It is important to realize that in order for students to make connections to mathematical concepts and to retain the skills that they are taught from year to year, they must have some meaningful purpose. Mathematical Practice 4 addresses this need by having students model real world situations. The numerical quantities that students are representing should be related to concrete objects in their lives and interpreted in a concrete, pictorial, or symbolic way.
How will these resources help?
These videos and articles provide examples of mathematical situations where representation is essential to understanding. In these resources students are shown choosing their own solution methods and their own representations. In some cases a variety of representation options are offered to students as guidelines but it is ultimately up the student to show their understanding in a chosen manner. Allowing students to choose their own method of representation is essential to understanding Mathematical Practice 4; by allowing students to model problems in their own pictorial, conceptual, and concrete ways there will be more discussion about these representations, to clarify that the student understands the math concept and has drawn the appropriate conclusion. Two specific articles in this collection address this concern: “Engaging Reluctant Problem Solvers” and “It’s Not Just Notation: Valuing Student’s Representations”.
The importance of Conceptual Learning
In order to gain mastery of any topic, students must first understand all of the components involved. In math, this means the scaffolding progression between concrete models, pictorial models, and symbolic models must be incorporated into topics throughout the K-12 education process. It is unlikely that all three representations will be utilized in every unit, especially with younger students who may rely heavily on manipulatives and pictures as their representations. Practice Standard 4 requires all students to model their understanding in a manner that they are comfortable with by representing the problem and solution. Younger students may choose to draw pictures or use manipulatives to represent the problem and to explain their solution, while older students will likely be writing number sentences, equations, or paragraphs to explain their thinking.
In this blog entry Michelle Flaming describes what the implementation of MP 4 looks like in the classroom. She lists student actions, teacher actions, and open-ended questions that teachers can ask students to elicit the kind of reasoning called for in MP 4.
This article discusses the importance of abstraction and generalization in being able to use models and diagrams effectively, whether children are creating their own models to represent a situation and explain their thinking, or interpreting a provided model. The resource offers examples to illustrate these ideas as well as teaching strategies and activities in the areas of parity, mental addition, equivalent fractions, and multi-digit multiplication.
This brief article discusses the importance of young children creating their own informal graphical representations of their mathematical thinking and problem solving. As distinguished from formal recording of a completed process, these early markings and symbols enable children to develop understanding and make meaning as well as communicate their thinking. The article includes a list of references, including the authors' research on which the article is based.
This qualitative research article examines teachers' choices of instructional representations and their bearing on student learning. Through the context of teaching fractions, the author examines the importance of, and complexities in, teacher choices when representing mathematical concepts in the classroom in order to teach those concepts. Teachers' pedagogical content knowledge is explained as the basis for these choices.
This session, the first one in the Annenberg series "Patterns, Functions, and Algebra" enables the teacher to discover what it means to think algebraically, and to learn to use algebraic thinking skills to make sense of different situations. Topics covered include describing situations through pictures, charts, graphs, and words; interpreting and drawing conclusions from qualitative graphs; and creating graphs to match written descriptions of real-life situations. The session includes sequentially organized problems, video viewing, interactive activities, readings, and homework. The materials may be used on your own, in a study group, or as a facilitated online course for graduate credit, which is offered at a reasonable cost.
In this 24-minute professional development video, students apply their problem solving and measurement skills to estimate the amount of money, in dollar bills, that will fit into a given suitcase. Educators can observe the teamwork, communication, and the problem solving strategies students use and hear the instructor's comments. Support Materials include a Measurement: Million Dollar Giveaway Lesson Plan.
This one-page article describes and illustrates how arrays can be used to represent many number concepts, including building multiplication facts, commutativity, parity (odd/even), and exploring factors, prime numbers, and square numbers.
In this 5-minute video third grade teacher Jean Saul demonstrates how she uses problem solving tasks to create a classroom climate that fosters persistence, independence, responsibility, and risk-taking. Students are asked to find three different methods for solving each problem and to record them on a Choose Three Ways graphic organizer. Through collaboration and presentation of their work to peers, students develop math language and discourse skills. A side bar provides reflection questions. Supporting materials include a transcript of the video (doc), the graphic organizer (doc), and two samples of student work (pdf).
In this series of videos (16 clips, 1-10 min each), math coach Becca Sherman works with a fourth grade class in developing concepts of equal groups, multiplication, division, and the relationships among them. She employs multiple models, including the "Singapore Bar Model," and facilitates student discussion, allowing her to assess students understanding of the language and the math involved. Included are teacher commentary, video transcripts, student work samples, and a video of debriefing among colleague observers.
These 4 activities, part of the Mathematics Developmental Continuum of the State of Victoria, Australia, are intended to introduce learners to the problem solving skills of clarifying questions, making assumptions and choosing a solving strategy. Students are challenged to investigate problems in household math, open-ended planning projects, and Fermi questions. Supporting documents and progress indicators are included.
This professional development video clip shows students engaged in the Common Core Practice Standard #4—Model with mathematics. In this video clip a second grade teacher uses a literature book,"Caps for Sale" to create eight story-based centers. Additional resources include a video transcript, teaching tips, and a link to a professional development reflection activity based upon the video.
This article provides information from research on the benefits and cautions associated with using manipulatives to teach children. The authors urge that much care be used in assessing whether the learner regards the activity with concrete objects in the same context as was intended by the instructor. The article looks at the comparative advantages of computer manipulatives and gives advice on the selection and use of both physical and computer manipulatives. An extensive reference list is included.
In this 13-minute video Teacher Suney Park's students apply their knowledge of area and perimeter of rectangles in the context of a dinner table. The resource includes a transcript of the video, extension activities, and reflection questions for teachers.
In this 5 minute video, Dedra Wright demonstrates how to teach students about mean as fair share. She uses unifix cubes to model how cooperative learning groups can equally share their cubes. The video uses two data sets to model this strategy and make the connection to the algorithm. A companion video, Mean as the Balance Point, is cataloged separately as a related resource.
This page of videos is designed to showcase classrooms in which the NCTM Process Standards are evident. Scroll to video #45, and select the "VoD" box to view the 20-minute Cranberry Estimation video. Second-graders in Massachusetts estimate the number of scoops of cranberries that will fit in a jar. They report, graph, and discuss group estimates with the class as the concepts of range, mode, and median emerge. Observe as the teacher leads the class in developing strategies, learning about new concepts, and sharing and organizing results.
This article describes ways children represent their work with several tasks, a first grade activity involving making tens and a fourth grade task using fraction bars. The authors demonstrate the importance of listening to the children describe their thinking rather than imposing the teacher’s own interpretation which may or may not be accurate.
This article describes the success that a math specialist has using a pattern-block configuration to help a small group of fifth graders understand that fractional parts of a whole unit must be equal in size. The authors also draw attention to the importance of the specialist’s deep understanding of rational numbers as she identifies and addresses the students’ misconceptions. Some details about the Virginia program for training math specialists are included.
In this article the authors advocate the use of “tasks without words” to engage students who are usually reluctant to attempt solving problems. Several examples of these tasks are described, and examples of student work are included.
This investigation from NCTM is a three-lesson set of activities designed to help students understand equivalent fractions. The article includes the goals of the activity, the prerequisite skills, a list of materials, and a description of each lesson.