It is important to realize that even young students are capable of identifying the structure of things around them; for example, students can recognize patterns in a story, sequences of events from their daily lives, and identify differences among basic geometric shapes. The vocabulary for these observations may not be very advanced but the skillset still exists; therefore it should be expected that all students with the proper instruction should be able to achieve success with Practice Standard 7 as they look for and make use of structure in mathematics. Two of the articles presented with this collection address how to make the connections between familiar stories and their patterns.
How will these resources help?
These video examples and articles depict classroom experiences of students engaged in Practice Standard 7. They provide suggestions for how to guide students towards the discovery of structure so that they may use that structure to expand a pattern, read or complete a graph, classify geometric shapes, or build a sense of number, operations, and place value through structural models.
The importance of Visual Learning
In order to identify the structure of a mathematical problem, students often need to engage in some form of visual learning or visualization. In early algebra this may mean that students are making a pattern or displaying data in a table to determine where a pattern exists. Structures are used to build place value and number sense. Students use various models in order to understand numbers and how they can be composed and decomposed. With data analysis and statistics students use the structure of the graph or other data presentation to make sense of the problem and find a solution. It is therefore essential that students learn multiple ways to represent and analyze data, numbers, and shapes in order to determine and understand the structure of any problem.
This web page from the Inside Mathematics initiative aims to guide and support educators in understanding the seventh CCSS Practice Standard. Instances of teachers engaging students to be mathematically proficient with the capacity to "look closely to discern a pattern or structure" are described in Classroom Observations and illustrated in a video excerpt of fourth grade students discussing how to multiply 26 times 4 and in another video clip of fifth grade students discussing the different ways to write a rule. The video clips are excerpts from public lessons: "Number Operations" and "Numerical Patterning" (both cataloged separately).
This webpage from the state of Victoria, Australia, discusses the importance of students understanding the meaning of the equals sign in number sentences. It illustrates common misconceptions and their causes and proposes strategies and activities that help teach and reinforce equality concepts. Downloadable game cards (pdf) and a student worksheet (doc) are available. The page includes a list of references and a link to a page defining stages and indicators in the progression of developing these concepts.
The author of this one-page article discusses the importance of developing a "sense of ten" as a foundation for place value and mental calculations. She suggests teaching strategies using ten-frames to promote this sense, including instructions for simple games. The article includes a list of references and a link to a related article, "Developing Early Number Sense" (cataloged separately).
In this 6-minute video kindergarten teacher Stephanie Latimer describes and models techniques for developing children's number sense and visual recognition of number combinations. After quickly displaying groups of objects on a ten frame, she asks her students to describe the ways that they see the objects grouped. The resource includes reflection questions for viewers and a transcript of the video (doc).
This session, the third one in the Annenberg series "Patterns, Functions, and Algebra", enables the teacher to investigate algorithms and functions. Topics covered include the importance of doing and undoing in mathematics, determining when a process can or cannot be undone, using function machines to picture and undo algorithms, and recognizing that functions produce unique outputs. 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 video- and web-based course K-8 teachers examine the three main categories in the Number and Operations strand of Principles and Standards of School Mathematics (NCTM): understanding numbers, representations, relationships, and number systems; the meanings of operations and relationships among those operations; and reasonable estimation and fluent computation. The course covers the real number system, place value, the behavior of zero and infinity, meanings and models of basic operations, percentages, modeling operations with fractions, and basic number theory topics (factors, multiples, divisibility tests). The course consists of 10 approximately 2.5 hour sessions, each with video programming, problem-solving activities, and interactive activities and demonstrations on the web. Participants can work through the sessions on their own, in a study group, or as part of a facilitated, face-to-face, graduate-level course for credit.
This brief article advocates the use of arrays to model the commutative and distributive properties as well as the inverse relationship between multiplication and division. The author explains how arrays also help children form mental pictures that support their memory and reasoning.
The author of this article uses four NRICH investigations to illustrate how a teacher might extend the results of students' work as a springboard for developing deeper mathematical understanding. He models good prompting, e.g., "Tell me what you notice about the result" and "I wonder what would happen if ...", and warns against trying to lead students' thinking. [The problems discussed are cataloged separately.]
In this 2-minute video teacher Jen Saul describes and models a technique for helping students develop number sense and mental math skills and learn complements of 10. As prompted by the teachers, students repeatedly form groups of 0-10 from 10 popsicle sticks. Viewer reflection questions are provided.
This 5 minute video shows Heather Zemanek's 3rd grade class learning about bar graphs. The video shows short excerpts of the lesson and commentary by the teacher. Along with the video are downloadable attachments: the lesson plan and transcipt as Word Documents and two graph examples in PDF form.
This 3-page document (pdf) offers numerous strategies that children can use to perform addition, subtraction, and multiplication mentally. These strategies help develop fact fluency, number sense, operation sense, and use of patterns.
This blog page offers more than 25 activities and games that use a Hundred Chart to develop students' skills and concepts in a variety of math topics: basic operations, number sense, patterns, number theory, fractions, decimals, and logic. The suggestions include links to materials and to other websites.
In this brief article the numerous uses of the number line are detailed: counting, measurement, addition, subtraction, decimals, and fractions. The article contains visual representations of the some of the concepts and links to related topics.
The author, Christine Losq, explains why 10-frame tiles offer an effective alternative to base-10 blocks for teaching place value. The article, from TCM, describes how teachers can use the 10-frame model to help children develop understanding of number and place value concepts and skill with computation that is effective not only with special needs students, but with all students.
In this article from the “Links to Literature” department of Teaching Children Mathematics, the authors describe how primary grade children begin to develop the foundations of algebraic thinking. Debra Johanning, William B. Weber Jr. , Christine Heidt , Marian Pearce, and Karen Horner explain how they used “The Polar Express” to construct a cross-disciplinary lesson in which the mathematics builds capacity for numerical pattern recognition using function tables.
In this article from the “Early Childhood Corner,” author Portia Elliott describes how children’s literature can be used to incorporate foundations of algebraic thinking into the curriculum. Specific stories are used to illustrate strategies that teachers can use to support students’ early understanding of algebraic structures.