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.

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.

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.

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.

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.

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.

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."

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.

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.

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.

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.