## Terms in this set (44)

Students also viewed, mtt quiz #1.

## MTT - Quiz #1

## MTT 202-03 Quiz 1

## MTT 202-03 Quiz 2

Recent flashcard sets, topic 9 control systems (up to mock exam).

## community quiz (chp. 28)

## orthodontics 3

## urzedy rzymskie

## Other sets by this creator

The rate of vocabulary memorization of the average student in a foreign language course is given by

d v d t = 40 t + 1 \frac{d v}{d t}=\frac{40}{t+1} d t d v = t + 1 40

## Recommended textbook solutions

## Social Psychology

## Myers' Psychology for the AP Course

## Learning and Behavior

## Fundamentals of Psychology: Perspectives and Connections

## Biology Final Questions

## 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

## K-5 Math Centers

K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, math problem solving 101.

## Read the Problem Without Numbers & Ask Questions:

After reading the problem (without numbers) to the students, I asked the following questions:

- Can you describe what is happening in your own words?
- What is the main idea of the problem?
- How could you act this out?

## Make a Plan & Ask Questions:

- Sample Answers include- We know that Kai has some goldfish. Kai donated or gave away some of the goldfish.
- Sample Answers include – We need to know how many goldfish Kai has. We also need to know how many he gave anyway. We also need to know how many bowls there are.
- Sample Answers include- We need to find out how many fish belong in each bowl.

## Team Work Counts

## Benefits to Using this Process:

- Students understood what the problem is asking them to do
- Students are required to think and communicate as a team
- Students avoid making errors that can come with only using keywords
- Students are required to record their math reasoning using the problem solving template
- After using this process a couple of times, students get used to explaining and justifying their answers
- You become the facilitator of the learning by asking more questions, thereby making students independent thinkers

## Things to Consider Include:

- This process in NOT quick. It requires TIME. You should not rush the process and expect to have it completed in 20 – 30 minutes in one day.
- This process is not a one time lesson. Students may not get it the first time. It should be seen a routine that can be used when solving word problems.

Be sure to let me know how this process works in your classroom in the comments below.

## You might also like...

## Reflect and Reset: Tips for Becoming a Better Math Teacher

## Student Math Reflection Activities That Deepen Understanding

## 5 Math Mini-Lesson Ideas that Keep Students Engaged

## A Rigorous Elementary Math Curriculum for Busy Teachers

## What We Offer:

## The Lesson Study Group

## Teaching Through Problem-solving

## What is Teaching Through Problem-Solving?

## Why Teaching Through Problem-Solving?

As students build their mathematical knowledge, they also:

- Learn to reason mathematically, using prior knowledge to build new ideas
- See the power of their explanations and carefully written work to spark insights for themselves and their classmates
- Expect mathematics to make sense
- Enjoy solving unfamiliar problems
- Experience mathematical discoveries that naturally deepen their perseverance

## Phases of a TTP Lesson

## WHAT STUDENTS DO

- Understand the problem and develop interest in solving it.
- Consider what they know that might help them solve the problem.

## WHAT TEACHERS DO

- Show several student journal reflections from the prior lesson.
- Pose a problem that students do not yet know how to solve.
- Interest students in the problem and in thinking about their own related knowledge.
- Independently try to solve the problem.
- Do not simply following the teacher’s solution example.
- Allow classmates to provide input after some independent thinking time.
- Circulate, using seating chart to note each student’s solution approach.
- Identify work to be presented and discussed at board.
- Ask individual questions to spark more thinking if some students finish quickly or don’t get started.
- Present and explain solution ideas at the board, are questioned by classmates and teacher. (2-3 students per lesson)
- Actively make sense of the presented work and draw out key mathematical points. (All students)
- Strategically select and sequence student presentations of work at the board, to build the new mathematics. (Incorrect approaches may be included.)
- Monitor student discussion: Are all students noticing the important mathematical ideas?
- Add teacher moves (questions, turn-and-talk, votes) as needed to build important mathematics.
- Consider what they learned and share their thoughts with class, to help formulate class summary of learning. Copy summary into journal.
- Write journal reflection on their own learning from the lesson.
- Write on the board a brief summary of what the class learned during the lesson, using student ideas and words where possible.
- Ask students to write in their journals about what they learned during the lesson.

## How Do Teachers Support Problem-solving?

## Additional Readings

## Can’t find a resource you need? Get in touch.

- What is Lesson Study?
- Why Lesson Study?
- Teacher Learning
- Content Resources
- Teaching Through Problem-solving (TTP)
- School-wide Lesson Study
- U.S. Networks
- International Networks

- News & Calendar
- Career Center
- Get Involved
- Notice and Wonder
- The Year Game
- Back to School Resources
- Math Sightings
- Video Lessons
- Reasoning & Sense Making Task Library
- Student Explorations in Mathematics
- Problems of the Week Resources
- Problems to Ponder
- Illuminations
- Figure This!
- Continuing the Journey
- Activities with Rigor & Coherence - ARCs
- Designing Innovative Lessons and Activities

- Featured Books
- Mathematics Teacher: Learning and Teaching PK-12
- Journal for Research in Mathematics Education
- Mathematics Teacher Educator
- Legacy Journals and Blogs
- “Best of” Issues
- Institutional Subscriptions
- Write, Review, Referee
- Rights and Permissions
- JRME Editor Search
- Principles to Actions
- Common Core State Standards
- Principles and Standards
- Standards for Mathematics Teacher Preparation
- Curriculum Focal Points
- Focus in High School Mathematics
- More NCTM Standards
- Position Statements
- Catalyzing Change
- Policies and Recommendations
- Advocacy Toolkit
- Advocacy and Legislation
- ESSER - District Solutions
- Every Student Succeeds Act - ESSA Toolkit
- NCTM Social Justice and Equity Resources
- Research Briefs & Clips
- Linking Research & Practice
- Research Conference
- Research Monographs
- NCTM Annual Meeting and Exposition
- Professional Development Resources
- Regional Conferences & Expositions
- Virtual Conferences
- Past and Future Events
- Be a Speaker
- Professional Services
- Webcast Library
- Exhibit, Advertise, Sponsor
- NCTM District Solutions
- NCTM Teacher Education Program Review Training
- Implementing the Common Core Standards for Mathematical Practice
- Funding Opportunities
- About Mathematics Education Trust
- Browse All Grants
- Giving Opportunities
- Special Events
- Lifetime Achievement Award
- Social Justice and Mathematics
- Individuals
- Schools and Districts
- Group Content Access

Problem Solving: An Approach to Understanding and Critiquing Our World

Problem Solving: An Approach to Understanding and Critiquing Our World July 2022

- supports making connections across disciplines;
- prepares students for future professional opportunities;
- develops students’ positive mathematical identity;
- is a matter of equity and access;
- builds students’ confidence, persistence, flexibility, creativity, perseverance, and curiosity;
- gives students voice and promotes discussion;
- shifts math authority to students; and
- has a positive effect on learning.

(NCTM 2000; 2014, 2018, 2020a, 2020b)

Trena Wilkerson NCTM President @TrenaWilkerson

Goris, Brittany. 2017. Hypatia: Explorer of Geometry. Seattle, WA: Girls Rock Math.

## Mathematics as a Complex Problem-Solving Activity

By jacob klerlein and sheena hervey, generation ready.

“Problem-solving is not only a goal of learning mathematics, but also a major means of doing so.”

## Learning to problem solve

## Beliefs underpinning effective teaching of mathematics

- Every student’s identity, language, and culture need to be respected and valued.
- Every student has the right to access effective mathematics education.
- Every student can become a successful learner of mathematics.

## Why is problem-solving important?

- The ability to think creatively, critically, and logically
- The ability to structure and organize
- The ability to process information
- Enjoyment of an intellectual challenge
- The skills to solve problems that help them to investigate and understand the world

## Problems that are “Problematic”

- Are accessible and extendable
- Allow individuals to make decisions
- Promote discussion and communication
- Encourage originality and invention
- Encourage “what if?” and “what if not?” questions
- Contain an element of surprise (Adapted from Ahmed, 1987)

- Understand and explore the problem
- Find a strategy
- Use the strategy to solve the problem
- Look back and reflect on the solution

## Pólya’s Principals of Problem-Solving

Students move forward and backward as they move through the problem-solving process.

## Getting real

## Planning for talk

## People also looked at

- 1 Department of Education, Uppsala University, Uppsala, Sweden
- 2 Department of Education, Culture and Communication, Malardalen University, Vasteras, Sweden
- 3 School of Natural Sciences, Technology and Environmental Studies, Sodertorn University, Huddinge, Sweden
- 4 Faculty of Education, Gothenburg University, Gothenburg, Sweden

## Introduction

## The Present Study

a) What is the effect of CL approach on students’ problem-solving in mathematics?

## Participants

FIGURE 1 . Flow chart for participants included in data collection and data analysis.

TABLE 1 . Background characteristics of classes and teachers in intervention and control groups.

## Intervention

## Implementation of the Intervention

## Control Group

## Tests of Mathematical Problem-Solving

## Measures of Peer Acceptance and Friendships

## Statistical Analyses

## What Is the Effect of the CL Approach on Students’ Problem-Solving in Mathematics?

## Is Social Acceptance and Friendships Associated With the Effect of CL on Students’ Problem-Solving in Mathematics?

## Limitations

## Implications

## Data Availability Statement

## Ethics Statement

## Author Contributions

The project was funded by the Swedish Research Council under Grant 2016-04,679.

## Conflict of Interest

## Publisher’s Note

## Acknowledgments

We would like to express our gratitude to teachers who participated in the project.

## Supplementary Material

CrossRef Full Text | Google Scholar

PubMed Abstract | CrossRef Full Text | Google Scholar

Received: 15 May 2021; Accepted: 09 August 2021; Published: 24 August 2021.

*Correspondence: Nina Klang, [email protected]

## IMAGES

## VIDEO

## COMMENTS

The six steps of problem solving involve problem definition, problem analysis, developing possible solutions, selecting a solution, implementing the solution and evaluating the outcome. Problem solving models are used to address issues that...

The answer to any math problem depends on upon the question being asked. In most math problems, one needs to determine a missing variable. For instance, if a problem reads as 2+3 = , one needs to figure out what the number after the equals ...

To calculate percentages, convert the percentage to a decimal and multiply it by the number in the problem. For example, to find 40 percent of 50, change it to 0.40 times 50, which gives you the result of 20.

Problem solving in elementary mathematics focuses on: engaging students in interesting, meaningful problems. Common Core State Standards includes all of the

b) meant to be used to solve math problems when everything else fails

Question 13 / 3 ptsProblem solving in elementary mathematics focuses on:students answering computational problems quickly.teaching problem solving

focus on teaching strategies and conceptual understanding.

al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving

Typically some students focus solely on keywords when solving word problems, but I do not advise using this approach exclusively. With math problems

In Teaching Through Problem-solving (TTP), students learn new mathematics by solving problems. Students grapple with a novel problem, present and discuss

A classroom environment that promotes problem solving provides opportunities for creativity, supports developing a positive mathematics identity for all

They imply that solving mathematics problems is a procedure to be memorized, practiced, and habituated. 4. They lead to an emphasis on answer getting. These

Students acquire their understanding of mathematics and develop problem-solving skills as a result of solving problems, rather than being taught something

Mathematical problem-solving constitutes an important area of mathematics instruction, and there is a need for research on instructional