Author: Julia Burdulis

On April 25th, 2019, I visited the Environmental Sciences Magnet School at Mary Hooker (ESM) to guest-teach a sixth grade math lesson on volume in real world situations. Prior to my visit, the students had been practicing solving the volume of rectangular prisms using cubic units with fractional edge lengths.

The learning objective of my lesson was as follows:

Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V= lwh and V= bh to find volumes of right rectangular prisms with fractional edge lengths in the context in the context of solving real-world and mathematical problems.

These objectives are outlined in the Eureka Math Instructional Guide, which is the math curriculum used by ESM.

My lesson plan which I carried out was the following:

1. 3 Act Task – Act 1

Three Act Task Presentation

Three Act Task Worksheet

I opened the class with only a brief introduction, explaining to the students that we would be watching a video two times. I instructed them to write down things they notice and things they wonder about the video in the boxes designated on their worksheet.

2. Notices and Wonders

I proceeded to write two columns on the board labelled “Notice” and “Wonder” I asked the students to share some of the things they had written down. I had the goal of the students “wondering” the main question for the day which was “How many Rubik’s cubes are needed to fill the box?” and to my delight, it was the first of the wonders that the students shared

3. Main Question and Estimates

Next, I asked the students which of our wonders they thought would be the best main question to investigate. Again, they were right on their first try, probably because they knew this was a volume lesson, and selected “How many Rubik’s cubes are needed to fill the box?”

Once this had been decided, I asked the students to make an estimate of what would be too high, too low, and about right for how many Rubik’s cubes would fit. The students took a lot of time on these, examining the actual box and Rubik’s cubes from the video.

4. 3 Act Task- Act 2

The next question I asked the class “What information do we need to answer our main question?”. I made a list on the board again as they called out answers, which were all in the realm of saying that the dimensions of the box and Rubik’s cube were needed. I then put up the slide displaying these measurements, and Mr. Smith suggested to the class that they sketch the box and the cube and label the side measurements. From there, I told the class that they had all the information they needed to answer the main question, and I asked them to start trying out solving.

5. Student’s Explanations of their work

Once students had had a couple minutes to work on solving without help, I began to move around to check-in with the students and draw their attention to potential mistakes. In my first two experiences with this particular class, it seemed like most kids had a fairly thorough understanding of the lessons’ content and were very practiced at the skills in the objective. This time, the kids were definitely in different places in terms of their understanding of volume, so I had some students finish in under 10 minutes, who were instructed by Mr. Smith to do it again without the calculator. As further assessment beyond just me checking in with students, I asked students who completed the problem to head to my camera in the back of the room, and hold up their sheet and explain how they know how many cubes will fill the box.

6. 3 Act Task- Act 3

Admittedly, we did not formally reach Act 3, which would have involved revealing the answer and reviewing the estimates. Unfortunately, due to students being in different places as I previously mentioned, some of the students were working on the 3 Act task until the very end, and I did not want to give away the answer. Instead, once students had completed their calculations and explained their work to my camera, I quietly let them know they had successfully answered the main question.  Below is some of the students completed work.

7. Extension Activity

Only a handful of students reached the extension activity, and those who did probably only worked on it for approximately 5 minutes towards the end of the lesson. It essentially extended the 3 Act Task situation to ask “How many boxes of Rubik’s cubes will fill a trailer truck? How many Rubik’s cubes would be in the trailer?

Shipping Extravaganza

Equity

My lesson was equitable in that the items used in the problem being addressed were objects familiar to the students universally, as the 3 Act Task video displayed them in the same way to all, and furthermore, the actual objects were equally available for all students to look at during class time. Thus, none of the items involved in the situation made the problem inequitable for students due to a lack of familiarity. Furthermore, I made a point to never call on the same student twice during the beginning portion of class, calling on nearly every student and writing down every response, giving value to each one of them. Finally, when I was going around the room helping students, I often found myself answering the questions to many of the same students. I noticed however that there were a handful of students who were not asking me questions, but seemed to be struggling, so I tried to help them just as much as the more outgoing students by asking them questions to try to guide their work in a direction they might understand better. Overall, I feel my lesson did a good job of being related to a familiar topic to all students, and that I did well in conducting the lesson in a way inclusive and helpful to all kinds of student.

Assessment

My lesson had formative, individualized, captured, assessment which was directly connected to the learning goal. It was formative in that much of the assessment took place throughout the lesson through me asking student questions about their work, and seeing and drawing their attention to mistakes made in the process. This was individualized in that I had one-on-one conversations with students, and additionally when students would go explain their work to the camera, they went mostly alone, or in pairs under the instruction that they both were expected to talk the same amount. My assessment was captured in that I have both video evidence as well as student work on their worksheets to serve as evidence of learning. Assessment was directly connected to the learning goal of using cubic units to find the volume of rectangular prism as this was exactly the solving needed to answer the main question.

Sources

EUREKA MATH -TEACHER GUIDE

Reflection

While this lesson was a success, I do feel that it was less creative due to the students seemingly finding it less fun than my lesson before. However, a three act task, compared with the activity I conducted last time is an entirely different type of teaching, and it is important to have a repertoire of teaching techniques which I’m comfortable to move into, so at the end of the day, it was good to try this out in order to expand my repertoire.

There were a couple of mistakes in the flow of the lesson. While I did improve on my main self-critique in the past, which has been speaking too fast, there were other pieces of my presentation that should have been more polished. The first, thing that could have gone better was the estimates. We ended up spending 10 minutes on students guessing too high, too low, and about right estimates for how many Rubik’s cubes would fill the box, because many students were too meticulous. The estimate really should have been a one minute task where the students took a quick guess. We ended up not reaching the point of revisiting the estimates, which would have given the lesson both closure and fun as it would have allowed students to see how good their estimates actually were, which is quite exciting for them. But, in part due to taking too long making estimates in the first place, there was never time for this. Another mistake I made was in the initial presentation itself. With the technology at ESM, I had to email my powerpoint to Mr. Smith before I arrived. However, I had actually changed the side length labels slightly on the box (the students would never actually measure the box itself so approximating measures was ok), so that the numbers were easier to work with. However on one of the slides in my presentation, the old measurements were labelled, causing much confusion obviously. I should have easily prevented this mistake by checking my slides more stringently before the lesson.

As a whole, many students found this lesson difficult. I think there were two reason for this. The first is that I sense some of the students still didn’t feel practiced enough with this skill, and were not confident enough in the concept of volume to be able to formulate a method to use it in a real life situation. The second is that many students were likely not used to the format of a three act task in which they are only solving one problem. I think that the typical structure of their lessons is likely more along the lines of solving a multitude of problems, and following the same steps somewhat blindly in order to find a correct answer. Finding correct answers in this typical view also likely serves are reinforcement to students that they are doing something right, helping them build confidence that they can solve this type of problem. However, this is not to say that this format of a lesson is better than a three act task. Particularly because the objective involved solving real world problems, I think the format of a three act task is excellent because, just like in the real world, students have to discover the question themselves, and not be told how exactly to get to an answer, and thus discover the process themselves as well. These students were likely less comfortable with this setup because they are so used to one where the steps they must take are handed to them. I think that with more practice with more diverse types of lessons like three act tasks, these students might become more comfortable with discovering the components of a problem, and this would make them better real-world mathematicians.

On April 4th, 2019, I visited the Environmental Sciences Magnet School at Mary Hooker (ESM) to guest-teach a sixth grade math lesson on area in real world situations. Prior to my visit, the students had been practicing solving the area of triangles, with an emphasis on right triangles, as well as parallelograms. I was in contact with the class’s regular math teacher, Mr. Smith, in the days leading up to my workshop, to clarify what the students’ needs would be. He informed me that reviewing shapes and area formulas would not be necessary, as the students had done much practice with it over about a week and should be very comfortable with solving for area of triangles and parallelogram. This was excellent as it gave me more time during my workshop time to do take the students further with a skill they had already been practicing.

The learning objective of my lesson were as follows:

• Find the area of right triangles, other triangles, special quadrilaterals, and polygons, by composing into rectangles or decomposing into triangles and other shapes: apply these techniques in the context of solving real-world and mathematical problems.

These objectives are outlined in the Eureka Math Instructional Guide, which is the math curriculum used by ESM.

My lesson plan which I carried out was the following:

1. Launch

Grab the class’s attention by asking them about some of the basic features of zoos. Have they been to the zoo before? What kind of places are the animals kept? What is different about the places each animal lives in at the zoo? I wanted to get the students excited and engaged with the idea of the zoo for the first moments of the lesson before introducing the math of it.

2. Introduce the activity

I transitioned into explaining the activity I had planned. It was essentially a group activity in which students designed a map of their own “zoo” on a sheet of poster paper, making animal enclosures out of shapes. I used this Zoo Area powerpoint as an aid to show the steps of the activity, and below is the slide I projected as a visual aid while I explained the rules of the activity.

3. Show an example

I showed the class a list of instructions for what they were expected to do in making each animal enclosure. I then showed photos (pictured below) of the steps I took to make my own zoo map, which I had posted on the board for the duration of the lesson.

With this I introduced the next step. Relating back to my launch when we established that different animals need different sizes of enclosures, I passed out a sheet listing all the options of animals the students could choose from to put in their zoo, along with the minimum area one of each animal needed in its enclosure. I explained that once the students had found the area for an enclosure in their zoo, they could call me over to check their work and give them them  a paper cut-out of the type and amount of the animal of their choice.

Next, I projected the instructions for the lesson onto the board, which I also left up, as a reminder for what was expected in the process of making each zoo enclosure.

With that, each group was given their larger sheet of paper, took out their rulers and calculators, and started designing their zoos.

The kids all did really well with this activity, and at one point I even heard a few students talking amongst themselves call my activity fun! Mr. Smith was right in that the kids generally did not need much guidance as they clearly all knew how to find the areas of the shapes we were working with. I found that whenever mistakes were being made, most of what I had to do was make them aware of them rather than give much explanation. Most of the mistakes I saw were

1. Forgetting to label the units or to write units squared
2. Using the hypotenuse in a right triangle to solve for area instead of the legs
3. Mixing up the formulas for area of a triangle and area of a parallelogram (b x h vs ½ b x h)

In all these errors I found that all around I only needed to point out that there was something missing, or remind them to check their work for them to quickly realize what they had missed, and promptly fix it and call me back over to give them their animals

Below are some clips of the students working on the activity. Unfortunately due to all of the groups talking during the class, it is difficult to hear most of the audio in the clips.

The animals proved to be a big motivator for the class, because they had to find the area and do it correctly in order to get them. By the time the activity was in full swing, I was having a hard time keeping up with all the requests to check work and get the animals for groups. Below are a few of the groups maps- most did not fill the whole page but finished several of the enclosures within class time.

Equity

My lesson was equitable in that the subject matter was something that all of my students had knowledge and interest in- animals. Some of my students actually did not say they had been to the zoo, so perhaps I could have addressed that. However, all of the students did say they were familiar with the idea of the zoo, and in making their animal’s enclosures, seemed to understand that different animals needed different amount of space as their habitat. For this reason the lesson was equitable.

Assessment

Formative assessment was built into my lesson in the form of handing out the animals for the students to put on the page. I had a calculator in hand and looked at each groups enclosures and area calculations as they went, checking their work and directing their attention to mistakes to fix in order to earn their zoo’s animals.

Sources

Eureka Math Instructor Guide

Reflection

Compared to my last workshop, I think that this one went far better with much fewer issues. I’m happy with how engaged the students were, to the extent that it seemed like they didn’t even mind doing the math if it earned them different animals for their zoos.

The area I think needed improvement the most was in my launch and introduction of the activity. I was worried that I still spoke too fast, and did less engaging of the students by asking questions. I think this was a result of there being many factors to be explained of the activity, and I was too focused on making sure I did not forget to say anything than on making my launch interesting and ensuring the students understood. There was a moment when I had finished explaining that I was feeling very nervous that none of the student had understood me and that I had designed a lesson too complicated for them to complete. the students did have a couple of questions following my explanation of the instructions, and watching it back, I realized that I probably should made the small improvement of repeating the students’ questions before answering them. The rest of the students likely would have benefitted from hearing the questions, because I ended up getting the same questions later as students worked on the activity. 

I think I missed an opportunity to address equity at the moment in my launch when I asked if everyone had been to the zoo before and not everyone said yes. While I can not change that not every student had been to a zoo before, I definitely could have spent a moment leading a discussion with the students about the features of a zoo. I worry that while all of the students had few troubles with the actual math of the activity, some may have missed out on the imaginative aspect of designing a zoo that engaged some of the kids. For example, I had one group call me over and ask to exchange their deer for another animal, because they said the deer kept running away and hiding, and it was too boring for the guests at their zoo. I was happy they got so into the activity and really imagined what the experience of visitors to their zoo would be like, and I wish I had done more to try to promote this fun side of the activity for all of the students, making it more “real world”.

On February 28th, 2019, I visited the Environmental Sciences Magnet School at Mary Hooker (ESM) to guest-teach a sixth grade math lesson on substituting to evaluate addition and subtraction problems. Prior to my visit, the students had been learning about variables and expressions for about two weeks. A day before I came to the class, the regular class teacher, Mr. Smith,  notified me that he had introduced the students to substitution in addition and subtraction expressions, as well as multiplication and division expressions, and furthermore had already completed the workbook pages associated with my lesson topic. Thus, I had to adapt my initial plan so my lesson would serve primarily as a review and practice of skills rather than an initial introduction to it. The students were already very familiar with the topic which allowed me to facilitate more advanced activities than I had originally planned for with them.

The learning objectives of my lesson were as follows:

• “Write, read, and evaluate expressions in which letters stand for numbers”
• “Use variables to represent numbers and write expressions when solving a real world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specific set

These objectives are outlined in the Eureka Math Instructional Guide, which is the math curriculum used by ESM.

Most of what I had originally planned for the students to be doing was pattern finding from tables. This was to help them see the basis of the expressions they were writing and to help them understand the concept of the variable as represented by the values in the table. As I previously said, this was shifted slightly due to the students having more comfort with pattern finding in tables and expressions. Therefore, I loaded more on solving real world mathematical problems, to put this skill they had already practiced in context.

In my original lesson plan I had outlined my lesson time as follows:

Launch

This was my plan for the introduction to my lesson. It included:

• A review of the concept of a variable
• Modeling a problem similar to those in the workbook:
  • Completing a table comparing student ages to my age as a class
  • Facilitating a discussion to look for the patterns in the table
  • Interactively writing the expressions represented by the table, modeling what each variable represents

Click here to view the table and questions that I prepared to go over during the launch.

Collaborative workbook practice

The students sit in desk clusters of 2-3, and during my observation I saw that they frequently complete group work within their seating groups. I planned to instruct the students to solve their workbook section on substituting for addition and subtraction expressions following the steps in the model problem we had completed in the launch. I also planned to circulate the classroom guiding students and answering their questions.

Below is a photo of the workbook problems the students had in their workbook.

From Eureka Math student workbook

Challenge Activity and Exit Ticket

For any students who completed the workbook effectively and quickly, I had a challenge activity planned in which students were to write their own word problem and corresponding addition and subtraction expressions. Then, I planned to select the most thoughtful of these word problems and present it to the class, asking them to create the table for the given expression and word problem as their exit ticket.

Click here to view  original design for this activity.

Note that this was my initial plan for my lesson, which in a short time before my teaching had to be altered due to the students having more experience with the topic than I had anticipated. The lesson summary that follows reflects the changes I made to this plan to adapt.

Lesson Summary

Launch

To begin the lesson, I briefly reviewed the original lesson which the students had begun two days prior as a refresher. Additionally, Mr. Smith had provided me with a powerpoint of one of the problems from the student Eureka problem set corresponding to this lesson, so I called on students to model this problem on the board as well.  The students were very interactive and generally attentive, reflective of their prior experience substituting to evaluate addition and subtraction expressions and with variables. It seemed to come easily to the students!

See Mr. Smith’s Problem Set PowerPoint from the Eureka Math workbook.

Challenge Activity

Below is the updated edition of my challenge activity. I had shared my original design with Mr. Smith, and he suggested most of the edits which were made to create the final version of it which I used in my lesson.

Challenge Problem

My introduction of the activity:

Nearly all of the students actively collaborated moving through the worksheet as myself and Mr. Smith circulated the room checking in with them as they went. I periodically called the class’s collective attention for a moment to review the instructions for a section at a time when it seems that several students were ready to move on, however I always emphasized that students should move at their own pace and make sure their work was complete and high quality before moving to the next step.

Here are some clips of the groups working on the challenge activity, including my “check-ins” with the group seated closest to the camera:

Rather than selecting one problem for the entire class to complete as an exit ticket, since every student had completed the challenge problem I used the remaining time to allow the students to present their situations on posters and display them for their peers. As a form of formative assessment, I collected the students’ completed challenge activities. Below are some examples of student work on this activity:

Equity

My lesson addressed equity to the very best of my abilities. While I did not ask enough questions during my launch for every student in the classroom to answer one, I deliberately never called on the same student twice, aiming to give as many students opportunities to speak as possible. Additionally, in the bulk of my time in the classroom during which I was circulating throughout the room to provide support, I paid close attention to who was asking me questions and who wasn’t. When I sensed students might be behind in understanding any parts of the instructions, I made sure to go over that section again with their group. For this I looked at the worksheets themselves as students progressed. Some students seemed to actually just be copying their peer’s work rather that understanding it for themselves, so I made a point to ask individuals questions about their group’s work to guide them to understand and be engaged in it. This helped to give less outgoing students who might be hesitant to express their misunderstanding the opportunity for scaffolding just as much as students who either understand or are more willing to ask direct questions.

Assessment

The aspects of checking in on group work as well as looking at individuals within the group’s own efforts, and the collection of completed worksheets serve as formative assessments of the student’s understanding.

Sources

Eureka Math Instructor Guide

Eureka Math Lesson 19

Credit to Adam Smith for edits to worksheets

Reflection

My greatest overarching critique of my own teaching was my lack of preparedness in regards to adapting the lesson to the student’s level of knowledge on this topic. I will take into account in the future that Mr. Smith prefers to introduce any topic I’ll be teaching to his class so that their first exposure to it doesn’t come from me, as he expects me to serve rather as a review for his class to reinforce and advance what they’ve already learned.

With this taken into account, I feel my launch, which I left relatively unchanged, was likely too simple. I think it could have been more effective to additionally model a more advanced step of the basic skill being taught, as I only focused on the types of problems which the class had already been working with rather than doing a sort of activity which I would subsequently be asking the students to complete as an activity, in this case meaning practice coming up with real life situations in which there are patterns from which expressions can be written. Additionally, having reviewed the video of myself teaching, I think it would have been helpful to slow down my speaking, which was difficult as I was admittedly nervous to be teaching. My fast pace of talking led me to misspeak several times and also potentially move too fast for some students to understand.

The biggest change from my original lesson plan was the elimination of the student workbook practice problems as group work, as the students had already completed these before my workbook. My challenge activity thus became the primary focus of my lesson, leaving me with fewer activities planned. From this I learned that I should absolutely check in with Mr. Smith about a week prior to my scheduled visit, as finding out a day before did not allow sufficient time for me to effectively design new activities catered to what the students had already done.

I think the challenge activity, though probably less challenging considering this was the class’s second exposure to this lesson topic, was an effective learning activity which deepened their understanding and allowed them to apply the mathematical practice in a more advanced way. The student’s completed worksheets which I collected all had accurate pattern relationships written as real life situations and expressions representing them, indicating their successful understanding. While students certainly benefited from my answering questions and guidance whilst checking in with their groups, I didn’t sense that any students were entirely at a loss, as most of their questions for me were relatively specific about the activity.

I regret here not having further activities planned, as I don’t feel that asking the class to rewrite their problems as a poster presentation necessarily added to their learning in any form, as it was just copying what they had already written on their challenge activity sheets. However, I unfortunately had run out of activities at that point and had to fill the remainder of the lesson time somehow. In the future I will ensure that the entire duration of my lesson is going towards enhancing learning.

On February 14th, I visited the Environmental Science Magnet School to observe Adam Smith’s 6th grade math class. I really liked what he was doing with the students when we arrived. The students had a their math workbooks out and were looking at the practice problems for that day’s assignment. However, rather than working on solving them, Mr. Smith was facilitating an activity which was aiming to ensure all the students understood the question. He had instructed the students to circle the words which were confusing to them, and was going word by word through the question, asking what the problem was asking for and calling on different students around the room.

As they worked on this, I looked around Mr. Smith’s classroom to try to get an idea of the environment. The students were seated in clusters of 3, as they were frequently expected to work in these small groups with their neighbors. The front of the room was primarily handwritten posters of formulas and example problems and solutions. The side walls had a mix of more of these as well as some motivational posters. Mr. Smith’s desk was in the back, but he stayed up front by the white board the entire time he was teaching. He was utilizing the projector to display the instructions the students were looking at in their workbook, but also utilized a blank stretch of board to draw or write examples. Mr. Smith also travelled around the room most of the time, checking the students’ papers to make sure they were focusing on circling unknown words. He emphasized that he really wanted to make sure they cleared up any confusion before the students left the room.

Mr. Smith went a little over the normally scheduled class period. He passed out exit tickets for the students to complete. Initially he emphasized that he wanted everyone to complete the exit ticket prior to leaving, but upon realizing the time, he excused those in the class who had not completed it.

Mr. Smith was very eager and helpful when meeting with Lexi and myself once the students left. He shared the workbook pages that would be assigned on the day we would be teaching our first lesson, and also allowed us to take photos of the standards outlined in his teacher copy of the book. He seemed very flexible, telling us that we should feel free to conduct our lesson however we see fit to get the information to us. I’m very excited to work with him and his class, and I thought observing his teaching was a very insightful way for me to see what I’ll be doing in just a few short weeks.