MTHED430:

Introduction

In this lesson, we will consider what is unique and valuable about mathematical thinking:

• What does it mean for you, as a teacher, to be thinking mathematically?
• How about your students?
• How do you know your students are thinking "mathematically" as opposed to some other way of thinking and reasoning?

The goal in this lesson is for us to develop some shared language and for each of us to develop a personal, working definition of what it means to think mathematically. But we're not just going to be working on instinct! Many experts have discussed what makes thinking mathematical, so we're going to familiarize ourselves with several ways of thinking about this issue.

Stop and Think!

Remember that the purpose of Thought Questions is not to evaluate your understanding of course content but rather to provide you the space and time to reflect on your initial ideas about a topic. Your answers to Thought Questions are seen primarily by your instructor; however, your instructor may use some of your responses for discussion or demonstration purposes. The quality of your answers to Thought Questions is used to evaluate your participation in the course.

Objectives

At the end of this lesson, you should be able to

• compare recent and influential documents that describe mathematical thinking and
• synthesize a variety of perspectives on mathematical thinking into a personal, working understanding.

Road Map

The purpose of the lesson road map is to give you an idea of what will be expected of you for each lesson. You will be directed to specific tasks as you proceed through the lesson.

Making Connections

How Is Mathematical Thinking Portrayed in the Common Core State Standards?

How familiar are you with the Common Core State Standards? Across the United States, states and school districts have implemented the Standards to varying degrees. Some of you may have been teaching with the Common Core Standards in mind for a few years now, and others of you may only be newly familiar with them. Access the Common Core State Standards official website and explore its components. In the Mathematics section, there are content standards listed by grade level, high school course, and mathematical domain (e.g., geometry, counting, or functions). Please spend a few minutes examining the Standards as listed by the grade level or content domain that interests you most. (Keep in mind that states that have or will be adopting the Core Standards make modifications to these areas.)

Within the Common Core Standards, the Standards for Mathematical Practice are often overlooked. The Common Core authors have developed a list of eight of these practices, which relate to the ways students should interact with mathematical content across grade levels and across content domains. We can think of these eight practices as the authors' description of what it means to think mathematically. Please read the Standards for Mathematical Practice.

As the introduction to the Standards for Mathematical Practice states, these Standards are descendants of earlier documents that addressed what it means to do mathematics and to think mathematically.  You may have heard of some of the documents: the National Council of Teachers of Mathematics (NCTM) Process Standards, published in 2000, and Adding It Up: Helping Children Learn Mathematics, published by the National Research Council in 2001. You will read more about these—and other authors' ways of describing mathematical thinking—in the readings for this lesson.

Let's begin by reading a selection from the book Smarter Than We Think: More Messages About Math, Teaching, and Learning in the 21st Century. It was written for educators, school administrators, parents, and anyone interested in K–12 math teaching. It was written by Cathy Seeley, a former president of the NCTM. Are you familiar with this organization? If not, you should be. It's the primary national professional organization for mathematics teachers. If you're not familiar with the Council, please spend a few minutes looking around the NCTM website. The organization offers membership packages for teachers, teachers in preparation, and schools. All of these packages offer benefits, such as access to online resources, web-based videos on a variety of professional development topics, and a subscription to one of the organization's magazines (Teaching Children Mathematics, Mathematics Teaching in the Middle School, or Mathematics Teacher). Whether or not you are or may consider becoming a member, it's important for you to be familiar with this important organization, which has done a lot to professionalize and advance the field of mathematics teaching.

Smarter Than We Think was written as a series of short "messages." For this lesson, you will be reading Message 31, which offers what I consider one of the best summaries of the Common Core Standards for Mathematical Practice. The piece connects the Common Core Standards to the NCTM Process Standards and the Five Strands of Mathematical Proficiency from Adding It Up.

As you read Message 31, note the terms and ideas you find most interesting and compelling. You should incorporate these into your concept map--one of your assignments for this course which you will work on at the beginning and the end of this course. (Please read the description of the Concept map assignment now by referring to its description on the Activities page) You will notice that Figure 31.1 on page 250 is a sort of concept map; you might use it to inform your own (but note that the author herself writes on page 251 that her personal graphic would be different from that one!).

Relational and Instrumental Understanding

Have you ever heard the phrase "Ours is not to reason why, just invert and multiply"? If you're anything like me, your reaction to that is probably something like Ugh! None of us who care about students learning and truly understanding mathematics would ever want to teach using mnemonics, rhymes, or other "tricks." On the other hand, most of us can probably understand or be sympathetic to reasons teachers sometimes fall back on using shortcuts. Can you think of times when you've resorted to "teaching through tricks"?

The next article talks more about the issue of teaching through tricks. The article, by Richard Skemp, was first written in 1976, but was reprinted in the magazine Mathematics Teaching in the Middle School in 2006. Skemp does a great job discussing the drawbacks of teaching through tricks, but he is also honest about some of the benefits. The terms relational understanding and instrumental understanding are no longer used very much, but Skemp's descriptions of these two contrasting types of understanding is still quite illustrative. (Consider adding the terms relational understanding and instrumental understanding somewhere in your concept map.)

One interesting site where mathematics teachers discuss the appropriate use of teaching tricks is Nix the Tricks. I encourage you to look around and want to point out that, in addition to publishing a book, this project maintains a publicly accessible, living document that you can read and contribute to. You are welcome to join any of the discussions if you care to engage, but please do note that this a Google document open to the public. Any information you provide, including personally identifiable or work-related information, will be available to the general public.

Procedural and Conceptual Understanding

The terms relational and instrumental are no longer commonly used to describe types of understanding. More common in educational research, curriculum materials, and standards documents are the terms procedural and conceptual understanding. This is not to say that there is widespread agreement about the definitions of these terms and whether/how much they overlap—some people consider procedural and conceptual understanding to be very distinct (with conceptual considered more important than procedural), while other people consider the two to overlap quite a bit.

All this is to say that the debate over what constitutes mathematical understanding is alive and well!

For the final reading, we’re going to go back in history a little further with a 1973 article by a researcher named Stanley Erlwanger. He studied the mathematical understanding of a sixth grader named Benny. Benny’s mathematics instruction consisted of Individually Prescribed Instruction (IPI), a type of program that was popular at the time, in which students worked individually through cards or booklets that presented mathematics lessons and then took mastery tests to confirm whether or not they could progress. Some things about this program may seem old-fashioned to us now, but in fact this model is similar to many computer-based self-guided instructional programs used today. Benny’s understanding of mathematics topics is quite interesting and revealing. As you are reading, you should again be noting terms, descriptions, or ideas that you might want to incorporate into your concept map.

Activities