TeachME Professional Development

Improving Mathematical Problem Solving

Introduction

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.

A. True

B. False


2. Mathematical problem solving involves reasoning and analysis, __________________, and the development of innovative strategies.

A. Verbalization of thought processes

B. Corrective feedback

C. Argument construction

D. Guided practice


3. The practice guide expert panel defined the characteristics of problem solving to include each of the following EXCEPT:

A. Students can learn mathematical problem solving, as it is neither an innate talent nor happenstance that creates skilled problem solvers

B. Mathematical problem solving is relative to the group for which it is being taught, and what is challenging or non-routine for one student is generally the same for the group

C. Mathematical problem solving need not be treated like just another topic in the pacing guide, but instead, it can serve to support and enrich the learning of mathematics concepts and notation

D. Often more than one strategy can be used to solve a problem, and may enable students to think more flexibly when presented with a problem that does not have an obvious solution


4. Problem solving includes more than working word problems and should include algebraic and visual-spatial problems.

A. True

B. False


Summary of the Recommendations

5. When working with struggling math students, experts recommend that teachers teach one strategy that can be used to solve multiple problems, as this will be less likely to overwhelm the students.

A. True

B. False


Use of Research

6. Studies of problem-solving interventions in the past 20 years have consistently shown that teaching students to persist in solving challenging problems is important to math success.

A. True

B. False


Table 2. Recommendations and Corresponding Levels of Evidence

7. Which of the following recommendations for math problem solving is supported by a strong level of research evidence?

A. Assist students in monitoring and reflecting on the problem-solving process.

B. Prepare problems and use them in whole-class instruction

C. Expose students to multiple problem-solving strategies

D. Help students recognize and articulate mathematical concepts and notation


Recommendation 1: Prepare Problems and Use Them in Whole-Class Instruction

8. 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.

A. True

B. False


How to Carry out the Recommendation--Definitions of Routine and Non-Routine Problems

9. Teachers must consider students' ____________________________ to determine which problems will be routine or non-routine for them.

A. Ability to retrieve basis math facts

B. Level of motivation

C. Comfort with verbalizing the steps to solve a problem

D. Previous experience with problem solving


10. If the goal is to help students understand the meaning of an operation or mathematical idea, teachers should choose non-routine problems for their students.

A. True

B. False


11. 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.

A. True

B. False


Addressing Issues Students Might Encounter with the Problem's Context or Language

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

A. Choose problems with familiar contexts

B. Choose challenging problems that will increase students' confidence in problem solving

C. Clarify unfamiliar language and contexts

D. Reword problems, drawing upon students' experiences


Consider Students' Knowledge of Mathematical Content When Planning Lessons

13. As teachers plan problem-solving instruction, they should remember that problems that align with the current unit often draw on skills taught in prior units or grade levels, and that they may need to clarify the language in the problem before expecting students to solve it.

A. True

B. False


Potential Roadblocks and Solutions

14. 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.

A. True

B. False


Assist Students in Monitoring and Reflecting on the Problem-Solving Process

15. During problem solving, monitoring and reflecting helps students think about what they are doing and why they are doing it, evaluate the steps they are taking to solve the problem, and connect new concepts to what they already know.

A. True

B. False


Summary of Evidence: Strong Evidence

16. Studies that examined students' mathematics achievement in different content areas showed that:

A. Providing students with a task list that identified specific steps to solving problems resulted in better student achievement

B. Using a self-questioning checklist improved achievement

C. Additional instructional components such as visual aids and multiple-strategy instruction may contribute to positive results

D. All of the above


How to Carry out the Recommendation-Provide Students with a List of Prompts

17. Experts recommend that students be encouraged to explain and justify their response to each mathematical prompt orally, and preferably in a group setting rather than individually.

A. True

B. False


18. As students become more comfortable with their reasoning abilities and take greater responsibility for monitoring and reflecting during problem solving, teachers can gradually withdraw the amount of support they provide.

A. True

B. False


Example 7. Sample Task List

19. 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 problem.

A. Getting feedback from peers

B. Distinguishing substantive information from superficial information

C. Identifying underlying structural features of each problem

D. Using a visual to represent and solve the problem


Use Student Thinking About a Problem to Develop Students' Abilities

20. 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.

A. True

B. False


Potential Roadblocks and Solutions

21. Ideally, whenever students work on solving problems, they should be able to learn from unsuccessful explorations and conjectures by reflecting on why they were unsuccessful.

A. True

B. False


Recommendation 3--Teach Students How to Use Visual Representations

22. A major task for any student engaged in problem solving is to translate the quantitative information in a problem into a ______________ necessary for solving the problem.

A. Graphic illustration

B. Symbolic equation

C. Distinct diagram

D. Allegorical statement


23. Schematic diagrams are visual representations that use rectangles to represent quantities presented in the problem.

A. True

B. False


Summary of Evidence: Strong Evidence

24. Studies indicate that both general education students and students with learning disabilities performed better when taught to use visual representations.

A. True

B. False


Select Visual Representations that are Appropriate

25. While percent bars are appropriate for percent problems, strip diagrams are best suited for ratio and proportion problems.

A. True

B. False


Use Think-Alouds and Discussions to Teach Students how to Represent Problems Visually

26. 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.

A. True

B. False


Example 12. Variations in Story Contexts for a Proportion Problem

27. Teachers may promote discussions by asking students guiding questions as they practice representing problems visually, such as each of the following EXCEPT:

A. What kind of problem is this and how do you know?

B. What is the relevant information in this problem and why is it relevant?

C. Which problem solving proficiencies did you use when you solved this type of problem last time?

D. What would you do next? Why?


Potential Roadblocks and Solutions

28. If a student is not capturing the relevant details in a problem, teachers can help them improve their representations by building upon their thinking and by asking questions that will help students clarify and refine their representations.

A. True

B. False


Recommendation 4-Expose Students to Multiple Problem-Solving Strategies: Summary of Evidence: Moderate Evidence

29. 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.

A. True

B. False


Provide Instruction in Multiple Strategies

30. 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.

A. True

B. False


Provide Opportunities for Students to Compare Multiple Strategies in Worked Examples

31. Teachers can use worked examples to facilitate comparison of strategies with interesting contrasts and not just minor differences, with the added benefit of allowing for certain concepts to be highlighted.

A. True

B. False


Ask Students to Generate and Share Multiple Strategies for Solving a Problem

32. When demonstrating the use of multiple strategies, teachers should randomly call on students to share their strategies, rather than select students purposefully based on the strategies they have used to solve the problem.

A. True

B. False


Potential Roadblocks and Solutions

33. In order to have enough time in their math class for students to present and discuss multiple strategies, teachers can:

A. Have students do a problem-solving task at the beginning of the math class as a warm-up activity and devote 5-10 minutes to sharing and discussion

B. Purposefully select the two most effective strategies for sharing and discussing

C. Encourage students to share their strategies by writing them on the board for classmates to see

D. Have students take turns sharing their own strategies for 15-20 minutes at the beginning of class


34. For students who lack the prerequisite knowledge or do not remember certain skills, exposure to multiple strategies will likely help them because they will have a variety of options to use in solving the problem.

A. True

B. False


Help Students Recognize and Articulate Mathematical Concepts and Notation

35. Mathematical concepts and notation provide students with familiar __________ for organizing information in a problem and helping students understand and think about the problem.

A. Formats

B. Frameworks

C. Structures

D. Configurations


How to Carry out the Recommendation-Describe Relevant Mathematical Concepts and Notation

36. Students tend to enter school with ____________________ constructed ways of making sense of math, so teachers can turn problem-solving activities into learning opportunities by connecting students' intuitive understanding to formal mathematical concepts and notation.

A. Nonconventional and globally

B. Ambiguous and individually

C. Inaccurate and subjectively

D. Informal and personally


37. Teachers may need to draw attention to mathematical ideas and concepts by directly instructing students in them before engaging the students in problem solving.

A. True

B. False


Ask Students to Explain Each Step Used to Solve a Problem in a Worked Example

38. Students should be given the opportunity to explain the process used to solve a problem and to articulate mathematically valid explanations that provide justifications for their methods.

A. True

B. False


Help Students Make Sense of Algebraic Notation

39. Which of the following is NOT one of the expert recommendations for helping students understand symbolic notation used in algebra?

A. Provide familiar arithmetic problems as an intermediate step before asking students to translate a problem into an algebraic equation

B. Revisit students' earlier knowledge of simple arithmetic to help connect what they already know with new information

C. Have students explain each component of an algebraic equation by having them link the equation back to the problem they are solving

D. Introduce concepts at a quick pace in order to facilitate familiarity and increase comfort level


Potential Roadblocks and Solutions

40. To help students explain their thoughts in more detail, teachers can ask them specific questions about how a problem was solved and how they thought about the problem, as well as having them create a:

A. Poster to be displayed in the classroom that provides explanations

B. Reason sheet of mathematical rules

C. Graph or drawing that expresses mathematical ideas

D. None of the above


41. 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.

A. True

B. False


Conclusion

42. 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.

A. True

B. False


43. Teachers should _______________________ to connect the problem-solving activity to relevant mathematical concepts and notation.

A. Debrief each step

B. Use diagrams and pictorial information

C. Have students model approaches

D. Include motivational strategies


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