Ep 134: How do you know you're in a mathematizing classroom?

Pam: 0:00

Hey, fellow mathematicians. Welcome to the podcast where Math is Figure-Out-Able. I'm Pam.

Ashley: 0:07

And I'm Kim.

Pam: 0:08

And you found a place where math is not about memorizing and mimicking, waiting to be told or shown what to do. But it's about making sense of problems, noticing patterns, and reasoning using mathematical relationships. We can mentor mathematicians as we co-create meaning together. Not only are algorithms and step-by-step procedures not particularly helpful in teaching mathematics. But rotely repeating steps actually keep students from being the mathematicians they can be.

Ashley: 0:37

I'm really excited about today's episode, Pam.

Pam: 0:40

Woah! Yay!

Ashley: 0:41

So, this week, we want to tackle something that is a reality for so many of our teachers.

Pam: 0:46

Yeah, absolutely.

Ashley: 0:47

This one is for you, leaders, and coaches, and administrators, on behalf of teachers everywhere. So, teachers have come to us saying that they're working on building a quote "Math is Figure-Out-Able" classroom or we could say a mathematizing classroom. Where they have students experience more and more mathematics but also have to be careful because they are very aware that they have administrators who will be doing walkthroughs and they'll be getting evaluated. And we have heard, "If only my admin or my coach knew what I was trying to do here."

Pam: 1:21

Yeah, we have heard loud and clear from so many teachers that we really respect that if we can help administrators, and coaches, and anyone who's in sort of a position of doing an evaluation or helping teachers become better at their craft. If we could help them understand what are the kinds of things that you would see in a mathematizing classroom? What are the things that you want to see, and how could you actually help teachers get better at their craft in a way that's helpful and purposeful and not less helpful because you don't understand what the teacher's going for/where we're heading in a mathematizing or Math is Figure-Out-Able classroom? Totally. Often, we hear of instances. And for sure, Kim, you and I both had it when we were in the classroom. We had a sort of checklist systems where whoever was evaluating us would come through, and they had a certain kind of list of things that they were looking for. I had one principal that would script everything I did, but then he would go back and label everything according to a certain checklist. I had a different principal that had a real short checklist system. There were just a few things that he was looking for at certain times, and sometimes that list would change. But I didn't have input into that list. Many of our teachers don't have input in the list. Many of our principals and administrators don't necessarily have input in that list. But if we had a way of having some common understandings of the goals of a Math is Figure-Out-Able teacher, I think we could all help each other understand how we can handle this. It's going to happen.

Ashley: 3:05

Sure.

Pam: 3:06

We're realistic. We're not saying, "So, no more evaluation." Like, that's not even helpful. We're not suggesting it would be helpful to not evaluate. We actually want your help. We want to invite you that we can have a more productive help if we all have a more common understanding of what things mean... That's not quite the way I want to say that. Of what we are hoping for, and aiming for, and working toward. That's probably the best word I could say. What we're working toward in a math is Figure-Out-Able classroom. So, our goal is to help you understand what you might see or want to see in that Math is Figure-Out-Able, mathematizing classroom. And to do that, we are going to focus on the NCTM, effective teaching practices. But we might do it with a little bit of a twist. So, if you're a leader out there, and you just tuned out. You're like, "Oh, I've seen those." Maybe stay. Stay focused a little bit. We're going to give you, definitely, our Math is Figure-Out-Able bent on those effective teaching practices. So, let's do it.

Ashley: 4:12

So, there are eight practices, and we're going to take just a minute to talk about each one of those and what we think that they mean. So, the first one...

Pam: 4:20

Ooh, Kim. I'm sorry. Can I just really, really fast?

Ashley: 4:22

Yeah.

Pam: 4:23

Just for a little, tiny bit more background. The NCTM... No, that's not true. The Common Core State Standards came out with a mathematical practice standards, which describe standards for students in a mathematizing, Math is Figure-Out-Able classroom, at least (unclear). Those are practices that we would want to see in a mathematizing, Math is Figure-Out-Able classroom for students. And then, NCTM, NCSM, and leadership got together and said, "If those are the kinds of student practices we want to see, what are the effective teaching practices to make those happen?"

Ashley: 4:57

Yeah.

Pam: 4:58

And I think that was a brilliant move. Well done. And those are the teaching practices that we're going to focus on today.

Ashley: 5:03

Yeah. So, the first one is "Established mathematics goals to focus on learning." Which sounds fabulous, right? So, so fantastic.

Pam: 5:14

Yeah.

Ashley: 5:14

But what I think a lot of people take that to mean is,"Let me go through my standards, and let me copy down the 3.2.a standard onto the board, on a chart, on a flip chart, on my dry erase board, and maybe read it aloud, maybe just posted each day."

Pam: 5:34

Maybe make students write it in their notebook for the day. (unclear).

Ashley: 5:40

Yeah.

Pam: 5:40

"So, at the end of today, I will..." And then, they kind of, maybe even have written that standard in their own "I will" language, "I can" language.

Ashley: 5:48

Yeah.

Pam: 5:48

And we're suggesting that"establishing mathematics goals to focus on learning" is a very necessary and important thing for teachers to do to focus their lesson. But that does not translate into proclaiming that standard in front of students at the beginning of the lesson. We do not find those to be equivalent statements. So, do we want to help teachers establish goals? Absolutely. Do we want to force them to write that objective on the board at the beginning of the class? We are saying no. We're saying that that actually can be counterproductive for classrooms.

Ashley: 6:28

Yeah. So, what it does mean is that kids walk away knowing what they got out of the time that you spent together.

Pam: 6:37

Yeah. So, let's put the emphasis on, we need the teacher to know ahead of time what the mathematics goal is; we need the student to walk out of the classroom being really clear on the objective of the day and what they were able to accomplish, what the goal was, and what they're maybe still working towards. So, the goal could be a work in progress. But we do need students to have clarity around the fact that they were working... I don't want to say working hard. That sounds like it was all about effort. It's not. That they were developing something specific during that class, that day. So, we want to maybe help leaders and teachers flip the timing, that it's not about students.... We want students to be really clear they're going to learn in class today. But we don't want to... I don't want to say "spoil the surprise", Kim. Help me put better words around it.

Ashley: 7:34

Well, there are definitely some standards that if you announce ahead of time,"This is what we're going to do today," then students are deprived of the opportunity.

Pam: 7:48

Of the sense making.

Ashley: 7:49

Of the sense making that they would be doing in class. Yeah.

Pam: 7:52

Yeah, they're deprived with a sense-making, and grappling, and by definition, learning that they could be doing in class. And it will also nudge them too much towards just doing. "Oh, if that's the thing, then I'm going to do it, and get an answer, and be done, and I'm not going to engage in the grappling." It can have the effect of trivializing and making everything too much about getting an answer and not about developing the learner as a mathematician.

Ashley: 8:21

Yeah.

Pam: 8:22

Cool. Alright. So, that was number one. Number two, the next effective teaching practice is... Oh, by the way. Maybe I should just say it. We're not going to fully flesh all these out. Our goal of this podcast is to give you some ideas for each of these essential teaching practices to help you, help your teachers become the mathematician, the math teacher they want to be. So, we're just going to mention the ones that are sparking the most for us right now. We're not pretending that we're giving all of these a full treatment.

Ashley: 8:52

So, can I just jump in real quick? So, what that means for leaders, and administrators, and coaches is that you might have teachers who have actually thought about the fact that they want kids to be thinking about goals for the day, but they recognize that writing it on the board tells them what they're going to do, rather than giving them the opportunity to think and process. So, rather than, you know, "Ding! You didn't write 3.2.a"...

Pam: 9:18

"...on the board."

Ashley: 9:18

Right. Maybe a conversation will help give you a vision for why they didn't.

Pam: 9:25

Ah, so in other words, we're encouraging leaders, have the conversation with teachers."Tell me about your goal for the day.""Help me understand that you had established a

Ashley: 9:31

Yeah. mathematics goal for this lesson." Cool. And then, how are students going to walk out of here being clear on that goal? Yeah, nice. Alright. Number two"Implement tasks that promote reasoning and problem solving." Mm, that's a good one.

Pam: 9:50

We like that one a lot.

Both Pam and Kim: 9:51

Yeah.

Pam: 9:52

But we also recognize it's not trivial. It's not trivial for teachers to find those tasks. It's also not trivial for administrators, evaluators to evaluate the task itself. So, that also might benefit from a conversation about why particular tasks were chosen. Especially in what way were they going to lead towards that established goal that we just talked about. Yeah. We would suggest that a thing that you'd be looking for is, that it's not about direct teaching. Now, I've been reading some tweets lately that... I'm so clear that language is so tricky because there are some bright, articulate people who are pushing back on the pushback on direct teaching. Because they've said, "Well, it depends on what you mean by direct teaching." So, it does. It absolutely does. We are suggesting that it's not about giving students a step-by-step procedure that you clearly outline, "This is what you do. Here, here, here. Now, go practice it 29 times." That does not fulfill the effective teaching practice of implementing tasks that promote reasoning and problem solving. We would look for kids grappling. We would look for kids working together. We would look for kids mimicking and say that's not a good... if we see mimicking, then we don't(unclear) the task hit the target. Is there anything you'd add to that?

Ashley: 11:22

Yeah. I would say that if kids are really problem solving, that's messy work, and so there's probably going to be some conversations. This might come up again later. But some conversations and some... You said "grappling", but it might not be so cleanly tied up with a bow at the end of your five minute bell ring, ding time. So, like, some of it will spill over. Some of it will provide conversation. And that's perfectly fine.

Pam: 11:54

It's desirable.

Ashley: 11:54

It's expected and desirable.

Pam: 11:58

And so, when we said that we want students to walk out at the end of the class being clear on what they learned, that might not be them saying "I learned this procedure."

Ashley: 12:08

Yeah.

Pam: 12:08

We don't want it to be, "I learned this procedure" or "I mimicked such and such." But it also might not. They might be able to say, "I worked on this problem, and I worked hard on it, and there are some things that I'm grappling with about that problem."

Ashley: 12:19

Yeah.

Pam: 12:20

That could be not only acceptable, but a desirable outcome. At some point, yes, there'll be closure on that problem. But it might not be in that part of the lesson that you observed that day. But it would be good to have a conversation with the teacher. "You know, I noticed that students left here still grappling with some things. Tell me about that." And then, that becomes a conversation that I think could be really productive for everyone.

Ashley: 12:48

Yeah. Okay, number three is "To use and connect mathematical representations."

Pam: 12:56

Love! Models and modeling is such a huge thing for us. Making thinking visible. We say that a lot. And maybe we'll just leave it at that. In a mathematizing, Math is Figure-Out-Able classroom, there should be modeling of thinking, making thinking visible. Yep.

Ashley: 13:16

Yeah. I appreciate especially that connection, the word "connect mathematical representations". Because it's not just about,"What's the one way you solved it, and the one way you represented it," but the connection between student thinking and seeing their thinking in other people's work that's so valuable. I think maybe it doesn't get enough credit.

Pam: 13:41

Yeah, and similarly but slightly different is, it's not just about multiple representations. It's about the connections between multiple representations.

Ashley: 13:48

Yeah.

Pam: 13:49

Nice. Cool. Alright. The next one. Number four:"Facilitate meaningful mathematical discourse". Which you kind of brought up just a minute ago, Kim. Yeah. Definitely.

Ashley: 13:59

Discourse is so valuable as a mathematician. And, you know, we spend so much time, I think, hearing about what classrooms should look like. And you and I are both not really into compliance necessarily. You know, we like community. And kids want to be a part of the community when we set those up meaningful. And this "meaningful mathematical discourse" does not mean, you know, there's arguments or there's, you know, a riot happening. But kids justifying their thinking, and arguing their points, and ruminating on what's going on in their heads, and then grappling with it out loud to others. Learning is happening in all that.

Pam: 14:46

Absolutely. And if you'd like more on that, we have an episode on Do, Say, Represent where we really talk about acquisition of language, and how language can help learning occur, and how it's the grappling with being able to put things into words that can. And then, representing that thinking and that process of... Well, we call it Do, Say, Represent. So(unclear) check out that episode, if you'd like more on why mathematical discourse is so important.

Ashley: 15:15

Yeah.

Pam: 15:15

Cool.

Ashley: 15:16

Alright. The next one is"posing purposeful questions".

Pam: 15:20

Yeah, that has been something that I've been thinking about for a very long time. I was always very interested in better questioning even before I knew about Real Math. But then, when I dove into the research, and into my kids classrooms, and figured out what it meant to do real math, all of a sudden, questions became this whole other gamut of, it's not just about me questioning you through a procedure; it's not just me questioning to see if you got the thing that I was having you mimic. In fact, it's not those things at all. Now, it's me questioning, so that you learn through the questioning. It has everything to do with that discourse that we just talked about. Yeah. Kim, you wanted to say something about"funneling"? Do you remember what you were going to say about that?

Ashley: 16:04

Oh, gosh, no, I don't think I do.

Pam: 16:06

I think we were talking about the difference between. And NCTM in the discussion about these standards does a very nice job of parsing out focusing questions versus funneling questions. Which actually, the first time I had seen that I was a young, super young teacher. I just started teaching. And it had come out in an article in The Mathematics Teacher, where they had parsed out those two, is where I really began thinking about the different kinds of questions and how we want to help students focus when we question. Focusing versus funneling talk about patterns of questioning. And so, it's not just about a single question, having the right question in a moment. But it's really about your pattern of questioning. So, if you picture... Focusing questions is all about helping students focus on the question, focus on what they're thinking, focus on the relationships. Versus, a less effective pattern called a "funneling pattern". And a funneling pattern.... And it's funny, I'm doing it right now with my finger. I'm like swirling a tornado in to be a funnel in the air right now. Because a funneling pattern sort of takes kids through this tornado, this funnel, where you ask a question, they answer it, and that leads to a next question, and the answer that leads to a next one. And by the end of it, you've reached the end of the funnel. They've gotten an answer. But without that funnel, the kids lost. Like, if you don't re-funnel them through that, then the kid can't reason through the question. And so, we want to lean away from funneling patterning of questioning to a focusing pattern of questioning.

Ashley: 17:40

Well, and I think. I'm so glad that you talked about that because I think that it could feel like, you know, as a teacher or as a leader, you're watching your teachers. It could feel like it might be helpful because you'd see a teacher circulating to a student, talking with them for a few minutes, then walking over to the next student, talking with them for a few minutes. But that time that is so valuable in every single classroom is maybe helping a student get an answer to a problem. But if they're asking...

Pam: 18:12

If you're only funneling.

Ashley: 18:14

...funneling type questions. Then, yeah. That hasn't grown the learner and help them do anything other than get an answer to that one problem.

Pam: 18:23

And what we are looking for is to help students develop their mathematical reasoning and their mathematics, so that they can reason through all sorts of problems, not just get an answer to one kind of problem.

Ashley: 18:35

Yeah.

Pam: 18:35

Yeah, nice. Cool.

Ashley: 18:37

Alright, we got a few more. So, the next one is "build procedural fluency from conceptual understanding." And I know you have lots to say about this one.

Pam: 18:48

And this would be the one that I completely disagree with. So, I like the other eight, but... Or, the other seven. Sorry. There's eight total. I like the other seven. So, I'm going to tweak number six, so that it works for us. I want to"build fluency with conceptual understanding". So, the standard reads "build procedural fluency from conceptual understanding", and I think the word"procedural" gets in the way. I've talked to the writers of the standards, and they're all like, "No, no, no. We just want kids to, you know like, have a sense of what to do. We want them not to like sit there and have no no idea and just fumble around." Well, absolutely. I couldn't agree with that more. I don't want students to fumble unnecessarily. I don't want them to have no idea what to do. I don't want them to sit there stymied with no. And they're like, "Yeah, see, we want fluency." So, I'm okay if the definition of "fluency" is, is that when students see a problem, they dive in, and then they let what pings for them work. They let relationships like, "Oh, yeah. I could use that relationship." Or, "Ooh, I could go that way." To me, that could define fluency. That students have the wherewithal to jump in productively using relationships they know, solving problems based on their understanding. Which then begs that we would have conceptual understanding. So, do I want conceptual understanding? Absolutely. The part that I have a problem with is that there would ever be a goal to get fluent with rote memorize procedures. And writers have said to me, "No, no, no. That's not what we met." Except then that's what the paragraph underneath kind of says. And I understand. My bet is that when the writers got together... Can I say this? I'm going to say it. I bet that when the writers got together. I'm well aware of the writing group that created these standards. And, ya'll, you did a fantastic job...almost, for all of it. But in this one case, I think this was a an example of negotiating. I think there was some negotiation, and some people didn't really want this procedural fluency idea in there and others did, and so they kind of compromised. And I think this one was the compromise. And I'm not going to compromise. And I'm going to say, "Nope, we do not need to just start with a little conceptual understanding because our goal is to get kids fluent at the step-by-step procedures." No. What we do want is understanding because we are developing mathematicians, and mathematicians fluently think and reason using relationships. So, that would be my push on six. Alright, let's move on, and let's. Kim, do on anything to six or did I just pontificate enough about that one?

Ashley: 21:30

[laughs]. I think we're good.

Pam: 21:33

Okay, cool. So, the next one: "Support productive struggle in learning mathematics."

Ashley: 21:39

Oh, this one is so near and dear for me. So, "productive struggle" is, for me, this idea...and we've talked about it before...that we want kids to grapple a little bit. But this for me screams, "Do you need time or help?" And we've done a whole episode on that, and so, you know, we don't need to take a lot of time here. But kids just stymied and going nowhere is not helpful. But we don't want to supply too much. We don't want to give them too much that they're not growing, they're not learning. And so that the key word for me here is to "support", support the productive struggle, to step in in the right moment and help guide, rather than give too much.

Pam: 22:27

And we like this one so much that we're going to actually spend a future podcast on this very idea, so stay tuned for more about "supporting productive struggle in learning mathematics". And then last,"Elicit and use evidence of student thinking." And this might be the hardest one for teachers who have a background of being taught sort of traditionally, and it was all about memorizing step-by-step procedures. What does it mean for students to even think? And so, we look for in a mathematizing, Math is Figure-Out-Able classroom. We look for problems, like we said, implement tasks that promote reasoning and problem solving. We look for problems that are worth thinking about. Meaning, students have access to them. In fact, can I just say? I heard something on Twitter the other day that...I want to do a whole episode about this, so I'll say a little bit about it now...where someone said, "the most damaging idea in math education right now is that you can give students problems, and that they will then learn math on their own." And the teacher, the reply in the tweet was, "If they don't know what they're doing, it's stupid to just have them sit there confused and frustrated, trying to figure out how to do it." And I so understand that perspective IF math is about memorizing step-by-step procedures that someone else gives you. But if Math is Figure-Out-Able, then we can give students problems that by thinking about them help them develop more mathematics, and that they can, that we can be... It's not about handing them random problems that kids have no idea how to do. At all!

Ashley: 24:08

Yeah.

Pam: 24:08

It's about being very judicious about handing them problems that are just on the edge of their zone of proximal development. That by grappling with those problems, the students develop as mathematicians, and they develop mathematics. And the teacher is an active participant in that. Yeah, go ahead.

Ashley: 24:24

I'm so glad you clarified that because I think that it could be heard as, "I'm a new teacher. I've got fourth grade students, and my job is to just give them stuff and say,'Go'." But I love that you said it's just outside their zone of proximal development. They have something There.

Pam: 24:43

On the edge. Just on the edge.

Ashley: 24:44

Just on the edge. That you support, and nudge, and encourage, and provide the support that they need. And then, you up the ante. Do the next bit.

Pam: 24:53

With very purposefully chosen problems. It's not about random frustration of trying to guess what's in the teacher's head. At all! So, in order to have those kinds of tasks. When we do, then as you're evaluating, as you're using these to help you have a conversation with the teacher that you're observing, you're looking for. One thing you're looking for is that they are eliciting student thinking. And then, they're using that evidence of student thinking in some way to move the math forward. Alright. So, the eight essential teaching practices are wonderful with a little tweak on number six. And we highly recommend that that could be a place for you to consider as you look for a mathematizing, Math is Figure-Out-Able classroom.

Ashley: 25:42

Yeah. So, leaders and administration, I would love to invite you. We have a challenge. The "You Can Change Math Class" challenge is next week. Registration is currently open. You can register now at mathisfigureoutable.com/change. Grab your teachers. Teachers, if you're listening, grab your leaders and coaches. You can do it together and have some great conversation about what a Math is Figure-Out-Able classroom can look like.

Pam: 26:11

Absolutely. So, we will continue that conversation and help more and more make a Math is Figure-Out-Able classroom. The dates are January 18th through the 20th with an extra bonus day on the 21st where we will meet in the evenings at 7 p.m., Central time on Zoom. People from all around the world will be joining in. If you register and can't make those times there are recordings available. It is a super fun, engaging, productive way for us to all make math more and more figure-out-able. That also means that registration for our online workshops is opening soon. We only open registration three times a year, and that's coming up, so get your funding ready. Alright. Thank you 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. Let's keep spreading the word that Math is Figure-Out-Able!