How do we know when students understand? Can we assume that a right answer means they comprehend both conceptually and procedurally the mathematics being taught? Can we assume they don't understand when they answer incorrectly? How can we determine or assess students' conceptual and procedural understanding? This wiki explores how students think and reason about the mathematics they are learning. Through this in-depth analysis, we can analyze many components of effective teaching and learning which will help us diagnose and plan effective instruction. Some of these components include:

Conceptual Understanding
Conceptual understanding is one of the five strands of mathematical proficiency, the overall goal of K-12 mathematics education, recommended by the National Research Council's 1999-2000 Mathematics Learning Study Committee in their report titled Adding It Up: Helping Children Learn Mathematics, published by the National Academy Press in 2001. Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts, tricks, or methods. They understand why a mathematical idea is and can connect these ideas to other strands of mathematics. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know.Adding It Up: Helping Children Learn Mathematics, National Academy Press, 2001. For example, multiplying fractions is procedurally simple computation requiring only a simple rule to memorize. However, how many students or adults know why the product is smaller when multiplying fractions? Can students represent multiplication of fractions using a model or visual representation? Can the student use this model or representation to communicate comprehension or apply this understanding to other types of multiplication of fractions? This comprehensive and in-depth understanding is crucial to the development of conceptual understanding.

Procedural Fluency
Procedural fluency refers to the development of procedures or algorithms for solving mathematical exercises. For example, the "way" you learned to add and subtract is most likely quite procedural. You may have been told to "borrow" in order to subtract or "carry" the "one" to the next column because you can't have two digits in one place. Procedures are important to mathematics teaching, but have often replaced conceptual understanding. The two types of understanding need to be taught with purpose and explicitly to develop sustained and powerful student mathematical comprehension.

Student Disposition
Students develop beliefs about their own mathematics ability from societal influences and classroom experiences. They bring these beliefs to the classroom every day as they work on comprehending the mathematics they are learning. Teachers and parents can and do inadvertently reinforce these inherent beliefs through classroom activities, expectations, ability-grouping or other experiences. Key dispositions for successful mathematics understanding include, but are not limited to, perseverence, persistence, precision, and communication. These dispositions are reflected in the Common Core Standards for Mathematical Practices , 2010.

Teacher Practice
Teacher practice includes invented strategies or methods, manipulatives, visuals, "tricks", activities, vocabulary, and/or procedures for teaching mathematics. A common example is the popular "alligator" or "pacman" visual used to teach the concept of greater and less than. Students draw teeth, act out the alligator sounds, and pretend to "eat" the larger number in order to determine which of two numbers are larger. The practice can be effective or ineffective depending on how the conceptual understanding is developed. Examples of teacher practice can be found in every classroom. The key to teacher practice is the connection of the practice to student understanding and mathematical comprehension.

Common Core State Standards
The Common Core State Standards Mission Statement states, "The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy," Common Core State Standards , 2010.

Use this form to analyze the pencast interviews. Choose to focus on one or more of the elements included or those assigned!

You will find student pencasts organized on the pages of this wiki. Some of the pencasts are found on several different pages because they illustrate different components of the diagnostic process.

This prompt illustrates a student's fractional understanding. Note the mathematical content understanding of fractions and dispositions exhibited by the student.

Maya
This third grader demonstrates her understanding of addition and place value.

By Beth Kobett, Stevenson University

How do we know when students understand? Can we assume that a right answer means they comprehend both conceptually and procedurally the mathematics being taught? Can we assume they

don'tunderstand when they answer incorrectly? How can we determine or assess students' conceptual and procedural understanding? This wiki explores how students think and reason about the mathematics they are learning. Through this in-depth analysis, we can analyze many components of effective teaching and learning which will help us diagnose and plan effective instruction. Some of these components include:Conceptual UnderstandingConceptual understanding is one of the five strands of

mathematical proficiency, the overall goal of K-12 mathematics education, recommended by the National Research Council's 1999-2000 Mathematics Learning Study Committee in their report titledAdding It Up: Helping Children Learn Mathematics, published by the National Academy Press in 2001. Conceptual understandingrefers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts, tricks, or methods. They understand why a mathematical idea is and can connect these ideas to other strands of mathematics. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know.Adding It Up: Helping Children Learn Mathematics, National Academy Press, 2001. For example, multiplying fractions is procedurally simple computation requiring only a simple rule to memorize. However, how many students or adults know why the product is smaller when multiplying fractions? Can students represent multiplication of fractions using a model or visual representation? Can the student use this model or representation to communicate comprehension or apply this understanding to other types of multiplication of fractions? This comprehensive and in-depth understanding is crucial to the development of conceptual understanding.Procedural FluencyProcedural fluency refers to the development of procedures or algorithms for solving mathematical exercises. For example, the "way" you learned to add and subtract is most likely quite procedural. You may have been told to "borrow" in order to subtract or "carry" the "one" to the next column because you can't have two digits in one place. Procedures are important to mathematics teaching, but have often replaced conceptual understanding. The two types of understanding need to be taught with purpose and explicitly to develop sustained and powerful student mathematical comprehension.

Student DispositionStudents develop beliefs about their own mathematics ability from societal influences and classroom experiences. They bring these beliefs to the classroom every day as they work on comprehending the mathematics they are learning. Teachers and parents can and do inadvertently reinforce these inherent beliefs through classroom activities, expectations, ability-grouping or other experiences. Key dispositions for successful mathematics understanding include, but are not limited to, perseverence, persistence, precision, and communication. These dispositions are reflected in the Common Core Standards for Mathematical Practices , 2010.

Common Core Mathematical Practices

Teacher PracticeTeacher practice includes invented strategies or methods, manipulatives, visuals, "tricks", activities, vocabulary, and/or procedures for teaching mathematics. A common example is the popular "alligator" or "pacman" visual used to teach the concept of greater and less than. Students draw teeth, act out the alligator sounds, and pretend to "eat" the larger number in order to determine which of two numbers are larger. The practice can be effective or ineffective depending on how the conceptual understanding is developed. Examples of teacher practice can be found in every classroom. The key to teacher practice is the connection of the practice to student understanding and mathematical comprehension.

Common Core State StandardsThe Common Core State Standards Mission Statement states, "The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them. The standards are designed to be robust and relevant to the real world, reflecting the knowledge and skills that our young people need for success in college and careers. With American students fully prepared for the future, our communities will be best positioned to compete successfully in the global economy," Common Core State Standards , 2010.

Pencast AnalysisUse this form to analyze the pencast interviews. Choose to focus on one or more of the elements included or those assigned!

You will find student pencasts organized on the pages of this wiki. Some of the pencasts are found on several different pages because they illustrate different components of the diagnostic process.

Student ExamplesEstimating Multiplication

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Place Value 29

## My Dad Taught Me!

A first grader shares a strategy for addition with regrouping.## Where Does 29 go?

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A first grader decides where 29 should go on a number line by using a specific technique.

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Sharing Cookies## This prompt illustrates a student's fractional understanding. Note the mathematical content understanding of fractions and dispositions exhibited by the student.

MayaThis third grader demonstrates her understanding of addition and place value.

Many Ways to Represent 52This third grader demonstrates multiple representations for 52.

The Birthday Party

A fourth grader solves a real-life word problem.

## Division with two-digit divisors

A fifth grade student demonstrates her understanding of division.How Many Sets of Ten?This third grader explains her reasoning for finding how many sets of ten are in 237.

Rectangles

A first grader determines which of the shapes presented are rectangles.

.Subtracting Fractions

A fifth grader subtracts fractions using a unique strategy.

Presents and balloonsA first grader solves a join word problem.

SubtractionA second grader explains how to subtract with regrouping

Fractions on a Numberline

A fourth grade student places fractions on a number line.

Conner and the Candies

Conner shares how he compares two amounts to find the difference.

Using touchpoints to add

A young student uses the technique of touchpoints to regroup in addition.

Kindergarten Problem Solving

What is a triangle?

A first grader decides which shapes are triangles.

Estimating Multiplication

Blank x Blank

A middle schooler explains how he would find missing products.