# Teaching Reasoning in Math: Types & Methods

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• 0:03 Reasoning, Defined
• 0:36 Types of Reasoning
• 2:35 More About Connective…
• 3:51 Teaching Math Proficiency
• 4:54 Lesson Summary
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Lesson Transcript
Instructor: Kerry Gray

Kerry has been a teacher and an administrator for more than twenty years. She has a Master of Education degree.

This lesson will introduce several types of reasoning, including: connective, inductive, deductive, abstract, quantitative, and adaptive. Further, we will discuss methods for teaching students to use connective reasoning to solve mathematical problems.

## Reasoning, Defined

Has this ever happened to you? Last week, Kim went through the drive-through at my favorite Mexican fast food place, ordered two burritos, and was told that her total was \$30.23. She and the cashier laughed about how good the burritos must be for that price, but they both used their reasoning skills to see that there was a mistake somewhere.

Reasoning is making sense of mathematical processes by investigating and justifying new information based on its connection with existing schema. Let's look at some mathematical reasoning skills your students need to know.

## Types of Reasoning

There are many types of reasoning your students will use as they become successful mathematicians. Here are a few:

• Inductive reasoning is when a student creates a rule from a few specific cases. For example, there are 31 days in July and August, therefore every month has 31 days. As you can see from this example, inductive reasoning is not always correct. However, in some situations it is. For example, 0 x 5 = 0 and 0 x 6 = 0, therefore any number multiplied by 0 equals 0.

• Deductive reasoning is starting from a rule and then applying it to specifics. For example, if the rule is that any number multiplied by 0 equals 0, then 0 X 111 = 0.

• Abstract reasoning, also called conceptual reasoning, is the ability to recognize patterns and make logical theoretical connections to solve problems. For example, students use abstract reasoning to figure out the next numbers in the following sequence: 5, 9, 17, 33, ..., (Hint: Multiply the previous number times 2 and then subtract 1.)

• Quantitative reasoning is the ability to apply mathematical principles and statistical data to real world problems. For example, Zachary drives from Dallas to Austin, which is a distance of 195 miles. If he averages 60 mph, what is the latest time he can leave to make it to a 3:00 P.M. meeting?

• Adaptive reasoning is the ability to not only develop logical solutions, but to be able to defend those solutions. For example, a student knows that 12 x 3 = 36 and can defend it by putting 36 counters in 3 groups of 12.

• Connective reasoning combines adaptive reasoning, conceptual understanding, strategic competence, productive disposition, and procedural fluency to create proficient mathematicians.

## More About Connective Reasoning

We already learned about adaptive reasoning and conceptual understanding. Let's take a closer look at the other three strands of mathematic proficiency under connective reasoning:

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