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Ellerman, David Brain functors: A mathematical model of intentional perception and action. Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens.

If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets.

Math Teacher Training Through Instruction Reflection – Part 3

By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions essentially a formulation of universal mapping properties using hets can then be recombined in a new way to define the notion of a brain functor that provides an abstract model of the intentionality of perception and action as opposed to the passive reception of sense-data or the reflex generation of behavior.

More information and software credits.

If leaders discover other common concerns, they may want to add these to the tool in the future, especially if these areas relate to a school or district initiative. At this stage, teachers may again ask for help from an instructional leader in an area they selected for growth. For instance, a fourth-grade teacher who is having difficulty matching models to instruction, might benefit from watching the series of six ORIGO One videos about fraction models.

The first one is shown below. If a teacher asks for help in finding appropriate methods for promoting fluency in multiplication the leader might direct the teacher to this webinar.

Math with My Scholastic Teachables & Giveaway!

Leaders who are training or coaching can make the process more powerful by using a similar tool for feedback after walkthroughs or other observations. This tool may also be utilized by a growth-partner, another teacher who teams with a colleague to observe and offer feedback. Lesson Feedback. Using the tool below will help focus the conversations. Ideally this process will be repeated multiple times during the year.

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Teachers will record themselves and self-reflect. At the end of the year, teachers will gather the feedback and reflect on the growth during the year and its impact on instruction and student success.

They will begin the process of choosing goals for the following year. Yearly Summary Reflection. During the course of these articles, I have mentioned the importance of training by supporting teachers in accomplishing their goals when they need assistance in achieving mastery.

Hopefully, the strategies and tools provided help to cultivate productive partnerships that positively impacts student achievement. Thank you to Dr. Tammy Heflebower, contributing author to Becoming a Reflective Teacher , for helping to confirm and clarify my thoughts about supporting teacher growth.

Click HERE for the downloadable resources for this article! ORIGO Education is dedicated to making learning meaningful, enjoyable and accessible for all students with Pre-K and Elementary print and digital instructional materials, as well as professional learning for mathematics. Share this article.

By Way of Introduction: Moving beyond show & tell: Intentional math discourse

Math Teacher Training Through Instruction Reflection — Part 3 This is the third in a series of articles about training and supporting teachers in math instruction: In the first article, we discussed instructional change in the math classroom In the previous article on math leadership , I discussed the importance of effective planning and provided tools and tips for teachers and administrators.

In this article, I am providing tools for teachers to reflect on their instruction and for administrators to cultivate intentional conversations about the instruction. Teacher Post-Lesson Reflection and Goal Setting At this stage, teachers may again ask for help from an instructional leader in an area they selected for growth.