CHAPTER 3 Teaching Through Problem Solving Elementary and Middle School Mathematics Teaching Developmentally Ninth Edition Van de Walle, Karp and Bay-Williams Developed by E. Todd Brown /Professor Emeritus University of Louisville

Problem Solving Teaching for problem solving — teaching skills, then providing problems to practice those skills (explain-practice- apply) Teaching about problem solving — Polya’s four-step problem-solving process (understand, devise a plan, carry out the plan, look back) Teaching through problem solving — teaching content through real context, problems, situations, models, and exploration

Teaching with a Problem-Based Approach Which of the following can be used in a problem-based approach to teaching? 42 − 19 = ____ How many times can a hiker fill a 0.5-liter water bottle from the 10-liter supply tank? 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.   h   b h b

Features of Worthwhile Tasks Worthwhile tasks are problematic and poses a question for students. They: Have no prescribed or memorized rules or methods to solve Do not have a perception that there is one “correct” solution Offer boundaries or constraints Have the potential to provide new mathematical insights or knowledge

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? 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?

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? 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?

Levels of Cognitive Demand

Examples of Multiple Entry Points

Children’s Literature: A Rich Source for Problem Solving Create high-cognitive-demand tasks Multiple tasks from one story Resonates with children’s experiences and imagination

What is a Worthwhile Task or Problem for Learning Mathematics?

Drill and Practice: A new or different perspective Practice is the opportunity to: develop conceptual ideas and more elaborate and useful connections Try out alternative and flexible strategies Encourage more students to understand Get the message that mathematics is about figuring things out Drill can provide students with: Increased facility with a procedure already learned A review of facts or procedures for retention Limitations of drill: Focus on a singular method Exclusion of flexible alternative methods False appearance of understanding Rule-oriented view of mathematics

Productive Talk for Supporting Classroom Discussions

Questioning Considerations Level of question- various models but key idea is to ask higher-level question in mathematics teaching Type of targeted understanding- questions must address procedural and conceptual- both are important Patterns of questions-initiation-response- feedback (IRF) do not lead to class discussions. Focusing is a pattern that uses probing questions to negotiate class discussions Who is thinking of the answer?-use strategies to be sure everyone is accountable to think of the answer. Ask students to “talk to a partner” about the question. Employ the talk tools How you respond to an answer-When you confirm a correct solution, rather than use one of the talk tools above, you lose an opportunity to engage students in meaningful discussions about mathematics, and thereby limit the learning opportunities.

How Much to Tell or Not Tell Researchers suggest three things that teachers DO need to tell students: Mathematical conventions- symbols( + =), terminology and labels Alternative methods- an important strategy that does not emerge naturally from students should be introduced as “another” way, but not the only Clarification or formalization of students’ methods- help students interpret their ideas and point out related ideas- i.e. 38 + 5 =. a student adds 2+ 38 = 40 and 3 more is 43- utilized the make 10 strategy

Writing to learn mathematics Importance of student writing about mathematics — a rehearsal for classroom discussion — can serve as a written record that remains long after the lesson __ focus students on the need for precise language in mathematics __ written product provides evidence of student understanding __ engages students in reflecting on what strategy makes sense

I-THINK framework supports metacognitive skill development Individually think about the task. Think about the problem. How can it be solved? Identify a strategy to solve the problem Notice how your strategy helped you solve the problem. Keep thinking about the problem. Does it make sense?

Frequently Asked Questions How can I teach all the basic skills I have to teach? Why is it okay for students to “tell” or “explain” but not for me? Is it okay to help students who have difficulty solving a problem? Where can I find the time to cover everything? How much time does it take for students to become a community of learners and really begin to share and discuss ideas? Can I use a combination of student-oriented, problem-based teaching with a teacher-directed approach? Is there any place for drill and practice? What do I do when a task bombs?