In various ways, the notion of interconnection between all things takes on importance for all aspects of Indigenous life, and therefore, writers in the field of Indigenous education often allude to the priority of interconnection for teaching and learning. The theoretical lens on interconnection, in Indigenous writing, tends to fall into two camps: one, that the world is comprised of distinct entities that are nevertheless connected; and, the other, that one thing is constituted by the entire world. In both cases, Indigenous theories of interconnection can be contrasted with, and even galvanized by, Western rationality, which overwhelmingly tends to fragment things in the world from each other. Education itself for the Indigenous participant may then be more a reflection of the fact of all things—its constitution of the self and all other things—than simply a transmission of knowledge. In this sense, the problem of “education” for Indigenous peoples may not lie only in the fact that education is separated out from other disciplines in dominant Western practice, but also that its attitude towards the world, with its focus on the mind, and with the clarity that fragmented things bring, does not reflect interconnection. It is unlikely that dominant Western modes of education can fully incorporate the values and ethics of Indigenous interconnection. In both pre-tertiary and tertiary education, however, some advances towards holistic and interconnected approaches are possible. In pre-tertiary, a focus on the development of the rational mind (which, from an Indigenous perspective, sits unspoken at the base of Western education) can be moderated somewhat by looking to the human self as a culmination of the world (and vice versa); and, in tertiary education, participants may revise notions of ethics and proper writing to incorporate those things that exist beyond human knowledge.
Critical thinking is active, good-quality thinking. This kind of thinking is initiated by an agent’s desire to decide what to believe, it satisfies relevant norms, and the decision on the matter at hand is reached through the use of available reasons under the control of the thinking agent. In the educational context, critical thinking refers to an educational aim that includes certain skills and abilities to think according to relevant standards and corresponding attitudes, habits, and dispositions to apply those skills to problems the agent wants to solve. The basis of this ideal is the conviction that we ought to be rational. This rationality is manifested through the proper use of reasons that a cognizing agent is able to appreciate. From the philosophical perspective, this fascinating ability to appreciate reasons leads into interesting philosophical problems in epistemology, moral philosophy, and political philosophy. Critical thinking in itself and the educational ideal are closely connected to the idea that we ought to be rational. But why exactly? This profound question seems to contain the elements needed for its solution. To ask why is to ask either for an explanation or for reasons for accepting a claim. Concentrating on the latter, we notice that such a question presupposes that the acceptability of a claim depends on the quality of the reasons that can be given for it: asking this question grants us the claim that we ought to be rational, that is, to make our beliefs fit what we have reason to believe. In the center of this fit are the concepts of knowledge and justified belief. A critical thinker wants to know and strives to achieve the state of knowledge by mentally examining reasons and the relation those reasons bear to candidate beliefs. Both these aspects include fascinating philosophical problems. How does this mental examination bring about knowledge? What is the relation my belief must have to a putative reason for my belief to qualify as knowledge? The appreciation of reason has been a key theme in the writings of the key figures of philosophy of education, but the ideal of individual justifying reasoning is not the sole value that guides educational theory and practice. It is therefore important to discuss tensions this ideal has with other important concepts and values, such as autonomy, liberty, and political justification. For example, given that we take critical thinking to be essential for the liberty and autonomy of an individual, how far can we try to inculcate a student with this ideal when the student rejects it? These issues underline important practical choices an educator has to make.
Fostering self-direction in students has long been an aim for both educators and parents as they fear the potentially coercive influence of peer pressure and the many sources that compete to influence what we think and what we do. These fears have motivated educational philosophers to explore the contours of what such self-direction or autonomous thought and action entails on the demands of individual thinking and behavior but also on the types of educational environments needed to foster its emergence. Likewise, educational philosophers have also argued the merits of promoting autonomy in public schools out of fears that some forms of autonomy may limit the ranges of conceptions of the good life that are available to students; many are concerned that promoting autonomy may inspire students to reject family and community ways of life. Despite those concerns, drawing upon thought that traces back to the ancient Greeks, contemporary educational philosophers continue to debate the contours of and justifications for an autonomy promoting education.
Kristie J. Newton, Christina A. Barbieri, and Julie L. Booth
Mathematics learning encompasses a broad range of processes and skills that change over time. Magnitude and equivalence are two fundamental mathematical ideas that students encounter early and often in their mathematics learning. Numerical magnitude knowledge is knowledge of the relative sizes of numbers, including whole numbers, fractions, and negative numbers, within a given scale. Understanding mathematical equivalence means understanding that two or more specific quantities with the same value can be represented in a variety of ways and remain equal and interchangeable. A major area of research on equivalence is knowledge of the equal sign. Both equal sign knowledge and magnitude knowledge are foundational in that they predict later learning in mathematics, including algebra. Implications for practice include the use of number lines and more variation in the way that arithmetic problems are formatted.