Teaching Mathematics Through Problem-Solving in K-12 Classrooms
So I wrote a book. It’s called Teaching Mathematics Through Problem-Solving in K-12 Classrooms, due out October 1, 2018, from Rowman & Littlefield.
The 1980s, by many accounts, were the “decade of problem-solving”. I came to realize, over 16 years of teaching and learning, that we haven’t truly defined what “problem-solving” means, let alone, “teaching through problem-solving”, so I set out to flesh that out. (There are many, many, resources and books out there that address problem-solving, but I haven’t really found a clear, concise, and to the point explanation of what teaching through problem-solving actually means).
Teaching through problem-solving is a muddy, slippery concept. Like many concepts in the education world, it is both over, and under-theorized. Figure that one out.
Early inspiration for this book came from the mathematical symbol for “there exists”, which calls in to being a “thing”, or a class of “things”. My essay “There Exists An Elegant, Lovely, and Inspiring School Mathematics” is my aspirational, not technical, vision for mathematics teaching and learning, and to this day is the most read of around 350 Medium pieces I have written.
So we have this “thing”, that could, and should exist, but how can we make it so? I decided to proceed from axioms for teaching through problem-solving, used these to frame each chapter, and attempt some sort of “proof” that they work. These are not mathematical proofs, obviously, as my own story as a learner and teacher is necessarily interwoven. There is no QED intended at the end of this book-it is intended to generate and spark debate.
Here are the axioms I settled on. Please feel free to quibble and quarrel with them in the comments, or on Twitter.
If the book does inspire you, I hope it is to create your own axioms. Know what you stand for, know what works for you. Know also the research-I cited over 30 sources, but the book is not intended to be comprehensive in that sense. I leave that to others.
These are not axioms in the immutable, mathematical sense. Like all things outside of mathematics, they are open to interpretation and debate. (#4 is one that gets people fired up, for example, as some believe it is, well, not axiomatic at all.)
Axioms for Teaching Through Problem-Solving
- There exists a school mathematics that is inspiring, interesting, playful, and elegant, where students work on interesting problems, together and alone, and share their own thoughts, ideas, conjectures, and reasoning.
- Problems are tasks or questions which are interesting, and contain big and interesting mathematical ideas.
- Interesting problems are the mainstay of instruction in mathematics classrooms. Problem-solving is not a reward, special event, or a “once in a while” thing.
- All students can be taught to think mathematically.
- Problem-solving is an iterative, ongoing process of thought.
- Representation in mathematics classrooms is both a mental and physical activity-one of thinking about, and “bringing into being”.
- Basic skills and problem-solving (procedural and conceptual understanding) are not at odds, but rather are mutually complementary.
- Full instructional guidance is a necessary condition of problem-solving classrooms.
- Mathematics classrooms are places of talking, thinking, conjecturing, and wondering about the mystery and beauty of mathematics.
- Mathematics should be playful.
The last axiom has its own subset of axioms. I drew on ideas from Cathy Bruce and Francis Su and Dan Finkel, and a little TEDx I did last Fall. I am happy to see NCTM going in the “math is play” direction.
Overall, this book is the culmination of 16 years of thought. It represents where I was at in my thinking in Fall 2017-Spring 2018. I am proud to release it into the world, and most proud of all if it sparks debates, arguments, math fights, Twitter spats, and so on.
Pick up on one of the axioms and comment, if you like, here, or on Twitter. Happy to start a conversation.
Q.E.D. (Just kidding!)
Twitter: Matthew Oldridge
Link to the book on Amazon: https://www.amazon.com/Teaching-Mathematics-through-Problem-Solving-Classrooms/dp/1475843321/ref=sr_1_1?ie=UTF8&qid=1534513404&sr=8-1&keywords=oldridge+math
