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The conceptual framework of mathematical modeling (e.g., Lesh & Doerr, 2003) is a vital area in mathematics education research, and its implementation has potential for deeply involving children in integrated and meaningful learning. In mathematical modeling learners are active agents in content-integrated, real-world problem solving. This emphasis on integrating multiple content areas to answer big questions, the pursuit of mathematical modeling, descends from Dewey’s work. We present the definition, principles, and design of modeling practices for readers who may be familiar with early childhood curriculum but less so with using modeling for learning. We explore the application of mathematical modeling to early childhood classrooms and its compatibility with early childhood pedagogies and philosophies. Young children may often be underestimated, assumed to be unable to pose big questions that can be answered through activity, experience, and data; but we discuss how young children can be engaged in problems through mathematical modeling. Finally, as preservice teacher educators, we discuss preparing preservice and in-service teachers for modeling in their classrooms. We offer examples and guidance for early childhood teachers to engage in authentic practice – meeting children where their interests are and creating integrated problem-solving experiences.

In his discussion of early lexical development in children Barrett (1995) notes that utterances can be of the following types: an expression of an internal state, a response to a specific context, a social-pragmatic utterance and a referential utterance. A referential utterance can be thought of in the following way: the thing referred to, a mental representation, a word representation and a word sound. An utterance may refer to an object, an action, an attribute or an event. Some utterances are used as the names of classes of objects while other utterances are used as the proper names of individual objects. Looking at this in abstract we might say that, in the early years of childhood, language is used to refer to elements, sets, functions and relations – in other words to the mathematical structures which were discussed in Chapter 2. Of course although early language is used to refer to mathematical objects, the character of the language itself takes the form of ordinary language.

Problems with learning disabilities are life affecting (Murray, C., Goldstein, D. E., & Nourse, S. (2000). The postsecondary school attendance and completion rates of high school graduates with learning disabilities. Learning Disabilities Research and Practice, 15, 182–186; Westby, 2000; Rojewski, 1999a; Hall et al., 2002). The impact of poor mathematical skills on employment prospects is even bigger than the influence of poor reading skills (Dowker, 2005). After an introduction on the definition, prevalence and impact, gender and birth order, subtypes, comorbidity and assessment of cognition and metacognition in mathematical learning disabilities, we will focus on the features of mathematical learning disabilities in adolescence and adulthood and on the STI(mulation), CO(mpensation), R(emediation) and DI(spensation) (STICORDI) devices to help students with mathematical learning disabilities. With such devices “reasonable” adjustments are provided to ensure that disabled students are not placed at a substantial disadvantage compared to non-disabled students.

Being numerate involves the ability to use mathematical knowledge meaningfully across multiple contexts allowing us to order our day, optimise our health and well-being, and function in technology rich environments. Addressing numeracy from the early years of learning, and across all areas of the education curriculum, is key to lifelong learning and quality of life. Being numerate, however, is more than mathematical knowledge; the language that underpins it heavily impacts how we become numerate. This chapter examines numeracy, or mathematical literacy, investigating how literacy can include, and exclude, students from opportunities to learn at school and beyond. This chapter will also examine how numeracy can be used to provide access to educational curricula and personalised goals for students with diverse learning needs in ways that many have ignored.

The purpose of this paper is to explore students' mathematization through a flow of lessons using the Thailand Lesson Study Incorporated Open Approach (TLSOA) to improve the excellence of instruction.

A total of 16 Grade 4 students were selected because they have been taught using the TLSOA model for four years. Six Lesson Study (LS) team members participated, and two instruments were utilized, namely student worksheets, and field notes. An ethnographic research design was employed.

The results revealed that the students' mathematical ideas were developed from the real world to the mathematical world through a flow of lessons based on the four phases of the Open Approach (OA).

Firstly, the students demonstrated their ability to represent the real world independently when the teacher posed an open-ended problem. Secondly, the students demonstrated their ability to use semi-concrete aids to develop their ideas while self-learning. Thirdly, the students showed how they developed their ideas to solve the open-ended problem using relevant objects or related concepts as part of a whole-class discussion and comparison exercise. Finally, the students demonstrated their abilities to represent the mathematical world using numbers and symbols to communicate their ideas when they were required to make a summary by connecting their mathematical ideas.

This study adds new insight to the literature on students' mathematization using the TLSOA model.

The purpose of this paper is to examine how lesson study can lead to teacher learning and improvement of instruction by developing teachers’ mathematical-task knowledge and by supporting teachers’ selection, modification, and implementation of mathematical tasks. Mathematical-task knowledge includes knowledge needed to use tasks that require a high level of thinking and reasoning.

A qualitative case study design was used to explore the learning and instruction of three teachers as they went through the process of lesson study, developed knowledge around mathematics tasks, and made changes to their instruction. Methods included direct observation of lessons, semi-structured interviews, and participant observation.

This lesson study project supported teachers in developing mathematical-task knowledge and in making change to instructional practice. The teachers discussed in this paper added to their understanding of mathematical tasks and changed how they implemented tasks after lesson study. The teachers began to challenge students to go beyond memorizing or executing procedures to deepen the students’ understanding of mathematical concepts. Teachers developed key insights and understandings of mathematical tasks, triggering shifts in their thinking, and changes to instruction. Collaboration and reflection altered the selection, modification, and implementation of mathematical tasks.

This study reveals the connections between features of lesson study and the pathways that lead to learning and improvements to instruction. Limitations included the demands of the school district’s pacing guide and curriculum, and a limited number of interviews and observations were conducted after lesson study.

There are many different ways lesson study has been implemented in the USA, yet the effectiveness of many lesson study projects is still unclear. This study reveals more about the lesson study process, what features are important, and how these features lead to development of knowledge and practice. This study examines how teachers within the same lesson study group added to their knowledge and practice and how different features of lesson study prompt them to make changes.

This work is devoted to increasing the effectiveness of a mathematical modelling lesson with the help of mobile devices. It verifies the authors' hypothesis which states that enabling students to solve mathematical models on mobile devices improves their academic results in the discipline.

The paper describes an experiment conducted among 38 college students in an extracurricular mathematical club where they solved mathematical models with the help of their own smartphones. The authors describe the mathematical models assigned to students, analyse their academic performance and gather their opinions.

The usage of mobile devices in the mathematical modelling class positively affects students' scores and interest in the subject. The percentage of positive grades among students working on mobile devices is higher than among students working on desktop computers.

The authors discover that in the context of the college-level mathematical modelling course, mobile devices can be successfully used as an alternative replacement for traditional desktop computers.

The purpose of this paper is to explore three different types of digital environments for mathematics learning that may support mathematical play and the failure and feedback mechanics present in each.

Interaction analysis and the lenses of failure, feedback and mathematical play are used to analyze the mathematical interactions afforded by three different digital environments.

Each digital environment supports or restrains the potential for mathematical play through mathematical representations, failure and feedback.

The primary contribution of this paper is to highlight different ways in which digital failure and feedback designs can influence the emergent experience of mathematical play.

Seligman noted four topics that Rogers investigated in this pamphlet: the principles that regulate the exchange value of commodities; the wage theory; the incidence of taxes on agricultural products and an analysis of the economic consequences of a commutation of the tithe. This last topic Rogers treated mathematically. Seligman asserted that the appearance of Malthus's Principles of Political Economy in 1820…[gave] rise to an active discussion on some of the fundamental topics in dispute between Ricardo, Say and Malthus…. Most of the essays of the time, however, were concerned with the discussion of the nature and measure of value, and of these the majority based themselves on the theory advocated by Ricardo and McCulloch. (1903, pp. 351–352)