3,350 results
Search Results
2. Visualizing Math: How Number Lines Can Empower Problem-Solving
- Author
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Tiffany Berman and Casey Hord
- Abstract
Research has shown the importance of helping students, especially those with mild-to-moderate learning disabilities, to offload information during problem-solving. When students can get their thoughts onto paper, number line strategies can help them develop a firm foundation in mathematical problem-solving while understanding the relationships between mathematical operations. These strategies are helpful for the development of addition, subtraction, multiplication, division, and later, fractional mathematics. In this article, we describe the progression of number lines as a supportive strategy for elementary students and those with developmental delays in mathematics to improve mathematical understanding. This strategy is based on students being able to show their work and think about what they have written on paper or how they have used manipulatives.
- Published
- 2024
3. The Prime Number Theorem as a Mapping between Two Mathematical Worlds
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Norton, Anderson and Flanagan, Kyle
- Abstract
This paper frames children's mathematics as mathematics. Specifically, it draws upon our knowledge of children's mathematics and applies it to understanding the prime number theorem. Elementary school arithmetic emphasizes two principal operations: addition and multiplication. Through their units coordination activity, children construct two mathematical worlds: an additive world and a multiplicative world. Understanding how children might map between their additive and multiplicative worlds provides insights into the prime number theorem. It also helps us appreciate the power of children's mathematics, constructed through the coordination of their own mental actions. [For the complete proceedings, see ED630210.]
- Published
- 2022
4. Computing Science: Pencil, Paper, and Pi
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Hayes, Brian
- Published
- 2014
5. Fraction Addition through the Music
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Maria T. Sanz, Carlos Valenzuela, Emilia López-Iñesta, and Guillermo Luengo
- Abstract
This study examined the effects of an academic intervention, associated with music, on the conceptual understanding of musical notation and arithmetic of fractions of first-year students of high school from a mixed Spanish multicultural and socioeconomic public school. The students (N = 12) had previous concepts about musical instruction, as well as operations with fractions, particularly addition. This is an observational study in which a battery of four tasks was administered before and after an instruction based on a musical environment, music being a semiotic function. The instruction included 9 sessions of 50 minutes each. The results prior to the intervention show deficiencies in a concept that was not new to the students, however, after the intervention the students were competent in addition with fractions. [For the complete proceedings, see ED657822.]
- Published
- 2023
6. 'Leave No One Behind': A Systematic Literature Review on Game-Based Learning Courseware for Preschool Children with Learning Disabilities
- Author
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Zolkipli, Nurzamila Zasira, Rahmatullah, Bahbibi, Mohamad Samuri, Suzani, Árva, Valéria, and Sugiyo Pranoto, Yuli Kurniawati
- Abstract
The rapid growth of multimedia pedagogy in the education sector has brought about a game-based technological approach that is shaping the learning of children nowadays. The focus of the approach is to encourage active participation from children; the effect can be seen in their interaction, involvement, and engagement throughout the learning content. Active participation in a learning environment would have a substantial impact on children's self-development and academic achievement, particularly for those with learning disabilities (LD). Therefore, the purpose of this paper is to systematically analyze research conducted on Game-Based Learning (GBL) courseware to support the education of children with LD. A systematic literature review was undertaken, following the PRISMA framework for paper selection. A total of 109 articles were retrieved from the Scopus and Science Direct databases by using a specific keywords search. 14 articles were finalized at the end of the screening based on the inclusion and exclusion criteria. Results reveal the trend of publications, approaches used, and research themes of the selected papers, which include the courseware requirement, student adaptation, and impact of the implementation. The findings demonstrate that GBL is one of the effective methods to be applied in preschool education. It has a significant impact on the development of cognitive skills and assists children who have difficulty in reading, writing, and arithmetic. The findings in this paper can be used as a guide in developing GBL courseware that is developmentally appropriate and effective for children with LD.
- Published
- 2023
7. Gender Differences in How Students Solve the Most Difficult to Retrieve Single-Digit Addition Problems
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Mathematics Education Research Group of Australasia (MERGA), Russo, James, and Hopkins, Sarah
- Abstract
Despite curriculum expectations, many students, including a disproportionate number of girls, do not 'just know' (retrieve) single-digit addition facts by Year 3. The current study employed structured interviews to explore which strategies Year 3/4 students (n = 166) used when solving more difficult addition combinations. Results revealed that students preference the near-doubles strategy when the difference between the addends was one, the bridging-through-10 strategy when one of the addends was a nine, and the count-on-from-larger strategy when a derived strategy was more effortful. Moreover, whereas boys were more inclined to use derived strategies, girls were almost three times more likely to use the count-on-from-larger strategy.
- Published
- 2023
8. Tap or Swipe: Interaction's Impact on Cognitive Load and Rewards in a Mobile Mental Math Game
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Jost, Patrick, Rangger, Sebastian, and Künz, Andreas
- Abstract
With the growing prevalence of mobile apps for self-directed learning, educational games increasingly find their place in everyday routines, becoming accessible to a broad audience. Despite the growing ease of content creation by artificial intelligence computing, the challenge of designing effective and engaging Serious Games remains, particularly in managing cognitive resources and ensuring quality engagement, notably influenced by the game's interaction modalities. This study explores these challenges within the context of a casual mobile mental arithmetic game, investigating the differential impacts of tap and swipe interaction variants on cognitive load and reward-based engagement. The study presents the findings of an international field study on Google Play. In a between-group design, the two casual interaction paradigms were compared regarding their impact on practice performance, cognitive load and effect on classic casual game rewarding represented through points, leaderboards and badges. The findings show that tap interaction can optimise cognitive load with a better mental math practice performance than the more indirect swipe gesture. A combination of elementary tap interaction with point rank and interaction precision badges indicates to enhance practice motivation. The results are synthesised into interaction design recommendations for casual mental math mobile games. [For the full proceedings, see ED636095.]
- Published
- 2023
9. Exploring Insights from Initial Teacher Educators' Reflections on the Mental Starters Assessment Project
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Rosemary D. L. Brien and Sharon M. Mc Auliffe
- Abstract
Background: The Mental Starters Assessment Project (MSAP) seeks to address poor performance in Grade 3 mathematics. The programme focusses on eliminating inefficient counting methods and promoting strategic mathematical skills, including numerical sense, mental calculation and rapid recall skills. Additionally, MSAP supports teachers' professional growth by providing them with a toolkit of effective calculation strategies to bridge the performance gap and enhance mathematical education. Aim: This paper explores the insight gained from reflections of final year Bachelor of Foundation Phase (FP) initial teacher educators (ITE) students in South Africa. Setting: Grade 3 classrooms. Methods: The ITE students were given training and materials to implement the MSAP, and this occurred over a 4-week teaching practicum, after which they completed a reflective task on the implementation. A total of 20 students were selected from a cohort of 138 based on their academic performance. Results: The analysis of the reflections showed that ITE students benefitted from reflecting on their practice and highlighted important elements of their professional learning. The reflections raised issues related to challenges in their pedagogical content knowledge (PCK) as well as their confidence and competence to teach mathematics and manage the classroom context. Conclusion: With a multi-dimensional model of reflection, ITE students can achieve a deeper understanding of mathematics teaching and learning when building learners' mental strategies, fostering professional growth and elevating the overall quality of mathematics education. Contribution: Overall, the findings provide insight into the benefits of reflective practices for ITE students' professional development and the improvement of mathematics education.
- Published
- 2024
10. Using Interviews with Non-Examples to Assess Reasoning in F-2 Classrooms
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Mathematics Education Research Group of Australasia (MERGA) and Copping, Kate
- Abstract
The development of mathematical reasoning is a key proficiency for mathematics within the Australian Curriculum. However, reasoning can be difficult for teachers to assess, particularly with pen and paper tests. In this study, interview tasks were designed across three curriculum areas at three different levels to assess student reasoning through the use of examples and non-examples. Non-examples can be used to assist in building boundaries and deepening conceptual understanding. Through the interview, teacher and student dialogue can help students to demonstrate reasoning and clarify concepts through explanation and justification.
- Published
- 2021
11. Construction of Arithmetic-Algebraic Thinking in a Socio-Cultural Instructional Approach = Construction d'une pensée arithémico-algébrique dans une approche socioculturelle de l'enseignement
- Author
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Hitt, Fernando
- Abstract
We present the results of a research project on arithmetic-algebraic thinking that was carried out jointly by a team in Mexico and another in Quebec. The project deals with the concepts of variable and covariation between variables in the sixth grade at the elementary level and the first, second, and third years of secondary school--namely, children from 11 to 14 years old. We target secondary students (first year or K7) in this article. Our objective relates to the development of a gradual generalization in arithmetic-algebraic thinking in a socio-cultural approach to the learning of mathematics. We experimented with investigative situations using a paper-and-pencil approach and technology. We analyze the emergence, in this context, of a visual abstraction, the production of institutional and non-institutional representations, a sensitivity to contradiction, and, finally, the concepts of variable and of covariation between variables. [For the complete proceedings, see ED629884.]
- Published
- 2020
12. Justifications Students Use when Writing an Equation during a Modeling Task
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Roan, Elizabeth and Czocher, Jennifer
- Abstract
Literature typically describes mathematization, the process of transforming a real-world situation into a mathematical model, in terms of desirable actions and behaviors students exhibit. We attended to STEM undergraduate students' quantitative reasoning as they derived equations. Analysis of the meanings they held for arithmetic operations (+, -, ·, ÷) provided insight into how participants expressed real-world relationships among entities with arithmetic relationships among values. We extend the findings from K-12 literature (e.g., using multiplication to instantiate a rate) to STEM undergraduates and found evidence of new ways of justifying the usage of arithmetic operations (e.g., using multiplication to instantiate an amount). [For the complete proceedings, see ED630210.]
- Published
- 2022
13. Middle School Students' Mature Number Sense Is Uniquely Associated with Grade Level Mathematics Achievement
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Kirkland, Patrick K., Guang, Claire, Cheng, Ying, Trinter, Christine, Kumar, Saachi, Nakfoor, Sofia, Sullivan, Tiana, and McNeil, Nicole M.
- Abstract
Students exhibiting mature number sense make sense of numbers and operations, use reasoning to notice patterns, and flexibly select the most effective and efficient problem-solving strategies (McIntosh et al., 1997; Reys et al., 1999; Yang, 2005). Despite being highlighted in national standards and policy documents (CCSS, 2010; NCTM, 2000, 2014), students' mature number sense and its nomological network are not yet well specified. For example, how does students' mature number sense relate to their knowledge of fractions and their grade-level mathematics achievement? We analyzed 129 middle school students' scores on measures of mature number sense, fraction and decimal computation, and grade-level mathematics achievement. We found mature number sense to be measurably distinct from their fraction and decimal knowledge and uniquely associated with students' grade-level mathematics achievement. [For the complete proceedings, see ED630210.]
- Published
- 2022
14. Going Online: A Simulated Student Approach for Evaluating Knowledge Tracing in the Context of Mastery Learning
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Zhang, Qiao and Maclellan, Christopher J.
- Abstract
Knowledge tracing algorithms are embedded in Intelligent Tutoring Systems (ITS) to keep track of students' learning process. While knowledge tracing models have been extensively studied in offline settings, very little work has explored their use in online settings. This is primarily because conducting experiments to evaluate and select knowledge tracing models in classroom settings is expensive. To fill this gap, we introduce a novel way of using machinelearning models to generate simulated students. We conduct experiments using agents generated by the Apprentice Learner Architecture to investigate the online use of different knowledge tracing models (Bayesian Knowledge Tracing, the Streak model, and Deep Knowledge Tracing). An analysis of our simulation results revealed an error in the initial implementation of our Bayesian knowledge tracing model that was not identified in our previous work. Our simulations also revealed a more fundamental limitation of Deep Knowledge Tracing that prevents the model from supporting mastery learning on multi-step problems. Together, these two findings suggest that Apprentice agents provide a practical means of evaluating knowledge tracing models prior to more costly classroom testing. Lastly, our analysis identifies a positive correlation between the Bayesian knowledge tracing parameters estimated from human data and the parameters estimated from simulated learners. This suggests that model parameters might be initialized using simulated data when no human-student data is yet available. [For the full proceedings, see ED615472.]
- Published
- 2021
15. Measuring Mental Computational Fluency with Addition: A Novel Approach
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Mathematics Education Research Group of Australasia, Russo, James, and Hopkins, Sarah
- Abstract
Measuring computational fluency, an aspect of procedural fluency, is complex. Many attempts to measure this construct have emphasised accuracy and efficiency at the expense of flexibility and appropriate strategy choice. Efforts to account for these latter constructs through assessing children's computational reasoning using structured interviews (e.g., MAI), are necessarily time-intensive. In this paper, we introduce a novel measure of Mental Computational Fluency with Addition (MCF-A) that attempts to incorporate these aspects by requiring children to reason from the perspective of another child. We describe results of a pilot study using the MCF-A with 169 Year 3 and 4 students.
- Published
- 2018
16. Rote versus Rule: Revisiting the Role of Language in Mathematical Thinking
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Qin, Jike and Opfer, John
- Abstract
Language is often depicted as the sine qua non of mathematical thinking, a view buttressed by findings of language-of-training effects among bilinguals. These findings, however, have been limited to studies of arithmetic. Nothing is known about the potential influence of language on the ability to learn rules about the relations among variables (e.g., algebra). To test whether arithmetic and algebraic thinking differ, Chinese-English bilinguals were trained to solve arithmetic and algebra problems in either Chinese or English and then tested on new and old problems in both languages. For arithmetic problems, solution times were always longer for English than Chinese; in both languages, solution times dropped during training; after training, solution times continued to drop for old problems, but returned to pre-training levels for new problems. In contrast, for algebra problems, solution times did not differ across language; solution times dropped during training; after training, gains in speed were preserved for both old and new problems. These findings suggest that the contribution of language to mathematical thinking may be limited to the areas of mathematics that are learned by rote and not by rule. [This paper was published in: "Proceedings of the 40th Annual Conference of the Cognitive Science Society," edited by T. T. Rogers et al., Cognitive Science Society, 2018, pp. 918-923.]
- Published
- 2018
17. Productive Seeds in Preservice Teachers' Reasoning about Fraction Comparisons
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Whitacre, Ian, Findley, Kelly, and Atabas, Sebnem
- Abstract
Reasoning about fraction magnitude is an important topic in elementary mathematics because it lays the foundations for meaningful reasoning about fraction operations. Much of the research literature has reported deficits in preservice elementary teachers' (PSTs) knowledge of fractions and has given little attention to the productive resources that PSTs bring to teacher education. We surveyed 26 PSTs using a set of 9 fraction-comparison tasks. We report the frequency of complete strategy-arguments and the perspectives (ways of reasoning) used for each item. We further examine incomplete strategy-arguments, noting substantial evidence for productive seeds of reasoning. Using data from interviews with 10 of these PSTs, we identify evidence suggesting these seeds are, in fact, productive in that they provide foundations for further development. We argue that this type of research is needed in order to further mathematics teacher education. [For the complete proceedings, see ED629884.]
- Published
- 2020
18. An Approach to Density in Decimal Numbers: A Study with Pre-Service Teachers = Un acercamiento a la densidad en los números decimales: un estudio con profesores en formación
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Suárez-Rodríguez, Mayra and Figueras, Olimpia
- Abstract
Researchers, who have studied the understanding of the density property in the set of decimal numbers, have shown that the student uses the property of the discrete of natural numbers to solve tasks related to density. So, a restructuring of concepts is necessary, that is, a conceptual change from "the discrete" to "the dense". This report presents evidence from ten pre-service teachers from Mexico City that this change can initiate through the implementation of a didactic sequence. The pre-service teachers managed to visualize that a decimal number can be found in an interval. Consequently, they were able to conceive an infinity. However, several of them persisted with the idea of the existence of a successor in the set of decimal numbers. [For the complete proceedings, see ED629884.]
- Published
- 2020
19. Are Approximate Number System Representations Numerical?
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Pickering, Jayne, Adelman, James S., and Inglis, Matthew
- Abstract
Previous research suggests that the Approximate Number System (ANS) allows people to approximate the cardinality of a set. This ability to discern numerical quantities may explain how meaning becomes associated with number symbols. However, recently it has been argued that ANS representations are not directly numerical, but rather are formed by amalgamating perceptual features confounded with the set's cardinality. In this paper, we approach the question of whether ANS representations are numerical by studying the properties they have, rather than how they are formed. Across two pre-registered within-subjects studies, we measured 189 participants' ability to multiply the numbers between 2 and 8. Participants completed symbolic and nonsymbolic versions of the task. Results showed that participants succeeded at above-chance levels when multiplying nonsymbolic representations within the subitizing range (2-4) but were at chance levels when multiplying numbers within the ANS range (5-8). We conclude that, unlike Object Tracking System (OTS) representations, two ANS representations cannot be multiplied together. We suggest that investigating which numerical properties ANS representations possess may advance the debate over whether the ANS is a genuinely numerical system.
- Published
- 2023
20. Moving beyond Test Scores: Analyzing the Effectiveness of a Digital Learning Game through Learning Analytics
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Nguyen, Huy Anh, Hou, Xinying, Stamper, John, and McLaren, Bruce M.
- Abstract
A challenge in digital learning games is assessing students' learning behaviors, which are often intertwined with game behaviors. How do we know whether students have learned enough or needed more practice at the end of their game play? To answer this question, we performed post hoc analyses on a prior study of the game "Decimal Point," which teaches decimal numbers and decimal operations to middle school students. Using Bayesian Knowledge Tracing, we found that students had the most difficulty with mastering the number line and sorting skills, but also tended to over-practice the skills they had previously mastered. In addition, using students' survey responses and in-game measurements, we identified the best feature sets to predict test scores and self-reported enjoyment. Analyzing these features and their connections with learning outcomes and enjoyment yielded useful insights into areas of improvement for the game. We conclude by highlighting the need for combining traditional test measures with rigorous learning analytics to critically evaluate the effectiveness of learning games. [For the full proceedings, see ED607784.]
- Published
- 2020
21. Towards a Critical Mathematics
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Savich, Theodore Michael
- Abstract
The goal of this paper is to express necessary conditions for arithmetic in ways that are compatible with the unity of being and knowing understood within first-person experience. In psychological literature, this experience of unity is discussed as flow, but the epistemological and ontological unity is prior to the observer's position from which psychology unfolds, making this project essentially non-psychological. Instead of an externalized psychological conceptualization of mind, the idea explored is internal to first-person experience that experiences consciousness as movement, with key elements of that movement explicable as inference and algorithm. The notion of first-person experience is often taken to be transparent and irrelevant, so an exercise for orienting readers to the I-feeling and its connection to movement is elaborated. A mathematical notion of history as algorithmic elaboration is introduced in order to explicate how to move from talk about judgments in general to doing arithmetic. In this preliminary and exploratory paper, I forgo many possible linkages with modern mathematics education research literature to focus on articulating what I think are some of the foundational elements of a critical mathematics. The purpose of this preliminary work is to orient readers to an ongoing project in critical mathematics, not to provide instrumental notions of how one might improve teaching or research, though brief discussions of how this mathematical theory could be applied in mathematics education research and practice are included.
- Published
- 2020
22. A Conceptual Analysis of the Equal Sign and Equation -- The Transformative Component
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Ying, Yufeng
- Abstract
Mathematics education scholars have generally classified students' conception of the equal sign as either operational or relational. Adding to these conceptions, Jones (2008) introduced the idea of substitutional conception. Building off these scholars, I introduce a form of understanding the equal sign that includes a transformative equivalence component and extends the conceptions of the equal sign to conceptions of equations. [For the complete proceedings, see ED629884.]
- Published
- 2020
23. Strategies Used by Grade 6 Learners in the Multiplication of Whole Numbers in Five Selected Primary Schools in the Kavango East and West Regions
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Ilukena, Alex Mbonabi, Utete, Christina Nyarai, and Kasanda, Chosi
- Abstract
This research paper reports strategies used by Grade 6 learners in multiplying whole numbers in five selected primary schools in Kavango East and West regions. A total of 200 learners' mathematics exercise books were analysed in order to identify the commonly used strategies by learners in multiplying whole numbers. A total of ten teachers teaching grade 6 mathematics were also requested to complete a questionnaire which required them to indicate the strategies that they employed in class when teaching multiplication of whole numbers. The teachers indicated that they used a variety of strategies including repeated addition, complete-number (including doubling), partitioning and compensation to teach multiplication of whole numbers. The results also disclosed that the majority of the learners' mathematics exercise books reflected the use of the traditional method of repeated addition contrary to the teachers' claims. It was also found that a few of the learners used other strategies such as long method, short method and learner "invented" strategies. Additionally, the mathematics curriculum for upper primary learners (Grade 4-7 mathematics syllabus) requires learners to use paper and pencil algorithms to carry out multiplication of whole numbers without calculators (Ministry of Education, Arts & Culture [MoEAC], 2015, p. 2). However, at Grade 6, learners were expected to use paper and pencil algorithms to multiply numbers within the range 0-100000. Analysis of the learners' exercise books indicated that the majority were not able to multiply a two digit by a single digit, a two digit by a two digit and a three digit by a two digit number.
- Published
- 2020
24. Investigating the Variety and Usualness of Correct Solution Procedures of Mathematical Word Problems
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Samková, Libuše
- Abstract
The contribution focuses on issues related to the implementation of formative assessment methods into inquiry based teaching, by means of issues related to solving twelve multiple-step arithmetic word problems based on operations with natural and rational numbers. These word problems have multiple correct solution procedures and the presented qualitative exploratory empirical study investigates how varied and how usual might be correct solution procedures provided by diverse groups of solvers -- future primary school teachers attending diverse university mathematics courses of diverse forms and/or time extent. According to written data collected from 149 solvers, six notions are introduced in the paper: majority, minority and even solution procedures, and majority, minority and mixed solvers. Issues regarding minority solvers are recognized as an important element for implementing formative assessment methods. All the six notions are illustrated in the paper by samples of solution procedures and diagrams of relative frequency. Implications are given for formative assessment within any kind of education involving multiple-step word problems, regardless of the extent of implemented inquiry.
- Published
- 2020
25. Varieties of Numerical Estimation: A Unified Framework
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Qin, Jike, Kim, Dan, and Opfer, John
- Abstract
There is an ongoing debate over the psychophysical functions that best fit human data from numerical estimation tasks. To test whether one psychophysical function could account for data across diverse tasks, we examined 40 kindergartners, 38 first graders, 40 second graders and 40 adults' estimates using two fully crossed 2 × 2 designs, crossing symbol (symbolic, non-symbolic) and boundedness (bounded, unbounded) on free number-line tasks (Experiment 1) and crossing the same factors on anchored number-line tasks (Experiment 2). This strategy yielded 4 novel tasks to assess the generalizability of the models. Across all 8 tasks, 90% of participants provided estimates better fit by a mixed log-linear model than other cognitive models, and the weight of the logarithmic component ([lambda]) decreased with age. After controlling for age, the weight of the logarithmic component ([lambda]) significantly predicted arithmetic skills, whereas parameters of other models failed to do so. Results suggest that the logarithmic-to-linear shift theory provides a unified account of numerical magnitude estimation and provides uniquely accurate predictions for mathematical proficiency. [This paper was published in: "Proceedings of the 39th Annual Meeting of the Cognitive Science Society," edited by G. Gunzelmann et al., Cognitive Science Society, 2017, pp. 2943-2948.]
- Published
- 2017
26. The Development of Addition and Subtractions Strategies for Children in Kindergarten to Grade 6: Insights and Implications
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Mathematics Education Research Group of Australasia, Gervasoni, Ann, Giumelli, Kerry, and McHugh, Barbara
- Abstract
This paper provides insight about the development of addition and subtraction strategies for nearly 22,000 Australian primary school children in 2016. The children were each assessed by their teacher using a task-based assessment interview that identified the strategies they used to mentally perform addition and subtraction, and matched these to a growth point framework. The findings highlight the broad distribution of strategies used by children in each grade level and suggest that few children, including those in Grade 6, reach Growth Point 6 that involves the mental calculation of two-digit and three-digit numbers. These findings have important implications for classroom teaching and professional learning.
- Published
- 2017
27. Hypothesis of Developmental Dyscalculia and Down Syndrome: Implications for Mathematics Education
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Mathematics Education Research Group of Australasia and Faragher, Rhonda
- Abstract
In this paper, the hypothesis that Developmental Dyscalculia (DD) is a characteristic of Down syndrome (DS) is proposed. Implications for the hypothesis are addressed: If it were to be confirmed that DS implies DD, what would be the consequences for the mathematics education of learners with DS? The use of prosthetic devices to overcome the impaired calculation capabilities of the brain is essential. Fortunately, electronic calculating devices are readily available. Their routine use opens the possibility of studying areas of mathematics that were once inaccessible.
- Published
- 2017
28. The Strategy the Use of False Assumption and Word Problem Solving
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Eisenmann, Petr, Pribyl, Jirí, and Novotná, Jarmila
- Abstract
The paper describes one problem solving strategy -- the Use of false assumption. The objective of the paper is to show, in accordance with Phylogenesis and Ontogenesis Theory, that it is worthwhile to reiterate the process of development of the concept of a variable and thus provide to pupils one of the ways helping them to eliminate usual difficulties when solving word problems using linear equations, namely construction of the equations. The paper presents the outcomes of a study conducted on three lower secondary schools in the Czech Republic with 147 14-15-year-old pupils. Pupils from the experimental group were, unlike pupils from the control group, taught the strategy the Use of false assumption before being taught the topic Solving word problems. The tool for the study was a test of four problems that was sat by all the involved pupils three weeks after finishing the topic "Solving word problems" and whose results were evaluated statistically. The experiment confirmed the research hypothesis that the introduction of the strategy the Use of false assumption into 8th grade mathematics lessons (14-15-year-old pupils) helps pupils construct equations more successfully when solving word problems.
- Published
- 2019
29. Students' Mental Model in Solving the Patterns of Generalization Problem
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Prayekti, N., Nusantara, T., Sudirman, and Susanto, H.
- Abstract
Mental models are representations of students' minds concepts to explain a situation or an on-going process. The purpose of this study is to describe students' mental model in solving mathematical patterns of generalization problem. Subjects in this study were the VII grade students of junior high school in Situbondo, East Java, Indonesia. This study was conducted using the qualitative descriptive method to investigate students' mental model during the process of solving the problem of generalization patterns. Students were required to speak aloud what he or she was thinking about solving problems ("think-aloud"). Based on the data obtained, the students' mental models in solving the patterns of generalization problem can be classified into two kinds of mental models, namely: (1) direct mental model and (2) indirect mental model. Students classified using direct mental model were using the only generalization of algebraic in solving the problem. Meanwhile, students classified using indirect mental model were using a combination of both generalizations of algebraic and arithmetic. Based on the results of the study, teachers are required to take into account students' mental model in solving the generalization patterns problem to achieve better learning process. [Paper presented at ICEGE 2018.]
- Published
- 2019
30. Learning from Lessons: Teachers' Insights and Intended Actions Arising from Their Learning about Student Thinking
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Roche, Anne, Clarke, Doug, Clarke, David, and Chan, Man Ching Esther
- Abstract
A central premise of this project is that teachers learn from the act of teaching a lesson and that this learning is evident in the planning and teaching of a subsequent lesson. We are studying the knowledge construction of mathematics teachers utilising multi-camera research techniques during lesson planning, classroom interactions and reflection. This paper reports on the learning of two Year 7 teachers, one in Melbourne and one in Chicago, teaching the same initial lesson focusing on division, remainders and context. Both teachers claimed to have learned about their students' mathematical thinking after teaching the initial lesson, but found planning a second lesson to accommodate this learning challenging.
- Published
- 2016
31. The Language Used to Articulate Content as an Aspect of Pedagogical Content Knowledge
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Mathematics Education Research Group of Australasia and Chick, Helen
- Abstract
Mathematical knowledge in classrooms is mediated through the use of both technical and informal language. This paper is a report of a study of the language use of teachers as they examine students' work and discuss teaching for the topic of fraction operations. This provides a window on their pedagogical content knowledge and also on the way in which language is used to make sense of mathematical knowledge, either personally or for students. It was found that some mathematical knowledge appeared to be "taken as understood," perhaps because the expected words were used.
- Published
- 2015
32. A Conventional and Digital Mathematical Board Game Design and Development for Use by Students in Learning Arithmetic
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Fathurrohman, Maman, Nindiasari, Hepsi, and Rahayu, Ilmiyati
- Abstract
This paper reported the design and development of a conventional and digital mathematical board game for use by students in learning arithmetic. At the time of research, there is no significant indication that a mathematical board game is available in scientific and published patent documentation. The availability of mathematical board games for students' drills and practice in arithmetic, especially in mathematical statement construction, would benefit them, as this competency is an essential life skill. This research was conducted through the design and development research method with the procedure of users' need analysis, researcher as developer capability analysis, product design, product development, field testing in its natural setting environment, and the prototype. The board game prototype was developed in both conventional printed and digital versions. The field testing for the conventional printed version was conducted at secondary school classes with 34 and 36 students, respectively, while for the digital by selected participants. The field testing shows that the developed mathematical board game can work as expected in its natural setting environment.
- Published
- 2022
33. Preschoolers' Ways of Experiencing Numbers
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Björklund, Camilla, Ekdahl, Anna-Lena, Kullberg, Angelika, and Reis, Maria
- Abstract
In this paper we direct attention to 5-6-year-olds' learning of arithmetic skills through a thorough analysis of changes in the children's ways of encountering and experiencing numbers. The foundation for our approach is phenomenographic, in that our object of analysis is differences in children's ways of completing an arithmetic task, which are considered to be expressions of their ways of experiencing numbers and what is possible to do with numbers. A qualitative analysis of 103 children's ways of encountering the task gives an outcome space of varying ways of experiencing numbers. This is further analyzed through the lens of variation theory of learning, explaining why differences occur and how observed changes over a prolonged period of time can shed light on how children learn the meaning of numbers, allowing them to solve arithmetic problems. The results show how observed changes are liberating new and powerful problem-solving strategies. Emanating from empirical research, the results of our study contribute to the theoretical understanding of young children's learning of arithmetic skills, taking the starting point in the child's lived experiences rather than cognitive processes. This approach to interpreting learning, we suggest, has pedagogical implications concerning what is fundamental to teach children for their further development in mathematics.
- Published
- 2022
34. A Complicated Relationship: Examining the Relationship between Flexible Strategy Use and Accuracy
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Garcia Coppersmith, Jeannette and Star, Jon R.
- Abstract
This study explores student flexibility in mathematics by examining the relationship between accuracy and strategy use for solving arithmetic and algebra problems. Core to procedural flexibility is the ability to select and accurately execute the most appropriate strategy for a given problem. Yet the relationship between strategy selection and accurate execution is nuanced and poorly understood. In this paper, this relationship was examined in the context of an assessment where students were asked to complete the same problem twice using different approaches. In particular, we explored (a) the extent to which students were more accurate when selecting standard or better-than-standard strategies, (b) whether this accuracy-strategy use relationship differed depending on whether the student solved a problem for the first time or the second time, and (c) the extent to which students were more accurate when solving algebraic versus arithmetic problems. Our results indicate significant associations between accuracy and all of these aspects--we found differences in accuracy based on strategy, problem type, and a significant interaction effect between strategy and assessment part. These findings have important implications both for researchers investigating procedural flexibility as well as secondary mathematics educators who seek to promote this capacity among their students.
- Published
- 2022
35. Kindergarten children's symbolic number comparison skills predict 1st grade mathematics achievement: Evidence from a two-minute paper-and-pencil test
- Author
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Lisa M. D. Archibald, Daniel Ansari, Nadia Nosworthy, and Zachary Hawes
- Subjects
Arithmetic ,Number line estimation ,4. Education ,05 social sciences ,Neurosciences ,050301 education ,Sample (statistics) ,Non-symbolic number ,Variance (accounting) ,Pencil test ,050105 experimental psychology ,Arabic numerals ,Early mathematics achievement ,Education ,Task (project management) ,Number line ,Symbolic number ,Developmental and Educational Psychology ,Mathematics education ,Psychology ,0501 psychology and cognitive sciences ,The Symbolic ,0503 education ,Incremental validity - Abstract
© 2018 Elsevier Ltd Basic numerical skills provide an important foundation for the learning of mathematics. Thus, it is critical that researchers and educators have access to valid and reliable ways of assessing young children's numerical skills. The purpose of this study was to evaluate the concurrent, predictive, and incremental validity of a two-minute paper-and-pencil measure of children's symbolic (Arabic numerals) and non-symbolic (dot arrays) comparison skills. A sample of kindergarten children (Mage = 5.86, N = 439) were assessed on the measure along with a number line estimation task, a measure of arithmetic, and several control measures. Results indicated that performance on the symbolic comparison task explained unique variance in children's arithmetic performance in kindergarten. Longitudinal analyses demonstrated that both symbolic comparison and number line estimation in kindergarten were independent predictors of 1st grade mathematics achievement. However, only symbolic comparison remained a unique predictor once language skills and processing speed were taken into account. These results suggest that a two-minute paper-and-pencil measure of children's symbolic number comparison is a reliable predictor of children's early mathematics performance.
- Published
- 2019
36. Teacher Knowledge and Learning 'In-situ': A Case Study of the Long Division Algorithm
- Author
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Takker, Shikha and Subramaniam, K.
- Abstract
The aim of the study reported in this paper was to explore and enhance experienced school mathematics teachers' knowledge of students'thinking, as it is manifested in practice. Data were collected from records of classroom observations, interviews with participating teachers, and weekly teacher-researcher meetings organized in the school. In this paper, we discuss the mathematical challenges faced by a primary school teacher as she attempts to unpack the structure of the division algorithm, while teaching in a Grade 4 classroom. Through this case study, we exemplify how a focus on mathematical knowledge for teaching 'in situ' helped in triggering a change in the teacher's well-formed knowledge and beliefs about the teaching and learning of the division algorithm, and related students' capabilities. We argue that in the context of educational reform, an analysis of knowledge demands placed on the teacher helps in understanding and supporting teachers' work.
- Published
- 2018
37. Relationships between Units Coordination and Subitizing
- Author
-
MacDonald, Beth L.
- Abstract
This proposal explores relationships between young children's unit development/coordination and young children's subitizing. In particular, this theoretical commentary considers students' degrees of abstraction, students' development of actions on units, and students' operations with units when subitizing. As a result of this commentary, this author offers questions regarding how subitizing may elicit actions on units and perceptual/figurative material in a different order. These questions indicate possible alternative means in which comprehensive operations with natural numbers may develop. Possible educational implications and future research around these questions are also discussed. [For the complete proceedings, see ED606531.]
- Published
- 2018
38. Zooming-in on Decimals
- Author
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Mathematics Education Research Group of Australasia, Gorman, Amelia, and Way, Jennifer
- Abstract
This study investigated how six Year Four (9-10 years old) students interacted with a dynamic (zoomable) digital number line to demonstrate knowledge and understandings of decimal fractions. Results from a task-based interview indicated that the zoomable number line is able to assist students in developing conceptual understandings of decimal density, place value and relative size. The app proved to be a powerful mathematical representation of decimal fractions because of its dynamic affordances that allowed the students to 'see' concepts that would otherwise be 'unseeable' when using a traditional static number line.
- Published
- 2018
39. Information presentation through a head-worn display (“smart glasses”) has a smaller influence on the temporal structure of gait variability during dual-task gait compared to handheld displays (paper-based system and smartphone).
- Author
-
Sedighi, Alireza, Ulman, Sophia M., and Nussbaum, Maury A.
- Subjects
- *
GAIT in humans , *MOTOR ability , *PHYSICAL activity , *STANDARD deviations , *ENTROPY - Abstract
The need to complete multiple tasks concurrently is a common occurrence both daily life and in occupational activities, which can often include simultaneous cognitive and physical demands. As one example, there is increasing availability of head-worn display technologies that can be employed when a user is mobile (e.g., while walking). This new method of information presentation may, however, introduce risks of adverse outcomes such as a decrement to gait performance. The goal of this study was thus to quantify the effects of a head-worn display (i.e., smart glasses) on motor variability during gait and to compare these effects with those of other common information displays (i.e., smartphone and paper-based system). Twenty participants completed four walking conditions, as a single task and in three dual-task conditions (three information displays). In the dual-task conditions, the information display was used to present several cognitive tasks. Three different measures were used to quantify variability in gait parameters for each walking condition (using the cycle-to-cycle standard deviation, sample entropy, and the “goal-equivalent manifold” approach). Our results indicated that participants used less adaptable gait strategies in dual-task walking using the paper-based system and smartphone conditions compared with single-task walking. Gait performance, however, was less affected during dual-task walking with the smart glasses. We conclude that the risk of an adverse gait event (e.g., a fall) in head-down walking conditions (i.e., the paper-based system and smartphone conditions) were higher than in single-task walking, and that head-worn displays might help reduce the risk of such events during dual-task gait conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
40. Review Paper on High Speed Area Efficient Linear Convolution in Different Adder
- Author
-
Nashrah Fatima, Paresh Rawat, and Shubhi Shrivastava
- Subjects
021110 strategic, defence & security studies ,Adder ,Theoretical computer science ,law ,Computer science ,Transistor ,0211 other engineering and technologies ,Multiplier (economics) ,02 engineering and technology ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Arithmetic ,law.invention - Abstract
On this Technical era the excessive velocity and low area of VLSI chip are veryvery crucial elements. Each day quantity of transistors and different active and passive elements are drastically developing on a VLSI chip. All of the processors of the gadgets adders and multipliers are playing an essential position. An adder is a pleasing element for the designing of fast multiplier. Ultimately here want a fast adder for excessive bit edition. In this paper, they carried out of linear convolution are based on ripple carry adder and array multiplier. Offering common Boolean common sense (CBL) adder presents much less additives, less path delay and better pace compare to different present CBL adder and different adders. Right here, we're evaluating the linear convolution of differentextraordinary word length from different adders. The design and experiment may be executed by way of the useful resource of Xilinx 6.2i Spartan device circle of relatives.
- Published
- 2016
41. Review Paper on Discrete Cosine Transform using Different Types of Adder
- Author
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Paresh Rawat, Aarti Ranji, and Nashrah Fatima
- Subjects
Adder ,Theoretical computer science ,Computer science ,Hardware description language ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,02 engineering and technology ,Integrated circuit ,Inverse discrete cosine transform ,020202 computer hardware & architecture ,law.invention ,Computer Science::Hardware Architecture ,law ,Computer Science::Multimedia ,VHDL ,0202 electrical engineering, electronic engineering, information engineering ,Discrete cosine transform ,Lapped transform ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Arithmetic ,computer ,Transform coding ,computer.programming_language ,Data compression - Abstract
power design is one of the most significant challenges to maximize battery life time in portable devices and to save the energy during system process. Discrete Cosine Transform (DCT) is widely used in image and video compression process. Here in this paper, we review on low power Discrete Cosine Transform architecture by using varies methods. Discrete Cosine Transform (DCT) is most popular method used today in video compression systems. A number of algorithms have been proposed for implementation of the DCT. Loeffler (1989) has specified a new class of 1D-DCT using only 29 additions and 11 multiplications. To implement such an algorithm, one or more than one multipliers have to be integrated. This process requires a high occupation of silicon area. Arithmetic distribution is generally used for such algorithms. The coding for reconfigurable 8-point Discrete Cosine Transform (DCT) has been done using VHDL, under Xilinx platform. KeywordsCosine Transform (DCT), Inverse Discrete Cosine Transform (IDCT), Very High Speed Integrated Circuit Hardware Description Language (VHDL)
- Published
- 2016
42. Predicting Mathematics Achievement from Subdomains of Early Number Competence: Differences by Grade and Achievement Level
- Author
-
Devlin, Brianna, Jordan, Nancy, and Klein, Alice
- Abstract
This study investigated the relative importance of three subdomains of early number competence (number, number relations, and number operations) in predicting later mathematics achievement in cross-sequential samples of pre-K, kindergarten (K), and first graders (N=150 at each grade). OLS-regression analyses showed that each subdomain predicted mathematics achievement at each grade level, controlling for the other two subdomains as well as background variables. All of the subdomains explained a significant amount of variance in later mathematics achievement. Unconditional quantile regression analyses examined relations between number competencies and mathematics achievement at quantiles representing low (0.2), intermediate (0.5) and high (0.8) achievement. The subdomain of number operations was highly related to mathematics achievement for high achievers. For low achievers, number and number relations abilities were most highly related to future mathematics achievement in the pre-K sample, and number relations abilities in the K and first grade samples. Findings highlight the unique importance of all three subdomains of early number competence for later mathematics achievement, but show some of the relations are contingent upon achievement level. [The paper will be published in "Journal of Experimental Child Psychology."]
- Published
- 2021
43. Correcting a paper on the Randi{\'c} and geometric--arithmetic indices
- Author
-
Britto Antony Xavier, Suresh Elumalai, Mohammad Rostami, and Toufik Mansour
- Subjects
010304 chemical physics ,010201 computation theory & mathematics ,General Mathematics ,0103 physical sciences ,0102 computer and information sciences ,Arithmetic ,01 natural sciences ,Mathematics - Published
- 2017
44. Review Paper on Radix-2 DIT and DIF Fast Fourier Transform
- Author
-
Nashrah Fatima, Kausar Ali, and Paresh Rawat
- Subjects
Decimation ,Computer science ,Speech recognition ,Fast Fourier transform ,Radix ,Video processing ,Arithmetic ,Chip ,Field (computer science) ,Sign (mathematics) - Abstract
With the arrival of recent generation in the field communication, there is also an ever growing demand for low power consumption and minimum consumed It is also a well-known reality that the chip location and maximum combinational path delay (MCPD) unit forms an indispensable a part of processor layout. Due to this regard, high speed and low area architectures grow to be the want of the day. A fast Fourier transform (FFT) is any rapid algorithm for computing the DFT. The decimation (DIT) fast Fourier remodel (FFT) very often has advantage over the decimation-in-frequency (DIF) FFT for most real valued programs, like speech/photo/video processing, biomedical sign processing, and time-collection evaluation, and so on., since it does now not require any output reordering.
- Published
- 2016
45. The State of Primary School Third-Grade Pupils' Making Sense of the Concepts of '0' and '1'
- Author
-
Önal, Halil, Çekirdekci, Sitki, and Yorulmaz, Alper
- Abstract
This paper focuses on determining the opinions of primary school third-grade pupils about the conceptual meaning and use of the numbers "0" and "1". The current study employed the case study design, which is one of the qualitative research methods. In the selection of the sample, the criterion sampling method, one of the purposive sampling methods, was used. The study was conducted with the participation of a total of 114 third-grade pupils (58 girls and 56 boys) attending two state primary schools located in the city of Ankara in the fall term of the 2019-2020 school year. As the data collection tool, a semi-structured interview form developed by the researchers in relation to the concepts of "0" and "1" was used. In the analysis of the data obtained in relation to the concepts of "0" and "1", content analysis was used. The primary school third-grade pupils made sense of the number "0" in association with the categories of four operations, ineffective element, absorbing element, number, natural number, meaningless, valueless, absence, even number, beginning, and letter. The categories in which most thoughts about the meaning of the number "1" are gathered are number, ineffective element, and uniqueness. It was determined that the pupils are more successful in the use of the numbers "0" and "1" in the addition and subtraction operations than in the multiplication and division operations.
- Published
- 2021
46. Information presentation through a head-worn display ('smart glasses') has a smaller influence on the temporal structure of gait variability during dual-task gait compared to handheld displays (paper-based system and smartphone)
- Author
-
Alireza, Sedighi, Sophia M, Ulman, and Maury A, Nussbaum
- Subjects
Adult ,Male ,Adolescent ,Physiology ,Vision ,Cognitive Neuroscience ,Equipment ,Social Sciences ,Walking ,Young Adult ,Cognition ,Neuropsychology ,Ocular System ,Medicine and Health Sciences ,Humans ,Psychology ,Gait ,Neuropsychological Testing ,Communication Equipment ,Arithmetic ,Biological Locomotion ,Cognitive Neurology ,Cognitive Psychology ,Biology and Life Sciences ,Biomechanical Phenomena ,Neurology ,Stroop Test ,Physical Sciences ,Engineering and Technology ,Cognitive Science ,Eyes ,Female ,Sensory Perception ,Smartphone ,Cell Phones ,Anatomy ,Gait Analysis ,Head ,Psychomotor Performance ,Mathematics ,Research Article ,Neuroscience - Abstract
The need to complete multiple tasks concurrently is a common occurrence both daily life and in occupational activities, which can often include simultaneous cognitive and physical demands. As one example, there is increasing availability of head-worn display technologies that can be employed when a user is mobile (e.g., while walking). This new method of information presentation may, however, introduce risks of adverse outcomes such as a decrement to gait performance. The goal of this study was thus to quantify the effects of a head-worn display (i.e., smart glasses) on motor variability during gait and to compare these effects with those of other common information displays (i.e., smartphone and paper-based system). Twenty participants completed four walking conditions, as a single task and in three dual-task conditions (three information displays). In the dual-task conditions, the information display was used to present several cognitive tasks. Three different measures were used to quantify variability in gait parameters for each walking condition (using the cycle-to-cycle standard deviation, sample entropy, and the “goal-equivalent manifold” approach). Our results indicated that participants used less adaptable gait strategies in dual-task walking using the paper-based system and smartphone conditions compared with single-task walking. Gait performance, however, was less affected during dual-task walking with the smart glasses. We conclude that the risk of an adverse gait event (e.g., a fall) in head-down walking conditions (i.e., the paper-based system and smartphone conditions) were higher than in single-task walking, and that head-worn displays might help reduce the risk of such events during dual-task gait conditions.
- Published
- 2017
47. Integration of Computational Thinking with Mathematical Problem-Based Learning: Insights on Affordances for Learning
- Author
-
Cui, Zhihao, Ng, Oi-lam, and Jong, Morris Siu-Yung
- Abstract
Grounded in problem-based learning and with respect to four mathematics domains (arithmetic, random events and counting, number theory, and geometry), we designed a series of programming-based learning tasks for middle school students to co-develop computational thinking (CT) and corresponding mathematical thinking. Various CT concepts and practices articulating the designated mathematical problems were involved in the tasks. In addition to delineating the design of these learning tasks, this paper presents a qualitative study in which we examined 74 students' learning outcomes and characterized their CT and mathematical thinking co-development as they accomplished the tasks. The research results demonstrate the codevelopment of both mathematics- and CT-related concepts and practices in the four mathematics domains. Two types of interactions are identified: (i) applying mathematical knowledge to construct CT artifacts and (ii) generating new mathematical knowledge with CT practice. The new insights provided by the present work are threefold. First, from a mathematical learning perspective, the nature of the solution processes of the designed problems should not be immediately obvious. Second, from a technology-enhanced learning perspective, the dynamic representations and immediate visual feedback afforded by the programming tool are beneficial to student learning. Third, from a pedagogical perspective, the room for customization offered by both the designed problems and programming tools can provide affordances for learning.
- Published
- 2023
48. (IR)Reversability in Mathematics
- Author
-
Norton, Anderson
- Abstract
In this theoretical paper, I consider reversibility as a defining characteristic of mathematics. Inverse pairs of formalized operations, such as multiplication and division, provide obvious examples of this reversibility. However, there are exceptions, such as multiplying by 0. If we are to follow Piaget's lead in defining mathematics as the science of reversible mental actions, such exceptions must be examined. We consider the case of multiplying by 0 by adopting Davydov's model of multiplication as a transformation of units and by investigating the underlying mental actions. Results of this investigation have implications for breaking down the barriers between various domains of mathematics. [For the complete proceedings, see ED583608.]
- Published
- 2016
49. NOTES ON THE PAPER 'A CRITERION FOR BOUNDED FUNCTIONS' [BULL. KOREAN MATH. SOC. 53 (2016), NO. 1, 215-225]
- Author
-
Jin-Lin Liu and Yi-Ling Cang
- Subjects
Algebra ,General Mathematics ,Bounded function ,010102 general mathematics ,0101 mathematics ,Arithmetic ,01 natural sciences ,Mathematics - Published
- 2016
50. Contemporary Problems of Teaching and Learning in Mathematics Education
- Author
-
Iwuanyanwu, Paul Nnanyereugo
- Abstract
The way mathematics teaching and learning activities are presented to learners can make them hate or like the subject. The question of accomplishing the mathematics education of the learner from primary to post-secondary school levels is one which necessarily tasks, not only the teacher's stock of mathematical knowledge but also his skill, his method of approach and finally his handling of the processes of feedback mechanisms. This paper outlines some of the fundamental problems confronting mathematics teachers and learners, and by means of previously published work, it proposed some useful suggestions which have gained support through classroom implementation.
- Published
- 2021
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