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2. Tablets Instead of Paper-Based Tests for Young Children? Comparability between Paper and Tablet Versions of the Mathematical Heidelberger Rechen Test 1-4
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Hassler Hallstedt, Martin and Ghaderi, Ata
- Abstract
Tablets can be used to facilitate systematic testing of academic skills. Yet, when using validated paper tests on tablet, comparability between the mediums must be established. Comparability between a tablet and a paper version of a basic math skills test (HRT: Heidelberger Rechen Test 1-4) was investigated. Five samples with second and third grade students participated. The associations between the tablet and paper version of HRT showed that these modes of administration were comparable for three arithmetic scales, but unacceptable for a pictorial counting scale. Scores were lower on tablet. Test-retest reliability for arithmetic scales on tablet was satisfactory, but was inferior for a low-performing sample. The overall convergent validity was satisfactory. No effect of test administrator was found. Arithmetic scales can potentially be transferred to tablet with good comparability and maintained test-retest reliability. Precautions are necessary when transferring pictorial scales into tablet. Separate norms for tablet are needed when interpreting scores.
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- 2018
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3. Rupture or Continuity: The Arithmetico-Algebraic Thinking as an Alternative in a Modelling Process in a Paper and Pencil and Technology Environment
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Hitt, Fernando, Saboya, Mireille, and Zavala, Carlos Cortés
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Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers.
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- 2017
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4. Comparing the Use of the Interpersonal Computer, Personal Computer and Pen-and-Paper When Solving Arithmetic Exercises
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Alcoholado, Cristián, Diaz, Anita, and Tagle, Arturo
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This study aims to understand the differences in student learning outcomes and classroom behaviour when using the interpersonal computer, personal computer and pen-and-paper to solve arithmetic exercises. In this multi-session experiment, third grade students working on arithmetic exercises from various curricular units were divided into three groups. The first group used personal computers (netbooks), the second group used an interpersonal computer (ie, one projector with a screen, one computer and one mouse per child) and the third group used pen-and-paper. The results of the experiment indicate that all three groups achieved an increase in learning, as shown by the pretest and posttest scores. No significant difference was found between the interpersonal computer and personal computer groups. This suggests that the key characteristic shared by the two groups is the provision of feedback. The format that such feedback takes, either private (through a personal screen) or public (through a shared screen), is shown to make no difference. However, the results significantly favour groups that are provided with instant feedback (interpersonal computer and personal computer) as opposed to delayed feedback (pen-and-paper).
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- 2016
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5. Visualizing Math: How Number Lines Can Empower Problem-Solving
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Tiffany Berman and Casey Hord
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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.
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- 2024
6. Extendible functions and local root numbers: Remarks on a paper of R. P. Langlands.
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Koch, Helmut and Zink, Ernst‐Wilhelm
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SOLVABLE groups , *SET functions , *ARITHMETIC , *L-functions , *ARTIN algebras , *PROFINITE groups - Abstract
This paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number Δ=Δ(E,χ)$\Delta =\Delta (E,\chi)$, where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of E×$E^\times$, can be appropriately extended to a local ε‐factor εΔ=εΔ(E,ρ)$\varepsilon _\Delta =\varepsilon _\Delta (E,\rho)$ for all virtual representations ρ of the corresponding Weil group WE$W_E$. Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of εΔ$\varepsilon _\Delta$ will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role. [ABSTRACT FROM AUTHOR]
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- 2024
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7. Mismatch of the South African Foundation Phase Curriculum Demands and Learners' Current Knowledge
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Fritz, Annemarie, Long, Caroline, Herzog, Moritz, Balzer, Lars, Ehlert, Antje, and Henning, Elizabeth
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Against the background of the low mathematical performance of South African learners in international panel studies, there is an urgent need to improve mathematical education. In particular, the curriculum and its structure raise questions. It is logical that the prescribed curricula should align with learners' developmental trajectories. Given the hierarchical structure of mathematics, the curricular requirements should pay attention to learners' current knowledge of mathematical concepts. The aim of this study was to compare the curricular requirements as defined by the CAPS with the conceptual current knowledge of South African learners. South African Grade 1 learners (N = 602) were assessed on a test of numeracy concepts, based on a theoretically informed and empirically validated model of developing mathematical proficiency. The content of the CAPS for Grade 1 was aligned to the model levels by two experts (Cohen's [kappa] = 0.753, p < 0.001). Results show that the curricular requirements go far beyond the current knowledge required to engage with these new concepts of the vast majority of South African Grade 1 learners. The mismatch may to some extent be responsible for the unsatisfactory results in international comparison studies. These results show that the intended curriculum is beyond the grasp of most South African Grade 1 learners. These children are unlikely to develop new arithmetic concepts based on their lack of required foundation knowledge. We therefore argue that the intended curriculum for Grade 1 should focus more on counting skills, ordinal relations between numbers and--most importantly--set-based number representations and part-part-whole relations.
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- 2020
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8. Computing Science: Pencil, Paper, and Pi
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Hayes, Brian
- Published
- 2014
9. The Prime Number Theorem as a Mapping between Two Mathematical Worlds
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Norton, Anderson and Flanagan, Kyle
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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.]
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- 2022
10. 'Leave No One Behind': A Systematic Literature Review on Game-Based Learning Courseware for Preschool Children with Learning Disabilities
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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.
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- 2023
11. 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
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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.
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- 2023
12. 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.]
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- 2023
13. Effects of an Adaptive Math Learning Program on Students' Competencies, Self-Concept, and Anxiety
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Hilz, Anna and Aldrup, Karen
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Studies on math learning programs are lacking that consider a wide set of outcome variables, and students' practice behavior. Therefore, we investigated whether an adaptive arithmetic learning program fosters students' math performance (addition and subtraction), math self-concept, and a reduction of math anxiety, and how practice behavior (practiced tasks and practiced weeks) affect the investigated variables. We used a pre-post control group design with a total of 366 fifth grade German students. Randomization took place on the class level. Students in the experimental condition used the program for 22 weeks. Math self-concept only improved in the experimental group. Students' subtraction performance improved as a function of practiced tasks, and addition performance improved as a function of practiced weeks.
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- 2023
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14. 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
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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
15. Construction of Arithmetic-Algebraic Thinking in a Socio-Cultural Instructional Approach = Construction d'une pensée arithémico-algébrique dans une approche socioculturelle de l'enseignement
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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
16. The Evolution from 'I Think It plus Three' towards 'I Think It Is Always plus Three.' Transition from Arithmetic Generalization to Algebraic Generalization
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María D. Torres, Antonio Moreno, Rodolfo Vergel, and María C. Cañadas
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This paper is part of broader research being conducted in the area of algebraic thinking in primary education. Our general research objective was to identify and describe generalization of a 2nd grade student (aged 7-8). Specifically, we focused on the transition from arithmetic to algebraic generalization. The notion of structure and its continuity in the generalization process are important for this transition. We are presenting a case study with a semi-structured interview where we proposed a task of contextualized generalization involving the function y = x + 3. Special attention was given to the structures evidenced and the type of generalization expressed by the student in the process. We noted that the student identified the correct structure for the task during the interview and that he evidenced a factual type of algebraic generalization. Due to the student's identification of the appropriate structure and the application of it to other different particular cases, we have observed a transition from arithmetic thinking to algebraic thinking.
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- 2024
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17. Justifications Students Use when Writing an Equation during a Modeling Task
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Roan, Elizabeth and Czocher, Jennifer
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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
18. 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.
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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.]
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- 2022
19. Students' Mental Model in Solving the Patterns of Generalization Problem
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Prayekti, N., Nusantara, T., Sudirman, and Susanto, H.
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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.]
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- 2019
20. 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.
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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
21. Teaching Redundant Residue Number System for Electronics and Computer Students
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Somayeh Timarchi
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This paper describes a study for teaching number system in a computer arithmetic course and addresses the existing gap in the current course by focusing on the characteristics of applications. A redundant residue number system (RRNS) is an efficient and innovative number system that inherits the interesting properties of both residue number system (RNS) and redundant number systems. It breaks the operands into residues by considering a moduli-set and then signifies the residues by a redundant representation. So, by limiting the carry propagation chain inside and outside the moduli, RRNS proposes more efficient arithmetic units which could be employed in digital signal processing (DSP) applications. Despite the applicability of RRNS, there is not well-organized teaching on RRNS covering the recent achievements. Besides, there is an important question for students and researchers what is the most appropriate number representation for each application category? In this paper, we present a step-by-step education process for RRNS that includes basic concepts of RNS and the redundant number system. Then we address characteristics of different applications that RRNS is suitable to be employed for them. So, the method gives a well-organized proceeding to computer arithmetic designers and students.
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- 2023
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22. Productive Seeds in Preservice Teachers' Reasoning about Fraction Comparisons
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Whitacre, Ian, Findley, Kelly, and Atabas, Sebnem
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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.]
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- 2020
23. 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
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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
24. Measuring Mental Computational Fluency with Addition: A Novel Approach
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Mathematics Education Research Group of Australasia, Russo, James, and Hopkins, Sarah
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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
25. Rote versus Rule: Revisiting the Role of Language in Mathematical Thinking
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Qin, Jike and Opfer, John
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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
26. Are Approximate Number System Representations Numerical?
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Pickering, Jayne, Adelman, James S., and Inglis, Matthew
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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
27. 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.
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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
28. Towards a Critical Mathematics
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Savich, Theodore Michael
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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
29. 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
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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
30. A Conceptual Analysis of the Equal Sign and Equation -- The Transformative Component
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Ying, Yufeng
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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
31. Investigating the Variety and Usualness of Correct Solution Procedures of Mathematical Word Problems
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Samková, Libuše
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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
32. 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
33. 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
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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.
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- 2017
34. Hypothesis of Developmental Dyscalculia and Down Syndrome: Implications for Mathematics Education
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Mathematics Education Research Group of Australasia and Faragher, Rhonda
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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.
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- 2017
35. (IR)Reversability in Mathematics
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Norton, Anderson
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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.]
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- 2016
36. Unfolding Algebraic Thinking from a Cognitive Perspective
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Chimoni, Maria, Pitta-Pantazi, Demetra, and Christou, Constantinos
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Little is known about the cognitive effort associated with algebraic activity in the elementary and middle school grades. However, this investigation is significant for sensitizing teachers and researchers to the mental demands of algebra learning. In this paper, we focus on the relationship between algebraic thinking and domain-general cognitive abilities. The sample of the study comprised 591 students from grades 4 to 7. The students' abilities in algebraic thinking were assessed through a test that involved three task categories: generalized arithmetic, functional thinking, and modelling languages. Test batteries were used to assess students' domain-general cognitive abilities in terms of analogical, serial, spatial, and deductive reasoning. The results of structural equation modelling analysis indicated that: (1) analogical reasoning predicts students' abilities only in generalized arithmetic; (2) serial reasoning predicts students' abilities only in generalized arithmetic; (3) spatial reasoning predicts students' abilities in functional thinking and modelling languages; and (4) deductive reasoning predicts students' abilities in all three categories. Differences between students across grades are also discussed.
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- 2023
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37. The Role of Working Memory Updating and Capacity in Children's Mathematical Abilities: A Developmental Cascade Model
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Hu, Qiong, Liang, Zhanhong, Zhou, Yanlin, Feng, Shanshan, and Zhang, Qiong
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Background: Previous studies indicated that working memory (WM) updating and WM capacity play essential roles in mathematical ability. However, it is unclear whether WM capacity mediates the effect of WM updating on mathematics, and whether the cascading effects vary with different mathematical domains. Aims: The current study aims to explore the longitudinal mediating role of WM capacity between WM updating and mathematical performance, and how the relations change with the age and domains. Sample: A total of 131 Chinese first-graders participated the study. Methods: Participants were required to complete tasks on WM updating and WM capacity in Grade 1 and Grade 2, as well as paper-and-pencil tests on mathematics achievement in Grade 3. The role of WM updating and capacity in the development of pupil's mathematical achievement was examined. Results: Results revealed that verbal WM updating in Grade 1 predicted basic arithmetic and logical-visuospatial ability in Grade 3 via its cascading effect on verbal WM capacity in Grade 2. Moreover, visuospatial WM updating in Grade 1 predicted visuospatial WM capacity in Grade 2. Visuospatial WM capacity in Grade 1 predicted logical-visuospatial ability in Grade 3 instead of basic arithmetic ability in Grade 3. Conclusions: The findings suggested that WM updating exerts effect on pupil's mathematical performance via WM capacity, meanwhile, this effect depends on children's mathematics domain.
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- 2023
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38. How Do Third Grade Students Read the Clock?
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Gürefe, Nejla and Özdil, Seyma
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Students require knowledge of fraction, arithmetic and integer to tell the time. The purpose of the current study is to discover the conceptual knowledge related to hours, how the students read the clock and use half and quarter concepts to read the time in Turkey. A qualitative research method with a case study approach was used for this research. The participants of the study consisted of 12 students studying at the third grade of a public school located in the west of Turkey. The data of the study were collected through semi-structured interviews. A form consisted nine questions was used in the interviews. The data were then analysed using content analysis. As the result of analysis, it was determined that the students used fraction knowledge, number of minute and absolute time while telling the time, but mostly for fraction knowledge and some students could not associate the knowledge in the subject of the fractions with the reading the clock. [This study was presented as an oral presentation at International Congress on Science and Education (ICSE-2018), Turkey.]
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- 2019
39. The Strategy the Use of False Assumption and Word Problem Solving
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Eisenmann, Petr, Pribyl, Jirí, and Novotná, Jarmila
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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.
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- 2019
40. Kindergarten children's symbolic number comparison skills relates to 1st grade mathematics achievement: Evidence from a two-minute paper-and-pencil test.
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Hawes, Zachary, Archibald, Lisa, Ansari, Daniel, and Nosworthy, Nadia
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MATHEMATICS , *MATHEMATICAL ability in children , *MATHEMATICAL ability testing , *LONGITUDINAL method , *SYMBOLISM of numbers , *ARITHMETIC , *KINDERGARTEN , *CHILDREN - Abstract
Abstract 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 (M age = 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. Highlights • We tested the validity and reliability of a recently developed Numeracy Screener. • Performance on the symbolic comparison task predicted kindergarten children's arithmetic performance. • Performance on the symbolic comparison task in kindergarten predicted school mathematics achievement one year later. • The Numeracy Screener may be a useful tool for early mathematics researchers and educators alike. [ABSTRACT FROM AUTHOR]
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- 2019
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41. The Teachers' Views on Soroban Abacus Training
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Altiparmak, Kemal
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Soroban abacus training is called as mental arithmetic training in our country. It is known for mental arithmetic to increase the ability of four mode operations. Besides this, how is the situation for the students which are having Soroban abacus training in the terms of problem solving abilities, creativity, development of concepts, attraction towards mathematics, success in other lessons and their social relations with their peers? In order to find an answer to this question, this work studies the answers provided from the teachers of 14 students those are having mental arithmetic training which are aged between 7 and 12. In order to reach these students, the snow ball method which is a kind of improbable sampling methods had been used. In this study for the students which are having Soroban abacus training are observed for the effect of Soroban abacus training for their problem solving abilities, creativity levels, understanding of the concepts, their attraction towards mathematics lesson, their success status for the lessons other than mathematics and their social relations with their peers. For the areas mentioned above, the answers and comments from mathematics and form teachers of the 14 students that are having Soroban Abacus training are collected. Descriptive analysis had been conducted for the results obtained from these interviews. As a result, relying on the reviews and comments obtained from the teachers, it can be said that the students which are having Soroban abacus training are performing better than their peers in terms of the areas mentioned above. [Paper presented at the International Conference on Education in Mathematics, Science & Technology (ICEMST) (2015).]
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- 2016
42. 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
43. Teacher Actions to Facilitate Early Algebraic Reasoning
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Mathematics Education Research Group of Australasia and Hunter, Jodie
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In recent years there has been an increased emphasis on integrating the teaching of arithmetic and algebra in primary school classrooms. This requires teachers to develop links between arithmetic and algebra and use pedagogical actions that facilitate algebraic reasoning. Drawing on findings from a classroom-based study, this paper provides an exemplar of one teacher's journey in shifting her practice to integrate early algebra into her everyday mathematics lessons. The findings highlight the importance of addressing different areas including algebraic content, task development and enactment, and the classroom and mathematical practices to facilitate algebraic reasoning.
- Published
- 2015
44. 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.
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- 2015
45. 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
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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.
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- 2022
46. Preschoolers' Ways of Experiencing Numbers
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Björklund, Camilla, Ekdahl, Anna-Lena, Kullberg, Angelika, and Reis, Maria
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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.
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- 2022
47. 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.
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- 2022
48. Predicting Mathematics Achievement from Subdomains of Early Number Competence: Differences by Grade and Achievement Level
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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
49. Investigating the Role of Shared Screen in a Computer-Supported Classroom in Learning
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Shaikh, Rafikh Rashid, G., Nagarjuna, and Gupta, Ayush
- Abstract
Networked computers can potentially support classrooms to be more interactive. It can help students share representations amongst themselves and work together on a shared virtual activity space. In research on the role of shared screens or shared virtual workspace in learning settings, there has been less attention paid to contexts where learners are co-located. This paper looks at the impact of the shared screen in a computational game environment on mathematics learning and practices and the construction of learners' emotions and social status in classroom interactions. We designed two versions of a simple arithmetic game: a solo version in which the student played the game alone and a multi-player version in which the screen was shared, and the players could see the arithmetic moves of the other players. We implemented these two versions of the game in a 4th-grade classroom in a suburban school in a large metropolis in India. Classroom sessions were video recorded, computer logs were collected, and field notes were taken. Focus group sessions were held with the students. We coded a portion of the data to get at patterns of classroom interactions. Then we drew on qualitative video analysis tools to analyze specific episodes to understand the fine timescale dynamics of dominant interaction patterns in each setting. Our analysis shows that the shared screen served as a shared memory device, keeping a record of all the students' posts, and was entangled in the moment-to-moment dynamics of self- and peer- assessments of arithmetic. These findings suggest that thoughtful integration of networked digital tools in computer-supported learning environments can increase student-student interactions and support disciplinary learning.
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- 2023
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50. Teacher Knowledge and Learning 'In-situ': A Case Study of the Long Division Algorithm
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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
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