### 数学代写|数学建模代写math modelling代考|CONCEPTUAL MODEL-BASED PROBLEM SOLVING

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数学建模代写math modelling代考|Algebra Thinking in Problem Solving

Although American students are struggling with many aspects of mathematics, the National Mathematics Advisory Panel has identified “algebra as a central concern” (National Mathematics Advisory Panel, 2008, p. xiii). Interestingly, American students tend to enjoy school mathematics during the early elementary grades. However, they begin to experience difficulty in and come to dislike mathematics after fourth grade when learning becomes more abstract or symbolic and involves more algebraic thinking (Cai, Lew, Morris, Moyer, Ng, \& Schmittau, 2004). In particular, students with learning disabilities or difficulties in mathematics (LDM) are falling further behind their normal achieving peers as they move from elementary to secondary schools. A majority are essentially failing the secondary math curriculum. According to the Panel, mathematics achievement in the U.S. decreases significantly in the late middle grades when students are expected to learn algebra, which raises the essential question: How can students, including those with LP, “be best prepared for entry into algebra?”(Panel, p. xiii). No doubt, the Panel’s report underscores the importance of algebra-readiness instruction.

The purpose of this curriculum book is to present a Conceptual Model-Based Problem Solving (COMPS) approach to the teaching of elementary mathematics problem solving. It emphasizes the teaching of big ideas in mathematics problem solving and making connections between mathematical ideas including the connection between arithmetic and algebra learning.

In this chapter, I will first briefly characterize algebraic thinking in problem solving. Next, I will present a framework for mathematical modeling. Then, I will introduce the COMPS approach that emphasizes mathematical modeling involving algebraic thinking and readiness. Finally, I will provide a brief review of relevant research in word problem solving with students with LDM, and illustrate the distinctive features of COMPS and its advantages with the support of scientificbased research.

## 数学代写|数学建模代写math modelling代考|Mathematical Modeling

Recently, Blum and Leiss (2005) provided a framework for modeling (see Figure $\mathrm{Cl}-1$ ). In this modeling cycle, one must (1) read and understand the task, (2) structure the task and develop a real situational model, (3) connect it to and/or represent it with a relevant mathematical model; (4) solve and obtain the mathematical results, (5) interpret the math results in real problem context; and (6) validate the results (either end the task or re-modify the math model if it does not fit the situation). In light of research in mathematics education, many students have difficulties in making the transition from a real situational model to a mathematical model; and it is a weak area in students’ mathematical understanding (Blomhøj, 2004).

In short, modeling involves translation or representation of a real problem situation into a mathematical expression or model. Mathematical models are an essential part of all areas of mathematics including arithmetic and should be introduced to all age groups including elementary students (Mevarech \& Kramarski, 2008). It should be noted that engaging students in the modeling process does not necessarily mean engaging students in the discovery or invention of mathematical models or complex notational systems; however, according to Lesh, Doerr, Carmona, and Hjalmarson (2003), it does mean that when such models or systems are given to the students, “the central activities that students need to engage in is the unpacking of the meaning of the system” (p. 216), representation of the real problem situation in a mathematical expression or model, and the flexible use of the model to solve real world problems.

## 数学代写|数学建模代写math modelling代考|Theoretical Framework: Conceptual Model-based Problem Solving

Contemporary approaches to story problem solving have emphasized the conceptual understanding of a story problem before attempting any solution that involves selecting and applying an arithmetic operation for solution (Jonassen, 2003). Because problems with the same problem schema share a common underlying structure and hence require similar solutions (Chen, 1999; Gick \& Holyoak, 1983), students need to learn to understand the structure of the mathematical relationships in word problems and should develop this understanding through creating and working with a meaningful representation of the problem (Brenner et al., 1997) as well as mathematical modeling (Hamson, 2003).

The representation that models the underlying mathematical relations in the problem, that is, the conceptual model, facilitates solution planning and accurate problem solving. The conceptual model should drive the development of a solution plan that involves selecting and applying appropriate arithmetic operations. According to Lesh, Landau, \& Hamilton (1983), a conceptual model is defined as an adaptive structure consisting of the following primary components: (a) a within concept network of relations; (b) a between-concept system that links and combines within-concept networks; (c) a system of representations (e.g., written symbols, pictures, and concrete materials); and (d) systems of modeling processes. The first two components address students’ understanding of the idea or underlying structure of the concept. The third component concerns different representation systems, and the fourth component deals with modifying the situation to fit the existing model or changing existing model to make it applicable to a given situation. Based on Lesh et al. (1983), in applied problem solving, important translation and /or modeling processes include (a) simplifying the original problem situation by ignoring irrelevant information in the problem, and (b) “establishing a mapping between the problem situation and the conceptual models used to solve the problem” (p. 9).

Building on metaanalysis (e.g., Xin \& Jitendra, 2009) and cross-cultural curriculum evaluation (e.g., Xin, 2007), as well as empirical studies of intervention strategies (Xin, 2008; Xin et al., 2011; Xin, Wiles, \& Lin, 2008; Xin \& Zhang,2009), I have developed the Conceptual Model-based Problem Solving (COMPS) program that is consistent with the theoretical framework of mathematical modeling and conceptual models (e.g., Blomhøj, 2004; Lesh et al., 1983). One distinguishable difference between the COMPS approach and prior research in word problem solving by students with LD (e.g., schema-based instruction $[\mathrm{SBI}])$ is that the former focuses on representing the word problem in a defined mathematical model (the stage of “mathematical model” as it is presented in Blum and Leiss’s mathematical modeling cycle, see Figure Cl-1), which is expressed in an algebraic equation that directly drives the solution plan. In the next section, I will provide a brief review of intervention research with students with LDM using SBI and more recently Conceptual Model-based Problem Solving (COMPS) in facilitating elementary students’ ability to solve mathematics word problems.

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