What is the role of the teacher in strengthening a student’s problem-solving abilities?
Anita L. Gordon, Former ACSI Math Services Consultant, Math Instructor—University of Colorado, Colorado Springs
The ability to apply concepts to solve problems and the ability to reason in a logical manner are two of the strands of mathematical proficiency (Kilpatrick, Jeremy, and Jane Swafford, eds. 2002. Helping children learn mathematics. Washington, DC: National Academy Press). These are necessary skills for a student to be truly proficient in mathematics. Make your classroom one in which students feel free to express their ideas about solving problems. If you facilitate a discussion in which children are willing to offer an explanation of why they picked a certain strategy or to explore alternate methods suggested by classmates, students will be more successful in their development of problem-solving proficiency. This can be done effectively by asking the right questions of your students to guide them in developing good mathematical practices and reasoning methods. Sometimes asking a question rather than offering a direct comment will keep the discussion going in the right direction. Try using questions like the following:
- Does the answer seem realistic to you?
- Did anyone else arrive at this same answer?
- What were you thinking about when you decided to multiply?
- If you had to do this problem again, what would you do differently?
- Have we solved other problems similar to this one?
- Would this statement be true for every example that you can think of?
- If your classmate needed help with this problem, how would you explain it?
- Is this always true? Can we make a rule that would apply in all cases?
These are the types of questions that good problem solvers learn to ask themselves. A good teacher must be able to assess what is happening in the mind of the student on the basis of what the student is doing. If the student is struggling with a problem because of lack of knowledge about a procedure or a vocabulary word, be ready to offer some remedial work in that area. For example, if the problem has been properly analyzed and a good strategy has been employed, but the student is having a problem changing a mixed number to a fraction in order to multiply, provide instruction in that specific area.
As you observe the thinking patterns and strategies of your students, try to help them develop the connections between their concept knowledge, arithmetic computation skills, and representations of a problem such as manipulatives or a drawing (Small, Marian et al. 2004. Navigating through problem solving and reasoning in grade 2. Reston,VA:The National Council of Teachers of Mathematics). Even the student who may be reluctant to share ideas should be encouraged to participate, whether working in a small group or with the entire class.
Each grade level of the Purposeful Design Mathematics series contains a problem-solving guide (called a problem-solving path in grades 1 and 2). This is to serve as a blueprint for analyzing, solving, and checking word problems. It can be used for a wide variety of problems and strategies. Students are encouraged to use math journals (inexpensive spiral notebooks for recording warm-up exercises, sample problems solved in class, group project work, and a student-generated glossary) in every grade level of the series. At the end of this article, you will find blackline masters to use in class and/or as journal pages when teaching the skill of problem solving. Each page corresponds with the appropriate problem-solving guide in the student textbook and teacher edition. These pages may be used for problems in the Directed Instruction, the student text, or additional problems such as the proposed samples. Encourage students to write their own problems for a given situation or set of data.
- My Problem-Solving Record (Grades 1 and 2)
- Teacher Edition Sample Problem for Grade 2
- Problem Solving Record (Grades 3 through 6)
- Teacher Edition Sample Problem for Grade 4
- Problem-Solving Record (Intermediate Courses A and B)
- Teacher Edition Sample Problem for Course A
Check out the Navigation through Problem Solving and Reasoning books, available at each grade level from prekindergarten through grade 4.The investigations in these publications can be used in conjunction with the pages provided above. Not only do these books contain rich classroom investigations, but they include instruction for best teaching practices.The Navigations series, published by the National Council of Teachers of Mathematics, highlights the different content areas of mathematics in grade-band specific volumes.The accompanying CD-ROMs feature interactive electronic activities, master copies of activity pages, and additional material.
Problem solving is not easy, especially for students who lack reading skills. Recognize that developing this skill will take time and effort. Create an environment in which students feel free to take a risk in exploring various methods.You may be surprised at the reasoning skills of your students when they are encouraged to tackle problems far beyond their procedural knowledge.
*Examples are available in the attached PDF.
Math by Design volume 1 number 2