## Hosted by Peter Liljedahl & Sarah Schaefer

The grand question asked in the session: How do we create and implement tasks for thin-slicing?

Peter answered this by listing out a four step process.

- Identify what students already know.
- Identify what you want students to accomplish by the end of the concept.
- Develop and group the tasks to fill this gap.
- Sequence tasks.

In his book *Building Thinking Classrooms*, Peter shares the principles that Marton and Tsui discovered regarding variation theory; one of the theories that contributed to Peter’s development of thin-slicing.

**Principle 1: **Variation in thin-slicing questions begins on the foundations of what students know. This may take a few questions of the same variation with in the sequence.

**Principle 2: **Only one thing can change from problem to problem. Too much change will inhibit students’ flow in thinking and stop them from utilizing connections from the previous problems to successfully work through the current one.

### What could this look like in a curriculum?

The following is an example of how the think!Mathematics curriculum aligns with these principles in the development of the Guided Practice.

**The concept: Subtraction with regrouping.**

Notice the slight variation in number 1 from regrouping to subtract a 1-digit number, to now applying a second step of subtracting the tens.

Next, students are continuing to regroup and subtract, including from the tens, beyond the value of 10.

Lastly, we extend students application of renaming beyond regrouping. The set in question 3 lend themselves to the use of reasoning strategies like rounding up to the nearest ten, subtracting, then adjusting back. Furthermore asking the student to reason, versus solve.

*Guided Practice source: think!Mathematics Second Edition Grade 2A*

## View Transcript

– Good evening, everybody. Welcome. It’s a little different format. We’ve got a lot of folks with us so this is exciting. My name is Mark Griffis and I am one of the team members here at Mathodology. And tonight is just going to be a terrific episode. Peter is waiting in the wings, and he’ll be discussing the idea of thin slicing, the what, the how, the where, the whys, all of it. So we’re really excited about that. Now, given the big audience that we have, we did go with a webinar format, so that makes it a whole lot more inclusive. We can get a lot of more people here, but, unfortunately, we’ll just kinda limit the discussion mostly to Sarah and Peter. But if you have a question or a comment, plug it in there. If we don’t get to it tonight, we’ll make it part of the follow up round table notes and links that we always send out. And we also have some terrific giveaways, but you have to be present to win. So be sure to hang around till the end of the event tonight. Now, with so many newer people here for this episode, some of you may not be familiar with Sarah Schaefer. And quite honestly, Sarah is the passion behind everything we do here at Mathodology. She has a master’s in mathematical education. She’s been teaching K through 12 mathematics in both the public school and the private school setting for over three decades. And because of this experience, everything she develops and what she prepares for has two people in mind. And that is you, the instructor, and that child that’s sitting across from you. Another interesting thing about Sarah is that she was actually a finalist in the Florida Times-Union EVE Awards, and this is given annually to women in the North Florida, South Georgia area who have contributed most to the community with efforts that make truly lasting improvements. She speaks nationally and internationally about mathematics. She’s a contributing author to the think!Mathematics curriculum. She’s the lead author of the pre-K and the kindergarten Developing Roots. And in addition, she’s written six other support books for parents and teachers. She’s tireless in her pursuit, which quite honestly is why we have round tables, and these are great opportunities. And those of us who work with Sarah can attest to her tireless pursuit because we routinely get emails from her before 5:00 AM, and it’s not a rare thing. So, Sarah, without any other ado, the podium is yours.

All right, thanks Mark. Thankful for our team who’s gonna monitor your questions and answers today and for the crazy hours, but also for you guys, the teachers and students or professionals for being here. I love working day in and day out with you in making a difference. The picture I have up here, quick story. In 2019, I was asked to present at the Building Math Moment Summit, and I did a session on integer operation. I’m always driven to watch and learn from other industry leaders. And you see, I spent the rest of the two days during this summit listening and learning, and it was an interview with Peter Liljedahl that caught my attention. I had to know more. When you hear greatness, can’t stop. So afterwards, I shot Peter an email request and he was gracious enough to respond. Since then, he continues to respond. So Peter, countless hours of talking through my ideas, he’s spoken at my events. We’ve even flown to small little towns to work together with groups of teachers. And recently, we were connecting to plan our summer event and found out through our texts back and forth, we were literally right down the street from each other. So few hours later, Peter, his wife, my husband Ban Har were sitting around a table in Honolulu having dinner. So Peter’s most down to earth relatable person I’ve ever met. As Mark stated, my passions about helping teachers, and in order to do this, I constantly have to take my game to the next level, and I need to work with people that are at the front of the pack, and that’s why Peter’s here with us tonight. So excited for you to hear from Peter as we unpack the art of thin slicing. A little bit about Peter. He’s got his PhD in mathematics education. He’s a professor of mathematics education at SFU. He’s the winner of the Cmolik Prize for Enhancement of Education, which is Canada’s biggest education prize. And he’s the winner of the Fields Award for Enhancement of Mathematics Education, which is a pretty prestigious award. It’s the same fields that gives out the Fields Medal for Mathematics. And it’s given every four years on the occasion of the International Congress of Mathematicians to recognize outstanding achievement and work, existing work, and the promise of future achievement. Peter probably is best known by many of you as being the author of “Building Thinking Classrooms.” I’m so excited for you to hear tonight from Peter, and I know every event I walk away as a better teacher, presenter, and really a better person. So Peter, welcome to the Mathodology round table and we are ready to talk about the art of thin slicing.

Thanks, Sarah. I gotta say that’s probably one of the best introductions I’ve ever had. That definitely was not written by ChatGPT.

No, it was not.

No, that one came from the heart.

[Sarah] It comes from the heart, yeah.

Thank you for that. I do get a lot of introductions that I’m pretty sure ChatGPT wrote. Yeah.

So wait, Peter, before actually we get totally into art of thin slicing, you just said you’re at home probably not training. What does a great day look like outside of training? What does Peter Liljedahl do when he’s not training?

When I’m not training. So by training, you mean working with teachers?

Yes.

Well, there aren’t many of those days either that that happens on a weekend when I’m writing and catching up on the three to 500 emails I get every week, or the last few weekends when we were in Hawaii, so I was able to lay under a palm tree somewhere, read a book, or I’m on a plane somewhere, going somewhere. But I also do carve out a fair bit of time every year to go skiing. So my wife and I, our whole family, actually, are avid skiers. So we have a place in the mountains and we go downhilling. Actually, we were skiing for five days, and then, the very next day, we were on the beach in Waikiki. So it’s one thing or another. Less time zone challenge and more temperature zone challenges there. Yeah, but try to find that time for skiing, that’s one of my passions.

Excellent. All right. So Peter, I have a bunch of questions people will put questions in, and Fran, and Donna, and Mark will monitor that, but the title is “The Art of Thin Slicing.” And in regards to “Building Thinking Classrooms,” what is thin slicing?

Okay, so thin slicing is really just this term I came up with to describe a carefully sequence set of tasks that moves a student cognitively and experientially from one place in their understanding to a different place in their understanding. So that’s in its simplest form, and if we think about every single math lesson you have ever designed ever in your life, that has been your goal. It’s my students are coming in with this knowledge. I want ’em to leave with this knowledge. How am I gonna take ’em on a journey from here to there? And there’s lots of ways to create that journey. We can have a discovery exploration inquiry type session. For very, very many years, it was a lecture or direct instruction that we would try to get the learner from there to there. Thin slicing is just another way to do that. And it’s a way of, I guess you could say it’s a guided discovery in a way. It’s I’m gonna ask a question and then that question is going to lead you to think a certain way. And then I’m gonna ask another question that’s gonna continue getting you to think that certain way. And then I’m gonna ask another question that’s gonna extend your thinking a little bit. And then I’ll ask another question that extends your thinking even further, and so on and so forth. And then before you know it, the kids are figuring things out. So I can give you a really concrete example. One of the lessons I co-taught when I was in Hawaii two weeks ago, we were looking at angles. It was a middle school lesson. I’m picking that one because it’s sort of right in the middle. The students knew how to measure an angle. They knew what an angle was, but they had… And they knew some properties of angles. They knew that a right angle was 90 degrees. They knew that a straight angle is 180 degrees. So this is where we started off with that idea. So the very first question was, I have a 90-degree with an angle in it. If I tell you that this angle is 25 degrees, what is that angle? And that gets ’em thinking a particular way. And like in the launch, we reviewed that right angle and a straight angle were 90 and 100 respectively. So that gets ’em thinking a certain way, and then we asked another question that pushed their thinking a little further. Then we asked a question about a straight angle. Then we asked a question where you had two, we had one known angle and then two other angles, but those two angles were congruent. So I taught them the congruency symbol so that they had to think about, okay, so if this is 30 and this is a straight angle, then 150 is what’s left. We gotta divide that in two, and that’s 75 each, things like this. Get ’em thinking, thinking, thinking, thinking. Before you knew it, we had them looking at intersecting lines, and they were able to figure out. They actually discovered vertically opposite angles were the same because they were able to do that sort of straight angle twice. And we just keep pushing, and pushing, and pushing until they’re coming out of the lesson with a much more advanced knowledge and understanding of something than they did at the beginning, and none of that is being facilitated by me just telling them directly. It’s just very small incremental steps just to pull them along, just to pull them along with what’s going on. Oh, you’re muted, Sarah,

As you talk about pulling them along, how do you make sure you’re not pulling along too fast, but you’re giving enough repetition before you increase the load?

So there’s redundancy in this. And that’s redundancy. I call it filling. So there’s a process here, but the redundancy is, okay, so now I have the sort of different types of tasks, but I need to fill in redundancy there. And so we may have four or five to six tasks that are very similar, and not every group is gonna do all of those. So there could be a group that’s, after two, they’re really confident with this. We move them on to that next harder one. Another group may need three or four in order to really start to feel comfortable with this. And I saw there was a question in the Q&A about how do we differentiate. This is differentiation, except it’s real time differentiation. It’s not that sort of differentiation that comes where I make assumptions about what students are capable of. It’s based on real time data. This group needs another one just like it. And guess what, they’re gonna need another one after that. And I’m gonna have to move the marker, and this student’s gonna have to… I need all of you to do this. Whereas this group here, you can explain it after two. I’m moving you on. So it’s sort of like a sequence of tasks that get progressively harder with parallel tasks. And what the journey of a individual group will be will depend very much on how quickly they’re catching onto this. So it’s like one group might do two and then move up, another group may do four and then move up.

Yeah, so-

And it’s not self-directed in that regard. It’s we’re leading that.

So with that idea differentiation though, they’re in a group.

Yes.

How are you making sure that the individuals in that group, because sometimes we all have that kid that’s not joining in or the ones that you know have it, but how do you move those people along in the group knowing that there’s some in the group, or do they eventually catch up?

So the reality is that there’s way more harmony within a group than we think there is, even when we have students of hugely varying degrees of abilities. So one of the lessons I did last week, there was a teacher, the opening conversation was, “I have such a huge variety of students. I teach grade seven. They’re from grade two to grade seven. How is this gonna work?” I say, “Why don’t we just run it and see how it goes? And then we can talk about it.” And then we ran the last setting, and we got through all of solving o ne and two-step equations in 60 minutes, and like all the kids. And the teacher’s like, “How is this possible?” Well, it’s possible because kids learn very effectively from each other, especially when empathy is present, when the kids care about each other. One of the things that “Building Thinking Classrooms” does really well through its random groups and its frequent random groups, is it builds community. And when community forms, empathy is unlocked. The kids start to care about each other. They care that they’re understanding and they start to take care of each other. So that’s part of it. Another part of it is that, with the redundancy, one of the things that we have to remember is that the human mind doesn’t work the way mathematics works. Mathematics is very logical. It’s just this… I like to think of mathematics as perfect in many ways. When it’s laid out correctly, mathematics is perfect. Concept A leads to concept B which leads to concept C. From a logical perspective, it’s all just, like, “Yeah, that makes sense, that makes sense, that makes sense. I get it all, I get it all, I get it all.” And that would be nice, but the human mind doesn’t work that way. It’s kinda like, “Okay, we went from A to B,” and I’m like, “Hold on a minute, what happened there?” I gotta see. I need to see a B.1, and a B.2, and a B.3, and now I’m starting to see the pattern, and I’m starting to recognize what’s going on. Oh, okay, I get it now, and now we can go onto A to a C, and so on and so forth. The human mind doesn’t grasp… Well, for some kids it does, but not very many. It doesn’t grasp it like that. So the nice thing about thin slicing is that all the students are in a state of discovery. They’re all in this state where they’re learning, and figuring, and spotting patterns, and thinking. Some are doing it faster than others, but they’re all in this space of meaning making. And some are making meaning a little faster than others, but it’s still this messy space where they’re all in it and they’re all in a state perpetually in a state of trying to figure things out. So there’s greater equity within those spaces because there’s so much tentative knowledge happening, One of the things that I talk about in my first book, and it very subtly so, but it’s the difference between tentative and absolute knowledge. So absolute knowledge is when I tell you something. This is Pythagorean theorem. This is how we factor a quadratic. Here’s what the tangent ratio is. This is how we solve a two-step equation. Here’s the rule for parallel lines with a transversal. I’m gonna give you this knowledge, and this knowledge is absolute. That’s it. You have students in the class who are like, “Got it.” And other students are like, “What is he talking about?” And the inequity in that space is huge, but if this knowledge is tentative, in that meaning-making process, the kids are going, “I think this would just be, wouldn’t those angles be equal? Isn’t that what that means? Isn’t that…” And that’s very different from me coming along and saying, “Those angles are equal.” So when they’re in that tentative space that’s much more inviting, the kids are engaged, they’re talking to each other, there’s just so much synergy in those spaces. So yeah, there’s diversity, but there’s less than we think there is. Having said that, we still have to react to the diversity that’s in there as teachers. The good news is you got a whole career worth of experience having done that. That’s still relevant. We’re still gonna have to step into that space and say, “Donna, can you explain this to me?” And then I’m analyzing, I’m diagnosing. Okay, Donna, she’s pretty good. All right, how about Mark? No, Mark is… I don’t know what Mark is doing here. So, okay, Mark, you’re gonna hold the marker and you guys are gonna do another one. And I’m gonna come back and check to make sure that everybody in the group understands, and I’m gonna move along, and I’m gonna come back, and then I’m gonna check with Mark. And Mark’s like… “You get it, Mark?” And he’s like, “Yep. Fran did a really good job explaining, I got it.” Okay, let’s do one more. Let’s check that. Or, I come back and Mark’s eyes are still spinning around in his head. Then I’m like, okay, “Hey Fran, let’s talk through this one. Explain to me so we’re talking together.” I’m taking the hit as the person who doesn’t understand so that Mark has a chance to catch up. All that teacherly craft of how we differentiate in that moment is still necessary.

So I’m seeing in the chat there too, a few other things as far as if we go back to that layering and how you said sometimes you stay and there’s not quite the variable or the change. How do you determine the list of… How do you build the thin slicing list that you’re going to use?

So there’s sort of a four-step process. It’s one of my favorite things to do with teachers. Teachers usually come into workshops with me thinking that this is so . And usually what I do is I… So the first stage is called identifying. So the way I do that is I say, “Okay, tell me something the kids know how to do from last week inside the same content area.” And they’ll give me a question that they know how to do. Okay, that’s probably where we’re gonna start because we need to start with something that’s familiar territory with ’em. One of the things we’ve learned about thin slicing is you gotta back up to something they’re confident with, start there and then move forward. So the next day we’re gonna back up and move forward, back up and move forward. There’s that sort of that constant micro spiraling. Okay, so you start off by identifying. So we’ve identified our starting point. Then I’ll say, “Give me an example of a question that you want the students to be able to answer at the end of this week inside the same topic,” assuming that the unit doesn’t break at some point, and they’ll give me that question. I say, okay, there’s kind of our end point. We have our starting point, the place where we want to enter, and we have our end point. And then I’ll say, “Okay, now let’s talk about all the different types of tasks that go in this space, in between these two. What are the possibilities? What are the possible things that we could be doing?” So I was doing a lesson on the big island. and we were subtracting fractions. Okay? It was a group of grade fives. We’re subtracting fractions. So what are the possible things that could happen? So the very entry is 5/7 minus 2/7. Okay, that’s where we’re entering. That’s it. That’s as pretty much as low as you can go with subtracting fraction. Where do we wanna get to at the end of this week? And it’s like, well, we wanna be able to do 5 1/7 minus 3 4/5. So different denominators. We’re subtracting mixed numbers. There’s gonna be have to be a regrouping. Ooh, there’s a lot of… That’s a lot of ground to cover, but is that it? Is that the only two types of tasks that there are. No, there’s other tasks in that space. There’s subtracting inside of one. All the fractions are less than one and we have common denominators. We have friendly denominators, like fourths and eighths, where one is a multiple of the other. We have different denominators, like fifths and sevenths, but we’re gonna do that subtracting inside of that space. But those are three different types of tasks. But now we also have subtracting a mixed number, so a mixed number minus a proper fraction, a mixed fraction minus a proper fraction, so with common denominators. So 5 /6/7 minus 1/7 is an example of that. And then we have the same thing where we’re subtracting with friendly denominators and different denominators. But then we have a mixed fraction minus a mixed fraction where we can actually just subtract the whole from the whole and then the fraction from the fraction, like it works out nicely. And again, we can do that with common denominators, friendly denominators, different denominators. And then we get to the subtracting where the fraction portion of the subtrahend is too big to take away from the fraction portion of the bigger number. And so we’re gonna have to do some regrouping, and we may have to change 4 1/5 to 3 6/5. I’m a fan of doing that as opposed to turning everything into an improper fraction, but that’s sort of regrouping that, borrowing one sort of thing. Well, look at how we’ve just populated that space in there. Those are all the different types of fractions, fraction subtraction that we’re gonna encounter. So that’s called identifying. We’ve identified the rainbow, all of the things that we’re going to need the students to encounter. So the next phase is now grouping. So this is where we’re gonna have to step back and say, okay, well, are these all really that different? Do we wanna make each one of these a category, a type of its own, or do we wanna lump some things together, or maybe we want to take some that we’ve created and say, “You know what, we gotta pull that one apart.”? But we need to do some grouping here. I got a lot of categories. I need to group some of those together ’cause that’s too many. And so now we gotta decide. is doing something like 4 4/5 minus 1 1/5 more like subtracting proper fractions with common denominators, or is it more like mixed fractions? Is it a category of its own? Do we keep all the mixed fractions together and all the proper fractions together? And now we gotta deal with this conceptually. Is that one more similar to like the last one I said, 5 1/7 minus 3 4/5? Is it more like that or is it more like subtracting with common denominator? I hope you’re saying it’s more like common denominator. The common denominator is the challenging part here, not the mixed part. So now we’re doing this grouping. We’re taking these types and we’re saying, “Well, these ones belong together. Yeah, there’s a progression here. that’s a little more conceptually advanced, but it’s still inside of that group.” So we’re creating the groups, and we can reduce subtracting fractions down to kinda like three groups. And then once we’ve done that, we gotta get ’em in the right order because now I’ve kind of lumped things together. I gotta decide which group is conceptually easier than the other one. Sometimes this is challenging, but 99% of the time it’s clear. It is really clear that we’ve gotta do this. And basically, the groups that we came up with was, look, there’s fractions where we don’t have to do anything to the denominator, and we can actually take this up to really complex things, including regrouping, but the denominators are good to begin with. The next one, the next category is where one denominator is a multiple of the other. And again, we lumped everything into that category, and that’s more conceptually advanced because we’re gonna have to do some sort of common denominator stuff. And then the last category is where you gotta multiply both denominators in order to find a common. They’re relatively prime to each other, or maybe they’re not relatively prime, but one is not a factor of the other or multiple of the other. So now we’ve got our groupings, we got our sequencing, now we gotta fill. So inside of that, each group has to get sequenced, but now we gotta fill. Like I said, we gotta put in all those redundant questions. We gotta create some redundancy in there so that the students can encounter multiple examples if they need to until they get confident, and then we move them up, and now we have. We actually have way more than one lesson plan. Thin slicing is not bound to a lesson. It’s content development in reality. So now we have this trajectory, and we’re gonna start here, and let’s see how far we get today, and then tomorrow maybe we’ll go back to the beginning, and go, skip, skip, and then go further. And the next day skip, skip, skip, and go further. But that’s in essence what we do, and it’s a lot of fun to do with other people. I know it’s a lot of fun. Here I am saying, “Oh yeah, it’s lot of fun.” Thin slicing is fun. You should do that Friday night. Get a math party going. You can have som queso and some chips. It’ll be fun, but it’s very satisfying. And one of the things, one of the nice byproducts of it, every time I do it, I learn more about the topic that I’m teaching than I did going in. It’s just, for me, personally, it’s such good professional development to think about math in this way.

So you said the identifying the grouping, sequencing, and filling.

Yeah.

Yeah. Talk to me about how often. In your book, you have your thinking tasks and then you have this idea of thin slicing. And so I always say it’s a never an always and it’s never a never. How do you balance in your classroom how often you’re doing thinking tasks versus these thin slicing which seem to be more practice-based type of-

Yeah, well, they certainly have practice and redundancy built into them, but they’re not practice. They’re concept development. They look like practice because they kinda look like the tasks that we have in our textbook, which are definitely for practice, homework type questions, and they’re often even called practice, but this is not practice. This is about concept development, and attainment, and thinking, and so on and so forth. But I know what you’re saying. When do we thin slice? When do we use a more rich problem-solving task? And when do we thick slice, which is something I think we’re gonna talk about a little bit later? I’ll give you an example. All right, so let’s go back to the angle example I gave you. So I actually did two different lessons with two different teachers and two different groups of students on angle measurement with grade five students two weeks ago, last week and then the week before. And the first lesson was the one I already described. The students had already learned how to measure angles, and now we were really just working on developing some real competency with angles and additive and subtractive properties of angles, and so on and so forth. And that’s an ideal thing for thin slicing. It’s you’re gonna take a little bit of knowledge and you’re gonna expand on that knowledge through logic, through that thin slicing sequence. But in the other lesson, the kids had not yet learned how to measure an angle. What they knew was that this is a 90 degree and this is a 180. That’s all they knew. And most of them knew it from skateboarding and they knew it from… They had been doing some Lego programming robots and they were making them turn, so they knew how to make a turn 90 and 180 and so on and so forth, which are practical things when you’re programming a robot, but they didn’t really know anything else. So what we did was we had a whole class set, like every group had one of those gigantic protractors, a kind that is magnetic and sticks to the whiteboard. So what we did was we gave them a huge sheet of paper that we just magnet it up on their whiteboards, and it had a straight line and a perpendicular. And we said, “Okay, so what is this angle?” And they’re like, “90 degrees.” And I said, “And what is this angle?” And they said, “180.” And I said, “Here’s a protractor. Figure out how to use a protractor.” So they’re standing there looking. These grade fives, they’re looking. They thought this was so cool, and they’re looking at this protractor, and they’re going, “Hey, there’s 90 and there’s 180.” And they’re trying to figure out like how to make this protractor work by putting it on the paper. They knew that this had to give 90 degrees and you can actually kinda just put a protractor up and move it up and down. It’s always gonna say 90 degrees. But how do we… Oh no, we also had a 45 on there. So we also had a 45 on there. And they’re like, “That’s 45. how do we make this thing tell me it’s 45 degrees, and tell me this is 180, and tell me this is 90?” So they’re fiddling with this. And it took some groups 10 minutes to figure this out. Other groups were like, it took ’em five minutes to figure out, but they’re in this really thinking exploratory phase, and that was the right thing to do with them. Yes, I could’ve taught them how to do this, but how many times have you had to teach kids how to use a protractor over, and over, and over again? So we decided to do it as an exploration thinking activity. And it really stuck with them. By the end of the lesson, they were measuring things that were between 90 and 180, and they were measuring angles that were not coming off of this where they have to rotate the protractor. And then when we took out the class set of small ones, they thought they were the cutest things on the planet ’cause of course we usually start with kids with small ones, and then the big one just seems so cool. Here we start with the big ones. And then when they got the small one, they’re like, “That is so cool.” And they wanted to figure out how to use the small, clear plastic one and so on. But there’s an example where we can go into thinking exploration and still be a curricular activity. So one of the things that math education has done really well in the last 10 years is develop a whole bunch of really rich problem-solving activities that help students understand curriculum. And there’s tons of resources out there that help ’em. There’s problem-solving, problem-based, there’s project-based, three-act tasks, which one doesn’t belong. All of these things are great thinking tasks that help students understand curriculum. Open Middle is one of my favorites. They’re thinking… These are these problem-solving tasks and yet they’re hidden curriculum. And then there is what we call non-curricular tasks, which we use primarily to build a culture of thinking. It disarms students. It’s safe. We’re just having fun. So I was doing some of these with kids as well who had never experienced thinking in classrooms, and I was in a primary class, and they were like… The task we did was a Farmer John problem so it was like, “Hey, back home in Canada, I live on a farm.” I don’t really, but I’m telling the kids I live on a farm. “And on this farm, we have pigs and we have chickens.” And I’ve drawn a pig and I’ve drawn a chicken. And of course they’re grade ones and kindergarten, so they’re super critical of everything I draw. And they’re thumbs up on the pig. The chicken they said looked like a duck. But anyway, I drew a pig and I drew a chicken, but they didn’t have any legs. And then I said, “So this is one of my pigs. How many legs do they have?” And they’re all yelling four, and so I draw four legs. “And this is my chicken.” They’re like, “That’s a chicken? It looks like a duck.” I said, “It’s a chicken.” And then they’re like… “How many legs does it have?” And they’re like, “Two.” So I drew two legs. I said, “You know, on my farm, we have two pigs and two chickens. How many legs do we have?” And everyone yells, “Four.” Because they’re five and six years old. And I said, “Nope, off you go.” And now they’re at the boards, and they’re trying to figure out how many legs do we have? And the reason we call this non-curricular, because, to them, it doesn’t feel like we’re doing math. It just feels like they’re playing and so on and so forth. And yeah, by the end of the lesson, it’s like, “I got one pig and 22 legs. How many chickens do I have?”

Awesome. We’ve got a bunch of questions here around language and the thin slicing when problems tend to be heavy or rich in language. When do you attach the vocabulary? When do you let them figure out the vocabulary? But also I’m thinking more of like word problems. Are there certain types of tasks that are better for thin slicing?

You can thin slice pretty much anything because that’s how learning happens. It’s through that exploration thinking, pulling yourself forward a little bit further. I’ll give you an example of one. All right, so one… I don’t know, when I was there in February, I think that’s when we had dinner together, it was like everybody in Hawaii was on word problem week. And it was no matter where I went, everybody wanted to do word problems. And I’m like, “What is going on in Hawaii this week?” But everyone wanted to do word problems. Even the kindergarten teachers, we gotta do word problems. Can your kids read? And they’re like, “No, but we gotta do word problems.” And I’m like, “Okay, so what is it about word problems?” Because word problems are heavy in words. So the first thing we gotta understand about word problems, and just stay with me here for a little bit. There’s a couple of unique characteristics of word problems. I’m not a fan. Anybody who’s read my book knows I’m not a huge fan. But would you agree… And I would say that your entire career, all four years built around this, our job as teachers is to take things that are complex and try to make them easier to understand, and accessible, and simpler. Our goal as teachers is to take complex things and bring them to students so that they can understand them. That’s our job as teachers. Word problems do the exact opposite. They take things that are already simple to understand, and then they make them more complex by adding words. And so it’s like, okay, so… And that’s why it feels so awkward sometimes for teachers to do this because it feels like it’s going in the opposite direction to who I wanna be as a teacher. The second thing about word problems that we have to understand is that word problems actually live in the same category as an essay in my mind. So because they were both created for the purpose to see if students could do them. We’ve created word problems to see if students can do word problems. And we’ve created essays to see if students can write essays. Nobody’s ever gonna write an essay after they leave school. It’s just not a functional piece of… You learn things about argumentation and so on, but you’re never gonna write an essay. You’re never gonna solve a word problem after you leave school. You’re gonna encounter problems. We just got through the weekend, three kids, four sports, one car. How are you gonna solve that problem? But nobody wrote that on a piece of paper and with a soccer ball drawn on it and handed it to you, and said, “Here’s your word problem for this weekend.” Nobody did that. This problem comes to us in more authentic ways. It doesn’t come to us as a word problem. Word problems exist only for the purposes of seeing if students can do word problems, and it has this sort of weird thing. So one of the… There’s some a whole bunch of challenges with word problems. And one of the ways that we’ve overcome it by thin slicing is by verbalizing them as much as possible. So for example, I did a task in a grade three classroom recently, and I started out. kids are gathered around me and I held up my phone. First, I said, “You know, at home in Vancouver, I’m a professor. I work at a university, but I also own a sporting goods store.” Also not true. “But I own a sporting goods store. And one of the things that we… We sell a lot of sporting equipment and there’s one piece of equipment that I’ve been buying a lot of lately because I think it’s gonna really catch on. And that’s these.” And I hold up my phone and I show ’em this picture. Every time I’ve done this, there’s a kid in the room who’s going, “Hey, that’s a moon ball.” “Yeah, that’s a moon ball, that’s a moon ball, highest bouncing ball in the world. And it has sort of odd shape so that it doesn’t always bounce straight. So it’s kind of fun to play with. Yeah, I’ve been waiting for ’em to become popular, so I’ve been buying whenever I can get my hands on ’em. I know when I left Vancouver, we had 112 of them in the store, but I just got a message from my manager that we just got six boxes. Each box has 12 moon balls in it. How many moon balls do we have now?” And now off they go to the boards. And this is a word problem. It’s just I haven’t made them read it yet. And then when they’re working away and they’ve solved it, I’m like, “Okay, and we just got another shipment, 10 boxes, five balls each. How many do we have now?” And they’re multiplying and adding. And then it’s like, “Oh, good news. Finally, it’s starting to sell. We just sold to another store. We sold 10 boxes each with seven balls in it.” And they’re doing two-step word problems here. And we just keep this narrative going. And then all of a sudden, I come along and I go, “The moon ball thing’s been going so well. We’ve decided we’re selling golf balls.” except now I hand ’em a slip of paper That has the word problem about golf balls on it, except it’s sort of the same story. We have so many in the store. We just bought this many boxes of so many, and now we’re doing T-shirts, and they come in five different colors. And now we’re selling chocolate, and I sold five boxes. We had 80 this morning, and now we have 40 chocolate bars left, but we sold five boxes. How many in a box? And the kids, they’re just working. But these are all written as word problems now. But instead of starting off with the word problems, we’re thin slicing into it. We’re starting with a simple task and we’re getting harder, and harder, and harder. And then one of those extensions, one of those groups is a word problem in text form, but now they’re not so intimidated by it. So we stay with the same narrative, and we build up that competency, and now here comes a word problem. They’re decoding in groups. And it’s one of the ways that we can work with word problems is we just have these slips of paper.

So it seems like you just layered in that, you secretively got it in there.

Yeah.

Yeah.

Hey, Sarah.

Yeah. if I could just interject for a second. We have about 20 minutes left, so just wanna make sure everybody’s aware of that, but we had a great question that I wanted to ask Peter from Michelle, because it’s kind of a practical approach to kinda what we’re looking at here. Her question was, “When I use thin slice tasks, I have trouble managing the pacing of all six of my groups at once. Any logistical ideas to help us manage this?”

Okay, so it depends on the task. So if it’s word problems or geometry problems where there’s lots of diagrams and so, we use slips of paper. One of the ways to do it, when it’s just coming at the tail end of a bunch of verbal tasks, I just walk around with slips of paper between my fingers and I’m just, “here you go, here you go, here you go.” It looks like I’m doing the 50/50 lottery at the hockey game or whatever. When it’s a lot of tasks, like the geometry ones where we had 20 geometry tasks and they were going from really simple to really hard and the kids are flying through ’em, what we do is we designate one spot in the room and we lay out sort of the first… We give every group the first one, but then we tell ’em, “Go to the desk. When you’re ready, go get the next one.” And so there’s three that are there in order, and they just take the next one. And when everybody has taken, the easiest one of those, that pile disappears, we add another pile at the bottom end. So it’s like every time they go, there’s three, but they got to… They’re not being overwhelmed by the number and they’re not accidentally jumping ahead too far. If there’s another individual group that’s racing ahead, I’ll just it to ’em as they work. So that’s one practical way. It’s important that they don’t see all of them because then they start racing, and it’s also important that we don’t number them. That’s fascinating. The minute you number ’em, you got groups racing each other. You don’t number ’em, they don’t care. But what if I’m doing something like the one we talked about earlier, which was subtracting, doing subtraction. For that one, we use something we call the banner. So the banner is like the top six to eight inches of the whiteboard. And the rule is whatever task you’re working on, it’s written in the banner. You do the work below the banner. And as soon as you’ve done your task, don’t wait for us to give you another one. Just look at everybody else’s banner. Find a new task and then do that one. So you erase your old one. You write the new one up on the banner, you do it. And it’s like crowdsourcing. So I only have to give out that task once to one group, and then it spreads around the room, and they’re stealing from each other. And again, we gotta keep the variance relatively small, three or four different ones, so that if somebody takes ’em out of order, they’re not all of a sudden jumping way ahead to something that’s really hard. They’re staying within the realm of achievable for them. So if there’s a group that’s really far ahead, I’ll give them the next task, but they’re not allowed to put it on the banner. And likewise, if there’s a group that’s really far behind, I’m there giving them the tasks as they finish. So we’re kind of managing it. 80% of it manages itself, and then we just gotta work on the fringes. So it’s not so frantic.

It’s the slips of paper kinda where they can come, you have three at a time, or the banner. And the banner, you said that they have the problem up and then they solve it. Do they erase their solutions after or do those solutions-

They can if they want.

Okay.

But-

How do they correct?

[Peter] The banner only has the question.

Okay.

Some teachers really wanna see the students work, but the problem is we don’t want students to work at the pace of the teacher because then there’s too much downtime.

Yeah. Yeah.

So some teachers that I work with, what we’ll do is we’ll partition the board into four. So you still have the banner, but you do the first question in this quadrant, you do the second one there, third, fourth, fifth, sixth. So the work kinda lives on the board for longer so the teacher can come around and see it, but the kids can still move along without having to wait for the teacher to see it. That’s also a really good strategy when you want ’em to spot a pattern in something. They need to… If they keep erasing their work, they’re never gonna see the pattern because the pattern lives in what’s the same in all of their solutions. So that sometimes I did one where we divided the board into eight and they were just working, working, working, working. And then it’s like, “Okay, well now you gotta… This one’s really hard. We’ll look back at what you’ve done previously, see if you can spot a pattern.” And then, “Oh yeah, okay.” So it’s like it slows down the work. It keeps the work alive longer when you do that. These are just micro moves. These are just pragmatic ways to help us achieve the positive effect of that thin slicing.

And what would a pragmatic move be for giving solutions or making sure that kids get it right so they’re not waiting for you.

Right, so this comes back to the power between tentative and absolute knowledge. I’d rather keep ’em in that tentative knowledge space for as long as possible. But more than that, there’s a huge difference between students having an answer and knowing they have the answer. And many people will argue that math hasn’t really begun until they know they have the answer, and I don’t mean know because a teacher said yes or put a happy face on their board. I mean, they know because they’ve talked it through, they’re confident, we got it. We actually want to get our students to that space where they’re like, “Yeah, this is right,” Even if it’s wrong, I’d rather have ’em be in that space. I can come in and correct things and help them see where they made mistakes, but I want ’em to build up that confidence in themselves, not so that they’re uncritical, but that they are critical and derived confidence from being critical and reflective. So I want them to self-evaluate if they think they have it, and then move on. And then I can always come in and correct them and sort them out later. And we get the value of both of those things, especially if we’re slowing down their work by having the quadrant. But even if we don’t, if a group’s making a mistake on question three, they’re still making it on question five. Maybe I didn’t catch it on question three, but I’ll eventually catch it, and then we can back up and do a few more and so on and so forth. This is not autopilot. The teacher is still active in the room, and you still need to be active and using all that stuff you always do in the classroom, monitoring, redirecting, correcting, questioning, challenging, extending. We’re still doing all of those things. We’re verbing all over the place.

Absolutely. So we’re gonna pause for a moment here because we have about 10 minutes and I want to answer. One of the questions here was about your book, and we are going to give away a book tonight as well. So I’m gonna bring that up here. Can you tell us a little bit about your book that’s coming? It said May, right? ‘Cause I pre-ordered it.

Yeah.

Okay.

So it’s coming in May, but if you’re pre-ordering on Amazon, I think the date they’re giving now is June 4th, but they’re being really conservative. I think it’s gonna be middle of May. It is so close to going to the printers right now. So I co-wrote it with Maegan Giroux who is a fabulous elementary educator from Central Canada. And if any of you have an opportunity to bump into her, just engage her in conversation. So what it is… So it’s organized around tasks. Everywhere I go, people are like, “I want more tasks, we need more tasks.” I’m like, “You do not need more tasks.” The world is full of tasks. Every book is full of tasks. If you Google problem of the day, I think you get 7 trillion hits. It’s just ridiculous how many tasks are out there. The world doesn’t need more tasks. So we decided we’re gonna write a book about tasks. So what it is, is tasks are the foundation of it, but it’s now, what do we do with it? So it’s got all of the new research in it, the front matter. The book is broken into four parts. Part one is what’s new and in review. So what are the new things in the BTC research and what are the things we need to review so you can understand what happens in this book? So it’s got all the new research on consolidation, and meaningful notes, and check your understanding questions, and so on. Then part two are 20 non-curricular tasks, except it’s not just the task. It’s got the launch script. It tells you how to differentiate it and extend it. It’s got all the extension scripts, and then it also has everything you… It’s got everything you need to know to close the lesson. What would a consolidation look like for this task? What would check your understanding questions look like for this task? And so on and so forth. Part three is 30 curricular tasks. So these are tasks from the curriculum from kindergarten to grade five. And again, it’s the full lesson. Here’s the task. Here’s everything you need to know about it. What content does it touch on? Who’s it relevant for? We can’t really say grades because curriculum moves so much across the world, depending on where topics land. So we just talked about which topics it covers. And then how do you launch it? What’s your launch script? What is the thin slice sequence of tasks? All right there. How do we consolidate it? What does meaningful notes look like with this? What are the check your understanding questions? And then notes from us all over the place so that you can really just take one of these and go, “This is how I want to teach this lesson.” And there it is. But there’s also got a space for notes because, every lesson we’ve ever done, we change it after we’ve done it. And it doesn’t matter how many times I’ve done it. And so there’s place where you can add notes and so on. And then part four has got more resources if you want more tasks. It has links to templates where you can enter in, build your own tasks inside if you want, whether they’re for yourself or to share. It also has the art of thin slicing. It has a whole chapter on that. So everything we talked about today is there. It also talks about thick slicing, how to thin slice word problems and so on. So yeah, it’s pretty comprehensive,

Are super, super, super excited. So Mark, how are we giving away this book? What is the plan?

Well, the plan is very easy. It’s a math plan. I simply need Peter to pick a number between one and 10.

Seven.

Leslie Gardner from The Hockaday School. Leslie, if you are here, you’re supposed to be here, just hold on till the end, and we’ll get a little information from you.

Awesome.

All right. And the next thing I wanted to talk about, I know it’s again coming down. There’s a ton of questions in there. If we didn’t get to your question, we always follow up our sessions with a recap. We’ll put the questions in there. I’ll reach out to Peter, we’ll get those answered, and you’ll get an email from us. In addition to the book, Wipebooks, which are the standing… This picture doesn’t show it, but the idea is the vertical whiteboards, that you can put there. And let me just maybe grab-

Amazing product. You can put ’em anywhere,

You can put ’em anywhere.

You can cut to ’em to shape. You can put them wherever you want.

And we will be… Peter and I are doing a session this summer in Jacksonville, so that was the other picture of small groups, and we’re gonna have Wipebooks everywhere, but we’re gonna give away some classroom sets of Wipebooks. So Peter, go ahead and pick some… Mark, what numbers does he choose from this time?

This time we’re narrowing it down to one to seven, since he picked seven last time.

Okay, two, my favorite number, two.

All right, Jody, I believe it’s pronounced Mongello, from Branford. Jody, would you hold on as well?

And then, Fran, can you go ahead and put this link. If you did not win, you can continue to win. You can put this… If you click on this code, it’ll put you on a spinner and Wipebook tonight is going to spin it a few times and give us a few more winners. So wanted to make sure that we gave away those as well. Peter, let’s look at another question. Let’s take another question as we’re finishing up here. I didn’t know if you had a chance to look at any of them.

I was browsing, but I was… I also wanna say that if you want more BTC, like an unlimited quantity of BTC, join one of the Facebook groups. But we’re also having the BTC Conference in Phoenix, Arizona, July 1, 2. And it is 48 hours of BTC. Over a hundred presenters all presenting on BTC.

So that is… When did you say that was?

July 1, 2. I’m gonna find the link and put it in here.

Yeah, put that in there. And then mid-July, again, in Jacksonville, we have smaller groups. You’ll work in a small group. Peter, talk to us about what those two days will look like. Small groups around tables. What could they expect to experience?

Well, you’re gonna live and breathe a thinking classroom. You’re gonna experience a thinking classroom, what it can feel like, including having multiple experiences of what thin slicing is like to experience learning new math through a thin slicing lesson, a complete lesson. You’re gonna, multiple times, will model a lesson, but also opportunities to work together to do that sort of stuff with support from me and Sarah, who is an expert on this as well.

And some people are gonna be lucky enough to get to eat lunch with you and pick your brain

Ah, yes.

During lunch, yeah.

Well, I’ll be lucky enough to get to eat lunch with them.

Yeah, well, that’s great. So in wrapping up here, Mark, do you wanna finish us off here? We’ve got a few minutes.

Sure. And this has been a great evening. It’s one of those examples of when you want to get better at something, you surround yourself with people who are champions in that field, and we’ve certainly been able to do that tonight. And a virtual atmosphere is great, but as Sarah just mentioned, we really love to do in-person. And in Jacksonville Beach, this summer, we’ll have three events. We just talked about the one with Peter. There’ll also be the Essentials in Mathematics, which is two days with Dr. Ban Har. If you’ve never experienced Ban Har in a session like that, you are missing a treat. And I promise you, take this phrase back to your classroom, “Is today a clever day?” Okay, that will be with you forever. And then the other one that’s in person, there is a three-day immersion into our Developing Roots kindergarten and pre-K program. It’s called Developing Roots at the Early Elementary Levels. And Sarah and the team of instructors turn a conference room into a giant kindergarten classroom. There are stations, there’s materials, there’s manipulatives on the floor. I mean it’s just an amazing experience of what goes on there. So if you have an interest in any of those, go to Mathodology.com, hit the Training tab, and then drop down to either summer workshops or summer institutes is what it would be listed there. And of course, continue to check your emails from us. We try to keep you apprised of all the things going on. And if your school is looking for a curriculum that supports the type of thinking that we’re talking about tonight, and you’d like to explore, think!Mathematics or Developing Roots, please email me, or just simply go to the website and hit the contact us, and I’ll email you. We’d love to share with you about it. So finally, next month is the last round table of the year, and it’s gonna be one of those look ahead kind of sessions. Fran, Julian, and Sarah Schaefer will be bringing some more insights into the new edition of think!Mathematics that will be coming out for the fall. And again, there’s a place where you’ll find many of the bigger ideas that Peter is talking about, and they’re already in there for you. The other thing that’ll be touched on is the Splash of Math series. kindergarten will be available this summer, correct?

Correct.

Fingers crossed. Yes, all right, so we will be having kindergarten available. That’s a great supplemental for your students and for planning for the next year. So thank you again for joining us. If you’re a winner, please hang around. And if you have any questions, just direct it to any of us, and we’ll make sure it gets to the right person.

Alrighty, so as we opened up today thanking you for everything that you do in taking the time tonight, we hit a little bit on thin slicing, just kind of the top layer. If you wanna know more about thinking classrooms and “Building Thinking Classrooms” initial ideas, you can find the other videos that Peter has recorded with us at that Mathodology site as well. The Jacksonville session is geared to a K-6 audience. We only have a few seats left. We have a cap of about between 60 and 70 in the room. So it’s meant to be a small, intimate group where you really get to work one-on-one with Peter and myself. So that’s a little bit more about that. These other questions, there’s great questions here. We couldn’t get to all of them, but as I said in our follow-up email with Fran… Fran, this is gonna be a long one. We got lots of questions to answer. So Peter, we’ll need your help to answer some of these to make sure we get it right. But thanks for coming tonight. Peter, thanks for your time as always. I know you’re super busy and fact that you just got home this weekend, and you’re sitting here on Monday night talking to us means a lot to us.

Well, I’m not even home. We were home for 24 hours. We’re on the road again,

The time, the time, so I appreciate it. Looking forward to seeing you in Jacksonville. Tell Theresa we said hello and enjoy. Enjoy your week.

All right, likewise.

All right.

And thanks to all the participants. I think this is… People just don’t realize how dedicated teachers are to their profession. To give up your Monday evening, to come out here to do professional development in a way that only benefits your students and you is the height of professionalism. So thank you for that.

Absolutely. All right. Have a great Monday and a week ahead. Make it a good one. We’ll see ya.