Chapter 3 Teaching Through Problem Solving Copyright © Allyn and Bacon 2010 This multimedia product and its contents are protected under copyright law. The following are prohibited by law: any public performance or display, including transmission of any image over a network preparation of any derivative work, including the extraction, in whole or in part, of any images any rental, lease, or lending of the program
Copyright © Allyn and Bacon 2010 Problem Solving Teaching for problem solving — teaching skills, then providing problems to practice those skills Teaching about problem solving — teaching strategies Teaching through problem solving — teaching content through problems (focus of this chapter & theme of the book!)
Copyright © Allyn and Bacon 2010 Teaching Through Problem Solving Most, if not all, important mathematics concepts and procedures can best be taught through problem solving
Copyright © Allyn and Bacon 2010 “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.” From J. Hiebert et al., “Problem Solving as a Basis for Reform in Curriculum and Instruction: The Case of Mathematics,” Educational Researcher, 25(4)(1996), p. 12.
Copyright © Allyn and Bacon 2010 What Is a Problem? A Problem: 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. From J. Hiebert et al., Making Sense: Teaching and Learning Mathematics with Understanding (Portsmouth, NH: Heinemann, 1997).
Copyright © Allyn and Bacon 2010 It must begin where the students are. The problematic or engaging aspect of the problem must be the mathematics that students are to learn. It must require justifications and explanations for answers and methods. Features
Copyright © Allyn and Bacon 2010 Teaching Using a Problem-Based Approach Which of the following can be used in a problem based approach to teaching? A.42 − 19 = ____ B. How many times can a hiker fill a 0.5-liter water bottle from the 10-liter supply tank? C. The area of a rectangle is given by the relation A = b × h, where A = area, b = base, and h = height. Find the formula for the areas of any triangle, any parallelogram, and any trapezoid. You may use any tools you like.
Copyright © Allyn and Bacon 2010 Problems and Tasks for Learning Mathematics Begin where the students are The problematic or engaging aspect of the problem must be due to the mathematics that the students are to learn Require justifications and explanations for answers and methods
Copyright © Allyn and Bacon 2010 A Shift in the Role of Problems Away from: Explain, practice, then story problems One way (the teacher’s way) Show and tell approach with the student as a passive learner Problem-solving problems as a separate activity Expecting explicit directions for how to solve problems
Copyright © Allyn and Bacon 2010 A Shift in the Role of Problems Good problems start with ideas that students already have. Students learn mathematics as a result of problem solving. Children are learning mathematics by doing it!
Copyright © Allyn and Bacon 2010 The Value of Teaching Through Problem Solving The focus of the students’ attention on ideas and sense making Develops the belief in students that they are capable of doing mathematics and that mathematics makes sense! Provides a context to help students build meaning for the concept Allows an entry point for a wide range of students
Copyright © Allyn and Bacon 2010 Provides ongoing assessment data that can be used to make instructional decisions, help students succeed, and inform parents Allows for extensions and elaborations Engages students so that there are fewer classroom discipline problems Develops “mathematical power” It is a lot of fun! The Value of Teaching Through Problem Solving (Continued)
Copyright © Allyn and Bacon 2010 Selecting or Designing Problem-Based Tasks Multiple Entry Points Provides the opportunity for students to work on the problem using their own ideas at their own level Allows for more than one correct way to work the problem
Copyright © Allyn and Bacon 2010 Examples of Problem-Based Tasks Procedures and Processes Students develop procedures via a problem- solving approach. They are engaged in the process of figuring out mathematics, not just accepting procedures blindly.
Copyright © Allyn and Bacon 2010 Examples of Problem- Based Tasks Concept: Partitioning Grades: K–1 Think about the number 6 broken into two different amounts. Draw a picture to show a way that six things can be broken in two parts. Think up a story to go with your picture. Concept: Estimating Fractions Greater Than 1 Grades: 4–6 Place an X on the number line about where 11⁄8 would be. Explain why you put your X where you did. Perhaps you will want to draw and label other points on the line to help explain your answer. |———————————| 0 2
Copyright © Allyn and Bacon 2010 Concept: Comparing Ratios and Proportional Reasoning: Grades: 6–8 Jack and Jill were at the bottom of a hill, hoping to fetch a pail of water. Jack walks uphill at 5 steps every 25 seconds, while Jill walks uphill at 3 steps every 10 seconds. Assuming constant walking rate, who will get to the pail of water first?
Copyright © Allyn and Bacon 2010 Selecting or Designing Problem- Based Tasks (Continued) Creating Meaningful and Engaging Contexts Allows integration of children’s literature to build context Makes it possible to integrate natural links to other disciplines
Copyright © Allyn and Bacon 2010 How to Find Quality Tasks and Problem-Based Lessons A Task Selection Guide How is the activity done? What is the purpose of the activity? Can the activity accomplish your learning goals? What must you do? Good tasks: check out
Copyright © Allyn and Bacon 2010 How to Find Quality Tasks and Problem-Based Lessons (Continued) A standards-based curriculum provides an increased emphasis on learning through problem solving Traditional textbooks can be used to adapt a non-problem-based lesson by using the best lessons or using the main ideas of the chapter
Copyright © Allyn and Bacon 2010 Activity Evaluation and Selection Guide STEP 1: How Is the Activity Done? Actually do the activity. Try to get “inside” the task or activity to see how it is done and what thinking might go on. How would children do the activity or solve the problem? What materials are needed? What is written down or recorded? What misconceptions may emerge?
Copyright © Allyn and Bacon 2010 Activity Evaluation and Selection Guide STEP 2: What Is the Purpose of the Activity? What mathematical ideas will the activity develop? Are the ideas concepts or procedural skills? Will there be connections to other related ideas?
Copyright © Allyn and Bacon 2010 Activity Evaluation and Selection Guide STEP 3: Can the Activity Accomplish Your Learning Goals? What is problematic about the activity? Is the problematic aspect related to the mathematics you identified in the purpose? What must children reflect on or think about to complete the activity? (Don’t rely on wishful thinking.) Is it possible to complete the activity without much reflective thought? If so, can it be modified so that students will be required to think about the mathematics?
Copyright © Allyn and Bacon 2010 Activity Evaluation and Selection Guide STEP 4: What Must You Do? What will you need to do in the before portion of your lesson? How will you activate students’ prior knowledge? What will the students be expected to produce? What might you anticipate seeing and asking in the during portion of your lesson? What will you want to focus on in the after portion of your lesson?
Copyright © Allyn and Bacon 2010 Teaching about Problem Solving This is an important part of teaching through problem solving
Copyright © Allyn and Bacon 2010 Four-Step Problem-Solving Process Understanding the problem Developing a plan Carrying out the plan Looking back
Copyright © Allyn and Bacon 2010 Four-Step Problem- Solving Process 1. Understanding the problem. This means figuring out what the problem is about— identifying what question or problem is being posed. 2. Devising a plan. This is thinking about how to solve the problem. Will you want to write an equation? Will you want to model the problem with a manipulative?
Copyright © Allyn and Bacon 2010 Four-Step Problem- Solving Process 3. Carrying out the plan. This is the implementation of the plan. 4. Looking back. This phase, arguably the most important as well as the one most often skipped by students, is the moment you determine whether the answer from step 3 answers the problem as originally understood in step 1. Does the answer make sense?
Copyright © Allyn and Bacon 2010 Problem-Solving Strategies Draw a picture Look for a pattern Guess and check Make a table or chart Try a simpler form of the problem Make an organized list Write an equation
Copyright © Allyn and Bacon 2010 Teaching in a Problem-Based Classroom Let students do the talking How much to tell and not to tell The importance of student writing Metacognition Disposition Additional goals
Copyright © Allyn and Bacon 2010 Teaching in the Problem-Based Classroom Let the students do the talking How did you solve the problem? Why did you solve it that way? Why do you think your solution is correct and makes sense?
Copyright © Allyn and Bacon 2010 How much to tell Mathematical conventions Carefully suggest alternative methods Clarification of students’ methods And not to tell Establish teacher- preferred methods Teacher thinking Teaching in the Problem-Based Classroom
Copyright © Allyn and Bacon 2010 Teaching in the Problem-Based Classroom Importance of student writing — a reflective process — can be a rehearsal for the discussion period — can serve as a written record that remains long after the lesson. Tools for writing — text editing — wikis — blogging tools
Copyright © Allyn and Bacon 2010 Metacognition— being aware of how and why you are doing something Disposition— attitudes and beliefs about (in this case) mathematics Teaching in the Problem-Based Classroom Attitudinal goals for student — gaining confidence — taking risks — enjoying doing math Attitudinal goals for teacher — build in success — praise effort and risk taking — listen to all students
Copyright © Allyn and Bacon 2010 A Three-Part Lesson Format Before: Getting Ready Activate useful prior knowledge. Be certain the problem is understood. Establish clear expectations. During: Students Work Let go! Avoid stepping in front of the struggle. Listen carefully. Provide appropriate hints. Observe and assess. After: Class Discussion Encourage a community of learners. Listen! Accept student solutions without evaluation. Summarize main ideas and identify future problems.
Copyright © Allyn and Bacon 2010 The Before Phase Activate useful prior knowledge Be certain the problem is understood Establish clear expectations
Copyright © Allyn and Bacon 2010 The During Phase Let go! Listen actively Provide appropriate hints Provide worthwhile extensions
Copyright © Allyn and Bacon 2010 The After Phase Encourage a community of learners that includes ALL children Listen actively without evaluation Summarize main ideas and identify future problems
Copyright © Allyn and Bacon 2010 Creating a Mathematical Community of Learners Encourage student to student dialogue. Request explanations to accompany all answers, both correct and incorrect. Call on students for their ideas, often calling first on the children who tend to be shy or lack the ability to express themselves well. Encourage students to ask questions.
Copyright © Allyn and Bacon 2010 Creating a Mathematical Community of Learners Be certain that your students understand what you understand. Occasionally ask those who understand to offer explanations for others. Move students for more conceptually based explanations when appropriate.
Copyright © Allyn and Bacon 2010 Frequently Asked Questions How might you respond to the following questions about using a problem-based approach to teaching mathematics? 1. How can I teach all the basic skills I have to teach? 2. Why is it okay for students to “tell” or “explain” but not for me?
Copyright © Allyn and Bacon 2010 Frequently Asked Questions 3. Is it okay to help students who have difficulty solving a problem? 4. Where can I find the time to cover everything? 5. How much time does it take for students to become a community of learners and really begin to share and discuss ideas?
Copyright © Allyn and Bacon 2010 Frequently Asked Questions 6. Can I use a combination of student- oriented, problem-based teaching with a teacher directed approach? 7. Is there any place for drill and practice? 8. What do I do when a task bombs?
Copyright © Allyn and Bacon 2010 Online Resources Annenberg/CPB: MathSolutions Lessons from the Classroom: wp9&crid=56 wp9&crid=56 ENC Online (Eisenhower National Clearinghouse): Writing and Communication in Mathematics: g_in_math g_in_math