To us everything begins with creating the perfect problem for our students. We strive to give students open ended, rich math problems that allow students to think critically and creatively, and lead them to discover or see the need for a mathematical concept we intend to teach them. Often when people come to observe our classrooms they remark on how well our students collaborate in groups and carry on mathematical conversations among each other. This happens because of the problem we have designed for our students. We see the problem and the intentional group work structure as a yin-yang relationship. One does not exist without the other. Below, we outline a few of the guiding principles we use when creating or modifying problems for our students.
The Necessity Principle
During the planning period we begin by first thinking about what is the “need” for students to learn the mathematics we intend to teach them, “for students to learn the mathematics we intend to teach them, they must see a need for it, where ‘need’ means intellectual need, not social or cultural need” (Harel, 2008b, p. 900). For example, what is the mathematical need for students to learn solving systems of equations? The need is not because it’s in our standards or because they need to know it for a future math classroom. Instead, we think about why this method was created by mathematicians and how it make our lives easier. Dan Meyers has a series on his blog called, “If Math Is The Aspirin, Then How Do You Create The Headache?" He was also inspired by Harel’s ideas and the importance of showing students the necessity for the mathematics we intend to teach them.
Low Floor-High Ceiling:
We are borrowing this term from Jo Boaler. As the name implies the purpose of this is to construct the problem in such a way so that it is accessible to all students, but is expandable higher levels of mathematics. Jo gives a great example of this with the pattern problem in this video: https://www.youtube.com/watch?v=hKmypL2yQAI.
In the video Jo references the pattern below and suggests to begin the problem by asking students, "how they see pattern growing" instead of asking "how many tiles are in figure (blank)". Starting the problem in this manner opens up the problem to a variety of ways of thinking and makes it accessible to all types of learners. In the video Jo demonstrates that people often see the pattern changing in variety of ways.
What we appreciate about this way of starting is that it allows students to process the pattern and perhaps even understand its structure. If we had started the problem with the "how many.." question then those students that are quick at calculating would reach their answer and those that are not would feel the anxiety of not having enough time to think about it. We have also seen how this approach of starting the problem helps even the most reserved students to come up and share their thinking. This problem can be extended to higher levels by asking student to figure out the number of tiles in higher figure numbers or giving them some number of tiles and asking them if it would be possible to construct a specific figure number.
This concept also comes from Harel’s teaching framework. A holistic problem refers to a problem where one must figure out from the problem statement the elements needed for its solution—it does not include hints or cues as to what is needed to solve it. A non-holistic problem, on the other hand, is one which is broken down into smaller parts, each of which attends to one or two isolated elements. Often each of such parts is a one-step problem. Part of developing students’ mathematical thinking skills is having them develop their own methods and approaches. When we create a problem we make sure not break down into steps or provide any hints or suggestions as to how students should approach it.
There is nothing worse than a problem telling students exactly what to do: make a table, graph, or equation etc. That takes all the thinking out of it, but it also stifles students’ problem solving skills and most importantly their creativity! When students come up with their own approaches to solving a problem we name the method after them. This is powerful for our students because they feel that they are part of the mathematics we are learning. This builds their confidence and our classroom community as other students begin to refer to each others' methods when solving future problems.