- that are perplexing so that students are engaged and want to use mathematics to explore the topic
- that have multiple entry points for students of varying abilities
- that are rich enough so that the exploration can drive the entire lesson
- that allow students to solve a problem in many different ways and then explain and prove their results.
As I read this book it really made me think about what makes a good math question and how I could adapt and improve anchor questions that I have used in order to make them more open ended and accessible to all students. I decided to go through the book and pull out all the questions that I could use in my sixth grade math classroom and organize them by the topics that we cover, in the order that we cover them, giving me ready access to some model questions that I can use or adapt throughout the school year. This was a very valuable process for me because by reviewing all the questions, especially those aligned with the sixth grade common core standards, I was able to more clearly understand what makes a good question and how to create these type of questions for other topics we cover.
As the TIMSS video study found, Japanese teachers use a structured problem solving format for most of their lessons where the teacher poses a complex thought-provoking problem, students struggle with the problem, various students present ideas or solutions to the class and then the teacher leads a class discussion using the various solution methods provided by the students and summarizing the class' conclusions (from The Teaching Gap by James Hiebert and James Stigler and CT Regional School District #10 Math Program FAQ Page). This lesson format has proven to be an extremely effective method of teaching mathematics and a model I have tried to use in my classroom. But finding rich problems that create the set up for a student centered lesson for each of the topics covered in our curriculum can be a challenge. Japanese educators are part of a teaching community that uses "lesson study" as a way for teachers to share and refine the problems and questions they pose at the beginning of a lesson that create the conditions in the classroom where students are engaged in deep mathematical thinking. This book is one resource that teachers in the US can use to find good thought provoking questions.
Here are a few of my favorite open questions and parallel tasks from the book:
1) Percent: 72 is _____% of ______
2) Percent: Choose something you have been wanting to buy that costs more than $50. Imagine you have $30 saved. What discount does the store need to offer before you can afford it?
3) Algebraic Expressions: An expression involving the variable k has the value 10 when k = 4. What could the expression be?
4) Algebraic Expressions: Order these values from least to greatest. Will your order be the same no matter what the value of n is? Explain.
- Choice 1: n/2 3n n^2 3n + 1 10 - n
- Choice 2: 4n 3n 10n 3n + 1 5n + 2 -n