This framework may be the most difficult to outline, but it is arguably the most important for math teachers. Below, I tried to break this post into four main sections, recognizing that this is only a draft and will certainly expand.
Three (now four?) Acts of a Mathematical Story
I am sad that someone of Dan Meyer’s caliber is no longer in the classroom, but I am very grateful for his Three Acts of a Mathematical Story, publicly available collection of Three-Act Math tasks, and his team’s production of the Desmos Activity Builder. I am inclined to agree with Meyer’s statement below:
Many math teachers take act two as their job description. Hit the board, offer students three worked examples and twenty practice problems. As the ALEKS algorithm gets better and Bill Gates throws more gold bricks at Sal Khan and more people flip their classrooms, though, it’s clear to me that the second act isn’t our job anymore. Not the biggest part of it, anyway. You are only one of many people your students can access as they look for resources and tools. Going forward, the value you bring to your math classroom increasingly will be tied up in the first and third acts of mathematical storytelling, your ability to motivate the second act and then pay off on that hard work.
The general public, including students and many teachers, view mathematics as a body of knowledge that has always existed and is complete with no room for growth. On the contrary, mathematics is an ever expanding field full of humans with fascinating personal stories pushing against the boundaries of current mathematical knowledge. By divorcing the teaching of mathematical skills and theorems from the very human stories and motivations that led to their creation/discovery, we limit students’ ability to connect with this creative endeavor. By telling mathematical stories that motivate questions and the learning of new skills, we make students participants in the practice of mathematics. Although school mathematics has a beginning, middle, and end, this does not mean we cannot introduce students to the mathematical stories that still only have a beginning, although that beginning may be hundreds of years old. As teachers, we have the ability to construct these narratives, thus I highly recommend reading Three Acts of a Mathematical Story.
Students are Individuals with Unique Motivations
I am incredibly grateful that educators like Sam Shah exist because they come up with resources like Explore! Math! and student research journals. Students have a wide variety of interests, yet teachers have a very limited time with them to cover a vast amount of content. Therefore, there need to be easy ways to encourage independent investigation so that students can connect mathematics to their own interests. Many mathematicians, myself included, entered mathematics as a happy accident rather than as part of a planned path. I’ll never forget wandering into a discrete mathematics course and finding out the field of mathematics rested on a foundation of beautiful proofs. The more opportunities for discovery we provide students, the more chances they have of stumbling upon the branch of math they find beautiful and relevant.
Mathematics does not have to be experienced within the walls of the classroom, especially since the majority of most peoples’ lives are lived outside of the classroom. Isaac Newton famously discovered the laws of gravity while sitting below an apple tree (maybe not entirely accurate) and developed calculus in order to model the motion of planets. Cartography and the mathematics of modern navigation were only developed because people were traveling across oceans, through forests, and over mountains and needed a way to gauge their own position in relation to a destination. Examples of the use of mathematics are all around us in architecture, transportation, etc. Even Euler, a famous mathematician, was inspired by getting outdoors. The entire field of graph theory traces its origins to a question about whether it was possible to travel an entire city by crossing its seven bridges once, and only once.
Slow Release of Information
The inherent flaw of textbooks is that they have no choice but to show all of their cards immediately: practice, examples, topics, answers, etc. There is no opportunity to motivate the problems, introduce the uncertainty of no available answers, and follow the unexpected paths that student questions may lead. The research tasks have an opening question that is designed to set up the proposed line of reasoning. They are not meant to simply be printed and handed out, recognizing the student questions that arise from the seed question may lead down a different mathematical path.