Showcase Math Lesson

For this lesson, I led the first phase, which consisted of launching the task, monitoring

student progress and leading part of the whole group discussion. We worked with a group of 27

fifth grade students, in a math class taught in English.

This instructional activity engaged a group of students in a whole class discussion where

different student solutions were compared and discussed. Through discussion, students

developed an understanding of key features related to measurement, such as the zero point and

its flexibility, and iteration.

The task contained two questions in which students were asked whether it is possible to

measure two lines with a broken ruler. In Part A the line was shorter than the ruler, and in Part B

the line was longer than the ruler.

We started the lesson by activating prior knowledge; students were asked what they know

about measurement. We then introduced the task to the students with a skit and students began

working on it individually. Next, students were led in a whole class discussion, focusing on a

few carefully sequenced student responses in order to reach the mathematical goals. Finally, we

compared student responses, focusing on their similarities and differences.

The lesson was sequenced in a way to target the following instructional goals:

1. To develop an understanding that when we measure, the point of origin begins at zero,

which we call the zero point.

2. To develop an understanding that the starting point is flexible and can be shifted to

measure. This entails understanding that linear measure is a measure of distance traveled.

3. To develop an understanding that units can be iterated, meaning that they can be reused

to measure.

In addition, students were asked to justify their thinking when giving the solution they came up

with, and to use mathematical language when providing solutions and justifications.

Throughout our discussion, several student solutions were presented. This gave me a

chance to elicit and respond to student thinking. In addition, by presenting student solution, I

was able to have student understand their peers’ ideas and to discuss them. When comparing the

similarities and differences between student solutions, we represented their ideas on the board,

creating a public record.

Reflection of a moment intended to target instructional goals:

In this moment, my instructional goal was for students to understand that the starting

point of the measurement begins at zero. We started with this solution, since this student did not

seem to understand our first instructional goal; the understanding that a measurement starts at the

zero point. Thus, we began with this student’s solution since it would be elucidated in the course

of the following students’ solution. This student started from the 3 (at the start of the broken

ruler) and simply read the number at the end of the line, which was 4. However, the line was

actually 1 cm.

The following was his solution to the question “could you use the broken ruler to

measure the chocolate bar and how long do you think it is.”

*Student later

changed his answer from 4 cm to 3.7 cm.

Student: “The length of the chocolate bar is 4 cm.”

Me: “Can you explain how you got your solution?”

Student: “I put the chocolate bar against my pencil and marked the length on my pencil, then I

measured the length using the broken ruler and it showed me that the chocolate bar is 4 cm.”

Though this student’s solution was not correct, it was important for the other students to

understand his solution, in order to develop an understanding that when we measure, the point of

origin begins at zero. The student’s solution was unclear, so I further elicited his thinking in

order to understand (and for students to understand) exactly what he did to get his solution.

Me: “Could you come up to the board and to show us how you solved the problem?”

Up on the board was a blown up version of the broken ruler and of the chocolate bar. The

student came up and showed the class exactly how he came to his solution, using the

manipulatives. This ensured that other students understood his thinking and were then able to

take part in clarifying his misunderstanding.

Reflection of a moment I would like to change:

As the students were working on their solutions, I used the time to gather various student

solutions and sequence them, in order to reach my instructional goals. The second student

solution I chose said “We can’t measure the chocolate because we’re missing the 0.” This

student understood that measurement must start at a zero point, but did not understand the

flexibility of the starting point. I chose this solution since it would help elucidate the

misunderstandings of the previous student regarding the zero point.

However, when I asked this student to share his solution with the class, he’d now

changed his answer to “we can’t solve it because we’re missing digits before and after the 3, 4,

and 5, therefore you can’t even start the measurement.”

At this point, I elicited his thinking and asked him: “What numbers are we missing?”

Student: “We’re missing the 1 and 2 before the 3.”

This threw me off, since he did not mention the zero and therefore, it would not solidify the

concept that that when we measure, the point of origin begins at zero. I decided to ask the

students to discuss this idea with a partner, in the hopes that this student’s peers would clarify his

error.

If I could rewind and go back in time, I would give this student a regular ruler and ask

him to show me how to measure the chocolate with a whole ruler. I would then ask the student

at what number he started. When the student answered “0”, I would then say so do you still

stand by your solution that “we can’t solve it because we’re missing 1 and 2”? In doing this, the

student would realize that not only are 1 and 2 missing on the ruler, but more importantly 0 is

also missing. This would reinforce my instructional goal that measurements must begin at the

zero point.

Choices in Sequencing to Structure Discussion:

We chose to begin with a student whose solution was “yes you can measure the chocolate

bar, it is 4 cm.” We started with this solution, since this student had misunderstandings of our

first two instructional goals; the understanding that a measurement starts at the zero point and

that this point is flexible. Thus, we began with this student’s solution since both

misunderstandings would be elucidated in the course of the next students’ solutions.

Next we chose a student who believed we could not measure the chocolate bar because

the ruler was missing the zero. This student understood that measurement must start at a zero

point but did not understand the flexibility of the starting point. This solution helped demonstrate

to the previous student that measurement must start at a zero point.

The final solution we chose to discuss was “Yes, we can start on the three and pretend the

three is a zero, the 4 is a one, and the 5 is a two.” This student understood that measurement must

start at zero and that there is a flexible starting point in measurement. This solution will help

explain the idea that the start point is flexible, as well as solidify the understanding that

measurements must begin at a zero point.

The discussion was sequenced as such because each solution builds on the understanding

of the previous one. With each solution, the students elucidated a mathematical goal of the

lesson. The focus was first on the idea that measurement starts at a zero point, since it is easier to

understand. Once this was understood, we focused on the flexibility of the starting point.

A Reflection of How my Math Teaching Has Changed:

I began struggling with math concepts in grade five and it only got worse, as the concepts

became more challenging. Since math concepts build on each other, as the class progressed, I

found myself farther and farther behind. When I first began teaching, I taught math in the same

way it was taught to me. Though I knew this type of procedural, direct instruction had not

worked for me, I did not know any other way of teaching the concepts.

Finally, in university, I was introduced to a new way of teaching math, which appealed to

my innate desire to make sense of the world. This model of instruction is based on the Principles

and Practices of High-Quality Teaching, as seen in the table below.

The goal of this model is to support students in their development of meaningful and lasting

understandings of mathematics.

In order to treat children as sensemakers and follow the practices, I now teach math

through rich math problems that require high cognitive demand; often through inquiry and

exploration. These problems are open-ended, have multiple entry points and solutions and

connect to students’ real life experiences. They are accessible to all students and simultaneously

challenge students to reach their full potential. Thus, students establish strong foundations for

future learning and avoid falling behind.