Problem solving is a way of thinking through which one is engaged in a variety of appropriate strategies. Prior knowledge, curiosity, and confidence are used in unfamiliar situations in which the solution is not known in advance. New mathematical knowledge is introduced, solidified and extended through investigations of real life examples. The selection of worthwhile problems is crucial to helping students develop mathematical ideas.
The key to good problem solving is an environment where problem solving is fostered across real world and curricular lines. Self-reflection of the problem solving process will allow students to reflect on their work and make the necessary and appropriate adjustments. Problem solving should include the five content areas of data and probability, measurement, geometry, algebra and number operations.
According to the NCTM, problem solving should enable all students to:
- Build new mathematical knowledge through problem solving
- Solve problems that arise in mathematics and in other contexts
- Apply and adapt a variety of appropriate strategies to solve problems
- Monitor and reflect on the process of mathematical problem solving
Reasoning and Proof
Mathematical reasoning and proof are ways of developing and expressing insights into various concepts. This is a habit that must be developed and should be consistent from pre-K to 12th grade. Mathematical reasoning is the ability to justify the process to develop a solution. Mathematical reasoning and proof allows students the opportunity to compare their ideas with others. It also allows students to work together to discover diverse approaches to various situations.
Another aspect of mathematical reasoning and proof is to involve students in discovery with conjecture. Students should be able to, in their own language, explore, compare, and justify their answers and thought patterns. Some examples of teacher questioning would be "Why do you think this is true?", "What do you think will happen next?", "Is this always true?", "Does anyone think the answer is different, and why do you think so." These types of questions should allow students to investigate, support, and / or argue their reasoning and then apply it to diverse situations. As students progress, they should be able to go from trial and error, to proof by contradiction, to increasingly sophisticated written forms.
The communication standard focuses on students’ ability to communicate mathematics by speaking, writing, reading, and listening. These activities should be centered on worthwhile mathematical activities that are carefully selected by the classroom teacher to promote discussion. Discussions will occur in small group and whole group settings that require students to present mathematically accurate arguments both symbolically and verbally. It is essential for the classroom teacher to develop a community in which students will feel free to express their ideas.
The communication standard requires the student to take on two unique roles: a presenter of mathematical ideas and a critical listener. The presenter justifies their argument in a concise and logical order using special mathematical language that allows the audience to understand the student’s ideas. The listener then examines the problems from other students’ perspectives by using questioning and probing techniques that allow the student to develop a deeper understanding. The collaboration process of students discussing their ideas and defending their positions will act as a catalyst to build and develop new reasoning.
Mathematics relates to all subjects as well as personal interests. It can be used and applied to our students real-life, everyday experiences. Mathematics cannot be viewed as a set of disjointed topics and concepts taught in isolation. It is crucial that mathematics is taught conceptually and abstractly at all levels. The students become responsible for their learning from year to year to avoid the practice of reteaching previously taught concepts. Rather, teachers are charged with building upon previous learning to produce a coherent body of ideas/concepts that meet current grade level expectations.
The term representation refers both to process and to product. Process referring to a person's ability to learn and express a mathematical concept through symbolic language. Product referring to the symbolic form itself. This can be done through diagrams, graphical displays and symbolic expressions. Representations should be treated as essential elements in supporting students' understanding of mathematical concepts.
As technology grows, this representation becomes even more important. The three measures of representation are:
- To be able to express, explain, and show understanding of a mathematical idea.
- Select, apply, and translate among mathematical representations to solve problems.
- Use representations to model and interpret physical, social, and mathematical phenomena by using mathematical modeling to understand the events.