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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.

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.

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.

It is well documented in literature that rational number is an important area of understanding in mathematics. Therefore, it follows that teachers and students need to have an understanding of rational number and related concepts such as fraction multiplication and division. This practitioner reference paper examines models that are important to elementary and middle school teachers and students in the learning and understanding of fraction multiplication and division.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

Mental brackets constitute an idiosyncratic use of brackets sometimes used to evaluate arithmetic expressions and are closely connected with students' structure sense. The relevant literature describes the use of mental brackets focusing on primary school students and in the context of arithmetic. Using 181 high school students' solutions to seven tasks, this paper attempts to broaden the scope of mental brackets. It investigates high school students' mental brackets use and their presence in different mathematical domains. The findings show that high school students use mental brackets in the context of arithmetic while evaluating arithmetic expressions, and in the context of algebra, mainly when substituting numerical values in the variables. Moreover, the analysis provides further insights into the use of mental brackets. Sometimes they are applied to the whole expression (global mental brackets) and other times just to parts of it (local mental brackets).

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.

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.]

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.

This paper resulted from an unfolding of an ongoing doctoral research that deals with the characterization of a specific type of arithmetic in times of Brazilian industrial education, that is, between 1942 and 1968. The objective is to analyze knowledge produced by Roberto Mange for training, aiming not only to recognize them as such, but also to situate the social place of the producing agent, Roberto Mange, an "expert" in industrial education. The concepts mobilized are in line with the studies by Hofstetter, Schneuwly and Freymond (2017), and the methodological guidelines of Morais (2020). The research reveals that Roberto Mange introduced in Brazil new ways of teaching Mathematics, using the Occupational Methodological Series, the Instruction Sheets, and the Individualized Instruction Method. The conclusion is that Roberto Mange is an "expert" in industrial education.

The teaching guides that complement textbooks have key importance in the assessment of competence in problem solving, because these materials contain the assessment tools that teachers frequently use to quantify the achievements of their students. In this paper, we set two aims: to analyze which curriculum contents are given priority in the assessment tests of the teaching guides; and to check to what extent these tests assess the steps of the mathematical problem solving process. For this, an analysis of the initial and final assessment tests of six Spanish publishers was conducted. The results show that the distribution of mathematical tasks by type of content does not fully conform to the theoretical framework proposed by TIMSS. In addition, only one of the six publishers considered the problem-solving process as evaluable.

This paper aims to characterise an indicator of the development of the number sequence scheme among students at the level of Compulsory Secondary Education (14-16 years old students). To do so, we use a scheme development proposed by the APOS theory to characterise students' use of relations between mathematical elements when solving a mathematical task. We use a qualitative methodology and the data collection instruments are a written questionnaire and a semi-structured interview. In this work we show the questionnaire task that provides analytical expressions and ask students to determine which of them numbers sequences are. We find that students' use of logical implication when solving tasks related to number sequences is an indicator of the development of the scheme. This indicator helps to locate the transition mechanisms between the levels of development of the number sequence scheme. Moreover, our research shows that arithmetic and geometric progressions play a key role as an indicator of the development of the number sequence scheme.