The Importance of Implementation

A recent NPR article entitled, 5 Lessons Education Research Taught Us in 2014, seems to have a lot of definitive answers about our currently polarized educational climate.  The article mentions a research paper which encourages the use of teacher-directed, explicit instruction of mathematical computation skills for procedural fluency with students with mathematical difficulties.  To me this read as favoring explicit instruction and direct modeling of mathematics for students with disabilities over the project or problem-based, hands-on (manipulatives), collaborative (student-led), investigative style instruction that makes up some “reform” mathematics curriculum.

As a counterpoint to the NPR article, the National Council of Teachers of Mathematics lists procedural fluency as just one part of what is referred to as Mathematical Proficiency.  In chapter 2 of the book, Achieving Fluency: Special Education and Mathematics, mathematical proficiency is discussed as including the following four components: procedural fluency, conceptual understanding, strategic and adaptive mathematical thinking, and productive disposition.  Together these four components lead to mathematically proficient students which lead to mathematically proficient adults, disabilities or not.

1. Procedural fluency involves using basic skills such as facts, procedures, and formulas efficiently (i.e., quickly and accurately). It also entails knowing when to use them and, if necessary, how to adapt them. In other words, procedural fluency is skill in carrying out routines appropriately and flexibly as well as efficiently.

2. Conceptual understanding is knowledge of facts, generalizations, or principles underlying the comprehension of concepts (categories), relations (between categories), or operations (actions or events involving categories).

3. Strategic competence involves the ability to formulate, represent, and solve mathematical problems, and adaptive reasoning entails the capacity for logical thought, reflection, explanation, and justification.

4. Productive disposition entails believing that mathematics makes sense and is useful, that learning it requires diligence, and that everyone is capable of significant mathematical learning.

Since the NPR article about educational research only references one paper specifically about mathematics instruction, you only get one point of view.  This NCTM book provides another viewpoint of what are important goals for mathematics lessons with struggling students.

Simple Prompts Can Lead to Complex Mathematical Thinking

This post is inspired by chapter 8 of Steve Leinwand’s book Accessible Mathematics.  If you haven’t read this book, do it!  Leinwand is a leading voice in the push for math instruction that makes sense to students and will lead to longer lasting mathematical understanding.  Chapter 8 is entitled, “How Big, How Far, How Much?” and in it Leinwand encourages this instructional shift:

Tie the math to such questions as How big? How much? How far? to increase the natural use of measurement throughout the curriculum.

He goes on to say that measurement as a mathematical skill is often a “skipped chapter,” but is also one of the most pervasive life skills in the mathematics curriculum.  Leinwand goes on to encourage teachers to incorporate measurement as “an ongoing part of daily instruction and the entry point for a larger chunk of the curriculum” (p. 46).

Now you may be saying, “But I have a million other goals and standards and expectations and test prepping and whatnot that I have to do before I teach the kids to use a ruler!”

Well, lucky for you the Standards for Mathematical Practice also have you covered.