Math Matters

Its a new year and that brings lots of hope for the potential of what can come from the efforts we are willing to make. The new year always brings a chance for renewal - an opportunity to remake yourself, change a habit, or even begin to think differently and behave differently than you have before. This renewed hope and positive attitude swept in with the winter winds can also be an opportunity to improve the way students think about mathematics. By placing a new focus on mathematical processes and reasoning, we can show students that Math really does Matter!

According to the Software Engineering Institute at Carnegie Melon University, a process defines "what" needs to be done and procedure defines "how" to do the task. Consider this in a mathematical problem solving situation. The processes we use describe how the task is analyzed and the steps and procedures we should take are identified whereas the procedures come second, as we solve the problem, and employ the procedure through a series of steps to derive the solution. I think about process thinking as mathematical reasoning - the sum total of interpreting, analyzing, solving, justifying, and explaining along the problem solving journey. In this way procedural thinking is just one part of the whole process. When we attempt to develop procedural thinking first, we limit students to a series of steps that aren't always universal to use and we resort quickly to tricks and gimmicks to help students remember the various procedures and when they would apply. But rather than this, if we focus on developing a sense of process, then we enable students to harness the power of mathematics because they begin to see patterns, explore when, how, and why certain procedures work, and can explain their thinking and back up their explanations with solid mathematical understanding.

Process skills pervade the mathematics standards, both nationally and in individual states such as Texas. The National Council of Teachers of Mathematics has included standards on problem solving and process thinking since publishing An Agenda for Action in 1980. Using the NTCM Principles and Standards as a foundation, the Common Core State Standards for Mathematics along with specific state standard documents such as the Texas Essential Knowledge and Skills, have increased a focus on the inclusion of process skills that should overarch the teaching and learning of content skills.

The current version of the Texas Essential Knowledge and Skills for Mathematics, adopted in 2012 and implemented in 2014 showcase standards for process thinking at the forefront, and then layer in specific content objectives using the process skills as a foundation of strong problem solving and reasoning. All of the knowledge statements that comprise the standards that follow the process skills begin with the phrase “the student applies mathematical process standards to…” illustrating a newfound emphasis on the use of these skills in working with all mathematical content to be learned. The process standards found within the TEKS, like those found in the NTCM Principles and Standards and the Standards for Mathematical Practice in the CCSS, describe ways in which students are expected to engage in learning and working with mathematics content throughout their courses of study.

Read more in my paper about the importance of these process skills and find a reference chart highlighting the process standards found within the NCTM Principles and Standards, the CCSS, and the TEKS.

Making the shift to a focus on process thinking over procedural thinking doesn't have to be hard, it just means we have to design learning around what we want students think about and discover about the mathematical ideas in our content rather than only teaching them the rote procedures to apply. This means designing work that allows students to make connections about math content across concepts, to generalize, find patterns, analyze solutions, and explain/justify what they did and what they were thinking about in the problem solving cycle. A few ideas are:

• Provide students a series of problems that are already solved. Have them identify correctly worked solutions and those that are incorrect. For all incorrect work, have them identify where the mistake occurred and describe why that person may have made that error.
  
• Have students match representations together or to display there solution using more than two representations - such as verbal, graphical, tabular, algebraic, etc. Students should be able to explain how their solution is represented in all the views and explain why they selected each. Are some representations better than others?
• For a problem with a series of steps, or a proof with several steps, provide students all the steps or component parts and have them determine the order they should be in. For example, for order of operations problem solving, put each step in the solution on a separate strip of paper. Provide a few extra steps that are incorrect based on common errors. Then have students put the steps in the correct order and try to identify the extraneous steps. Have them explain their thinking as they made decisions about which steps were to be used and which were not - using mathematical justifications.

Math Cut Ups are designed as a hands on way to engage students to do more math. No one wants to work problems on a boring worksheet. Math Cut Ups take the problems off the worksheet and make them interactive and more fun for students.

Thinking cards are an easy way to keep the focus on mathematical reasoning and process thinking. Thinking card sets contain questions that scaffold and support students to reason, explain, analyze, and summarize their ideas while they work. Each set of cards is designed be used within a small group of students but certainly you could provide each student their own set. As students problem solve with the concepts presented in class, they use the cards to ask and answer higher order thinking questions and prompt thinking about the task and how it might be solved. Promote the use by assigning students or groups to share out responses to certain card questions, or have different groups share out different thoughts using the card letters to identify which questions are to be explained.

Thinking cards come in 4 types. Process Thinking Cards focus on the mathematical process skills while Algebraic Thinking Cards, Functional Thinking Cards, and Geometric Thinking cards focus specifically on the mathematics processes and reasoning within narrower strands of the standards. Each package of Thinking Cards contains 10 sets of the question cards, ideal for small group instruction. All cards are pre-printed in color on cardstock and for just $5 a set, comes ready to laminate, cut, and use!

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K-12 Mathematics Specialist

Mathematics Enthusiast

Math Cut Ups creator

Edusmart Mathematics Director

Email: integralmathematics@gmail.com
Website: www.mathcutups.com
Phone: 214-471-5760
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