Number Sentences Research Articles

Exploring Pre-Service Teachers' Understanding of Join and Separate Word Problems in the Foundation Phase (Grades 1-3).

This study explores pre-service teachers’ understanding of different types of join and separate word problems, focusing on their ability to construct word problems from given number sentences. It is a descriptive qualitative study. The participants were second-year pre-service foundation phase teachers taking mathematics learning courses at a public university. Data were collected through number sentence tasks and analysed using a qualitative content analysis procedure. The findings of this study revealed that pre-service teachers demonstrated challenges in correctly formulating word problems from change-unknown and start-unknown scenarios. The findings further reveal that pre-service teachers conflate additive and subtractive reasoning. The findings suggest that pre-service teachers rely on rote memory rather than a deeper understanding of mathematical relations. In addition, the study revealed common misconceptions, particularly in framing “change-unknown” and “result-unknown” problems, where pre-service teachers struggled to model real-life contexts accurately. Findings suggest the need for teacher education programs to focus more explicitly on developing a deep understanding of word problem structure. Initial teacher education programs should include word problem formulation and problem-solving tasks in all five mathematics learning areas, thereby underlining the importance of this research

Flexibility across and flexibility within: The domain of integer addition and subtraction

To better understand the role that flexibility plays in students’ success on integer addition and subtraction problems, we examined students’ flexibility when solving open number sentences. We define flexibility as the degree to which a learner uses more than one strategy to solve a single task when prompted, as well as the degree to which a learner changes strategies when solving a range of tasks to accommodate task differences. We introduce the categorizations of flexibility within and flexibility across to distinguish these two ways of operationalizing flexibility. We examined flexibility and performance within and among three groups of students — 2nd and 4th graders who had negative numbers in their numerical domains, 7th graders, and college-track 11th graders. Profiles of five students are shared to provide insight in relation to the quantitative findings.

Thinking about the equal sign: What do students see about the equal sign?

The equal sign seems to be interpreted differently by students depending on the learning experiences they have had in the early grades. In fact, the interpretation of the equal sign as a relational symbol does not seem easy or fast for many students to understand. This study aims to explore how students from elementary school to college students describe their understanding of the equal sign. The Qualitative Comparative Analysis can be used to analyze several cases in complex situations so that it fits the purpose of this study. The process of collecting data through the method of written assignments, semi-structured interviews, and documentation was carried out in one time period. This study involved 30 participants in Bandung, Indonesia. The results show that although there are substantial differences in viewing the equal sign, there are similarities in terms of dependence on computational methods in drawing conclusions. This is related to how their experience of number sentences in lower grades places more emphasis on rules than on the meaning of concepts.

Middle School Students’ Understanding of Equal Sign

The present study investigates the understanding of the equals sign among middle school students. The study data were gathered from a total of 433 middle school students (111 in the fifth grade, 106 in the sixth grade, 95 in the seventh grade and 121 in the eighth grade) using a ten-item equals sign test developed by the researcher. The study findings suggest that most students have an operational understanding of the equal sign. The students were most successful in answering true/false number sentences and successful lowest in equivalent equations. It was observed that those who applied a relational strategy to the solving of problems understood the underlying structure of the equation, while those who used an operational strategy were observed to make such errors as “the answer comes next” and “extend the problem”.

Relational Thinking And Problem Solution Strategies Beginning Algebra High School Students

Many students' difficulties in learning early algebra are associated with deficiencies in learning during the student's transition from arithmetic to algebra. relational thinking; a way of thinking that can be a gateway to algebraic thinking, is touted as one of the solutions to these difficulties. This study aims to determine the profile of relational thinking and early algebra problem solving strategies for junior high school students. Through the phenomenological design, the researcher selected six participants who fit the research criteria. The data analyzed refers to several theoretical frameworks on relational thinking from several experts. This study focuses on the indicators of interpreting the equals sign , using the relationship between numbers and applying the basic properties of number operations, as well as the methods students use in solving initial algebraic problems (algebraic forms and linear equations of one variable). As a result and its implication, students' relational thinking is seen in several items, namely the first 3 indicator items: each item is seen in S1, S4, and S6; The second 2 indicator items are visible on the S4; and the third 3 indicator items are seen in S1. The majority of subjects do computing dominantly and or apply the nature of operations based on procedural knowledge. Similarly, in solving early algebra problems, it was found that the dominant tendency of students to consider algebraic expressions from a procedural perspective rather than a structural perspective (based on mathematical concepts or structures). From these results, it is deemed necessary to develop relational thinking since students study arithmetic in elementary school, including by familiarizing students with number sentences in various forms and encouraging understanding of mathematical structures (relationships between elements and understanding of the nature of number operations).

Relational Thinking And Problem Solution Strategies Beginning Algebra High School Students

Many students' difficulties in learning early algebra are associated with deficiencies in learning during the student's transition from arithmetic to algebra. relational thinking; a way of thinking that can be a gateway to algebraic thinking, is touted as one of the solutions to these difficulties. This study aims to determine the profile of relational thinking and early algebra problem solving strategies for junior high school students. Through the phenomenological design, the researcher selected six participants who fit the research criteria. The data analyzed refers to several theoretical frameworks on relational thinking from several experts. This study focuses on the indicators of interpreting the equals sign , using the relationship between numbers and applying the basic properties of number operations, as well as the methods students use in solving initial algebraic problems (algebraic forms and linear equations of one variable). As a result and its implication, students' relational thinking is seen in several items, namely the first 3 indicator items: each item is seen in S1, S4, and S6; The second 2 indicator items are visible on the S4; and the third 3 indicator items are seen in S1. The majority of subjects do computing dominantly and or apply the nature of operations based on procedural knowledge. Similarly, in solving early algebra problems, it was found that the dominant tendency of students to consider algebraic expressions from a procedural perspective rather than a structural perspective (based on mathematical concepts or structures). From these results, it is deemed necessary to develop relational thinking since students study arithmetic in elementary school, including by familiarizing students with number sentences in various forms and encouraging understanding of mathematical structures (relationships between elements and understanding of the nature of number operations).

How Not to Analyse Number Sentences

ABSTRACT Number and Count Sentences like ‘The number of Martian moons is two’ and ‘Mars has two moons’ give rise to a puzzle. How can they be equivalent if only the truth of Number but not that of Count Sentences requires the existence of numbers? Proponents of Linguistic Deflationism seek to resolve this puzzle by arguing that on their correct linguistic analysis the truth of Number Sentences does not require the existence of numbers. In this paper, I argue that Katharina Felka’s recent attempt to vindicate this strategy by analysing Number Sentences as so-called specificational sentences is philosophically dissatisfying and sketch a promising alternative.

Learner-generated drawings by students with mathematical learning difficulties in finishing open number sentences

Although MLD students do not have good mathematical performance in completing addition and subtraction operations of integers, MLD students have suggestive ideas in the form of drawings produced in solving open number sentences questions. This study aims to classify the types and identify changes in the drawing produced by MLD students in solving open number sentences questions. This research method is qualitative with a micro generic study approach to understand students' thinking individually and explore drawing changes in solving open number sentences questions between sessions. The research subjects were 2 out of 20 MLD grade 5 elementary school students who produced the most varied drawings in solving open number sentences questions. Data collection techniques used are giving questions and interviews. The results showed that MLD students produced: discrete object drawings by focusing on the cardinality of the quantity of a number; transitions from objects to the number line by focusing on the magnitude of numbers; partitioning the number line using magnitude reasoning; number sentences; and others using verbal reasoning. Changes in the drawings produced by MLD students between sessions indicate the development of students' understanding towards a better direction in interpreting symbolic representations to visual representations. The results of this study contribute to the theory that although MLD students have low mathematical performance. However, MLD students can produce variations and changes in drawings with rich mathematical idea information representing integer operations.

The Value of Incorporating Indigenous Games in the Teaching of Number Sentences and Geometric Patterns

The Value of Incorporating Indigenous Games in the Teaching of Number Sentences and Geometric Patterns

The dimensions of prospective elementary and middle school teachers’ problem posing for integer addition and subtraction

Elementary and middle school prospective teachers participated in semi-structured problem posing for integer addition and subtraction. The prospective teachers (n = 98) posed a variety of different stories, but this paper focuses on the temperature stories they posed. Results include descriptions of their posed temperature stories through the lens of the various dimensions (i.e., problem types, realism, consistency, correctness). Prospective teachers posed mainly state-translation-state problem types and rarely posed state-state-distance, state-state-translation, or translation-translation-translation problem types. They often changed the structure of their number sentences. Although they posed mostly realistic and mathematically correct temperature stories, the stories compromised realism or consistency in order to use state-translation-state problem types. Coordinating of the various dimensions (e.g., problem types, consistency) when problem posing requires flexibility with problem types. This work highlights the complexity of posing temperature stories, and coordinating the various dimensions highlights the need for prospective teachers to experience problem posing. Implications for problem posing with integers and temperature are extended to all contexts that inherently support translation and relativity. In the discussion, we coordinate the different problem types with various number sentences and dimensions. Unpacking the various dimensions illuminates prospective teachers’ thinking and offers a way of considering integers and contexts.

The role of balance scales in supporting productive thinking about equations among diverse learners

ABSTRACT This research focuses on ways in which balance scales mediate students’ relational understandings of the equal sign. Participants included 21 Kindergarten–Grade 2 students who took part in an early algebra classroom intervention focused in part on developing a relational understanding of the equal sign through the use of balance scales. Students participated in pre-, mid- and post-intervention interviews in which they were asked to evaluate true-false equations and solve open number sentences. Students often worked with balance scales while solving these tasks. Interview analyses revealed several categories of affordances of these tools for supporting students’ productive thinking about equations.

Development of Secondary School Students’ Relational Thinking Skills with a Teaching Experiment

Purpose: Since there are a limited number of studies on how to develop relational thinking in secondary school students in mathematics education literature, this study will contribute to the field both in theoretical terms and concerning the implications for in-class applications. In this respect, this study aims to examine how to develop the relational thinking skills of 5th-grade students. Research Methods: The participants of this study, which was adopted as a research design of the teaching experiment, were six students attending 5th grade in secondary school. The teaching process was eight sessions per week with one session. The main data source of this study was in-class teaching videos. The data were analyzed descriptively. Findings: The questions, the in-class dialogues directing relational thinking and activities in each session of the teaching experiment conducted with the fifth-grade students were presented under related themes. Implications for Research and Practice: The most general result was that at the end of the teaching process based on numbers, relationships between numbers, operations and properties, the students made use of equality axioms to evaluate the true/false and open number sentences without any calculation. It was also seen that the students made connections between addition-subtraction, addition-multiplication and multiplication-division and that they made effective use of commutative, associative and distributive properties.

Conceptual models for integer addition and subtraction

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories – the conceptual models for integer addition and subtraction. These four conceptual models for thinking about and using integer addition and subtraction include Bookkeeping, Counterbalance, Relativity, and Translation, and are generated from the students’ posed stories. We also provide profiles of conceptual model use for two of the 6 students that describe how the students posed stories to accommodate conceptual model use, such as posing unconventional or unrealistic stories or changing the structure of the number sentences. The conceptual models and descriptions of how the students used them provide perspective into student thinking about integers and contexts, highlighting the mathematics of the students, and calling for a re-examination of contexts used in school mathematics.

Integer Numbers and Temperature Problems

When solving temperature problems with negative integers, number sentences may be mathematically equivalent but not contextually equivalent.

Grade 5 children’s drawings for integer addition and subtraction open number sentences

Three Grade 5 children participated in a microgenetic study embedded in 12-week teaching experiment on integer addition and subtraction. They solved open number sentences in four individual sessions across the 12-weeks and produced drawings. Through the lens of learner-generated drawings and qualitative analysis, these drawings provide perspective into the children’s thinking about integer addition and subtraction. The following categories are described: Single and Double Set of Objects, Number Sequences, Empty Number Lines, Number Lines, Number Sentences, Sign Emphasis, and Answer in Box Only. One student drew sets of objects frequently and the other students drew number lines more. Descriptions of how use of their drawings changed over time are provided. Implications point to a re-examination of integer instructional models and insight into potential learning progressions.

The relationship between mental computation and relational thinking in the seventh grade

Relational thinking involves understanding equivalence and numerical relationships. The present study examined the relational thinking of seventh graders before and after a 15-day mental mathematics intervention in the context of whole number arithmetic. Using two intact seventh-grade classes and a staggered treatment design, students were assessed at three time points on their (a) ability to solve equivalence problems, and (b) reasoning abilities about true–false number sentences. The results indicated that the students in the class that received the intervention first (the Intervention First group) improved their performance on both measures after the intervention, and a similar pattern was found for the second class (the Intervention Second group), indicating that each group improved immediately following the mental mathematics intervention. Students in the Intervention First group were able to maintain their scores on the test of equivalence problems 4 weeks after the conclusion of the intervention. As the results suggest a link between mental mathematics and relational thinking, it is recommended that mental mathematics play a more prominent role in the seventh-grade mathematics classroom.

Analysis on effect of T/F discussion learning of number sentences for developing relational thinking

Analysis on effect of T/F discussion learning of number sentences for developing relational thinking

Pictorial representations of simple arithmetic problems are not always helpful: a cognitive load perspective

At the start of mathematics education children are often presented with addition and subtraction problems in the form of pictures. They are asked to solve the problems by filling in corresponding number sentences. One type of problem concerns the representation of an increase or a decrease in a depicted amount. A decrease is, however, more difficult to represent in paper pictorially than an increase. Within the Cognitive Load Theory framework, we expected that commonly used decrease problems would be harder and slower to solve than increase problems because of higher intrinsic cognitive load. We also hypothesised that combining the pictorial information with auditory information would reduce this load and, as a result, would lead to improved performance. We further expected that these effects would be most prominent in children who score below average on a general mathematics test. We conducted an experiment with sixty children attending the first grade of primary school, who were divided into a higher and lower mathematics-achieving group. As expected, children performed worse on the decrease problems compared to the increase problems. Also, the combination of pictorial and auditory information reduced the accuracy lag of the decrease problems compared to the increase problems. With response time this effect occurred only in the group of lower mathematics achieving children. These results are in line with the cognitive load theory framework, but we also offer an alternative explanation regarding attention.

Elementary Students’ Understanding of the Equal Sign in Number Sentences

Introducción. Movidos por nuestro interés en la propuesta curricular de integración del pensamiento algebraico en las matemáticas de educación primaria, analizamos la comprensión de igualdades y sentencias numéricas de un grupo de estudiantes de primaria y su evolución cuando trabajan en la resolución de igualdades abiertas y sentencias numéricas verdaderas y falsas.Método. Se llevó a cabo un experimento de enseñanza, con un grupo de estudiantes de entre ocho y nueve años, que comparte las características de la investigación de diseño. Consiste en seis sesiones de una hora, a lo largo de un año, en las que se propuso a los estudiantes actividades escritas individuales, discusiones de gran grupo y entrevistas individuales.Resultados. Los estudiantes pusieron de manifiesto cuatro significados diferentes del signo igual los cuales empleamos aquí para distinguir niveles en su comprensión. Estos niveles nos permiten analizar la evolución de la comprensión de las igualdades y sentencias numéricas, por parte de los estudiantes, a lo largo del experimento.Discusión. Los estudiantes emplearon multiplicidad de significados del signo igual y algunos mostraron cierta inestabilidad en su comprensión a lo largo de las seis sesiones. Identificamos los factores específicos que explican esta inestabilidad y extraemos algunas consecuencias didácticas. Los resultados obtenidos son contrastados con los de estudios previos.

Component processes in arithmetic word-problem solving and their correlates.

Arithmetic word-problem solving is an important component of elementary mathematics curricula that links school mathematics to real-life problem solving. The present 3-year longitudinal study examined children’s arithmetic word-problem solving through understanding its 2 component processes: number-sentence construction and computation. Chinese first graders (n = 153) were tested on their arithmetic word-problem solving, in which they wrote down the number sentences before they solved the problems. They were also given a parallel test of arithmetic computation. Various cognitive predictors and mathematical outcomes were assessed. It was found that the children’s difficulty in solving arithmetic word problems lay more with writing number sentences rather than in computation. The results from path analysis showed that word reading and various numerical-magnitude processing and domain-general skills significantly predicted arithmetic computation whereas only domain-general skills significantly predicted number-sentence construction. Both number-sentence construction and computation significantly predicted future arithmetic computation and mathematics achievement even after controlling for previous arithmetic computation. Theoretical and practical implications are discussed.