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# Content: Connecting Concepts and Procedures

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## Purpose

The purpose of this activity is to give you a chance to learn about two concepts related to subtraction and then think about how that conceptual knowledge could be emphasised along with procedural practice.

## Background

There are two conceptual understandings of subtraction. The first is when you have one set of objects and you subtract from that set. Sometimes this is referred to as “separating” or “taking away.” Look at the figure below. This shows subtraction as a separating for the problem 6-4=2. You start with a set of 6 objects Then, you subtract or separate 4 of those objects by crossing them out with a red slash. After subtracting, 2 objects remain. So, 6 minus 4 equals 2.

figure: Subtracting as Separating

The second conceptual understanding of subtraction is when you compare two sets of objects to find the difference between the sets. Sometimes this is referred to as “comparing.” Look at the next figure. This shows subtraction as comparing for the problem 6-4=2. You start with a set of 6 green cubes. Then, you compare the 6 tree cubes to 4 yellow cubes. The difference is outline in red. The difference between 6 and 4 is 2 So, 6 minus 4 equals 2.

Subtracting as Comparing

Once students have a firm understanding of the concepts of subtraction, then it can be time to engage in other types of conceptual practice alongside practice with the procedures of subtraction.

## Procedures

1. Grab a set of household objects (e.g., spoons, pencils, paperclips). You will want at least 20 of these objects.Â
2. Explain out loud using words and your objects (to yourself or to someone else) how to subtract 11 â€“ 5 by separating. One thing that is often helpful when emphasizing concepts is to describe a real-life story (e.g., I had 11 cookies and I gave 5 to you). Create a real-life story to tell as you describe 11 â€“ 5 by separating.Â Â
3. Explain out loud using words and your objects (to yourself or to someone else) how to subtract 11 â€“ 5 by comparing. Create a real-life story to tell as you describe 11 â€“ 5 as comparing.Â
4. After focusing on the two concepts of subtraction, brainstorm two ways to practice the procedures of subtraction. Think about ways to practice concepts and procedures side by side.Â Â

#### Module: Helping Students Do Math

• Unit 1: What is Math?
• Unit 2: Experiences with and Attitudes Toward Math
• Unit 3: Math Instruction: The Big Picture
• Unit 4: Learning from a Math Textbook
• Unit 5: Math and Technology
• Unit 6: Asking Questions That Help Students Think
• Unit 7: Problem Solving
• Unit 8: Math and the Common Core
• Unit 9: Number Sense (Math Facts, Memorization, and Practice)
• Unit 10: Visual Math
• Unit 11: Putting it All Together for Paraprofessionals
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