Procedures and their implementation

Mathematics Education

The objective of this work is to describe five specific methods of questions and strategies that encourage students to discuss their ideas, procedures, rules and definitions that they used to solve a problem and to discuss at least four ways in which justification of solutions to improve students' relational understanding of mathematics.

The work of Jones (2000) entitled: "Instructional Approaches to Teaching Problem Solving in Mathematics: Integrating Theories of Learning and Technology" states that: "Problem solving is defined by Kantowski as 'a situation for which the individual confronting it has no readily accessible algorithm that will guarantee a solution." (2000) NCTM standards define problem solving as "the process by which students experience the power and usefulness of mathematics in the world around them." (Jones, 2000)the stages of problem-solving are stated to be:

Understanding the problem;

Making a plan;

Carrying out the plan;

4) Looking back. (Jones, 2000)

There are five strands of mathematical proficiency, which are stated to be those as follows:

conceptual understanding;

procedural fluency;

strategic competence;

adaptive reasoning; and Productive disposition. (Taplin, nd)

Conceptual understanding of mathematics involves comprehension of mathematical concepts, operations and relations. Procedural fluency involves skills in carrying out procedures in a flexible, accurate, efficient and appropriate manner. Strategic competence involves the ability to formulate, represent, and solve mathematical problems. Adaptive reasoning involves a capacity for logical though, reflections, explanation and justification. Finally productive disposition involves the habitual inclination to see mathematics as sensible, useful and worthwhile, coupled with a belief in diligence in ones' own efficacy.

Five specific strategies that teachers may use for encouraging students to discuss their ideas, procedures, rules and definitions that they used to solve a problem include those as follows:

1) the teacher provides just enough information to establish the intent of the problem;

2) the teacher accepts right or wrong answers in a non-evaluative manner;

3) the teacher guides, coaches and asks insightful questions;

4) the teacher intervenes when appropriate and when not appropriate the teacher allows the students to make their own way;

5) the teacher encourages students to make generalizations about rules and concepts.

Balacheff (1987) described four levels of justification, which are those as follows: (1) Native empiricism;

2) Crucial experiment;

3) Generic example; and 4) Thought experiment. (Taflin, nd)

Naive empiricism is stated to be "an assertion based on a small number of cases." (Taflin, nd) Crucial experiment is stated to be "an assertion based on a particular case, but the case was used as an example of a class of objects." (Taflin, nd) the generic example is stated to be "...an assertion based on a particular case, but the case was used as an example of a class of objects." (Taflin, nd) Thought experiment is "an assertion detached from particular examples and begins to move toward conceptual proofs." (Taflin, nd)