The Knowledge Quartet as an Organising Framework for Developing and Deepening Teachers’ Mathematics Knowledge

In a previous study of 2 schools in England that taught mathematics very differently, the first author found that a project-based mathematics approach resulted in higher achievement, greater understanding, and more appreciation of mathematics than a traditional approach. In this follow-up study, the first author contacted and interviewed a group of adults 8 years after they had left the 2 schools to investigate their knowledge use in life. This showed that the young adults who had experienced the 2 mathematics teaching approaches developed profoundly different relationships with mathematics knowledge that contributed towards the shaping of different identities as learners and users of mathematics (Boaler & Greeno, 2000). The adults from the project-based school had also moved into significantly more professional jobs, despite living in one of the lowest income areas of the country. In this article, we consider the different opportunities that the 2 school approaches offered for longterm relationships with mathematics and different forms of mathematical expertise that are differentially useful in the 21st century (Hatano & Oura, 2003).

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Enhanced Language Model with Hybrid Knowledge Graph for Mathematical Topic Prediction

10.22541/au.163491250.03226531/v1 ◽

2021 ◽

Author(s):

Minghui Wu ◽

Canghong Jin ◽

Wenkang Hu ◽

Yabo Chen

Keyword(s):

Language Model ◽

Real Data ◽

Mathematical Concept ◽

Fine Tuning ◽

Knowledge Graph ◽

Mathematics Knowledge ◽

Hybrid Knowledge ◽

Model Set ◽

Set Up ◽

Mathematical Topic

Understanding mathematical topics is important for both educators and students to capture latent concepts of questions, evaluate study performance, and recommend content in online learning systems. Compared to traditional text classification, mathematical topic classification has several main challenges: (1) the length of mathematical questions is relatively short; (2) there are various representations of the same mathematical concept(i.e., calculations and application); (3) the content of question is complex including algebra, geometry, and calculus. In order to overcome these problems, we propose a framework that combines content tokens and mathematical knowledge concepts in whole procedures. We embed entities from mathematics knowledge graphs, integrate entities into tokens in a masked language model, set up semantic similarity-based tasks for next-sentence prediction, and fuse knowledge vectors and token vectors during the fine-tuning procedure. We also build a Chinese mathematical topic prediction dataset consisting of more than 70,000 mathematical questions with topics. Our experiments using real data demonstrate that our knowledge graph-based mathematical topic prediction model outperforms other state-of-the-art methods.

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What counts in preschool number knowledge? A Bayes factor analytic approach towards theoretical model development

10.31234/osf.io/kn7j6 ◽

2017 ◽

Author(s):

Yi Mou ◽

Ilaria Berteletti ◽

Daniel C. Hyde

Keyword(s):

Individual Differences ◽

Model Comparison ◽

Bayes Factor ◽

Model Development ◽

Language Abilities ◽

Numerical Ability ◽

Selection Strategies ◽

Bayesian Model Comparison ◽

Mathematics Knowledge ◽

Number Knowledge

Preschool children vary tremendously in their numerical knowledge and these individual differences strongly predict later mathematics achievement. To better understand the sources of these individual differences, we measured a variety of cognitive and linguistic abilities motivated by previous literature to be important and then analyzed which combination of these variables best-explained individual differences in actual number knowledge. Through various data-driven Bayesian model comparison and selection strategies on competing multiple regression models, our analyses identified five variables of unique importance to explaining individual differences in preschool children’s symbolic number knowledge: knowledge of the count list, non-verbal approximate numerical ability, working memory, executive conflict processing, and knowledge of letters and words. Further our analyses revealed that knowledge of the count list, likely a proxy for explicit practice or experience with numbers, and non-verbal approximate numerical ability were much more important to explaining individual differences in number knowledge than general cognitive and verbal abilities. These findings suggest that children bring a diverse set of number-specific, general cognitive and language abilities that are involved in children’s learning of mathematics knowledge, and further suggest that number-specific abilities overshadow more general ones in their contribution to children’s early learning of symbolic numbers.

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Student’s thinking to identify concave plane based on Gregorc model

Beta Jurnal Tadris Matematika ◽

10.20414/betajtm.v12i2.338 ◽

2019 ◽

Vol 12 (2) ◽

pp. 109-121

Author(s):

Surya Enjang Krisdiantoro ◽

Erlina Prihatnani

Keyword(s):

Mathematics Learning ◽

Structured Interview ◽

Thinking Styles ◽

Thinking Style ◽

9Th Grade ◽

Mathematics Knowledge ◽

Thinking Process ◽

Thinking Processes ◽

The Difference ◽

Constructing Knowledge

[English]: Mathematics learning should facilitate students' construction of knowledge. In constructing mathematics knowledge, students involve various types of thinking processes and styles. This qualitative research aimed to describe the process of students’ thinking in identifying concave plane based on Gregorc’s model of thinking style. It involved thirty-three 9th-grade students with a different style of thinking. Data were collected through tests, questionnaire, and non-structured interview then descriptively analyzed to reveal students’ thinking process and styles. The present study found two different thinking styles, namely Sequential Concrete (SC) and Random Abstract (RA) from students who successfully identified the concave plane as a kite. There were different thinking processes in the development of definition, opinion, and conclusions from subjects with different thinking styles. However, the difference in the thinking process from each thinking styles do not hamper students’ success in constructing knowledge. Keywords: Thinking process, Concave place, Thinking style, Gregorc model [Bahasa]: Pembelajaran matematika seharusnya memfasilitasi siswa membangun pengetahuan sendiri. Dalam membangun pengetahuan, siswa melibatkan beragam proses dan gaya berpikir. Penelitian kualitatif ini bertujuan untuk mendeskripsikan proses berpikir siswa dalam mengidentifikasi bangun datar concave berdasarkan gaya berpikir model Gregorc. Subjek penelitian adalah 33 siswa kelas IX SMP yang memiliki gaya berpikir berbeda. Data dikumpulkan melalui tes, angket, dan wawancara non-terstruktur kemudian dianalisis secara deskriptif untuk mengungkap gaya dan proses berpikir siswa. Penelitian ini menemukan dua gaya berpikir berbeda yaitu Sekuensial Konkret dan Acak Abstrak dari siswa yang berhasil mengidentifikasi bangun datar concavesebagai layang-layang. Terdapat perbedaan proses berpikir dalam pembentukan pengertian, pembentukan pendapat, dan penarikan kesimpulan dari siswa dengan gaya berpikir berbeda. Namun demikan, perbedaan gaya berpikir dari setiap proses berpikir tidak membatasi keberhasilan siswa dalam mengkonstruksi suatu pengetahuan. Kata kunci: Proses berpikir, Gaya berpikir, Bangun concave, Model Gregorc

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Ações colaborativas na formação de professores: investigação dos conceitos de padrão e generalização

Revemat revista eletrônica de educação matemática ◽

10.5007/1981-1322.2020.e73606 ◽

2020 ◽

Vol 15 (2) ◽

pp. 1-18

Author(s):

Maria Auxiliadora Vilela Paiva ◽

Tatiana Bonomo de Sousa

Keyword(s):

Teacher Training ◽

Elementary School Teachers ◽

Cultural Context ◽

School Teachers ◽

Teaching Knowledge ◽

Knowledge For Teaching ◽

Mathematics Knowledge For Teaching ◽

Mathematics Knowledge ◽

Mathematics For Teaching

Esse artigo traz reflexões sobre uma formação continuada, parte de uma pesquisa qualitativa, que teve por objetivo investigar os saberes docentes (re)construídos por professores do Ensino Fundamental, por meio do estudo de padrões e generalizações com enfoque de uma matemática para o ensino. Destaca-se nessa pesquisa o papel dos saberes que emergem da prática para construção de saberes próprios da profissão docente. O estudo baseou-se em teorias que valorizassem a apropriação de um saber matemático para o ensino, em um processo coletivo e colaborativo de formação. Os relatos dos professores nas discussões coletivas revelaram que eles, em sua maioria, se apropriaram de uma cultura matemática referente ao conteúdo de padrões e generalizações, pois conceitos relacionados a esses conteúdos e às ideias subjacentes surgiram das reflexões da prática docente e das discussões das problematizações propostas. Ao enfatizar processos de colaboração e investigação, essa formação continuada proporcionou, dentro de um contexto histórico, social e cultural, a (re)construção de novos saberes de uma Matemática para o ensino da Álgebra. This article brings reflections on continuing teacher training, part of a qualitative research that aimed to investigate the teaching knowledge (re) constructed by elementary school teachers through the study of patterns and generalizations focusing on mathematics for teaching. Stands out In this research, the role of the knowledges that emerge from practice, for the construction of specifics knowledges of teacher profession. The study based on theories that value a mathematics knowledge for teaching, in a colletive and colaborative process. The teachers reports in the colletive discussion revealed that them, in their majority, appropriated of a mathematics culture referring to the content of patterns and generalizations, since concepts related to these contents and the underlying ideas emerged from their teachers practice reflections and from proposed problematizations discussions. To emphasize collaboration and investigation this teacher training process provided, within a historical, social, cultural context, the (re) construction of new mathematics knowledge for teaching.

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Mathematics teachers’ knowledge for teaching problem solving

LUMAT International Journal on Math Science and Technology Education ◽

10.31129/lumat.v3i1.1049 ◽

2015 ◽

Vol 3 (1) ◽

pp. 19-36 ◽

Cited By ~ 8

Author(s):

Olive Chapman

Keyword(s):

Problem Solving ◽

Mathematics Teachers ◽

Mathematical Problem ◽

Research Literature ◽

Mathematical Problem Solving ◽

Knowledge For Teaching ◽

General Ability ◽

Mathematics Knowledge ◽

Teaching Problem ◽

Problem Solvers

In recent years, considerable attention has been given to the knowledge teachers ought to hold for teaching mathematics. Teachers need to hold knowledge of mathematical problem solving for themselves as problem solvers and to help students to become better problem solvers. Thus, a teacher’s knowledge of and for teaching problem solving must be broader than general ability in problem solving. In this article a category-based perspective is used to discuss the types of knowledge that should be included in mathematical problem-solving knowledge for teaching. In particular, what do teachers need to know to teach for problem-solving proficiency? This question is addressed based on a review of the research literature on problem solving in mathematics education. The article discusses the perspective of problem-solving proficiency that framed the review and the findings regarding six categories of knowledge that teachers ought to hold to support students’ development of problem-solving proficiency. It concludes that mathematics problem-solving knowledge for teaching is a complex network of interdependent knowledge. Understanding this interdependence is important to help teachers to hold mathematical problem-solving knowledge for teaching so that it is usable in a meaningful and effective way in supporting problem-solving proficiency in their teaching. The perspective of mathematical problem-solving knowledge for teaching presented in this article can be built on to provide a framework of key knowledge mathematics teachers ought to hold to inform practice-based investigation of it and the design and investigation of learning experiences to help teachers to understand and develop the mathematics knowledge they need to teach for problem-solving proficiency.

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Application of Information and Communication Technology to Create E-Learning Environments for Mathematics Knowledge Learning to Prepare for Engineering Education

Cases on Research-Based Teaching Methods in Science Education - Advances in Educational Technologies and Instructional Design ◽

10.4018/978-1-4666-6375-6.ch015 ◽

2014 ◽

pp. 345-373

Author(s):

Tianxing Cai

Keyword(s):

Engineering Education ◽

Learning Environments ◽

Teaching And Learning ◽

Engineering Application ◽

Mathematical Practice ◽

Mathematical Methods ◽

Mathematics Knowledge ◽

Mathematical Problems ◽

Information And Communication ◽

E Learning

The standards for mathematical practice describe varieties of expertise that mathematics educators should develop in their students, including NCTM process standards (problem solving, reasoning and proof, communication, representation, and connections), NRC's report “Adding It Up” (adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition), common core state standards in mathematics (ICT application) to support mathematics teaching and learning. There is a need to provide effective ways that technology can be integrated into mathematics classrooms. Mathematical methods and techniques are typically used in engineering and industrial fields. It can also become an interdisciplinary subject motivated by engineers' needs. Mathematical problems in engineering result in rigorous engineering application carried out by mathematical tools. Therefore, a solid understanding and command of mathematical knowledge is very necessary. This chapter presents the introduction of currently available ICTs and their application of to create e-learning environments to prepare for the students' future engineering education.

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