This reference is a chapter from a book. It is concerned with skills related to teaching and learning of mathematical problem-solving. Mayer identifies four aspects of mathematical problem-solving. They are:

1. Translation training, which involves teaching students how to transform each sentence of a problem into an internal representation.
2. Schema training, which involves teaching students how to integrate the information into a coherent representation.
3. Strategy training, which involves teaching students how to devise and monitor solution plans, and
4. Algorithm automaticity, which involves teaching students to effortlessly use arithmetic and algebraic procedures.

As Mayer points out, learning to solve mathematical problems requires the acquisition of large amounts of domain-specific knowledge.

Translation training involves the student having the ability to understand the English language and having some factual knowledge about objects or events. In his research, Mayer found that this step was a major source of errors for many students because they held misconceptions about objects or did not make proper relationships between objects.

In the second step, schema training, Mayer found that skilled problem solvers have extensive knowledge about problem types, and this helps them categorize problems in order to begin to solve them. Many novices, on the other hand, may lack a schema for a problem thereby making their problem representation inaccurate.

The next step is to develop a solution plan. Several researchers have proven that general problem-solving strategies usually fail, but that domain-specific strategies are more successful.

Finally, the problem solver must carry out the mathematical operations and computations. Interestingly, Mayer found that students use several different strategies, even for something as simple as counting, and that several students employ procedures which contain "bugs", which are inaccurate or faulty representations.

What can be done about students who do not possess these strategies? Mayer suggests several ideas within each aspect.

Examples of Translation Training:

1. Restate the problem "givens" in other words.

2. Represent the problem sentence as a picture or diagram.

3. Represent the problem sentence as an equation.

4. Ask the student to paraphrase the problem.

Examples of Schema Training:

1. Provide direct instruction on problem types.

2. Learn to recognize relevant and irrelevant information.

3. Represent the entire problem as a number sentence, equation, or diagram.

Examples of Strategy Training:

1. Establish subgoals.

2. Identify necessary operations.

3. Draw conclusions.

4. Have students compare their own process to a worked-out "model."

Examples of Algorithm Automaticity:

1. Students need experience in computing.

2. Students' procedural knowledge can be analyzed for counting methods or "buggy" procedures.

3. Students should achieve high levels of automaticity on component skills before moving on to more demanding algorithms.

These strategies can be helpful to teachers who are trying to instruct students in a wide range of math classes. Some of these strategies may also be applied to the teaching of other subject matter.