Active Solving and Strategy Use/Impact of Attention, Sequential Ordering, Memory, and Higher Order Cognition

Computation is a vital component of math. But students need to focus on more than just accuracy in calculation. The ability to reason through a word problem, to think critically about all aspects of the mathematical situation, is an increasingly important goal in math instruction.

Active problem solving skills are strongly tied to a student’s attention abilities. Students must closely attend to problem details to determine what the question is, what kind of answer to look for, and what information will be salient or important when solving the problem. Students must analyze the problem, possibly breaking it down into a logical sequence of smaller steps (instead of reacting impulsively or ‘jumping’ to a conclusion). In addition, the ability to preview, or estimate likely outcomes within a problem, enables the student to make quantitative and strategic predictions, such as what amounts will likely be involved or what strategies are likely to be used.

Students use their long-term memory to help determine whether a word problem reflects a familiar pattern. They search in memory for prior knowledge, learned rules, or relevant skills that have worked in the past for that type of problem, and then, apply that knowledge in the new situation.

Effective problem solving also requires flexible thinking. Students may need to use, evaluate or change strategies; at times, they may need to consider several alternative strategies. And, finally, throughout the problem solving process, students must be able to monitor the outcomes of their calculations, and refine their solutions when necessary.

Here are some strategies to help students become active problem solvers in math.

Helpful Hints

  • Model problems for the class and explain each step when teaching students how to be active problem solvers. Think out loud for students as you reason through a problem, choose a strategy to use, decide if the strategy is working, etc. Have students talk through problems with each other as well.  
  • To promote strategy use and adjustment, ask students guiding questions as they solve problems, e.g., "Is there an easier way to do that‘", "Will that strategy always work‘", etc.  
  • Have students communicate their understanding of a problem through both oral discussion and written explanation.  
  • Have a brainstorming session with students to discuss the types of behavior or steps are involved in problem solving, characteristics of ‘good problem solvers’, etc. Some ideas may include reasoning, looking for patterns, patience, persistence, hypothesizing, stating the obvious, creativity, etc.  
  • Encourage students to explore multiple strategies that could be used for solving a math problem. For example, ask students to find the length of the diagonal of a 12" x 16" rectangle. Students will likely recognize that the rectangle is made up of two right triangles, and apply the Pythagorean theorem. One approach might be to calculate by hand or to use a calculator for the computation (12² + 16² = ‘) to eventually come up with the answer of 20 inches.  
  • However, an alternate view of the problem makes it even easier to solve. A student might notice that the 12" and 16" sides are both divided evenly by 4, resulting in a triangle with sides of 3" x 4", respectively. The student will likely recognize the missing diagonal length to be 5", making the standard 3" x 4" x 5" triangle. Then, simply multiplying the 5" back by 4 would give the answer to the diagonal: 20 inches. No lengthy computations would be needed. (Adapted from Brumbaugh, Ashe, Ashe & Rock, 1997).  
  • Have students practice selecting what strategies might be appropriate for solving a given problem. For example, in each case, would it be helpful to act out the problem, make a model, draw a picture, make a chart or graph, use logic, guess and check, break it into parts, etc.’  
  • Promote students’ flexible thinking by presenting situations in which there is more than just one right answer. For example, have students take out a piece of paper, fold the paper in half, then fold the paper in half again. Ask students to count how many rectangles have been formed. Answers will vary depending upon how the second fold was made in the paper, if students count the whole piece of paper as one of the rectangles, etc. Have students discuss how different folding approaches resulted in different ‘answers’. (Adapted from Brumbaugh, Ashe, Ashe & Rock, 1997).  
  • Give students practice estimating the answers to problems. Have them move from estimation to calculation, then back again to estimation. Help students develop their sense, before starting calculations, of what a general solution to the problem might be, and also to take time to examine their answers, after calculation, to see if they seem credible. Students may require guidance in useful strategies for estimation, e.g., rounding numbers, creating a visual image, etc.