First, thanks so much for sharing the video. The vulnerability that you express matches what I feel often in the classroom when I have to make decisions in the moment, especially when a student offers an idea and I want to acknowledge the student’s participation and thinking, support the student’s agency, but the idea isn’t correct. It happened to me recently in a second-grade class and I’m still smarting a bit about how I handled it.

When the boy said that each circle was equal to 1 ½ squares, it was a Huh? moment for me. After that, you asked, “Anyone have a comment about that, agree, disagree?” That’s when I stopped the video and took the time to make my own drawing and check it out. I’m not familiar with this lesson, so maybe you quickly saw that it made sense, but I had to make sense of it for myself. And I suspect that it wasn’t something that was obvious or clear to other students.

I think it might have been helpful, after asking the others if they agreed or disagreed, to give them time to talk about it in their groups. Maybe present it as a conjecture:

S thinks that each of the circles equals 1 ½ squares. Talk in your groups about whether you agree or disagree, and why.

By moving on quickly to his conclusion, “So your answer is that it won’t balance,” I’m concerned that some students were left in the dark. During many of my college math classes as an undergraduate, many times I was left in the dark and had to find/crawl my way back after class. One of the commitments I made early on as a math teacher was to try and not leave students behind. It’s hard to live up to that commitment as a teacher, I’ve learned. Giving students time to think takes time, and the pressure of classroom teaching looms. (Henry Picciotto gave an Ignite talk about the tyranny of time in the classroom. https://www.youtube.com/watch?v=QoX6lnqzXUs

A quote from one of my educational heroes: "But putting ideas in relation to each other isn't a simple job. It's confusing and this confusion does take time. All of us need time for our confusion if we are to build the breadth and depth that give significance to our knowledge." -- Eleanor Duckworth, The Having of Wonderful Ideas and Other Essays on Teaching and Learning.

Once students see his idea, the next math challenge is for them to use that information to decide that B won’t balance. That, too, won’t be obvious to all and, again, students may need time. I think it would be time well spent.

This stuff is so hard in the moment, but the chance to reflect together like this is so great! I wonder if instead of you being first to respond, a quick turn and talk about how the shared strategy was the same or different than their own would get that nice classroom buzz going? Then bring them back together and ask: 1) did anyone (or their partner) think about it similarly and/or 2) if there were any clarifying questions for the student at the board. Really appreciate the zooming into small, specific places in the lesson with a focused question! Thanks!

100% Agreed. It's so much easier to be at the back of the room watching or in this case watching a video. It is HARD to be in the moment and make these kind of decisions on the fly.

First of all, I would like to thank you for this bit-sized PD opportunity!! This would be a great strategy to use with teachers as they implement Desmos curriculum.

A good "go-to" is to frame a question for partner discussion. Using a question such as, discuss how this strategy would or would not work for B, would allow all students to participate in the lesson.

I like the suggestion to turn and talk. I don't think I would have thought of this question on my feet, but what if you asked kids to turn and talk about, what would the original hanger need to look like if hanger B *is* balanced? As in, what would they need to add to the right side of the original hanger to make it consistent with hanger B?

> I don't think I would have thought of this question on my feet, but what if you asked kids to turn and talk about, what would the original hanger need to look like if hanger B *is* balanced?

Love the question. And to your first comment there, yeah, I think 90% of my trouble here was my teaching is rusty. But there's also another 10% where it's just easier to think of the right question when your brain isn't trying to process the two billion bits of sensory info a classroom sends you every second of a class.

This is such a great video! Thanks for sharing. One of my “go to’s” is. Whoa this is interesting and different. Let’s all take a moment and try to explain to your elbow partner what “insert student name” just said. If no talk… “st. Can you break that down for us one more time”. Then give students to practice retelling to a partner. Gives students time to decide if they understood what was just shared.

Dan, I appreciate you opening up your teaching practice for a virtual #Observeme learning opportunity. This mini bite-sized PD is also providing slow PD as I have been thinking about it for a week now.

There are a lot of great comments here about good teacher moves that we can all apply to improve our teacher practice. I think it is equally important that we focus on the learners and their “right now” learning needs. Math education has come a long way from expecting students to generate the perfect right answer to now encouraging problem solving, wrong answers and iteration. However, the culture of teaching–with its rigid teacher evaluation system–still expects and rewards a perfect, neatly wrapped up 50 minute lesson. Instead of thinking about how to teach the next lesson better, I’m more interested in what teacher moves you would make tomorrow to continue this lesson and ensure student learning of this specific objective. If this were a Wordle, the student at the board would be a green square for his innovative thinking and ability to show his understanding to the class. Your attempt to generate classroom dialog around the half squares is a yellow square because you recognize the opportunity to help students think in a new way about this topic. The rest of the class would be gray squares because we really don’t know what they are thinking. What warm-up would you do with these students the next day to help them understand? How would you modify lesson plans for the next day to end up with all green squares related to this learning target? From my understanding, you are not teaching this class on a daily basis so the idea of iterating has less application in your context. However, I couldn’t pass up the opportunity to discuss the important question: “How can I help students improve?” in addition to your reflection question: “How can I improve?” As a profession, how can we adopt a problem solving approach to teaching that applies reflection on our lesson missteps (which are bound to happen regularly) and iterates constantly to reach our end goal of helping students learn the given objective?

I think I'd like to get other students on record about the board student's work. Not agree / disagree. Rather to present them with another hanger and ask them how "square splitting" tells them whether or not it balance. That might help me understand who's yellow or green rather than just gray.

Thank you for sharing. What a great way to start my day... with a snippet of PD :) We've all had crickets in the lesson! I like the idea of turn and talk to see if you agree or disagree with the strategy. Then, "Can someone else summarise S's strategy for the class?" Then perhaps you could say, "Pretend you are S. How could you approach problem C or D using this same strategy." Give the kids a chance to try that strategy on a few other problems and have another student present C or D using the same strategy.

One thought: Beyond asking whether they think it balances or not, you could ask something with more texture. Three options: does it balance, or is the LHS heavier, or is the RHS heavier? Then send them to talk with their partners, seeking agreement and explanation. The ceiling is a bit higher on that question and the issue of which side is heavier might offer access to some students who have lost the thread.

Focus your question around Error Analysis and/or a Misconception so students have something to think about instead of a a yes/no question: For example: Why might someone say B is balanced and identify the error? (Half of 3 is 1.5. Didn't take into account the square on the left side)

Oo nice. A binary question offers only two possibilities and one of them is probably the wrong one. "Why might" offers a much larger answer space and each one can be argued.

I like this. Another option in this vein would be for you (the teacher) to argue the "incorrect" answer, "i.e. I think B is balanced. If I make the circle worth two squares, then it is balanced." Doing this has a couple of positive consequences: 1. It lets students know they can argue with teachers and "prove" them wrong. 2. If a student is too shy to speak out against the perceived "smart kid," they can feel emboldened by the teacher arguing against them.

Apr 26, 2022·edited Apr 26, 2022Liked by Dan Meyer

My grandson loves math and I can see why if it is taught like this. My question would be, do you just give them one (a) balance sample.? I would give five or six to keep students engaged and so they might discover different sequences and solutions.

I was also thinking that if you had the student explain how to balance 2 more purple circles (so 4 purple circles altogether), that might alleviate some "fraction" confusion and possibly elicit more discussion from the class. Then once that was discussed, the small groups could then work together with the 2 purple circles alone. Sometimes I find when you say the word fraction, students shut down.

Apr 26, 2022·edited Apr 26, 2022Liked by Dan Meyer

I just thought you cut off the student—the creator of the move— who was about to provide an insight to your question (“Who told you that you can split up a square?” which I think was meant to be humorous and something we all do for a laugh or an exaggeration) but then missed that the student was ready to reply and provide more insight but didn’t get the chance. He might not thought of 2 to 3 ratio on the previous demos but rather as 1 to 1.5 ratio. Next time, after you let the student reply, play off his nugget by making a teacher move and ask a variation on the question, even ask how many are it as 2 to 3 abs how many see it as 1 to 1.5 (which can then lead to the idea of a unit rate).

Like the lesson progression, though, from playing with hangers to solving static riddle questions

Apr 26, 2022·edited Apr 27, 2022Liked by Dan Meyer

Did students talk about all of them before explaining their described one? I like to start with “raise your hand if you thought of it in the same way/also used splitting for yours” or “fist to 5 how much sense does it make to you to split the shape like that” and then like Scott maybe a quick- “talk to your partner about why/how he split the shapes, and if you can do that” or maybe “about how his way of doing that connects to your way of figuring it out”

I think you got stuck because the question more about is he right or wrong (ambiguous about is the method right or wrong vs his final answer that the hangar is unbalanced) vs what made him think that/why his method does or doesn’t work.

Also thinking about the goal here- is it that you suspect this was novel thinking of a crucial idea and students need more time to process it? Is it to shift a norm around who we “present” to and what kind of thinking or response the class is expected to do when people are sharing “answers” at the front of the room?

> Did students talk about all of them before explaining their described one?

Yep. One fun part of the lesson was noticing the initial disagreement on the teacher dashboard for that screen before our 12 minute conversation and then consensus afterwards.

> I like to start with “raise your hand if you thought of it in the same way/also used splitting for yours” or “fist to 5 how much sense does it make to you to split the shape like that”

Really like how this takes the temp down. Showing something with your hands alongside all of your classmates seems very different than being the only person verbally staking out a claim.

> I think you got stuck because the question more about is he right or wrong (ambiguous about is the method right or wrong vs his final answer that the hangar is unbalanced) vs what made him think that/why his method does or doesn’t work.

My standard move when I don't get any (or few) hands, is to ask students to check in with their partner. "Discuss what (the student) just did. Do you agree or disagree with his reasoning? Why?" After a minute or two discussion, when I bring it back to the whole class I usually have more buy in and can have a more robust class discussion.

I'm thinking about how to tweak the "agree/disagree" prompt to focus students' attention on the thinking (as opposed to getting the right answer.)

What it if were a regular move in class, after seeing a student's work, to ask students to turn and talk with a more specific prompt. As a pair, they have to agree to respond to one of three possibilities:

1. Agree - why do you agree?

2. Disagree - why do you disagree?

3. Ask a specific question.

Not sure if this would work, but it feels like it would give everyone good options:

* The third options gives students who are confused an out - and focuses their attention on the reasoning.

* It allows you as the teacher to sequence the next part of the convo by starting with the questions.

First, thanks so much for sharing the video. The vulnerability that you express matches what I feel often in the classroom when I have to make decisions in the moment, especially when a student offers an idea and I want to acknowledge the student’s participation and thinking, support the student’s agency, but the idea isn’t correct. It happened to me recently in a second-grade class and I’m still smarting a bit about how I handled it.

When the boy said that each circle was equal to 1 ½ squares, it was a Huh? moment for me. After that, you asked, “Anyone have a comment about that, agree, disagree?” That’s when I stopped the video and took the time to make my own drawing and check it out. I’m not familiar with this lesson, so maybe you quickly saw that it made sense, but I had to make sense of it for myself. And I suspect that it wasn’t something that was obvious or clear to other students.

I think it might have been helpful, after asking the others if they agreed or disagreed, to give them time to talk about it in their groups. Maybe present it as a conjecture:

S thinks that each of the circles equals 1 ½ squares. Talk in your groups about whether you agree or disagree, and why.

By moving on quickly to his conclusion, “So your answer is that it won’t balance,” I’m concerned that some students were left in the dark. During many of my college math classes as an undergraduate, many times I was left in the dark and had to find/crawl my way back after class. One of the commitments I made early on as a math teacher was to try and not leave students behind. It’s hard to live up to that commitment as a teacher, I’ve learned. Giving students time to think takes time, and the pressure of classroom teaching looms. (Henry Picciotto gave an Ignite talk about the tyranny of time in the classroom. https://www.youtube.com/watch?v=QoX6lnqzXUs

A quote from one of my educational heroes: "But putting ideas in relation to each other isn't a simple job. It's confusing and this confusion does take time. All of us need time for our confusion if we are to build the breadth and depth that give significance to our knowledge." -- Eleanor Duckworth, The Having of Wonderful Ideas and Other Essays on Teaching and Learning.

Once students see his idea, the next math challenge is for them to use that information to decide that B won’t balance. That, too, won’t be obvious to all and, again, students may need time. I think it would be time well spent.

This stuff is so hard in the moment, but the chance to reflect together like this is so great! I wonder if instead of you being first to respond, a quick turn and talk about how the shared strategy was the same or different than their own would get that nice classroom buzz going? Then bring them back together and ask: 1) did anyone (or their partner) think about it similarly and/or 2) if there were any clarifying questions for the student at the board. Really appreciate the zooming into small, specific places in the lesson with a focused question! Thanks!

Yeah, a turn and talk! I definitely needed to lower the risk of contributing here and some rehearsal with a neighbor would have helped.

100% Agreed. It's so much easier to be at the back of the room watching or in this case watching a video. It is HARD to be in the moment and make these kind of decisions on the fly.

I love what Marilyn Burns said recently. "It’s hard to teach and think at the same time." Felt that! https://marilynburnsmath.com/what-was-rebecca-thinking/

First of all, I would like to thank you for this bit-sized PD opportunity!! This would be a great strategy to use with teachers as they implement Desmos curriculum.

A good "go-to" is to frame a question for partner discussion. Using a question such as, discuss how this strategy would or would not work for B, would allow all students to participate in the lesson.

I like the suggestion to turn and talk. I don't think I would have thought of this question on my feet, but what if you asked kids to turn and talk about, what would the original hanger need to look like if hanger B *is* balanced? As in, what would they need to add to the right side of the original hanger to make it consistent with hanger B?

> I don't think I would have thought of this question on my feet, but what if you asked kids to turn and talk about, what would the original hanger need to look like if hanger B *is* balanced?

Love the question. And to your first comment there, yeah, I think 90% of my trouble here was my teaching is rusty. But there's also another 10% where it's just easier to think of the right question when your brain isn't trying to process the two billion bits of sensory info a classroom sends you every second of a class.

This is such a great video! Thanks for sharing. One of my “go to’s” is. Whoa this is interesting and different. Let’s all take a moment and try to explain to your elbow partner what “insert student name” just said. If no talk… “st. Can you break that down for us one more time”. Then give students to practice retelling to a partner. Gives students time to decide if they understood what was just shared.

Dan, I appreciate you opening up your teaching practice for a virtual #Observeme learning opportunity. This mini bite-sized PD is also providing slow PD as I have been thinking about it for a week now.

There are a lot of great comments here about good teacher moves that we can all apply to improve our teacher practice. I think it is equally important that we focus on the learners and their “right now” learning needs. Math education has come a long way from expecting students to generate the perfect right answer to now encouraging problem solving, wrong answers and iteration. However, the culture of teaching–with its rigid teacher evaluation system–still expects and rewards a perfect, neatly wrapped up 50 minute lesson. Instead of thinking about how to teach the next lesson better, I’m more interested in what teacher moves you would make tomorrow to continue this lesson and ensure student learning of this specific objective. If this were a Wordle, the student at the board would be a green square for his innovative thinking and ability to show his understanding to the class. Your attempt to generate classroom dialog around the half squares is a yellow square because you recognize the opportunity to help students think in a new way about this topic. The rest of the class would be gray squares because we really don’t know what they are thinking. What warm-up would you do with these students the next day to help them understand? How would you modify lesson plans for the next day to end up with all green squares related to this learning target? From my understanding, you are not teaching this class on a daily basis so the idea of iterating has less application in your context. However, I couldn’t pass up the opportunity to discuss the important question: “How can I help students improve?” in addition to your reflection question: “How can I improve?” As a profession, how can we adopt a problem solving approach to teaching that applies reflection on our lesson missteps (which are bound to happen regularly) and iterates constantly to reach our end goal of helping students learn the given objective?

You're singing my song with Wordle right here!

I think I'd like to get other students on record about the board student's work. Not agree / disagree. Rather to present them with another hanger and ask them how "square splitting" tells them whether or not it balance. That might help me understand who's yellow or green rather than just gray.

Thank you for sharing. What a great way to start my day... with a snippet of PD :) We've all had crickets in the lesson! I like the idea of turn and talk to see if you agree or disagree with the strategy. Then, "Can someone else summarise S's strategy for the class?" Then perhaps you could say, "Pretend you are S. How could you approach problem C or D using this same strategy." Give the kids a chance to try that strategy on a few other problems and have another student present C or D using the same strategy.

One thought: Beyond asking whether they think it balances or not, you could ask something with more texture. Three options: does it balance, or is the LHS heavier, or is the RHS heavier? Then send them to talk with their partners, seeking agreement and explanation. The ceiling is a bit higher on that question and the issue of which side is heavier might offer access to some students who have lost the thread.

Texture! Love thinking about the texture of a question.

Focus your question around Error Analysis and/or a Misconception so students have something to think about instead of a a yes/no question: For example: Why might someone say B is balanced and identify the error? (Half of 3 is 1.5. Didn't take into account the square on the left side)

Oo nice. A binary question offers only two possibilities and one of them is probably the wrong one. "Why might" offers a much larger answer space and each one can be argued.

I like this. Another option in this vein would be for you (the teacher) to argue the "incorrect" answer, "i.e. I think B is balanced. If I make the circle worth two squares, then it is balanced." Doing this has a couple of positive consequences: 1. It lets students know they can argue with teachers and "prove" them wrong. 2. If a student is too shy to speak out against the perceived "smart kid," they can feel emboldened by the teacher arguing against them.

edited Apr 26, 2022My grandson loves math and I can see why if it is taught like this. My question would be, do you just give them one (a) balance sample.? I would give five or six to keep students engaged and so they might discover different sequences and solutions.

All kinds of hanger situations! At one point students CREATE their own hanger problems for each other. It's a treat. Check this one out.

https://teacher.desmos.com/activitybuilder/custom/5f1af3878031017692e8f666?collections=5f8a43db06b0d9a8bd84c3cf%2C5f8a446f06b0d9a8bd84c3de#preview/ee2d3f82-752d-4929-9552-f844c0e3b714

Fantastic. Love the opportunities to wonder and speak that are created.

I was also thinking that if you had the student explain how to balance 2 more purple circles (so 4 purple circles altogether), that might alleviate some "fraction" confusion and possibly elicit more discussion from the class. Then once that was discussed, the small groups could then work together with the 2 purple circles alone. Sometimes I find when you say the word fraction, students shut down.

edited Apr 26, 2022I just thought you cut off the student—the creator of the move— who was about to provide an insight to your question (“Who told you that you can split up a square?” which I think was meant to be humorous and something we all do for a laugh or an exaggeration) but then missed that the student was ready to reply and provide more insight but didn’t get the chance. He might not thought of 2 to 3 ratio on the previous demos but rather as 1 to 1.5 ratio. Next time, after you let the student reply, play off his nugget by making a teacher move and ask a variation on the question, even ask how many are it as 2 to 3 abs how many see it as 1 to 1.5 (which can then lead to the idea of a unit rate).

Like the lesson progression, though, from playing with hangers to solving static riddle questions

edited Apr 27, 2022Did students talk about all of them before explaining their described one? I like to start with “raise your hand if you thought of it in the same way/also used splitting for yours” or “fist to 5 how much sense does it make to you to split the shape like that” and then like Scott maybe a quick- “talk to your partner about why/how he split the shapes, and if you can do that” or maybe “about how his way of doing that connects to your way of figuring it out”

I think you got stuck because the question more about is he right or wrong (ambiguous about is the method right or wrong vs his final answer that the hangar is unbalanced) vs what made him think that/why his method does or doesn’t work.

Also thinking about the goal here- is it that you suspect this was novel thinking of a crucial idea and students need more time to process it? Is it to shift a norm around who we “present” to and what kind of thinking or response the class is expected to do when people are sharing “answers” at the front of the room?

> Did students talk about all of them before explaining their described one?

Yep. One fun part of the lesson was noticing the initial disagreement on the teacher dashboard for that screen before our 12 minute conversation and then consensus afterwards.

> I like to start with “raise your hand if you thought of it in the same way/also used splitting for yours” or “fist to 5 how much sense does it make to you to split the shape like that”

Really like how this takes the temp down. Showing something with your hands alongside all of your classmates seems very different than being the only person verbally staking out a claim.

> I think you got stuck because the question more about is he right or wrong (ambiguous about is the method right or wrong vs his final answer that the hangar is unbalanced) vs what made him think that/why his method does or doesn’t work.

Super helpful analysis.

My standard move when I don't get any (or few) hands, is to ask students to check in with their partner. "Discuss what (the student) just did. Do you agree or disagree with his reasoning? Why?" After a minute or two discussion, when I bring it back to the whole class I usually have more buy in and can have a more robust class discussion.

I'm thinking about how to tweak the "agree/disagree" prompt to focus students' attention on the thinking (as opposed to getting the right answer.)

What it if were a regular move in class, after seeing a student's work, to ask students to turn and talk with a more specific prompt. As a pair, they have to agree to respond to one of three possibilities:

1. Agree - why do you agree?

2. Disagree - why do you disagree?

3. Ask a specific question.

Not sure if this would work, but it feels like it would give everyone good options:

* The third options gives students who are confused an out - and focuses their attention on the reasoning.

* It allows you as the teacher to sequence the next part of the convo by starting with the questions.