As a math educator, how often do we hear gifted students stating, “I know how to do this.”? With the advent of the common core, the duality of mathematics knowledge is highlighted and provides an avenue to explore this claim. The Common Core State Standards not only delineate the content of the discipline of mathematics, but it also describes the content of a student’s mathematical character. Both parts are integral to substantiating a student’s claim to know.

The CCSS can be used to examine and extend a student’s claim of knowing a particular concept. This examination can occur by changing one word in the statement – I know how to do this. Changing one word assesses multiple knowledge domains (i.e. conceptual, procedural, metacognition) and targets the development of the standards for mathematical practice. If we focus on the function words of how, why, where, and when, an interesting cognitive shift occurs that unveils a student’s comprehensive understanding of a concept.

**Do you know how to do this?**

Students that know *how* to do a particular concept will be able to represent that concept in multiple ways. They will be able to demonstrate concrete verbal and pictorial representations, numeric representations, and abstract symbolic representations. This demonstrates a student’s application of mathematical practice (MP) 2. Students will be able to reason abstractly and quantitatively.

Students that know *how* to do a particular concept will know what tools are needed to negotiate the task or scenario. Math practice 5 notes that students will be able to strategically choose tools that will aid in problem-solving.

**Do you know why to do this?**

When students begin to understand the value of a math concept, the likelihood that they will continue to use and retain understanding of that concept beyond the classroom increases. Students that know *why* are able to provide a rationale for the selection of an algorithm or law used to negotiate a problem. An explanation of why triggers the development of multiple math practices including MP3 and MP1. Students will be able to make meaning of a problem and construct a viable argument as a result.

**Do you know where to do this?**

Mathematics is a discipline that describes patterns, relationships, and changes in everyday life. Math practice 4, modeling, captures the essence of why we study mathematics – to be able to interpret the world we live in. Students that know how to apply a math concept should be able to explain the context in life that calls for *where *that particular application is needed.

**Do you know when to do this?**

Students that know will be able to explain *when* a concept relates to other concepts and *when* that concepts may be called upon in a logical progression of thought. One of the instructional shifts introduced by the common core is that of coherency. Knowing when a concept is needed and how it relates to other concepts being employed demonstrates a coherent understanding of the discipline. The ability to look for and make use of structure, generalize, and to know when to use these generalizations (Math Practices 7 and 8) demonstrates a comprehensive understanding of the math concept.

#### Application Example:

**Do you know how to find the area of the shape? If so, what is the area of the shape?**

**Do you know why you are applying this concept?**

The shape presented is a composite of some more basic shapes, namely a triangle and a semi-circle. The formulas selected provide the area for a circle and a triangle. I divided the area of the circle in half to represent the total area of the semi-circle. In order to use the triangle formula, I needed to determine the value of the base and height. In a triangle, the height is the perpendicular line segment from any side (base) and the opposite vertex. Therefore, either 9 in or 6 in (the diameter of the circle) can serve as the base. The fact that the shape is on the coordinate plane indicates the 90° relationship between the two values. I added the two individual areas to find the area of the composite shape. The value of π remains in the answer to demonstrate the precise value.

**Do you know where to apply this concept**?

This problem can be applied to solve any area problems in real life that involve the combination of many basic shapes known such as triangles, squares, rectangles, circles, etc. I can use this line of thought to calculate the total area needed for carpeting, painting, and landscaping specifications.

**Do you know when to apply this concept?**

I applied the area formula after I was able to determine needed information in the scenario such as the radius of the circle, type of triangle present (i.e. right, equilateral, etc.). Once it was determined that the base and height of the triangle were known (no need to use the Pythagorean Theorem), I was able to calculate the total area. This concept can be applied building upon the existing understanding of the areas of basic shapes and how these basic shapes can be combined to create composite shapes. Finding the areas of composite shapes leads to finding the area of irregular shapes not composed of basic shapes.

#### Application Example

David reads two pages every five minutes. Do you know how many pages he will read after twenty-five minutes?

#### David’s Reading:

**Based on the pattern that follows, David will have read 10 pages after 25 minutes. 2 x minutes = 5 x pages**

**Do you know why you are applying this concept?**

There is a constant rate of change comparing the number of minutes read to the number of pages. For every change in time of five minutes there is a change in the number of pages by two. Recognizing this structure, lead to the application of the concept.

**Do you know where to apply this concept?**

This concept can be applied when determine distance or time traveled after maintaining a constant speed. It can also be applied to calculate total savings based on a constant amount saved over a period of time. It can also be used to make scale models and drawings.

**Do you know when to apply this concept?**

After I determined that there is a consistent rate of change between minutes and pages, I was able to use this pattern to calculate the resulting number of pages. This concept relates to other concepts such equivalent fractions and proportions.

#### Application Activity:

Think of a math concept that you know and have taught in your practice. Ask yourself the following questions:

- Do you know
*how* - Do you know
you are applying this concept?*why* - Do you know
to apply this concept?*where* - Do you know
to apply this concept?*when*

Do you have the evidence to validate your claim of knowing the concept?

What did you experience as a learner? (i.e. cognitive shift, anxiety)

What are the implications for your classroom instruction and curriculum design?

As we continue to change the paradigm in mathematics education of what it means to be a student of mathematics, we will continue to hear the response of “I know how to do this.” Redefining this idea of knowledge calls upon us as educators to ask the questions of not only Do you know *how* to do this?, but also Do you know *why* to do this?, Do you know *where* to do this?, and Do you know *when* to do this? The precise (Math Practice 6) answers to these questions using the language of the discipline will help assess a student’s claim of knowing and provide support, resulting in students developing as mathematicians.

**Dr. Jared DuPree** is a regional Mathematics Coordinator for the Los Angeles Unified School District. He holds a Doctorate in Education from the University of Southern California. He also holds a Bachelors of Science in Mathematics and a Masters of Arts in Educational Administration. As a current secondary math educator, he strives to present the objective nature of mathematics in a subjective manner providing access for students at all levels. Jared is a member of the California Association for the Gifted (CAG), National Council for Teachers of Mathematics (NCTM), Western Association of Schools and Colleges (WASC), and the California Mathematics Council (CMC). He is a demonstration teacher for the CAG and USC Teacher Institutes. Jared has presented at local, state, and national conferences on topics such as Depth and Complexity in Secondary Mathematics, the Problem Solving Process in Algebra, Strategic Questioning in Mathematics, Thinking Like a Mathematician, and Mathematics Differentiation. His doctoral research focused on district level strategies that impact student outcomes in algebra.

#### What would you like to read now?

- Fall 2013 Contents Page
*U**sing the Common Core Mathematical Practices to Assess Math Knowledge*by Jared DuPree*Fostering Intellectualism in Gifted Students*by Robert Grubb*Where in a Lesson*by Sandra Kaplan*Building a Back to School Foundation*by Helene Solomon*Using Bulletin Boards to Differentiate the Classroom Environment*by Jessica Manzone*Lessons Learned from a 20 Dollar Green Screen*by The Bui*Maximizing the Social and Emotional Growth of Gifted Children*by Ann Smith*Book review: If I’m so Smart Why Aren’t the Answers Easy?*review by Elaine Wiener*Book review: College@13,*review by Elaine Wiener*Book review: The Ultimate Guide to Internet Resources for Teachers of Gifted Kids,**review by Elaine Wiener*

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