observations in maths assignment

COMPLETE MATH

George mason university, 2. making assumptions.

Make assumptions and focus the problem with information and mathematics.

• What information do we already know/need to know to make a model?
• What assumptions do we need to make to build a model?
• What are some important quantities in your situation?
• What mathematical tools could
• you use in your model?

                                            KWI CHART

        Gathering Information

If i knew______, i could figure out_______.

(For example, If I knew the cost of a school bus for a field trip,  then I could figure out the cost of the transportation.

If I knew how many students and chaperones were coming on the trip , then I could figure out the total number of school buses we would need. )

Standard for Mathematical Practices:Reason abstractly and quantitatively.

Module 4: Probability and Statistics

4.7 observational studies and experiments, learning outcomes.

• Identify the differences between observational studies and experiments

Read with a pencil in hand

Continue to read these sections with a pencil in hand. Make note of definitions, but also make note of keywords that will help you to identify an observational study versus an experiment. Work the TRY IT and EXAMPLES out by hand to gain the experience you’ll need to correctly identify these on a test.

What you’ll learn to do: Examine the methods for sampling and experimentation and how bias can affect the results

As we mentioned previously, the first thing we should do before conducting a survey is to identify the population that we want to study. In this lesson, we will show you examples of how to identify the population in a study, and determine whether or not the study actually represents the intended population. We will discuss different techniques for random sampling that are intended to ensure a population is well represented in a sample.

We will also identify the difference between an observational study and an experiment, and ways experiments can be conducted. By the end of this lesson, we hope that you will also be confident in identifying when an experiment may have been affected by confounding or the placebo effect, and the methods that are employed to avoid them.

Observing vs. Acting

So far, we have primarily discussed observational studies – studies in which conclusions would be drawn from observations of a sample or the population. In some cases these observations might be unsolicited, such as studying the percentage of cars that turn right at a red light even when there is a “no turn on red” sign. In other cases the observations are solicited, like in a survey or a poll.

In contrast, it is common to use experiments when exploring how subjects react to an outside influence. In an experiment, some kind of treatment is applied to the subjects and the results are measured and recorded.

Observational studies and experiments

• An observational study is a study based on observations or measurements
• An experiment is a study in which the effects of a treatment are measured

Here are some examples of experiments:

A pharmaceutical company tests a new medicine for treating Alzheimer’s disease by administering the drug to 50 elderly patients with recent diagnoses. The treatment here is the new drug.

A gym tests out a new weight loss program by enlisting 30 volunteers to try out the program. The treatment here is the new program.

You test a new kitchen cleaner by buying a bottle and cleaning your kitchen. The new cleaner is the treatment.

A psychology researcher explores the effect of music on temperament by measuring people’s temperament while listening to different types of music. The music is the treatment.

These examples are discussed further in the following video.

Is each scenario describing an observational study or an experiment?

a. The weights of 30 randomly selected people are measured

b. Subjects are asked to do 20 jumping jacks, and then their heart rates are measured

c. Twenty coffee drinkers and twenty tea drinkers are given a concentration test

a. Observational study

b. Experiment; the treatment is the jumping jacks

c. Experiment; the treatments are coffee and tea

This is the end of the section. Close this tab and proceed to the corresponding assignment.

• Revision and Adaptation. Provided by : Lumen Learning. License : CC BY: Attribution
• Experiments. Authored by : David Lippman. Located at : http://www.opentextbookstore.com/mathinsociety/ . Project : Math in Society. License : CC BY-SA: Attribution-ShareAlike
• lab-research-chemistry-test. Authored by : PublicDomainPictures. Located at : https://pixabay.com/en/lab-research-chemistry-test-217041/ . License : CC0: No Rights Reserved
• Experiments. Authored by : OCLPhase2's channel. Located at : https://youtu.be/HSTTKzsdHEw . License : CC BY: Attribution
• Confounding. Authored by : OCLPhase2's channel. Located at : https://youtu.be/SrCm12HZay0 . License : CC BY: Attribution
• Controlled Experiments. Authored by : OCLPhase2's channel. Located at : https://youtu.be/UkCHUeqMb5Y . License : CC BY: Attribution
• Blind Experiments. Authored by : OCLPhase2's channel. Located at : https://youtu.be/7BFZVGCxeYc . License : CC BY: Attribution
• Question ID 6736, 6728, 6914. Authored by : Lippman, David. License : CC BY: Attribution . License Terms : IMathAS Community License CC-BY + GPL

• Geometric reasoning
• Mental computation
• Using the part-whole model
• Using the measure model
• Using the division model
• Using the operator model
• Working with ratio
• Unequal areas
• Teaching equal parts
• Paper folding
• Cassowary fractions
• Wrong number
• Expanding the view
• Fraction fiddle
• Shifting wholes
• Fixing wholes
• Cut and find
• Unit and non-unit fractions
• Strategy testing
• Sense of size
• Comparing non-unit fractions
• Visualisation
• Fractions as numbers
• Fractions of collections
• Sharing as division
• Linear models
• Grids and arrays
• Same denominator
• Related denominators

Observation

• Task-based interview
• Digital assessment tools
• Choosing the fraction model
• Closed or open tasks
• Choosing the response type
• With or without context
• Cake fractions
• Comparing unit fractions
• Divide it up
• Dividing pancakes
• Dynamic fractions
• Fraction strips
• Fraction wall game
• Fraction wheel
• Grids and jumps
• Hit the apple
• Number lines
• Overlay grids
• Part and wholes
• Reach the target
• Sequencing and counting
• Acknowledgements

Home > Fractions > Assessment > Assessment approaches > Observation

Teachers constantly use informal observation to monitor student responses to classroom experiences. However, to be an effective assessment approach, observations need to be more purposeful and deliberate.

Observation situations should be planned. Some possibilities include:

• targetted student-centred learning activities, particularly where the students will be engaged in observable actions, such as modelling mathematical processes
• tasks for pairs or small groups that involve discussion and collaboration
• consolidation or application tasks that require students to apply knowledge and skills developed in previous lessons.

Observations need to be focussed. For example, the focus could be on:

• the individual's choice and use of representations (drawings and materials)
• the use of mathematical language in a small group
• individual confidence and motivation levels
• strategies for solving problems (visible evidence might need to be confirmed by an informal discussion).

Observations need to be recorded in some form. They could include:

• anecdotal records — jottings of notable achievements or concerns
• checklists of specific indicators of learning
• digital records — photos or audio recordings.

Observation example

The combination of a photo and anecdotal notes is used to record observations of a pair of students playing a fraction matching game.

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Classroom observation and mathematics education research

• Published: 14 September 2019
• Volume 24 , pages 5–31, ( 2021 )

Cite this article

• Jonathan Bostic   ORCID: orcid.org/0000-0003-2506-0491 1   na1 ,
• Kristin Lesseig 2   na1 ,
• Milan Sherman 3   na1 &
• Melissa Boston 4   na1  

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Classroom observations have become an integral part of research related to mathematics education. In this qualitative study, we describe the current state of the mathematics education field with regard to the use of classroom observation. The research question was: How is classroom observation being used to measure instructional quality in mathematics education research ? In all, 114 peer-reviewed manuscripts published between 2000 and 2015 that involved classroom observation as part of an empirical study were examined using a cross-comparative methodology. Seventy (61%) did not use a formalized classroom observation protocol (COP), 21 (18%) developed their own COP, and 23 (20%) used a previously developed COP. Of the implemented COPs, 44% have published validity evidence in a peer-reviewed journal. We perceive the great variety of research approaches for classroom observation as necessary and potentially challenging in moving mathematics education forward with respect to research on instructional contexts.

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Classroom observation frameworks for studying instructional quality: looking back and looking forward.

Anna-Katharina Praetorius & Charalambos Y. Charalambous

Theoretical and methodological challenges in measuring instructional quality in mathematics education using classroom observations

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On Classroom Observations

Alan H. Schoenfeld, Robert Floden, … Anna Zarkh

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Acknowledgements

We would like to share our sincere appreciation to Timothy Folger, Maria Nielsen, and Davis Gerber at Bowling Green State University, and Dan Chibnall at Drake University for their assistance throughout this project.

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Jonathan Bostic, Kristin Lesseig, Milan Sherman, and Melissa Boston have contributed equally to this manuscript.

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Bowling Green State University, Bowling Green, OH, USA

Jonathan Bostic

Washington State University Vancouver, Vancouver, WA, USA

Kristin Lesseig

Drake University, Des Moines, IA, USA

Milan Sherman

Duquesne University, Pittsburgh, PA, USA

Melissa Boston

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Correspondence to Jonathan Bostic .

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Bostic, J., Lesseig, K., Sherman, M. et al. Classroom observation and mathematics education research. J Math Teacher Educ 24 , 5–31 (2021). https://doi.org/10.1007/s10857-019-09445-0

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DOI : https://doi.org/10.1007/s10857-019-09445-0

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Topics: Assessments Implementation Support Early Education Student Achievement Eureka Math Squared Kindergarten

Observational Assessments in K–2: Making Student Thinking Visible

by Jada Singleton

every child is capable of greatness.

Posted in: Aha! Blog > Eureka Math Blog > Assessments Implementation Support Early Education Student Achievement Eureka Math Squared Kindergarten > Observational Assessments in K–2: Making Student Thinking Visible

Observational assessments are a great way to gather information about what students currently understand because observational assessments feel more like a conversation and less like a formal test. This encourages students to talk more about their thinking. When you observe students and ask them questions, you learn the mathematical errors they are making and misconceptions they have and why students are making those errors. You can spend less time worrying about formal assessments and more time focusing on what’s important, which is how to help each student broaden and deepen their understanding of mathematics. Let’s look at how to use Eureka Math 2 ® ’s observational assessments effectively.

Take inventory of all the assessments in the module.

For kindergarten, review the Observational Assessment Recording Sheet for the module as well as the Module Assessments, which are designed to be delivered as one-on-one interviews with your students. These can be found in the back of Teach or on the digital platform, in the Module Overview under the Resources tab. In grades 1 and 2, in addition to the Observational Assessment Recording Sheets and Module Assessments, review the Exit Tickets and Topic Tickets. You’ll find Exit Tickets and Topic Tickets embedded within each lesson in both Teach and in the online platform. Like kindergarten, Module Assessments and Observational Assessment Recording Sheets are found in the back of Teach or on the digital platform in the Module Overview under the Resources tab. Reviewing all these assessments helps you get a better sense of what is expected of students by the end of each lesson, topic, and module. Knowing what’s in each assessment helps you make decisions about which assessments to use.

Use the Observation Assessment Recording Sheet daily.

Since you are always observing and assessing students, you can note their proficiency journeys during daily lessons. Some teachers carry the observation recording sheets with them during the lesson, noting what they see in real time. Others choose to take a minute at the end of the lesson to jot down notes. In kindergarten, the observational assessments may be the primary form of assessment. In grades 1 and 2, they are an option to use instead of, or in addition to, written assessments. But how do you know what to look for in each lesson?

In Teach , the first page of each lesson lists the Achievement Descriptors associated with the lesson. These Achievement Descriptors are also on your observation sheets. For kindergarten, there’s a picture of the module recording sheet highlighting the Achievement Descriptors for the lesson. You’ll also see possible activities within each kindergarten lesson directly aligned to the descriptors. These will give you great opportunities to pause and gather some evidence. Another place to look is on the page following your Observation Assessment Recording Sheet where you’ll find a nice at-a-glance graphic showing you what you can observe throughout the module.

On the Great Minds Digital Platform, use the expand toolbox button located on the bottom left of any lesson in the teacher’s guide. The expanded menu includes a choice to show you exactly which Achievement Descriptors your students will be working toward in the lesson.

Use additional assessment tools as needed.

The Module Assessment for kindergarten is an interview assessment. While you could interview every student one-on-one, you probably don’t have that kind of time. Instead, you may choose to use the Module Assessment to gather assessment information on a student if you don’t already have enough evidence from your daily observation. And don’t feel like you must use every single item with a student. Just use the items that give you the additional assessment data you need. Daily observational assessments save you time in the long run because they limit the need for the Module Assessments.

With first and second grade students, you’ll have more opportunities for students to show their thinking by using paper and pencil. For that reason, you can be more strategic about who you observe and when. Consider prioritizing observational data for students who still struggle with reading and writing. You’ll probably also notice students who are struggling with their Exit Tickets or Topic Tickets, but you may not be quite sure why. Using a mix of observational and pencil-and-paper assessments will help you maximize the effectiveness of assessment in your classroom, making student thinking visible.

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Jada Singleton

Jada Singleton is a Eureka Math Implementation Leader. She is a former administrator, instructional math coach, and teacher from Lafayette Parish School District in Louisiana.

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More blogs on Assessments:

Achievement Descriptors and Student Understanding

Make Student Learning Visible with High-Quality Assessments

Kindergarten Assessment Made Easier

Making the Most of Eureka Math Affirm

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Using Positive Feedback in Math Classrooms

Providing math students with positive feedback can help them clarify their thinking, take risks, and apply concepts in new contexts.

If you were a “good” math student, your teacher may have filled your papers with a checkmark next to each correct answer. But handing back “perfect work” with a slew of checkmarks is a missed opportunity for math teachers. Every time we provide a student with feedback, we have a chance to send a message. We can signal what we value and the strengths we see in the student’s work, providing insight into areas for growth and further learning. In math we value clarity and logic, creative solutions, perseverance, and curiosity. Used strategically, positive feedback can reinforce these cornerstones of the discipline. Although the examples I provide in this article are geared toward middle and high school math classes, positive feedback can be used at a variety of grade levels.

Reinforcing Effective Communication

Math teachers frequently ask students to show their work. When students give us insight into their thought processes by writing enough on the page, we can provide positive and specific comments in return. Our reinforcement helps students learn what specific aspects of their work were effective at clarifying their logic.

Here are some examples:

• Your explanation here helps me see how you got from one step to the next.
• The picture you drew helped me to understand how you’re thinking about this.
• When you defined the variable before using it, I was able to follow your reasoning.
• The sentence at the end helped me to see that this is your final answer.

All of these comments acknowledge the time that students put into explaining their reasoning. Providing positive reinforcement when students effectively convey their thought processes helps them develop into effective communicators of mathematics.

Recognizing a Mathematician’s Craft and Choices

Math teachers constantly remind their students that “there is more than one way to solve a problem.” This is true—the art of problem-solving allows students to discover elegant paths to a solution. Math talks have become popular because of their emphasis on multiple problem-solving techniques. The comments that we make on student work can reinforce the value of mathematical thinking.

You might write something like these:

• I didn’t think of this strategy!
• This is a clever implementation of factoring.
• This step reminds me of the example we looked at when _____.
• I like how you adapted the idea from _____ to this problem.

These comments will help students see their technique within the space of many pathways to a solution. When we reference the strategy that they used and contextualize it, we help them connect their problem-solving process to the content.

Celebrating Growth and Perseverance

Math is hard! Math teachers must find ways for students to grapple with the content and engage in productive struggle. If we give students credit for their progress and perseverance, we can encourage them to push through challenges the next time they arise.

Here are some examples of how you might emphasize growth and praise perseverance:

• Great job catching the mistake here.
• I can tell this was a long and messy computation. By keeping your work organized and sticking to your plan, you persevered.
• I noticed that you had trouble with this skill in the last unit, but you have mastered it now! Great job sticking with it.

Students don’t always notice their own growth. When we can point it out to them, they see that their hard work and struggle is worth it.

Encouraging Reflection

When students perform at a high level, positive comments can push their thinking beyond the standard content. With feedback, we can inspire their curiosity and encourage deeper thinking.

Here are some reflective comments you could make:

• What about this problem helped you realize that you needed this particular strategy?
• How did you check your work?
• What strategies did you use to keep track of all the steps needed to solve this problem?
• How will you remember the connection that you made here?
• What do you think would happen if the problem were slightly different? How would you have to adapt your approach?

All of these questions send the message that learning doesn’t stop at a perfect test score. We should continue to ask questions, pursue our curiosity, and find new and novel ways to apply what we’ve learned.

I hope that these ideas inspire you to provide comments beyond the simple check mark. We should be putting just as much care and attention into the feedback we provide to high-achieving students as what we give to students who need more practice.

While some may think that math teachers have it easy when it comes to grading, we shouldn’t take the easy way out. Writing a few comments might add a minute or two to the time it takes to mark a paper, but I assure you that the gains you will see in student confidence, motivation, creativity, and understanding will be worth it. By taking the time to provide meaningful, detailed, specific, and positive feedback, we provide all of our students the opportunity to grow.

Note: I wrote this piece after reading Alex Shevrin Venet’s “ How to Give Positive Feedback on Student Writing ,” because I was struck by how many of the guidelines that she provides are applicable to the math classroom. I recommend taking a look if you haven’t already!

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Classroom Observation

See this video in the context of an entire classroom lesson deb negrete videos, jaguanana lathan videos, resources for leadership and change: building a school culture of mathematical thinking.

Theory of Action

The First in Math Consortium is a subset of districts in SVMI that was chosen to work deeper and more systemically in changing learning experiences for students with the ultimate goal of improving students’ access to and success in higher mathematics. The theory of action is the frame for the comprehensive approach to working in concert with all the district-employed stakeholders.  Download

Pathways of Influence

This graphic shows the stakeholders within a school system and how learning is influenced. The math department is positioned in a very influential location to address and support teacher and student learning. This graphic communicates the importance of developing a math department into a learning community. It also illustrates how others can influence the work.  Download

Tracking and Assessing Cultural Shifts Ver. 1 & Ver. 2

A major emphasis of the initiative is to develop, support, and sustain professional learning communities as a structure for improving teaching and learning. This tool assists principals in assessing how their teachers are moving from traditional independent agents to learning communities that focus on student learning. There are two versions. The first has a more comprehensive view of the school, while the second is designed to focus primarily on the work of the math department.  Download Ver. 1  +  Download Ver. 2

Resources for Developing Faculty Capacity in Mathematics

Role Group Expectations

The document describes the roles and expectations each district stakeholder must meet to function in a systemic approach to improve math instruction. Making the expectation explicit is at the heart of working cohesively in a district organization.  Download

Tasks for Role Groups

This checklist accompanies the Role Group Expectations. It lays out the duties and jobs for the stakeholders. In this manner, everyone knows what they are expected to accomplish and how their duties align with the other stakeholders – where jobs/duties overlap and where they reside in relation to other work.  Download

Planning for Site-Level Work – Continuum of Roles

This instrument assists principals in planning their strategy and work in promoting professional learning communities within their school. It helps them look at the different change agents and assess where they are on the continuum. Utilizing personnel to support change is a critical leverage point for promoting change.  Download

Resources for Mathematics Engagement and Intervention

Re-Engagement Protocol

Use the steps outlined in the file linked here to re-engage students in the mathematics of a particular lesson.  Download

Intervention Worksheet

This instrument is built on Phil Daro’s work to describe the different levels that students struggle in school, in this case specifically in math class. The chart includes columns that describe what students need and suggest some interventions. There is space for principals to add information about intervention activities at their school. It works both as a needs assessment and for planning for the future. It supports principals in analyzing their school's intervention programs. Download

Reflection-Observation

This tool helps principals and site administrators reflect on observing mathematics classroom and department meetings. The classroom tool was designed to encourage the observer to focus more on students, their thinking, and mathematics. This tool is one device to make observing a classroom more than a mere “walk-through.” It shifts the focus away from examining artifacts (bulletin boards, students' seating arrangements) or evaluating teacher moves to a focus on student thinking and learning. Similarly, the department tool is used to gauge what the math department’s discussion is focused on.  Download

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All About Teacher Observations: How to Get Them Right

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More than a decade after being recognized as the Arkansas 2007 teacher of the year, Justin Minkel still found himself flustered when his principal slipped into the back row of class. “When my principal walks in with her laptop or a clipboard and pen, I’m instantly afflicted by a crippling self-doubt I haven’t felt since junior high,” the teacher wrote in a 2018 Opinion essay. “I scan the room with the alert panic a gazelle must feel when scanning the savannah for predators.”

Five years later, his jitters over observations— and his four tips for “surviving” them —continue to hit home for classroom teachers.

Earlier this school year, when the essay was reshared on Facebook, teachers flocked to the comments to affirm that teacher observations remain a perennial concern. In a lively conversation of 280 comments , readers volunteered their own success stories of the observation process working well and commiserated over their shared frustrations.

“I personally don’t mind them,” wrote Facebook commenter Lacey Peters, “because I am a self criticizer and usually the admin is saying much more positive things about my teaching even when I think the lesson went horribly!”

“I don’t have feelings of self-doubt,” another commenter, Rebecca Salomonsson, wrote, “I have feelings of resentment that someone is in my room actively taking notes on me. What other profession does this to its professionals? My husband is an engineer. He is trusted to do his job.”

In many commenters’ impressions of being observed, the deciding variable seemed to be how much they trusted the administrators observing them. How to build a bridge of support rather than judgment between teacher and principal has long been a source of inquiry for educators writing in Edweek’s Opinion pages.

In her 2022 essay “ The Most Important Thing Principals Can Do in a Teacher Observation ,” English teacher Kelly Scott charts the lasting impact of a single moment of encouragement in her first year of teaching. That memorable observation started with just one word: “ Wow!”

“He knew that what I really needed—more than professional development, more than goal setting and professional standards—was someone to cheer me on,” she recalled of her administrator’s enthusiastic feedback during that vulnerable first observation.

Leading with enthusiasm isn’t the only advice teachers have to offer the observers coming into their classrooms. Two years ago, when teacher blogger Larry Ferlazzo asked his peers for best practices when administrators (or other teachers) observe their lessons, 19 contributors shared their own ideas. His four-part series on the topic rounded up a slew of their actionable guidance and emotional reflections:

• 18 Ways to Improve Teacher Observations
• How to Make Teacher Observations (Almost) Stress-Free
• Throw Out the Protocol for Teacher Observations. Use Common Sense Instead
• How to Create a Positive Atmosphere for Teacher Observations

It’s not just teachers with a stake in the observation process; administrators have had their say as well.

Last year, Atlanta Assistant Principal NaTasha Woodey-Wideman explained that not every professional learning effort has the same goal—but they all reflect a leader’s instructional values.

In “ How You Deliver Professional Learning Says a Lot About You ,” she urged principals to be intentional about the goals of a specific professional learning effort and then use teacher observations in service of those goals: “If the focus of a session is to provide teachers with tools for formative assessment, the lens of subsequent teacher observations should be formative assessment. After a session on building a strong classroom culture, walk-throughs should focus on culture.”

Soon after, Woodley-Wideman joined principal-turned-leadership-coach Opinion blogger Peter DeWitt for a live online discussion to consider how educator professional learning can move beyond the “sit and get” model.

In the discussion, her guidance began with a reminder that professional learning efforts should put an emphasis on the learning : “We tend to forget that teachers are also learners.”

She concluded her advice by flipping that formulation for school leaders. “Never forget you are a teacher,” she reminded building leaders. “Your classroom is that entire building.” (You can watch the full discussion on-demand for free here .)

Nearly a decade before their conversation, DeWitt was already beating the drum for principal introspection, asking readers: “ Leaders: Are Your Teacher Observations Active or Passive? ”

He cautioned against a box-checking approach to teacher observations, noting, “It is often seen as a process to get done ... instead of a process to get done right.” Principals need to structure the process less like distant evaluators and more like instructional coaches, DeWitt proposed.

That’s a call to action that has been echoed by other educators since, including in David Edelman’s “ Teacher Evaluation That Goes Beyond Check Boxes .” The most helpful post-observation feedback from his years in the classroom, he wrote in the 2016 Opinion essay, came from an informal collaboration with a fellow teacher who engaged deeply with his instructional practice rather than merely handing out a rating.

In the not-so-distant future, some of those moments of professional coaching may not just come from fellow teachers—or even fellow humans. Drawing on their work designing a natural-language-processing tool to provide teachers immediate feedback after a lesson, researchers Jing Liu, Dora Demszky, and Heather C. Hill invited readers to “imagine a world where we could harness the power of AI to provide teachers with automated, valuable feedback.”

That world shouldn’t come at the expense of interpersonal relationships in schools but rather work in service of building even stronger ones, they argued in “ AI Can Make Education More Personal (Yes, Really) ” this past summer.

Whether tech-assisted or otherwise, one thing remains true: There’s no ignoring the emotional vulnerability of the teachers being observed.

After all, to return to Justin Minkel’s appraisal of the observation process, the stakes can feel high. “It’s not just our professional competence that’s wrapped up in an observation,” he reminded readers, “but a sense of our worth as human beings.”

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