Mathematics Benchmark 10.B.ECa

IELDS benchmark performance descriptors give a general sense of what a child might be able to do related to this benchmark if

• her experience is at the beginning level (exploring),
• she has some experience and has begun to construct her understanding (developing), or
• her understanding is more solid (building).

The performance descriptors for IELDS Benchmark 10.B.ECa are:

EXPLORING	DEVELOPING	BUILDING
Organize materials with teacher support to prepare for graphing (e.g., sort leaves by color, sort fruit by type).	Participate in creating a data display using concrete objects or pictures with teacher support (e.g., organize children’s favorite fruit in rows to demonstrate whether more children prefer apples or oranges).	Compare numerical information derived from graphs to find answers to questions with teacher support as needed (e.g., use information depicted on a chart or graph to describe which classroom games are most popular).

When we say that we want children to be able to “Organize, represent, and analyze information using concrete objects, pictures, and graphs” with teacher support, what do we mean?

Let’s Break It Down

What do these terms mean?

• Information
  In this benchmark, the word information means “what children have found out about something.” Information is “facts” or data that can help children answer questions about an object or event—questions that begin with words such as what, when, where, who, how, and why. Information can include location, size, shape, texture, quantity, time, and other important facts and details about the object or event.
• Organize
  Preschoolers organize information when they collect, label, sort, or sequence items that meet certain characteristics.
• Represent
  When children “represent” information, they show in an intentional way what they have observed or found out about something. They might display their knowledge using drawings, diagrams, words, photographs, audio recordings, or other media.
• Analyze
  Preschoolers analyze information when they explain or discuss what they have noticed, focusing on similarities and differences among objects or ideas. Analysis can involve:

  
  

  • comparing quantities, sizes, weights, shapes, colors, etc. (“I counted five people wearing socks today. Five people have no socks on. That’s the same number.” “I can lift the bucket of snow. I can’t lift the bucket of water. It’s too heavy.”)
  • comparing ordinal position or location (“Jamar is first in line. Aisha is next.” “The X is on top, and the Y is under it.”
  • telling about a sequence of events (“First the turtle looked at the beetle. Then he stuck his neck out. He grabbed with his mouth. Then he chomped it.”)
  • comparing predictions to observations (“I predicted that today would be rainy, but it’s not. It’s snowing.”)

The 10.B.ECa benchmark description mentions several ways that children might represent information—using concrete objects, creating pictures, and making graphs.

Concrete objects

Concrete objects are “real” things that a person can see, touch, smell, etc. Concrete objects might include specimens or artifacts that children collect, organize, and label. For example, some children who are studying doors might collect keys, sort them, and help the teacher make labels with information about each key. For example, a label might say, “key to front door” or “antique key.” Collecting and labeling these concrete objects help the children keep track of what they have found out and represents their findings for other people. Figure 1 shows a collection of keys that has been organized in a container but not labeled.

Models are another type of concrete object that children might use to show what they have learned about something. Models can be made from clay, blocks, “boxes and junk,” wood, and other materials. Models may be small enough to fit on a shelf or large enough for children to play in. The teacher can guide and promote children’s efforts to analyze information by asking them to identify the various parts of the model, drawing, etc., and explain how it works.

For example, Figure 2 shows a preschool boy’s representation of a car wash made from blocks after his class made a site visit to a local car wash. His construction includes a roof, driveways leading to and from the building, and a conveyer made of arch-shaped blocks placed upside down.

Figure 3 shows a preschool child’s model of a car wash (shown here without a roof), using a shoe box and other found materials, based on data collected during a site visit to a car wash. The model has an entrance for cars, an exit, and red and green signal lights. It also includes the washing machine that moves over and around the cars.

Figure 4 shows a large representation that some preschoolers made of the mail truck that delivered mail to their school. They have included windshield wipers, headlights, turn signals, the grille, and other important details. Their mail truck, made from an appliance box, is large enough that two to three children can play inside it.

Children might also use concrete objects to convert the dramatic play area into a place that they have been investigating—a hospital, a pet store, a fire station, etc. For example, Figure 5 shows a “pizza restaurant” created by a preschool class. After learning about local pizza restaurants, the children designed their restaurant using ideas from notes and sketches they had made during field work. They rearranged furniture; added props such as pizza boxes, tablecloths, and a toy cash register; and constructed a model of a pizza oven from a cardboard box. They then took roles as workers operating the business based on what they had learned about pizza restaurants.

Pictures

The pictures that children use to represent their data might be sketches, drawings, photographs, paintings, or other flat representations of objects or activities. The pictures are not “art”—they are meant to convey information about the objects to others.

For example, Figure 6 shows a child’s field sketch of a shopping cart she observed when her class visited a local store.

The tempera painting in Figure 7 was made by a preschool boy to show what he had found out about a car wash. His painting shows the darkness inside the building as well as the lighting and some construction details.

Figure 8 is a series of digital photographs taken by some preschool children. The children organized the photographs to show the growth sequence of some daffodils they planted. With some help from the teacher, they mounted and labeled their pictures.

Pictures can also include videos taken by children.

Graphs

A graph is a diagram that helps to compare quantities or amounts of things. Children may use a variety of graphs to show certain kinds of data. Graphs that use groups of real objects are called “real graphs.” The real graph in Figure 9 shows the different subjects of postcards in a collection: three that depict people, four that focus on buildings, and three that depict animals.

Pictographs use pictures (drawings, magazine cut-outs, etc.) to show data. The pictograph in Figure 10 records the number of seeds found in an apple, a peach, a cherry, and an orange. This pictograph allows children to quickly compare the quantities.

Children can also use tally marks, as in Figure 11, to represent numerical data. The tally chart below is a record of the number of seeds counted in four kinds of fruit—an apple, an orange, a peach, and a banana. (It’s a good idea to show preschoolers that they can represent five items with one diagonal mark crossing four vertical marks.)

Surveys can function as graphs, too. Figure 12 shows a survey constructed as a T-chart, which is easy for preschoolers to use when they ask questions with only two possible answers.

Let’s Summarize

Benchmark 10.B.ECa says that we want children to be able to “organize, represent, and analyze information using concrete objects, pictures, and graphs, with teacher support.” In other words, a child meets Benchmark 10.B.ECa if she can

• arrange information about an object or event in a way that makes sense to her;
• show others what she knows about the topic by creating a picture, a graph, a model, a set of related objects, or some other type of representation; AND
• explain what she has found out to peers or adults.

There are many different ways you can help preschool-age children address Benchmark 10.B.ECa. The following are some teacher-tested suggestions:

Planning

• Explore teacher resources to find interesting ways to involve children in organizing and representing their data. (For some examples, see the “Resources” section of this module.)
• Arrange for classroom experiences and investigations when children can collect, explore, and represent a variety of data and artifacts. Be sure they have something worthwhile to “collect data about.” Project work—long-term investigations of nearby places and things—can provide many such opportunities. (For more ideas, see the IEL Tip Sheets Anticipating What Children Might Learn or Project Approach: Phase 1 – Choosing a Topic to Investigate.)
• Practice organizing and representing information yourself. Make sketches, build models, and work with a variety of media (e.g., clay, paint, found objects) so you can draw from your experience with those activities when you talk with the children about what they are doing.

When Children Gather Information

• Invite children to collect objects related to what they are investigating. Depending on topic, these items could include tools, machine parts, seeds, buttons, flyers, etc. Help them create labels for the collection and for parts of the items they have collected. Write their words or type them into the computer and mount a card with the child’s words next to the item.
• Help children become careful observers. One way to do this is to encourage them to make sketches and drawings of what they are learning about. They might draw things they see on site visits or things they have collected and brought to the classroom. It can be especially helpful to have them revisit an object once or twice. Most children notice more details each time.
• Talk with the children about the equipment and materials they will need to organize and represent the information they collect. What will a child need to take a survey? Make a drawing? Tally the number of red cars in the parking lot?

When Children Sort and Summarize Information

• Invite several children to work together to group artifacts or other items they have collected according to similarities and differences. Ask them to talk about what helped them decide how to group the items.
• Help children make simple numerical summaries of data. Show them how to use tally marks. Involve them in making Venn diagrams or graphs with real objects, which can lay the groundwork for understanding and creating more complex graphs. “T-chart” surveys with two possible responses are another way to introduce graphing. Tables and bar graphs are also good ways for children to compare some of the data. (For more ideas, see the IEL Tip Sheets Path to Math: Real Graphs for Preschoolers or Children Taking Surveys.)

When Children Display Knowledge

• Help children think about ways they might show others what they have found out about a topic. Suggest using drawings as the basis for making murals or three-dimensional representations from clay, cardboard, wood, wire, and “found” materials. What materials does a child want to work with to make a mural, a sculpture, or a working model of something—paint, clay, cardboard, wood?
• Give children time to practice using these materials before they get involved in a complex representation.

When Children Explain and Discuss Findings

Schedule times for children to tell you or their classmates about their drawings, photos, models, collections, graphs, etc. Large class meetings or smaller “study group” meetings are good times to let children talk with each other about their representations of data related to their topics. Encourage them to ask each other questions about their data and their representations. (For more ideas, see Cooperation in the Preschool Classroom: Class Discussions or Math Lesson Addressing Benchmark 10.B.EC.a.)

When You Need to Modify and Accommodate to Meet Individual Needs

• Be aware of how children in your class may differ in experience or developmental level. For instance, some children will not be fully ready to count, but they can make tally marks for a classmate to count. Others may not be ready for sorting and classifying objects by more than one characteristic, but working with a partner or small group of peers to label parts of a collection can provide them with needed experience. (For more ideas, see Including Every Child or Math Lesson Addressing Benchmark 10.B.EC.a.)
• Bear in mind that some children may be challenged by the fine-motor work involved in tasks related to organizing and representing data. They may become frustrated. On the other hand, if they feel a sense of purpose about expressing what they know, they may work especially hard to create representations that classmates will recognize and understand. Some children will just need more time and encouragement from teachers and peers. Working with a partner may also be helpful for a child with such issues.
• Similarly, oral questioning and sharing information may be difficult for some children, but again, a child’s wish to be understood by peers can encourage her to “take risks” in conversation. This can be true for dual language learners and for children with speech and language delays or disabilities. (For more ideas, see Discussing Project Work with Children in Inclusive Classrooms.)

Documentation

Documentation can provide evidence of how well a child has learned a skill or understood a particular concept. It can also show how children have worked together during an activity, giving you an idea of how well they collaborate on tasks and problem-solving. Documentation includes

• your notes and anecdotal records, photographs, recordings, transcripts, etc., regarding a child’s activities
• samples of the child’s work that you have collected over time: drawings, paintings, photographs, collections, dictations, lists, models, and other representations of his or her knowledge and skills.

Assessment

“Assessment” refers to finding out what a child understands or knows how to do or how well she has learned a concept or skill. The documentation you collect can be useful with both formative and summative assessments of children’s progress in organizing, representing, and analyzing information.

• Formative assessment tells you how a child is doing “in the moment.” What does he seem to understand about a topic or a process? What does he do that suggests that he is working on, mastering, or having trouble with a skill or concept?
• Summative assessment helps you find out what a child knows, understands, or is able to do at the end of a particular time. How thorough and accurate is her understanding of the topic at the end of a project on tools people use? What skills does he have? What does she know now that she didn’t know at the beginning of the project?

This benchmark includes several points that the teacher can document and assess:

• Can the child—with or without adult help—organize information or items he has gathered about a topic, in at least one way, and explain what he has done?
• Can the child represent that information by drawing, making a model, or other method?
• Can the child analyze (make sense of) information in a way that allows her to explain it clearly to others?

The table below provides examples of what a child might do at each level: exploring, developing, building.

EXPLORING	DEVELOPING	BUILDING
Organize materials with teacher support to prepare for graphing (e.g., sort leaves by color, sort fruit by type).	Participate in creating a data display using concrete objects or pictures with teacher support (e.g., organize children’s favorite fruit in rows to demonstrate whether more children prefer apples or oranges).	Compare numerical information derived from graphs to find answers to questions with teacher support as needed (e.g., use information depicted on a chart or graph to describe which classroom games are most popular).

The following ideas may help you assess a child’s progress toward meeting this benchmark.

Keeping Documentation for Assessment

It’s a good idea to keep a documentation file for each child, with every item dated. Your documentation is a key to assessing the child’s ability to organize, represent, and/or analyze information.

A child’s file can include:

• Notes on your observations of the child relevant to the benchmark—what he does and how he does it, what he says to peers or adults, how he works through problems, etc.
• Video and audio recordings (or transcripts of relevant portions) of the child involved in related activities such as class discussions or field work. You may also want to include notes on the child’s comments about what he was doing and thinking when the recordings were made.
• Photographs of children in activities related to the benchmark. For additional insight, it’s a good idea to ask a child to tell you what is going on in a particular photograph.
• Samples of the child’s work sketches and drawings, photographs, graphs, charts, models, etc., in chronological order. It’s a good idea to include notes about changes in the details and accuracy in the child’s representations over time. You might also include any comments the child makes about a particular work sample.
• Samples of items the child worked on with peers (e.g., a collection, a list, a graph, a model) along with your notes about what you observed as they interacted.

Assessing What the Child Can Do

Child’s Ability to Organize Information

Note how the child organizes a set of items. Does she consistently group items according to specific characteristics? (For example, does she group a set of buttons by size, shape, or color?) Can she consider more than one characteristic when grouping items? (For example, can she group the buttons by color and shape, or by color and the number of holes?)

If the child creates a collection, is it logically and neatly organized? Are the relationships among the objects clear? (For example, are all of the items parts of a bicycle, or types of leaves?)

Note how a child reconstructs events (e.g., by placing a set of photographs into the proper sequence). Does he seem to recognize the order in which events occurred? (“First we planted beans. Then we waited a long time. Then we watered them. Then they sprouted.”)

Child’s Ability to Represent Information

Compare what the child depicts in a drawing, model, or other representation with what actually exists. What does it show about his knowledge and understanding of related concepts? Did he depict the most important parts of the object? Does he seem to grasp the relative size of the parts and their placement? If he labeled the parts, did he name them correctly?

Note how effectively she uses tools and art media (pencil and paper, clay, found objects, and fasteners) in his representations.

Note whether his graphic organizers, such as Venn diagrams, charts, and graphs, clearly show what he wanted to express.

Compare a child’s later or final work samples with earlier samples to see how her understanding or skills have changed. What details do you notice in later work that were absent in the first sample?

Compare a series of the child’s representations of the same object as a summative assessment, making a rubric.

Sample Rubric for Assessing 2- and 3-D Representations

Activity: Month-long project on “tools people use”

The teacher took photographs of the child’s representations of a hammer and dated them. The columns with dates include teacher’s comments on the type of representation as well as comments about the representation.

Child’s Ability to Analyze Information

Listening to a child’s responses to questions about information that he or his classmates have organized and represented is essential to understanding his ability to analyze information. For example, you might ask the child to:

• Explain her process of creating a collection: “What made you decide to put that leaf into this pile instead of that pile?” “What’s another way you could have organized these things?”
• Explain his drawing, model, or other representation. Questions might include, “What is this called? What is it made of? What does this thing do? How do these parts work?” “What did you use to make this model? How did you decide to use that material? Did you run into any problems when you were making this?”
• Explain what is shown by a survey, graph, chart, or diagram. Questions might include “What is being compared here?” “How are these things like each other/different from each other?” “What do you see the most of?” “What can the class do with this information, now that we know about it?”

How clear and thorough are the child’s responses? Look also at her dictations and conversations about the topic to see what she has said and asked about the topic.

It’s also a good idea to listen to children’s comments and questions about classmates’ representations. Are they asking for details? Correcting a friend’s misunderstanding? Showing that they misunderstand something?

Individual benchmarks may be derived from research or from what is generally understood to be the basis for good practice for teaching and learning. Benchmarks frequently are related to principles and standards developed by professional organizations concerned with the specific content area, such as those of the National Council of Teachers of Mathematics (NCTM), which are themselves based on both research and professional knowledge.

Illinois Early Learning Benchmark 10.B.ECa is related to specific standards developed by the National Council of Teachers of Mathematics (NCTM) and to recommendations in a joint position statement of NCTM and the National Association for the Education of Young Children (NAEYC). Parallels to Benchmark 10.B.ECa can also be found in the Head Start Child Outcomes Indicators.

• Head Start Child Outcomes Framework
  • Domain: Science, Scientific Skills & Methods: Collects, describes, and records information through discussions, drawings, maps, and charts.
• National Council of Teachers of Mathematics (NCTM)
  • Representation Standard for PreK–2: Instructional programs … should enable all students to … create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; use representations to model and interpret physical, social, and mathematical phenomena.
  • Process Standard: Create and use representations to organize, record, and communicate mathematical ideas.
  • Data Analysis and Probability Standard: PreK–2 Expectations: Represent data using concrete objects, pictures, and graphs.
• NAEYC/NCTM Joint Position Statement
  • Example of Typical Knowledge and Skills, From Age 3 to Age 6: Organizes and displays data through simple numerical representations such as bar graphs and counts the number in each group (p. 21).

NAEYC and NCTM refer frequently to the importance of having children represent data in a variety of ways in their joint position statement, Early Childhood Mathematics: Promoting Good Beginnings, on early childhood mathematics (NAEYC, 2002, 2010). For example:

• “In high-quality mathematics education for 3- to 6-year-old children, teachers and other key professionals should … use curriculum and teaching practices that strengthen children’s problem-solving and reasoning processes as well as representing, communicating, and connecting mathematical ideas” (p. 3).
• “Experiences and intuitive ideas become truly mathematical as the children reflect on them, represent them in various ways, and connect them to other ideas” (p. 6).
• “When children pursue a project or investigation, they encounter many mathematical problems and questions. With teacher guidance, children think about how to gather and record information and develop representations to help them in understanding and using the information and communicating their work to others” (p. 7).

The following academic and professional publications also give support for or recommend the kinds of activities described in Benchmark 10.B.ECa:

• Clements, D. H. (2004). Major themes and recommendations. In D. H. Clements & J. Sarama (Eds.), Engaging young children in mathematics: Standards for early childhood mathematics education (pp. 7–72). Mahwah, NJ: Lawrence Erlbaum.
  Based on understanding of development of mathematical concepts in early childhood, the author recommends that children initially use physical objects to make graphs, progressing to picture graphs, then line plots, and, finally, bar graphs.
• DeMarie, D. (2001). A trip to the zoo: Children’s words and photographs. Early Childhood Research & Practice, 3(1).
  The author analyzed photographs taken by preschoolers and older children who were given cameras on a trip to the zoo. Unlike the older children, preschoolers tended to photograph familiar animals such as chipmunks rather than zoo animals, as well as clouds, the ground, and other items not unique to the zoo.
• Evangelou, D., Dobbs-Oates, J., Bagiati, A., Liang, S., & Choi, J-Y. (2010). Talking about artifacts: Preschool children’s explorations with sketches, stories, and tangible objects. Early Childhood Research & Practice, 12(2).
  In a study of children’s exploratory learning, the authors found that preschoolers’ interaction with tangible objects generated more talk and exploration, and provided more insight into their ideas about possible functions of artifacts than did interaction with sketches or stories about similar objects.
• Hoisington, C. (2010). Picturing what’s possible: Portraits of science inquiry in early childhood classrooms. Collected Papers from the SEED Conference.
  Among other aspects of drawing, the author noted a variety of ways in which children used drawing to express the speed and direction of water flow.
• McGuire, P. (2010). Supporting high quality teacher-child interactions in Pre-K mathematics (Doctoral dissertation).
  The author investigated use of “five-frames” as instructional scaffolds (e.g., assisting children in counting, partitioning, and tagging concrete objects) and as ways to represent numerical quantities and help them connect different numerical representations.
• Sgroi, L. A., Gropper, N., Kilker, M. T., Rambusch, N. M., & Semonite, B. (1995). Assessing young children’s mathematical understandings. Teaching Children Mathematics, 1(5), 275–277.
  The authors discuss the opportunities presented by graphing activities for assessing children’s knowledge. Graphing is appealing to young children, complex enough to afford opportunities for questioning and can be linked to subject areas other than mathematics.
• Sharples, M., Davison, L., Thomas, G., & Rudman, P. D. (2003). Children as photographers: An analysis of children’s photographic behaviour and intentions at three age levels. Visual Communication, 2, 303–330. doi:10.1177/14703572030023004
  The authors compare how children’s behavior as photographers and the photographs they create may vary across ages.

The following standards and credentialing requirements may be addressed in this module.

Illinois Professional Teaching Standards

The following is a list of Illinois Professional Teaching Standards that may be addressed by completing the Benchmark 10.B.ECa MELBA (for use by instructors and students in teacher education programs).

• 2F—knows how to access the tools and knowledge related to the latest findings (e.g., research, practice, methodologies) and technologies in the discipline
• 3A—understands the Illinois Learning Standards, curriculum development process, content, learning theory, assessment, and student development and knows how to incorporate this knowledge in planning differentiated instruction
• 3G—understands how research and data guide instructional planning, delivery, and adaptation
• 5B—understands principles and techniques, along with advantages and limitations, associated with a wide range of evidence-based instructional practices
• 5E—knows techniques for modifying instructional methods, materials, and the environment to facilitate learning for students with diverse learning characteristics
• 6D—understands writing processes and their importance to content learning;
• 6I—knows appropriate and varied instructional approaches, including those that develop word knowledge, vocabulary, comprehension, fluency, and strategy use in the content areas
• 7B—understands that assessment is a means of evaluating how students learn and what they know and are able to do in order to meet the Illinois Learning Standards;