### 数学代写|数学建模代写math modelling代考|COMPS PROGRAM

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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|A Cognitive Heuristic DOTS Checklist

A cognitive heuristic DOTS checklist was developed to help students’ self- regulation of the problem solving process (please refer to the DOTS checklist in Unit 7 ). According to Polya (1957), such general heuristic procedures prepare students to develop good mental habits in the problem solving process. In the context of solving basic arithmetic word problems, it is important that students will first read and understand the problem as a whole. Based on their understanding of the problem, the learner needs to first detect whether the story or word problem is an additive structure (part-part-whole structure) or multiplicative problem structure (multiple equal groups) to which different mathematical models would apply. Although it is important to know the strategies, it is more important to know when to use what strategies and how to apply the strategy correctly.

After students Detect the problem structure and apply an appropriate mathematical model, the rest of the problem solving process is about mapping information from the problem to the diagram. As only two basic models are necessary for most of the elementary arithmetic word problems that involve four operations (add, subtract, multiply, and divide), the WP story grammar described above will help students $\text { Organize or represent the information (from various structured additive or }$ multiplicative problems) in either the additive or multiplicative model diagrams. It is important that students represent the problem in the diagram equation on the basis of a thorough understanding of the problem; this is where the WP story grammar plays a critical role in facilitating the conceptual understanding. After that, all a learner needs to do is Transform the diagram equation into a real algebraic equation (by “peeling off” the boxes and labels in the COMPS diagrams). The last step in the DOTS strategy is to Solve for the unknown quantity in the algebraic equation, provide a complete answer to the question, and check the accuracy (and meaningfulness) of the answer.

## 数学代写|数学建模代写math modelling代考|Additive and Multiplicative Word Problem Structure and its Variations

The additive problem structure includes a range of Part-Part-Whole and Additive Compare problem structures. A Part-Part-Whole (PPW) problem describes an additive relation between multiple parts and the whole (i.e., parts make up the whole). It includes problems such as combine (e.g., Christine has 5 apples. John has 4 apples. How many apples do they have together?), change-join (e.g., Christine had 5 apples. John gave her 4 more apples. How many apples does Christine have now?), and change-separate (e.g., Christine had 9 apples. Then she gave away 4 apples. How many apples does she have now?) (Van de Walle, 2004). Placement of the unknown can be on the part or on the whole (see eight variations of $P P W$ problems in Table C2-1). An Additive Compare (AC) problem compares two quantities and it involves a compare sentence that describes one quantity as “more” (AC-more) or “less” (AC-less) than the other quantity (e.g., “Christine has 9 apples. She has

5 more apples than John. How many apples does John have?” or “Christine has 9 apples. John has 4 less apples than Christine. How many apples does John have?”). Placement of the unknown can be on the big, small, or difference quantity (see six variations of $A C$ problems in Table $\mathrm{C} 2-1$ ).

The most basic multiplicative problem structure includes various Equal Groups problem structures and various Multiplicative Compare (MC) problem structures. An Equal Groups (EG) problem describes a number of equal sets or units. The placement of the unknown can be on the unit rate (# of items in each unit or unit price), number of units or sets, or on the product (see three variations of $E G$ problems in Table C2-2). A Multiplicative Compare (MC) problem compares two quantities and it involves a compare sentence that describes one quantity as a multiple or part of the other quantity. Placement of the unknown can be on the compared set, the referent set, or the multiplier (i.e., multiple or part) (see three variations of $M C$ problems in Table C2-2). It should be noted that the MC problems in Table 2 b only include those with multiple NOT part relations such as ” $2 / 3$.”

## 数学代写|数学建模代写math modelling代考|Instructional Phases

Instructions to carry out COMPS will be delivered in two parts: problem structure representation and problem solving. During the instruction of problem structure representation, word stories with no unknowns will be used to help students understand the problem structure and the mathematical relations among the quantities. Specifically, students will learn to identify the problem structure and map the information from the problem to its corresponding COMPS diagram equation (see Figure Cl-2 for an example: unit rate $x$ # of units = product). During that stage,

as all quantities are given in the story (no unknowns) students will be able to check the “balance” of the equation to shape and reinforce the concept of “equality” and the meaning of an equal sign.

Problem representation instruction will be followed by problem solving instruction. During problem solving instruction, word problems with an unknown quantity will be presented. When representing a problem with an unknown quantity in the COMPS diagram, students can choose to use a letter (can be any letter they prefer) to represent the unknown quantity. Students are encouraged to use the DOTS checklist (see Unit 7 ) to guide the problem solving process.

Overall, the instruction requires explicit strategy explanation and modeling (see the Appendix for modeling worksheets for students to follow along during the instruction), dynamic teacher-student interaction, guided practice, performance monitoring with corrective feedback, and independent practice. During independent practice, students will be provided with an independent worksheet to solve either additive or multiplicative word problems (see the Appendix for independent worksheets) they have just learned. It is suggested that the COMPS model equations be provided on all modeling and guided practice worksheets, or even on independent practice worksheets in the beginning stage of the instructional program. However, they should be gradually faded out on the worksheet once students have internalized the models.

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## MATLAB代写

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