Observing Student Reasoning and Proof
 Introduction | Sums of Numbers | Problem Reflection #1 | Products of Numbers | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal

Think about the student work you've just observed and answer the following questions. Once you've formulated your own answers, select "Show Answer" to see our response.

 Question: How does this problem help students improve their understanding of conjectures? Show Answer
 Sample Answer: Even though Robert's conjecture seems simple, it has value because a student proposed it and it is not too hard for other students to think of a variety of ways to investigate it. All students can at least test the conjecture with a few numbers. Testing a lot of numbers, however, does not constitute a proof. Testing is just a method of investigation, and it cannot be used to say that something is always true, unless there is a very limited number of possibilities to test. Instead, students can draw on their prior experiences with even and odd numbers to help them informally prove that the conjecture is true for any whole number.
 Question: What evidence did you see of a student making a convincing argument? How did students justify their answers? Show Answer
 Sample Answer: The students based their argument on an already understood and generally accepted definition of "even." For example, the explanation using linking cubes relates "even" to physically being able to break a number of items into two equal parts. The explanation about the organized list also supports the conjecture that all sums of a number added to itself will be even, because "the sums always go up by 2 . . . because you add 1 to each of the two numbers every time you move to the next row."
 Question: How would you help students who are having difficulty reasoning through this problem? Show Answer
 Sample Answer: Those students probably need to do some more work around the definitions of odd and even numbers. Also, working with various models and forms of representation, particularly those related to addition, may be helpful in strengthening their understanding. Finally, students may also need to work on recognizing and discussing number patterns and developing some reasoning skills around those patterns.

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