<p> Mathematical Concepts and Concept Descriptions</p><p>Identifying Key Concepts</p><p>One important aspect of writing worthwhile mathematical tasks is learning to identify and describe mathematical concepts. Mathematical concepts are the fundamental building blocks of the learning goals that drive the selection and implementation of worthwhile mathematical tasks. They are generally NOT procedures but are the principles that underlie why a procedure works. Once a mathematical concept has been identified, describing the concept is also important.</p><p>Writing Concept Descriptions</p><p>Description Formation Each concept description must have three parts: 1. A short heading 2. A general statement of the mathematical idea or concept; and 3. A specific example of the concept</p><p>Example of a Concept Description The following is an example of a concept description for the topic of division of fractions: Measurement Meaning of Division of Fractions. One way to interpret what division by a fraction means is to think of the number of the divisor that fit in the dividend. This is called the measurement model of division.</p><p>1 Example: 3 ¸ means the number of one-halves in 3 wholes. 2</p><p>Note that this concept description contains a heading, a description, and an example. You should attempt to make all three of these as complete and yet clear and succinct as possible. </p><p>Definition of a Concept Here is what a concept is NOT. It is not a step-by-step method for DOING something. That’s a procedure. It is not a teaching method. It is something that you want your students to understand. Concepts deal with meaning, why something works, ways of imagining or seeing things, and connections.</p><p>Non-Examples of Concept Descriptions  I want students to understand what slope means.  I want students to understand why you divide when finding slope. In both cases above, the person talks about a concept, but did not actually describe the concept. If you want to describe the concept of what slope means, then actually say what it means. If you want to describe why you divide when finding slope, then talk about meanings for division and why they are appropriate for slope situations. The Concept of Comparing Fractions</p><p>Non-Examples (Below are several descriptions of the concept of comparing fractions that were written by students but are lacking in different ways.)  If something is divided into parts, when two individual parts are compared, one may be greater than, less than, or equal to the other part.  A fraction can be represented in an infinite number of ways. It is always possible to determine whether one number is larger than another. Fractions (i.e. Rational numbers) can be compared and ordered.  Equality and Inequality. What does it mean for a fraction to be greater than another fraction?  To help students better understand how different fractions compare and in what ways we use them in the real world.  How to figure out the relationship between different fractions.  How fractions of different proportions can add or subtract value or how fractions are inequal.  By the end of the lesson the student will be able to tell which of 2 fractions is greater. They will also be able to find the common denominator of 2 fractions.  We can compare fractions even if their denominators vary. A common denominator allows us to more accurately compare fractions.  A whole can be divided into many kinds of fractions (e.g. 1/4, 1/3/ 1/12 etc.) –Given two fractions with the same or different denominators, the fraction can be compared as less than, greater than or equal to.</p><p>Example (Below is a description of the concept of comparing fractions that more accurately captures the concept.) To compare two fractions, you need to know the relative size of the pieces in each fraction and the number of pieces in each fraction. This can be aided by having both fractions with the same size pieces or possibly the same number of pieces.</p>