PurposesEffective teaching; Professional development activity; Student collaboration; Reasoning

DescriptionObjective:

This analytic can be used for professional development to provide both pre-service and in-service teachers with an opportunity to observe fourth grade students’ reasoning of fraction concepts where students are observed exhibiting some of the Common Core State Standards (CCSS) for Mathematical Practice.

Main Description:

This analytic provides viewers with the opportunity to observe a mathematics lesson where the instructor’s moves are used to facilitate students’ reasoning in learning about fractions. The video data for this analytic originate from a 1993 Rutgers University study of fourth grade children from Colts Neck. Throughout this lesson, fourth grade students exhibit some of the CCSS for Mathematical Practice as they learn to compare unit fractions and place fractions on the number line.

Comparing unit fractions and placing fractions on the number line are content standards included in the CCSS. In the CCSS, there are five content domains with specific expectations for students in the fourth grade. These domains include: operations and algebraic thinking, number and operations in base ten, number and operations-fractions, measurement and data, and geometry (CCSSI, 2010). The first purpose of this analytic is to identify instances where the fourth grade class is working with mathematics content located in the domain: Number and Operations-Fractions (4.NF) from the cluster "Extend understanding of fraction equivalence and ordering" (CCSSI, 2010, p. 30). The following is the fourth grade Common Core State Standards for Mathematical Content observed in this analytic.

"Standard 1. Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Standard 2. Compare two fractions with different numerator and different denominators, e.g., by creating common denominators or numerators or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record and justify the conclusions, e.g., by using a visual fraction model" (CCSSI, 2010, p. 30).

The significance of focusing on reasoning is supported by several published studies on attending to students’ reasoning (Francisco & Maher, 2011; Francisco, Maher, Powell, & Weber, 2005; Maher, Landis, & Palius, 2010; Martino & Maher, 1999; National Council of Teachers of Mathematics (NCTM), 2009). In fact, Davis (1992) posits that "mathematics is a way of thinking that involves mental representations of problem situations and of relevant knowledge" (p. 226). In addition, Davis (1992) stresses "the real essence is something that takes place within the student’s mind" (p. 226). This analytic will give the viewer some insight into the reasoning that is going on inside the heads of the fourth grade students at Colts Neck by observing the students’ behaviors and listening to the students’ reasoning.

This insight is the foundation for the second purpose of this analytic. This second purpose of the analytic includes comparing how the observed practices of the students and teachers during the mathematics tasks used in the professional development intervention illustrate the Common Core State Standards for Mathematical Practice.

Throughout this lesson, fourth grade students exhibit some of the eight CCSS for Mathematical Practice as they learn to compare unit fractions and place fractions on the number line. Teachers are expected to incorporate as many of the eight CCSS for Mathematical Practices as possible within their lessons (CCSSI, 2010). The CCSSI (2010) states the following eight Standards for Mathematical Practice:

1) Make sense of problems and persevere in solving them.

2) Reason abstractly and quantitatively.

3) Construct viable arguments & critique others reasoning.

4) Model with mathematics.

5) Use appropriate tools strategically.

6) Attend to precision.

7) Look for and make use of structure.

8) Seek & Express regularity in repeated reasoning (pp. 6-8).

Although the practices are listed separately, they may overlap when incorporated into lessons. Also, not all eight practices are illustrated in this analytic. The CCSS for Mathematical Practice illustrated in this analytic include the following:

"1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they had a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and--if there is a flaw in an argument--explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1,2) with slope 3, middle school students might abstract the equation (y-2)/(x-1) = 3. Noticing the regularity in the way terms cancel when expanding (x-1)(x+1), (x-1)(x2+x+1), and (x-1)(x3+x2+x+1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results" (CCSSI, 2010, pp. 6-8).

Facilitating the classroom discussion and mathematical tasks is researcher Carolyn Maher. The viewer should note the questions asked of the students, the pauses after each question giving enough time to allow students to think, the allowance of having students consult with a partner, and how researcher Carolyn Maher circulated throughout the classroom. The researcher also praised and encouraged students to make sense of the problems by using multiple representations to express their reasoning, critique their classmates’ reasoning, and provide convincing arguments.

References:

Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved from: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Davis, R. B. (1992). Understanding "Understanding". Journal for the Research in Mathematics Education, 11, 225-241.

Francisco, J. M. & Maher, C. A. (2011). Teachers attending to students’ mathematical reasoning: Lessons from an after-school research program. Journal of Mathematics Teacher Education, 14(1), 49-66.

Francisco, J. M., Maher, C. A., Powell, A., & Weber, K. (2005, May). Urban Teachers Attending to Students’ Mathematical Thinking: An Emergent Model of Professional Development. Paper presented at the fifteenth ICMI Study Conferences: The Professional Education and Development of Teachers of Mathematics, Àguas de Lindòia, São Paulo, Brazil.

Maher, C. A., Landis, J. H. & Palius, M. F. (2010). Teachers attending to students’ reasoning: Using videos as tools. Journal of Mathematics Education 3 (2), 1-24.

Martino, A.M. & Maher, C. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us. Journal of Mathematical Behavior, 18, 53-78.

National Council of Teachers of Mathematics. (2009). Focus in high school mathematics:

Reasoning and sense making. Reston, VA: Author.

This analytic can be used for professional development to provide both pre-service and in-service teachers with an opportunity to observe fourth grade students’ reasoning of fraction concepts where students are observed exhibiting some of the Common Core State Standards (CCSS) for Mathematical Practice.

Main Description:

This analytic provides viewers with the opportunity to observe a mathematics lesson where the instructor’s moves are used to facilitate students’ reasoning in learning about fractions. The video data for this analytic originate from a 1993 Rutgers University study of fourth grade children from Colts Neck. Throughout this lesson, fourth grade students exhibit some of the CCSS for Mathematical Practice as they learn to compare unit fractions and place fractions on the number line.

Comparing unit fractions and placing fractions on the number line are content standards included in the CCSS. In the CCSS, there are five content domains with specific expectations for students in the fourth grade. These domains include: operations and algebraic thinking, number and operations in base ten, number and operations-fractions, measurement and data, and geometry (CCSSI, 2010). The first purpose of this analytic is to identify instances where the fourth grade class is working with mathematics content located in the domain: Number and Operations-Fractions (4.NF) from the cluster "Extend understanding of fraction equivalence and ordering" (CCSSI, 2010, p. 30). The following is the fourth grade Common Core State Standards for Mathematical Content observed in this analytic.

"Standard 1. Explain why a fraction a/b is equivalent to a fraction (nxa)/(nxb) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

Standard 2. Compare two fractions with different numerator and different denominators, e.g., by creating common denominators or numerators or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record and justify the conclusions, e.g., by using a visual fraction model" (CCSSI, 2010, p. 30).

The significance of focusing on reasoning is supported by several published studies on attending to students’ reasoning (Francisco & Maher, 2011; Francisco, Maher, Powell, & Weber, 2005; Maher, Landis, & Palius, 2010; Martino & Maher, 1999; National Council of Teachers of Mathematics (NCTM), 2009). In fact, Davis (1992) posits that "mathematics is a way of thinking that involves mental representations of problem situations and of relevant knowledge" (p. 226). In addition, Davis (1992) stresses "the real essence is something that takes place within the student’s mind" (p. 226). This analytic will give the viewer some insight into the reasoning that is going on inside the heads of the fourth grade students at Colts Neck by observing the students’ behaviors and listening to the students’ reasoning.

This insight is the foundation for the second purpose of this analytic. This second purpose of the analytic includes comparing how the observed practices of the students and teachers during the mathematics tasks used in the professional development intervention illustrate the Common Core State Standards for Mathematical Practice.

Throughout this lesson, fourth grade students exhibit some of the eight CCSS for Mathematical Practice as they learn to compare unit fractions and place fractions on the number line. Teachers are expected to incorporate as many of the eight CCSS for Mathematical Practices as possible within their lessons (CCSSI, 2010). The CCSSI (2010) states the following eight Standards for Mathematical Practice:

1) Make sense of problems and persevere in solving them.

2) Reason abstractly and quantitatively.

3) Construct viable arguments & critique others reasoning.

4) Model with mathematics.

5) Use appropriate tools strategically.

6) Attend to precision.

7) Look for and make use of structure.

8) Seek & Express regularity in repeated reasoning (pp. 6-8).

Although the practices are listed separately, they may overlap when incorporated into lessons. Also, not all eight practices are illustrated in this analytic. The CCSS for Mathematical Practice illustrated in this analytic include the following:

"1. Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they had a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and--if there is a flaw in an argument--explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

5. Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1,2) with slope 3, middle school students might abstract the equation (y-2)/(x-1) = 3. Noticing the regularity in the way terms cancel when expanding (x-1)(x+1), (x-1)(x2+x+1), and (x-1)(x3+x2+x+1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results" (CCSSI, 2010, pp. 6-8).

Facilitating the classroom discussion and mathematical tasks is researcher Carolyn Maher. The viewer should note the questions asked of the students, the pauses after each question giving enough time to allow students to think, the allowance of having students consult with a partner, and how researcher Carolyn Maher circulated throughout the classroom. The researcher also praised and encouraged students to make sense of the problems by using multiple representations to express their reasoning, critique their classmates’ reasoning, and provide convincing arguments.

References:

Common Core State Standards Initiative (CCSSI). (2010). Common Core State Standards for Mathematics. Retrieved from: http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Davis, R. B. (1992). Understanding "Understanding". Journal for the Research in Mathematics Education, 11, 225-241.

Francisco, J. M. & Maher, C. A. (2011). Teachers attending to students’ mathematical reasoning: Lessons from an after-school research program. Journal of Mathematics Teacher Education, 14(1), 49-66.

Francisco, J. M., Maher, C. A., Powell, A., & Weber, K. (2005, May). Urban Teachers Attending to Students’ Mathematical Thinking: An Emergent Model of Professional Development. Paper presented at the fifteenth ICMI Study Conferences: The Professional Education and Development of Teachers of Mathematics, Àguas de Lindòia, São Paulo, Brazil.

Maher, C. A., Landis, J. H. & Palius, M. F. (2010). Teachers attending to students’ reasoning: Using videos as tools. Journal of Mathematics Education 3 (2), 1-24.

Martino, A.M. & Maher, C. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us. Journal of Mathematical Behavior, 18, 53-78.

National Council of Teachers of Mathematics. (2009). Focus in high school mathematics:

Reasoning and sense making. Reston, VA: Author.

Created on2013-07-05T13:04:27-0400

Published on2015-06-22T10:34:31-0400

Persistent URLhttp://dx.doi.org/doi:10.7282/T3445P71