Learning Goal Artifacts
1. Learning Goals for Lesson on Triangle Sum Theorem
Art. 3: Triangle Sum Lesson Plan with Learning Goal
This first artifact is a lesson plan that I created for a lesson on the Triangle Sum Theorem. Highlighted in yellow in section (B) at the top of the plan is the learning goal I wrote for this lesson. I could have very easily just taught a lesson on this topic centered on a performance goal and simply told my students that the interior angles of a triangle add up to 180 degrees, but learning goals support my vision of You-do We-do I-do. According to Smith and Stein in order "to ensure that a discussion will be productive, teachers need to set clear goals for what they want students to learn from the lesson, and they must select a task that has the potential to help students achieve those goals."
What stands out in this lesson plan is that the learning goal I created starts with "Students will understand...". I use this verbiage to reinforce the idea that my goal for the lesson was for students to learn why the Triangle Sum Theorem made sense in order to apply it, rather having my sole aim to be for students to do or perform something with the Triangle Sum Theorem. Literature from Principles to Actions supports this by saying: "To use mathematics effectively, students must be able to do much more than carry out mathematical procedures... Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results." What might seem surprising is that I also include two SWBAT goals in my lesson. I did this to the request of my host teacher and administrators. Their reasoning behind SWBAT goals is that they are more immediately measurable than a learning goal.
The learning goal I created shows the understanding about the Triangle Sum Theorem that I was hoping for students to leave the lesson with. What this artifact fails to show is how I evaluate whether students met this goal. I chose to focus this learning goal on understanding because "conceptual understanding establishes the foundation and is necessary for developing procedural fluency" (National Council of Teachers of Mathematics, 2). With this learning goal in mind I chose a Desmos task which allowed students to explore relationships among angle relationships, and understand the math behind why the interior angles of any triangle add up to 180 degrees. Since I had a clearly defined learning goal in mind, as I monitored student progress on the task, I was able to focus my attention on asking students probing questions like "How do you know that makes sense" and "What is the mathematical relationship between those angles?". By the end of the lesson, many students were able to explain in their own words why the Triangle Sum Theorem made sense. The disadvantage of devoting so much time in a lesson to making sure students had deep understanding was that there was little time for students to practice applying this theorem to problems. What I found, though, was that students quickly caught on to the application problems after having this deep understanding for why the theorem made sense. This is supported by Principles to Actions as it's stated that "when procedures are connected with the underlying concepts, students have better retention of the procedures and are more able to apply them in new situations."
This first artifact is a lesson plan that I created for a lesson on the Triangle Sum Theorem. Highlighted in yellow in section (B) at the top of the plan is the learning goal I wrote for this lesson. I could have very easily just taught a lesson on this topic centered on a performance goal and simply told my students that the interior angles of a triangle add up to 180 degrees, but learning goals support my vision of You-do We-do I-do. According to Smith and Stein in order "to ensure that a discussion will be productive, teachers need to set clear goals for what they want students to learn from the lesson, and they must select a task that has the potential to help students achieve those goals."
What stands out in this lesson plan is that the learning goal I created starts with "Students will understand...". I use this verbiage to reinforce the idea that my goal for the lesson was for students to learn why the Triangle Sum Theorem made sense in order to apply it, rather having my sole aim to be for students to do or perform something with the Triangle Sum Theorem. Literature from Principles to Actions supports this by saying: "To use mathematics effectively, students must be able to do much more than carry out mathematical procedures... Mechanical execution of procedures without understanding their mathematical basis often leads to bizarre results." What might seem surprising is that I also include two SWBAT goals in my lesson. I did this to the request of my host teacher and administrators. Their reasoning behind SWBAT goals is that they are more immediately measurable than a learning goal.
The learning goal I created shows the understanding about the Triangle Sum Theorem that I was hoping for students to leave the lesson with. What this artifact fails to show is how I evaluate whether students met this goal. I chose to focus this learning goal on understanding because "conceptual understanding establishes the foundation and is necessary for developing procedural fluency" (National Council of Teachers of Mathematics, 2). With this learning goal in mind I chose a Desmos task which allowed students to explore relationships among angle relationships, and understand the math behind why the interior angles of any triangle add up to 180 degrees. Since I had a clearly defined learning goal in mind, as I monitored student progress on the task, I was able to focus my attention on asking students probing questions like "How do you know that makes sense" and "What is the mathematical relationship between those angles?". By the end of the lesson, many students were able to explain in their own words why the Triangle Sum Theorem made sense. The disadvantage of devoting so much time in a lesson to making sure students had deep understanding was that there was little time for students to practice applying this theorem to problems. What I found, though, was that students quickly caught on to the application problems after having this deep understanding for why the theorem made sense. This is supported by Principles to Actions as it's stated that "when procedures are connected with the underlying concepts, students have better retention of the procedures and are more able to apply them in new situations."
2. Learning Goals Visual Pattern Activity
Art. 4a: Learning goals for activity highlighted in yellow (See full activity plan here)
This is an artifact related to a learning goal I created for students. Art. 4a shows the learning goals I created for this activity. The curriculum I was using required me to teach a more direct lesson in order to provide students with plenty of opportunities to practice a skill. As a result, in order to be able to do a richer task that allowed students to make sense of multiple representations, I only had time to squeeze it into a Do Now activity. Student voice was compensated.
What Art. 4a shows about the learning is that I place an emphasis on students being able to flexibly use multiple representations to represent linear functions. This helps build deeper understanding because "when students learn to represent, discuss, and make connections among mathematical ideas in multiple forms, they demonstrate deeper mathematical understanding and enhanced problem-solving abilities" (National Council of Teachers of Mathematics, 24). Because of the time crunch I ended up doing more of the talking than the students during this task, and as a result, it was hard for me to know whether all students met the learning goal for this task. What this tells me is that a task like this, with a multi-part learning goal, can not be engaged with in a meaningful way in only 10 minutes. In the future I would dedicate a larger chunk of class time to a task such as this one.
What I discovered as I monitored student progress during this task was that students felt much more comfortable making a table than writing an equation. As a math educator this makes sense for students who are still in the early stages of the development of their algebraic thinking. A table is a much more concrete representation of a linear relationship than an equation with variables which leads me to a thought about how I might sequence topics on linear relationships in the future. My thought for future teaching is that I will introduce students to tables earlier, and then build on this representation to make clear more abstract representations such as a graph or equation.
What Art. 4a shows about the learning is that I place an emphasis on students being able to flexibly use multiple representations to represent linear functions. This helps build deeper understanding because "when students learn to represent, discuss, and make connections among mathematical ideas in multiple forms, they demonstrate deeper mathematical understanding and enhanced problem-solving abilities" (National Council of Teachers of Mathematics, 24). Because of the time crunch I ended up doing more of the talking than the students during this task, and as a result, it was hard for me to know whether all students met the learning goal for this task. What this tells me is that a task like this, with a multi-part learning goal, can not be engaged with in a meaningful way in only 10 minutes. In the future I would dedicate a larger chunk of class time to a task such as this one.
What I discovered as I monitored student progress during this task was that students felt much more comfortable making a table than writing an equation. As a math educator this makes sense for students who are still in the early stages of the development of their algebraic thinking. A table is a much more concrete representation of a linear relationship than an equation with variables which leads me to a thought about how I might sequence topics on linear relationships in the future. My thought for future teaching is that I will introduce students to tables earlier, and then build on this representation to make clear more abstract representations such as a graph or equation.
Analyzing these two artifacts together, allows me to see clearer connections to the Learning Goals component of my vision. Art. 3 included both a learning goal and performance goals, while Art. 4a contained only a learning goal. Based on how much easier of a time I had assessing student understanding and mastery in Art. 3's lesson, I'm getting the idea that a balance of both types of goals can help provide the best outcomes for student learning. Something that I am curious to investigate deeper is the balance between both types of goals in math. Although "the practice of establishing clear goals that indicate what mathematics students are learning provides the starting point and foundation for intentional and effective teaching" (Smith and Stein, 17), I have come to understand how tricky it can be at times to measure which students have met the learning goal when teaching with limited time. I would consider performance goals more measurable than learning goals, so I am also curious to learn more concrete strategies for tracking student learning of learning goals.