Teaching Practices

Welcome to Coaches Corner

Coaches Corner with Amy Belik mathodology

Meet Amy Belik-Math Instructional Coach FXW School- Chicago

Coaching Vision

I coach to empower both teachers and students to see math as an open, visual subject that is full of connections. I coach teachers and leaders to teach in a way that empowers learning to take an active role in making sense of math and take ownership of their own learning. Through self-reflection and collaboration, we can all improve in our craft. I coach people to find their own power and to empower others so that we can transform FXW, our community, and our world. Over the next few months, we will follow Amy as she sets goals with teachers and works to support each teacher to meet their goals. What goals do you have for 2020?

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Measurement & Math

As we work through our measurement chapter, students are learning to be more precise by watching out for gaps and not overlapping. These are skills that will sharpen with time and exposure. Allowing students to work together to tackle how they will measure and what they will measure with, also provides invaluable skills.

When students are given the opportunity to work together on tasks, they are forced to use mathematical language and reasoning skills to explain their thinking with their partner/group.

In the pictures below students were placed in groups to measure different paths. They had to agree as a group which tool they would use (paper clips, tiles, straws, string or paint stirrers). As they traveled from path to path, their measurements became more precise because they collaborated and discussed their strategies for measuring. The lesson to be learned here is that with time, collaboration and the teacher stepping to the side, students can figure out everything we want them to. Know what the key ideas allow the learning to happen!

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Building Visualization in Students

2019 Lesson Study Take-Aways

mathodology ban har sarah schaefer

Class Discussions / Structured Exploration

In our study on visualization, I learned that as a teacher I should have a less is more approach. Specifically, teachers should focus on the ideas presented in discussions and not just the answers. We should allow the students to provide details in problem-solving strategies and methods. To promote more visualization, teachers can slow down, ask questions to prompt thinking and allow students the time to discover the answer. Teachers should ask the big questions and provide structure for exploration. Students need time for verbalization of ideas, but without structure, the time is not used effectively.

Physical Movement

Carpet time allowed children to move about the room and partner with someone not in their discussion group. Additionally, moving to carpet or floor helped eliminate distractions. Students were then able to reflect on the pages of the textbook, discussing the multiple methods and (if relevant) could then make connections between the textbook and methods discussed in class. Students were able to better visualize the textbook concepts than at desks with base tens and other distractions.

Questions To Prompt Visualization

○ “Can you imagine?”
○ “Is it possible?”
○ “Is there a different way?”
○ “I wonder…

One is my biggest take a way’s from this lesson study was the power of visualization when teaching and exploring mathematical concepts. What does it mean to visual something? How formidable visualizing is in creating and solving problems. Visualization is a powerful cognitive tool in problem-solving and enhancing this process in children is paramount in helping them solve and create.  I enjoyed collaborating with the teachers to give the children better tools to visualize numbers concepts. 

Our lesson study with Ban Har and Sarah was as valuable as any professional development I’ve ever attended. The ability to plan, create, and revise a lesson with such an experienced group, was beneficial in all subject areas. The small group setting allowed the collaboration to be thorough and specific to the needs of our students.

The in-depth discussions leading up to the lesson planning were worthwhile and relevant. Being able to have different grade levels involved in the planning provided useful information. It was eye-opening for all of us to talk through the skill sets taught in the grade level (2nd) before the intended lesson(3rd) and the grade level after (4th). Being able to see this progression was important. 

I found the lesson study to be a more productive form of professional development. Watching the teaching of the lesson gave me such insight into ways to improve my own teaching. Observing the students as they worked through the lesson, showed us ways we could improve for the next session. 

The discussions (from our lesson study group) following the first lesson were helpful to dissect everything from student strategies used, journal entries, ways to improve the lesson. Everyone brought unique ideas to the lesson and the collaborative sessions highlighted important information. 

Being part of the lesson study is a commitment for the members and requires support from administration (allowing for substitutes/meetings etc. . .); however, it was such a positive experience that provided me valuable insight. The goal is to teach best practices to the students in all subject areas . . . the lesson study project certainly helped me be a better teacher.

I think the lesson study was a great way to examine in depth how to best teach a lesson using strategies to help the children dig deeper and not only solve the problem, but more importantly be able to share and put into words what they learned to see if the concept was truly mastered or if the teachers needed to go a little further. I really liked how the teacher input was very limited in the actual teaching of the lesson. It was by using manipulatives and talking and discussing with their peers and with very strategic open-ended questioning by the teacher that helped the children continue to dig deeper and truly understand the concept.

As we prepared for the 3rd-grade math lesson study, it was noticeably beneficial for the teachers involved to become very familiar with the related previous lessons, which were building blocks for the focus lesson. Just as beneficial was researching the lessons that would follow the target lesson. This helped us more fully understand the future skills and goals for the students. As the lesson was prepared, we also strived for a clear presentation of the lesson, allowance during the lesson for perseverance to explore with concrete materials, multiple methods for problem-solving, and time for students to explore efficient strategies for problem-solving. Along with the goal of the academic unit skill mastery, we strived to promote student discussions leading to a deeper understanding of the skill and visualization. 

After the first lesson was observed and we had ample time to share our observations, we planned strategies we believed would help to improve the lesson. These strategies included offering the students a more detailed explanation of the problem to be solved, encouraging verbalization and visualization before manipulatives were provided, and allowing more time for exploration with manipulatives.

After the lesson, we reviewed the students’ journal entries showing specific problems to be solved. This observation step provided even more opportunities for teacher reflection and validation of the success of the lesson.


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What role do we as teachers play in setting an example of the growth mindset?

As educators, it’s very likely that we have encountered the idea of “growth mindset” for our students. Over 30 years ago researcher Carol Dweck began exploring the link between student’s attitudes toward academic achievement and the comparison between a “fixed mindset” and a “growth mindset”. The research was revolutionary, finding that when students believe that they can get smarter, they understand that effort makes them stronger. This then leads to students putting in extra time and effort, and that leads to higher achievement.

Since this research, many schools, school districts, and state Departments of Education have made the idea of “growth mindset” a cornerstone of academic achievement, rightfully so. This leads to the question, as teachers, can we learn from our students? How can we create a classroom environment conducive to consistent self-assessment, but also apply that idea to who we are professionally, fulfilling the best of who we want to be in our classrooms each and every day?

What is hiding in the grade book?

It might help to first have an honest conversation…we’ve all been there, end of the grading period, grades aren’t exactly up to par so we give a “quiz” or last-minute “open book” review for no other reason than to appease our inner teacher conscience or to hit a “grade quota” set by the district. It’s nothing to be ashamed of, it’s normal, human, but are we being fair to ourselves and our students? While this example may seem small and very relatable is it aligned to our own “teacher growth mindset”?

As educators, one freedom that most of us can still relish in is our ability to deliver standards, curriculum, and content in the way and environment that we chose to create. With so many challenges facing teachers, the ability to bring our own “growth mindset” to our classrooms might be our most valuable tool. The confidence to find your own teaching voice, incorporate techniques that work best for your class, and even on the “hard days” to preserve and trust that the standards you have set, will lead to amazing academic growth and classroom culture that reflects your confidence and delivery of instruction.

With Great Freedom, Comes Great Responsibility

Questions that we should ask ourselves and be willing to answer honestly might start here; In delivering content, what are my strengths and weaknesses? What would I like to change or improve? Most importantly, are you creating a learning environment that is “student-focused” and if you’re not, how can you change that? Another element to focus on could be assessments. Do you have purposeful assessments? What do you learn from their outcomes? Do they help guide your instruction? If they don’t help you and your students, what purpose do they serve?

Unchartered Waters Can Be Scary

Venturing into the “growth mindset” of teaching can be uncomfortable for adults, but the positive outcomes can be tremendous. Revel in the knowledge that all your hard work is purposeful and goal-oriented, that you are not only expecting 110% from your students, but you are leading by example, which will not go unnoticed by the kids in your classroom.

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How Do We Build Metacognitive Thinking Through Self-Assessment

We can all agree that metacognitive thinking is obviously beneficial. The environment in which a student feels comfortable enough to be a self assessor can be tricky. What does that look like? What is the goal?

Getting away from the time-honored question, “Is this the right answer?!”

As former students, this is probably the question that motivated most adults as young learners. The response of, “Yes, you are correct!” or “No, you need to work harder.” always seemed to hold a finality of the lesson.

That is where we all decided that we were “great at math” or that we would be lifelong math strugglers. We all have those memories.

So how do we as teachers change that narrative? How do we facilitate students in becoming their own teachers? How do we give them a voice and the confidence to speak to their own capabilities and shortcomings and how is that beneficial?

Learning Goals and Scales

Reflecting back to John Hattie’s book, “Visual Learning”, there are questions that guide a student through the self-assessing process.

Relating to the content each student should ask themselves:

  • Where am I going?
  • How am I going to get there
  • Where to next?

Keeping those guiding questions in mind each student is given a Learning Goals and Scales chart for each chapter studied.

The goals and scales are based on a combination of Common Core Standards and State of Florida Item Descriptors.

Seen within the items are leveled abilities. This is important to guide the learners in their self-evaluation. It is also very helpful for the students to keep the chart throughout the chapter.

This is not a summative assessment or checklist to use at the end of the chapter. The students should have access to help guide their thinking and goals. The goals and scales are written in “kid-friendly” language as “I can” statements.

This is done to make ownership of the goals easily relatable to elementary-age students. In reading each statement the student is then able to designate the color that best fits their comfort level for that item.

As it relates to their feelings about a topic:

  • Red indicates that the student feels that they are not confident in this skill. They feel that they require more work in this area.
  • Yellow indicates that the student feels somewhat comfortable with the statement. They can be effective when leading in whole group scenarios, or with partner support, but may still have questions.
  • Green indicates that the student feels very comfortable with this skill. Not only could they complete the tasks at hand, they could also explain them to a classmate.

Support through Discussion

Learning Goals and Scales and Think!Mathematics US

The students’ responses to their Scales and Goals can be seen below. Those are then used to guide the students through self-assessment by applying the goals to different examples that have been either created in their Math Journal or through concrete examples that the students can demonstrate and explain.

Examples below show the students reflecting on their goals for their first chapter in their mathematics classroom this year.

Students Reflecting on Their Goals and Think! Mathematics US

The students were allowed to use their goals and scales to keep track of their thoughts and their own accountability and then relate those directly to examples.

These discussions are truly the cornerstone of metacognition for the students. Being told that their thinking is more valued than a simple score is the first step in supporting a student in seeing themselves as not just as a spectator in their education but as their own teacher.

Initial Teacher Takeaways

Having a specific routine for this method of accountability and reflection is crucial. There must be a procedure put in place that helps the student guide themselves independently. If these steps are not taken students become distracted in the implementation of the task and the quality of the actual reflection is lost. 

Providing the students with the Scales and Goals (I can statements) is the first step, but the teacher contribution in creating those goals must be instructionally sound and purposeful.

Something as simple as incorporating the math topic in the journal headings allows students to organize their thoughts for their accountability discussions with their teacher.

For example, if the first three “I Can” statements relate to place value and the student feels that they can prove that they are comfortable with that topic, looking back in their journal headings that address Place Value will give them a place to reflect and prove their thinking.

It should also be said that in some of the discussions and pictures seen below you can see that through the interactions the students actually change their ability levels from their initial thinking. That might be the most profound takeaway with these strategies early in the school year.

Accountability and Reflection and Think!Mathematics US

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Assessments: Setting Routines for Self Reflection

Setting Routines for Self Reflection

MJ Kinard

Assessments and Mathodology

Precision teaching is the idea that relates to the book, Visible Learning by John Hattie. The premise behind Hattie’s research is the correlation between surface, deep, and transfer learning.

The findings go on to address the influence that different aspects of the education system have on our students and the math classroom. Basically, the idea of getting the “most bang for your buck”.

Every school district, school, and classroom teacher wants a magic formula for success. With limited time and funding, how do we ensure our academic goals are being met in the math classroom?

As math teachers, how do we know if our students are learning?

Setting Routines

Setting routines are one of the cornerstones of a successful classroom. It is important for teachers to understand the difference in a compliant classroom and an environment conducive to learning.

When beginning the school year it is crucial to ask ourselves a few questions when considering routines and procedures.

  • Do your students know what they are supposed to be learning? And more importantly why?
  • Are your students benefitting from your expertise?
  • Are your students capable of connecting their learning to what they’ve already learned and are they able to see where this learning goal will lead?
  • Are your students able to manage their own learning?
  • At the end of the lesson, are your students able to hold themselves accountable for what they have learned? (leading into self-assessment)
Pictures of students during our “admiration tour”. You can see them and the work they have produced.


Self-assessment has one of the largest effect sizes for student learning (Hattie, 2009). The process of self-assessment requires several different elements implemented in the classroom.

One example of self-assessments in the math classroom is the use of Math Journals. Beyond a journal prompt containing the “content goal” teachers have an amazing opportunity to go beyond a right/wrong answer by providing students with a voice for self-reflection.

Below is an example of a pre, mid, and post year self-assessment used in a fourth-grade classroom to gauge confidence and attitude toward mathematics.

This same questioning is also used in this classroom in the aforementioned Math Journal activity. The example seen below addressed the math topic of Place Value. The students were asked to describe the value of the 8 in the number 88,888.

They could use pictures and words. The prompt was actually shared at the beginning of class as an informal discussion. Students were asked to read the prompt and think about what they already knew about place value, what they remember from previous lessons that would help them with today’s goal, if they felt like the topic was going to be difficult and if so, why?

They were given time to think about their responses and write them in their journal, then after the lesson, they returned to their journal to work on the prompt and reflect on their thoughts from the beginning of class. Once journals were complete, they were allowed to share the journals with their group.

The outcome of this part of the activity is to foster a safe space for math discourse, more importantly, mathematical talk is proven to lead to the ultimate level of transfer of learning and metacognitive awareness.

Self Evaluation/Assessment Form


If self-assessment is the goal and math journaling is one tool, what is the payoff? How do these techniques help spark success in our math students? Metacognition is the ability to think about our thinking (almost sounds too simple).

While it may sound simple it is the cornerstone of all actual and true learning. Metacognition happens when students practice self-reflection on their level of thinking. In addition, when students can relate this to a target it becomes powerful enough to increase understanding and motivation.

This knowledge promoted in the classroom is invaluable. As teachers, we all dream of that day when our students are intrinsically motivated. A moment in the classroom when the value of the content rises above a state standard or an arbitrary notion but becomes a self-fulfilling desire to learn and grow. But students need guidance and tools to develop their metacognitive awareness and become confident in the ability to self-question.

Formative Evaluation

Formative evaluation is the process of gathering evidence to inform instruction. In simpler terms, it is the process for teachers and students to communicate about the learning that is taking place and the direction that instruction should go. Formative evaluation should drive instruction decisions, gathering real-time data is crucial in guiding how the teacher will proceed with the delivery of the lesson.

In the book, Visual Learning, Hattie speaks of several internal questions that drive learners:

  • Where am I going? What are my goals?
  • How am I going there? What progress is being made towards my goal?
  • Where to next? What activities need to be undertaken next to make better progress?

Consequently, these are the same three questions that teachers must ask about instruction as they make adjustments based on the data they gather from students. Some examples shown below, include response cards (whiteboards), and exit tickets, journal entries may also serve as a way to inform instruction.

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We are Community!

Why Should We Work So Hard to Build Community?

Class Math Agreements and Community and Mathodology

Building community beginning on the first days helps to create new relationships and strong bonds that will last throughout the year. Creating a shared vision of the expectations, developing a common understanding of classroom limits, and fostering a love of learning are only a few of the characteristics you might have in mind as desired outcomes. Ultimately, achieving mutual respect and a spirit of collaboration creates an ideal working environment for the classroom.

When community exists, each child feels valued. A sense of shared purpose unites the group and working together to accomplish goals becomes a priority. Our goals are BIG and require the effort of all of our members. The uniqueness that each student provides as a member of the community must be valued and each individual strength will make the community stronger and better. As children develop a sense of duty to the community, self-discipline is likely to emerge more naturally and from the child’s (intrinsic) motivation rather than from external or reward-based methods (extrinsic).

Early in the year, creating purpose in the child’s movement and activity is desired and we balance the freedoms offered within the environment, the needs of the young child to move, and the constraints of the environment. Providing structures and routines will help to create order as well as ensure a safe environment for your children. A strong sense of community is one of the most effective ways to teach how to use individual freedoms.

How do we build community?

We play games and have fun together. We share lunch and work with each other, mixing-up our groups with an emphasis on getting to know new friends. We interview and find out more about each other by sharing experiences, stories, traditions, and the accomplishments we are proud to have achieved. We make time to appreciate each other and learn how to recognize others, as well as ourselves. 

In our community, we learn to problem solve, developing the skills necessary to take care of ourselves and others. When solutions are found and conflicts resolved with little or no direction or intervention by an adult, students feel great pride! Creating a class agreed-upon list of rights and responsibilities with the students allows them to partner in holding others accountable and enforcing your shared vision of community.

Grace and courtesy work also play a role in learning how to act in a community. A firm handshake and smile in the morning set a respectful tone for the day. Allowing students to have the role of a “class greeter” is a great way to have students serve in a leadership role as they create personal and inviting welcomes to the community. Practicing how to greet visitors with a cup of tea and a special chair or preparing a class snack are other ways students can assume responsibility. Modeling ways to ask for help, challenge other student’s ideas and even how to say “no thank you” respectfully are tools your students will need to have in order to work effectively in their community.

A natural extension of building community within our classrooms is to reach outward. The work that starts within our classroom might find opportunities in other areas within the school. Participating in the work of the larger community helps the students feel proud and invested.  Students experience, on a small scale but in a real way, that they can create change. We can act individually or as a group – and we DO make a difference! 

Specific Ideas to try at the beginning of the year might include:

  • Toss a ball in a group to help learn names
  • Learn a favorite food of a new friend
  • Create a scavenger hunt in the room to learn a new environment
  • Share with a friend something you like about yourself
  • Work together to line up without talking
  • Offer lessons on classroom jobs
  • Provide lessons and model grace and courtesy
  • Make a list of “Classroom Rights and Responsibilities” WITH your students and have them initial or sign
  • Have a procedure or place in the classroom for resolving conflicts –create a “Peace Table” or “Peace Corner”

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Report Cards

In emphasizing the core competencies in mathematics, schools have added indicators in giving feedback to their parents.  Here are questions I typically receive.

We are trying to determine how to modify the indicators on our report card.  Have you seen any schools use the indicators/categories from the Singapore Core competencies?  Any insight?

Below is a sample of what one school developed.


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Is it possible?

Is it possible?

Participants were asked at the 2017 Fall Mathodology Institute if it was possible to use the digits 0-6 only once to make a true statement?

Is it possible?  Participants worked and worked and did not find a way to solve the problem with the given conditions.  The teacher allowed the task to move on without closure for a purpose.  When students work to become confident problem solvers we want them stating, “this is not possible, I have exhausted every situation.” Listen to their reasoning, and look for a systematic way to prove it is not possible. All too often we focus on finding the solution, and we can deepen the process and allow for practice when we alter the task.

Try giving a task where it is not possible and see how your students handle it.  Follow-up by asking, “if it is not possible, can we change one of the symbols to make it a true statement?”

When we do not provide immediate closure, we allow students to continue to explore.  Many participants emailed after the institute because they would not quit until they figured it out.  Below are a few of the pictures they sent.

Great perseverance by participants who refused to give up?

This is what we want from students…next question for Shelly…can I change any symbol to make it a true statement?  If so is there only one way?


Ok…..I’ve worked literally for hours and I’ve come to the conclusion that this problem is NOT possible. If the lesson was to teach us to let kids struggle so they work harder like a ???? I get it. I’ve done at least 100 fraction problems over the weekend. If you would’ve given us the answer I would’ve never thought about this problem again……but you’ve let us struggle and now it’s just painful. Please tell me it’s not possible ???? my sanity is at stake.

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Grade 1 Task

Grade 1 Task- Not enough information

At 2017 Mathodology Fall Institute participants worked on a task from think!Mathematics which required them to use the given shapes to make a composite figure. Figure C posed a problem for many.  Is it possible?  Participants worked and worked and did not find a way to solve the problem with the given shapes.  The teacher allowed the task to move on without closure for a purpose.  When students work to become confident problem solvers we want them stating, “Figure C is not possible with the given shapes.  I would need an additional… to complete this shape.”  We have built perseverance and another way to assess them.  Do they know what to ask?  “I need two more pieces of one shape.”

When we do not provide immediate closure, we allow students to continue to explore.  Many participants emailed after the institute because they would not quit until they figured it out.  Below are a few of the pictures they sent.

Try giving a task where there is not enough information and see how your students handle it.  Will they be confident to state this is not possible, and know what to ask?

Great perseverance by participants who refused to give up?

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CPA Number Bonds

CPA Progression

Can you see the CPA progression?  Modeling with actual items in making the lesson come to life first.  Then move to pictorial representation with pictures of the objects.  Last student model with cubes and record in a number bond.

Modeling with actual items in making the lesson come to life first.  Then move to pictorial representation with pictures of the objects.  Last student model with cubes and record in a number bond.

Then move to pictorial representation with pictures of the objects.  Last student model with cubes and record in a number bond.

Lastly, student model with cubes and record in a number bond.


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