Unit 3: Reasoning and Explaining (MP2 and MP3)

# 3.3 Identifying Flaws in Reasoning

Students learn about mathematics by exploring their own and others' reasoning in problem-solving situations. Exploring the reasoning of self and others allows for flaws in thinking to be revealed and corrected. Opportunities for students to reveal their thinking and for their peers to evaluate and contribute to the improvement of student thinking can lead to stronger mathematical understanding. As students become more mathematically proficient and their reasoning skills increase they should be able to identify flaws in their own and others' thinking; thus prompting revision of thinking that leads to better problem solving."

Daro, 2012

**Common Flaws with Odds and Evens**

Several common flaws emerged as the 5th-grade students made their arguments part of the public conversation.

**Definitions:**

The most frequent issue encountered was competing meanings of “even”, as indicated in the definitions below:

- A number that can be divided evenly by any divisor without a remainder
- A number that is a multiple of 2

The belief that “even” means divided evenly (without a remainder) is invalidated by the equation 15 ÷ 3 = 5. Furthermore, if both definitions of “even” are used, then a number can be both odd and even (e.g., 15 ÷ 3 = 5 [even] and 15 ÷ 2 = 7 R-1 [odd]).

**Pronoun referents:**

In some student explanations, there is a lack of clarity as to what a specific pronoun refers to. Consider Max’s statement, “Odd numbers if you add it, you’d have to add two different numbers, not like the same; and if you divide it, you get 1 remainder.” It is not clear what “it” refers to.

**Partial understanding or explanations:**

Some students demonstrated partial understanding, such as:

- “Odd means there is a number left out when you count by 1s, 3s, 5s, 7s, or 9s”
- “I think 2 decides, because 2 is always even”

This 5th-grade community is novice in its reasoning, and as the students learn to handle their flaws for odd and even, more sophisticated reasoning skills can become the focus of instruction.