4th Grade Lesson Study – Cuisenaire Rods on the Number Line March 7, 2017

Research Question: What important mathematical ideas will students discover when using Cuisenaire Rods with the number line early in a fractions unit?

This lesson study team found that it was helpful to anticipate the actual

models/representations that students would use and which ones we were planning on using, displaying and having students discuss. Having anticipated those representations thoroughly before-hand helped to support decisions that would be made throughout the lesson. Though many misconceptions were visible today, the team felt that it seemed less alarming than in a regular lesson (i.e. not in a lesson study lesson) because we established a clear mathematical goal to stay focused on. Also important to note is the fact that, before the lesson started only 2 out of 9 students said there are numbers between zero and one, but all 9 said “yes” to the same question at the end of the lesson, clearly indicating a change in reasoning as a result of the lesson. Below are the important discoveries the team observed students making during the lesson:

• Using the Cuisenaire Rods in conjunction with the number line helped students see that fractions are numbers, that they can be placed on the number line, and that fractions fall between zero and one on the number line. These were all considered to be important foundational concepts outlined in the research phase of the lesson study cycle.

• Students showed evidence that they came to understand the size of the whole can be changed, while the corresponding fractional parts would also change in size. Using the rods clearly facilitated this opportunity as students could focus on the relationship between the parts and whole despite the size of the whole. Students were willing to

select different size rods to use as a whole, and then discover the fractional parts of that whole.

• could tell where the fractions would go on the number line, but had trouble describing why, or using the rods to prove a precise location. It seemed that this came from many students having trouble seeing “length” on the number line or in the rods.

• It became evident that students discovered that fractions are numbers on the number line, yet the types of unit fractions in this case were limited to those discussed in class (halves and fourths).

• Some students used the rods to discover that four fourths make a whole. Students had difficulty connecting their current understanding of area models with fractions to the rods and the number lines. However, they showed that they were trying to make those connections evidenced by the fact that some drew area models to make sense of the number line without being prompted to do so.

• Several students reasoned that since it took 4 light green rods to make a whole, that one light green is must be one-fourth and that half of a half is one-fourth.

• One student who was clearly productively struggling during this lesson discovered and conjectured “there’s always a half to everything”.

Given these discoveries, the team felt these are the most important implications for teaching: • Because the lesson was planned to support students in connecting different representations

(rods and number line) students were able to use their current understanding of fractions with area models to connect to the number line because they saw the rods somewhat as an area model. It seemed that this supported the mathematical goal of seeing fractions as a number. Therefore, using and connecting student representations is a critical component to advance their reasoning.

• Students should be introduced to the number line early in their work with fractions, and Cuisenare Rods allow for transitional understanding between area models and length models.

• Students need to see that a fraction is as a number early in their experiences with fractions (not just a picture of a part-whole relationship).

• Misconceptions that surfaced during this lesson will serve as the launching point for further student discourse. Therefore, allowing students to manifest their misconceptions through open tasks in the beginning of the unit provide valuable insight in to students’ current understandings. Further, an attempt to rush towards procedural fluency early in a unit is a missed opportunity to link students’ beginning understandings to new learning.

• Using a number line that provides significant visible space between 0 and 1 (12 cm in this case) is helpful in allowing students to realize there are numbers between 0 and 1, that a fraction is a number and to make connections between different representations of fractions.

• Using a number line in coordination with the rods is important to make sense of which fractions should be placed in which location on the number line through reasoning about the length of units or unit fractions.

• Students should be given opportunities to see and discover that the position of a fraction on a number line is determined by the distance of the sum of the lengths of its unit fractions from zero.