Tasks Artifacts
This page analyzes a few high-level tasks I enacted with my students during my student-teaching. Under each artifact I briefly summarize each task. Then at the bottom of the page I provide a joint analysis for both artifacts.
1. Staircase Task (taken from mathematician Fawn Nyguen at http://fawnnguyen.com/staircase-steepness/)
1. Staircase Task (taken from mathematician Fawn Nyguen at http://fawnnguyen.com/staircase-steepness/)
Art. 5: Staircase Task
This is a task I enacted with my classes in which students were to explore and gain a deep understanding of slope as steepness. I used this task with my students in a lesson early in a unit on linear relationships. Rather than simply tell my students that slope could be thought of as the "steepness" and could be calculated by "rise over run", I wanted to give them an opportunity to mathematically discover this for themselves, because as cited earlier, "students have better retention of procedures when they have first built a strong conceptual understanding" (National Council of Teachers of Mathematics, 42).
Within the TAG, I would categorize this task as Doing Mathematics because I would argue that it met the following criteria:
In #1-5 students examined a set of 6 staircases, and made predictions on which staircase is the steepest, before ever getting a formal definition of slope. After this exploratory phase, students had opportunities to measure the different components of the staircase (which we universally defined as a class after #5). Once all the staircases were measured, students needed to use these measurements and the correct order of staircases in #5, in conjunction with their reasoning skills to propose a formula for slope.
A theme that comes up in the work that this task required students to do was making predictions/conjectures. Research suggests that "over time, the cumulative effect of the use of mathematics tasks is students' implicit development of ideas about the nature of mathematics - about whether mathematics is something that they personally can make sense or and how long and how hard they should have to work to solve any mathematical task" (National Council of Teachers of Mathematics, 24). In other words, the way students engage with math tasks affects their ideas of what mathematics is about. With this in mind, since mathematicians regularly make predictions and conjectures in their work, I wanted to give students opportunities to experience this for themselves in the task. This is an example of how this Tasks component is connected to the Norms/Expectations component. Students seemed to have an easier time with #1 because they could simply guess. #d required reasoning and critical thinking skills to propose a formula to calculate slope, which gave students trouble because it is rare that we ask them to come up with a formula themselves. Often, especially in the GRR model, students are just given the formulas to use.
This is a task I enacted with my classes in which students were to explore and gain a deep understanding of slope as steepness. I used this task with my students in a lesson early in a unit on linear relationships. Rather than simply tell my students that slope could be thought of as the "steepness" and could be calculated by "rise over run", I wanted to give them an opportunity to mathematically discover this for themselves, because as cited earlier, "students have better retention of procedures when they have first built a strong conceptual understanding" (National Council of Teachers of Mathematics, 42).
Within the TAG, I would categorize this task as Doing Mathematics because I would argue that it met the following criteria:
- Requires complex and non-algorithmic thinking: There is no strategy or procedure that is suggested for students to use.
- Requires students to explore and understand the nature of mathematical concepts and relationships: Students must study the relationship between the height and the length of the different staircases to be able to determine that the ratio for steepness is rise over run.
- Requires students to analyze the task and actively examine task constraints: Since in #5 I revealed the order of staircases from least steep to steep, this presented a constraint for students as they tested different relationships between the components of the staircases. Their calculations for their proposed formula had to give them answers that matched the order that I revealed.
- Requires considerable cognitive effort: Being that there was no specified method or procedure to follow, this required a great deal of thinking on the students' parts to consider the relationship between the rise and the run of the staircase.
In #1-5 students examined a set of 6 staircases, and made predictions on which staircase is the steepest, before ever getting a formal definition of slope. After this exploratory phase, students had opportunities to measure the different components of the staircase (which we universally defined as a class after #5). Once all the staircases were measured, students needed to use these measurements and the correct order of staircases in #5, in conjunction with their reasoning skills to propose a formula for slope.
A theme that comes up in the work that this task required students to do was making predictions/conjectures. Research suggests that "over time, the cumulative effect of the use of mathematics tasks is students' implicit development of ideas about the nature of mathematics - about whether mathematics is something that they personally can make sense or and how long and how hard they should have to work to solve any mathematical task" (National Council of Teachers of Mathematics, 24). In other words, the way students engage with math tasks affects their ideas of what mathematics is about. With this in mind, since mathematicians regularly make predictions and conjectures in their work, I wanted to give students opportunities to experience this for themselves in the task. This is an example of how this Tasks component is connected to the Norms/Expectations component. Students seemed to have an easier time with #1 because they could simply guess. #d required reasoning and critical thinking skills to propose a formula to calculate slope, which gave students trouble because it is rare that we ask them to come up with a formula themselves. Often, especially in the GRR model, students are just given the formulas to use.
2. Amusement Park Task(Adapted from State Fair Activity from Corwin Inc.)
Art. 6 Amusement Park Task
I used the following task for a lesson on initial value and writing the equation of a line from a graph. I chose this task because I wanted to provide students with an opportunity to understand what an initial value meant in context. Originally, the context of this task was a state fair, but I altered it to an amusement park because I felt more of my students would be able to relate to an amusement park rather than a state fair. I noticed a large amount of my students seemed to become engaged with this task as we talked about some of our favorite amusement park rides when I was introducing the task. Although it helped engage more of my students, amusement parks might not have been relevant to all students, which has made me wonder how I could've adapted the context so relevance was not an issue for any of my students.
While a high number of my students were able to get started on the task, many of them got stuck on question #1 where they were asked to find the price of 4 ride tickets. Many students fell into the trap of thinking it would cost $34, since by inspection of the graph, there is a point at (1, $8.50). When asking students that used this strategy if their answer made sense when considering the other data points, it often led to frustration or disengagement on the end of the student. They seemed to lack understanding of an admission fee. Questions I asked to help these students get back on track were: "What does the y-axis represent? What are some different things you could spend money on at an amusement park? "How much have you spent when you've bought zero ride tickets? Why is it not $0 at zero tickets? I chose to ask questions such as these because at no point do I give away information to the student in these questions. The questions require students to do their own thinking and come up with an answer.
A specific reason I picked this task was because I thought it would be a good cognitive challenge for my students. Since the time when I enacted this task, I am wondering if the cognitive challenge of this task lies in the mathematics, or if this task presents a challenge to students because of a contextual misunderstanding. According to Hiebert, "What is problematic about the task should be the mathematics rather than other aspects of the situation." I worry that this task could be problematic for students who don't understand how an admission fee works, which according to Hiebert, could prevent this task from being considered high-quality.
What might be surprising in Art. 6 is that none of the directions or questions specifically ask the students to use a particular method. Had something like that been included this task could possibly be categorized on the TAG as Procedures without Connections. Instead, I would categorize this as a Procedures WITH Connections task because it meets the following criteria:
I used the following task for a lesson on initial value and writing the equation of a line from a graph. I chose this task because I wanted to provide students with an opportunity to understand what an initial value meant in context. Originally, the context of this task was a state fair, but I altered it to an amusement park because I felt more of my students would be able to relate to an amusement park rather than a state fair. I noticed a large amount of my students seemed to become engaged with this task as we talked about some of our favorite amusement park rides when I was introducing the task. Although it helped engage more of my students, amusement parks might not have been relevant to all students, which has made me wonder how I could've adapted the context so relevance was not an issue for any of my students.
While a high number of my students were able to get started on the task, many of them got stuck on question #1 where they were asked to find the price of 4 ride tickets. Many students fell into the trap of thinking it would cost $34, since by inspection of the graph, there is a point at (1, $8.50). When asking students that used this strategy if their answer made sense when considering the other data points, it often led to frustration or disengagement on the end of the student. They seemed to lack understanding of an admission fee. Questions I asked to help these students get back on track were: "What does the y-axis represent? What are some different things you could spend money on at an amusement park? "How much have you spent when you've bought zero ride tickets? Why is it not $0 at zero tickets? I chose to ask questions such as these because at no point do I give away information to the student in these questions. The questions require students to do their own thinking and come up with an answer.
A specific reason I picked this task was because I thought it would be a good cognitive challenge for my students. Since the time when I enacted this task, I am wondering if the cognitive challenge of this task lies in the mathematics, or if this task presents a challenge to students because of a contextual misunderstanding. According to Hiebert, "What is problematic about the task should be the mathematics rather than other aspects of the situation." I worry that this task could be problematic for students who don't understand how an admission fee works, which according to Hiebert, could prevent this task from being considered high-quality.
What might be surprising in Art. 6 is that none of the directions or questions specifically ask the students to use a particular method. Had something like that been included this task could possibly be categorized on the TAG as Procedures without Connections. Instead, I would categorize this as a Procedures WITH Connections task because it meets the following criteria:
- Suggest pathways to follow that are broad general procedures that have close connections to underlying conceptual ideas: While there is no narrow formula that students are required to use, as mentioned above, the graph and table on the opening page give students an idea of two strategies that could be helpful to solve this problem.
- Usually are represented in multiple ways: Students could make a table, use the graph, write a description in words or create an equation to help make sense of this situation.
- Requires some degree of cognitive effort: The main challenge for students in this problem is figuring out that there is an admission fee of $8. Once students can pass this hurdle, the rest of the task is relatively less cognitively demanding.
Joint Analysis
One thing these high-level tasks exposed me to were student reactions to tasks that require considerable cognitive effort. For students that were used to procedural, low cognitively demanding tasks in the GRR model, these tasks proved to be a great challenge. Student reactions differed. (1) Some students embraced the challenge, revising their thinking and trying other strategies multiple times. (2) Students who attempted the task and revised their thinking only one time before disengaging from the task (3) Students who sat waiting for me to guide them through the problem (4) Students who completely disengaged from the task - head down, going to the bathroom, playing with supplies/manipulatives like rulers and pencils. Using a growth mindset lens to analyze these behaviors, one might conclude that students with behavior (1) have a growth mindset, while behaviors (2)-(4) represent students without growth mindset. These behaviors are nothing new. Research from the NCTM, warns that although productive struggle is necessary, "some students will simply shut down in the face of frustration, proclaim 'I don't know', and give up" (National Council of Teachers of Mathematics, 50). While I don't strive for my students to be frustrated, and I would like all my students engaged on the task, I would not compensate this for less opportunities for productive struggle. According to Principles to Actions, "[effective mathematics] instruction embraces a view of students' struggles as opportunities for delving more deeply into understanding the mathematical structure of problems and relationship among mathematical ideas, instead of simply seeking correct solutions. Teaching that embraces and uses productive struggle leads to long-term benefits, with students more able to apply their learning to new problem situations" (48).
While a growth mindset lens is one way to analyze student reactions, there could be other possible reasons for these reactions beyond purely having a growth mindset or not. Applying Gorski's Equity Literacy lens, I ask myself:
"Were there barriers that prevented students access to this task?" Perhaps students weren't sure how to use the ruler or students with lower reading levels couldn't make out what the directions read. Words like 'measurement', 'prediction', and 'steepness' could be problematic for students who's reading levels are low. Using Gorski's lens I also ask: "Were students basic needs met?", "Did they have healthy food in their stomachs?", "Could students have been tired because they had to wake up early due to lack of close proximity to efficient public transit which meant it took them over an hour to get to school?" Other questions I consider:
One thing these high-level tasks exposed me to were student reactions to tasks that require considerable cognitive effort. For students that were used to procedural, low cognitively demanding tasks in the GRR model, these tasks proved to be a great challenge. Student reactions differed. (1) Some students embraced the challenge, revising their thinking and trying other strategies multiple times. (2) Students who attempted the task and revised their thinking only one time before disengaging from the task (3) Students who sat waiting for me to guide them through the problem (4) Students who completely disengaged from the task - head down, going to the bathroom, playing with supplies/manipulatives like rulers and pencils. Using a growth mindset lens to analyze these behaviors, one might conclude that students with behavior (1) have a growth mindset, while behaviors (2)-(4) represent students without growth mindset. These behaviors are nothing new. Research from the NCTM, warns that although productive struggle is necessary, "some students will simply shut down in the face of frustration, proclaim 'I don't know', and give up" (National Council of Teachers of Mathematics, 50). While I don't strive for my students to be frustrated, and I would like all my students engaged on the task, I would not compensate this for less opportunities for productive struggle. According to Principles to Actions, "[effective mathematics] instruction embraces a view of students' struggles as opportunities for delving more deeply into understanding the mathematical structure of problems and relationship among mathematical ideas, instead of simply seeking correct solutions. Teaching that embraces and uses productive struggle leads to long-term benefits, with students more able to apply their learning to new problem situations" (48).
While a growth mindset lens is one way to analyze student reactions, there could be other possible reasons for these reactions beyond purely having a growth mindset or not. Applying Gorski's Equity Literacy lens, I ask myself:
"Were there barriers that prevented students access to this task?" Perhaps students weren't sure how to use the ruler or students with lower reading levels couldn't make out what the directions read. Words like 'measurement', 'prediction', and 'steepness' could be problematic for students who's reading levels are low. Using Gorski's lens I also ask: "Were students basic needs met?", "Did they have healthy food in their stomachs?", "Could students have been tired because they had to wake up early due to lack of close proximity to efficient public transit which meant it took them over an hour to get to school?" Other questions I consider:
- Did students have some sort of bad past experiences with math?
- How did my identity as a white, middle-class male play in to my black and brown student's engagement in the class? Did my black and brown students, who didn't engage, have negative past experiences with white teachers?
- Did the student experience something traumatic? Was there something that took place that day, in or out of math class that triggered a past traumatic experience?
- How do I positively redirect a student who disengaged from a task because of the cognitive challenge the task presented?
- How do my identities come into play with the identities of my students and affect their behaviors?
- What trauma-informed practices I could implement in the classroom to provide all students with a safe learning environment?
- What scaffolds I could provide students in the future that help them engage with the task without lowing the level of cognitive demand required?