THE FIRST FOUR STANDARDSThe First Four Standards  Grades 34OverviewIn 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 34 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 problemsolving 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 wellfed mealworm crawls in the same amount of time. Third and fourthgraders 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 counterexamples to those incorrect generalizations. For example, if fourthgraders 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. OnLine Resources
The First Four Standards  Grades 34Vignette  Tiling a FloorStandards: 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 twoday 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 twoday 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 foursided 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. The First Four Standards  Grades 34Vignette  Sharing CookiesStandards: In addition to The First Four Standards, this vignette highlights Standards 6 (Number Sense) and 8 (Numerical Operations). The problem: The fourthgrade 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 First Four Standards  Grades 34IndicatorsThe 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 gradelevel 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 K4 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 34 will be such that all students: 1. Use discoveryoriented, inquirybased, and problemcentered approaches to investigate and understand mathematical content appropriate to early elementary grades.
2. Recognize, formulate, and solve problems arising from mathematical situations and everyday experiences.
3. Construct and use concrete, pictorial, symbolic, and graphical models to represent problem situations.
4. Pose, explore, and solve a variety of problems, including nonroutine problems and openended problems with several solutions and/or solution strategies.
5. Construct, explain, justify, and apply a variety of problemsolving strategies in both cooperative and independent learning environments.
6. Verify the correctness and reasonableness of results and interpret them in the context of the problems being solved.
7. Know when to select and how to use gradeappropriate mathematical tools and methods (including manipulatives, calculators and computers, as well as mental math and paperandpencil techniques) as a natural and routine part of the problem solving process.
8. Determine, collect, organize, and analyze data needed to solve problems.
9. Recognize that there may be multiple ways to solve a problem.
Building upon knowledge and skills gained in the preceding grades, experiences in grades 34 will be such that all students: 1. Discuss, listen, represent, read, and write as vital activities in their learning and use of mathematics.
2. Identify and explain key mathematical concepts, and model situations using oral, written, concrete, pictorial, and graphical methods.
3. Represent and communicate mathematical ideas through the use of learning tools such as calculators, computers, and manipulatives.
4. Engage in mathematical brainstorming and discussions by asking questions, making conjectures, and suggesting strategies for solving problems.
5. Explain their own mathematical work to others, and justify their reasoning and conclusions.
Building upon knowledge and skills gained in the preceding grades, experiences in grades 34 will be such that all students: 1. View mathematics as an integrated whole rather than as a series of disconnected topics and rules.
2. Relate mathematical procedures to their underlying concepts.
3. Use models, calculators, and other mathematical tools to demonstrate the connections among various equivalent graphical, concrete, and verbal representations of mathematical concepts.
4. Explore problems and describe and confirm results using various representations.
5. Use one mathematical idea to extend understanding of another.
6. Recognize the connections between mathematics and other disciplines, and apply mathematical thinking and problem solving in those areas.
7. Recognize the role of mathematics in their daily lives and in society.
Building upon knowledge and skills gained in the preceding grades, experiences in grades 34 will be such that all students: 1. Make educated guesses and test them for correctness.
2. Draw logical conclusions and make generalizations.
3. Use models, known facts, properties, and relationships to explain their thinking.
4. Justify answers and solution processes in a variety of problems.
5. Analyze mathematical situations by recognizing and using patterns and relationships.

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