Math Leadership Coaching Interview, Part 2
By Shan Jiang
CHAPTER 3 Content Knowledge and Worthwhile Tasks
Developing Fluency
How does the information on worthwhile tasks mesh to what you already know?
What is a take away for you on the development of fluency?
The information about worthwhile tasks connects well with what I already know about promoting higher-level thinking in the classroom. Worthwhile tasks involve high-level thinking that engages students in meaningful problem solving. When designing or selecting these tasks, teachers need to consider what types of problems will truly engage students and encourage deep thinking. As mentioned in Chapter 3, “procedures with connections” tasks and “doing mathematics” tasks are examples of high-level tasks that promote cognitive demand and support deeper mathematical understanding. These types of tasks help students make meaningful connections and think critically about mathematical ideas.
A key takeaway for me about the development of fluency is that procedural fluency involves four important components: accuracy, efficiency, appropriate strategy selection, and flexibility. Among these, I think efficiency is especially important for teachers to consider when planning instruction. Although there are many strategies that can be used to teach a concept, it is essential for teachers to choose the strategies that best fit the needs of their current group of students. Selecting the most appropriate strategy not only helps students learn more effectively but also makes the lesson more meaningful and efficient.
Discuss the Coaching Considerations on p. 50-51. What resonates with you?
What really resonates with me are points C and D on page 50. Point C highlights that students naturally see the connection between concepts and procedures, and this reminds me how important it is for teachers to help make those connections visible. When solving math problems, I find that using visual representations, such as diagrams, can bridge the gap between abstract ideas and concrete understanding. For example, when students draw a diagram to represent a problem, they begin to see how numbers relate to real situations, which helps them grasp the bigger mathematical concepts more naturally.
Point D also stands out to me because it emphasizes the value of open-ended prompts as higher-level questions. When teachers ask questions like “What do you notice?” or “What do you wonder?”, students are invited to think more deeply and engage creatively with the content. I’ve noticed that these kinds of questions not only deepen students’ understanding but also encourage them to take ownership of their learning and explore new ideas based on what they already know. It reminds me that meaningful learning happens when students are given the freedom to think, question, and discover.
Discuss the Coaching Questions
We have discussed questions 1,3,5,7 and 9 on page 55-56. In Mrs. Young's class, she always tries to spiral content standards into lessons so that students are constantly reminded of what they’ve learned. For instance, if the students are working on character change over time, she may include a theme standard along with it so that students aren’t just taught a standard once, but have opportunities to revisit and apply it repeatedly in different contexts.
Discuss the Coaching Questions
She always models what she wants students to do so they can see what good thinking and writing look like. They talk about what it looks and sounds like, and then a student models it for the class.
Discuss the Coaching Questions
During the lesson, she gives students time to talk, share ideas, and find evidence in the text. At the end, they go back to the “I can” or big question statement that they discussed at the beginning.
During the lesson, they always build background knowledge by discussing vocabulary and making connections through hand movements, synonyms, definitions, and using the words in sentences. We also talk about what we already know about a topic. If a task is too easy, she challenges students with higher-order thinking questions; if it is too difficult, she breaks it into smaller, more manageable steps.
3.5 Analyzing level of cognitive demand
Describe your overall evaluation of whether this task/lesson has the potential to engage students in high-level thinking.
This lesson has strong potential to engage students in high-level thinking. In Mrs. Morgan’s class, students were not only practicing subtraction but also exploring the relationships between numbers and discussing why those relationships worked. The tasks encouraged students to notice patterns, such as how the answers changed when the subtracted numbers increased, and to explain their reasoning. By using manipulatives like counters and discussing multiple strategies, students were able to connect concrete models with abstract ideas. Mrs. Morgan’s open-ended questions and guided discussions supported deeper understanding and analysis rather than simple recall. Overall, the lesson effectively promoted high-level thinking through exploration, reasoning, and reflection.
What adaptation can you make to the task or lesson to increase its higher-level thinking potential?
I think Mrs. Morgan’s class already included high-level thinking tasks because the students were highly engaged in the lesson. However, to further increase the higher-level thinking potential, I would suggest allowing students to create their own problems based on what they learned. In the current lesson, all the tasks came from the lesson plan or textbook, but giving students the opportunity to design their own subtraction problems would help them think more deeply. This would not only show that they understand how to solve problems but also demonstrate their ability to apply and extend their learning. Creating their own problems encourages creativity and critical thinking, which aligns with the “create” level in Bloom’s Taxonomy—the highest level of cognitive thinking.
Thank you!
Math Leadership Coaching Interview, Part 2
shan jiang
Created on November 4, 2025
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Transcript
Math Leadership Coaching Interview, Part 2
By Shan Jiang
CHAPTER 3 Content Knowledge and Worthwhile Tasks
Developing Fluency
How does the information on worthwhile tasks mesh to what you already know?
What is a take away for you on the development of fluency?
The information about worthwhile tasks connects well with what I already know about promoting higher-level thinking in the classroom. Worthwhile tasks involve high-level thinking that engages students in meaningful problem solving. When designing or selecting these tasks, teachers need to consider what types of problems will truly engage students and encourage deep thinking. As mentioned in Chapter 3, “procedures with connections” tasks and “doing mathematics” tasks are examples of high-level tasks that promote cognitive demand and support deeper mathematical understanding. These types of tasks help students make meaningful connections and think critically about mathematical ideas.
A key takeaway for me about the development of fluency is that procedural fluency involves four important components: accuracy, efficiency, appropriate strategy selection, and flexibility. Among these, I think efficiency is especially important for teachers to consider when planning instruction. Although there are many strategies that can be used to teach a concept, it is essential for teachers to choose the strategies that best fit the needs of their current group of students. Selecting the most appropriate strategy not only helps students learn more effectively but also makes the lesson more meaningful and efficient.
Discuss the Coaching Considerations on p. 50-51. What resonates with you?
What really resonates with me are points C and D on page 50. Point C highlights that students naturally see the connection between concepts and procedures, and this reminds me how important it is for teachers to help make those connections visible. When solving math problems, I find that using visual representations, such as diagrams, can bridge the gap between abstract ideas and concrete understanding. For example, when students draw a diagram to represent a problem, they begin to see how numbers relate to real situations, which helps them grasp the bigger mathematical concepts more naturally.
Point D also stands out to me because it emphasizes the value of open-ended prompts as higher-level questions. When teachers ask questions like “What do you notice?” or “What do you wonder?”, students are invited to think more deeply and engage creatively with the content. I’ve noticed that these kinds of questions not only deepen students’ understanding but also encourage them to take ownership of their learning and explore new ideas based on what they already know. It reminds me that meaningful learning happens when students are given the freedom to think, question, and discover.
Discuss the Coaching Questions
We have discussed questions 1,3,5,7 and 9 on page 55-56. In Mrs. Young's class, she always tries to spiral content standards into lessons so that students are constantly reminded of what they’ve learned. For instance, if the students are working on character change over time, she may include a theme standard along with it so that students aren’t just taught a standard once, but have opportunities to revisit and apply it repeatedly in different contexts.
Discuss the Coaching Questions
She always models what she wants students to do so they can see what good thinking and writing look like. They talk about what it looks and sounds like, and then a student models it for the class.
Discuss the Coaching Questions
During the lesson, she gives students time to talk, share ideas, and find evidence in the text. At the end, they go back to the “I can” or big question statement that they discussed at the beginning. During the lesson, they always build background knowledge by discussing vocabulary and making connections through hand movements, synonyms, definitions, and using the words in sentences. We also talk about what we already know about a topic. If a task is too easy, she challenges students with higher-order thinking questions; if it is too difficult, she breaks it into smaller, more manageable steps.
3.5 Analyzing level of cognitive demand
Describe your overall evaluation of whether this task/lesson has the potential to engage students in high-level thinking.
This lesson has strong potential to engage students in high-level thinking. In Mrs. Morgan’s class, students were not only practicing subtraction but also exploring the relationships between numbers and discussing why those relationships worked. The tasks encouraged students to notice patterns, such as how the answers changed when the subtracted numbers increased, and to explain their reasoning. By using manipulatives like counters and discussing multiple strategies, students were able to connect concrete models with abstract ideas. Mrs. Morgan’s open-ended questions and guided discussions supported deeper understanding and analysis rather than simple recall. Overall, the lesson effectively promoted high-level thinking through exploration, reasoning, and reflection.
What adaptation can you make to the task or lesson to increase its higher-level thinking potential?
I think Mrs. Morgan’s class already included high-level thinking tasks because the students were highly engaged in the lesson. However, to further increase the higher-level thinking potential, I would suggest allowing students to create their own problems based on what they learned. In the current lesson, all the tasks came from the lesson plan or textbook, but giving students the opportunity to design their own subtraction problems would help them think more deeply. This would not only show that they understand how to solve problems but also demonstrate their ability to apply and extend their learning. Creating their own problems encourages creativity and critical thinking, which aligns with the “create” level in Bloom’s Taxonomy—the highest level of cognitive thinking.
Thank you!