Interacting with Indeterminate Quantities through Arithmetic Word Problems: Tasks to Promote Algebraic Thinking at Elementary School

The internal representation of arithmetical word problem sentences: Frontal and posterior-injured patients compared

Solving the Arithmetic Word Problems (AWPs) using AI techniques has attracted much attention in recent years. We feel that the current AWP solvers are under-utilizing the relevant domain knowledge. We present a knowledge- and learning-based system that effectively solves AWPs of a specific type—those that involve transfer of objects from one agent to another (Transfer Cases (TC)). We represent the knowledge relevant to these problems as TC Ontology. The sentences in TC-AWPs contain information of essentially four types: before-transfer, transfer, after-transfer, and query. Our system (KLAUS-Tr) uses statistical classifier to recognize the types of sentences. The sentence types guide the information extraction process used to identify the agents, quantities, units, types of objects, and the direction of transfer from the AWP text. The extracted information is represented as an RDF graph that utilizes the TC Ontology terminology. To solve the given AWP, we utilize semantic web rule language (SWRL) rules that capture the knowledge about how object transfer affects the RDF graph of the AWP. Using the TC ontology, we also analyze if the given problem is consistent or otherwise. The different ways in which TC-AWPs can be inconsistent are encoded as SWRL rules. Thus, KLAUS-Tr can identify if the given AWP is invalid and accordingly notify the user. Since the existing datasets do not have inconsistent AWPs, we create AWPs of this type and augment the datasets. We have implemented KLAUS-Tr and tested it on TC-type AWPs drawn from the All-Arith and other datasets. We find that TC-AWPs constitute about 40% of the AWPs in a typical dataset like All-Arith. Our system achieves an impressive accuracy of 92%, thus improving the state-of-the-art significantly. We plan to extend the system to handle AWPs that contain multiple transfers of objects and also offer explanations of the solutions.

This paper presents a novel approach to automatically solving arithmetic word problems. This is the first algorithmic approach that can handle arithmetic problems with multiple steps and operations, without depending on additional annotations or predefined templates. We develop a theory for expression trees that can be used to represent and evaluate the target arithmetic expressions; we use it to uniquely decompose the target arithmetic problem to multiple classification problems; we then compose an expression tree, combining these with world knowledge through a constrained inference framework. Our classifiers gain from the use of quantity schemas that supports better extraction of features. Experimental results show that our method outperforms existing systems, achieving state of the art performance on benchmark datasets of arithmetic word problems.

The issue of early algebraic thinking is of great importance for the school life of young students. In this paper, the Father Woodland, a puzzle-like environment, is presented aiming to facilitate the use of algebraic thinking in early schooling. Seventy grade 3 students participated and their work was examined within Radford’s (2011, 2014) framework for algebraic thinking. The findings show that the students were able to refer to the unknowns of the problems, to use a broad range of (not necessarily alphanumeric) symbolization and act upon the indeterminate quantities as if they were known numbers.

Arithmetic word problems remain one of the most difficult area of teaching mathematics. Learning by problem posing has been suggested as an effective way to improve students’ understanding. However, the practice in usual classroom is difficult due to extra time needed for assessment and giving feedback to students’ posed problems. To address this issue, we have developed a tablet PC software named Monsakun for learning by posing arithmetic word problems based on Triplet Structure Model. It uses the mechanism of sentence-integration, an efficient implementation of problem-posing that enables agent-assessment of posed problems. The learning environment has been used in actual Japanese elementary school classrooms and the effectiveness has been confirmed in previous researches. In this study, ten Indonesian elementary school students living in Japan participated in a learning session of problem posing using Monsakun in Indonesian language. We analyzed their learning activities and show that students were able to interact with the structure of simple word problem using this learning environment. The results of data analysis and questionnaire suggested that the use of Monsakun provides a way of creating an interactive and fun environment for learning by problem posing for Indonesian elementary school students.

Developing algebraic thinking in elementary school has gained consensus among mathematics educators. The objective of this study is to understand the developmental trajectory of early algebraic thinking in elementary school students so as to assist teachers and curriculum developers in implementing instruction that aligns with students' cognitive development. This study adopted a cross-sectional survey approach, involving 526 students from grades three to five in Shanghai, who were tested using a 12-item assessment that measured three aspects: "generalized arithmetic", "functional thinking", and "quantitative reasoning". Latent class analysis was used to analyze students' response strategies, and, in conjunction with individual interviews, this study identified potential developmental pathways in students' early algebraic thinking, progressing from "arithmetic thinking" to "concrete algebraic thinking", "generalized algebraic thinking", and finally to "symbolic algebraic thinking". As thinking levels advanced, significant differences in students' response strategies emerged, with notable improvements in "generalization abilities" and "symbolization abilities". This study suggests that educational practices should encompass content in elementary arithmetic curricula that fosters generalization abilities. Additionally, providing students with opportunities for diverse representations can effectively stimulate the development of early algebraic thinking.

Pattern exploration is A pivotal activity in all mathematics, indeed in all the scientific disciplines. Children who are attempting to express perceived patterns mathematically are in an excellent position to learn algebraic language and engage in algebraic activity. Principles and Standards for School Mathematics (NCTM 2000) acknowledges the relationship of pattern exploration and algebraic thinking by placing pattern work within the Algebra strand. Yet one can undertake considerable pattern exploration without engaging students in any algebraic thinking whatsoever and teachers may, themselves, be unclear about how patterns can be used to further algebraic thinking. Work with repeating patterns in the early grades, or teaching patterns as a “topic” in the middle grades, may not foster the development of algebraic thinking in students. In this article, we will address this question: How can teachers exploit pattern work to further algebraic thinking and introduce the formal study of algebra in middle school?

Cognitive precursors of word reading versus arithmetic competencies in young Chinese children

In this introduction to the Special Issue Early algebraic thinking: studies from different perspectives, approaches and regions we provide readers with the main motivation and interest underlying research on early algebraic thinking. We then introduce the five papers that make up this Special Issue. Each of these studies, coming from various countries around the world, has a different theoretical framework, methodological approach and perspective regarding teachers, students, instructional practices and curricula. In this introduction, we summarize some of the main themes throughout the different papers, including interviewers’ practices that can promote early algebraic thinking; the kinds of practices that teachers can implement to have a positive impact on students’ early algebraic thinking; and the use of physical models to explore and promote aspects of early algebraic thinking. The Special Issue also includes an additional Prospectives piece by Analúcia Schliemann and David Carraher, in which they examine the algebraic thinking research in elementary school in light of the current mathematics standards in the United States.

This research addresses the challenges elementary students face in developing algebraic thinking skills, particularly during the introduction of plane geometry topics such as squares and rectangles. Algebraic thinking, essential for identifying patterns, relationships, and generalizations, is often hindered by various learning obstacles. Using a qualitative approach with Didactical Design Research (DDR), data were collected through tests, interviews, and document studies. The research involved 30 second-grade students, with six selected for detailed unstructured interviews to confirm their responses. The findings highlight five major learning obstacles: (1) difficulty understanding word problems, (2) difficulty representing problems mathematically, (3) difficulty in measurement and calculations, (4) challenges in communicating solutions, and (5) inability to generalize solutions. These results emphasize the need for improved didactic designs that address learning obstacles and foster algebraic thinking. The study offers actionable insights for enhancing mathematics education in elementary schools.

7 - Children With and Without Mathematics Difficulties: Aspects of Learner Characteristics in A Developmental Perspective

The purpose of this study was to investigate the preparation for learning algebra of 2nd graders and to derive some didactic implications of transition from arithmetical thinking to algebraic thinking. 100 2nd graders in elementary school were selected as subjects of this study from H, J, K, W elementary schools in Seoul, and B elementary school in Gyeonggi-do. These students were found to be less than 1st level of arithmetical thinking as a result of the preliminary test according to the arithmetical thinking level test developed by Lim & Chang(2017). Questionnaires were prepared by adapting and reconstructing four word-problems for preparing algebraic learning which are proposed to be available to 1st and 2nd graders in elementary school by Caldwell et al.(2011), and Caldwell et al.(2014). Using the questionnaires, the students’ problem solving strategies were analyzed. From the results of the analysis, some meaningful implications for helping early algebraic learning and the elementary students’ transition from arithmetical thinking to algebraic thinking were found.

With this study, we proposed to map and discuss the Brazilian academic research that has as its object of study the development of Algebraic Thinking in Elementary School in the interstice from 2017 to 2023. The methodological approach employed is qualitative, of the mapping typology. The results reveal a growing body of research involving algebraic thinking in the context of the initial and continuing training of teachers who will teach and who teach mathematics in Elementary School, so further analysis of future research is needed to try to identify these realities, perhaps looking at the curricular components studied in the initial training courses and the profile of this teacher who will arrive at the school and also the courses offered to teachers who have already joined a school unit.

Algebraic thinking is a person’s ability to understand, analyze, and solve problems using algebraic concepts to simplify statements and find solutions. Currently, many prospective teachers still lack proficiency in applying algebraic thinking skills. Self-efficacy is one of the factors that influences algebraic thinking ability. This study aims to reveal the relationship between self-efficacy and algebraic thinking skills in pre-service mathematics teachers. In the context of solving math problems, especially algebraic ones, algebraic thinking skills are crucial. Using a qualitative method with a descriptive approach, the study employed interview guidelines, questionnaires, and tests as instruments. The results show a clear correlation between the level of self-efficacy and algebraic thinking ability. Pre-service teachers with high self-efficacy can effectively evaluate information, use symbols to represent variables, and solve algebraic equations well. They are also able to determine the values of unknown variables. On the other hand, participants with moderate self-efficacy can interpret and communicate information but are less systematic in selecting problem-solving steps that involve abstraction. Participants with low self-efficacy struggle to interpret information and cannot explain the relationship between the information in the problem and the question asked, leading to incorrect solutions. The conclusion of this study is that the higher the level of self-efficacy, the better one’s algebraic thinking ability. This indicates the importance of enhancing students’ self-efficacy to support more effective algebra learning.

Algebraic thinking skills play an important role in mastering mathematical concepts and problem-solving; however, Linear Algebra learning still faces difficulties, particularly in understanding proofs. Therefore, a proof-based Guided Inquiry strategy is needed to help students develop deeper algebraic thinking skills. This study aims to describe the implementation of the proof-based Guided Inquiry strategy, analyze the improvement and achievement of students’ algebraic thinking skills between classes using this strategy and those using conventional learning, and describe students’ responses to Linear Algebra learning through this approach. The research employed a mixed-method approach with a quasi-experimental design using a non-equivalent control group. The subjects were two classes of mathematics education students, each acting as an experimental and control class. The instruments consisted of pre- and post-tests of algebraic thinking skills, implementation observation sheets, and student response questionnaires. Quantitative data were analyzed using N-Gain and mean difference tests, while qualitative data were obtained from observations and student responses. The results showed that the implementation of learning achieved an average score of 92% (very good category). N-Gain analysis revealed that the experimental class achieved a moderate improvement (0.401), higher than the control class (0.246, low category). Moreover, student responses were positive, with an average percentage of 84.8%. These findings indicate that integrating inquiry exploration with mathematical proof effectively enhances students’ algebraic thinking skills and the quality of Linear Algebra learning.