This week the students were reviewing in chapter 1 of Go Math!. For one day, they focused on “1.3: Compare and Order Numbers.”
I adapted the “Expanded Lesson: Close, Far, and in Between” on pages 169-170 of Van de Walle1 to be a 10-15 minute teacher center.
On individual notecards, I wrote three numbers: 27, 83, and 62. On separate notecards, I wrote the questions:
- Order the numbers from smallest to largest. How can you prove it?
- Which two numbers are closest to each other? How can you prove it?
- Think about three other numbers: 50, 55, and 73. Which of the first three numbers is closest to each of these?
I similarly wrote three larger numbers – 219, 457, and 364 – each on an individual notecard. I wrote similar questions to those above.
As instructions, before I showed the students the first question or the first set of numbers, I told them: You will need to explain why your answers are correct. Use math tools to explain: words, numbers, a number line, pictures, diagrams – whatever makes sense to you.
I planned to ask them questions such as:
- Ask one student: What did you decide to do? How do you know? Can you prove it?
- Ask other students: Do you understand that strategy? What questions do you have? Who used the same strategy? Who used a different strategy?
- After a couple of strategies: Which strategy is most efficient (fastest)? Which strategy is most valid (correct)?
Before implementing the task, I wrote a plan for noticing and reflection, based on recommendations in the Van de Walle lesson.
- Noticing Strategies (e.g., counting by ones, fives, or tens; subtraction; visuals like number line, benchmark, other)
- Noticing whether students could think of only one strategy or more
Reflection on Task and Implementation
I led the center as one of four centers that students visited. Each group included 4-6 students. I felt generally that the task went well, but we did not have enough time to get past the first group of numbers.
Strategies for “Order the three numbers”: To order numbers, students used a few different strategies.
- Place value: Most students used place value (we had just gone over using place value to compare numbers in the first part of the math lesson). I felt that I saw some students who used place value procedurally (i.e., simply compared the first digit of each number and could only explain “6 is more than 2 so this number must be bigger”) and others who used it conceptually (i.e., compared the first digit of each number and explained something similar to “6 tens is more than 2 tens”).
- Counting: Many students counted or used counting as a justification: “I know because I count to 27 first and then 62 and then 83.”
- Diagram: One student drew the numbers with hash marks between them representing tens: 27”’62’83 (he counted: 27, 30, 40, 50, 62, 70, 83). This strategy was similar to the place value strategy, but a different way to visually show the place values.
- Number Line: One student drew a number line using time as a measure to determine length between the numbers. She was only able to compare 27 and 62, but explained she knew 62 was larger than 27 because of the number line. This strategy was a visual representation of her counting, because she drew the line with a tiny push forward for each number as she counted from 1 to 27 and then from 27 to 62.
Strategies for “Which of the first three numbers is closest to each of the second set?”: To order the second set of numbers, students used a few different strategies.
- Order and Connect: Several students ordered the second set of numbers (50, 55, 73) and then connected them directly to the ordering of the first set (27, 62, 83). So, they said 27 was closest to 50, 62 to 55, and 83 to 73. This strategy almost worked except that 50 is closest to 62 instead of 27.
- Subtraction: Some students subtracted and found that 50 and 55 are both closest to 62. (only 3 or 4 of the 22 students weren’t “tricked”)
Group Management: I noticed that some groups and some students were more engaged than others. I asked the teacher the next day:
- What are some strategies for keeping students on task and using tools appropriately? (some students were drawing pictures instead of comparing numbers or listening to other students’ strategies)
- What are some strategies for supporting students in thinking about more than one strategy and trying at least one strategy?
- I thought one strategy could be to choose the tools carefully and “force” different strategies by giving different students different tools.
1 Pages 169-170: Van de Walle, J. A., Karp, K. S., Lovin, L. A. H., & Bay-Williams, J. M. (2014). Teaching Student-centered Mathematics: Developmentally Appropriate Instruction for Grades 3-5 (Vol. 2). Pearson Higher Ed.