often more than one strategy can be used.”
(Ornstein & Lasley, 2004, p. 229)
“By providing knowledge of and practice in applying learning strategies, the teacher invariably indirectly enhances the self-concept and coping abilities of the students . . . which, in turn, provides the means for problem solving. Such confidence is essential for students to cope with minor frustration; to play with ideas; to take educated guesses; to delete, add or modify parts of problems; and to select a plan of action and carry it out. Various in-class support groups (e.g., peer tutors, sharing dyads, cooperative learning) can help relieve anxiety and stress associated with problem solving.”
(Ornstein & Lasley, 2004, p. 231)
Recent research on teaching math problem solving in both classroom-level studies and tutorial studies has revealed several important findings for the instruction of math problem solving. Outcomes for children across ability levels and for children with specific difficulties in mathe-
A problem-solving approach is essential for actively involving students in stimulating activities that present challenges and in which they use a variety of representations (concrete, visual, diagram models) to reach an in-depth understanding of mathematics.
The problems presented to students in a problem-solving approach are more than just word problems. They are situations that allow students to explore concepts, use prior knowledge, acquire new knowledge, reason, communicate ideas, and make connections in relevant and engaging contexts. “Problem solving is not only a goal of learning mathematics but also a major means of doing so” (National Council of Teachers of Mathematics, 2000, p. 52). Students often use concrete or visual representations to solve problems and communicate their explanations or strategies to others. To put this another way, teachers should encourage the use of open problems that enable the use of a variety of strategies and answers. “Metacognitive skills such as reflecting, organizing, and structuring … enhance student learning and student retention of important content because they help students create their own connections with knowledge” (Ornstein & Lasley, 2004, p. 281).
Important considerations
Before presenting the problem, consider:
• the context: Is it relevant, engaging, and familiar to students?
• strategies students will need to solve the problem: Are students familiar with at least one of the possible strategies that can be used to solve the problem?
• groupings: Mixed-ability groupings can support struggling students and challenge more able students.
When presenting the problem, consider:
• using visual prompts and modelling the use of appropriate math language; • repeating information and instructions several different ways;
• having students restate the problem to ensure that they understand it. While students are solving the problem, consider:
• working with students individually or in a small group to get them started;
• modelling an efficient problem-solving strategy and then asking students to solve the problem on their own;
• providing a checklist to help students stay on track;
• using think-aloud to help students reason through the process;
• modelling how students can represent their thinking using graphic organizers, pictures, lists, concrete materials, procedural writing, or verbal explanations;
• how students can best represent the process and solution, and allow for choice. When reflecting and consolidating with students, consider:
• using probing questions to help students communicate their understanding; • referring to previous problems and contexts to help students make connections; • using think-aloud and concrete materials to model the thinking process through the
Student areas of need that may have an impact on the effectiveness of instruction
• Prior knowledge and experience (i.e., Does the student have the math knowledge and skills needed to complete the task? Has the student completed a similar task before?)
• Language abilities (i.e., Does the student have the ability to process information and make connections?)
• Metacognitive abilities (i.e., Can the student keep track of the instructions and information?)
• Metacognitive abilities (i.e., Is the student able to identify and select appropriate strategies? Is the student able to select and use appropriate manipulatives?)
• Self-regulation (i.e., Is the stu- dent able to organize what he or she has done and then provide a recording of it?)
Considerations for implementation
• Refer and make connections to problems that students have experienced before.
• Use contexts that students are familiar with and are interested in.
• Refer to the math word walland math strategy wall* to remind students of previous learning. • Provide visual prompts to illustrate the task. • Repeat the information in different ways and ask students to describe the problem in their own words, using their own representations. • Provide a step-by-step checklist or graphic
organizer.
• Provide the instructions in manageable chunks. • Use think,pair,share* to generate strategies
and approaches.
• Consider grouping students to support different levels of ability:
– Pair students to model different strategies and approaches.
– Work with students individually or in small groups to get them started, or to model one possible strategy or solution. – Change groupings to meet needs. • Restate the problem and instructions. • Give students a strategy to try.
• Check in with students regularly to keep them on task.
• Provide an outline of how the solution is to be recorded. Use mind mapping and procedural writing* to help students organize their thinking.
• Model the use of manipulatives to represent the problem or solution. Provide alternative materials.
Table 6. How to Teach Through Problem Solving: A Concrete Example
Phases of effective instruction
1. Getting started The teacher:
• presents the problem and gives the informa- tion needed to solve the problem;
• gives instructions for completing the task; • activates prior knowledge; • engages students through contextual information or an interesting situation. At this point – just before they begin the task – the students understand the problem and the teacher’s expectations.
The teacher guides, assists, observes, asks questions, redirects, adjusts groupings, modifies the task.
Students:
• may be working inde- pendently, in pairs, or in small groups; • may be in flexible
groupings;
• are using materials/ manipulatives appro- priate for the task; • are trying/testing their
own strategies.
Student areas of need that may have an impact on the
effectiveness of instruction Considerations for implementation Phases of effective