The First Four Standards - Grades K-2

Overview

Young children enter school with informal strategies for solving mathematical problems, communication skills, ideas about how number and shape connect to each other and to their world, and reasoning skills. In grades K-2, students should build upon these informal strategies.

Early instruction in problem solving should focus on taking time to understand the problem before rushing to solve it. Kindergartners should begin, for example, by representing problems using physical objects. By second grade, students should begin to move away from dependence on physical objects towards the use of pictures and figures. One of the goals of problem solving in numerical situations is to move students toward the use of more efficient problem solving strategies - from modeling with concrete objects to counting methods to using number facts. Even kindergartners should have experience with multiple-step problems (Mary has 3 cookies. She eats one. Her mother gives her two more. How many cookies does she have now?) in order to focus their attention on understanding the problem and developing a plan for its solution. Students should be able to describe how they have solved a problem and justify their answer. They should also develop the habit of comparing problems to each other, noting how they are alike and different.

Communication activities in grades K-2, whether with individuals, small groups, or the whole class, initially emphasize oral (e.g., counting) and pictorial representations. Much time is spent, however, in introducing students to symbolic representations (e.g., numerals and symbols for operations). As students develop written communication skills, they also begin to communicate in writing about mathematics. At first, the teacher may write the students' responses on the board or on sentence strips in order to facilitate this written communication. Students use many concrete representations (e.g., base ten blocks, pattern blocks) and need to learn how to represent their work with these manipulatives through pictures. Students also begin to communicate mathematics using graphs and diagrams.

Many mathematical connections begin to be established in kindergarten. Students should connect thenumber three to triangles, for example, as well as to sets of three objects and the numeral 3. Especially important are quantification (how much? how many?), patterns, and representing quantities and shapes. Using children's literature to motivate and set a context for problem solving and learning mathematics is especially appropriate for K-2, as is illustrated in one of the following vignettes. Connections to social studies may involve using graphs to describe characteristics of the class, the school, or the community.

Many connections between science and mathematics can be established, from looking for patterns to developing specific skills in measurement and data collection. Children observe life cycles and cycles in nature, such as the seasons, and the growth and decay of plant forms. Children begin by using words to describe physical characteristics: color intensity (bright or dull), sound volume (loud or quiet), temperature (hot or cold), and size (longest or shortest). This allows them to make simple descriptive comparisons and to place objects in an order. They move on to using numbers to describe such characteristics. For example, students might measure the height of plants at different times, summarize their data in a table, and prepare a graph (bar or line) showing the height over time. They might repeat the experiment with different growing conditions, and then compare their graphs for the different conditions.

Students in grades K-2 should spend a great deal of time on inductive reasoning, looking for patterns, making educated guesses, generating hypotheses, and forming generalizations based on their experiences. They should also begin to develop some skill in drawing logical conclusions and justifying answers (deductive reasoning), perhaps by using manipulatives such as attribute blocks. They should continually strive to make sense of mathematics by using reasoning to predict answers and compare and contrast examples and problem situations.

In grades K-2, students build on what they already know as they develop their skills in problem solving, communication, mathematical connections, and reasoning. They begin to move from informal, intuitive strategies and processes towards more symbolic representations and more explicit recognition of their thinking strategies.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.

Standards: In addition to the First Four Standards, this vignette highlights Standards 6 (Number Sense), 7 (Geometry), 9 (Measurement), and 11 (Estimation).

The problem: The second grade was in the midst of a unit on dinosaurs when the teacher read to her class the book Danny and the Dinosaur by Syd Hoff (Harper & Row, 1958). After the first reading, the children re-examined some of the illustrations. One picture depicted the dinosaur larger than a block of homes, another showed the dinosaur almost completely hidden by one house. One picture showed the dinosaur taller than an apartment building and yet another showed the dinosaur not quite as tall as a lamp post. Students were intrigued by the idea that Danny's dinosaur friend did not seem to be of a consistent size. They voiced opinions about the dinosaur's actual size. Since students seemed to have a sustained interest in exploring the sizes of dinosaurs, the teacher presented students with this question: Do you think that a dinosaur could fit into our classroom?

The discussion: Brainstorming was encouraged by the teacher as questions such as the following were posed by students and by the teacher. What does it mean to "fit" in the classroom? What information would we need to get in order to determine if a dinosaur could fit in our classroom? Do you think all of our answers will be the same? Why? What do we know already that might help us? What materials do you think we would need?

Solving the problem: Students worked in groups of 3 over a period of several days. They began by choosing a specific dinosaur and then they used a variety of books and computer software in the classroom to find the size of their dinosaur. They determined the size of the classroom, choosing to measure with a trundle wheel or a tape, or by using estimation. Then they decided, by comparing the measures found in books with those made of the classroom, whether the dinosaur would fit into the classroom. Each group was responsible for creating a display and making a presentation to the class to answer the question. The displays made use of models, pictures, and text. Students with more than a few sentences to write were encouraged to make use of the word processor available in the classroom.

Summary: Students used their displays to make presentations to the class. There were a variety of answers. Those who had chosen one of the smaller dinosaurs, the velociraptors, for example, found that the dinosaur could walk through the doorway and several dinosaurs would fit in the room. Others, who had chosen larger dinosaurs, the stegosaurus, for example, found that if the dinosaur could have gotten through the doorway, several would have fit in the room. Still others, who had chosen very large dinosaurs, the brachiosaurus, for example, found that the dinosaur would not have fit into the room at all. As the presentations ended, several children suggested further explorations that might be interesting: Would the dinosaur I chose fit into the multi-purpose room? Was the dinosaur I chose as long as the driveway in front of the school? Was the dinosaur I chose taller than the school building?

Standards: In addition to the First Four Standards, this vignette highlights Standards 7 (Geometry), 11 (Patterns), and 14 (Discrete Mathematics).

The problem: The students in kindergarten had been involved in a unit that allowed them to explore their town. They had been exposed to a variety of activities, including building symmetric and non-symmetric block buildings, drawing neighborhood maps, and using letter-number ordered pairs (like A-2) to locate places on a grid. In this lesson, pairs of students were challenged to build towns with attribute blocks and loops based on a rule or pattern that they made up.

The discussion: With the class sitting on the carpet in a circle, the teacher placed a loop within everyone's sight. She explained that the loop was a town and that the blocks were buildings. Using blocks of different colors, she then placed several triangles inside the loop and several non-triangles outside the loop. Ideas about the rule used to build Town 1 were discussed: Tell me about the town. Describe a pattern that you see. Put this triangle on the carpet to follow the pattern. Put this circle on the carpet to follow the pattern. How could you tell someone else about our town so they could build one just like it? The verbalization was then called the rule for the town. Town 2 was created with two loops, blocks were placed inside and outside these loops, and similar questions were raised and discussed. Several reasonable rules were suggested. For example, one rule was: triangles in one loop, blue blocks in the other loop, other colors and shapes in the overlapping loop and outside the loop. Another rule was: triangles in one loop, blue blocks in the other loop, blue triangles in the overlapping loop, and all other blocks outside the loops.

Solving the problem: Students were given loops and some attribute blocks. They were challenged to work together to build a town that used a rule. At the end of the working time, each pair of students challenged the class to place other blocks in their town and then to verbalize the rule that was used to create the town.

Summary: Students worked independently to record their town designs using crayons and shapes cut from colored construction paper. Students described the rules that they used to build their towns.

The cumulative progress indicators for grade 4 for each of the First Four Standards appear in boldface type below the standard. Each indicator is followed by a brief discussion of how the preceding grade-level vignettes might address the indicator in the classroom in kindergarten and in grades 1 and 2. The Introduction to this Framework contains three vignettes describing lessons for grades K-4 which also illustrate the indicators for the First Four Standards; these are entitled Elevens Alive!, Product and Process, and Sharing a Snack.

Experiences will be such that all students in grades K-2:

1. Use discovery-oriented, inquiry-based, and problem-centered approaches to investigate and understand mathematical content appropriate to the early elementary grades.

• Will a Dinosaur Fit? uses the question Do you think a dinosaur would fit into our classroom? to launch an investigation involving measurement, geometry, estimation, and large numbers. Shapetown develops students' logical (deductive) reasoning skills using shapes (geometry), sorting (discrete mathematics), and pattern analysis.

2. Recognize, formulate, and solve problems arising from mathematical situations and everyday experiences.

• In Will a Dinosaur Fit?, students recognize and help to formulate the question they will investigate, based on a book they have read and its illustrations. In Shapetown, students develop their own logic problems in connection with a unit in social studies on their community.

3. Construct and use concrete, pictorial, symbolic, and graphical models to represent problem situations.

• Students in Will a Dinosaur Fit? begin with a pictorial model (the pictures in the book) and then use numerical models and graphs to represent the problem situation. Students in Shapetown use concrete materials (attribute blocks) to represent their problem situation and then record their "rules" using pictures.

4. Pose, explore, and solve a variety of problems, including non-routine problems and open-ended problems with several solutions and/or solution strategies.

• In Will a Dinosaur Fit?, each group investigates a different dinosaur, using their own strategies. Different groups have different answers, depending on the size of theirdinosaur. In Shapetown, pairs of students pose their own problems for the others to solve.

5. Construct, explain, justify, and apply a variety of problem-solving strategies in both cooperative and independent learning environments.

• The students in Will a Dinosaur Fit? work in groups of three, measuring the classroom and collecting data from books. The students in Shapetown work in pairs to develop the rules for their towns.

6. Verify the correctness and reasonableness of results and interpret them in the context of the problems being solved.

• Students in Will a Dinosaur Fit? present their results to the class for verification. The students in Shapetown verify their results by having other students solve their problems.

7. Know when to select and how to use grade-appropriate mathematical tools and methods (including manipulatives, calculators and computers, as well as mental math and paper-and-pencil techniques) as a natural and routine part of the problem-solving process.

• In Will a Dinosaur Fit? students select their own measuring tools and some use computers. They decide whether to estimate or measure and how to determine their answers (compare numbers or subtract mentally or with a calculator or with paper-and-pencil). The students in Shapetown use manipulatives (attribute blocks) to develop their rules.

8. Determine, collect, organize, and analyze data needed to solve problems.

• The students in Will a Dinosaur Fit? determine what information they need to know about their dinosaurs, collect that information, organize it and analyze it. The students in Shapetown organize and analyze the placement of objects in the town in accordance with the rules they were given and the rules they generated or discovered.

9. Recognize that there may be multiple ways to solve a problem.

• In their sharing, the students in Will a Dinosaur Fit? find out about the many different ways in which students address this problem. The students in Shapetown might explain how they figure out the "rules" their classmates use for their own towns.

Experiences will be such that all student in grades K-2:

1. Discuss, listen, represent, read, and write as vital activities in their learning and use of mathematics.

• In Will a Dinosaur Fit?, the students read a story, read information from books about their dinosaurs, represent their results using symbols and words, and explain their resultsorally. In Shapetown, the students listen to the teacher explain how to develop a "rule," discuss their rules in pairs as they develop them, and record their rules with a picture.

2. Identify and explain key mathematical concepts, and model situations using oral, written, concrete, pictorial, and graphical methods.

• The students in Will a Dinosaur Fit? model their problem situations using oral and written language. Some groups may also use pictorial and/or graphical methods. The students in Shapetown use concrete materials to model their problems and oral methods to solve them.

3. Represent and communicate mathematical ideas through the use of learning tools such as calculators, computers, and manipulatives.

• Some students in Will a Dinosaur Fit? use computers; others use trundle wheels or measuring tape. Students in Shapetown use manipulatives (attribute blocks).

4. Engage in mathematical brainstorming and discussions by asking questions, making conjectures, and suggesting strategies for solving problems.

• The teacher in Will a Dinosaur Fit? begins the discussion of the problem by having students brainstorm what it means for a dinosaur to fit in the classroom. The students in Shapetown discuss the problems posed by the teacher and make conjectures as they try to solve them.

5. Explain their own mathematical work to others, and justify their reasoning and conclusions.

• Students in Will a Dinosaur Fit? explain their work and justify their reasoning about their group's dinosaur. Students in Shapetown explain their work and justify their results as they challenge each other to solve their problem.

Experiences will be such that all students in grades K-2:

1. View mathematics as an integrated whole rather than as a series of disconnected topics and rules.

• In both vignettes, the students are investigating problems that involve several content standards.

2. Relate mathematical procedures to their underlying concepts.

• In Will a Dinosaur Fit?, students research the size of their dinosaurs, determine the size of their classroom by measuring, and compare the measures to see which is larger. InShapetown, students apply the fundamental concepts of Venn diagrams.

3. Use models, calculators, and other mathematical tools to demonstrate the connections among various equivalent graphical, concrete, and verbal representations of mathematical concepts.

• In Will a Dinosaur Fit?, students create a display and make a presentation to the class to support their conclusion. In Shapetown, the students verbalize the rule used for their town and then create an equivalent representation for their attribute block models using a picture.

4. Explore problems and describe and confirm results using various representations.

• The second-graders in Will a Dinosaur Fit? use a variety of representations (symbols and words) to record their results as they investigate the problem. The students in Shapetown use a pictorial representation to describe their results.

5. Use one mathematical idea to extend understanding of another.

• The teacher in Will a Dinosaur Fit? uses the students' understanding of relative size to extend their understanding of estimation and measurement. The students in Shapetown use their understanding of geometric shapes to build their "rules" as they learn more about logical reasoning.

6. Recognize the connections between mathematics and other disciplines, and apply mathematical thinking and problem solving in those areas.

• The dinosaur lesson involves applying mathematics to learn about dinosaurs (science). The Shapetown lesson builds upon a social studies unit in which students use mathematics to locate buildings, construct buildings, and draw maps.

7. Recognize the role of mathematics in their daily lives and in society.

• The students in Will a Dinosaur Fit? learn how mathematics is involved in the sizes of illustrations in the books that they read. The Shapetown students learn how mathematics is used in buildings, in determining locations, and in classifying and characterizing objects.

Experiences will be such that all students in grades K-2:

1. Make educated guesses and test them for correctness.

• The students in Will a Dinosaur Fit? could address this indicator by predicting whether their dinosaur will fit before measuring the classroom. The students in Shapetown are challenged to guess the rule for placing blocks on the carpet, and then to verbalize the rule they think is being used.

2. Draw logical conclusions and make generalizations.

• The students in Will a Dinosaur Fit? draw conclusions from the data they collect by measuring and using texts or the computer. They might also make some generalizations about dinosaurs collectively after discussing the results of all the groups. Drawing logical conclusions is the major focus of the Shapetown lesson.

3. Use models, known facts, properties, and relationships to explain their thinking.

• The students in Will a Dinosaur Fit? use models, known facts (from books and software), and relationships to explain how they know whether their dinosaur will fit. The Shapetown students use models to explain their thinking.

4. Justify answers and solution processes in a variety of problems.

• Students in both vignettes justify their answers and solution processes.

5. Analyze mathematical situations by recognizing and using patterns and relationships.

• The students in Will a Dinosaur Fit? solve their problems by comparing the sizes of the various dinosaurs with other sizes, such as the classroom and its doorway. The students in Shapetown recognize and use patterns and relationships as they pose and solve their problems involving attribute blocks.