**1. Experts believe that students who develop proficiency in mathematical problem solving early are better prepared for advanced mathematics and other complex problem-solving tasks later in life.
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**2. Mathematical problem solving involves reasoning and analysis, argument construction, and:
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**3. The practice guide expert panel defined the characteristics of problem solving to include each of the following EXCEPT:
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**4. Experts recommend teaching one strategy that can be used to solve multiple problems, as this will be less likely to overwhelm the students than teaching multiple strategies.
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**5. Which of the following recommendations for math problem solving is supported by a strong level of research evidence?
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**6. The expert panel believes instruction in problem solving must be an integral part of each curricular unit, and that teachers should use a variety of problems intentionally while ensuring that students have the language and mathematical content necessary to solve problems. **

**7. In order to determine which problems will be routine or non-routine for students, teachers must consider students’:**

**8. Cognitive demands are associated with solving non-routine problems, and generally much more time will be needed to interpret the problem and determine what information is relevant, as well as how it should be used. **

**9. Which of the following is NOT one of the recommendations for teachers in helping prepare lessons to ensure student understanding?**

**10. To make time during instruction for problem solving, teachers should consider balancing the number of problems students are required to solve during seatwork activities with worked examples students can simply study.
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**11. Studies that examined students' mathematics achievement in different content areas showed that:
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**12. Experts recommend that students be encouraged to explain and justify their response to each mathematical prompt orally rather than in written form, and preferably in a group setting rather than individually.
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**13. When responding to prompts, steps include identifying the givens and goals of the problem and the problem type, recalling similar problems to help solve the current problem, _____________________________, solving the problem, and checking the solution.
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**14. By building on students' ideas, teachers can help students clarify and refine the way they solve a problem, which may be particularly helpful for students who dislike working with teacher-provided prompts or who are having difficulty understanding these prompts.
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**15. A major task for any student engaged in problem solving is to translate the quantitative information in a problem into a:
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**16. Schematic diagrams are visual representations that use rectangles to represent quantities presented in the problem. **

**17. Thinking aloud involves having the teacher tell students what he or she is doing, and expressing thoughts while explaining what decisions he or she is making and why those decisions are being made.
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**18. Teachers may promote discussions by asking students guiding questions as they practice representing problems visually, such as each of the following EXCEPT:
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**19. The estimated effects of teaching multiple strategies on students' ability to solve problems correctly is known as operational knowledge, while awareness of mathematical concepts is known as theoretical knowledge.
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**20. When teaching multiple strategies, it is beneficial to periodically employ unsuccessful strategies and demonstrate changing to an alternate strategy to show students that problems are not always solved easily the first time.
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**21. When demonstrating the use of multiple strategies, teachers should randomly call on students to share their strategies, rather than selecting students purposefully based on the strategies they have used to solve the problem.
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**22. In order to have enough time in their math class for students to present and discuss multiple strategies, teachers can:
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**23. Teachers can turn problem-solving activities into learning opportunities by connecting students' intuitive understanding to formal mathematical concepts and notation, as students tend to enter school with ways of making sense of math that are:
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**24. Which of the following is NOT one of the expert recommendations for helping students understand symbolic notation used in algebra?
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**25. Teachers should encourage students to use logical variables that relate to the items in the problem rather than arbitrary ones, as this will help clarify the abstract role that variables play in representing quantities. **

**26. When teachers state a prompt in front of the class and describe how they used it to solve a problem, students see how prompts or items from a task list are used and how people think as they solve problems.
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**27. Each of the following is an accurate statement about procedural knowledge and flexibility EXCEPT:
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**28. Several studies, including some that also involved teacher modeling, prompted students to self-question or to complete tasks or steps while problem solving, and in all of these studies, the intervention’s effects were positive. **

**29. While studies clearly indicated that students with disabilities performed better when specifically taught how to use different visual representations for different types of problem-solving, the results were inconclusive for general-education students.
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**30. Solving intermediate arithmetic problems before representing them with algebraic notation may help students understand problem structure using the mathematical knowledge they already possess, and students can then use this existing knowledge to more easily determine algebraic notation.
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