day: Embed Content with Play

Can you remember your typical elementary school field day? In this article, we provide details on hosting a mathematics field day, focused on embedding rich mathematics into authentic fun-filled field day experiences.

The Subjects That the Pre-service Classroom Teachers Perceive as Difficult in Elementary Mathematics Curriculum

OECD-PISA which evaluates the education quality of Europe focused on mathematics, and Turkey was ranked 50 among 72 countries. When the mathematics literacy was analyzed in terms of years, it is seen that the performance of the students in Turkey in PISA 2015 was lower compared to PISA 2009 and PISA 2012. This shows that students have difficulty in comprehending the subjects in the mathematics curriculum. Therefore, learning experiences designed for the pupils to have rich mathematics experiences during elementary school is of higher importance. At this point, it is necessary to know the subjects that the pre-service teachers, who will be teachers of the future, are challenged in teaching mathematics. Therefore, the purpose of study is to determine the subjects in which elementary school pre-service teachers perceive teaching in elementary school mathematics as difficult and to examine the causes of these difficulties. The case study method is used in this study and the participants of the study are 120 pre-service classroom teachers who study in the fourth grade in a state university. Participants were administered a 19-item learning difficulty index covering mathematics learning areas of 1-4 grades (n = 120). The distribution of answers given by the pre-service teachers for each topic and the learning difficulty index for each topic were calculated. Interviews were done with the pre-service teachers in response to their responses to the questionnaire items to understand the causes of the difficulties. In interviews conducted one by one, the pre-service teachers were asked general mathematics lecture notes, mathematics fear level and the reasons for the topics they found difficult to teach.

You, y'all, we

“You, Ya'll, We is a lesson structure to approach open-ended math tasks that focus on inquiry where all students have an opportunity to engage in rich mathematics, share their thinking, and learn alongside others. Postscript items are designed as rich grab-and-go resources that any teacher can quickly incorporate into his or her classroom repertoire with little effort and maximum impact.

En/countering Inclusive Mathematics Education: A Case of Professional Learning

Despite the push for inclusive mathematics education, students with disabilities continue to lack access to, and achievement in, rich mathematics learning opportunities. We assert that mathematics teacher educators have a central role in addressing these contradictions. This role includes enacting facilitative moves during mathematics teacher professional learning to encounter and counter social forces, which we denote in this article as en/counters. As part of a larger study, we explored the extent to which the use of an inclusive education-oriented tool, developed and introduced during a teacher learning program, elicited en/counters that mediated participants' learning toward inclusive mathematics education. We discuss shifts in participants' conversational content and focus on surrounding practices that involved students with disabilities and features of the tool and processes that supported these shifts, including specific facilitative moves that helped redirect deficit-focused conversations.

Executive function in learning mathematics by comparison: incorporating everyday classrooms into the science of learning

ABSTRACTIndividual differences in executive functions (EFs) are well established to be related to mathematics achievement, yet the mechanisms by which this occurs are not well understood. Comparing representations (problems, solutions, concepts) is central to mathematical thinking, and relational reasoning is known to rely upon EF resources. The current manuscript explored whether individual differences in EF predicted learning from a conceptually demanding mathematics lesson requiring relational reasoning. Analyses revealed that variations in EF predicted learning when measured at a delay. Thus, EF capacity may impact students’ overall mathematics achievement by constraining their resources available to learn from cognitively demanding reasoning opportunities in lessons. To assess the ecological validity of this interpretation, we report follow-up interviews with mathematics teachers who raised similar concerns that cognitively demanding activities such as comparing multiple representations in mathematics may differentially benefit their high versus struggling learners. Broader implications for ensuring that all students have access to, and benefit from, conceptually rich mathematics lessons are discussed. We also highlight the utility of integrating methods in science of learning (SL) research.

Using The Internet In Teaching Algebra To Middle School Students: A Study Of Teacher Perspectives And Attitudes

During the past decades, technological resources have been improved to support the teaching of mathematics. While the improvement of technological resources, the World Wide Web provides teachers and students many resources that engage students in rich mathematics experiences. There are a vast number of educational resources available through the Internet for students and teachers, which should be implemented in the classroom to support and improve algebra instructions. This study seeks to identify the extent of the middle school teacher’s awareness of using the Internet to teach and develop students’ learning of algebra. The participant’s teachers in this study are teaching in schools located in a rural Appalachian region of a Midwestern state in the United States. The study found that the Internet’s function in the algebra classroom within this particular group of middle childhood teachers are used for purposes that do not contribute to algebra instruction.

Expected Shannon Entropy and Shannon Differentiation between Subpopulations for Neutral Genes under the Finite Island Model.

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.

Creating Cultures of Participation to Promote Mathematical Discourse

IntroductionAs a former eighth grade mathematics teacher, I was aware of the impact discourse had in shaping students' thinking and thus often implemented strategies that supported these efforts. However, a reality became clear when pursuing these National Board Certification, which demanded a 15-minute unedited video of me facilitating a whole-class discussion. While recording the lesson, which centered on an introduction to irrational numbers, I thought students were engaged in lively discussion. However, analysis of the video-taped lesson revealed that I was doing almost all of the talking! The perceptions of my practice were not aligned with my actual practice. After this revelatory experience, I was curious to learn more about how other teachers perceived the use of discourse to support adolescents reasoning in mathematics and particularly how they created cultures of participation to support equitable discursive interactions with adolescent students from diverse backgrounds.Consistent with my own experience, equitable access to mathematically rich and meaningful learning experiences continues as a critical need in the classroom (Cobb 8c Hodge, 2011; National Middle School Association [NMSA], 2010). Rich learning experiences also are fundamental in supporting and developing students' mathematical reasoning and sense making (Chapin, O'Connor, 8c Canavan Anderson, 2003), which is of particular concern during the middle grades as the level of abstraction in mathematics increases greatly. As a means of openly engaging in understanding meaningful and rich mathematics, discourse offers one avenue for teachers to create equitable and mathematically rich learning environments and interactions; this article demonstrates why an emphasis on mathematical discourse should be a common practice within the middle level classroom (Bartolini Bussi, 1998).Discourse requires students to evaluate and interpret the perspectives, ideas, and mathematical arguments of others as well as construct valid arguments of their own. That is, students develop deeper understandings of mathematics when they engage in meaningful social interactions such as whole class discourse (Cobb, Yackel, W NCTM, 1991; Rigleman, 2010). The adoption of the Common Core State Standards in many states reinforces the importance of creating classrooms that focus on reasoning, deep conceptual understanding, and the communication of mathematics (Larson, 2012;NCTM, 2000). In addition, the teacher's efforts and attitudes in creating a caring and well-structured classroom environment with high expectations for all students are more likely to increase student engagement (Klem 8c Connell, 2004), which can lead to greater achievement (Smith, Rook, 8c Smith, 2007). Such classrooms exemplify cultures of participation. With that said, knowing that discourse can support student learning and creating the structures to help discourse come alive are not the same thing. Creating a classroom culture of participation is a necessary first step in implementing meaningful discourse and creating equitable learning experiences.What follows is a compilation of strategies and classroom structures from 13 middle level teachers from highly urban communities with tremendous cultural, linguistic, and ethnic diversity. …

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207Conflict of interest: The authors have declared no conflicts of interest for this article.For further resources related to this article, please visit the WIREs website.