problem solving method in mathematics ppt

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

Module 1: Problem Solving Strategies

• Last updated
• Save as PDF
• Page ID 10352

Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.

Pólya’s How to Solve It

George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985.1

1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY

In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:

First, you have to understand the problem.

After understanding, then make a plan.

Carry out the plan.

Look back on your work. How could it be better?

This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.

Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!

Problem Solving Strategy 1 (Guess and Test)

Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.

Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet?

Step 1: Understanding the problem

We are given in the problem that there are 25 chickens and cows.

All together there are 76 feet.

Chickens have 2 feet and cows have 4 feet.

We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.

Step 2: Devise a plan

Going to use Guess and test along with making a tab

Many times the strategy below is used with guess and test.

Make a table and look for a pattern:

Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.

Step 3: Carry out the plan:

Notice we are going in the wrong direction! The total number of feet is decreasing!

Better! The total number of feet are increasing!

Step 4: Looking back:

Check: 12 + 13 = 25 heads

24 + 52 = 76 feet.

We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.

Videos to watch:

1. Click on this link to see an example of “Guess and Test”

http://www.mathstories.com/strategies.htm

2. Click on this link to see another example of Guess and Test.

http://www.mathinaction.org/problem-solving-strategies.html

Check in question 1:

Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)

Check in question 2:

Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)

Problem Solving Strategy 2 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric thinking visually can help!

Videos to watch demonstrating how to use "Draw a Picture".

1. Click on this link to see an example of “Draw a Picture”

2. Click on this link to see another example of Draw a Picture.

Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)

Gauss's strategy for sequences.

last term = fixed number ( n -1) + first term

The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.

Ex: 2, 5, 8, ... Find the 200th term.

Last term = 3(200-1) +2

Last term is 599.

To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2

Sum = (2 + 599) (200) then divide by 2

Sum = 60,100

Check in question 3: (10 points)

Find the 320 th term of 7, 10, 13, 16 …

Then find the sum of the first 320 terms.

Problem Solving Strategy 4 (Working Backwards)

This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.

Videos to watch demonstrating of “Working Backwards”

https://www.youtube.com/watch?v=5FFWTsMEeJw

Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?

1. We start with 11 and work backwards.

2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.

3. The opposite of doubling something is dividing by 2. 18/2 = 9

4. This should be our answer. Looking back:

9 x 2 = 18 -7 = 11

5. We have the right answer.

Check in question 4:

Christina is thinking of a number.

If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)

Problem Solving Strategy 5 (Looking for a Pattern)

Definition: A sequence is a pattern involving an ordered arrangement of numbers.

We first need to find a pattern.

Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?

Example 1: 1, 4, 7, 10, 13…

Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.

Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.

So the next number would be

25 + 11 = 36

Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.

In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5

-5 – 3 = -8

Example 4: 1, 2, 4, 8 … find the next two numbers.

This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?

So each number is being multiplied by 2.

16 x 2 = 32

1. Click on this link to see an example of “Looking for a Pattern”

2. Click on this link to see another example of Looking for a Pattern.

Problem Solving Strategy 6 (Make a List)

Example 1 : Can perfect squares end in a 2 or a 3?

List all the squares of the numbers 1 to 20.

1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.

Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.

How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?

Quarter’s dimes

0 3 30 cents

1 2 45 cents

2 1 60 cents

3 0 75 cents

Videos demonstrating "Make a List"

Check in question 5:

How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)

Problem Solving Strategy 7 (Solve a Simpler Problem)

Geometric Sequences:

How would we find the nth term?

Solve a simpler problem:

1, 3, 9, 27.

1. To get from 1 to 3 what did we do?

2. To get from 3 to 9 what did we do?

Let’s set up a table:

Term Number what did we do

Looking back: How would you find the nth term?

Find the 10 th term of the above sequence.

Let L = the tenth term

Problem Solving Strategy 8 (Process of Elimination)

This strategy can be used when there is only one possible solution.

I’m thinking of a number.

The number is odd.

It is more than 1 but less than 100.

It is greater than 20.

It is less than 5 times 7.

The sum of the digits is 7.

It is evenly divisible by 5.

a. We know it is an odd number between 1 and 100.

b. It is greater than 20 but less than 35

21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.

c. The sum of the digits is 7

21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.

Check in question 6: (8 points)

Jose is thinking of a number.

The number is not odd.

The sum of the digits is divisible by 2.

The number is a multiple of 11.

It is greater than 5 times 4.

It is a multiple of 6

It is less than 7 times 8 +23

What is the number?

Click on this link for a quick review of the problem solving strategies.

https://garyhall.org.uk/maths-problem-solving-strategies.html

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

5 Teaching Mathematics Through Problem Solving

Janet Stramel

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. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. 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 real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

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.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

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.

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

• 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

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

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

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

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

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

• 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

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

• 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.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

• Inside Mathematics Number Talks
• Number Talks Build Numerical Reasoning

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

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:

• “I think I can do this.”
• “I have an idea I want to try.”
• “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

• 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 ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

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

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

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

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

Share This Book

• My presentations

Auth with social network:

Download presentation

We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!

Presentation is loading. Please wait.

Problem Solving Strategies

Published by Tyler Perkins Modified over 8 years ago

Similar presentations

Presentation on theme: "Problem Solving Strategies"— Presentation transcript:

An introduction to Problem Solving Math 110 Iris Yang.

M ATHEMATICAL P ROBLEM S OLVING What is it? Why encourage it? How is teaching like a problem-solving endeavor?

Chapter 1 The Art of Problem Solving © 2008 Pearson Addison-Wesley. All rights reserved.

Kevin Cummins The maths toolbox is a set of strategies that students can put into place to solve mathematical problems. The purpose.

Polya’s Four Step Problem Solving Process

Welcome to MATH 302A Please find the index card with your name on it and sit there. On the other side of the index card, write: Name as you wish to be.

MAKING PROBLEM SOLVING LESS PROBLEMATIC

Exploration 1.1 With 24 people in class, including yourself, if each person shakes hands with every person, how many handshakes will there be?

Problem Solving The process of applying previously acquired knowledge to new and unfamiliar situations.

Correlating Go Math & Standards for Mathematical Practices

SASE Contextualised group work – teaching a broader mathematics curriculum to first year science students: Case study – Problem solving Jo-ann Larkins.

Problem Solving Tool: KFC.

Test Preparation Strategies

Survey of Mathematical Ideas Math 100 Chapter 1 John Rosson Tuesday January 23, 2007.

M ATH C OMMITTEE Mathematical Shifts Mathematical Practices.

PROBLEM SOLVING in Math What you need to know. OVERVIEW: Define “Problem” Where do I start… How can I solve problems… Trying strategies –Patterns –Tables.

Problem solving Math 123. Why problem solving? Essential for mathematics According to NCTM, one of the processes through which mathematics should be.

Helping Children with Problem Solving CHAPTER 6 Tina Rye Sloan To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons.

Helping Children with Problem Solving CHAPTER 6 Tina Rye Sloan To accompany Helping Children Learn Math10e, Reys et al. ©2012John Wiley & Sons.

Buckland CE Primary School

About project

© 2024 SlidePlayer.com Inc. All rights reserved.

Got any suggestions?

We want to hear from you! Send us a message and help improve Slidesgo

Top searches

Trending searches

solar eclipse

25 templates

education technology

180 templates

32 templates

28 templates

thanksgiving

38 templates

Problem-Solving Strategies - Math - 1st Grade

Problem-solving strategies - math - 1st grade presentation, free google slides theme and powerpoint template.

Download the "Problem-Solving Strategies - Math - 1st Grade" presentation for PowerPoint or Google Slides and easily edit it to fit your own lesson plan! Designed specifically for elementary school education, this design features vibrant colors, engaging graphics, and age-appropriate fonts; elements that capture the students' attention and make the learning experience more enjoyable and stimulating. Provide a cohesive visual identity of your lessons to create a sense of familiarity and organization with this editable Google Slides theme and PowerPoint template!

Features of this template

• 100% editable and easy to modify
• Different slides to impress your audience
• Contains easy-to-edit graphics such as graphs, maps, tables, timelines and mockups
• Includes 500+ icons and Flaticon’s extension for customizing your slides
• Designed to be used in Google Slides and Microsoft PowerPoint
• Includes information about fonts, colors, and credits of the resources used

How can I use the template?

Am I free to use the templates?

How to attribute?

Attribution required If you are a free user, you must attribute Slidesgo by keeping the slide where the credits appear. How to attribute?

Related posts on our blog.

How to Add, Duplicate, Move, Delete or Hide Slides in Google Slides

How to Change Layouts in PowerPoint

How to Change the Slide Size in Google Slides

Related presentations.

Premium template

Unlock this template and gain unlimited access

Mathematical Problem Solving

Apr 03, 2019

300 likes | 449 Views

Mathematical Problem Solving. What is it? Why encourage it? How is teaching like a problem-solving endeavor?. What Types of Questions Are Asked in Math Classrooms?. recognition/recall : a situation that can be resolved by recalling specific facts from memory

Share Presentation

• traditional textbook word problems
• instructional decisions
• typical authentic problems
• entry points

Presentation Transcript

Mathematical Problem Solving What is it? Why encourage it? How is teaching like a problem-solving endeavor?

What Types of Questions Are Asked in Math Classrooms? • recognition/recall: a situation that can be resolved by recalling specific facts from memory • drill/exercise: a situation that involves following a step-by-step procedure or algorithm • simple translation/application: traditional textbook word problems which involve translating the words into a single equation, the format of which has typically been taught directly • complex translation/application: similar to a simple translation, but it involves at least two steps

nonroutine/process problem: solutions require the use of thinking processes; a situation that does not have an apparent solution path available–there is no set format available for finding a solution • applied problem: real-world, or at least realistic, situations where solutions require the use of mathematical skills, facts, concepts, and procedures. • open-ended problem: a situation in which the student is often required to identify interesting questions to pursue and/or to identify...not only is there no apparent solution path available, often there is no direct question to purse

So . . . which of these types of questions are “problems?”

We first need to ask: What is a problem? A problem is a difficult question; a matter of inquiry, discussion, or thought; a question that exercises the mind. . . . . and

It is a situation or task for which: • the person confronting the task wants or needs to find a solution. • the person has no readily available procedure for finding the solution. • the person makes an attempt to find the solution.

So . . . not all categories of questions are problems . . . When is the math or task “Problematic?”

What It Means for Math to be “Problematic” “Allowing the subject to be problematic means allowing students to wonder why things are, to inquire, to search for solutions, and to resolve incongruities. It means that both curriculum and instruction should begin with problems, dilemmas, and questions for students.” “Problem Solving as a Basis for Reform in Curriculum and Instructions: The Case of Mathematics” Hiebert, et al. Educational Researcher May, 1996 (p. 12)

Why encourage problem solving?

The Values of Teaching With Problems • Places the focus of attention on ideas and sense making rather than on following the directions of the teacher. • Develops the belief in students that: • they are capable of doing mathematics. • mathematics makes sense. • Provides ongoing assessment data so that: • instructional decisions can be made. • students can be helped to succeed. • parents can be informed.

Develops “mathematical power.” All five + 1 process standards are likely to be employed. • Allows entry points for a wide range of students. Good problems have multiple entry points. • Engages students so that there are fewer discipline problems. • It’s just a lot of fun . . . for both teachers and students!!!!!

An Idea Worth Considering: • Most mathematical ideas can be taught via problem solving. • That is— • Tasks or problems are posed to engage students in thinking about and developing the intended mathematical ideas.

What is George Polya’s Problem Solving Framework? • Understand the Problem • Involves not only applying the skills necessary for literary reading, but determining what is being asked.

Devise a Plan • Involves finding or devising a strategy to help in solving the problem and answering the question.

Carry Out the Plan • Involves attempting to solve the problem with some chosen strategy, recording thinking to keep an accurate record of work.

4. Look Back • Involves interpreting the solution in terms of the original problem . . . Does it make sense? Is it reasonable?

5.Look Forward • Involves asking the “next question,” generalizing the solution, or determining another method of finding the solution.

How is teaching mathematics like a “Problem-Solving Endeavor?” • Understand goals for teaching—be clear about what you want to accomplish. • Key question: What mathematics do I want the children to understand and develop?

Plan the tasks—select problems, explorations, or activities that will most likely require children to wrestle meaningfully and thoughtfully with the math you identified as important. • Key question: How can this problem (or activity) be used to help promote reflective thinking about the math I want children to learn?

Implement your plan—facilitate activities and focus appropriately on the needs and strengths of your individual students. • Key question: What “teacher moves” and “math talk” might I use to help develop understanding?

Look back—through active listening in a student-centered classroom, you can learn what students know and how they understand the ideas being discussed. • Key question: Based on what I learn about what my students understand, what direction do I need to go in my future planning for instruction?

Problem-Solving Strategies • Guess and check • Look for a pattern • Make a list • Solve a simpler version • Draw a picture

Use symmetry • Make a table or chart • Do a simulation • Look for a formula • Consider cases

Model the situation • Work backward • Write an open sentence • Use logical reasoning • Consider all possibilities • Change your point of view

So . . . . . . how do “traditional problems” differ from those that are typically more authentic—more closely related to real life?

Differences Between Traditional Word Problems and Many Real-Life Problems Typical Textbook Problems • The problem is given. • All the information you need to solve the problem is given. • There is always enough information to solve the problem. Typical Authentic Problems • Often, you have to figure out what the problem really is. • You have to determine the information needed to solve the problem. • Sometimes you will find that there is not enough info to solve the problem.

Typical Textbook Word Problems Typical Authentic Problems • There is typically no extraneous information. • The answer is in the back of the book . . . • There is usually a right or best way to solve the problem. • Sometimes there is too much information--must decide what is needed. • You, or your team, decides whether your answer is valid. • There are usually many different ways to solve the problem.

Goals for Problem Solving • Strategy and Process Goals • Develop problem analysis skills • Develop and be able to select strategies • Justify solutions • Extend and generalize problems

MetacognitiveGoals • Monitor and regulate actions • Determine when to try a new strategy • Determine when a solution leads to a correct solution

Attitudinal Goals • Gain confidence and belief in abilities • Be willing to try and to persevere • Enjoy doing mathematics

• More by User

Problem Solving

Problem Solving Views of Problem solving Well-defined problems Much studied in AI Requires search Domain general heuristics for solving problems What about ill-defined problems? No real mechanisms for dealing with these

1.26k views • 33 slides

Enhancing Mathematical Problem Solving for At-Risk Students

Enhancing Mathematical Problem Solving for At-Risk Students. Lynn Fuchs Vanderbilt University December 10, 2008. Principles of Effective Instruction for At-Risk Learners. * Address all valued outcomes (don’t assume transfer will occur). * Incorporate explicit, systematic instruction.

592 views • 27 slides

Problem solving

Problem solving. Problem solving: cognitive processes focused on achieving a specific goal. Strategies of problem solving: Trial and error, algorithms and heuristics Ill-defined vs. Well-defined problems. Heuristics vs. algorithms.

955 views • 22 slides

Problem Solving. From Conceptual Blockbusting , 4 th Edition, by James L. Adams. Solve This Puzzle.

925 views • 17 slides

Mathematical Modeling and Engineering Problem solving

1. Mathematical Modeling and Engineering Problem solving. Chapra : Chapter-1. Mathematical Modeling and Engineering Problem solving. Requires understanding of engineering systems By observation and experiment Theoretical analysis and generalization

1.45k views • 37 slides

Problem Solving revisited (Worthwhile Mathematical Tasks)

Last time together October 23. Think back over your lessons form October 24 to December 10 and pick a lesson or part of a lesson that went really well. Problem Solving revisited (Worthwhile Mathematical Tasks). December 11, 2013 Algebra and 8 th Grade Kimberly Tarnowieckyi.

331 views • 17 slides

Problem Solving. Steps and Vocabulary Martha Rice. Problems are all around!. You can use the same problem solving methods to solve just about any problem, from word problems to logic problems to real-world problems in your own life. Step 1: Evaluate the evidence.

602 views • 17 slides

Introduction to problem solving and Mathematical Models

Introduction to problem solving and Mathematical Models. Wild About Harry.

258 views • 9 slides

Introduction to (Mathematical) Problem Solving

Introduction to (Mathematical) Problem Solving. Dindin Abdul Muiz Lidinillah , S.Si ., S.E., M.Pd . Elementary School Teacher Education Program Education University of Indonesia – Tasikmalaya Campus.

481 views • 32 slides

Chapter 1 Problem Solving with Mathematical Models

Chapter 1 Problem Solving with Mathematical Models. Mathematical Model . A mathematical model is the collection of variables and relationships needed to describe pertinent features of such a problem. Operations Research (OR) [1.1]

693 views • 45 slides

Problem Solving. Problem Solving Approach. Break it down into smaller, manageable pieces Procedural approach: Break it down into a sequence of steps Consider tasks to be accomplished. Problem Solving Approach. Break it down into smaller, manageable pieces Procedural approach:

285 views • 7 slides

Problem solving. Original Source : http://www.ftsm.ukm.my/zma/TK1914/05-Algorithms and Problem Solving.ppt. Problem Solving. Programming is a process of problem solving Problem solving techniques Analyze the problem Outline the problem requirements

1.22k views • 20 slides

Mathematical Problem Solving. 7 White Math 10/26/2006. Why Use Problem Solving?. *Develop your PS Skills - the more practice you get, the better you get! *It positions you for future problem solving success. *It makes math meaningful!. Solutions .

427 views • 20 slides

Chapter 1 Mathematical Modeling and Engineering Problem solving

Chapter 1 Mathematical Modeling and Engineering Problem solving. Requires understanding of engineering systems By observation and experiment Theoretical analysis and generalization

558 views • 32 slides

Mathematical Modeling and Engineering Problem solving Chapter 1

Mathematical Modeling and Engineering Problem solving Chapter 1. Every part in this book requires some mathematical background. Computers are great tools, however, without fundamental understanding of engineering problems, they will be useless.

335 views • 10 slides

Working Memory, Attention, and Mathematical Problem Solving:

Working Memory, Attention, and Mathematical Problem Solving: A longitudinal study of Grade 1 Children at Risk and Not at Risk for Serious Math Difficulties H. Lee Swanson University of California-Riverside June, 2010. Key Contributors. Dr. Margaret Beebe-Frankenberger, Project Director

327 views • 30 slides

Problem-Solving and Mathematical Reasoning

Problem-Solving and Mathematical Reasoning. K-5 Module. Your Opinion. What does it mean to do mathematics? How do you know if you are good at it?. Process is as much a part of doing mathematics as the content itself. Process Is Important. Content Standards Algebra Number Sense Geometry

724 views • 68 slides

Problem Solving and Programming – Problem Solving

Dr Scott Turner. Problem Solving and Programming – Problem Solving. Why do you needed to develop problem solving skills?. One definition of programming is it is applied problem solving You have a problem (E.g Need a program to calculate the area of the circle). What needs to be done?

683 views • 29 slides

Problem Solving. We now have enough tools to start solving some problems. For any problem, BEFORE you start writing a program, determine: What are the input values that are needed from the user? What results (outputs) do we need to determine? What other values are needed?

561 views • 39 slides

Problem Solving. Intro to Computer Science CS1510 Dr. Sarah Diesburg. Numeric representation. We usually work with decimal numbers with digits from 0 to 9 and powers of 10 7313 = (7 * 1000 + 3 * 100 + 1 * 10 + 3 * 1) Or (7 * 10 3 + 3 * 10 2 + 1 * 10 1 + 3 * 10 0 )

352 views • 12 slides

MAT 331 Mathematical Problem Solving with Computers

MAT 331 Mathematical Problem Solving with Computers. Spring 2010. Ctr-alt-F1 Creates account Crtl-alt-F7 back to initial screen. Choose bash. Instructor: Moira Chas. Assistant Professor Department of Mathematics Best way to contact me: [email protected]

172 views • 17 slides

ORIGINAL RESEARCH article

This article is part of the research topic.

Research on Teaching Strategies and Skills in Different Educational Stages

BIOTECHNOLOGY PROJECT-BASED LEARNING ENCOURAGES LEARNING AND MATHEMATICS APPLICATION Provisionally Accepted

• 1 Universidad Técnica Particular de Loja, Ecuador

The final, formatted version of the article will be published soon.

Project-based learning (PBL) is a promising approach to enhance mathematics learning concepts in higher education. Here, teachers provide guidance and support to PBL implementation. The objective of this study was to develop PBL-based biotechnological projects as a strategy for mathematics learning. The methodology design was applied to 111 university students from Biochemical, Chemical Engineering and Business Administration careers. Knowledge, skills, perceptions, and engagement were measured through questionnaires, workshops, rubrics, and survey instruments. As a result, the paired comparison between tests, questionnaires and project shows significant differences (P<.001) between the experimental group and the control group. It is concluded that the teaching of mathematics should be oriented to the development of competencies, abilities, and skills that allow students to generate real solutions and broaden their vision of the applicability of their knowledge using new learning strategies. Key words: Mathematical models, Biotechnology, Project based learning.Science, technology, engineering, and mathematics (STEM) education has become a crucial topic both inside and outside of school (Han et al., 2015). Currently, mathematics learning tends to be oriented towards textbooks, and students can only work on math problems based on what the teacher exemplifies; however, if given different contextbased problems, they will have difficulty solving them (Fisher et al., 2020). Likewise, the traditional classroom model does not encourage student's interest in STEM (Sahin, 2009) The research gap is between what students learn at the university and what they really need in the workplace (Holmes et al., 2015). Higher education institutions have been trying to provide students with both (i) hard skills, such as cognitive knowledge and professional skills (Vogler et al., 2018), and (ii) soft skills, such as problem-solving and teamwork (Lennox and Roos 2017). However, these skills are difficult to achieve through traditional learning. One learning that creates an active, collaborative atmosphere, and can increase selfconfidence in students is Project-based learning (PBL) (Cruz et al., 2022; Guo et al., 2022;Markula and Aksela 2022). The PBL method is applied as a teaching model that involves

Keywords: Mathematical Models, Biotechnology, Project Based Learning (PBL), Learning, Learning mathematics activities

Received: 02 Jan 2024; Accepted: 03 Apr 2024.

Copyright: © 2024 Vivanco and Jiménez-Gaona. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mr. Oscar A. Vivanco, Universidad Técnica Particular de Loja, Loja, Ecuador

People also looked at