The First Four Standards - Grades 3-4

Overview

In the third and fourth grades, students continue to develop their ability to solve problems, communicate mathematically, make connections within mathematics and between mathematics and other subject areas, and reason mathematically.

Students in grades 3-4 should continue to focus on understanding in their problem solving activities but should also begin to develop a repertoire of strategies for solving problems. These should include not only drawing a picture, using concrete objects, and writing a number sentence, but also drawing a diagram, working backwards, solving a simpler problem, and looking for a pattern. Students begin to spend more time developing a problem-solving plan, since they now have a greater variety of strategies to consider and select from. They also focus more on looking back, comparing each problem to ones they have solved previously.

Communication activities become more elaborate in third and fourth grade, as students become more comfortable with symbolic and written representations of ideas. Students should communicate with each other about mathematics on a daily basis, exploring problem situations and justifying their solutions. Different types of writing assignments may be used: keeping journals, explaining solutions to math problems, explaining mathematical ideas, and writing about the reasoning involved in solving a problem. Students continue to use manipulatives to explore new ideas and learn to relate different representations of an idea to each other. For example, after using base ten blocks to solve 7 x 36, students might provide a pictorial representation of these blocks (at left below) followed by a written explanation of what they did to get 7 x 36 = 252. Linking the use of concrete manipulatives to the pictorial and symbolic representations is critical to understanding the mathematical procedures.

         |||******   |||******   |||******         36     I laid out 7 groups of
                                                  x 7     3 tens and 6 ones.

         |||******   |||******                            I counted up 7 x 3 = 21 tens
                                                  210     and wrote down 210.

         |||******   |||******                            I counted up 7 x 6 = 42 ones
                                                   42     and wrote down 42.

                                                  252     I added those together to get 252.

Children in third and fourth grade continue to build mathematical connections. Within mathematics, the major unifying ideas continue to be quantification (how much and how many, especially with larger quantities), patterns, and representing quantities and shapes. For example, students need to see the relationship between the quantification that they do with measurement (using centimeters and meters) and that they do with base ten blocks (representing numbers in the hundreds). Literature and social studies continue to provide opportunities for using mathematics in context. Students are also able to use mathematics more in their study of science, doing computations with the measurements they have made (e.g., averages). Measurement and data analysis, in particular, offer good opportunities for integrating science and mathematics. For example, students might measure the distance a hungry mealworm crawls in 90 seconds and compare it to the distance a well-fed mealworm crawls in the same amount of time.

Third- and fourth-graders use both inductive reasoning (looking for patterns, making educated guesses, forming generalizations) and deductive reasoning (using logical reasoning, eliminating possibilities, justifying answers). Teachers should create situations in which students may form incorrect generalizations based on only a few examples, and should be prepared to provide counter-examples to those incorrect generalizations. For example, if fourth-graders think that multiplying by 100 always means they add two zeros to the right side of the number, then the teacher should ask them to multiply 0.5 by 100 on their calculators. Instructional activities should continue to emphasize that mathematics makes sense and that mathematical reasoning helps people both to understand their world and to make decisions rationally.

Students in grades 3 and 4 continue to develop more formal and abstract notions of problem solving, communication, mathematical connections, and reasoning. They begin to focus more on what they are thinking as their communication and reasoning skills improve. They solve a wider range of problems and connect mathematics to a greater variety of situations in other subject areas and in life.

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 7 (Geometry) and 10 (Estimation).

The problem: The third grade students toured the school and the playground to find and sketch the geometric shapes that they saw. On returning to the classroom, the class discussed names for each shape, compared the shapes, and talked about where each shape had been found. Several of the shapes had been copied from tiles on walls and floors. The teacher used the tiling idea to challenge students to decide which of the shapes could be used to tile a floor or a wall. (The use of shapes to form a tiling pattern is often referred to as "tessellation.")

The discussion: Questions such as these were examined by the teacher and students, to help clarify the task: What do you know about tiling a floor? Can shapes go on top of each other? Can there be spaces? What are the names of the shapes that we found? How could we check each shape to see if we could tile with it? Do you think we all have to solve this problem the same way? What materials could we use to make the shapes? How many copies of each shape do you think we will need?

Solving the problem: Students worked in pairs over a two-day period. Each pair selected three shapes to test for tiling. Before copying each shape, students wrote in their journals, naming each shape they selected, predicting whether each shape could or could not be used as a tile, and estimating how many would be needed to cover one sheet of paper. The names of some of the unfamiliar shapes were taken from a poster that was hanging in the classroom. Students selected a variety of materials for making copies of their shapes. Some selected plain paper and used rulers to draw copies of their shapes, others selected grid paper, still others selected square or triangular dot paper. Some pairs recognized their shapes in the container of pattern blocks and used them instead. Several pairs used a computer drawing program and were able to create many copies of their shapes quickly and easily. After making and cutting out 5 or more copies of each shape, they attempted to tile sheets of paper with the shapes. Successful and unsuccessful tilings were glued to construction paper. Students checked their previous predictions and continued their journal entries, reflecting on their predictions.

Summary: At the end of the two-day working period, the tilings were sorted into two groups: successful and unsuccessful. Discussion began with the successful tilings. Each tiling was labeled with the name of the shape. The teacher had students talk about the similarities and differences among the successful tilings. Students noticed that many of the successful tilings were made with four-sided shapes, that all the triangles led to successful tilings, and that there were many more shapes that were unsuccessful in tiling than were successful. Similar ideas were discussed for the unsuccessful tilings. Then students tried to verbalize why some shapes could be used as tiles and others could not. They were able to generalize that shapes that could be used for tilings were able to fit around a point without leaving spaces and without overlapping. To close the activity, students wrote in their journals about this generalization using their own words.

Standards: In addition to The First Four Standards, this vignette highlights Standards 6 (Number Sense) and 8 (Numerical Operations).

The problem: The fourth-grade teacher was ready to introduce students to experiences with fractions. This problem was posed as a way to gather information about the ideas that each student already had about fractions:

You have 8 cookies to share equally among 5 people. How much will each person get?

The discussion: Discussion began when the teacher posed the question Why is this a problem? With some prompting, students began to realize that there were not enough cookies to give each person 2 whole cookies, but if they gave each person just 1 cookie, there would be some left over. Students concluded that they would have to give each person 1 whole cookie and some part of another cookie. They had realized that finding that part was the "problem." The next major question for the students was What would you like to use to solve the problem? Students made many suggestions: get cookies and cut them, use linking cubes, draw a picture of 8 circles, use paper circles. The teacher provided construction paper circles, linking cubes, and cookies with plastic knives.

Solving the problem: Working in groups of 3 or 4, students were told that each group was to decide which materials to use to solve the problem, and that each group would explain its solution using pictures and numbers. Finally, they were told that they would be asked to share their solution with the whole class. Students worked in their groups, most choosing to use the real cookies, until they felt comfortable with their solutions. This was one solution: give each person 1 cookie, divide the rest of the cookies into halves, give each person one of these halves, divide the remaining half of a cookie into 5 equal pieces, and give each person one of those pieces. Another solution was: divide all the cookies into halves, give each person three halves, divide the remaining half into 5 equal pieces, and give each person one of those pieces. Students wrote number sentences describing the amount of each person's share, but most found that they were unable to simplify the number sentences to determine how much cookie each person gets.

Summary: The summary discussion centered on how much cookie each person got. The teacher found that students were able to determine the size of the smallest piece of cookie (1/10), but they were unable to determine how much one cookie, 1/2 of a cookie, and 1/10 of a cookie were altogether. The teacher extended the discussion so that the class was able to explore what made the problem difficult and how the problem could be changed to make it easier.

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 grades 3 and 4. 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.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

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

• In Tiling a Floor, students investigate tiling with different geometric shapes in the context of tiling a floor. In Sharing Cookies, students begin their study of fractions by considering a real-life problem.

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

• In both vignettes, the students begin with a problem that arises from everyday experiences.

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

• In Tiling a Floor, the students use concrete materials (copies of shapes) to represent the problem situation. In Sharing Cookies, the students use a variety of manipulatives (paper circles and real cookies) to help them understand the problem.

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 Tiling a Floor, the students use a variety of shapes in their exploration, although most use similar strategies (either concrete materials or a software program). The students in Sharing Cookies solve the same problem in a variety of ways. Although they recognize that the different solution methods lead to the same answer, they are unable to explain what fraction that is.

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

• The students in the tiling vignette work in pairs, applying different strategies to investigate which shapes would tile the floor. The students in the cookie vignette work in groups of three or four; they construct a strategy for solving the problem, explain it, and justify it.

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

• In both vignettes, the students share their results with the whole class. This sharing serves the purpose of verifying the correctness and reasonableness of these results.

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.

• Students in both vignettes are encouraged to select the tools they wish to use to solve the problems.

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

• The students in Tiling a Floor decide what shapes they want to examine, collect data about which ones work and which ones do not, organize this information, and begin the analysis of the results.

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

• In Tiling a Floor, the students select a variety of materials, including a computer program, to solve the problem. In Sharing Cookies, the students all use different methods to solve the same problem.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

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

• The students in Tiling a Floor discuss the problem both initially and at the end, represent the problem situation using manipulatives or the computer, represent their solutions pictorially, and write about the solutions in their journals. The students in Sharing Cookies discuss the problem before and after working in groups, represent the problem situation and their solution using manipulatives or a picture and a number sentence, and write about their solution.

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

• The students in the tiling lesson identify and explain the key concept of tiling from geometry. They use oral, written, concrete, and pictorial methods. In the cookie lesson, the students identify and explain the key concept of fractions (from number sense). They use oral, written, concrete, and pictorial methods.

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

• The students in Tiling a Floor use manipulatives and computers. The students in Sharing Cookies use manipulatives.

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

• In both vignettes, the students brainstorm to clarify the task. They also make conjectures and suggest strategies for solving the problem.

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

• In both vignettes, the students explain their solutions to the class and justify their work.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

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

• The students in Tiling a Floor tour the school and playground to find and discuss shapes, and then link these to discovering which shapes might be used for tiling a floor or wall. In Sharing Cookies, the students examine fractions as an application of a real-life problem rather than as an isolated topic in the book.

2. Relate mathematical procedures to their underlying concepts.

• In Tiling a Floor students are able to discover that tiles which fit around a point without leaving spaces can be used as a pattern for tiling the floor. The students in Sharing Cookies will, in the near future, be learning how to simplify their answers by using addition to find a single fraction that describes their answer.

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

• Students in Tiling a Floor might use a software program such as Tesselmania! to create their own successful tessellations. In Sharing Cookies, the students use models to demonstrate that their answers are the same even though the number sentences are different.

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

• The students in both vignettes use shapes to explore their problems and describe their results in pictures and words.

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

• The tiling vignette uses the idea of shape to extend understanding of tiling. The cookie vignette uses division to build understanding of fractions.

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

• The tiling vignette demonstrates the connection between mathematics and art. The cookie vignette connects mathematics to home economics where fractions are often used.

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

• Both vignettes illustrate the use of mathematics in daily life.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

1. Make educated guesses and test them for correctness.

• The students in Tiling a Floor predict whether each shape can or can not be used as a tile. The students in Sharing Cookies might investigate how many different cookie amounts they can find which can be shared equally among five people (or 4 people, or 7, 8, and so on) with none left over. They guess that every multiple of the number of people would work, and use models for several examples to show that each number they select can be separated into piles containing equal amounts with none left over.

2. Draw logical conclusions and make generalizations.

• The students in the tiling vignette make a generalization that shapes which can be used for tilings fit around a point without leaving spaces or being on top of each other. Students in the cookie vignette conclude that 8 cookies are not enough to give each of the 5 people twowhole cookies, but that each could have 1 cookie with some left over.

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

• Students in both vignettes use models to explain their thinking.

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

• Students in both vignettes explain their answers and how they got them.

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

• Students in the tiling vignette recognize that any triangle can be used as a tile. Students in the cookie vignette realize that each solution names the same fraction with a number sentence.