Unpacking the logic of mathematical statements

This study focuses on undergraduate students' ability to unpack informally written mathematical statements into the language of predicate calculus. Data were collected between 1989 and 1993 from 61students in six small sections of a “bridge" course designed to introduce proofs and mathematical reasoning. We discuss this data from a perspective that extends the notion of concept image to that of statement image and introduces the notion of proof framework to indicate the top-level logical structure of a proof. For simplified informal calculus statements, just 8.5% of unpacking attempts were successful; for actual statements from calculus texts, this dropped to 5%. We infer that these students would be unable to reliably relate informally stated theorems with the top-level logical structure of their proofs and hence could not be expected to construct proofs or evaluate their validity

Rules and principles in cognitive diagnoses

Cognitive simulation is concerned with constructing process models of human cognitive behavior. Our work on the ACM system (Automated Cognitive Modeler) is an attempt to automate this process. The basic assumption is that all goal-oriented cognitive behavior involves search through some problem space. Within this framework, the task of cognitive diagnosis is to identify the problem space in which the subject is operating, identify solution paths used by the subject, and find conditions on the operators that explain those solution paths and that predict the subject's behavior on new problems. The work presented in this paper uses techniques from machine learning to automate the tasks of finding solution paths and operator conditions. We apply this method to the domain of multi-column subtraction and present results that demonstrate ACM's ability to model incorrect subtraction strategies. Finally, we discuss the difference between procedural bugs and misconceptions, proposing that errors due to misconceptions can be viewed as violations of principles for the task domain

A mathematics teacher’s specialized knowledge in the selection and deployment of examples for teaching sequences

This paper explores the specialized knowledge mobilized by a mathematics teacher in the selection and use of examples for teaching sequences. Taking an experimental case study approach, we analyse the examples deployed in a series of third-year secondary level lessons on sequences and identify the different knowledge subdomains activated according to the mathematics teachers’ specialized knowledge analytical model. We will analyse active and passive examples, pointing out the mathematical entity that is being exemplified and the aspect of this entity which is being emphasized by the example. The results identify the different subdomains and categories which are drawn on in the selection and use of examples, along with the various interconnections across knowledge subdomains which interact in the process.The Ministry of Science and Innovation, Spanish Government, PID2021-122180OB-I00 and the Research Centre COIDESO (University of Huelva, Spain) supported this research

Examination of the relationship between generating examples and checking examples in children

Examples play an important role in teaching mathematics. Thus, the present research aims at study of the relationship between examples and understanding mathematics. To do so, the following questions are raised: when generating and checking examples, one aspects is given attention or different aspects? Does the learner’s ability in generating more examples make the learner not to check the examples?. To collect the data, the subjects were interviewed with all elementary students in grade three, four and five. were selected that through interview and tracking students’ intellectual method via frequent questions and controlling their drafts, their generated examples become identified. It should be mentioned names are unreal. The results show that there are differences between “generating example” and “checking examples”. Also, it looks that a strong relationship is available between generating example for a concept and understanding that concept

Pengembangan Soal Matematika Non Rutin Di SMA Xaverius 4 Palembang

Menurut Kurikulum Tingkat Satuan Pendidikan (KTSP), tujuan pembelajaran matematika antara lain menggunakan penalaran pada pola, membuat generalisasi, memecahkan masalah. Kurikulum Tingkat Satuan Pendidikan juga menyebutkan fokus dalam pembelajaran Matematika adalah pemecahan masalah terbuka dengan solusi tak tunggal, dan masalah dengan berbagai cara penyelesaian . Oleh karena itu, perlu dikembangkan soal yang dapat menimbulkan kemampuan siswa untuk mencapai tujuan tersebut. Penelitian ini bertujuan untuk menghasilkan soal non rutin yang valid dan praktis. Metode yang digunakan adalah metode penelitian pengembangan ( development research) yang terdiri dari analisis, desain, evaluasi dan revisi. Pengumpulan data dilakukan dengan tes tertulis untuk melihat valid dan praktis soal non rutin. Subyek penelitian ini adalah siswa kelas XI IPA SMA Xaverius 4 Palembang yang berjumlah 32 orang, dengan kesimpulan (1) prototip soal yang dikembangkan telah memenuhi kriteria valid dan praktis.(2) berdasarkan proses pengembangan diperoleh bahwa prorotip perangkat soal yang dikembangkan memiliki efek potensial terhadap kemampuan siswa mengerjakan soal matematika non rutin siswa kelas XI IPA SMA Xaverius 4 Palembang. Hal ini terlihat dari hasil tes siswa dengan rata – rata 31,94 pada rentang 12 sampai 48, dengan kategori baik. Oleh karena itu soal – soal yang dikembangkan dapat digunakan

Students’ Dichotomous Experiences of the Illuminating and Illusionary Nature of Pattern Recognition in Mathematics

Published ArticleThe concept of pattern recognition lies at the heart of numerous deliberations concerned with new mathematics curricula, because it is strongly linked to improved generalised thinking. However none of these discussions has made the deceptive nature of patterns an object of exploration and understanding. Yet there is evidence showing that pattern recognition has both positive and negative effects on learners’ development of concepts. This study investigated how pattern recognition was both illuminating and illusionary for Grade 11 learners as they factorised quadratic trinomials. Psillos’s fourconditions model was used to judge the reasonableness of learners’ generalisations in six selected examples. The results show that pattern recognition was illuminating in the first three examples where learners made use of localised pattern recognition. In one example, pattern recognition was coincidental but not beneficial in terms of conceptual understanding. In the last two examples localised pattern recognition was at the centre of learner confusion as they failed to extend its application beyond the domain of the examples that generated the pattern. Learners’ confusion with pattern recognition could be attributed to teachers’ failure to meet four important conditions for good generalisations. Results from this study confirm earlier studies showing that abduced generalisations developed out of a few localised instances might be illuminating at first but might not provide the best explanation when extended beyond the localised domain. Further studies are needed that assist in developing patternaware teachers

Perceptions on the Role of a Pre-service Primary Teacher Education Program to Prepare Beginning Teachers to Teach Mathematics in Far North Queensland

This paper employs a collaborative auto-ethnographic method to reflect on perceptions and design of a pre-service primary teacher mathematics education program in a regional university and the role of that program to prepare beginning teachers for classroom mathematics practice in Far North Queensland. A four-phase analysis that reflected on: a primary teacher education program at a regional university, literature on primary mathematics education, reflections of two teacher educators and a pre-service teacher on Explicit Teaching, and the possible modifications to the practice of teaching and learning in the mathematics education subjects was conducted. Three challenges that emerged from the thematic analysis include: need for critical reflection in using a single teaching approach; need to bridge different priorities existing between schools and university; and optimism to change the approaches to assist students. The paper then discusses possible modifications to the practice of teaching and learning in the mathematics education subjects

The Effect of Daily Fluency on Algebraic Procedural Fluency in Students

Abstract Algebra 1 students in the ninth grade struggled to follow procedural steps to answer elementary algebraic problems. An early pre-test found that the majority of pupils lacked the necessary fundamental knowledge to comprehend and investigate algebra\u27s abstract concepts, patterns, and relationships. Fifteen students from one of the researcher\u27s algebra one classes participated in a daily fluency skill-building intervention program to investigate whether daily fluency practice will affect students\u27 procedural fluency skills. They engaged in 50 minutes of daily fluency practice for five weeks. During each 50-minute session, students practiced specific skills to develop procedural fluency competencies, such as solid number sense and integer operations, using additive and multiplicative properties to solve elementary linear equations and inequalities. The researcher administered a post-test on the last day of the intervention plan to determine the effect of daily fluency practice on students\u27 procedural problem-solving abilities. The researcher used descriptive statistics to summarize and compare the initial pre-test and post-test mean scores to further assess the efficacy of the daily fluency skill-building practice. The results indicated a statistical improvement in the students\u27 ability to perform mathematical procedures with greater efficiency

Investigating students’ conceptualization of linear programming in mathematics, a case of a secondary school in Rachuonyo East Sub-County, Kenya

Mathematics is a compulsory subject in the Kenyan education system from pre-primary to secondary education. The ministry of education through the introduction of the 8-4-4 system of education in 1985 made Mathematics compulsory owing to its vast application in a wide range of spheres and professions. It is argued that proficiency in mathematics propels both economic and technological advancements. To further enhance delivery in this subject area, the ministry of education initiated SMASSE for science and mathematics teachers to help in simplifying the ‘high order’ concepts such as calculus, loci, and linear programming among others. Linear programming is applicable in various aspects of mathematical concepts and therefore its conceptualization is very vital to tackle the various applications involving this concept. The mitigation strategies availed for understanding linear programming concepts seem to have had very minimal impact due to the poor performance in KCSE examinations. It is against this backdrop that this study ventured into the investigation of students’ conceptualization of linear programming in Rachuonyo East sub-county in Kenya. A qualitative study approach with a case study design was applied to conduct the research. The study involved 12 students for the focus group discussion randomly sampled with each group consisting of six members, 2 teachers identified through purposive sampling for interviews, lesson observation being conducted, and document analysis. The findings of the study revealed that students conceptualize linear programming concepts based on the methodology employed by the teachers in the teaching process and therefore a systematic discourse and integration of different tools are plausible for enhancing conceptualization. The findings should be useful to the ministry of education, educators, and curriculum designers in modifying their approaches and developing methods for teaching linear programming to students for understanding and applicatio