Mathematics instruction for elementary school educators

¶ … globalization and the structures of testing in the "No Child Left Behind Initiative," it is becoming even more important that K-8 teachers be prepared to teach basic concepts of mathematics that adhere to their individual State standards, but also to a rigorous, diverse, and multicultural community. In the same way that a basic level of literacy is required before pursuing upper levels of schooling, certain mathematical constructs are vital in today's complex world of computerization, science, and the synergistic approach to many core courses. For too many people, mathematics stopped making sense somewhere along the way. Either slowly or dramatically, they gave up on the field as hopelessly baffling and difficult, and they grew up to be adults who -- confident that others share their experience -- nonchalantly announce, "Math was just not for me "or "I was never good at it" (Askey, 1999, 4).

There are four basic concepts covered in the course that particularly address the issue of relevancy in mathematical pedagogy: Mathematical Standards and Practices, Algebraic Thinking and Problem Solving, Numeration Systems and Number Theory, and Rational Numbers and Applications.

Mathematical Standards and Processes -- The National Council of Teachers of Mathematics, an international organization of teachers who are focused on improving the math curriculum globally, presented new standards in 2000 designed to improve curricula, teaching and assessment. Within their rubric, six principles were established to address themes that were valid regardless of the school culture:

Equity -- There must be high expectations and support for excellence in math education from all levels; teachers, administrators, school boards, and parents.

Curriculum -- More than a collection of problems or activities, a math curriculum should be focused, well-articulated, and flow from grade to grade.

Teaching -- Appropriate and effective math teaching requires not only an understanding of math principles but of what students need to understand, and how that should be effectively communicated to them.

Learning -- Students must learn math in a synergistic, step process- each previous module must present them with tools needed to move forward and actively build a knowledge base.

Assessment -- Assessment should support the learning aspect of math and be appropriate as a tool for understanding student needs; not simply as something easy to grade.

Technology -- Adapting technology is absolutely essential in learning mathematics (NCTM, 2009).

In addition to these overall principles, five more detailed standards and expectations were identified:

Problem Solving -- Building new knowledge through problem solving in math and other disciplines that involve mathematic calculations. Be able to apply and adapt problem solving skills.

Reasoning and Proof -- Establish initial understanding a rubric of reasoning out a problem, make and investigate mathematical conjectures, develop and evaluate mathematical arguments, and use appropriate levels of reasoning for different problems.

Communication -- Be able to communicate clearly verbally and in writing mathematical principles, equations, and solutions. Analyze the mathematical thinking of peers and others and use the language of math to express computational ideas.

Connections -- Understand the relevancy and connections among mathematical principles. Understand how mathematical ideas interconnect and form the basis for higher levels of computation in math and other scientific disciplines.

Representations -- Create and use representations to organize, record, and communicate mathematical ideas appropriately. Use the appropriate representations to model problems in the physical and social sciences (Ibid.)

Numeration Systems and Number Theory -- Number theory is a basis for all areas of mathematics. Number theory and sense are precludes to computation, to estimate, and to have an understanding of the ways numbers are represented and interrelated. Fluency of also understanding the way positive and negative numbers can be visually represented on a line, or how numerical values interrelate, are essential prior to moving toward higher level concepts (Kane, 2002).

Algebraic Thinking and Problem Solving -- Rather than viewing the subject of algebra as certain sets of problems, the appropriate way to introduce it into elementary levels is as the relationship among quantities, the use of symbols, the modeling of phenomena, and the study of change. Students should be able to understand patterns, relations, and functions and how numbers may be represented in different ways. Algebraic thinking is on par with the analysis of textual materials -- the what if, the why, and the what might be. Using these concepts, as early as possible, encourages transformative thinking and the ability to apply "what if" strategies at an earlier age (Blanton, 2008).