Algebraic Thinking Through Problem Solving
Exploring Problems That are Not Limited to Single Solution Answers
Jul 19, 2009
For students to make connections in algebraic thinking in problem solving situations, it is important they learn to use algebra symbolism to represent known and unknown information. To this end they must express problem solving situations as linear, quadratic, or exponential models. This way of thinking also requires students to model, represent, analyze, and generalize contextualized problems in a variety of problem solving situations as they begin to think algebraically.
To help students bridge the gap between single solution answers to answers that require more complex thought processes, they need practice in solving real world problems. Additionally, some may need additional guidance in how to think and represent problems that may have a multitude of possible answers.
Introduction to Algebra Using Word Problems
How much money is needed to buy 10 students ice cream cones that cost 89 cents each? There is only one possible answer – $8.90. However, when the number of students is not known, then the number of possible solutions increases.
Under the second problem, students may make a table of values, construct some portion of a graph, or invoke the use of variables. More important, they may identify, describe, and extend a pattern.
During the discussion of the two different problems, ask the following questions:
- What are the similarities between the problems?
- How is the second problem different?
- What strategies are needed to solve the second problem?
In solving the second problem, students' thinking may not have progressed beyond what was expected for the first problem. If this situation occurs, guidance is needed to show how to make connections with the second problem using tips for solving word problems. After additional instruction, provide students with similar type problems.
However, by posing the second ice cream cone problem, students have an opportunity to do more than simply generate specific solutions. Many will find the need to make a table, graph, or use variables to solve the problem. More important, they may identify, describe, and extend a pattern. In other words, the students are engaged in algebraic thinking.
Strategies for Helping Students Solve Algebra Word Problems
All classes in a middle school are making Valentine cards to sell to elementary school students in a school district. The cards are boxed in groups of 12 before being sent to elementary schools. Determine how many cards have been made when various numbers of boxes have been routed to the schools.
For students finding it difficult to begin the problem, pose the following:
- Find how many cards were made if 25 boxes were routed to the elementary schools?
- What if 40 boxes were routed?
- Describe how the problem was solved in each case.
- Write an explanation to describe the solution.
For students who have already begun to list the number of cards made for specific numbers of boxes, pose the following:
- Can the information be organized into a table or graph?
- Describe what type of table or graph selected.
For students who have already begun to organize a table or construct a graph, use the following:
- What if the district routed more boxes than those shown in the table or graph?
- Find a pattern that goes beyond the table or graph.
- Describe the pattern.
For students who have already begun to describe the pattern in words, use the following:
- What is the key aspect of the pattern described?
- What would happen if the pattern was expressed using a “b” to stand for the number of boxes and “c” for the total number of cards made for the elementary schools?
Making Connections in Algebraic Thinking
The ongoing interactions between the teacher and students capitalize on students' previous problem-solving experiences and extending student thinking beyond single solutions to problems. Students learn strategies that challenge math ways of thinking that require computation of specific numerical values, along with how to generalize beyond specific values to values not previously considered.
In essence, the intention is to help students recognize and describe thinking processes using a table, graph, or numerical patterns that lead to a more general description of the problem solution using algebraic equations. The most viable demonstration of this outcome is a representation that expresses the solution process in a symbolic way using variables. When they are able to construct such a symbolic relationship, thinking algebraically has begun.
This strategy must be repeatedly used with students to allow making connections using variables in an informative and concise description of a mathematical pattern in problems. Using an interactive problem solving website provides opportunity to integrate technology within students learning how to think algebraically. This strategy allows students to see these relationships when solving a range of related problems and to move to a more advanced stage in algebraic thinking.
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