Mathematical Reasoning Research Articles

Separate the Wheat from the Chaff: Model Deficiency Unlearning via Parameter-Efficient Module Operation

Large language models (LLMs) have been widely used in various applications but are known to suffer from issues related to untruthfulness and toxicity. While parameter-efficient modules (PEMs) have demonstrated their effectiveness in equipping models with new skills, leveraging PEMs for deficiency unlearning remains underexplored. In this work, we propose a PEMs operation approach, namely Extraction-before-Subtraction (Ext-Sub), to enhance the truthfulness and detoxification of LLMs through the integration of ``expert'' PEM and ``anti-expert'' PEM. Remarkably, even anti-expert PEM possess valuable capabilities due to their proficiency in generating fabricated content, which necessitates language modeling and logical narrative competence. Rather than merely negating the parameters, our approach involves extracting and eliminating solely the deficiency capability within anti-expert PEM while preserving the general capabilities. To evaluate the effectiveness of our approach in terms of truthfulness and detoxification, we conduct extensive experiments on LLMs, encompassing additional abilities such as language modelling and mathematical reasoning. Our empirical results demonstrate that our approach effectively improves truthfulness and detoxification, while largely preserving the fundamental abilities of LLMs.

When Do Program-of-Thought Works for Reasoning?

In the realm of embodied artificial intelligence, the reasoning capabilities of Large Language Models (LLMs) play a pivotal role. Although there are effective methods like program-of-thought prompting for LLMs which uses programming language to tackle complex reasoning tasks, the specific impact of code data on the improvement of reasoning capabilities remains under-explored. To address this gap, we propose complexity-impacted reasoning score CIRS, which combines structural and logical attributes, to measure the correlation between code and reasoning abilities. Specifically, we use the abstract syntax tree to encode the structural information and calculate logical complexity by considering the difficulty and the cyclomatic complexity. Through an empirical analysis, we find not all code data of complexity can be learned or understood by LLMs. Optimal level of complexity is critical to the improvement of reasoning abilities by program-aided prompting. Then we design an auto-synthesizing and stratifying algorithm, and apply it to instruction generation for mathematical reasoning and code data filtering for code generation tasks. Extensive results demonstrates the effectiveness of our proposed approach.

Frugal LMs Trained to Invoke Symbolic Solvers Achieve Parameter-Efficient Arithmetic Reasoning

Large Language Models (LLM) exhibit zero-shot mathematical reasoning capacity as a behavior emergent with scale, commonly manifesting as chain-of-thoughts (CoT) reasoning. However, multiple empirical findings suggest that this prowess is exclusive to LLMs that have exorbitant sizes (beyond 50 billion parameters). Meanwhile, educational neuroscientists suggest that symbolic algebraic manipulation be introduced around the same time as arithmetic word problems so as to modularize language-to-formulation, symbolic manipulation of the formulation, and endgame arithmetic. In this paper, we start with the hypothesis that much smaller LMs, which are weak at multi-step reasoning, can achieve reasonable arithmetic reasoning if arithmetic word problems are posed as a formalize-then-solve task. In our architecture, which we call SyReLM, the LM serves the role of a translator to map natural language arithmetic questions into a formal language (FL) description. A symbolic solver then evaluates the FL expression to obtain the answer. A small frozen LM, equipped with an efficient low-rank adapter, is capable of generating FL expressions that incorporate natural language descriptions of the arithmetic problem (e.g., variable names and their purposes, formal expressions combining variables, etc.). We adopt policy-gradient reinforcement learning to train the adapted LM, informed by the non-differentiable symbolic solver. This marks a sharp departure from the recent development in tool-augmented LLMs, in which the external tools (e.g., calculator, Web search, etc.) are essentially detached from the learning phase of the LM. SyReLM shows massive improvements (e.g., +30.65 absolute point improvement in accuracy on the SVAMP dataset using GPT-J 6B model) over base LMs, while keeping our testbed easy to diagnose and interpret, and within the reach of most researchers.

Reasoning Skills in Mathematics Teaching: A Meta-Synthesis on Studies Conducted in Turkey

This research aims to analyze the results of studies conducted in Turkey on reasoning skills in mathematics teaching and to reveal what kind of trend there is in this field. Within the scope of this study, databases were searched with the keywords "reasoning"(muhakeme, akıl yürütme) and "reasoning skill" (Muhakeme becerisi, akıl yürütme becerisi), and the results were examined in accordance with the inclusion criteria regarding mathematics teaching. One hundred sixty-three studies were included. Each of the studies included in the meta-synthesis study was analyzed descriptively according to type, year, method, sample type and size, data collection tools, statistical analysis, learning field, keywords, reasoning type and purpose. In addition, the studies’ results were content analyzed and tabulated by coding the differences and similarities between them with a holistic approach. It has been determined that studies on the learning fields in the mathematics curriculum are mostly carried out in the field of learning numbers and operations in the secondary school mathematics curriculum. When evaluated in terms of reasoning types, almost half of the studies were conducted on mathematical reasoning. It has been observed that after mathematical reasoning, the most focus is on proportional reasoning. When the aims of the studies included in the research were examined, it was determined that the most focused ones were "examining the factors affecting reasoning skills", "measurement of reasoning skills" and "the effect of teaching practices on reasoning skills". In the studies examined, it was seen that there were 33 teaching practices whose effects on reasoning skills were examined.

A framework for reasoning in school mathematics: analyzing the development of mathematical claims

This study introduces a framework for analyzing opportunities for mathematical reasoning (MR) in school mathematics, using MR-relevant claims and their derivation as the unit of analysis. We contend that this approach can effectively capture a broad range of opportunities for MR across various teaching situations. The framework, rooted in commognition, entails identifying necessary object-level narratives (NOLs) and the processes involved in their construction and substantiation. After theoretical development, the framework was refined through analyses of mathematics lessons in Norwegian primary school classrooms. Examples from the data illustrate how to utilize the framework in analysis and what such analyses can reveal in four typical teaching situations: the introduction of new mathematical objects, the introduction of procedures, work on exercise tasks, and work on problem-solving tasks. Drawing from the analysis of these examples, we discuss the value of the framework for analyzing MR in school mathematics and how such analysis can benefit teachers and researchers.

Role of Mathematics in Data Science - Machine learning

The mathematical concepts help with identifying patterns and algorithmic development. Mathematics is an important part of many modern scientific areas. It is also a field of major significance in data science. Mainly because of the concepts they propose and the tools they allow, statistics and arithmetic have such an essential role in data science. A specific type of mathematical reasoning is needed for analysing data, in addition to the basic concepts of calculus, discrete mathematics, and linear algebra. To understand how to apply such algorithms in data science, one must have a solid understanding of many different statistical and probability theory ideas. Modern data science methods have a strong mathematical basis for almost all of their techniques, including machine learning.

Robust fingerprint recognition approach based on diagonal slice of polyspectra in the polar space

Although fingerprint recognition is a mature technology and nowadays commercial state-of-the-art systems can be successfully used in a number of real applications, not all the problems have been solved and the research is still very active in the field. This paper presents a new approach for estimating the shift and rotation parameters between fingerprint images stored in the dadabase that operates in the third-order frequency-domain measure called the auto-bispectrum, which, allows us to estimate the shift and the rotation separately. The diagonal slices and their spectra of auto- and cross-bispectrum are proposed. The rotation parameters are estimated from the remaining polar sampled the third-order spectrum using cross-correlation, and then, after compensating for rotation, we may easily estimate the translational component, e.g., by using phase correlation.Experimental evidence of this performance is presented, and the mathematical reasons behind these characteristics are explained in depth. We compare our approach in a simulation to other frequency-domain fingerprint recognition algorithms. We find that our algorithm can better estimate shift and rotation parameters than the other methods.

Abstract strategy board game for mathematical reasoning: Opening the black box

Abstract strategy board games have potential to develop players’ mathematical skills, especially reasoning. In this study, the actions performed by the students in the reflected games were examined within the concept of milieu of the theory of didactical situations. Eight 7th grade students participated in the study and their games related actions were analyzed according to the reflected game functions. 13 indicators of reflected game functions were identified: 8 in the heuristic, 3 in the explanatory, and 2 in the checking. The interactions between these indicators were analyzed in terms of transition between perceptual and theoretical levels to observe the evolution of the game milieu. The findings of the study showed that the indicators of reflected game functions played a key role in the development of a strategy. These indicators were found to be critical in the evolution of the milieu as they allowed transitions between theoretical and perceptual levels.

The role of learning anxiety and mathematical reasoning as predictor of promoting learning motivation: The mediating role of mathematical problem solving

Mathematical problem solving stands as a fundamental aspect of 21st-century skills. Extensive educational studies have substantiated the cultivation of this competency, illustrating that exploring specific predictors and related variables can offer valuable insights for its enhancement. The study investigated how solving mathematical problems acts as a mediator between motivation for learning, anxiety about learning, and mathematical reasoning. A total of 345 primary school students participated in this study, responding to online questionnaires. Structural equation modelling was applied for data analysis, revealing satisfactory construct validity and instrument reliability. The findings highlighted a positive relationship between learning anxiety and mathematical reasoning, indicating their predictive role in improving learning motivation. In addition, the research identified mathematical problem solving as a mediator, playing a crucial role in reducing learning anxiety and fostering increased learning motivation. These outcomes have significant implications, underscoring the need to understand the intricate interaction between learning anxiety, mathematical reasoning, and motivation. This understanding could facilitate the development of targeted interventions and support systems, fostering a conducive learning environment that enhances student participation and academic achievement.

The Logic for Mathematics without Ex Falso Quodlibet

Abstract Informally rigorous mathematical reasoning is relevant. So too should be the premises to the conclusions of formal proofs that regiment it. The rule Ex Falso Quodlibet induces spectacular irrelevance. We therefore drop it. The resulting systems of Core Logic $ \mathbb{C}$ and Classical Core Logic $ \mathbb{C}^{+}$ can formalize all the informally rigorous reasoning in constructive and classical mathematics respectively. We effect a revised match-up between deducibility in Classical Core Logic and a new notion of relevant logical consequence. It matches better the deducibility relation of Classical Core Logic than does the Tarskian notion of consequence. It is implosive, not explosive.

Development of The STAD Cooperative Learning Model : Enhancing Mathematical Reasoning and Communication Skills in Higher Education

This study aims to develop the STAD cooperative learning model specifically designed for mathematics education in a higher institution to impact students' reasoning and communication skills. The study holds significance in its comprehensive analysis, encompassing (1) Plomp Stage 1 Schematic: Preliminary Study; (2) Plomp Stage 2 Schematic: Development Study; and (3) Plomp Stage 3 Schematic: Assessment Study of the developed STAD learning model. The method employed was Research and Development (R&D), utilizing questionnaires, tests, and observations as data collection techniques. The present research involved students at a university in Padangsidimpuan, North Sumatra, Indonesia. The analysis techniques used were inferential statistics and descriptive statistics. The findings indicated that: (1) The preliminary study of the model was conducted based on needs analysis incorporating the curriculum, student, and learning materials; (2) The model’s development study, encompassing six phases, yielded a significant validation result of 3.84, categorized as ‘Good’; (3) The assessment study, in terms of practicality, obtained a score of 4.60 (considered ‘Good’), and effectiveness demonstrated positive significance in affecting students' reasoning and communication skills. The overarching conclusion drawn is that the research and development of the STAD learning model revealed significant positive impacts on students' reasoning and communication skills.

Urban-Rural Differences in Early Arithmetic Performance are Accounted for by Phonological Processing

ABSTRACT Urban and rural children have different levels of performance in arithmetic processing. This study investigated whether such a residence difference can be explained by phonological processing. A total of 1,501 Chinese primary school students from urban and rural areas were recruited to complete nine cognitive tasks: two in arithmetic performance (subtraction and multiplication), one in phonological processing (word rhyming), two in quantity processing (numerosity and number comparison), one in visuo-spatial processing (mental rotation), as well as a mathematical reasoning task (number-series completion) as a contrast with arithmetic processing and two general cognitive tasks (choice reaction time and Raven’s progressive matrices). The results showed that Chinese urban children had superior performance in arithmetic, which was eliminated after controlling for phonological ability, but not after controlling for quantity or visuo-spatial performance or general cognitive factors. Urban children also showed better performance in mathematical reasoning than did rural children, but residence difference in mathematical reasoning was not eliminated by any of the cognitive factors. The above result suggests that phonological processing explains the relationship between residence and arithmetic performance.

Teaching Fifth-Grade Students With Specific Learning Disabilities to Explain Their Mathematical Reasoning Through Written Expression

The current study investigated the effectiveness of an intensive writing intervention focusing on mathematical reasoning through written expression. A group pretest–intervention–posttest comparison experimental design was used to implement a 12-lesson intervention, delivered through a combination of Google Classroom and in-person classroom teacher support, to fifth-grade students with specific learning disabilities ( n = 19). Although underpowered, our results indicated the intensive writing intervention led to all students in the intervention group ( n = 11) significantly outperforming the students in the control group ( n = 8) on subtraction and division word problems requiring an explanation of their solution in written form. Additionally, student treatment interviews revealed the strategy was both enjoyable and helpful in solving and explaining their word problem solutions and the teacher provided insight into potential additional scaffolds to support the most intensive students. Lessons learned and implications for research and practice are presented.

Layers of Student Understanding Based on Pirie Kieren's theory in Solving Story Problems in Terms of Cognitive Style

Objective: Understanding lines and angles serves as the foundation for further mathematical topics such as trigonometry, geometry, calculus, etc. Students may struggle with advanced mathematical reasoning and problem-solving without a firm grasp of these basic concepts. This study aims to investigate how students' mathematical understanding processes are based on the folding back theory. Method: The research utilizes a qualitative approach with descriptive eksploratif design. Two subjects were selected from 28 seventh-grade students, each representing the field-dependent and field-independent cognitive style. Data was collected through mathematical comprehension tests, GEFT tests, and interviews. Data were analyzed through data reduction, data presentation, and verification stages, with each subject being interviewed to verify the processes. Results: Based on the research results, students in the field-independent cognitive style were much more active and better understood the problem-solving process than those in the field-dependent. However, both subjects still required learning assistance. Novelty: This research explores the folding back theory in the mathematical understanding process based on cognitive styles, whereas previous studies have mainly focused on mathematical comprehension abilities. Therefore, further research would benefit from using instructional media to better engage students in understanding the material.

The Effect of Visual Reasoning on Arithmetic Word Problem Solving

First-grade students often encounter challenges in understanding and solving arithmetic word problems due to their limited reading comprehension abilities. Despite these difficulties, students may employ arbitrary strategies, such as combining numbers based on specific keywords, even if they lack a full understanding of the problems. Research suggests that effective mathematical reasoning involves the use of visual mental representations during the problem-solving process. To address this, some studies have explored methods to enhance students’ comprehension of word problems. Building on this, the current study explores the impact of first-grade pupils creating visual representations of problem situations on their comprehension and the number of correct solutions. In a typical math class, 45 first graders received a paper-and-pencil task, and, in a visual context, they solved similar problems after reading and illustrating the situation. The findings reveal that while most participants correctly represented the problem situations through drawing, about half struggled to determine the numeric solutions. Nevertheless, the visual context led to an increase in the number of correct problem solutions compared to the normal context, suggesting the potential benefits of incorporating visual representations in enhancing comprehension and problem-solving skills.

The relationships among working memory, inhibitory control, and mathematical skills in primary school children: Analogical reasoning matters

The current study examined the mediating role of analogical reasoning in the relationships among working memory, inhibitory control, and children’s mathematical skills. Two hundred fifty-one students from first to third grades were tested on visual-spatial working memory, verbal working memory, inhibitory control, analogical reasoning, and different mathematical skills (i.e., symbolic number processing and mathematical reasoning). After controlling for age and gender, analogical reasoning significantly contributed to both symbolic number processing and mathematical reasoning. The relationship between verbal working memory and symbolic number processing, as well as the relationship between verbal working memory and mathematical reasoning, were significantly mediated by analogical reasoning. The exploratory analyses further revealed that there was no significant age difference in the roles of analogical reasoning. These results highlight the important role of analogical reasoning in explaining the relationships among working memory, inhibitory control, and children’s different mathematical abilities. The findings also indicate that analogical reasoning, the ability of identifying and processing critical relational information, may be a potential avenue to improve children’s mathematical skills.

Meta-analysis: The effect of the discovery learning model on indonesian students’ mathematics reasoning

The ability of mathematical reasoning is one of the important skills for students. The discovery learning model is a suitable learning model to enhance students' mathematical reasoning abilities. The research uses meta-analysis methods through a review of articles in Indonesian national journals. From Google Scholar, Portal Garuda, and Sinta ristekbrin, the author obtained 10 studies that met the inclusion criteria for analysis on the Meta-Mar website to obtain a combined effect size. From the interpretation of this combined effect size, it can be concluded that the overall implementation of the discovery learning model has a moderate effect on students' mathematical reasoning abilities. The characteristics of this research study include educational level, year of research, and sample size. From the statistics, it was found that implementing the discovery learning model to improve students' mathematical reasoning abilities is influenced by educational level, year of research, and sample size.

Mathematical Reasoning of High School Students in Solving AKM Geometry and Measurements Problem Viewed from Multiple Intelligences

Mathematical Reasoning of High School Students in Solving AKM Geometry and Measurements Problem Viewed from Multiple Intelligences

Preservice teachers' mathematical knowledge for teaching: Focus on lesson planning and reflection

AbstractMathematics teacher educators have suggested preservice mathematics teachers' (PMTs') practices provide evidence of their mathematical knowledge for teaching (MKT). PMTs engage in approximations of practice to rehearse teaching. To explore connections between MKT and PMTs’ practices, we operationalized six MKT subdomains in terms of approximations of practice. We used that operationalization to investigate, which subdomains were evidenced in PMTs’ lesson plans and their lesson plan reflections. Eleven secondary PMTs participated in our study. Using a qualitative content analysis method, we found Knowledge of Content and Teaching most evidenced, suggesting that the PMTs demonstrated MKT when they described mathematical tasks and associated instructional activities. Also, the PMTs demonstrated Knowledge of Content and Students when they described how their students would engage with proposed mathematical tasks. However, the PMTs struggled to plan for activities by building on students’ potential mathematical thinking and reasoning. Only limited instances of Horizon Content Knowledge suggest that the PMTs had limited opportunities to demonstrate how their advanced mathematical knowledge connected with their mathematics teaching in the approximations of practice provided. We propose alternative forms of approximations of practice to optimize PMTs’ opportunities to demonstrate and develop MKT.

The use of circular functions applying right triangle geometry

The research aims to provide a game presenting methods and operations that favors mathematical education in elementary school, using easily accessible material combined on the elaboration and application of the activity called Dominó, which presents the subject of rooting and potentiation in domino pieces and in strategies of this board game. The activity observed by teachers during Mathematics Week, during the month of May 2023 at Colégio Ten. Rêgo Barros, was divided into two parts: the first, dedicated to the creation of the pieces using carpentry wood, in addition to involving the student with incentives to handle these materials; the second, dedicated to understanding the mathematics present in the construction of the pieces and the strategy of some moves, and, finally, putting this knowledge into practice in a match of the game. The sources used to write this article were the materials sent by the students, images of the buildings and materials used. The teachers who evaluated the work noted the construction of mathematical objects, providing a didactic transposition of mathematical subjects and understanding in a more reasonable way for all team members as well as classmates who learned and participated in the presentation. The evolution of the mathematical and deductive reasoning of the students who observed and integrated with others during the presentations was noticed, thus showing a new active tool for understanding mathematical concepts.