Principled Principles for Mathematics Educators and Curriculum Decision Maker
Principled procedures are descriptions of what teachers believe and guidelines based on those beliefs, ethical considerations, research, and wisdom of practice, which are used to make decisions facilitate learning.
1. Equity, believe and act as all students can learn. Challenge students by giving them autonomy to solve problems and support them with sufficient resources; technological, physical, emotional, and intellectual to develop each student's self-efficacy in mathematics with encouragement and high expectations according to each student's individual needs regardless of their personal backgrounds.
2. Understand students mathematical abilities and dispositions.
- Believe students learn best when they focus their thinking on problems or tasks that require solutions just beyond their edge of current understanding. At a point of cognitive dissonance, or disequilibrium, within their zone of proximal development.
- Believe and act on the idea people gain intrinsic satisfaction, increase their motivation, and self-efficacy as they solve problems and share in the process, which will increase their realization of the power mathematics for understanding and explaining the world.
- Believe motivation is increased when people build on their knowledge. That when students are challenged and supported and allowed to solve problems in their own flexible ways they will be successful and develop intrinsic motivation and dispositions to participate in and become mathematically literate.
- Believe mathematical self-efficacy is built when students are successful in taking real world physical objects and events through a visual and verbal process to create internal abstract representations for mathematical ideas, which can range anywhere from intuitive to a systematized formal kind of representation. Greater self-efficacy results as more systematic representations and greater ability to communicate them develop.
3. Facilitating learning is complex and requires numerous decisions with the best decisions being made by those who have a deep understanding of: mathematics and its uses in the world, how people learn mathematics, knowledge of students, how to assess what students know, and how to motivate students to learn.
- Facilitate learning by observing and listening to student's ideas and explanations and using this information to make informed decisions by reflecting and making good principled decisions that help students progress toward mathematical self-efficacy.
- Motivate students to learn powerful mathematical ideas with student centered inquiry where students regularly engage in deep meaningful thinking while solving challenging problems. Which requires intense ongoing observational information to make accurate inferences of student's ideas and explanations, relating this information to important mathematical ideas, seeking outside help when necessary, to insure sufficient information has been collected to reflect and make good principled decisions on the fly to adjust instruction in flexible ways that invite students to develop positive dispositions to participate in and learn that mathematics is coherent and connected and feel confident they can learn it and use it to solve real world problems.
- Procedures typically include: a setting (place, furnishings, grouping, materials) and sufficient time to solve problems and develop persistence and desire to think and reflect about mathematics: content, practices, processes, perspectives, and dispositions; problems or tasks for the students; directions or a process for learning that leads to a successful completion by looping through a learning cycle of tasks for students to generalize and connect their learnings.
- Learning cycle is a process that must begin with informal diagnostic assessment and move to formative assessment to inform the teacher and students of each students' current knowledge to facilitate his and her construction of mathematical knowledge so each can represent mathematical ideas, solve mathematical problems, provide reasonable explanations for their accuracy, develop positive dispositions for mathematizing, and make a wide range of connections among the practice of mathematics, mathematical ideas, and the real world.
- Problems or tasks are most beneficial when students value an active involvement of building new knowledge by connecting prior experiences and their present knowledge through problem solving, reasoning, and argumentation to achieve greater conceptual understandings 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 and explanations of their understandings to others.
4. Believe and act as assessment is an integral ongoing part of teaching and student learning 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 practices and 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 reflection and metacognition for self-assessment of their work and others.
5. Curriculum - Professional educators continually evaluate mathematical curriculum for all mathematical dimensions, comprehensiveness, the inclusion of powerful mathematics, the strength of the theoretical base claimed to support it, how practical it is to implement, it's coherence, its articulation across all grade levels, and its support by research and wisdom of practice.
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.