**Analysis of Lesson and Teaching**

**Lesson Context:**

7 students - 4 boys and 3 girls

Diverse ability levels in both math and reading skills

45 minutes - Last period of the day (they always pack up before last period)

**Summary of the Implemented Lesson:**

For the most part the lesson went as planned. I introduced the problem of unequal bags of apples using prepared props. I was proud of how my dramatic interpretation of a real life situation of feeling unbalanced walking home from the store hooked the students. We then thought about the problem in terms of a scale balance (which I had on hand). I demonstrated how the scale works and looks when it is balanced and unbalanced. We also got our of our chairs and used our bodies as a model for a scale balance. I informed students that I wanted help solving my problem of trying to balance my bags. Before beginning to work on the problem, we reviewed the various math strategies available to us if we need help making sense of the problem. Then I set them loose to "independently" try to solve the problem and balance my unequal bags. They were encouraged to use multiple strategies, including being allowed to conference with one another as long as they were on task. I was surprised to find that although I had provided materials for multiple approaches, the majority of the students relied heavily on drawing pictures and diagrams. I was able to walk around the room during this time and informally formative assess the students' strategies and skills. Also if I saw a student struggling or needing encouragement, I was able to quickly conference with them to identify what their struggle was and get them back on track and re-motivated. I am curious if I gave them too much time for this initial exploration because I knew time was passing quickly. It was just so encouraging to me to see them all so engaged, working with such dedication and doing such great thinking and talking.

Once I knew that all students had at least made sense of what the problem was asking and had started taking a shot at finding a solution, I pulled them back together to solve it together as a class. I attempted to have students share their thinking and strategies but that discussion fizzled out fast because my students have not had much discussion in class and were not used to being expected to reason, communicate or respond in that way. I also think I was distracted because I was aware of how much time had passed and knew we had a lot more to cover. After coming to a solution for balancing my bags, I shared on the overhead projector how the solution would be written mathematically (with number sentences). Then I reminded them of the balance scale and I modeled a few very simple examples of how we could compare two quantities with the drawn scale. At this point I also introduced the relational symbols of =, <. and >. I will admit that this felt very rushed. I had them model with their bodies as we went over the examples and we discussed how the "greater than" and "less than" symbols were like an alligator who always wants to eat the bigger number. Then I told them that I had some worksheets for them that required them to both think about what the scale would look like for these two numbers and what symbol we would use to compare them. I had them all start with the lowest level sheet and as they finished they came up to me to turn that one in and receive the next leveled sheet. Then unfortunately it was time for them to line up for dismissal and I was unable to do any wrap up or closing I had planned about real life applications of inequality and problems related to balance.

## The Four Dimensions of Teaching

## Tasks

## Initial Problem Exploration (Balancing Inequality)

I put a lot of thought into constructing this task. It was meant to hook my students in to the problem of inequality and give an application for balancing inequality that made sense to them. The problem was presented and worked on by students before much direct instruction took place

**.**This is in accordance with the**problem-centered**or inquiry based learning pedagogy that I have been interested in trying. I intended the problem to be more challenging and require more cognitive demand than the problems they would see on the later worksheets. That way they could spend some time with it and really think about what it means to be equal or balanced. If it was too easy for them then I wouldn't have had a chance to get a formative look at their strategies, strengths and struggles. This problem was designed to be right in the zone of proximal development for the above level students in my group, it meant that some students would finish this task before others. It also meant that some students could be frustrated and lost as to what they should do.**This task was meant to be more in line with "doing mathematics" and the worksheets to come could be classified as "procedures with connections".**What ended up working well was that I had scaffolding in place for the lower level students. In this way I could differentiate the cognitive demand as I attempted to do as well with the worksheets. Instead of just presenting the problem on the board or on a worksheet as a written task, it was like a performance, it was visual and kinesthetic. The other part of this task was that I required them not just to solve the problem but determine an effective way to communicate their thinking and solution to others. This gave the students who understood the problem right away a task to work on while the other students were still struggling through it. Please see the video "Math Strategies 2" at the bottom of this page to view students working hard on this problem. To see examples of the student work produced go to the**"Observations and Reflections Page"**.## Compare Quantity Using Scale Balance and Relational Symbols

The other task that students were asked to do was to compare two quantities by drawing what the scale balance would look like and then assigning the correct relational symbol to show the relationship between them. This was intended to build off of their thinking about balance and equality earlier in the lesson. I had modeled a few of these on the over head and we had demonstrated what happens to the scale with our bodies to reinforce the idea of tipping one way or the other. The students were very successful with the balance scale. However the introduction and instruction I provided of relational symbols was rushed and felt tacked on to the lesson. I think having students try to think about both of these things at once was too much for this first lesson. Although a couple of the students could handle it the others were confused by the symbols and should have just continued to really think about equality and balance. I think instead of asking them to think about greater than and less than, I could have instead just had them assign equal and unequal signs. I realized after the fact that this initial lesson was more about recognizing something as equal or not equal and thinking about equality in terms of balance. Worrying about which symbol to use was an additional layer to this task that they were not really ready for and took away from the deep understanding they were building around the concept of equality. I think it would've been an excellent follow up lesson on it's own where we could've built off this understanding of equality and balance to then really take time with the symbols. In this lesson they did not have enough time and instruction with the symbols for it to make sense for me to ask them to assign them independently yet. See the completed student worksheets on the

**"Observations and Reflections Page"**as evidence of this argument.## Tools

## Real Life Props

These props were key to engaging students in the initial problem solving. While I wanted them to struggle through the problem on their own initially, I also wanted to provide the right scaffolding and support so that my students who are not used to this type of pedagogy wouldn't become frustrated and not know where to begin. I used real apples and grocery bags so that they knew that this was indeed a real life problem that they could experience! The dots and numbers on the outside of the bag were meant as a visual reminder of what was inside the bags that students could (and did) reference throughout their problem solving. This worked well because it built off of the experiences with

**dot card number talks**we had done previously in class together. I intentionally used two different colors so that it would be more challenging and more to think about for my more advanced students. Also not providing the number of apples in each bag provided an opportunity for students to compose the quantities themselves first. I was curious to see who would add up each bag first, who would add up all the reds and then all the greens, who would just count, etc. The anchor chart for the math strategies, was an important reference tool and reminder for students that if they got stuck they should try a different strategy that might work better for them.## Scale Balance

## Our Bodies

At the beginning of the lesson we reviewed general math strategies for problem solving. I wanted to additionally provide them to with a strategies for remembering the concept of balance. My goal was to provide them with multiple access points to this concept, through introducing multiple models of balance. The first model of balance was the unequal grocery bags, the second was a physical scale balance that students were familiar with from using them in science, the third model attempting to bridge the bags and the scale balance by pretending our bodies are a scale balance. I believe this was successful in two ways. First it was awesome to see them making connections between this use of the concept of balance in math classes and then other applications in science (see the video "Introducing the Problem" at the top of the page to hear our brief discussion of this). Also, by getting them out of their seat and moving their bodies the students were more active and engaged during instruction. The "body scale" was also a go-to strategy that clearly stuck with some students because they put their arms out and thought about it when they were working independently (see the video "Math Strategies 2" at the bottom of the page to witness this). I am very interested in learning more about

**kinesthetic strategies, activities and learning**.## Paper and Drawing/Writing Tools

## Manipulatives

The drawing materials, paper and manipulatives were made available to students as part of their problem solving strategies. I think these various tools allowed for multiple access points for students to engage with the initial problem. I also believe that the paper having space for writing was helpful because some students could think about how they would communicate their thinking in addition to simply discovering a solution. I was disappointed in myself though for not remembering to also encourage them to use these tools during the independent work if they needed to. The students seemed to think that they had to do the problems in their head or by counting without additional support tools when it came to worksheets or independent work. Some of the students may have been more successful if they had took the time to use these tools and other associated strategies on their independent work. However it was interesting to assess who could solve these problems without that support.

## Overhead Projector

Simply put, the overhead projector allowed me to work through the solution to the initial problem with them, showing them one way to communicate the solution with number sentences (which none of the students had done). The three balance scales drawn were constructed together as shared/guided practice before moving on to the worksheets which required them to do this on their own. The overhead allowed me to share this information with all students at once easily. What was not been effective about this is that there was too much on the page at once. I think by the time I got to introducing the symbols they got lost in this large amount of visual information, some of which (the problem solution) we had already moved on from. I should have used a new transparency to do the guided practice and introduce the symbols, so students would be less distracted and overwhelmed. Also I turned the projector off during their independent work, but I think it would've been helpful to students to be able to reference our practice scales to remember what the task was and so I regret not leaving it up for them.

## Differentiated Worksheets

I have very mixed feelings about how the worksheets contributed to the lesson. I think they worked well as a way to differentiate expectations and cognitive demand but by introducing them as a progression to work through, the students got way too competitive about it. Also as discussed above, I believe asking them to think about both the balance and the symbols was confusing and too much for some students without a longer introduction to the task. Even though it was unintentionally competitive, I was glad that students got to experience and see multiple ways to think about equality and comparing quantity (number of objects, numerals, and expressions). This helps to reinforce how equality is more about "balance" than about "answer".

## Norms and Discourse

A very important norm that I established very explicitly was that for the initial problem they could use a variety of strategies to help them make sense and find a solution. It was also made clear that they should choose a strategy that works for them. Students were not expected to approach the problem the same way. I was worried at first that this kind of freedom was so new to them that they would be overwhelmed but they all quickly determined what they wanted to use and starting digging into the problem. However, when it came time to complete the worksheets at the end of the lesson, the students were not using strategies like before and I believe some of them would have been more successful if they had gone back to their strategies. I didn't do enough to encourage and reassure them that they could still use these multiple strategies on the worksheets if they needed to. I believe the strategy freedom during the initial problem exploration was what kept many of the students motivated and engaged instead of being frustrated or bored.

Most of the lesson was the traditional discourse model where I asked questions and had the students respond. However, I did try when we came back together to solve the initial bags problem to have a student share her approach and elicit responses from the students. This did not go well (see the video below). The student who was sharing was more concerned about telling me than her classmates and was annoyed to have to turn around and share her drawing with them. Also the other students seemed to take the attitude that they only really need to listen when the teacher is talking. When it's another student they are very distracted and surprisingly disrespectful. I believe I could have structured this better. I should have made the student sharing stand up at the front of the room. I also should've taken a moment to go over with students how they could learn from each other as much as me. Also rather than having her just share her drawing, I wish I had encouraged her to come to the overhead and redraw it while walking us through her steps. I think the other students would've paid more attention if it was more active sharing and not just looking at a small static drawing. After realizing that I was losing the engagement and focus of most students, I ended up reverting back to the more traditional discourse mode of question and answer. This lack of ability to have students effectively share ideas with one another was not a surprise because that kind of sharing is not an expectation/norm in the classroom these students operate in most of the time and I didn't do well introducing or setting up those new expectations in my lesson. What I found interesting is that I did overhear some very good sharing of ideas when they were conferencing during the period of problem exploration, but when they came together as a whole class, they seemed confused about the expectation to share their thinking and reasoning. Even though it did not happen successfully as a whole group, I was pleased to have seen in the more formative problem exploration time that they are in fact capable of it. They may just need me to structure the discussion differently and make my expectations for that time more clear.

There was one more implicit norm established during my lesson that was completely unintended and disappointed me when it became clear. Students were rushing and competing during the individual work when they were working through the worksheets. This was particularly upsetting because they had worked so diligently and taken their time with the initial problem at the beginning of the lesson. However, I believe I without meaning to, set up these students to interact with this activity in this manner. First of all I had started them all at the lowest level worksheet and explained to them that they could turn that one in for a slightly harder one. They really rushed through their worksheets, running up to me to exchange for the next one and using sloppy handwriting. Two students in particular were racing to see who could get to the hardest one and finish first. This unintentional spirit of competition seemed to engage them in the activity but also probably lead to some students feeling very anxious and to careless mistakes being made (which is frustrating because then I don't feel confident that the worksheets are an accurate assessment). Instead, I may have liked to start them where I knew their understanding level was and not tell them about the next level sheet until they come up to turn in the finished sheet. I would've rather had set the same norm as earlier in the lesson, that students should work at a pace that makes sense for them and to take time to think about and check their answers.

## The four dimensions during problem exploration...

Watch for students discussing the problem with each other, using various strategies and tools, asking for help, using their bodies as a reference to remember the concept of balance, and finding various ways to communicate their ideas. I was able to do a lot of formative assessment and some quick individual conferences.