by George Booker and Denise Bond
Successful math problem solving is more than just getting the answer. It's the processes students employ to get there.
Students who can analyze and carry out a plan to solve math problems have acquired deeper and more useful knowledge than simply being able to complete calculations, name shapes, use formulas to make measurements, or determine measures of probability and data analysis. Well-chosen problems encourage deeper exploration of mathematical ideas, build persistence, and highlight the need to understand thinking strategies. They also reveal the central role of sense-making in mathematical thinking. This may take the form of number sense, spatial sense, measurement sense, and data sense.
Number Sense
Number sense involves being able to work with numbers comfortably and competently. It is based on a full understanding of numeration concepts such as zero, place value, and the renaming of numbers in equivalent forms so that (for example) 207 can be seen as 20 tens and 7 ones as well as 2 hundreds and 7 ones. Number sense is the capacity to:
- understand relationships among numbers
- appreciate the relative size of numbers
- calculate and estimate mentally
- use fluent processes for larger numbers, including the adaptive use of calculators
- apply understanding of and facility with numeration and computation in flexible ways
Number sense is essential in allowing relationships within a problem to be revealed and taken into account when framing a solution.
Spatial Sense
Spatial sense is equally important, since the use of diagrams, tables, and other visual formats is often essential in developing conceptual understanding across all aspects of mathematics. Spatial sense is the capacity to:
- visualize and determine relationships among shapes and their properties
- link two-dimensional and three-dimensional representations
- present and interpret information in tables and lists
- use diagrams and models to visualize problem situations and apply understanding in flexible ways
Measurement Sense
Measurement sense is dependent on both number sense and spatial sense, since attributes that are one-, two-, or three-dimensional are quantified to provide both exact and approximate measures and allow comparison. Many measurements use aspects of geometry (length, area, volume), while others use numbers on a scale (time, mass, temperature). Money can be viewed as a measure of value and uses numbers more directly, while an understanding of how the metric system builds on place value, zero, and renaming is critical in both building measurement understanding and using it to solve many practical problems. Measurement sense is the capacity to:
- extend relationships from number understanding to the customary and metric systems
- appreciate the relative size of measurements
- use calculators and mental or written processes for exact and approximate calculations
- use understanding and facility with measurement in flexible ways
Data Sense
Data sense grows out of measurement sense and refers to an understanding of the way number, spatial, and measurement sense work together to deal with situations where patterns must be discerned among data or likely outcomes analyzed. Data sense is the capacity to:
- appreciate the relative likelihood of outcomes
- use calculators or mental and written processes for exact and approximate calculations
- present and interpret data in tables and graphs
- use understanding and facility with number combinations and arrangements in flexible ways
As students gain more experience in solving problems, an ability to see patterns in what is occurring will help them see the relationship between a new problem and one they have solved previously. Using pattern sense, students are able to identify similar problem types, even when the context appears to be quite different. It is this ability that often distinguishes a good problem solver from one who is more hesitant.
Tips for Teaching the Problem-Solving Process
Teamwork works best: When it comes to problem solving, collaborative work can be more productive than individual work. Students who may be tempted to quickly give up when working on their own can be encouraged to see ways of proceeding when discussing a problem in a group. This helps them build greater confidence in their capacity to solve problems as well as learn the value of persistence. When confronting a similar problem in the future, students are more likely to recall what they discussed with a group. The observations made in the group also increase the range of approaches that students can access.
Encourage discussion: Students who have an answer should be encouraged to discuss their solution with others who believe they have a solution, rather than tell their answer to another student or simply move on to the next problem. Explaining to others why they believe an answer is reasonable gets other students to focus on the entire problem-solving process rather than just quickly getting an answer.
Ask the right questions: Questions posed by the teacher must encourage students to explore possible means to a solution and try one or more of them, rather than point to a particular procedure. Questions can also help students see how to progress in their thinking rather than getting stuck in a loop where the same steps are repeated over and over.
Assess by listening: Taking time to listen to students as they try out their ideas, without comment or without directing them to a particular strategy, is also important. Listening provides a sense of how students? problem solving is developing, since assessing this area of mathematics can be difficult.
Adapt when necessary: It may be necessary to extend or adapt a given problem to ensure that students understand a particular problem-solving process and can use it in other situations, instead of moving on to a different problem in another area of math learning. This can help students see how a way of thinking can be adapted to other, related problems. Having students write their own similar problems is another way of both assessing them and bringing them to terms with the overall process.
Using a Problem-Solving Model
The cyclical model
Analyze?Explore?Try is a very helpful means of organizing and discussing possible solutions, as long as students realize it is not simply a procedure to be memorized and applied in a routine manner to every new problem.
Step 1: ANALYZE
- Students read through the problem for meaning. It may be helpful to state the problem in their own words to see what it is asking them to do.
- Students read through the problem a second, third, or fourth time to identify which information is needed, whether some is not needed, if other information must be gathered from the problem?s context (for example, data presented within a table accompanying the problem), and whether they must find other relationships among the information.
- Students think about the processes that might be needed and the order in which they might be used, as well as the type of answer that could occur.
- Students look beyond the problem?s expression or context to identify similarities to other problems they have already encountered and solved.
Step 2: EXPLORE
When necessary, students use materials (blocks, counters, and so on), diagrams, lists, or tables to think through the whole of the problem?s context. As these ways of thinking about the problem are understood, they can be included in the cycle of steps.
Step 3: TRY
Many students often try to guess a result. This can even be encouraged by talking about ?guess and check? as a means of solving problems. Changing this terminology to ?try and adjust? is more helpful in building a way of thinking and can lead to a very powerful way of finding solutions.
When using materials, a table, or diagram, identifying a pattern may reveal what is needed to try for a solution. At this point, it may also be reasonable to use pencil and paper or a calculator.
Step 4: ANALYZE
This is the point in the cycle when students assess an answer for reasonableness (for example, whether it provides a solution, is only one of several solutions, or there may be another way to solve the problem).
As students become successful problem solvers, mathematics becomes a subject that they readily engage with and feel in control of, instead of one in which they have to memorize and apply many rules devoid of meaning. This ability to think mathematically and the enthusiasm for learning it fosters will lead them to a search for meaning in new situations and allow them to apply mathematical ideas across a range of applications in school and everyday life.
