We need to:
- Challenge students to make sense of what they are doing to solve mathematics problems
- Pose questions that stimulate student’s thinking, asking them to justify their conclusions, strategies, and procedures.
- Have students evaluate and explain the work of others, and compare and contrast different solution methods for the same problem
- Ask students to represent the same ideas in multiple ways (symbolically, pictorially or with manipulatives)
So how do we begin… with the end in mind? A small but mighty
step is to allow students to explain their thinking. Having students explain
their mathematical reasoning on how they came up with the answer is so huge in
knowing what they know. Also for those
of us that work in a PLC (professional learning community) it is getting toward
question #2 “How will we know…” It is
more than student’s having the right answer it is knowing how the students came
up with the right answer. What process
did they use? Will that process work if the numbers are changed? Can they prove
how they came up with their answer using tools, drawings or manipulatives? When
students are asked to explain their thinking they have to be able to put their
thoughts into some type of organization. “First I did, and then I did…” We
often have students do this back at our desk one to one when they have gotten
too many problems wrong. But what if we do this as a whole class or small group
lessons on problems that students got correct? Think of the impact on the
students, their self-confidence, their willingness to communicate/participate,
the learning from each other not just the teacher just to name a few. No, this
doesn’t happen overnight. No, there will be messy situations in which you the
teacher are going to have to discuss with your math team what the child was
attempting to do, but isn’t that ok, isn’t it about the learning both ours and
our student’s? So as you start your math lesson planning try to work in some
opportunities for students to share their math reasoning. It will open many
doors to many great math opportunities in your classroom. To help you get going
here are five practices from the NCTM publication 5 Practices for Orchestrating Productive Mathematic Discussions by
Margaret Schwan Smith and Mary Kay Stein 2011 which help to have a
meaningful discussion not just a show and tell of math ideas or procedures.
- Anticipate student responses prior to the lesson
- Monitoring students’ work on and engagement with the tasks
- Selecting particular students to present their mathematical work
- Sequencing students responses in a specific order for discussion.
- Connecting different students’ responses and connecting the responses to key mathematical ideas.
Some additional resources on this subject are:
- Burns, M (2005). Looking at How Students Reason. Educational Leadership http://mathsolutions.com/wp-content/uploads/2005_How_Students_Reason.pdf
- Principles to Actions Ensuring Mathematical Success for All (2014) NCTM Executive Summary https://www.nctm.org/uploadedFiles/Standards_and_Focal_Points/Principles_to_Action/PtAExecutiveSummary.pdf Continuing its tradition of mathematics education leadership, NCTM has undertaken a major initiative to define and describe the principles and actions, including specific teaching practices, that are essential for a high-quality mathematics education for all students.
- Reinhart, S. (2000). Never say anything a kid can say. Mathematics Teaching in the Middle School, 5(8), 478–483.
- Smith, M. S., & Stein, M. K. (2011). 5 practices for orchestrating productive mathematics discussions. Reston, VA: National Council of Teachers of Mathematics.
