Analysis of Teaching
Students combined quantities ranging from 1 to 10 to sum 20, and worked on increasing their understanding of addition and subtraction up to 20. By creating different combinations of pizza toppings which had assigned values, students worked in pairs and then individually to discover possible combinations and strategies for addition up to 20, and used subtraction when necessary. The objective was for students to increase their fluency by making numeric combinations through the familiar and engaging framework of assembling personalized pizzas. Hiebert writes that “Setting goals in terms of students’ experiences and residues that might be left is a different way of thinking than the more traditional conceptions of goals as lists of instructional objectives” (Hiebert, 1997, p. 34). A goal was for students to take away meaning from the lesson by making a connection to the real-life application of ordering a pizza, with the residue being the connection between having a fixed amount of money and making food choices.
The pedagogical focus for the lesson was Selecting and using representations to make mathematics meaningful and draw connections between mathematical concepts. By giving students a graphic organizer with fixed values and paper materials to construct a personal pizza of their choice, my intention was that the concepts of addition and subtraction up to 20 would become more meaningful. The mathematical goal was for students to understand that there are multiple strategies for achieving a sum less than or equal to twenty. Using materials as tools for modeling these combinations, my objective was for students to combine dollar amounts, including 5, 10 and 2-5 to sum 20 or less, to improve their abilities to add these quantities, and to develop comfort articulating their addition strategies.
Tasks
The hook for the lesson was that we would do what I called pizza math. The first task was for students to fill in a price list and to come up with possible combinations of pizza toppings. I explained that students each had $20 and that they could spend up to that amount. The second task was that students would make personal pizzas that reflected the topping combinations of their choice. This dimension offered a structure to my lesson in that I was able assess student learning through observing the tasks.
For the first task, students worked in pairs and filled out the graphic organizers. They kept track of the desired quantities for each topping and used the work space to write the equivalent amount. The task focused on students’ ability to visually keep track of their work, writing numerals, and students’ understanding of numbers as symbols for addition. Engaging with this task prompted students to work together and to exchange opinions in a mathematical discussion. The graphic organizer was key in supporting my pedagogical and mathematical focus in that it allowed students to help make mathematics meaningful. Students made mathematical decisions by understanding assigned monetary values of toppings, how quantity affects cost per serving, and how to combine numbers, or prices per toppings, to sum 20.
The second task supported my pedagogical focus as well by acting as the visual representation of their problem solving. By selecting their favorite combination of pizza toppings to create a personalized pizza, each student produced an individualized combination of toppings that represented my mathematical goal of showing multiple ways to combine sums less than or equal to 20. While some students exceeded their $20 budget, their addition and recording of the price of toppings remained accurate throughout both tasks. Those students whose numeric combinations exceeded $20 were also conscious that they were exceeding their allotted budget. Since they were able to justify their decisions, I did not worry that they had gone outside of the parameters of my instructions.
Tools
I used the idea of building a personalized pizza as a foundational tool to engage students in what I called ‘pizza math.’ With pizza as something familiar and typically associated with a positive meaning, I emphasized the concept of choice in allowing students to design their own personalized pizzas. To keep track of the assigned values for each topping and quantity of each topping, students used pencils and graphic organizers (as seen below) as tools. The graphic organizers contained numerals whose symbolic meaning students were required to understand in order to effectively use them as tools for addition, or making combinations whose sum was less than or equal to 20. Hiebert writes that “The experiences that are represented by the symbols provide anchors for the symbols. Students can tie the symbols to something that already has meaning for them” (1997, p. 57).
The pedagogical focus for the lesson was Selecting and using representations to make mathematics meaningful and draw connections between mathematical concepts. By giving students a graphic organizer with fixed values and paper materials to construct a personal pizza of their choice, my intention was that the concepts of addition and subtraction up to 20 would become more meaningful. The mathematical goal was for students to understand that there are multiple strategies for achieving a sum less than or equal to twenty. Using materials as tools for modeling these combinations, my objective was for students to combine dollar amounts, including 5, 10 and 2-5 to sum 20 or less, to improve their abilities to add these quantities, and to develop comfort articulating their addition strategies.
Tasks
The hook for the lesson was that we would do what I called pizza math. The first task was for students to fill in a price list and to come up with possible combinations of pizza toppings. I explained that students each had $20 and that they could spend up to that amount. The second task was that students would make personal pizzas that reflected the topping combinations of their choice. This dimension offered a structure to my lesson in that I was able assess student learning through observing the tasks.
For the first task, students worked in pairs and filled out the graphic organizers. They kept track of the desired quantities for each topping and used the work space to write the equivalent amount. The task focused on students’ ability to visually keep track of their work, writing numerals, and students’ understanding of numbers as symbols for addition. Engaging with this task prompted students to work together and to exchange opinions in a mathematical discussion. The graphic organizer was key in supporting my pedagogical and mathematical focus in that it allowed students to help make mathematics meaningful. Students made mathematical decisions by understanding assigned monetary values of toppings, how quantity affects cost per serving, and how to combine numbers, or prices per toppings, to sum 20.
The second task supported my pedagogical focus as well by acting as the visual representation of their problem solving. By selecting their favorite combination of pizza toppings to create a personalized pizza, each student produced an individualized combination of toppings that represented my mathematical goal of showing multiple ways to combine sums less than or equal to 20. While some students exceeded their $20 budget, their addition and recording of the price of toppings remained accurate throughout both tasks. Those students whose numeric combinations exceeded $20 were also conscious that they were exceeding their allotted budget. Since they were able to justify their decisions, I did not worry that they had gone outside of the parameters of my instructions.
Tools
I used the idea of building a personalized pizza as a foundational tool to engage students in what I called ‘pizza math.’ With pizza as something familiar and typically associated with a positive meaning, I emphasized the concept of choice in allowing students to design their own personalized pizzas. To keep track of the assigned values for each topping and quantity of each topping, students used pencils and graphic organizers (as seen below) as tools. The graphic organizers contained numerals whose symbolic meaning students were required to understand in order to effectively use them as tools for addition, or making combinations whose sum was less than or equal to 20. Hiebert writes that “The experiences that are represented by the symbols provide anchors for the symbols. Students can tie the symbols to something that already has meaning for them” (1997, p. 57).
![Picture](/uploads/3/1/7/7/31776415/2303866.jpg?1418434375)
Additionally, I posted an enlarged graphic organizer on the wall (as seen to the right) for students to refer to, which contained pictures representing the different toppings along with their assigned values. In order for students to engage in their second task of creating personalized pizzas, construction paper represented pizza toppings. These tools were as essential part of my pedagogical focus because they enabled students to better understand addition with a sum up to 20 through an approachable manner. Using pizza as a metaphorical framework for making numerical combinations enabled students to expand upon their knowledge of addition and subtraction of various amounts including 5, 10, and 2-5 to sum 20 or less. This concept allowed students to individualize their decisions through the selection of different tools, and it also allowed students to change their minds while keeping track of their decisions, as they might do in the real world.
Discourse
This dimension provided an outlet for students to work out loud and to articulate their mathematical thought processes to one another. The dimension enabled them to work in pairs to create combinations of toppings and to problem solve through decision-making, both together and individually. Asking students to work in pairs while filling out the graphic organizer facilitated this dimension by encouraging student talk. They discussed their decisions for toppings and asked me to check their work. If students needed help making decisions, I would ask them, “What do you guys want on the pizza? That’s a good question to start with.”
My role within this dimension enabled me to ask questions that helped students to problem solve and to understand students’ thinking while problem solving, such as “How did you know that it was $6 for 3 servings?” I also asked questions that encouraged them to think hypothetically as well. Questions included “How many more combinations can you make if pizza dough is only $9?” and “If you spent $11, how much more do you have left over?” My role as facilitator enabled me to contribute to students’ practice of fluency by posing questions which asked them to think critically and to explain their mathematical thought processes.
Once all of the students had created personalized pizzas, I asked everyone to explain their selection process and toppings they chose. My objective was for students to listen to each other’s decision-making process and for this dimension to interact with my pedagogical focus of drawing connections between mathematical concepts. While students were able to articulate their decisions, they became a little restless as we went around the table. This particular discourse could have been more productive if I had structured it as a turn-and-talk instead of an around-the-table individualized discussion. Students would have maintained their attention better since they would have been required to lead the discussion in pairs. Even though the discussion felt long, students contributed positively to the discourse by sharing their insights and decisions for topping selection with their classmates.
Normative practices
The norm that I established at the beginning of our lesson was for students to give a thumbs up if they agreed with something another student or I said. Giving a thumbs up allows the lesson to move more fluidly and quickly, and it also positions students to be active listeners because they are processing whether or not their opinion or answer matches with that of a classmate. Raising one’s hand for asking questions and giving answers was a norm that prevailed. Although students raised their hands often, they also interrupted the discussion and each other at times. When students were talking over each other, I reminded them why it is important for us to listen to each other by reminding students that in order to listen to each other, we cannot be talking over each other.
Another norm that prevailed was students’ work in pairs. Working in pairs is a strategy that students are frequently asked to do in the classroom, specifically during math. They have experience collaborating on activities and making decisions together, and this previous experience in the classroom supported the norms necessary for a successful pair/share task. In addition to requiring students to work together to decide which toppings to use, students supported each other in making numeric combinations. The established norms for a pair/share activity allowed students to support each other’s fluency and to aide in the skill of double-checking results, which is a prominent feature as outlined in Russell's idea of accuracy.
This dimension provided an outlet for students to work out loud and to articulate their mathematical thought processes to one another. The dimension enabled them to work in pairs to create combinations of toppings and to problem solve through decision-making, both together and individually. Asking students to work in pairs while filling out the graphic organizer facilitated this dimension by encouraging student talk. They discussed their decisions for toppings and asked me to check their work. If students needed help making decisions, I would ask them, “What do you guys want on the pizza? That’s a good question to start with.”
My role within this dimension enabled me to ask questions that helped students to problem solve and to understand students’ thinking while problem solving, such as “How did you know that it was $6 for 3 servings?” I also asked questions that encouraged them to think hypothetically as well. Questions included “How many more combinations can you make if pizza dough is only $9?” and “If you spent $11, how much more do you have left over?” My role as facilitator enabled me to contribute to students’ practice of fluency by posing questions which asked them to think critically and to explain their mathematical thought processes.
Once all of the students had created personalized pizzas, I asked everyone to explain their selection process and toppings they chose. My objective was for students to listen to each other’s decision-making process and for this dimension to interact with my pedagogical focus of drawing connections between mathematical concepts. While students were able to articulate their decisions, they became a little restless as we went around the table. This particular discourse could have been more productive if I had structured it as a turn-and-talk instead of an around-the-table individualized discussion. Students would have maintained their attention better since they would have been required to lead the discussion in pairs. Even though the discussion felt long, students contributed positively to the discourse by sharing their insights and decisions for topping selection with their classmates.
Normative practices
The norm that I established at the beginning of our lesson was for students to give a thumbs up if they agreed with something another student or I said. Giving a thumbs up allows the lesson to move more fluidly and quickly, and it also positions students to be active listeners because they are processing whether or not their opinion or answer matches with that of a classmate. Raising one’s hand for asking questions and giving answers was a norm that prevailed. Although students raised their hands often, they also interrupted the discussion and each other at times. When students were talking over each other, I reminded them why it is important for us to listen to each other by reminding students that in order to listen to each other, we cannot be talking over each other.
Another norm that prevailed was students’ work in pairs. Working in pairs is a strategy that students are frequently asked to do in the classroom, specifically during math. They have experience collaborating on activities and making decisions together, and this previous experience in the classroom supported the norms necessary for a successful pair/share task. In addition to requiring students to work together to decide which toppings to use, students supported each other in making numeric combinations. The established norms for a pair/share activity allowed students to support each other’s fluency and to aide in the skill of double-checking results, which is a prominent feature as outlined in Russell's idea of accuracy.