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We should begin by helping teachers reflect on their current practices, identify areas for growth, and incorporate strategies that make math more relevant and engaging for all students.

A recommendation is to start with teacher reflection and goal-setting using tools like Analyzing Teaching and Learning (ATL).

Implementation would involve facilitating discussions where teachers assess their current practices, identify challenges, and set specific goals for incorporating the "New 3Rs" and ambitious teaching strategies. Real classroom scenarios and student data would guide the conversation.

This builds awareness and ownership, ensuring that changes are meaningful and targeted. Starting here, teachers can make intentional shifts that lead to more engaging, equitable, and effective math instruction.

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6th Grade: Students will be able to construct, interpret and solve one-step equations.

7th Grade: Students will be able to identify and interpret slopes given a graph.

8th Grade: Students will be able to interpret, construct, and solve equations in context.

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Assessing questions focus on what students understand. Advancing questions focus on taking students understanding and progressing them to the next level.

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I looked at the Calling Plans Task. I liked how Mrs. Brovey identified 3 desired outcomes for the lesson. Her identified outcomes were conceptual in nature, because students had to understand the meanings are behind the points on a line.

After reviewing the task, I read about assessing and advancing questions. The primary difference is that assessing questions focus on student understanding and advancing questions focus on using what students understand to progress them towards the goal of the lesson.

Mrs. Brovey engaged in both types of questioning. She asked, "Can you tell me what you're doing?" which was great! Sometimes a student does not approach a problem the way we would approach the problem and the teacher redirects them to his/her method. Asking the student what he/she is doing allows the student to explain their thought process.

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The text states, "Mathematics instruction that focuses solely on remembering and applying procedures favors students who are strong in memorization skills and may disadvantage students who are not. Moreover, because new procedures frequently rest on previously learned procedures, students who have struggled with mathematics and may not have strong procedural fluency at the start of instruction are more likely to be marginalized when instruction focuses solely on building new procedures." On the other hand, when instruction focuses on more of a conceptual approach it allows more solution paths for learners.

In summary, the text notes, "...the use of multiple entry points and representations for conceptual tasks makes it much more likely that each and every student will be able to start productively on a task and ultimately work his or her way toward procedural fluency."

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I would do tasks A and C prior to moving on to tasks B and D. Tasks A and C focus on the conceptual knowledge where students understand how we develop the formula for area of a triangle. Once students understand the "why" for finding the area of a triangle then they can move into the procedural tasks (B and D).

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On pages 60-62 we are given possible follow up tasks to Ms. Polosky's lesson. As I read this it was clear that you need to start with conceptual understanding and lead to procedural fluency. Again, if students simply enter a formula into Desmos they are not understanding the "why." Not understanding the "why" leads to struggles when students have to actually apply the knowledge in real world scenarios.

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Procedural fluency is knowing how to calculate. For example, procedural fluency would be memorizing multiplication facts. This should only be done after students have a conceptual understanding of what multiplication is.

This week I have been working with a teacher on teaching the pythagorean theorem. She showed images where 3 interconnected squares created a triangle in the middle. This allowed students to discover the pythagorean theorem because they could see a2 + b2 = c2. Once they had this conceptual understanding of how and why the formula works they are then able to practice procedural fluency with applying the formula. They now understand when and why to apply that formula.

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Conceptual understanding is when students understand the why behind the procedures. For example, 3 x 3 is the same as 3 groups of 3. Conceptual understanding builds number sense.

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Complete your feedback forms for the Ambassador Cycles. Click the link below to access the form.

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Complete your feedback forms for the Ambassador Cycles. Click the link below to access the form.

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Click the link below. Review the Chat section in the center of the NotebookLM. Review the notes in the bottom right column, including the study guide. Listen to the podcast audio overview. Then, reflect on what we have read and discussed so far, what are your thoughts about where we should begin when supporting our teachers with IMPLEMENTING these effective teaching practices? Provide your recommendation for where to start. Then share how and why you would start there.

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Click the link below. Review the Chat section in the center of the NotebookLM. Review the notes in the bottom right column, including the study guide. Listen to the podcast audio overview. Then, reflect on what we have read and discussed so far, what are your thoughts about where we should begin when supporting our teachers with IMPLEMENTING these effective teaching practices? Provide your recommendation for where to start. Then share how and why you would start there.

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To follow up on Ms. Polosky's initial lesson, I would use Task A and Task C. Task A provides students with the opportunity to make connections. Students are able to discover that the base of a triangle is arbitrary through connecting their right-triangle area formula with the area of the triangle. Task C has students to construct various triangles (right and non-right) from an initial right triangle. Task B and D lacks scaffolding support and doesn't directly support student's thinking of right triangles. However, students should be fluent in the concepts in Tasks B and D due to the conceptual understanding gained in the Tasks A and C. Conceptual understanding should come first as it supports procedural fluency, and encourages mathematical thinking. According to NCTM 2000, 2009; NGA Center and CCSO 2010, "asking students develop the initial conjecture, justify that conjecture, and then broaden it to all triangles supports the development of students' abilities to reason and prove, consistent wit curricular recommendations."

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Assessing questions determines what students are able to understand based on standards proficiency level rather advancing questions allows students to think more critically in "what if" situations at the distinguished level.

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Conceptual Understanding is not an equity issue but rather an opportunity if teachers are willing to listen and ensure ALL students are being heard. Ensuring students have a voice regardless if they are unable to speak the language or have a deficit in Mathematics, ensuring their voices are being expressed in multiple representations. Questioning is key when posed purposefully in a way that ALL students can understand and can reply.

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According to Polosky's Task Sequence, Task A allows students to explore a more critical way of why orientation of triangles does not change the area versus Tasks B and D which require students to apply a formula in order to find the area. Task C however is a spin off of Task A which allows students to derive the formula for several types of triangles. I would start with Tasks A and C in that order and move to Tasks B and D in that order to build procedural fluency.

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The next time a baby places the shapes into the slots they remember some of what they learned if not all how to place the pieces until it becomes procedural fluency.

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Conceptual Understanding is like when a baby takes a baby blocks shape sorter and explores how each shape fits into the slots with no assistance for the first time.

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Assessing questions elicit student thinking, check for understanding, focuses on what the student currently knows or understands, and clarifies, verifies, or confirms student responses. Advancing questions extend student thinking, deepens understanding, focuses on how the student can move forward in their reasoning, and encourages exploration, connections, and higher-order thinking. Both assessing questions and advancing questions are used in formative assessment during instruction, promote student engagement and reasoning, align with effective teaching practices to support student learning, and require teachers to listen carefully to students’ responses and respond accordingly. Assessing questions help determine where students are in their thinking and advancing questions help push their thinking to the next level. However, when both types of questioning are used strategically, they promote conceptual understanding, procedural fluency, and problem-solving skills.

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Conceptual understanding is an equity issue because all students deserve access to meaningful mathematical learning. Mathematical instruction that focuses only on procedural fluency favors students that have excellent memorization skills. Students who struggle with memorization fall further behind. According to Smith, Steele, & Raith, “When students learn mathematical procedures without a sound conceptual underpinning, they often misapply the procedures or forget them.” Pg.75. Conceptual teaching encourages discourse, justification, and critical thinking. This approach fosters equitable participation by valuing different ways of thinking, not just speed or accuracy.

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Procedural fluency is to the ability to apply procedures accurately and efficiently. It involves not only memorizing algorithms but also understanding when and how to use them appropriately in different contexts. It also allows students to solve problems effectively without relying on rote memorization, in which students can focus on higher-level reasoning and problem-solving. Procedural fluency is important because students are able to solve problems more quickly, adapt their approach to different problems and choose the most efficient strategy, builds confidence, and supports deeper understanding.

Conceptual understanding allows students to understand the "why" behind mathematical procedures, instead of just memorizing steps. This deeper understanding has a direct influence on procedural fluency, because students can adapt, modify, and apply procedures to new and complex problems. Studies showing that students who first develop conceptual understanding perform better in applying procedures. The National Council of Teachers of Mathematics (NCTM) emphasizes the balance between procedural fluency and conceptual understanding, as both are necessary for mathematical proficiency. Although, conceptual understanding must come first because it provides students with a deep foundation for reasoning, problem-solving, and long-term retention. When students understand the "why" behind mathematical concepts, they can apply their knowledge flexibly to new and complex problems instead of just memorizing steps without understanding. In addition, conceptual understanding must come first because it promotes meaningful learning, builds problem-solving skills, reduces reliance on memorization, supports procedural fluency, and encourages mathematical thinking.

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Conceptual understanding refers to a deep understanding of mathematical ideas, where one not only knows how to apply formulas and procedures but also understands why they work and how different concepts interconnect. It moves beyond rote memorization and procedural fluency to more emphasis on reasoning, problem-solving, and the ability to transfer knowledge to new situations.

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Mr. Donnelly's teacher moves that helped focus learning toward the goal were to ask questions that made the students think critically about the process(es) they were using to solve. He had previously given them strategies to help them solve so he asked them to refer back to those strategies as a way to gather their thoughts or organize their work. Mr Donnelly also looked at the approaches that students were using and found student groups that he wanted to ask to present their work because they were using the strategies that he was targeting.

The "student talk" that represented progress toward the goal was students breaking down their thinking in order to explain "why" they did what they did. Mr. Donnelly asked the students to compare the thinking of two different groups and see how they could both be right and this allowed the students to "see" the process being done two different ways, but arriving at the same result.

Mr Donnelly then closed out the lesson by using the different approaches in order to attach the strategy that was used to solve it and allowed the students to explain if that process was effective in achieving the intended goal.

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Patrick Donnelly employed an interactive teaching approach by actively engaging with students as they worked in groups.

He circulated around the room, listening to discussions, asking guiding questions, and encouraging students to revisit prior work and compare solutions. He identified effective strategies to showcase later, aligning student work with his lesson goal: understanding proportional relationships through scaling, scale factors, and unit rates.

Unlike some teachers who remain at their desks, Donnelly facilitated the learning process by observing and providing feedback. He emphasized high-level questioning and structured group discussions. To reinforce his objective, he selected cognitively challenging tasks with multiple entry points. He guided students in presenting and analyzing strategies, encouraging them to evaluate efficiency and connections among approaches.

Donnelly concluded the lesson by showcasing a group's solution, prompting students to assess its viability and justify their reasoning. His approach highlighted the value of strategic task selection, active facilitation, and fostering critical thinking through collaborative dialogue.

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1. Patrick Donnelly walked around the room, stopping at different groups to listen in on their conversations and to ask questions as needed. When students struggled to figure out what to do he encouraged them to look at the work from the previous day. He also encouraged students to make comparisons of the jars. He noted the strategies that students were using so that he could decide which groups he wanted to present their work.

2. Patrick Donnelly's teaching is similar to what some teachers do at our school. Some facilitate and circulate through their room as groups complete tasks. They ask questions and provide feedback where needed. They listen to academic discourse and see which students are on task. Some differences are that some teachers sit at their desks and the groups are working with no facilitator both on and off task.

3. I would want to see teachers observing and notating effective strategies. I want to see teachers using open-ended high-level questioning.

4. Mr. Donnelly's goal was for students to understand that quantities in a proportional relationship grow at a constant rate and that students can use three key strategies to solve problems of this type: scaling up, a scale factor, and a unit rate.

5. One teacher move that helped the focus on the goal was selecting a task that aligned with his goal, was cognitively challenging, and had multiple entry points. For student talk and work, he chose three groups that presented the strategies he aligned with his goal. During the discussion, he asked presenters to explain what their group did and why, and he invited other students to consider whether the approach made sense and to ask questions. Another teacher move that helped the focus on the goal was he labeled each of the three strategies, asking students which strategy was most efficient in solving this particular task and asking them questions that helped them make connections between the different strategies and the key ideas he was targeting. Another teacher move was at the end of the lesson, the teacher placed the solution that group 1 produced on the document camera and asked students to decide whether the approach was a viable one for solving the task and to justify their answer.

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My ambassador's goal is to keep her "hands in her pockets." She walks around helping her students but gives too much information instead of allowing her students to struggle productively. She writes notes to lead her students. Therefore, "hands in her pocket" is her metaphor for not giving answers, asking questions, and allowing the students to discover learning.

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Grade 6:

Students will solve a realistic problem by evaluating an expression when given the value of a variable.

Grade 7:

Students will determine whether two quantities presented in authentic problems are in a proportional relationship.

Grade 8:

Students will recognize that a solution to a system corresponds to the points of intersection and the values of that point satisfy two equations simultaneously.

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Always have a goal and be intentional about completing it. Plan with the goal in mind. This will guide you to better learning.

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I noticed that Ms. Culver had key features for her lesson from the start to the finish. She identified what the students should learn. She was intentional in how she planned and executed her activities for effective learning. Ms. Culver had a goal, made tools, and was strategic in sequencing the presentations, so they made sense to the learning. These are checklist points that teachers should use.

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Ms. Culver walked around, listened, asked questions of her students, and encouraged them to think at a higher level. She was making them connect and expand.

During observation, I have seen three different scenarios. Some teachers don't circulate to engage with students, some circulate but answer the questions students ask and explain any misunderstandings, and others circulate, ask questions, and engage, as Ms. Culver has done in her class.

I want the class environment to be mostly project-based. I want our students to be able to apply their knowledge to demonstrate understanding.

I would like our teachers to find their passion again for education.

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2. What does Patrick Donnelly do during the lesson to support his students' engagement in and learning of mathematics?

Student discussion, peer sharing, content connections, student reflection

What aspects of Patrick Donnelly's teaching are similar to or different from what you see during observations at your school?

Mr. Donnelly looked at the unit as a whole and saw the relationships among concepts rather than teaching the concepts in isolation.

What practices would you want to to se incorporated into the teaching practices of the teachers you support? I would like for them to look at the big teacher and create plans that allow students to make connections between content rather than teaching in isolation.

3. What were Mr. Donnelly's goals for the lesson that the case featured? He "wanted his students to understand that quantities that are in a proportional (multiplicative) relationship grow at a constant rate and that students can use three key strategies to solve problems of this type: scaling up, scale factor and a unit rate"

Identify the teacher moves that help focus learning toward the goal and the student talk and work that represents progress toward the goal.

Mr. Donnelly had students build on prior knowledge, he also had students share their thinking (the 3 different strategies), he labeled the strategies, etc.

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Patrick Donnelly demonstrated effective teaching by engaging with student groups, listening to their conversations, and asking questions to guide them. He encouraged students to refer to prior work and compare solutions to enhance their understanding. By observing strategies used, he selected groups to present their approaches. His goal was for students to understand that quantities in a proportional relationship grow at a constant rate and to apply strategies like scaling up, scale factor, and unit rate. Key strategies included selecting challenging tasks with multiple entry points, facilitating group presentations, labeling strategies, and prompting analysis of solutions.

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1. Patrick Donnelly walked around the room, stopping at different groups to listen in on their conversations and to ask questions as needed. When students struggled to figure out what to do he encouraged them to look at the work from the previous day. He also encouraged students to make comparisons of the jars. He noted the strategies that students were using so that he could decide which groups he wanted to present their work.

2. Patrick Donnelly's teaching is similar to what some teachers do at our school. Some facilitate and circulate through their room as groups complete tasks. They ask questions and provide feedback where needed. They listen to academic discourse and see which students are on task. Some differences are that some teachers sit at their desks and the groups are working with no facilitator both on and off task.

3. I would want to see teachers observing and notating effective strategies. I want to see teachers using open-ended high-level questioning.

4. Mr. Donnelly's goal was for students to understand that quantities in a proportional relationship grow at a constant rate and that students can use three key strategies to solve problems of this type: scaling up, a scale factor, and a unit rate.

5. One teacher move that helped the focus on the goal was selecting a task that aligned with his goal, was cognitively challenging, and had multiple entry points. For student talk and work, he chose three groups that presented the strategies he aligned with his goal. During the discussion, he asked presenters to explain what their group did and why, and he invited other students to consider whether the approach made sense and to ask questions. Another teacher move that helped the focus on the goal was he labeled each of the three strategies, asking students which strategy was most efficient in solving this particular task and asking them questions that helped them make connections between the different strategies and the key ideas he was targeting. Another teacher move was at the end of the lesson, the teacher placed the solution that group 1 produced on the document camera and asked students to decide whether the approach was a viable one for solving the task and to justify their answer.

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The text provides a few key messages. What is one key message you are taking from the reading?

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Share one of your Ambassador's goal statements. Provide constructive feedback for the goal statement of another coach's Ambassador. Include considerations for the type of mathematical work and thinking the goal expects of students.

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Reflect on Mr. Donnelly's goal and lesson. Share a goal statement of your own, or use Mr. Donnelly's goal. What type of mathematical work and thinking does the goal expect from students? What would need to be done before, during, and after the lesson to prepare students to engage in reasoning, thinking, and sense-making?

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Refer to Mr. Donnelly's lesson and goals. Identify the "teacher moves" that help focus learning toward the goal. Identify the "student talk" and work that represents progress toward the goal. Explain your response.

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Reflect on Ms. Culliver's or Mr. Donnelly's teaching practices. What effective teaching practices do you observe most often? What practices would you want to see incorporated into the practices of the teachers you support? Why? What support will teachers need to incorporate these practices more consistently?

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We currently use the gradual release model. However, the text notes that this model leads to practicing procedures with limited connection and meaning. Students also have few opportunities to reason and solve problems. The text notes, "Although they (students) may learn the procedure as intended, they often do not understand why it works and apply the procedure in situations in which it is not appropriate." If the research says the gradual release model is flawed for mathematics instruction, what should we use?

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I couldn't get the podcast audio to work, but I looked through all of the documents.

There is a lot to consider when selecting where to start with supporting teachers. Ultimately, I decided I would start with goal setting. Goal setting lays the foundation for your next steps. Goal setting also allows the teacher develop a clear understanding of what is expected of the teacher and student. If we start with goal setting we can then move into conceptual v procedural understanding to ensure the goals include conceptual understanding. From there, I would go into mathematical modeling. Modeling allows students to make inferences, estimates, predictions, and conclusions that inform action which are the thought processes we went to engage our scholars in.

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Compare and contrast assessing and advancing questions. How are these used to move mathematical thinking forward and understand student thinking?

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Questioning - Analyzing Teaching and Learning

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Questioning - Analyzing Teaching and Learning

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Answer as much as possible based on the scenario you read: What did you notice about the questions that the teacher asked? What did the teacher learn about the students? How did the teacher foster a discussion? Was there a difference in the types of questions asked? If so, what was the difference?

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How is conceptual understanding an equity issue?

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In what sequence might you use some or all of the tasks to follow up on Ms. Polosky's initial lesson? How does her sequencing support students with building procedural fluency?

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How does the sequence of activities help students develop a conceptual understanding of the volume formulas for rectangular prisms and cylinders? What next steps or follow-up tasks might support students in developing procedural fluency with the volume formulas for rectangular prisms and cylinders?

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What is procedural fluency and why is it important? Explain how conceptual understanding promotes procedural fluency. Why MUST conceptual understanding come first?

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In your own words share your interpretation of conceptual understanding in mathematics.