Principled Principles for Mathematics Educators & Curriculum Decision Makers

Principled Principles for Mathematics Educators &
Curriculum Decision Makers
2005

Principled procedures for a classroom are descriptions of what teachers and students will do in the classroom. They are based on beliefs and ethical considerations for the manner in which teachers and students interact with each other and everything in the classroom.

1. Equity for all students is provided with strong support of resources; technological, physical, emotional, and intellectual to develop each student's self-efficacy in mathematics through encouragement and high expectations with regards to each student's individual needs regardless of their personal backgrounds.

2. Curriculum is continually evaluated as a critical analytical exploration of the curriculum as a totality, a theoretical/conceptual and practical/empirical inquiry of all the many dimensions of the curriculum debate, planning, pedagogical theories and practices across time, that transcend mathematics and create a coherent curriculum of powerful mathematics that can be used inside and outside mathematics and articulated across all grades.

3. Teaching is a complex task that requires numerous decisions. Better decisions being made with a deep understanding of mathematics, how people learn mathematics, what students need to learn, how to assess what students know, and how to motivate students to learn.

Teachers understand their students' mathematical abilities and dispositions and make informed decisions to continually motivate students' learning of powerful mathematical ideas.

Facilitate learning is student centered inquiry where students regularly engage in deep meaningful thinking solving problems that are challenging, but attainable with facilitated learning. This requires intense ongoing observational information to make accurate inferences of student's ideas and explanations, relate this information to important mathematical goals, seek outside help when necessary, to insure sufficient information has been collected to reflect on past practices to make good principled decisions from their memory, and adjust instruction on the fly in flexible ways that invite students to develop a disposition to participate in and learn that mathematics is coherent and connected and feel confident they can learn it and use it to solve problems.

Procedures typically include: a setting (place, furnishings, grouping, materials) and sufficient time to solve problems so that they can develop persistence in thinking and reflecting on their mathematical understanding of content, process, dispositions and perspectives of mathematics; tasks for the students; directions (a series of steps to communicate a task); process to lead to successful completion; and looping through a learning cycle as additional tasks are selected extended for students to generalize and connect their learnings.

Learning cycle is informed by an initial informal diagnostic assessment and formative assessment while facilitating learning that informs the teacher and students of the students' current content knowledge, problem-solving skill, disposition for mathematizing, and range of connections being made.

Motivation is increased when people build on their knowledge. By challenging and supporting students' learning in flexible ways teachers help them develop a disposition to participate in and learn that mathematics is coherent and connected and feel confident that they can learn it and use it to solve problems. Therefore, teachers observe students, listen carefully to their ideas and explanations, relate this information to important mathematical goals, seek outside help when necessary, and use all this information to reflect on past practices to make good principled decisions that help students achieve mathematical self-efficacy.

Mathematical self-efficacy encourages and builds on the students' intuitive physical, visual, and verbal understandings as a basis for abstract understanding and a more formal kind of communication.

People gain intrinsic satisfaction from solving problems and the realization that the are making progress toward mathematical efficacy.

4. Learning is achieved when students focus their thinking on problems or tasks that require solutions just beyond their edge of current understanding. At a point of cognitive dissonance, or disequillibrium, or their zone of proximal development.

Problems or tasks are most beneficial when students value an active involvement of building new knowledge by connecting prior experiences and their present formal and informal knowledge through problem solving, reasoning, and argumentation to achieve greater understandings conceptual and ultimately procedural knowledge. Including the ability to communicate those understandings orally and in writing with sentences and or phrases, symbols and or equations, drawings, models, manipulatives, and drama along with arguments to support their reasoning of their understandings to others.

5. Assessment is an integral ongoing part of teaching and student learning that is conducted in a variety of ways to assure that students have opportunities to demonstrate clearly and completely their dispositions, reasoning abilities, problem solving ability, mathematical understanding of powerful ideas and their connections, as well as the ability to represent and communicate ideas in all of these areas. It informs and guides teachers as they make instructional decisions on their teaching and on each student's development. It helps students get a good sense of what mathematics is and how it can be used. Student's success enables them to learn how to set and achieve reasonable goals for their own learning, thereby becoming independent learners with a disposition and capacity to engage in self-assessment and reflection on their work and others.

6. Technology, specifically calculators and computers, are essential tools in teaching, learning, and doing mathematics. Mathematical ideas can be created and illustrated with models, equations, images, matrices, and other means that increase the power of doing mathematics and enrich students' mathematical understandings through engagement that is not possible using other methods, with increased speed, communication, focus on ideas or information, ease of working with large numbers, and generation of multiple solutions and possible solutions.

Dr. Robert Sweetland's Notes ©