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Simplify each expression. If the expression does not represent a real number, say so.

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x 2 − 6 x + 5 = 0 x^2-6 x+5=0 x 2 − 6 x + 5 = 0

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## Which statement about the teaching through problem solving approach is most accurate?

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## 5 Teaching Mathematics Through Problem Solving

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.

Key features of a good mathematics problem includes:

- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.

## Mathematics Tasks and Activities that Promote Teaching through Problem Solving

## Choosing the Right Task

- Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
- What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
- Can the activity accomplish your learning objective/goals?

## Low Floor High Ceiling Tasks

The strengths of using Low Floor High Ceiling Tasks:

- Allows students to show what they can do, not what they can’t.
- Provides differentiation to all students.
- Promotes a positive classroom environment.
- Advances a growth mindset in students
- Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

- YouCubed – under grades choose Low Floor High Ceiling
- NRICH Creating a Low Threshold High Ceiling Classroom
- Inside Mathematics Problems of the Month

## Math in 3-Acts

Act Three is the “reveal.” Students share their thinking as well as their solutions.

- Dan Meyer’s Three-Act Math Tasks
- Graham Fletcher3-Act Tasks ]
- Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

## Number Talks

- The teacher presents a problem for students to solve mentally.
- Provide adequate “ wait time .”
- The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
- For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
- Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

## Saying “This is Easy”

When the teacher says, “this is easy,” students may think,

- “Everyone else understands and I don’t. I can’t do this!”
- Students may just give up and surrender the mathematics to their classmates.
- Students may shut down.

Instead, you and your students could say the following:

## Using “Worksheets”

- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

20 seconds to 2 minutes for students to make sense of questions

## Share This Book

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## Quiz 3: Teaching Through Problem Solving

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## Initial Thoughts

Perspectives & resources, what is high-quality mathematics instruction and why is it important.

- Page 1: The Importance of High-Quality Mathematics Instruction
- Page 2: A Standards-Based Mathematics Curriculum
- Page 3: Evidence-Based Mathematics Practices

## What evidence-based mathematics practices can teachers employ?

## Page 5: Visual Representations

- Page 6: Schema Instruction
- Page 7: Metacognitive Strategies
- Page 8: Effective Classroom Practices
- Page 9: References & Additional Resources
- Page 10: Credits

## Research Shows

- Students who use accurate visual representations are six times more likely to correctly solve mathematics problems than are students who do not use them. However, students who use inaccurate visual representations are less likely to correctly solve mathematics problems than those who do not use visual representations at all. (Boonen, van Wesel, Jolles, & van der Schoot, 2014)
- Students with a learning disability (LD) often do not create accurate visual representations or use them strategically to solve problems. Teaching students to systematically use a visual representation to solve word problems has led to substantial improvements in math achievement for students with learning disabilities. (van Garderen, Scheuermann, & Jackson, 2012; van Garderen, Scheuermann, & Poch, 2014)
- Students who use visual representations to solve word problems are more likely to solve the problems accurately. This was equally true for students who had LD, were low-achieving, or were average-achieving. (Krawec, 2014)

## How does this practice align?

## CCSSM: Standards for Mathematical Practice

Definition : A straight line that shows the order of and the relation between numbers.

Common Uses : addition, subtraction, counting

Common Uses : addition, fractions, proportions, ratios

Definition : Simple drawings of concrete or real items (e.g., marbles, trucks).

Common Uses : counting, addition, subtraction, multiplication, division

Definition : Drawings that depict information using lines, shapes, and colors.

Common Uses : comparing numbers, statistics, ratios, algebra

Common Uses : algebra, geometry

## Elementary Example

Mrs. Aldridge ask her first-grade students to add 2 + 4 by drawing dots.

## High School Example

Mr. Huang asks his students to solve the following word problem:

## Manipulatives

If you would like to learn more about this framework, click here.

## Concrete-Representational-Abstract Framework

- Concrete —Students interact and manipulate three-dimensional objects, for example algebra tiles or other algebra manipulatives with representations of variables and units.
- Representational — Students use two-dimensional drawings to represent problems. These pictures may be presented to them by the teacher, or through the curriculum used in the class, or students may draw their own representation of the problem.
- Abstract — Students solve problems with numbers, symbols, and words without any concrete or representational assistance.

## For Your Information

Kim Paulsen, EdD Associate Professor, Special Education Vanderbilt University

## People also looked at

- 1 School of Mathematics and Statistics, Hunan Normal University, Changsha, China
- 2 School of Mathematics, Yunnan Normal University, Kunming, China
- 3 The High School Attached to Hunan Normal University, Changsha, China
- 4 School of Mathematical Sciences, East China Normal University, Shanghai, China
- 5 Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, Shanghai, China

## Introduction

1. What do preservice mathematics teachers know about mathematical problem solving and its teaching?

## Literature review

## Mathematical problem solving

## Preservice mathematics teachers

## Problem solving of preservice mathematics teacher

## Materials and methods

## Participants

## Data collection and analysis

## Research tools

Table 1. Typical student beliefs about the nature of mathematics ( Schoenfeld, 1992 ).

Table 2. Problem solving and its teaching cognition measurement framework.

The items of the entire questionnaire are as follows:

A. Strongly disagree B. Disagree C. Neutral D. Agree E. Strongly agree

• s2. Problem solving is calculating the result according to the conventional algorithm.

• s4. Problem solving is a skill that has little meaning after a student graduates.

• s6. Teaching students to solve problems is to teach students ready-made solutions.

• s8. A good problem solver must have a good knowledge structure.

• s9. Emotional attitudes have an important impact on problem solving.

• s10. Excellent problem solvers can constantly adjust their thinking to solve problems.

• s11. Good problem solvers use heuristics well and experiment with different strategies.

• s12. Excellent problem solvers have a good sense of experience.

## Reliability and validity

Table 3. Basic statistics for the recognition of in-service teachers.

Figure 1. Bar graph of in-service teacher scores.

Figure 2. Cognitive model of problem solving and its teaching.

Table 4. One-sample Kolmogorov–Smirnov normality test.

Table 5. Cognition of in-service teachers.

## Descriptive analysis

Table 6. Cognition of preservice mathematics teachers.

Table 7. Percentage of specific options for preservice mathematics teachers (%).

## Difference analysis

Table 8. Non-parametric independent samples tests (Mann–Whitney U test).

## Data availability statement

## Ethics statement

## Author contributions

## Conflict of interest

## Publisher’s note

CrossRef Full Text | Google Scholar

PubMed Abstract | CrossRef Full Text | Google Scholar

Polya, G. (1945). How to Solve It. Princeton: Princeton University Press.

Polya, G. (1962). Mathematical Discovery. New York, NY: Wiley.

Polya, G. (2002a). The goals of mathematical education: Part one. Math. Teach. 181, 6–7.

Polya, G. (2002b). The goals of mathematical education: Part two. Math. Teach. 181, 42–44.

Schoenfeld, A. H. (1985). Mathematical Problem Solving. Cambridge: Academic Press.

Received: 20 July 2022; Accepted: 18 October 2022; Published: 03 November 2022.

*Correspondence: Peijie Jiang, [email protected] ; Bin Xiong, [email protected]

## This article is part of the Research Topic

Psychological Studies in the Teaching, Learning and Assessment of Mathematics

## IMAGES

## VIDEO

## COMMENTS

Which statement about the teaching through problem solving approach is most accurate? Use of complex and non-algorithmic thinking. Selecting problem solving

2) Which statement about the teaching through problem solving approach is most accurate? A) Requires a four-step approach to problem solving.

Which statement about the teaching through problem solving approach is most accurate? A) It is closely associated with teaching for problem

2) Which statement about the teachingthroughproblem solving approach is most accurate?A) Requires a four-step approach to problem solving.

Chapter 3: Teaching Through Problem Solving Which statement best reflects the approach of teachingforproblem solving? It usually involves students

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students.

Which statement about the teaching through problem solving approach is most ... Which is the most accurate statement regarding posing a worthwhile problem?

Shifts have been made from the teacher talking most of time to. ... To teaching about and through problem solving... identify the shifts that are occurring

Teaching students to systematically use a visual representation to solve word ... to solve word problems are more likely to solve the problems accurately.

Preservice mathematics teachers' accurate understanding of mathematical problem solving and its teaching is key to the performance of their