Pam  0:01  
Hey, fellow mathers! Welcome to the podcast where Math is Figure-Out-Able! I'm Pam Harris, a former mimicker turned mather.

Kim  0:10  
And I'm Kim Montague, a reasoner who now knows how to share her thinking with others. At Math is Figure-Out-Able, we are on a mission to improve math teaching. 

Pam  0:18  
We

Pam  0:19  
know that algorithms are amazing human achievements, but they're not good teaching tools because mimicking step-by-step procedures can actually trap students into using less sophisticated reasoning than the problems are intended to develop.

Kim  0:33  
In this podcast, we help you teach mathing, building relationships with your students, and grappling with mathematical relationships.

Pam  0:40  
We invite you to join us to make math more figure-out-able. 

Kim  0:43  
Hi there. 

Pam  0:44  
Hey, what's

Pam  0:45  
up? 

Kim  0:46  
Oh, so much we're busy, busy, busy. Good stuff. 

Pam  0:49  
Happy, almost weekend. 

Kim  0:50  
Yeah, it's good stuff. 

Pam  0:52  
Let's do it.

Kim  0:53  
So, today we're going to have an interesting conversation because not too long ago, you were invited to be on a debate. And I want to give you a chance to kind of unpack that and think about, share maybe with our listeners some important things. One of our leader in our coaching programs said ahead of time. "You know, I really hope that this isn't like a I want to win the debate situation from both sides." And she mentioned that, "I hope it's not going to be filled with loaded words, and that it's not about one side wins." And I shared that with you ahead of time. So, why debate at all? What were you hoping to accomplish?

Pam  1:36  
You know, Kim, I got to tell you. I'm actually...nervous? A little nervous for this podcast episode. Which I'm not usually because it's just you and me chatting. (unclear). This matters to me. It matters to me. It's been really interesting that not only this particular leader, like you said, who's in our coaching group, but also several people reached out to me on social media. Or just said on social media, "Why the debate? We don't need this debate. This is old news." 

Kim  2:06  
Yeah. 

Pam  2:06  
And then with some things following that, like, "This debate is long over. Of course, we use both." I even heard someone say both. So, maybe I should be a little more clear. The name of the debate was Teaching Procedures vs Developing Reasoning. I think. Maybe it was Developing Reasoning vs Teaching Procedures. And we worked on those words a little bit because words. Words are so hard. And so, you might say, "Oh, yeah. This is the age old debate between like constructivist inquiry discovery versus explicit, or direct, or I Do, You Do, We Do. I mean, like, and all of those words I just said could mean so many different things to so many different people. And so, yeah. Maybe a debate where all we do is throw those words around isn't actually very helpful if we don't have some specific examples about what we mean. So, it's super interesting to me when I started reading a little bit of that pushback about why the debate at all. And one of the social media persons said something like, "Of course, you do both." Another person said, "It depends on the strength of the teacher. If the teacher is a really good teller, then they should just tell. If a teacher's more creative then they can do more inquiry." And I was like, "Wait, what?" Like so, so I'm going to say unequivocally I actually disagree with that last comment a lot. But I also want to invite our listeners to consider that I'm not actually on either of those maybe traditional sides of the debate, that I'm going to suggest we actually have a different message here. And our message is that math, mathematics, is figure-out-able, and that that then impacts the way that you teach. And I went into this opportunity to debate. So, first of all, I didn't set it up. I was asked. I was invited to the debate. The gentleman at Mathific that invited me said, "I think you could represent this. I could represent your position well, and I would invite you, you know, to debate this." I'm not actually sure what he meant as my position when I got invited. But I am very interested to bring into the world, into the sphere of discussion, ideas that I didn't have as a young teacher. As a student of math, you know, in school and then as a young teacher, I wish I had heard my now message. I wish I had been given an alternative that maybe isn't either one of these sides. Not maybe. It isn't either one of these kind of maybe traditional or what people were expecting when they came or saw that debate.

Kim  5:01  
Yeah. Yeah, it's interesting because I think sometimes everything is pitted as a either or. And you and I have talked quite a bit about the idea that there are different kinds of lessons and there are different things that we would be... I almost said explicit. You know, I try really hard to avoid the words "direct" and "explicit" and "inquiry" because of exactly what you said. It doesn't... What I mean is taken in by the person. You know, the words I use are taken in by the person that's hearing them, and it instantly gets turned into their understanding of that word. And so...

Pam  5:34  
It can't not, right? It can't not.

Kim  5:35  
Yeah. 

Pam  5:35  
It's our experience. Sure. 

Kim  5:37  
Right. Right, right. So, I generally try to avoid those words as well. But we have had some pretty long conversations about how, based on the math that we're doing, based on what we're trying to hoping to accomplish in the classroom, we might approach the way that we teach a little bit differently. Now, I think we're in agreement that we're never going to stand up the board and do I Do, We Do, You Do.

Pam  6:02  
(unclear).

Kim  6:02  
Unless... Yeah, go ahead.

Pam  6:02  
Do your unless, and

Pam  6:04  
then I'll describe. 

Kim  6:06  
Well, unless we're talking about social knowledge, if it's vocabulary that we have to tell kids. But teaching, the actual mathematics behind what we're doing is, you know, it's interesting because I think people think it's one or the other. And, like you said, we're talking about something a little bit different.

Pam  6:24  
Teaching styles is one or the other. 

Kim  6:25  
Yeah.

Pam  6:26  
Yeah.

Kim  6:26  
We're talking about experiencing math in a way that I think people haven't really experienced. Some people have, but not a lot of people have experienced what you're talking about. If they've come to a session, or they've come to done a workshop with us, they probably have a very clear picture for that portion of mathematics because they've experienced it with you. 

Pam  6:51  
Yeah,

Pam  6:52  
and let me just say people might have experienced real mathing. Like, they might be mathing in their mental actions are actually mathing. But they might not have experienced teaching, mentoring, helping students learn to do those mental actions of mathing. So, that's why we feel so strongly. Every workshop, presentation, discussion that we have, we begin with a common experience, so that people can go, "Wait, wait, wait, wait. That's what you mean?" It's not... Do you remember the conversation I told you about where I was just at a conference. You know, "Hi, how are you?" I didn't know this person. She didn't know me. And in the middle of the sessions, the facilitator said, "Turn and talk to your partner about some..." I don't remember it was. And the very nice teacher turned to me and said, "Yeah, you know, because that discovery stuff doesn't work." And I said, "Oh, tell me more about that." And she said, "Ha, I tried to. You know. I thought, I'll give this a go. I'll give it a try." And walked into her eighth grade classroom and said, "Alright, today, you guys are going to discover the Pythagorean Theorem. Go ahead, go." And then she kind of laughed, "Ha," you know at me. She goes, "Do you think that worked? Of course, not. So, then I said, well, that was a failed experiment. And then I just  told him the Pythagorean Theorem, and off we went."

Kim  8:04  
Mmhm.

Pam  8:04  
That's one interpretation of what could be on, you know, that side of discovery learning. It's not what we mean. So, if we have... So, ya'll, if you're listening to this podcast for the first time, we invite you to go listen to any of the mathing episodes, which I think most of them are, to gain a common experience with us, so that when we talk about what we're advocating. If the debates title was Developing Reasoning vs Teaching Procedures, then you have some idea of what I mean when I say "developing reasoning". And so, I felt really strongly in the debate that... In fact, I actually asked to tweak the questions. They  had told us the questions ahead of time, and I said, "Ooh, could we please have..." Well, one of them was inspired by you, Kim. The very first one was, how do you measure success? Like, what does success look like? What is your objective? What's your goal? And then I tried to be really specific with examples in that answer. And then also, what would a lesson look like, so that we could actually know what each other meant. And so, I did a quick Problem String. I kind of tried to give an example of high dosing with the major relationships, so that kids are using those to reason logically to solve problems. And I think... I don't exactly remember when she said it, but she said, "What would a lesson look like? Model, practice, model, practice, model practice." model, practice. That's a good lesson. And then, of course..." This is quoting her. "Of course, we would do all the other things, but really learning is going to happen when you model, practice, model, practice, model, practice." Well, if anybody's taken our mini workshop called Transforming Teaching, you know that we do a bit in there. Well, in fact, I think we do it in all of our workshops. We've done it in Developing Mathematical Reasoning books. Where we talk about what the word model means in mathematics education. And here's a great example of why I do that. Because when she says, model practice, model, practice, model, practice, she has a very specific format in mind. Now, you could be like, "Pam, what are you reading her mind? Well, I watched several webinars that she's put out that are  just widely... Widely. What's the word? Widely accessible? Is that the word I want? You can watch it out there.

Kim  10:08  
Easily accessible. 

Pam  10:12  
There we go. And, so I just, you know, grabbed what I could find when I Googled, and she had, you know, very... It's her in one of the demonstration videos. She's holding a script in her hand. And I don't remember exactly, but she wrote on the board the subtraction symbol. And she said, "Okay, class, this is called minus." And then she pointed "Kim." And then Kim, the student, said, "Minus." And then she repeated minus, and then she pointed at Steve. Said, "Steve." And Steve said, "Minus." And she said, "What is this called class?" Point at the class. They all said, "Minus." And then she wrote down "5 minus 3..." Or something like that. "...equals." And she said, "Okay, how do we say this class? Everybody with me. Ready? "5." And the points to the class. "5." "Minus." "Minus." "3." "3." If you could see me right now, I'm pointing back and forth between the board and the class. And then she said, "Okay, now everybody, now we're going to draw 5 lines, 1, 2, 3, 4, 5. Now, this says minus. What does that mean? We're going to get rid of them." Whatever her words were. "Do it with me. 1..." And then she crossed out 3 of them. Let's count. What's left over? Ready? 1, 2..." And they're all counting with her in this like coral kind of thing. "Okay, everybody. So, what is 5.." "5." "Minus." "Minus." "3." "3." "Is." "Is. "1, 2, 3. 2!" And then she writes down 2. And then that was the example of this model. Like, I'm modeling for you. Practice. Where I'm modeling for you, and we're practicing right there. She also mentioned, you know, that's a high response rate. We get students to respond often. And she used some, I don't remember the number, but you know like, we need to have students respond often, so that really gets it into long term memory. And maybe bores you to tears. I mean, could you imagine? Like, I don't know, Kim, did you ever have a class that was that scripted of repeat after me? Whoa.

Kim  11:53  
I mean, I might have at a very early age. I don't... You know, I don't remember that far back. I'm getting older.

Pam  11:59  
You're so old. Don't say that. You're 10 years younger than

Pam  12:02  
me. 

Kim  12:02  
In this example, though, there were two different parts that it seemed to be there was the part where she was naming minus, and having the kids say the word. And we would say that there is some

Kim  12:16  
direct teach to... 

Pam  12:18  
Telling kids, yeah. 

Kim  12:19  
Yeah. But

Kim  12:20  
then the 5 minus 3 part, the tally marks, the here's how you do it, I think is where we would push back that...

Pam  12:30  
We don't need to demonstrate that method. 

Kim  12:33  
Yeah. 

Pam  12:33  
Like, this is how you minus. You draw tally marks. You cross out this other number. You count what's left over. I mean, now everybody. And then the practice for her is going to be, "I'm going to give you a bunch of problems, and I expect to see exactly that."

Kim  12:48  
Mmhm.

Pam  12:48  
And at some point in one of the webinars, she said then she would circulate around and catch any misconceptions. And I got to hear that as, "I'm going to catch kids doing the steps wrong.

Kim  12:59  
Correct. That's how I would hear. 

Pam  13:00  
Then I'm going to tell kids what they're doing wrong, and I'm going to do that quickly and often. But the words they're using is like misconception. (unclear). Hey, Kim, when you just mentioned that we do have to tell kids minus. Let me just give kind of the opposite. What we're not saying is that we were going to say, "Hey, kids. See, this symbol? What do you think it's called?" That would be the extreme kind of discovery inquiry read wrong. 

Kim  13:23  
Yeah.

Pam  13:24  
Right? Like, understood like that's not what we mean. We don't. If something is social, then we tell kids. We're like, "Okay, this symbol right here." Now, we might do it in a kind of intriguing, tag it when kids need it, way. We might say, "Oh, I see what you're doing in that problem. We were just talking about how Kim had 5 oranges and she gave 3 to Pam. How many oranges does Kim have left over? I saw you like thinking about 5, and then you took away 3. You know, we could write that 5 minus 3." And I would write that down." And we say that as 5 minus 3. 5 subtract 3." Like, in that moment, I'm showing the symbol. I'm giving the kid a symbol. I'm not saying, "How do you think you write that? No, try again. No, try again. No." Like, it's not a guess what's in my head, guess what society has deemed to be that way. But only when it's when it's social. I'm going to try to do it when the kid needs it, so it can latch on their prior knowledge, and it makes sense in that moment. It's based on what they've been doing, but I'm not going to make kids guess that thing. Then, we're going to experience what it means to subtract. We're going to experience what it means to put two groups together. We're going to experience what it means to compare groups. I mean, we're using super young stuff here because I use that

Pam  14:39  
example.

Kim  14:40  
Yeah, I think the pushback sometimes from people who don't like the idea of a debate might push back because they hear that there are two camps, and it's either or. And what I what I've heard people

Kim  14:57  
say is...

Pam  14:57  
I have to win and squash you and walk away as champion.

Kim  15:01  
Yeah,

Kim  15:01  
and what I've heard people push back with is there's more of a continuum. And, you know, I heard, at some point, I don't know, within the last year or two, Dr Rachel Lambert talk about a continuum. And I would agree that there are different things that you want to do for different types of lessons. There might be some lessons where I lob this big problem out, and I have kids productively struggle and work, and I circulate, and I'm nudging.

Pam  15:29  
(unclear) grapple.

Kim  15:29  
Mmhm. And then there are some where it's maybe a little bit more structured. It's more... That seems like the thing that I just said was not structured, and it is. But a little bit more like short term, and it's maybe more of a reinforcement type lesson, and it's getting you...

Pam  15:44  
Quicker back and forth. 

Kim  15:45  
Familiarize. Familiarize with something that you've already grappled with, and it's more about fluency. So, there are different types of lessons, but I think we would say there's very few situations where we would be kind of on the, in my mind, right side, where it's a little bit more direct. We would err on the side of the...

Pam  16:05  
Don't be correct side when you said, right. You meant...

Kim  16:07  
Oh no.

Pam  16:08  
...this side over here.

Kim  16:09  
This side. I think we would, in general, live a little bit more towards the other side, but not so far to the end, where you just put some stuff out there. Stand back. Watch kids 

Pam  16:23  
Kids are on there own. 

Kim  16:24  
Yeah. 

Pam  16:25  
No help.

Kim  16:26  
Yeah.

Pam  16:27  
Yeah. 

Kim  16:27  
So, I don't think we think it's this or that other. I would say that we do agree that there's a continuum. It's just where do we spend most of our time, and in what circumstances, with what types of lessons? Because in all situations, the math that kids are doing, we want them mathing. We don't want them tell. We don't want to tell them the math. We want to raise experiences for kids where they are grappling with mathematics and supporting them in that. 

Pam  16:55  
So,

Pam  16:56  
there's an example of a different meaning, or the meaning that she meant of model. Model, practice, model, practice. But we're going to do that by modeling mathing behaviors. We're going to model curiosity. We're going to model stick-to-it-ness. We're going to model looking for patterns and using those patterns to try on other problems to see if they work. Sort of an inductive kind of approach. We're going to model...

Kim  17:21  
Becoming more efficient.

Pam  17:23  
Becoming more efficient. Sharing our ideas with others, so that we can verbalize the relationships that are we're creating in our heads, so that then we gain clarity as we verbalize. We're going to model all of those behaviors, and then we're going to encourage students to practice those behaviors as they dive in and solve problems. What were you? What?

Kim  17:44  
Yeah, when you say we're modeling those behaviors what we're not doing is saying, "Okay,

Kim  17:49  
students...

Pam  17:50  
Watch me. 1, 2, 3." (unclear). 

Kim  17:52  
We're going to

Kim  17:53  
be curious, and here's how you become curious. They are engaged in the relationships that we have as we are doing mathematics, and we are talking out loud, and thinking out loud, and involving them in the conversation. It's not that same type of model, practice where we state it. We represent it. Now, you do it.

Pam  18:14  
State it and act it out.

Kim  18:16  
Mmhm.

Pam  18:16  
We're not acting. We're not acting. We're mathing, and we're inviting them into that world where it's more of a mentorship of what it means to math.

Kim  18:27  
And I would just add on that in order for kids, anyone, people to math, you have to give them an experience where they're it's welcomed. And in that situation of model, practice, model, practice, they can't math because you've already told them what to do.

Pam  18:47  
Oh, that's nicely said. I mean, they could, but many don't. Like...

Kim  18:53  
Yeah. I mean, I guess I can ignore it and just mimic on my paper. (unclear).

Pam  18:57  
(unclear). Well, or you could ignore it like my personal son did,  and not write anything down and get not the best grades because... Well, honestly, he got fine grades because he was always right. The teachers very rarely knocked him off credit because he had the answers correct because he was reasoning. But that is super unusual. And we have kids out there who follow the rules, who do what they're told. I don't know that we want to knock that. I think we want to take advantage of it and say, "These are the rules here of mathing. This is what it looks like to actually do the mental actions of a mathematician." 

Kim  19:36  
Mmhm.

Pam  19:36  
One of the things that I wanted to bring up just in this conversation, if you don't mind, is while I was doing my research before the debate. And I was going to grab his name, and I forgot. But one of her colleagues had done a debate with on a Canadian podcast and said something like, "It doesn't really matter how good the lesson is because the learning is going to happen in the practice. So, we need repeated drilling over, and over, and over again. That's where the learning is going to happen." I find that really enlightening. So, first of all, I can just say I disagree with that completely. I disagree with the... That's not exactly what I mean. What I mean is the basis, what's behind that, what's under that, speaks to this idea that learning is rote memorizing something, and so it doesn't really matter how you were exposed to it the first time because I'm going to drill you over and over again. And if I just drill you on it, you're going to have it rote memorized. And then that's the definition of learning. I want to push back on that. Sue, who we work with, who we love. I'm not going to quote her right exactly either. She said something like, "It's almost like their definition of teaching..." No. "...of learning, is what the teacher's done, and it's a little bit less of the mental actions that a student's doing." Like, to actually have learned something. So, I'm teaching if I drill you on something, so that you have enough exposures to it, so you can have it rote memorized, but we would say you're teaching if the kids actually learning. And define learning. Learning is when they actually have mental connections and relationships that they can then logic through to reason for problems.

Kim  19:37  
Yeah, it reminds me of...

Pam  19:37  
Why are you laughing? 

Kim  19:37  
...working with teachers who say, "Well, I taught them." And in my head, I go, "Yeah, they haven't learned. So, have you taught them?"

Pam  19:37  
Like, the definition of teach, right? Yeah.

Kim  19:37  
It's tough because, you know, if your definition is you tell them, and then they're supposed to get it, then, you know, you might feel like you've done the job. But we have to take ownership of the fact that our job is to make sure that they have learned. 

Pam  19:37  
I

Pam  19:37  
got to tell you. I'm fussing with this continuum a little bit because I think I know what you mean, but I think I also might know how people might hear it. So...

Kim  19:37  
Ooh, yeah. Push back. 

Pam  19:37  
Tell me if I'm hearing you the way I think I'm hearing you. I wonder if people could hear, "So, it depends on the day or the way you feel. Or like sometimes you let kids kind of play around a little bit. But really the end goal on this, you know, you're going to keep the whole continuum, so that really you're going to tell them in the end. So, sure, we can do some conceptual understanding. We can throw some manipulatives at them, so that they kind of get, you know like, a feel for what's happening in the algorithm. But then we're going to tell them the algorithm." Like, I feel like that you could hear what you said about the continuum that way. I'll just say, from my perspective, I think of it a little... I think the continuum you meant was more there are some lessons that I'm going to let that are where the problem, the thing they're messing with, is rich enough that there's less quicker back and forth. That it's a bit more dig-in-able. That's more like what we would call a Rich Task. Which I know the terms. But then that's there are other lessons where in a Problem String, where I'm given a problem, and they're getting instant feedback, and  I'm representing thinking, and it's snappy, and we're going. But then there are other tasks we might do like I Have, You Need, where, bam, we're back and forth, back and forth, back and forth. Or even hint cards, where I'm saying, "Hey, from your own personal deck, here's 7 times 8. What do you got? And I'm not sure that hints working for you. Let's, maybe think of a different hint, so it works better. That's super like snappy. And we're actually wanting kids to travel that mental path often and quickly. Those are... Is that kind of the continuum you're

Pam  19:37  
thinking about?

Kim  21:06  
Yeah, I think so. Because when I think about like a really big Rich Task, I feel like I'm letting them grapple, and I'm observing, and I'm nudging, but I'm really just listening to see before maybe I interject too quickly. I'm nudging, for sure. And I'm... 

Pam  23:52  
You are interjecting.

Kim  23:53  
Yeah, it's just I feel like that's a little bit like, bigger, richer, deeper of a situation, and then... 

Pam  24:00  
Not

Pam  24:00  
unproductive. Not off without knowing what they're doing. Not, Whoa. Really, it's not that they're... Well, all the nots. Even in that richer, messier, everything you just described, kids are super clear on their task. It's not that they're all confused about their role. It's not they're confused about what they're to be doing. Not any of that at all. In that way, we're very, may I say, the word explicit. Like, we explicitly make sure they know their role, their task, what they can use, what their job, when they know they're done. All of that is very explicit. But the mathing itself is not, "Do this. Do this. Do this. Now, mimic me. I did it. Now, you do it. Now..." Wait, I do it, then we do, then you do it." I can't even get the order right.

Kim  24:46  
Yeah,

Kim  24:46  
I think it's interesting because when we talk about explicit, you and I are like, "Oh, we are very explicit. Like, we, they..." Listen, I...

Pam  24:57  
About their role.

Kim  24:57  
The structures, the systems, the. how do you know you're done type stuff. The actual math that's happening with any of these lessons is much more driven by where students are and where we are nudging them.

Pam  25:12  
And our goals and our standards that we have. Like, we are very explicit in the direction we're heading, and the what looks like success, and the...

Kim  25:23  
Which is interesting. 

Pam  25:24  
...correct answers are important.

Kim  25:26  
I think that's why this debate is important and why I was really happy to hear you include actual math and not just all these buzz words. Yeah. Because I think until you get to experience what we're talking about, then you, like you said, you might hear what we're saying and go, "That's what I think." Or, "That's what so and so thinks." But until you experience what we're talking about, the way that we mean it, by doing something with us, I think there's always going to be a little bit of a assumption or a little bit of...

Pam  26:00  
Well, basing what you heard hear on your past experience.

Kim  26:03  
Right. 

Pam  26:03  
So, we need to have a common experience, so that we're actually understand what each other are talking about, not just using words to say all the words, and then pretend that we're communicating with... Yeah. Yeah. Whew. Well, I'm glad it's over. I took lots of deep breaths. I really, really wanted to come across respectful. Kim, I don't know if you remember, one of the very first things that I said was that I invited everybody to when you say, "We're all going to agree..." Stipulate. "I'm going to stipulate right now we all want what's best for kids. We just have some differing ideas of how to make that happen. And let's have a debate, right now, on... Let's get those ideas out. Let's have examples, so that we know what we're talking about. And I'll just say that again. I think everybody involved in this conversation wants what's best for kids. We're all on the side of trying to help better kids lives. I think we could benefit from some examples of things that have helped us make that happen.

Kim  27:03  
Yeah. And I think I'll add that a lot of us have experienced the kind of classroom that was described from Tony. Many of us grew up in those classrooms. 

Pam  27:15  
How about most of us? 

Kim  27:17  
Yeah. But I don't know that many people have grown up in the classrooms like the one that you describe that you want people to experience, and so it can be very, very tricky for others who haven't experienced what we're talking about to be able to really understand, so... I mean, I think it's why we do the work we do. We're constantly putting out stuff, and workshops, and webinars, and challenges, and things like that, so that people have a vision for what, you know, what we're doing here.

Pam  27:45  
Absolutely. Alright, ya'll, hopefully that was crystal clear. Thanks for tuning in and teaching more and more real math. To find out more about the math is figure out able movement, visit mathisfigureoutable.com. And by the way, we will put the link to that debate, if you'd like to it, in the show notes. Let's keep spreading the word that Math is Figure-Out-Able!