Consolidation: Making Connections During Classroom Discourse

Consolidate learning by facilitating a student discussion. 

Students present their work one by one or group by group according to the sequence that the teacher determined during the Sense Making portion of the lesson. During student presentations, the teacher can promote discussion by facilitating commenting and question asking between classmates. The twin goals of discussion are to get students to engage one another and to generalize that place value strategies are an effective for adding and subtracting within 100 when starting from a number that is not a multiple of ten.

To promote student engagement and facilitate discussion, teachers can:

• Create a psychologically safe space to promote intellectual risk taking, new ideas, and broader participation during discussions,
• Provide guided practice in advance regarding how to present work, appropriate voice volume, how to listen, how to agree/disagree respectfully, and other behaviors  and practices to promote discussion.
• Make student work easily visible by using a document camera or other tool so that the image is large enough or all students to clearly see. so that the image is large.
• When an important point is made by a student, prompt other students to repeat/rephrase it. This encourages listening to one another and allows for an important point to be stated 2 – 3 times so that more people can hear it.
• Instead of answering questions asked by students, redirecting the question back to the class. This promotes student to student dialogue.
• While it is important to keep the consolidation purpose in mind, also being open to other important thoughts and ideas that arise. Honoring these rich insights builds excitement and students’ vision of themselves as mathematicians making discoveries.

To promote greater understanding of place value strategies, teachers can familiarize themselves with the following Place Value Strategies. As students explain their thinking during the class discussion, listen for and highlight their explanations and representations that demonstrate place value strategies.  

Look for and highlight examples of students using their understanding of tens and ones to help them flexibly solve the problem in a way that is efficient and accurate. This will resemble the work samples in Student Approaches  4, 5, or 6. In these approaches, students are decomposing the amount being taken away into tens and ones. Students can show this understanding using Base 10 block drawings, open number lines, or just equations. 

It is not necessary for students in your class to use each and every one of these approaches. Rather, utilize the examples that your students come up with as the examples of place value thinking that can be discussed by the whole group. By guiding the discussion in this direction with classroom based examples, teachers will help their class to construct their own understandings of adding and subtracting using place value strategies.

• Adding or Subtracting One Number in Parts (see Student Approach #4 for diagram)
  • Students who use this strategy for 54 – 26 will break the 26 into two parts by place, 20 and 6. Then they will subtract each part separately. Some students will first subtract the tens place number (54 – 20 = 34). Often, they will then decompose the 6 ones into 4 ones and 2 ones. They will first subtract 4 ones to get to the benchmark number (34 – 4 = 30). Then they subtract the remaining 2 ones to get the final answer (30 – 2 = 28) 

    Other students will decompose the 26 in the same way, but subtract in a different order. These students will first subtract 4 ones to get to the benchmark number (54 – 5 = 50). Then they subtract the 2 tens (50 – 20 = 30). Finally they subtract the remaining 2 ones to get the final answer (30 – 2 = 28).

• Adding Up (see Student Approach #6 for diagram)
  • Some students may be aware of the inverse relationship of addition and subtraction. These students know that if 10 – 3 = ?, then 3 = ? = 10. These students may find it easier to add up than subtract back. To solve 54 – 26 = ?, then rewrite it as 26 + ? = 54. They will then use their understanding of place value to add on tens and ones separately. Here are two ways students might add up.

• Making an Easier Problem
  • Some students like to take a problem such as 54 – 26 = ? by adjusting each number by subtracting 4 to make the easier 50 – 22 = ? that they can solve mentally. These students know that it is easier to subtract from multiples of tens. They also know that when each number in a subtraction problem is adjusted in the same way, the difference remains constant.

• Ask questions such as,
  • How did this student take away 26 from 54?
  • How is taking away 26 ones different from taking away 20 and then taking away 6?
  • Why did this student first take away 20, then take away 4, and then take away 2. Why not just take away 20 and 6?
  • Why not just take it all away at once? Why not just take away 26 ones?
  • How did the student make the problem easier by showing 54 as 5 sticks of ten and 4 singles?
  • Why did the student add 4 to 26 first? How did that help them to solve the problem?
  • How did this student use their understanding of place value to solve the problem?
  • How did the bar model diagram help some students to understand what was happening in the situation?

Have students present their strategies and reasoning for how to represent the number of cookies remaining in the box. Ask them to convince you and their classmates that their answer is correct by sharing their mathematical model. It can be very effective to project the student work with a document camera or project a photo of the student work from your computer so that everyone can easily see what the presenting student is referring to. When the other students can easily see the work sample, they remain more engaged and understand more deeply.

Discuss student strategies and elicit the questions and comments of other students during your consolidation. The goal is to help them to  build off of their current prior knowledge and understanding. We want students to be socially constructing their understanding in a way that makes sense for them. 

Here you are listening to this discourse with an ear for what is important for your students’ emerging understanding and application of place value strategies for addition and subtraction problems.