Mathematics Standards Research Articles

Making mathematics visible through narrative

The latest content and practice standards for mathematics have promoted the use of academic language and discourse in this content area. As such, the WIDA Consortium has evolved its guidelines to deepen teachers’ conceptual understanding of language in their content areas, and thus, enable them to provide more effective instruction to multilingual learners. Karen L. Terrell explores one of the primary “key language uses” in mathematics — the narrative genre. She also considers how visuals can be powerful scaffolds and means of expression for multilingual learners and discusses how visuals can be used for students to both receive and express knowledge. The use of narrative and visuals can make mathematics more accessible to all students.

On-the-Go Android-Based Learning Tool for Mathematics in Geographically Challenged Areas

One of the important challenges for mathematics educators during the Covid-19 pandemic was ensuring productive mathematics learning despite hindrances. This study explored the mathematics learning experiences of geographically challenged students during the pandemic and developed an on-the-go Android-based learning tool (OTG ALT) for mathematics learning in the challenged areas. The study employed educational design research (EDR) with three phases. A semi-structured in-depth interview guide and student satisfaction survey instrument were utilized. Six geographically challenged university students for interviews and forty students for the tryout of the OTG ALT were purposively selected as participants. Mean and standard deviation scores were used to analyze and interpret the quantitative data. The phenomenographic method was used to analyze the qualitative data. Results show that the experiences of the geographically challenged students in pandemic mathematics education were encapsulated in four different categories: 1) pandemic mathematics learning, a new experience, 2) teachers’ agency in mathematics, 3) tribulations and triumphs, and 4) mathematics that works. The developed learning tool for the course Mathematics in the Modern World (MMW) follows the i?SPARK model (introduce, study, practice, assess, reflect, and key to correction), with a video explainer per topic. Students were highly satisfied with the developed learning tool, in general and in terms of all indicators. Hence, the developed OTG ALT met the instructional standards for mathematics learning in the challenged areas. School administrators may improve the quality and effectivity of mathematics education in universities by considering minority perspectives. To make mathematics education inclusive and equitable, policymakers and curriculum designers can review the suitability of teaching materials and adapt them to the level and needs of the students.

대학 이공계열 신입생의 고등학교 수학과 성취기준 도달에 관한 연구

Objectives The purpose of this study is to analyze the understanding of the high school mathematics contents of freshmen majoring in science and engineering. Methods To do this, the degree of reaching achievement standards in high school mathematics, mathematics I, and mathematics II subjects was measured by freshmen majoring in science and engineering. In order to measure the degree of reaching the achievement standards of high school mathematics, an evaluation tool was developed based on the achievement standards and evaluation standards of the 2015 revised curriculum. A nonparametric test, Kruskal-Wallis, was used for analyzing the degree of reaching achievement standards according to the type of high school from which one came from and the mathematics subject, and the Mann-Whitney test was used as a posttest. Results The results of this study are as follows. First, the test results of 689 people for a total of 19 questions consisting of mathematics, mathematics I, and mathematics II averaged 8.53. Second, when analyzing the degree of reaching the achievement standard of the high school mathematics according to the type of high school from which they came from, freshmen who graduated from general high schools showed higher levels than freshmen who graduated from vocational high schools. Third, when analyzing the degree of reaching the achievement standard according to the subject, the degree of reaching the achievement standard of the mathematics subject was higher than that of the mathematics I. Fourth, when analyzing the degree of reaching the achievement standard according to the subject, the degree of reaching the achievement standard of the mathematics II subject was higher than that of the mathematics I. Conclusions Universities need to provide teaching and learning programs for freshmen majoring in the science and engineering so that they can understand the high school mathematics contents. In addition, as the curriculum changes, freshmen's understanding of the content of the subject may change, so it is necessary to establish a system to continuously check the main contents of the high school mathematics.

Connections within the Multiplicative Field: A Case Study of Adán’s Mathematical Thinking

ABSTRACT Mathematical coherence is a goal within the Common Core State Standards for Mathematics. One aspect of this coherence is how student mathematical thinking is developed across concepts. Unfortunately, mathematics is often taught as isolated ideas across grades. The multiplicative field is an area of study that needs to be examined as a space to develop connections across concepts. This paper uses a case study of mathematical thinking from one 5th-grade student over a school year to demonstrate the connections in his thinking across whole numbers, fractions, and graphing work. Analyzing his solution strategies with three frameworks detailing multiplicative, proportional, and functional thinking, his strategies show a developing understanding of proportional reasoning, and functional thinking as he is building his multiplicative thinking and connections from whole numbers to fraction problem-solving strategies to graphing. Examining this student’s mathematical thinking points to the opportunity for connections across the multiplicative field through teaching mathematics in a manner more responsive to student sense-making.

General inferential limits under differential and Pufferfish privacy

Differential privacy (DP) is a class of mathematical standards for assessing the privacy provided by a data-release mechanism. This work concerns two important flavors of DP that are related yet conceptually distinct: pure ε-differential privacy (ε-DP) and Pufferfish privacy. We restate ε-DP and Pufferfish privacy as Lipschitz continuity conditions and provide their formulations in terms of an object from the imprecise probability literature: the interval of measures. We use these formulations to derive limits on key quantities in frequentist hypothesis testing and in Bayesian inference using data that are sanitised according to either of these two privacy standards. Under very mild conditions, the results in this work are valid for arbitrary parameters, priors and data generating models. These bounds are weaker than those attainable when analysing specific data generating models or data-release mechanisms. However, they provide generally applicable limits on the ability to learn from differentially private data – even when the analyst's knowledge of the model or mechanism is limited. They also shed light on the semantic interpretations of the two DP flavors under examination, a subject of contention in the current literature.1

Spatial reasoning in mathematics standards: identifying how early elementary educators are systematically supported to teach spatial skills

Spatial reasoning is critical to early mathematics learning, but it is unclear how early elementary educators learn to teach and are supported in teaching its comprising skills. One view of the available supports can be found by examining the alignment of spatial reasoning skills and mathematics education standards, as standards provide the content of the intended curriculum children are expected to learn at each grade level. This study used content analysis methods to investigate how spatial reasoning might be taught through broadly adopted early elementary education standards in the United States, the Kindergarten through Grade 2 Common Core State Standards for Mathematics. The paper describes the frequency and degree of explicitness with which 38 spatial reasoning skills are therein represented. Findings indicate that most standards implicitly relate to some form of spatial reasoning through a pedagogical reach of teaching expertise, but few standards contain explicit spatial linkages. The implications and limitations of this analysis are discussed in relation to teaching spatial reasoning in early elementary grades and students’ opportunities to learn these critical skills.

Computationally efficient orthogonalization for pairwise comparisons method

Orthogonalization is one of few mathematical methods conforming to mathematical standards for approximation. Finding a consistent PC matrix of a given an inconsistent PC matrix is the main goal of a pairwise comparisons method. We introduce an orthogonalization for pairwise comparisons matrix based on a generalized Frobenius inner matrix product. The proposed theory is supported by numerous examples and visualizations.

Validation of four optimization evolutionary algorithms combined with artificial neural network (ANN) for landslide susceptibility mapping: A case study of Gilan, Iran

Landslides, the most significant geohazards in Iran, adversely affect the region's socioeconomic conditions and the environment. Landslide susceptibility mapping is crucial for proactive risk management, sustainable development, and protecting human lives, infrastructure, and the environment. It enables decision-makers to make informed choices, implement appropriate mitigation measures, and plan for potential landslide events, leading to safer and more resilient communities. Using a recommended subdivision methodology, this research has developed a landslide susceptibility map for a significant Gilan, Iran region. The production of the map involved utilizing an artificial neural network (ANN) model. This model incorporated sixteen causal components from several characteristics, including topographic and geomorphologic features, geological factors, land use patterns, hydrological aspects, and hydrogeological properties. Three hundred seventy instances were identified using multiple verified sources and analyzing aerial photographs to create the landslide inventory map. Examining and verifying the weights assigned to the causal elements were conducted per accepted mathematical standards, incorporating sensitivity analysis, earlier research, and empirical data on landslides. The area under the receiver operating characteristic curve (AUROC) was used to compare the various models. The study's findings indicated that the best swarm size value for COA-MLP11Cuckoo Optimization Algorithm (COA). is equal to 450, and the accuracy indices for this algorithm were 0.998 and 0.995 in training and testing datasets, respectively. Similarly, the AUC22Area under the curve (AUC). for the HS-MLP,33Harmony Search Algorithm (HS). SFS-MLP,44Stochastic Fractal Search (SFS). and TLBO-MLP55Teaching- Learning-Based Optiization (TLBO). were 0.997, 0.999, and 0.999 in training and 0.995, 0.996, and 0.995 in the testing dataset, respectively. Therefore, the use of optimization algorithms leads to an increase in the performance and accuracy of the neural network, and the high accuracy of the SFS-MLP model demonstrates the existence of a dependable criterion for delineating the susceptibility zones about forthcoming landslide occurrences. The model is a cost-effective and potentially indispensable tool for urban planners in growing cities and municipalities. Its effectiveness is demonstrated by comparisons with previous susceptibility analyses conducted in the region.

Exploring inquiry‐based STEM in preschool: The treehouse project

AbstractIntegrating STEM (science, technology, engineering, and mathematics) in authentic ways is often overlooked in early childhood spaces, despite young children having the ability to engage with the world around them through STEM practices. This piece showcases the work of children in a prekindergarten classroom as they explored an integrated STEM unit centering around the construction of a treehouse on their playground. Through the careful connection to content and standards in science, engineering, and mathematics, along with the integration of developmentally appropriate uses of technology, children learned about and constructed a treehouse on their school playground. Family and community partnerships also contributed to this work. We highlight the various phases of the project, associated standards, and tasks that children engaged with to help demonstrate the propensity of our youngest learners to authentically engage with STEM tasks.

The best pedagogical practices for teaching mathematics revisited: Using math manipulatives, children’s literature, and GeoGebra to produce math confident young people for a STEM world

Today math teachers can best reach their students and show them how math surrounds us by using manipulatives, children’s literature, and GeoGebra while teaching mathematics. These are some of the best pedagogical practices for teaching mathematics today. In our high-tech world, students need to be proficient in science, technology, engineering, and mathematics (STEM) fields. As endorsed by National Council of Teachers of Mathematics (2000) and stressed in common core state standards in mathematics, it is important that we teach with technology, address dispositions and math anxiety, and make the math that young people are learning pertinent and meaningful. Frequently, it may be best to start teaching young people geometry first as opposed to numbers, which are considered more abstract and difficult to learn. Geometry is one of the most tangible divisions of math and concentrating on this first can help students’ whole view of mathematics and their insolences towards learning the subject. Nowadays teachers also need to be cognizant and checking for attitudes and dispositions toward learning mathematics, as math anxiety is an issue in today’s classrooms. This paper will revisit the best pedagogical practices for teaching math the review of the use of math manipulatives, children’s books, and GeoGebra to help teachers create mathematically confident young people.

Interactive Homework: A Tool for Parent Engagement

Families have largely been excluded from the implementation of the Common Core State Standards in Mathematics (CCSSM), reducing their ability to extend their child’s mathematics learning. CCSSM emphasizes different instructional elements (e.g., pictorial representations, problem solving, multiple strategies for solving) that may differ greatly from how parents learned mathematics. In addition, many school officials have ineffectively engaged parents in the changes, further diminishing their capacity to participate in their child’s learning. This case study examined parent mathematics self-efficacy and parent mathematics knowledge for teaching, factors that influence the effectiveness of their engagement in their child’s mathematics learning. This study was also implemented to identify elements that the parent participant found helpful for their child’s mathematics learning. A thematic analysis was performed on the data sources, the interactive homework assignments, a survey, observations, a researcher’s journal, and an interview to conclude that the interactive homework assignments improved parent mathematics self-efficacy and parent mathematics knowledge for teaching. The parent participant also identified the assignments’ side-by-side examples, additional practice, and the easy access of the assignments as features of the intervention that enhanced her ability to support her child.

Educational Disparities in Primary School Students in Austria

Abstract: In this study, we analyze data from the Austrian assessments of educational standards in German and Mathematics, applying an intersectional perspective. We simultaneously consider the effects of gender, socioeconomic status (SES), and immigrant background on fourth-graders’ competencies in both subjects. Besides the expected main effects – girls perform better in German, boys better in mathematics, lower SES and immigrant background predict lower performance in both subjects – we found the following interaction effects: Second-generation immigrant students showed better achievement in German than their peers who had migrated to Austria later in life, except when their country of origin was Turkey. In students without an immigrant background, SES was more strongly related to achievement in mathematics in boys, while among students originating from former Yugoslavia, SES was more strongly related to achievement in German in girls. The study demonstrates that intersectional perspectives in quantitative educational research can help to identify groups with increased support needs.

Student learning and instructional tasks in different curricular contexts: A longitudinal study

This paper compares the longitudinal effect of instructional tasks on algebra learning that used a Standards-based curriculum [Connected Mathematics Project (CMP)] to that of classrooms that used a traditional curriculum (non-CMP). CMP was developed based on the National Council of Teachers of Mathematics (NCTM) Standards and can be characterized as a problem-based curriculum. CMP teachers were more than three times as likely to implement high-level instructional tasks than non-CMP teachers. Increases in the cognitive demand were associated with enhanced growth rates in problem solving, computation, and equation solving. Notably, when controlling for the cognitive demand of the instructional tasks, the advantage of the CMP curriculum over the non-CMP curricula on students’ growth in problem solving disappeared. However, non-CMP curricula had an advantage on students’ growth over the CMP curriculum after the cognitive demand of the instructional tasks was controlled.

Exploring Secondary School Principals’ Experiences and Leadership Practices in Building Professional Learning Communities: A Case Study in Klang, Malaysia

<p style="text-align: justify;">A study showed Malaysian students were unable to achieve the minimum standards in Mathematics and Science subjects, prompting the need for educational reformation. With this, educators need to inculcate Professional Learning Community (PLC) to enhance the teaching quality. This study explores secondary school principals’ PLC practices and experiences to improve student achievement in Klang, Malaysia. The qualitative case study was conducted using randomly selected two secondary school principals in Klang. The interview session comprised four interview questions and was conducted via an online platform. The results showed two common PLC practices which are annual meetings and observation. The challenges faced by school principals to implement PLC were time factors, lack of budget, and teacher attitudes. The factors hindering students' learning achievement were family background, lack of focus on education, and teachers' resistance to change. The roles of principals to curb these issues were collecting data on students' learning achievement, equipping teachers with knowledge and resources, being good role models, and building relationships with parents. This study highlighted that school principal 1 has shown unique ideas and approaches in improving students' learning achievement compared to school principal 2. The study suggests using descriptive analysis using more than one group of participants.</p>

The Journey Continues: Mathematics Curriculum Analysis from the Official Curriculum to the Intended Curriculum

In this paper, we share major lessons we have learned in our curriculum analysis explorations and provide suggestions for future mathematics curriculum research. Specifically, we will discuss the successes and challenges of using comparative methods and developing analytical frameworks. Using an old Chinese saying, follow the vine to get the melon (顺藤摸瓜), our journey starts with getting a holistic picture of standards and moves to identifying a specific topic in the textbooks. We first conducted a study to compare the geometry standards of the Common Core State Standards for Mathematics (CCSSM) and the Chinese Compulsory Education Mathematics Curriculum Standards (CMCS); then we analyzed the presentation of a specific topic, triangle congruence, in multiple geometry textbooks. For each comparative study we conducted, developing the analytical framework that can be used to guide future investigations constituted a critical step in the research methodology. We have learned lessons from adapting the well-known van Hiele model, which was created for the development of geometric reasoning, into a lens for a detailed curriculum analysis. We have also learned lessons from elaborating on the key constructs from the “Mathematics Curriculum as Story” framework for coding and analysis. These lessons as well as our future directions serve as the primary focus of this paper.

School Greenness and Student-Level Academic Performance: Evidence From the Global South.

Greenspace in schools might enhance students' academic performance. However, the literature-dominated by ecological studies at the school level in countries from the Northern Hemisphere-presents mixed evidence of a beneficial association. We evaluated the association between school greenness and student-level academic performance in Santiago, Chile, a capital city of the Global South. This cross-sectional study included 281,695 fourth-grade students attending 1,498 public, charter, and private schools in Santiago city between 2014 and 2018. Student-level academic performance was assessed using standardized test scores and indicators of attainment of learning standards in mathematics and reading. School greenness was estimated using Normalized Difference Vegetation Index (NDVI). Linear and generalized linear mixed-effects models were fit to evaluate associations, adjusting for individual- and school-level sociodemographic factors. Analyses were stratified by school type. In fully adjusted models, a 0.1 increase in school greenness was associated with higher test scores in mathematics (36.9 points, 95% CI: 2.49; 4.88) and in reading (1.84 points, 95% CI: 0.73; 2.95); as well as with higher odds of attaining learning standards in mathematics (OR: 1.20, 95% CI: 1.12; 1.28) and reading (OR: 1.07, 95% CI: 1.02; 1.13). Stratified analysis showed differences by school type, with associations of greater magnitude and strength for students attending public schools. No significant associations were detected for students in private schools. Higher school greenness was associated with improved individual-level academic outcomes among elementary-aged students in a capital city in South America. Our results highlight the potential of greenness in the school environment to moderate educational and environmental inequalities in urban areas.

Joint Effects of Heat Source and Magnetic Field on Unsteady Chemically Reacting Fluid Flow Towards A Vertically Inclined Plate in Addition of Cu-Nanoparticles

The primary goal of this evaluation task is to research the mathematical analysis for unstable, free convective incompressible viscous heat also mass transfer fluid movement across an inclined a plate that is vertically positioned in the occurrence of copper nanoparticles, Magnetism, thermal generator & chemical reaction in porous media. For this investigation, we assumed the effects of Cu-nanoparticles and Angle of inclination effects in the governing equations. Additionally, the effects of fluctuating temperature & concentration are studied. We established a set of basic equations for this fluid flow and translated nonlinear partial difference equations into linear incomplete comparisons, which were then answered using the implicit limited alteration technique. The impacts of several engineering fluid variables on flow variables such as velocity, temperature, & concentration profiles were explored in this research study via the use of graphs to show the findings. Along with the other findings, the mathematical standards of skin friction, heat transmission rate, & mass transmission constants are calculated and reported in tabular form. Finally, and perhaps most importantly, the mathematical consequences of the code validation programme are related to previously publish analytical results. In the instance of pure and nanofluids, the velocity profiles are shown to increase with rising values of the Heat transfer using the Grashof number, the mass movement Grashof number, the parameter for permeability, and the passage of time Increases in magnetic field component, the Schmidt number and the Prandtl number, the parameter for the heat source, the component of the chemical reaction, and the degree of inclination all result in a drop in the velocity profiles. With respect to temperature profiles, they have been on the rise with passing time, in contrast to the Prandtl number and the heat source parameter, for which the opposite trend has been seen. We discovered that the temperature and velocity profiles are both steeper for nanofluids than for pure fluids when the parameters are increased. The concentration profiles rise with increasing times, but the opposite is true for the Schmidt number. Moreover, increasing Chemical reaction parameter values result in decreasing profiles of concentrations.

Designing mathematics standards in agreement with science

To learn mathematics, historically students had no choice but to memorize fundamental facts and apply memorized algorithms. Since 1995 in the US, all states have adopted standards to govern K-12 mathematics instruction, and in most, standards have de-emphasized memorization and emphasized reasoning based on concepts. This change assumed the brain could reason in mathematics without relying on memorized knowledge. Scientists who study the brain have recently verified this assumption was mistaken. Due to stringent limitations in working memory (where the brain solves problems), mathematical problem-solving of any complexity requires applying well-memorized facts and procedures. A decade after the implementation of standards in most states, US young adults ranked last in testing in mathematics among 22 nations. Changes are proposed to state K-12 standards, which recent scientific research suggests would substantially improve student mathematics achievement.

From Logo Programming to Fab Labs: The Legacy of Constructionist Learning in K−12 Education Technology

Mens et manus, the motto that is embedded in the MIT official seal, is a simple Latin phrase that translates to “mind and hand.” As a continuation of MIT’s pioneering education and research, Seymour Papert, a renowned educator and learning theorist, established the foundation of constructionism, which emphasizes hands-on/minds-on learning. This article explores the evolution of constructionism in K-12 educational technology from the Logo computer language to Scratch block-based programming and the introduction of Fab Labs, which represent the third-generation platform for constructionist teaching and learning. The use of Fab Labs in K-12 education aligns with the Mens et manus philosophy by providing students with opportunities to apply theoretical knowledge to real-world applications through hands-on experiences with digital fabrication technologies. The adoption of the proposed Fab Labs constructionist framework can help prepare students for the demands of a rapidly changing world and aligns with educational standards for science and mathematics.

Three Principles of Intellectualization of Automated Systems

Three Principles of Intellectualization of Automated Systems