4th Graders Journey to Division of Fractions

Working with fractions is consistently a challenge for both younger and older students. Specifically, understanding the operation of division may be harder for younger students, who could be asked to imagine splitting a piece of something into yet smaller pieces. Unfortunately, the idea of division is often introduced to students as a rule of multiplying the initial value (numerator) by the reciprocal of the second value (denominator) in the number sentence, without understanding. In the more...

**Purpose**Reasoning

**Creator**Alexa Lindenberg (Rutgers University)

**Published**2017-07-19

**Persistent URL**http://dx.doi.org/doi:10.7282/T3251N0P

Author: Victoria Krupnik As educators and researchers, we frequently notice students’ struggles when they first encounter higher-order mathematical thinking, such as providing precise and convincing arguments for non-routine and challenging problems and developing formal, mathematical proofs. This occurs for several reasons, including, but not limited to, the following: 1. Math reasoning and argumentation requires more precision in its definitions and assumptions than is usually present more...

**Purpose**Effective teaching; Lesson activity; Student elaboration; Student engagement; Student model building; Reasoning; Representation

**Creator**Victoria Krupnik (Rutgers University)

**Published**2017-04-28

**Persistent URL**http://dx.doi.org/doi:10.7282/T34X5B6C

An Experiential Investigation of Algebraic Ideas Created and Implementedby Robert B. Davis

Objective: To demonstrate instances of experiential learning of 6th grade students with Professor Robert Davis as the facilitator. Students first solve single order equations and then build on their learning to solve and justify the solutions of quadratic equations. Solving second order equations is generally an 8th or 9th grade strand, but through their own reflections and without intervention by the facilitator, students are able to understand the "secret" behind their solution. more...

**Purpose**Effective teaching; Reasoning; Student model building; Student engagement

**Creator**Charles L. Silber (Rutgers University)

**Published**2013-10-09

**Persistent URL**http://dx.doi.org/doi:10.7282/T3NV9G6H

Analysis on Student Collaboration and Comparing Fractions

There is much focus on the environment in which students undertake problem solving. When properly established, a collaborative setting can be a key factor in the growth and development of students’ ability to reason and construct arguments. Mathematically proficient students develop conjectures and construct arguments and in collaborative settings, propose these ideas to others and respond to the reasoning of others (NGACBP & CCSSO, 2010). Students should be able to differentiate between more...

**Purpose**Effective teaching

**Creator**Margaret Molloy (Rutgers University)

**Published**2015-06-16

**Persistent URL**http://dx.doi.org/doi:10.7282/T3CZ38Z5

Ariel Constructing Linear Equations for "Guess My Rule" and the "Ladder" Problems

This analytic portrays a seventh grade student, Ariel, from Frank J. Hubbard Middle School in Plainfield, New Jersey, who works on several problems involving linear functions. Ariel is one of a number of middle school students who participated in the Informal Mathematical Learning project (IML). IML was an after-school, 3-Year National Science Foundation-funded longitudinal study (Award REC-0309062) conducted by the Robert B. Davis Institute For Learning at Rutgers University. This analytic more...

**Purpose**Professional development activity; Student elaboration; Student engagement; Student model building; Reasoning; Representation

**Creator**Mary Pierce (Rutgers University)

**Published**2015-06-16

**Persistent URL**http://dx.doi.org/doi:10.7282/T3NG4SD7

Objective: 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 more...

**Purpose**Effective teaching; Professional development activity; Student collaboration; Reasoning

**Creator**Phyllis Cipriani (Rutgers University)

**Published**2015-06-22

**Persistent URL**http://dx.doi.org/doi:10.7282/T3445P71

Greer and Harel (1998) state that: "We recommend that awareness of structure, including specifically the recognition of isomorphisms, should be nurtured in children as part of the general development of expertise in constructing representational acts" (p. 5). In this RUanalytic, Brandon, a 10-year old fourth grader, shares his reasoning on how he relates his solution of The Pizza Problem (below) to his solution of The Towers Problem. The Pizza Problem: A local pizza shop has asked us to more...

**Purpose**Student elaboration; Student engagement; Student model building; Reasoning; Representation

**Creator**Jeana Largin (Rutgers University)

**Published**2016-08-02

**Persistent URL**http://dx.doi.org/doi:10.7282/T3VH5R01

Building Multiple Models Using Recursive Reasoning

Over the course of the fractions unit, students transitioned from building a single model to justify an answer to building multiple models to support their ideas and represent a single problem. Many students used recursive reasoning, basing the structure of one model on a previous model they had built. Some students progressed further in their reasoning, using the recursive methods to predict other models, as well as form generalization which would describe all the models of a specific type. In more...

**Purpose**Student model building; Reasoning

**Creators**Baila Salb (Rutgers University); Carolyn Maher (Rutgers University)

**Published**2015-06-16

**Persistent URL**http://dx.doi.org/doi:10.7282/T3S46TRP

Building towers selecting from two colors--engaging two students in collaborative problem solving

Mediator/researcher Carolyn Maher of Rutgers University models effectively engaging two students to solve the problem of building towers four high selecting from cubes of two colors, moving from one student to two students to three students and then preparing for demonstration to the class. Carolyn uses questions, eye contact and participation in the problem solving to engage both students in convincing her and each other that "times two" is the answer to the more...

**Purpose**Effective teaching

**Creators**Grace J. Agnew (Rutgers University); Carolyn A. Maher (Rutgers University)

**Published**2011-09-21

Collaborative Learning Groups Construct Viable Arguments and Critique the Reasoning of Others

This Analytic demonstrates enactments of Common Core Mathematical Practice 3: To construct viable arguments and critique the reasoning of others. There has been much research and documentation which supports the benefits of collaborative group work in the classroom versus the traditional teacher led and fed instruction that we have seen in the past. This teaching philosophy has an even stronger impact with the implementation of the new Common Core Mathematical Practices, particularly more...

**Purpose**Effective teaching; Professional development activity

**Creator**Karen Kelly (Rutgers University)

**Published**2015-06-18

**Persistent URL**http://dx.doi.org/doi:10.7282/T3K9399C

Comparing 1/2 and 1/3: Confusion about the Unit

The data for this RUanalytic come from Sessions 5 (September 29th) and 6 (October 1st) of the fourth grade fraction intervention and continues the students’ investigation about which fraction is larger, 1/2 or 1/3 (Steenken, 2001; Yankelewitz, 2009). The particular focus is on the students’ reasoning as they propose, defend, and argue about precisely how much bigger 1/2 is than 1/3. This RUanalytic is the second in a series of three related RUanalytics. The first RUanalytic in the series more...

**Purpose**Student collaboration; Student engagement; Student model building; Reasoning; Representation

**Creators**Cheryl Van Ness (Rutgers University); Alice S. Alston (Rutgers University)

**Published**2015-09-17

**Persistent URL**http://dx.doi.org/doi:10.7282/T3ZW1NS9

Comparing Models and Justifying the Choice of Unit

This RUanalytic is the third in a series of three RUanalytics that show students’ investigation of how to use the rods to compare 1/2 and 1/3. In the first RUanalytic, “Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions,” after establishing the candy bar metaphor, students examine the different rods, attempting to establish that one of them can serve as an appropriate unit, referred to as “1” or at times “the whole” or the “candy bar”, so that it is more...

**Purpose**Student collaboration; Student engagement; Student model building; Reasoning; Representation

**Creators**Cheryl Van Ness (Rutgers University); Alice S. Alston (Rutgers University)

**Published**2015-09-22

**Persistent URL**http://dx.doi.org/doi:10.7282/T3XW4MQT

David, Erik, and Meredith Use Reasoning to Resolve Conjectures

Meredith rebuilds the previous day’s models for comparing 2/3 and 3/4. The longer model is 24 cm long; a train of a blue rod, a black rod, and a brown rod, representing 1. In this model, a dark green rod represents 1/4, a brown rod represents 1/3, a red rod represents 1/12, and a white rod represents 1/24. Meredith shows that the difference between three greens and two browns is one red (1/12) or two whites (2/24). The other model is 12 cm long; it consists of an orange rod plus a red more...

**Purpose**Student model building; Reasoning; Representation

**Creator**Elizabeth Uptegrove (Video Mosaic Collaborative)

**Published**2016-01-24

**Persistent URL**http://dx.doi.org/doi:10.7282/T3T43W4M

Developing Isomorphic Relationships in High School Mathematics

The fundamental concept of isomorphisms is often not formally introduced to students until more advanced, college-level mathematics courses such as calculus or real number analysis. However, this delayed presentation should not lead mathematics educators to assume that students are unable to build an understanding of the idea as they construct these isomorphic relationships earlier in their mathematical explorations. This analytic presents four eleventh-grade students - Amy Lynn, Robert, more...

**Purpose**Student collaboration; Student representation; Student reasoning; Student engagement

**Creator**Luis Leyva (Rutgers University)

**Published**2012-05-15

Developing Mathematical Precision in the Primary Grades

Learning to solve mathematical word problems is a complex endeavor, particularly in the primary grades when young students are simultaneously developing language proficiency to make sense of what the problems are asking to be solved. Word problems present the cognitive challenge of figuring out who is doing what, which numbers refer to which actions, actors or objects, and how to use the known number(s) to perform which mathematical operation to solve for the unknown. Sometimes manipulative more...

**Purpose**Student engagement; Reasoning; Representation

**Creators**Esther Winter (Rutgers University); Marjory Palius (Rutgers University)

**Published**2015-07-10

**Persistent URL**http://dx.doi.org/doi:10.7282/T3Q24213

Discovering Probability with Dice Games and the Evolution of a Convincing Argument

Students can discover through collaboration the principles of theoretical and experimental probability. This analytic focuses on a group of sixth-grade students as they take a simple dice game and open a world of questions that they themselves end up finding the answers to. The game seems simple: Roll two dice. If the sum of the two is 2, 3, 4, 10, 11 or 12, Player A gets 1 point. If the sum is 5, 6, 7, 8 or 9, Player B gets 1 point. Continue rolling the dice. The first player to get 10 points more...

**Purpose**Effective teaching; Homework activity; Lesson activity; Student collaboration; Student elaboration; Student engagement; Student model building; Reasoning; Representation

**Creator**Anthony Logothetis (Rutgers University)

**Published**2015-06-28

**Persistent URL**http://dx.doi.org/doi:10.7282/T3PN97F0

Eight-Grade Students explore Surface Area and Volume Problems: The Role of Representations

The purpose of this RUanalytic is to examine how students use different representations to illustrate their developing understanding of the concepts of surface area and volume. The events chosen for the RUanalytic focus on the following categories of external representations: manipulative/physical, written symbols, experiential, spoken language and pictures and diagrams (Lesh, Post and Behr 1987). The video data for this RUanalytic come from a longitudinal study, from first grade to twelfth, more...

**Purpose**Lesson activity; Professional development activity; Student collaboration; Student engagement; Student model building; Representation

**Creator**Kenneth Horwitz (Rutgers University)

**Published**2015-11-17

**Persistent URL**http://dx.doi.org/doi:10.7282/T3V40X46

Research has shown that, through argumentation, even young children can build proof-like forms of argument (Ball, 1993; Ball et al., 2002; Maher, 2009; Maher & Martino, 1996a, 1996b; Styliandes, 2006; 2007; Yankelewitz, 2009; Yankelewitz, Mueller, & Maher, 2010). The purpose of this VMCAnalytic is to illustrate events of a student involved in argumentation. The view of argumentation presented in this VMCAnalytic is consistent with Practice 3 of the Common Core Standards for Mathematical more...

**Purpose**Effective teaching; Homework activity; Lesson activity; Professional development activity; Student elaboration; Student engagement; Student model building; Reasoning; Representation

**Creator**Cheryl Van Ness (Rutgers University)

**Published**2015-06-18

**Persistent URL**http://dx.doi.org/doi:10.7282/T3FN180C

Research has shown that, through argumentation, even young children can build proof-like forms of argument (Ball, 1993; Ball et al., 2002; Maher & Martino, 1996a, 1996b; Styliandes, 2006; 2007; Maher, 2009; Yankelewitz, 2009; Yankelewitz, Mueller, & Maher, 2010). The purpose of this VMCAnalytic is to illustrate events of a student involved in argumentation. The view of argumentation presented in this VMCAnalytic is consistent with Practice 3 of the Common Core Standards for Mathematical more...

**Purpose**Effective teaching; Homework activity; Lesson activity; Professional development activity; Student collaboration; Student elaboration; Student engagement; Student model building; Reasoning; Representation

**Creator**Cheryl Van Ness (Rutgers University)

**Published**2015-06-22

**Persistent URL**http://dx.doi.org/doi:10.7282/T3QZ2CRF

Eleventh Graders Collaborating and Recognizing Isomorphisms in their Problem Solving

Initial Problem: A local pizza shop has asked us to help design a form to keep track of certain pizza choices. They offer a plain pizza that is cheese and tomato sauce. A customer can then select from the following toppings: pepper, sausage, mushrooms, and pepperoni. How many different choices for pizza does a customer have? List all the choices. Find a way to convince each other that you have accounted for all possible choices. Suppose a fifth topping, anchovies, were available. How many more...

**Purpose**Student collaboration; Student engagement; Student model building; Reasoning

**Creator**Jennifer Huereca (Rutgers University)

**Published**2015-07-31

**Persistent URL**http://dx.doi.org/doi:10.7282/T3HH6MXR

Establishing norms and creating a mathematical community

This analytic describes the representations, reasoning, and justification used by students to express their understanding of fraction ideas while building solutions to a set of tasks. These tasks were introduced during the first session of a research intervention that was conducted over twenty-five sessions to study how students build fraction ideas prior to their introduction through the school’s curriculum. Of these sessions, seventeen were focused primarily on building basic fraction more...

**Creator**Miriam Gerstein (Rutgers University)

**Published**2015-06-22

**Persistent URL**http://dx.doi.org/doi:10.7282/T30C4XH9

Extending Fraction Placements from Segments to Number Line: Obstacles and Their Resolutions

This RUanalytic focuses on a set of activities taken from a yearlong study designed to investigate how 4th grade students build fraction ideas (Schmeelk 2010). The activities related to comparing and ordering fractions have been partitioned into three analytics. The analytics preceding this one in the classroom investigations are entitled: (1) Imagining the Density of Fractions; and (2) Using Meredith’s Models to Reason about Comparing and Ordering Unit Fractions. These analytics focus on more...

**Purpose**Effective teaching; Professional development activity; Student collaboration; Student engagement; Reasoning; Representation

**Creator**Kenneth Horwitz (Rutgers University)

**Published**2015-08-31

**Persistent URL**http://dx.doi.org/doi:10.7282/T39Z96SR

Extending the Doubling Conjecture

In the previous day’s session, students conjectured that, once they had a model for comparing the fractions 2/3 and 3/4, it was possible to build another model twice the size of that original model. In this session, Alan uses the doubling conjecture to build three models to compare 1/2 and 2/5. The first model uses one orange rod to represent 1; the second uses two orange rods to represent 1; the third model uses four orange rods to represent 1. Alan conjectures that adding four orange rods more...

**Purpose**Student model building; Reasoning; Representation

**Creator**Elizabeth Uptegrove (Video Mosaic Collaborative)

**Published**2016-07-11

**Persistent URL**http://dx.doi.org/doi:10.7282/T3RR21CM

Fourth Grade Students’ Generic Reasoning while Exploring Fraction Comparison Ideas

The purpose of this RUanalytic is to illustrate instances of generic reasoning that occurred as fourth graders, participating in a longitudinal study on student reasoning, explored fraction comparison concepts. According to Yankelewitz (2009), generic reasoning occurs when a student reasons about the properties of a paradigmatic example that are representative of and can be applied to a larger class of objects in which it is contained and lends insight into a more general truth about that more...

**Purpose**Effective teaching; Lesson activity; Professional development activity; Student model building; Reasoning; Representation

**Creator**Cheryl Van Ness (Rutgers University)

**Published**2016-02-04

**Persistent URL**http://dx.doi.org/doi:10.7282/T3417044

Fourth Graders Analyses of Equivalence: 1/5 or 2/10?

Maher and Martino (1996) contend that when students are given sufficient time to work on a problem and are given the opportunity to discuss their solutions with their peers, they often express differences of opinion. These conflicts are not always resolved immediately; sometimes they are pushed off to a later class session, possibly being deferred over an extended period of time. This allows students to think about the problem and upon revisiting the identical or similar problem at a later more...

**Purpose**Effective teaching; Student model building; Reasoning; Representation

**Creator**Miriam Gerstein (Rutgers University)

**Published**2015-06-16

**Persistent URL**http://dx.doi.org/doi:10.7282/T3WW7KFN

Fourth graders build towards proportional reasoning

This analytic highlights students’ reasoning as they worked on five tasks which built towards the development of proportional reasoning. The events in this analytic are selected from a study of fourth grade students from Colts Neck, a suburban New Jersey district (Maher, Martino, & Davis, 1994). The three sessions depicted by this analytic were facilitated by Researchers Carolyn Maher and Amy Martino and took place consecutively in September of 1993 as part of a research intervention aimed to more...

**Purpose**Reasoning

**Creator**Esther Winter (Rutgers University)

**Published**2015-06-16

**Persistent URL**http://dx.doi.org/doi:10.7282/T35D8TMT

Fourth Graders Design a New Rod

This analytic describes the representations, reasoning, and justification used by students to express their understanding of fraction ideas while building solutions to a set of tasks. These tasks were introduced during the second session of a research intervention that was conducted over twenty-five sessions to study how students build fraction ideas prior to their introduction through the school’s curriculum. Of these sessions, seventeen were focused primarily on building basic fraction more...

**Creator**Miriam Gerstein (Rutgers University)

**Published**2015-06-18

**Persistent URL**http://dx.doi.org/doi:10.7282/T33B61X8

Fourth Graders Explore the Magnitude of Fractions and Make Comparisons

This analytic reveals how task design and the revisiting of tasks that involve concrete models allow children to build deep understandings of fundamental concepts that often receive superficial treatment in standard curricula. In this case, children internalize the concepts of “common denominators” without using the term. At the end of an eleven-session unit on fraction comparison, the children evidence their intuitive grasp of the ways in which fractions can be compared. In the first part more...

**Purpose**Lesson activity; Professional development activity; Student elaboration

**Creator**Miriam Gerstein (Rutgers University)

**Published**2015-06-18

**Persistent URL**http://dx.doi.org/doi:10.7282/T3736SQH

Fourth Graders Reason by Cases as They Explore Fraction Ideas

This analytic explores the various forms of reasoning employed by students as they share their solutions for the following task: "I’m going to call the orange and light green together one... Can you find a rod that has the number name one half?" This task, which does not have a solution, prompted the students to use multiple forms of reasoning to justify their claims; some of which are highlighted in this analytic (Yankelewitz, Mueller, & Maher, 2010). In particular, several students used more...

**Purpose**Reasoning

**Creator**Esther Winter (Rutgers University)

**Published**2015-06-18

**Persistent URL**http://dx.doi.org/doi:10.7282/T3Q2420N

Fourth graders’ argumentation about the density of fractions between 0 and 1

Research has shown that, through argumentation, even young children can build proof-like forms of argument (Maher & Martino, 1996a, 1996b; Styliandes, 2006; 2007; Ball, 1993; Ball et al., 2002; Maher, 2009; Maher & Martino, 1996; Yankelewitz, 2009; Yankelewitz, Mueller, & Maher, 2010). The purpose of this VMCAnalytic is to illustrate events of students involved in argumentation during which students are interacting with each other and a researcher. The view of argumentation presented in this more...

**Purpose**Effective teaching; Homework activity; Lesson activity; Professional development activity; Reasoning; Representation

**Creator**Cheryl Van Ness (Rutgers University)

**Published**2015-04-14

**Persistent URL**http://dx.doi.org/doi:10.7282/T39K4CZC

The goal of this analytic is to show the remarkable power of student collaboration and teacher questioning by following the progression of one student throughout an open ended problem situation presented by researchers. The structure of the environment allows for multiple student interactions. These interactions allow the student, Matt, to move from a passive listener to a confident, active participant who is able to articulate his understanding to others. This progression would not have more...

**Purpose**Student collaboration

**Creator**Michael Cimorelli (Rutgers University)

**Published**2015-07-24

**Persistent URL**http://dx.doi.org/doi:10.7282/T39C707G

From Taxicabs to Towers: An Analysis of Problem-Solving Strategies

Problem solving is a cornerstone to the mathematical learning experience that makes possible students' application of creative strategies and logical reasoning while working to complete particular tasks. This analytic is a detailed overview of various problem-solving strategies employed by four twelfth-grade students - Brian, Jeff, Michael, and Romina - when solving the Taxicab Problem, a challenging mathematical task involving combinatorial reasoning and ideas from coordinate geometry. Each more...

**Purpose**Student collaboration; Student elaboration; Student engagement; Student model building; Student reasoning; Student representation

**Creator**Luis Leyva (Rutgers University)

**Published**2012-05-15

In his analysis of student-to-student discursive interactions solving a mathematical problem referred to as the Taxicab Problem, Powell (2006) posited the notion of a socially emergent cognition. Powell described this type of cognition as "a process through which ideas and ways of reasoning materialize from the discursive interactions of interlocutors that go beyond those already internalized by any individual interlocutor." With that in mind, the purpose of this analytic is twofold: 1) to more...

**Purpose**Student collaboration

**Creator**Emmanuel Arguelles (Rutgers University)

**Published**2014-01-07

**Persistent URL**http://dx.doi.org/doi:10.7282/T32N5096

Imagining the Density of Fractions

This VMC Analytic focuses on a set of activities taken from a yearlong study designed to investigate how 4th grade students build fraction ideas (Schmeelk, 2010). The activities on this day of November 1, 1993 examine the placement of unit fractions on a number line segment between 0 and 1. This Analytic focuses on the November 1, 1993 session with fourth graders in a Colts Neck, New Jersey classroom as students are asked to extend their notions of fraction with rods as operator to the idea of more...

**Purpose**Professional development activity

**Creators**Suzanna Schmeelk (Video Mosaic Collaborative); Kenneth Horwitz (Rutgers University)

**Published**2015-07-10

**Persistent URL**http://dx.doi.org/doi:10.7282/T3FJ2JKN

James Finds the Difference Between 1/4 and 1/9

The class is investigating the difference between 1/4 and 1/9. In initial discussions, some students agree with Meredith, who says that the difference is 1/5 because 9 - 4 = 5. Researcher Maher indicates to the class that if they apply Meredith’s rule to 1/2 and 1/4, they would get an answer of 1/2, challenging their earlier reasoning that produced an answer of 1/4. James offers a model consisting of 3 orange rods and a dark green rod (36 white rods) that can be partitioned into ninths and more...

**Purpose**Effective teaching; Student engagement; Student model building; Reasoning; Representation

**Creator**Elizabeth Uptegrove (Video Mosaic Collaborative)

**Published**2016-01-24

**Persistent URL**http://dx.doi.org/doi:10.7282/T3PC34FQ

Learning through Collaborative Argumentation

From Piaget's perspective, interaction with peers is an important aspect of cognitive development and knowledge construction (Palincsar, 1998). From a constructivist perspective interaction is vital because it is through interactions with others that a learner, has his/her existing mental representations challenged. Interactions with peers provide opportunities for students to reconsider their existing thoughts and beliefs. In some cases, this reconsideration leads to new knowledge construction more...

**Purpose**Student collaboration

**Creator**Ashley Davis (Video Mosaic Collaborative)

**Published**2014-03-05

**Persistent URL**http://dx.doi.org/doi:10.7282/T3RX9962

My teacher's going to kill me: students collaborate to solve the pizza problem

Students bring forms of reasoning--guessing and checking, referencing a previous problem and recognizing a pattern, as well as previous knowledge (factorials, combination) and the calculator to the problem of how many possible combinations result from four or five pizza toppings. Students Stephanie and Michelle reinforce and support each other during the collaboration. "My teacher's going to kill me 'cause she knows I can do this." "She's never going to see more...

**Purpose**Student collaboration

**Creator**Grace J. Agnew (Rutgers University)

**Published**2011-09-16

This is an after-school session in which Jeff, Michael, Romina, and Brian volunteer to participate. This particular session is called the Night Session because it took place on an evening late in their junior year. These students have been working on many combinatorics problems since second grade, some of which are isomorphic to each other. In this session, they were discussing the meaning of combinatorial notation and discussing the addition rule of Pascal’s Identity in terms of that more...

**Purpose**Student representation; Student reasoning; Effective teaching

**Creator**Muteb Alqahtani (Rutgers University)

**Published**2011-10-26

Pascal’s Triangle and Pascal’s Identity: Contextualizing and Decontextualizing

According to the Common Core State Standards, the ability to contextualize and the ability to decontextualize are important mathematical skills for students to develop. Learners should have "the ability to decontextualize--to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have 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 more...

**Purpose**Effective teaching; Professional development activity

**Creator**Elizabeth Uptegrove (Video Mosaic Collaborative)

**Published**2015-06-16

**Persistent URL**http://dx.doi.org/doi:10.7282/T31N82WG

Questioning Elicits Reasoning and Representations for Combinatorics Tasks

This analytic contains 4 events. The purpose of this analytic is to assist teachers in acquiring pedagogical skill in supporting the development of students’ mathematical reasoning. More specifically, the analytic highlights researchers’ questioning to promote students’ engagement with new ideas, building models, and listening to input from other students. The analytic is also intended to answer the question: “In what ways do students respond to prompting and questioning?” The more...

**Purpose**Effective teaching; Professional development activity; Student model building

**Creator**Deidre C. Richardson (Video Mosaic Collaborative)

**Published**2015-06-22

**Persistent URL**http://dx.doi.org/doi:10.7282/T3M61N2W

Robert B. Davis Engages Students in Finding the Sum and Product Rule for Quadratic Equations

In a traditional math curriculum, the solving of quadratic equations is a topic for a first year algebra course, usually in the eighth or ninth grade. Solving quadratic equations typically follows other topics such as solving linear equations and factoring polynomials. Traditionally, the concept of solving quadratic equations is introduced through combining these two topics to find the solution(s). The purpose of this analytic is to illustrate a different approach to solving more...

**Purpose**Effective teaching; Lesson activity; Student collaboration; Student engagement; Reasoning

**Creator**Brad Poprik (Rutgers University)

**Published**2015-06-16

**Persistent URL**http://dx.doi.org/doi:10.7282/T3HQ41QB

Small Group and Whole Class Collaboration about Fractional Relationships Using Rod Models

This analytic is designed to show various forms of collaboration, which includes co-construction, integration, and modification, through small group and whole class discussions. Further, this analytic delves into how these three aspects of collaboration contribute to student’s understandings of the problems at hand. According to Elizabeth Uptegrove in her article “Shared communication in building mathematical ideas: A longitudinal study” (2015), “students build knowledge collaboratively more...

**Purpose**Lesson activity; Professional development activity; Student collaboration; Student elaboration; Student engagement; Student model building; Reasoning; Representation

**Creator**Megan Grocholske (Rutgers University)

**Published**2016-08-02

**Persistent URL**http://dx.doi.org/doi:10.7282/T3QR5094

Stephanie Uses Geometry To Explain Binomial Expansion

Researcher Carolyn Maher conducted interviews with math student Stephanie during her eighth grade year, exploring ideas about binomial expansion. Stephanie was also one of a group of children who participated in a longitudinal research study, doing math activities facilitated by the research team since first grade. Clips from these earlier years as well as the entire set of interviews are available for viewing on the Video Mosaic Collaborative (videomosaic.org). Researcher Maher met with more...

**Purpose**Lesson activity; Professional development activity

**Creators**Diane Kovac (Rutgers University); Alice S Alston (Rutgers University)

**Published**2013-04-19

Stephanie’s Journey with the Towers (Grades 3-8): A Metaphor for Making Connections

During this analytic, I hope to show that "learning is primarily metaphoric - we build representations for new ideas by taking representations of familiar ideas and modifying them as necessary, and the ideas we start with often come from the earliest years of our lives" (Davis, 1984, p. 313). Davis’ idea of teaching was centered on the idea that students should be provided with opportunities to build assimilation paradigms. According to him, assimilation paradigms were created when students more...

**Purpose**Effective teaching; Student elaboration; Reasoning; Representation

**Creator**Solaris Ortiz (Rutgers University)

**Published**2015-06-18

**Persistent URL**http://dx.doi.org/doi:10.7282/T3BV7JDG

Student Perseverance in Discovering Patterns: Guess My Rule with Robert B. Davis

The Common Core State Standards for Mathematics (2010) has identified eight varieties of expertise as Standards for Practice that students need to develop during their discovery and learning of mathematics in order to become mathematically proficient. These standards include: •Make sense of problems and persevere in solving them. •Reason abstractly and quantitatively •Construct viable arguments and critique the reasoning of others •Model with mathematics •Use appropriate tools more...

**Purpose**Student model building

**Creator**Katherine Makovec (Rutgers University)

**Published**2015-06-16

**Persistent URL**http://dx.doi.org/doi:10.7282/T3862J75

The class is investigating the difference between 1/4 and 1/9. In initial discussions, some students agree with Meredith, who says that the difference is 1/5 because 9 – 4 = 5. Researcher Maher indicates to the class that if they apply Meredith’s rule to 1/2 and 1/4, they would get an answer of 1/2, challenging their earlier reasoning that produced an answer of 1/4. Some students build one model to show ninths and a second model (of a different length) to show fifths. For example, Alan more...

**Purpose**Effective teaching; Student model building; Reasoning; English translation

**Creator**Elizabeth Uptegrove (Video Mosaic Collaborative)

**Published**2016-07-11

**Persistent URL**http://dx.doi.org/doi:10.7282/T3N018PN

Task Design Prompts Fourth Grade Students to Use Multiple Forms of Reasoning

This analytic demonstrates various forms of reasoning used by fourth grade students as they try to build a model with Cuisenaire rods to solve a fractions task for which there is no solution in the set. The events in this analytic are selected from a study of fourth grade students from Colts Neck, a suburban New Jersey district (Maher, Martino, & Davis, 1994). The session, facilitated by Researcher Carolyn Maher, took place in September of 1993, during the second session of a research more...

**Purpose**Reasoning

**Creator**Esther Winter (Rutgers University)

**Published**2015-06-18

**Persistent URL**http://dx.doi.org/doi:10.7282/T3ZK5JF0

Teachers Promoting Mathematical Discourse: Fraction Explorations by Fourth Graders

Developing and promoting discourse in mathematics has become a larger focal point in the math classroom in recent years. Although research pointing to the value of students’ constructing their own arguments and collaborating with others has been around for decades, it has only recently become part of the mainstream teaching standards. The Common Core State Standards now have included the constructing of arguments and critiquing of others’ arguments as the third of their eight Standards more...

**Purpose**Effective teaching; Lesson activity; Student collaboration; Student engagement; Reasoning

**Creator**Brad Poprik (Rutgers University)

**Published**2015-06-17

**Persistent URL**http://dx.doi.org/doi:10.7282/T34F1SHF

The Candy Bar Problem: Researcher Moves to Encourage 4th Grades Reasoning About Fractions

The students use Cuisenaire Rods to develop relational understanding (Skemp, 1978) of finding the difference of two fractions and explore, informally, comparison problems with fractions, developing an understanding of least common multiple. Students work on the following problems, “If we agree that the 8 students who received 1 and 1/4 pieces of candy got more than the 9 students who received 1 and 1/9 pieces, how much more did each of the students in the group of 8 receive? How much more...

**Creator**Stephanie Kulcsar (Rutgers University)

**Published**2016-08-02

**Persistent URL**http://dx.doi.org/doi:10.7282/T3M047M5

The development of upper and lower bound arguments while comparing fractions

This analytic demonstrates the development and use of upper and lower bound arguments by fourth grade students as they worked on fraction comparison tasks using Cuisenaire rods to build models. The events in this analytic are selected from a study of fourth grade students from Colts Neck, a suburban New Jersey district (Maher, Martino, & Davis, 1994). The four sessions depicted by this analytic were facilitated by Researchers Carolyn Maher and Amy Martino and took place consecutively in October more...

**Purpose**Reasoning

**Creator**Esther Winter (Rutgers University)

**Published**2015-05-06

**Persistent URL**http://dx.doi.org/doi:10.7282/T3ZS2Z8N

The Museum Problem: Seventh grader Ariel investigates linear functions

This RUAnalytic concentrates on a 7th grade boy, Ariel, in an urban middle school. Ariel, a volunteer in an after-school Informal Math Program, is introduced to the idea of linear functions. During this after school enrichment session, Ariel is presented with the Museum Problem. The Museum Problem states: A museum gift shop is having a craft sale. The entrance fee is $2. Once inside, there is a special discount tale where each craft piece costs $3. What would be the total amount that you more...

**Purpose**Student engagement; Reasoning

**Creator**Susan Hinds (Rutgers University)

**Published**2016-08-02

**Persistent URL**http://dx.doi.org/doi:10.7282/T3G73GWF

Beginning Ordering and Comparing (Clip1) -- In this clip the researcher begins by posing the question of, "which fraction is bigger?" She puts 1/2 , 1/3,1/4, and 1/5 on the board and asks the students to be able to prove which is bigger. Notice how David, without the rods, draws rods to illustrate which is bigger. Then in an attempt to move the students from a rod model to a number line, the researcher puts up a number line segment from 0 to 1 and then asks the students to place those same more...

**Purpose**Lesson activity; Student engagement

**Creator**Ken Horwitz (Rutgers University)

**Published**2011-10-26

Tracing Ariel’s Algebraic Problem Solving: A Case Study of Cognitive and Language Growth

While research has shown that understanding the concept of a function is essential for success in other areas of mathematics (Carlson, 1998; Rasmussen, 2000) students continue to struggle learning the concept (Vinner and Dreyfus, 1989). Research has revealed that young children, who are engaged in problem-solving activities designed to elicit justifications for their solutions, develop an understanding of fundamental algebraic ideas such as function (Maher, Powell & Uptegrove 2010; Kieran, more...

**Purpose**Student elaboration; Student model building; Reasoning; Representation

**Creators**Robert Sigley (Rutgers University); Louise Wilkinson (Rutgers University)

**Published**2015-02-23

**Persistent URL**http://dx.doi.org/doi:10.7282/T3N0186C

According to Davis (1984), children construct ideas by building on prior experiences. When students learn concepts in a meaningful way they construct "powerful representations" which can be used to solve new problems. The development of more advanced concepts is facilitated as students apply their previous knowledge and representations to understand new ideas. In a model which Davis terms an assimilation paradigm, learners assimilate the knowledge they have constructed from previous experiences more...

**Purpose**Representation

**Creator**Esther Winter (Rutgers University)

**Published**2015-06-23

**Persistent URL**http://dx.doi.org/doi:10.7282/T3SQ925Z

Unitizing Area Numerically and Algebraically; The Gap Between Arithmetic and Algebra

In this analytic, Researcher Carolyn Maher employs a very useful and effective strategy to facilitate Stephanie’s learning. The idea of “reasoning down” involves analyzing by cycling back to previous experiences or knowledge (Driscoll 1999). Through questioning, Researcher Maher discovers that Stephanie’s confusion with the meaning of the square of (a+b) is due to her difficulty with the meaning of a^2. A possible reason for this struggle lies in the leap to algebra, a means of more...

**Purpose**Effective teaching; Student model building; Reasoning; Representation

**Creator**Courtney Weitzer (Rutgers University)

**Published**2015-06-18

**Persistent URL**http://dx.doi.org/doi:10.7282/T3TT4SQ8

Using Meredith’s models to reason about comparing and ordering unit fractions

This VMCAnalytic focuses on a set of activities taken from a yearlong study designed to investigate how 4th grade students build fraction ideas (Schmeelk 2010). The activities on this day were partitioned into two analytics: the first focusing on the placement of fractions on a line segment and the second focusing on placement of fractions on a number line. The first task that is shown in this first VMCAnalytic involves comparing and ordering unit fractions and placing them on a number line more...

**Purpose**Effective teaching; Lesson activity; Professional development activity; Student collaboration; Student engagement; Student model building; Reasoning; Representation

**Creators**Kenneth Horwitz (Rutgers University); Suzanna Schmeelk (Rutgers University)

**Published**2015-05-06

**Persistent URL**http://dx.doi.org/doi:10.7282/T33J3FQG

Questions are an important part of learning and teaching; a form of communication to gain insight into students’ conceptual understanding (Roth, 1996). According to Wolff-Michael Roth (1996), “teacher questions are frequent, pervasive and [an] universal phenomena.” But although a classroom norm, its use to promote conceptual understanding amongst students is scarce. The misconception that frequent questioning brings about conceptual understanding is prevalent amongst both pre-service and more...

**Purpose**Effective teaching; Lesson activity; Student elaboration

**Creator**Simone Grey (Rutgers University)

**Published**2015-07-24

**Persistent URL**http://dx.doi.org/doi:10.7282/T34Q7WS9

Using video to teach about transfer: The case of Brandon

For many years, I used Brandon to teach about analogical transfer in a course in Educational Psychology and colleague Sharon Derry at University of Wisconsin used this in a course in the learning sciences as part of a hybrid problem-based learning course organized around video. After students worked on the tower and pizza problems themselves, they watched the Brandon video and were asked to solve the following problem as part of the STELLAR project: Knowing What Brandon Knows As a teacher, more...

**Purpose**Professional development activity; Student reasoning

**Creator**Cindy E.Hmelo-Silver (Rutgers University)

**Published**2011-12-12

Which is Larger, 1/2 or 1/3? An Introduction to Comparing Unit Fractions

The data for this RUanalytic were gathered on September 24th, 1993, during Session 3 of the fourth grade fraction intervention and the focus is on the students as they first begin specifically to investigate ways to compare fractions based on models constructed with the Cuisenaire rods (Steencken, 2001; Yankelewitz, 2009). This RUanalytic captures the beginning of several “visits” that the children have with the comparison task: Which is greater, 1/2 or 1/3, and by how much? This RUanalytic more...

**Purpose**Student collaboration; Student engagement; Student model building; Reasoning; Representation

**Creators**Cheryl Van Ness (Rutgers University); Alice S. Alston (Rutgers University)

**Published**2015-09-17

**Persistent URL**http://dx.doi.org/doi:10.7282/T3V40X3R