Where to begin...at the end of course

As many of us begin our school year today or within the near future it is important to think about where we want to end up in May/June instead of where to begin. As a math teacher I want my students to “know math.” Ok so what does that mean, for me it is a focus on helping students to develop a mathematical understanding and be able to put that understanding into practice daily. I know you are saying ok Christine, really something that big takes time, years. Yes your right it does that is why it is so important for us as educators to work collaboratively.  Focusing on a mathematical instructional paradigm shift where teachers focus on HOW concepts are taught, not What is taught. 

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.

1. Anticipate student responses prior to the lesson 
2. Monitoring students’ work on and engagement with the tasks
3. Selecting particular students to present their mathematical work 
4. Sequencing students responses in a specific order for discussion.
5. 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.