The testing of a task went horribly right.
Background:
Graham Fletcher (@gfletchy) tweeted an Open Middle (@OpenMiddle) prompt for comparing fractions.
Category: problem posing
My class always welcomes conjectures. This is made explicit on the very first day of the semester. This goes for everything from grade nine to grade twelve. As the grades advance, the topics have us venturing into increasingly abstract concepts, but conjectures are always honoured.
Certain class structures promote conjecturing more than others. Students offer questions during lectures, but they are often of a surface variety. They notice a pattern that has occurred in three straight examples, or think they have discovered a short-cut. I don’t like using tricks, but if they are “discovered” or “re-invented” (to borrow a term from Piaget and genetic epistemology), then we use them.
I have students in an enriched class that demand for me to give them more practice. I tell them that we practice mathematics with daily class activities. They don’t want practice, they want repeated practice. They are accustomed to receiving repeatable drills to cement understandings.
I have learned to compromise with this demand. I do believe there is a place for basic skills training in mathematics, and would raise an eyebrow at anyone who claims these unnecessary. I do, however, also believe that the heart of mathematics is problem posing, problem framing, and problem solving.
I consider reading an essential part of my professional development. I enjoy a morning glance through a chapter or two, and like to wind down a winter’s day with a book and a cup of coffee. Sometimes reading is the only way to relax my mind at the end of a day. (Naturally, some professional literature is better at putting me to sleep than others). To this point in my young career, no book has changed my perspective on the teaching and doing of mathematics more than The Art of Problem Posing: Third Edition by Stephen I. Brown and Marion I. Walter. The duo writes quite a bit for “Mathematics Teacher” (the high school journal for the National Council of Teachers of Mathematics) as well. The processes introduced in the book have been crucial to the penning of many posts on this blog. The book creates a framework from which creative mathematics flows.