Grundlage des Seminars ist der Standard 7, Reasoning & Proof der Standards 2000

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Overview          Standard 7: Reasoning and Proof

Mathematics instructional programs should focus on learning to reason and construct proofs as
 part of understanding mathematics so that all students-

• recognize reasoning and proof as essential and powerful parts 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 as appropriate.

Elaboration: Pre-K-12

Mathematical reasoning and proof offer powerful ways of developing and codifying insights about a wide range of phenomena. People who reason and think analytically explore the properties and structure of objects and systems. They note patterns or regularities in both real-world and symbolic objects; they ask if those patterns are accidental or if things have to be that way; and they conjecture and prove. Ultimately, a mathematical proof represents the formal codification of patterns of reasoning and justification.

Systematic reasoning of the type described above is a defining feature of mathematics. It is found in all content areas and, with different requirements of rigor, at all grade levels. In the area of number and operation, for example, first graders can note that even and odd counting numbers alternate; third graders can conjecture and justify that the sum of two even numbers will be even; students  in grade 6 can determine the likelihood of an even or odd product when two die are rolled and the numbers produced multiplied; and students in grade 10 can be expected to prove, in a variety of ways, that the square of an odd integer is always one more than a  multiple of eight. In geometry, elementary students can use manipulatives to determine areas of new shapes. Middle grade students can tear off the angles of a triangle to demonstrate that the sum of the interior angles of a triangle is a straight angle. High school students can prove these properties rigorously. Principled reasoning is at the core of all of mathematics.

Students at all grade levels can engage-in age-level-appropriate ways-in the kind of systematic thinking, conjecturing, and marshaling of evidence that are the precursors to formal mathematical argumentation. Primary grades teachers can make dramatic changes in what they teach once they become aware that the children in their classes are capable of highly sophisticated reasoning (Thompson 1998). By the time students are in secondary school they should be able to approach a problem or mathematical situation systematically, and suggest why what they think is true. Then they should be able to take the next fundamentally important mathematical step and make a compelling argument that shows it must be true. This is the sequence that Mason, Burton, and Stacey (1982) describe as follows: "Convince yourself; convince a friend; convince an enemy."

Recognize reasoning and proof as essential and powerful parts of mathematics
Part of the beauty of mathematics is that when things work, they work for good reason. Mathematics students should understand this. They should expect things to fit together, and they should expect there to be reasons for why things are as they are. Consider, for example, the following "magic trick" one might find in a book of mathematical recreations.

                           Write down your age.

                                    Add 5.

                                    Multiply the number you just got by 2.

                                    Add 10 to this number.

                                    Multiply this number by 5.

                                    Tell me the result.

                                    I can tell you your age.

The procedure given to find the answer is, "Drop the final zero from the number you are given and subtract 10. The result is the person's age."

The magical "answer" begs the question, "Why does it work?" This question is a mathematical one. Students at all grade levels can explore and explain problems such as this one. For example, a young student can respond to the question, "I'm thinking of a number. If I double it I get 22. What's my number?" and middle grade students can justify the "magic trick" described above using reasoning and informal algebraic techniques.

In grades pre-K-2, students can develop the precursors of formal reasoning and the understanding that it is important to have reasons for what they say. Questions such as "Why do you think it is true?" and "Does anyone think the answer is different and why do you think so?" can establish the expectation that statements need to be supported or refuted by evidence. Research by Resnick and Omanson (1984) highlights the developmental nature of reasoning about addition. When young children are asked to find "3 + 5," they will typically do so by laying out two sets of three and five objects (possibly their fingers), respectively, and then counting  them all: "1, 2, 3, 4, 5, 6, 7, 8." Somewhat older children will recognize that it is not necessary to count the first set, and they will spontaneously start counting from four. At some point, students will-often without instruction-start with the larger number and count from five. This action is based on the recognition that 3 + 5 = 5 + 3 and that it is more efficient to count starting from the larger number. The teacher who sees this development can raise the issue of what the student is doing and why it is justified. The class  discussion that follows can solidify the students understandings; it also can demonstrate that when things work, it is because there is a  reason.

 Similar kinds of experiences take place at all grade levels. Students in grade 7, for example, can explore properties of a triangle.  They notice various relationships such as the congruence of two angles in any isosceles triangle, and they can convince others of the  truth of their discovery. Or, they might use properties about triangles to develop and prove conjectures about quadrilaterals, such as  "What is the sum of the angles of any quadrilateral?" "Is it always the same?" Students in grades 9-12 can see that the various arithmetic and symbolic manipulations they perform are well justified, as in the case of the area diagram for computing (a + b + c)², given in figure 3.6 (adapted from Gelfand and Shen 1993, p. 38). The goal is that students make it a matter of habit to ask questions  and look for justification.
  

Such conjectures are so interesting that they often are named after the people who made them and have passed into mathematical folklore. Mathematicians have spent countless hours verifying or refuting them, often motivated by nothing more than curiosity. The  point is that doing mathematics involves discovery. Conjecture-that is, informed guessing-is a major pathway to discovery. Teachers  and researchers agree that students can learn to make, refine, and test conjectures in elementary school. For example, one lesson  from a third-grade class includes conversations among students as they looked for patterns when adding even and odd numbers. The  students conjectured that the sum of two even numbers will always be even, as will the sum of two odd numbers. They developed  representations of even and odd numbers, such as the following representation of the odd number 9:

 

Representations such as this make it clear how the "extra" unit in 9 can combine with the extra unit in another odd number, yielding an even number (Ball 1989). Similarly, Lampert (1990) analyzes a class session of fifth graders in which the students' task was to figure out the last digit in each of 5⁴, 6⁴, and 7⁴ without doing the multiplication. In considering the same example, Yackel (1998) notes,

The analysis shows that students moved back and forth between inductive and deductive arguments in the course of the lesson. By the end of the lesson, each student had either made an assertion about a pattern, provided a proof that the pattern would continue, and/or given an interpretation of another student's assertion. This is compelling evidence that elementary students can and do engage in sophisticated mathematical reasoning (p. 8)

At all levels, all branches of mathematics provide opportunities for reasoning and conjecture. Often, a simple shift in the way a task is  posed can spark such opportunities. For example, instead of saying, "Show that the mean of a set of data doubles when all the values  in the data set are doubled," a teacher can pose the following question: "Suppose all the values of a sample are doubled. What  change, if any, is there in the mean of the sample? Why?"

 Properly used, technology also opens up the potential for structured exploration. Calculators and computers can now perform with   ease operations that were once very costly in terms of time and effort. Dynamic geometry programs, for example, allow students to  explore transformations or to examine large numbers of cases where it was previously possible to explore only a few. Graphing  programs allow students to explore parameter changes. A variety of computer-based calculus courses now exist in which students  explore a wide range of phenomena, such as limits and convergence, before the results are formally introduced.

Develop and evaluate mathematical arguments and proofs
In chapters 4-7, each of the grade-band sections on Reasoning and Proof provides examples that are appropriate for that grade level. In grades pre-K-2, children can demonstrate their reasoning using concrete models. For example, a child might show the  teacher by counting out blocks that it does not matter which of two numbers is added first. In grades 3-5, students make  mathematical predictions based on observations and begin to provide mathematical justifications for claims they make. For example, a fourth grader might claim that a particular triangle and rectangle have the same area, because each was formed by dividing one of  two equivalent rectangles in half. In grades 6-8, students should be able to perceive and explain more complex patterns, as in the  case of "figurate numbers" described in chapter 6. In grades 9-12, students can be expected to construct relatively complex chains of  reasoning.

Hanna (1998) discusses research about proof, and notes that "Moore states that most postsecondary students have difficulty with  formal proof because they 'begin their upper level mathematics courses having written proofs only in high school geometry and having seen no general perspective of proof or methods of proof' (Moore 1994 p. 249)". Asking "why" provides the opportunity for  reasoning and proof in many areas and should be a recurring theme throughout the entire curriculum.

 An essential component of learning to reason mathematically is learning to evaluate mathematical arguments. It is important that students learn to question assumptions and to make sure that chains of reasoning are justified. Sometimes students make  inappropriate generalizations, such as "multiplication makes things larger." Sometimes reasoning mistakes can be subtle, as in the statement, "This number is divisible by 6 and by 4, so it's divisible by 24." Or students may confuse statements with their converses.  It is not effective to catalogue the wide range of mistakes students might make and warn them against them, although some errors,  such as expanding (a + b)² to obtain a² + b², occur frequently enough that it is useful to alert students to them. In any case, both   plausible and flawed arguments that are offered by the students themselves should create an opportunity for discussion. Classrooms where students are encouraged to present their thinking and in which everyone is encouraged to contribute by evaluating other students' line of thinking provide rich opportunities for developing and evaluating mathematical arguments.

Select and use various types of reasoning and methods of proof as appropriate
There are various kinds of logical arguments. There are traditions for presenting proof in various branches of mathematics, as in, for  example, narrative arguments or two-column proofs in geometry or truth tables in logic. Classification plays a major role in various  arguments (e.g., "This is an X, so it has the properties of all X's"). There are various forms of algebraic and geometric reasoning,  proportional reasoning, probabilistic reasoning, statistical reasoning, and so forth. There are various forms of argument, including  proof by analysis of all cases, disproof by counterexample, and proof by application of general results to specific cases. Students  need to encounter and build proficiency in all of these at levels of increasing sophistication as they move through the curriculum.

 One major goal of the mathematics curriculum is to support the development of reasoning capacity in all students, and the acquisition of tools for constructing proof at appropriate points in their mathematical careers. Students should always be encouraged to think things through carefully, to understand, and to be able to explain. As students' arguments grow more sophisticated, the explanations should increasingly be conveyed in the formal language of mathematics.

Throughout their years in school, students should be provided with opportunities to produce and defend their own chains of  reasoning and to examine and critique those produced by others. In the early years, their arguments may be informal and the chains of reasoning short. Later, reasoning and proof are formalized and increasingly complex.

More broadly, it should be stressed that exploring, conjecturing, representing, and proving are all deeply connected aspects of  mathematical thinking. "Reasoning" and "Proof" should not be thought of as separable from the bulk of mathematical activity. They  cannot simply be taught in a single unit on logic, for example, or by "doing proofs" in a geometry course. Rather, reasoning and proof must be a consistent part of students' mathematical experience in grades pre-K-12. Reasoning mathematically is a "habit of mind,"  and like all habits, it must be developed through consistent use.