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Standards for grades Pre-K–12
Number Standard for grades Pre-K–12
Algebra Standard for grades Pre-K–12
Geometry Standard for grades Pre-K–12
Measurement Standard for grades Pre-K–12
Data Standard for grades Pre-K–12
Problem Solving Standard for grades Pre-K–12
Reasoning Standard for grades Pre-K–12
Communication Standard for grades Pre-K–12
Connections Standard for grades Pre-K–12
Representation Standard for grades Pre-K–12
Electronic Examples for grades Pre-K–12




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Reasoning and Proof Standard for Grades 9–12

Instructional programs from prekindergarten through grade 12 should enable all students to—
  • recognize reasoning and proof as fundamental aspects of mathematics;
  • make and investigate mathematical conjectures;
  • develop and evaluate mathematical arguments and proofs;
  • select and use various types of reasoning and methods of proof.

Mathematics should make sense to students; they should see it as reasoned and reasonable. Their experience in school should help them recognize that seeking and finding explanations for the patterns they observe and the procedures they use help them develop deeper understandings of mathematics. As illustrated throughout this chapter, opportunities for mathematical reasoning and proof pervade the high school curriculum. Students should develop an appreciation of mathematical justification in the study of all mathematical content. In high school, their standards for accepting explanations should become more stringent, and they should develop a repertoire of increasingly sophisticated methods of reasoning and proof.


   

What should reasoning and proof look like in grades 9 through 12?

Reasoning and proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions, no matter what topic is being studied. In mathematically productive classroom environments, students should expect to explain and justify their conclusions. When questions such as, What are you doing? or Why does that make sense? are the norm in a mathematics classroom, students are able to clarify their thinking, to learn new ways to look at and think about situations, and to develop standards for high-quality mathematical reasoning (Collins et al. 1989).

Consider the following hypothetical classroom scenario:
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Mr. Hamilton's class at Manorville High School has established e-mail contact with a high school class in Osaka, Japan. The two groups of students begin by gathering and sharing information about their classes. They exchange data about the number of people in each student's family, about how far each student lives from the school, and about the number and type of pets each student has. The Japanese students have heard that houses and apartments in North Americaare very large, and they want to compare the living areas of the students in Manorville with their own. Each class makes a list of the floor areas in their families' houses or apartments. They compute the mean, the median, the mode, the range, and the standard deviation for their data and share them electronically. »

When the data arrive from their Japanese friends, Mr. Hamilton's students realize that they will have to do a bit more work before a comparison can be made. It has not occurred to them that the information from the Japanese students would be reported in square meters whereas all their measurements are recorded as square feet. At first Angela thinks that they will need to get the original data about the Japanese room sizes, convert the measurements to feet, and recompute the size of each living area before they can determine the statistics. Shanika points out that a spreadsheet would do the conversions quickly but thinks that they will still have to ask for the original data. Mr. Hamilton suggests that before asking the Japanese students to enter all those extra data, they might work with the summary statistics they have and see if they can find a way to compare them directly with their own.

The class decides to focus first on the mode. They know that the mode of the areas corresponds to the measurement of an actual living area. They think that if they figure out how to convert that value from square meters to square feet, they might be able to get started on the other statistics. Jacob observes that a meter is roughly equal to 3.3 feet. He proposes that they multiply the mode of the Japanese data by 3.3 to convert square meters to square feet. Mr. Hamilton asks the class if they agree. Several students nod, but Shanika objects. She points out that they aren't thinking about the fact that the mode is given in square meters and that 1 square meter is equal to about 10.9 square feet. Angela says she doesn't understand where that number comes from so Shanika draws her a diagram (see fig. 7.32) and explains, "See, suppose this is 1 meter by 1 meter. We could make it 3.3 feet by 3.3 feet, then when we multiply to find the area, we get about 10.9 square feet." Her reasoning makes sense to the rest of the class, so they convert the mode value.

Figure

Fig. 7.32. Shanika's diagram for explaining the conversion from square meters to square feet

Mr. Hamilton asks them to think about which of the other statistics could be converted in the same way. Angela says that they can use the same method to convert the median. At that point, two more students join the conversation.

Chuen: You can do it with the average of two numbers. The first number would get multiplied by 10.9, and so would the second, then you average them. But if you just factor out the 10.9, you get the average of the two numbers you had times 10.9.

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Robert: If it works when we average two numbers that way, it ought to work when we average more numbers. »

Chuen tries the mean of three numbers and announces that the method works in that case also. Mr. Hamilton asks the class to show it would always work. With some help, they argue that for any values x1, x2, and x3,

10.9x1 + 10.9x2 + 10.9x3 = 10.9(x1 + x2 + x3).

Shanika then observes, "It doesn't matter how many x's you have. You can always factor out the 10.9."

The discussion of standard deviation is similar. Damon says he tried the method with some simple numbers and it worked, so he thought it should be true. "OK," says Mr. Hamilton. "So we think it's likely to be true. But how do we know it will be? Are there any hints in anything we have written down?" Damon then suggests that they write out the formula for standard deviation and replace all the values of x with 10.9x. It takes a while to work through the details, but the class ultimately shows that if every number in a data set is multiplied by a constant, the standard deviation of the resulting data set equals the standard deviation of the original data set multiplied by the same constant.

An important point in this example is that reasoning and proof enabled students to abstract and codify their observations. Chuen's initial observation was that if each of two numbers is multiplied by 10.9, the mean of the resulting numbers is 10.9 times the mean of the original numbers. The reasoning he used ultimately produced the argument that if every number in a data set is multiplied by a constant, the mean of the resulting data set equals the mean of the original data set multiplied by that constant. The fact that this argument could be made algebraically furnished a mechanism for making a similar argument about the standard deviation. In this way, results with similar justifications can emerge.

Sometimes developing a proof is a natural way of thinking through a problem. For example, a teacher posed the problem of finding four consecutive integers whose sum is 44. The students tried the task and decided it was impossible. The teacher responded, "OK, so you couldn't find the integers. How do you know that someone else won't be able to find them?" The students worked quietly for a few minutes, and one student offered, "Look, if you call the first number n, the next three are n + 1, n + 2, and n + 3. Add those four numbers and set them equal to 44. You get 4n + 6 = 44, and the solution to that equation is n = 9 1/2. So no whole number does it." Here the proof works nicely to explain why something is impossible.

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The habit of asking why is essential for students to develop sound mathematical reasoning. In one class, imagine a student wants to divide an 8 1/2 inchtimes11 inch sheet of paper into three columns of equal width. The student is ready to measure off lengths of 2 5/6 inches, but the teacher says, "Let me show you a carpenter's trick." He places a 12-inch ruler at an angle on the page so that the 0-inch and 12-inch marks on the ruler are on the left-and right-hand edges, respectively, and makes marks at the 4-and 8-inch points on the ruler. He then repeats the procedure, with the ruler farther down the page. Drawing lines through the 4-inch marks and the 8-inch marks divides the page neatly into three equal parts. The teacher then says, "Carpenters use this trick to divide boards into thirds (see fig. 7.33). My questions to you are, » Why does it work? Can you find similar procedures to divide a board into four, five, or any number of equal parts?

Figure

Fig. 7.33. A carpenter's method for trisecting a board

The repertoire of proof techniques that students understand and use should expand through the high school years. For example, they should be able to make direct arguments to establish the validity of a conjecture. Such reasoning has long been at the heart of Euclidean geometry, but it should be used in all content areas: consider, for example, the number-theory arguments discussed in the "Number and Operations" section of this chapter or the arguments given about the effect on some statistics of a data set of multiplying every element in the data set by a given constant. Students should understand that having many examples consistent with a conjecture may suggest that the conjecture is true but does not prove it, whereas one counterexample demonstrates that a conjecture is false. Students should see the power of deductive proofs in establishing results. They should be able to produce logical arguments and present formal proofs that effectively explain their reasoning, whether in paragraph, two-column, or some other form of proof.

Because conjectures in some situations are not conducive to direct means of verification, students should also have some experience with indirect proofs. And since iterative and recursive methods are increasingly common (see, e.g., the "drug dosage" problem discussed in the "Algebra" section in this chapter), students should learn that certain types of results are proved using the technique of mathematical induction.

Students should reason in a wide range of mathematical and applied settings. Spatial reasoning gives insight into geometric results, especially in two-and three-dimensional geometry. Probabilistic reasoning is helpful in analyzing the likelihood that an event will occur. Statistical reasoning allows students to assess risks and make generalizations about a population by using representative samples drawn from that population. Algebra is conducive to symbolic reasoning. Students who can use many types of reasoning and forms of argument will have resources for more-effective reasoning in everyday situations.
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What should be the teacher's role in developing reasoning and proof in grades 9 through 12?

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To help students develop productive habits of thinking and reasoning, teachers themselves need to understand mathematics well (Borko and Putnam 1996). Through the classroom environments they create, mathematics teachers should convey the importance of knowing the reasons for mathematical patterns and truths. In order to evaluate the » validity of proposed explanations, students must develop enough confidence in their reasoning abilities to question others' mathematical arguments as well as their own. In this way, they rely more on logic than on external authority to determine the soundness of a mathematical argument.

As in other grades, teachers of mathematics in high school should strive to create a climate of discussing, questioning, and listening in their classes. Teachers should expect their students to seek, formulate, and critique explanations so that classes become communities of inquiry. Teachers should also help students discuss the logical structure of their own arguments. Critiquing arguments and discussing conjectures are delicate matters: plausible guesses should be discussed even if they turn out to be wrong. Teachers should make clear that the ideas are at stake, not the students who suggest them. With guidance, students should develop high standards for accepting explanations, and they should understand that they have both the right and the responsibility to develop and defend their own arguments. Such expectations were visible in the Manorville High School episode: Informal reasoning and a few calculations suggested to students how a summary statistic given in one unit seemed to be related to the same statistic given in a different unit. In that classroom, however, informal reasoning and supporting examples were a starting point rather than an end point. In a supportive environment, students were encouraged to furnish a carefully reasoned argument for verifying their conjecture that would meet the standards of the broader mathematics community.

   

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