## Overview

This mathematical activity was designed to scaffold students’ ability to formulate an understanding of how to define attributes for measurement, to use direct measurement that includes these attributes to compare objects, and to understand the importance of a common origin for measure. The lesson began with a whole-class discussion where students came up with attributes that described the objects. Then, students were split into small groups to measure the objects with the ribbons provided and determine which object was the biggest. Them, in a whole group setting, the teacher selected three different solutions and had students explain their reasoning for why a certain object was the biggest.

Mathematical Content Goals

Mathematical Content Goals

- Through this activity, students will begin to formulate an understanding of how to define attributes for measurement and start to use direct measurement to compare objects.
- Students will also be introduced to early conceptual understandings of measurement, including describing and defining attributes of measurement, understanding that attributes can be ordered by direct comparison, and understanding the importance of a common origin for measure.

## Targeting the Instructional Goals

After students completed their poster-solutions, students were invited to share their solutions with the class. This portion of the lesson was a key time to target the instructional goals. For instance, at 3:09 I asked a student "Which container was the biggest?" This is important as it targets the instructional goal of understanding that attributes can be ordered by direct comparison. In this case, ensuring that the student recognized that the object with the biggest measure was, in fact, the biggest object by comparison to other objects measured in the same way.

To further develop the idea that attributes can be ordered by direct comparison, at 3:48, I orientated students to each other's thinking by asking members of the class to restate the student's contribution. Specifically, I asked the students "Who can tell me what [student] just said?"; asking them to restate the student's explanation of how the biggest object was determined.

To further develop the idea that attributes can be ordered by direct comparison, at 3:48, I orientated students to each other's thinking by asking members of the class to restate the student's contribution. Specifically, I asked the students "Who can tell me what [student] just said?"; asking them to restate the student's explanation of how the biggest object was determined.

## A Moment to Change

From 3:30-3:45 I attempted to probe the student's reasoning with regards to how they concluded which object was the biggest. However, feeling the pressure of time, I realize that much of my inquiry were leading questions; effectively having the student express their reasoning in the way that I had envisioned rather than allowing them to express their reasoning in their own way. For example, I asked the student "Is the length of the ribbon the same as the height of the bottle?" If I were to repeat the activity, I would ask a more open question such as "how is the length of the ribbon related to the height of the bottle?" or, even more open, "Is the length of the ribbon related to the height of the bottle? In what way?"

## How the Discussion was Structured

**Solution 1: Height**

We chose to include this solution to introduce the early conceptual understandings of measurement. Students from this group demonstrates were able to confidently define the attribute, height, as a form of measurement which could be used through direct comparison with the other objects to determine the biggest.

**Solution 2: Exterior of the container**

This solution was selected to highlight the difference between measuring height (from bottom to top) versus the exterior of the container (from the bottom on one side to the top and then back down the bottom). Although this is not a usual form of measurement, this solution demonstrated that the students understood that the attribute of the exterior of the container could be a defining attribute in directly comparing the size of each container and demonstrates that these students were able to use a systematic form of measurement.

**Solution 3: Width**

This solution was chosen to highlight a different measurement attribute, width, that could be used to directly compare the containers to determine the biggest container. Although this may be unclear from the image, the students did understand the importance of beginning at the origin point. This understanding can be evidenced by the fact that students explained that D’s ribbon had accidentally been cut to short and should have been much longer than the others.

## How my Math Teaching has Evolved

I am far more conscious of how my choice of student solutions has pedagogical implications for our class discussion. For example, to target an instructional goal of objects being arranged by attribute, it is necessary to present examples of student work which display that attribute explicitly. In general, this means that I now select student solutions which directly target my instructional goals rather than simply sharing for sharing's sake.

Moreover, I find that I have improved much more in deep probing of students' reasoning. While it is easy to overlook a correct, or incorrect, response as an indication that the student comprehends the underlying principles, this is a dangerous way of thinking which may propagate students' misconceptions in mathematics. Indeed, I am currently tutoring a student in math who, when I ask him "How do you know [the answer]?", will say "I just know." I have learned to not take "I just know." as an answer and, instead, I insist that my students must show evidence for their reasoning; this allows me, as a teacher, to informally assess their understanding of mathematical principles and intervene when appropriate.

Moreover, I find that I have improved much more in deep probing of students' reasoning. While it is easy to overlook a correct, or incorrect, response as an indication that the student comprehends the underlying principles, this is a dangerous way of thinking which may propagate students' misconceptions in mathematics. Indeed, I am currently tutoring a student in math who, when I ask him "How do you know [the answer]?", will say "I just know." I have learned to not take "I just know." as an answer and, instead, I insist that my students must show evidence for their reasoning; this allows me, as a teacher, to informally assess their understanding of mathematical principles and intervene when appropriate.