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Episode 30: Building Math Coherence in Your School - An Interview w/ Karen K., Barbara D., Sarah B.



Desiree Harrison 0:00

Our beliefs dictate our thoughts, and our thoughts influence our actions. Even with a shared curriculum, do you really know what your colleagues think about the teaching and learning of math? If you're a parent, do you really have an understanding of the teaching philosophy of your child's teacher? What about how others believe word problems should be approached? About using different strategies and models? Or even in how someone would define these terms? Have you shared your beliefs about these concepts with others? When's the last time you reflected on them for yourself?


Today's guests explore why it's essential for educators to have these conversations that lead to an increase in equitable actions, teaching practices, and coherence with all stakeholders, including parents.


Desiree Harrison 2:54

Today on the podcast, we have the authors of The Math Pact: Achieving Instructional Coherence Within and Across Grades. Welcome, Karen Karp, Barbara Dougherty, and Sarah Bush to the podcast.


Karen Karp 3:36

Thank you, Desiree!


Barbara Dougherty 3:38

Aloha, Desiree!


Sarah Bush 3:41

Thanks so much for having us, Desiree!


Desiree Harrison 3:41

Alright! This is really, truly, my pleasure. Thank you for doing this and sitting down and talking with us about this math pact. And, I have read this book cover to cover and I've actually gotten two of my colleagues in my district - we're doing a book study with this, so I have the elementary version, we have a middle school coach that's reading the middle school, and then our curriculum coordinator is reading the high school version, and we're really talking through the chapters, and the points, and all the reflections that you all give in the book, and we're talking about core values, which this book really helps educators to think about core values and to challenge our core values. And to bring teams of teachers and team of educators together.


I know I work very closely with them, but sometimes we're still - we still get in those silos.


So you all have this plan in - this agreement. So can you please tell us, what is the Mathematics Whole School Agreement and why do schools need it?



A professional picture of Dr. Karen Karp outside posing next to a tree trunk
Dr. Karen Karp

Karen Karp 4:42

Thank you. Thank you for asking that. It's what we care deeply about so we're happy to talk about it. The Math Whole School Agreement is really a structure to help people move from an individual person, showing up at school and doing their own thing, to be a team. A member of a team that's working together. Where everyone actually has collaborative game plans. We know what we're gonna do, we know how we're gonna say it, and we're gonna work on it as a group.


We're actually even going to work up and down. So if I'm a 4th grade teacher, I'm working with the 3rd to make sure it's a cohesive experience. We want to know what 5th grade is doing, and it's all aligned.


So, what we want to do is have students from the very beginning, walk out of the building, with the math knowledge and processes they need to be successful. We really want them doing math, and we want them building really long-lasting and deep understanding.


So, the plans that we put in place involve several components that we travel through, you know, as we talk about how to do this in your own district. And so those plans involved:


-Shared Mathematical Language, we're all using the same language, we're not using borrowing and carrying, we're not using reducing fractions, we've got an agreement about what we're going to say.


-We want precise notation


-We really would like to see people using multiple representations, and we want the same representations to be revisited, so kids are not learning new things every year, but it's a building up of images, of drawings, of concrete materials.


-We want them to avoid rules that expire and we want them to instead build generalizations.


And so, that's the plan. That's how people walk through and everyone's involved. That's the key - all stakeholders: administrators, coaches, substitute teachers, paraprofessionals, families, if there's any student teachers, or students coming from methods courses in the universities nearby - everyone has to be on board. Even to the point of passing a handout out when you get a sub in the building so they know what to say and what not to say.


So, what happens though when we don't do this? Well, a kid can go - what's their path to 4th grade. Well, in 1st grade they learned flip-flop property, 2nd grade they learned something and it's now called the "ring around the rosy" property, 3rd grade they learn the commutative property, and by 4th grade everyone's confused.


And so, and each 4th grade teacher gets a different mix of kids from different teachers who have taught different things. So as you get older, children moving in the grades, there's more combinations of all different information that they can be carrying with them. I just want to say- that's not good.


And so, what we want to do is create excellent schools where we're not perpetuating confusion, where we have a unified classroom management system that we can put in place, well we can surely put into place a unified math teaching approach. -that's equity.


Desiree Harrison 8:01

What you just ended with, like, that is equity, was really powerful how you said that because we talk so much, especially now, about equitable instruction, that's a huge buzzword in education and on social media, and in other forms of media, right now too, and about leveraging opportunities, and you're talking about all stakeholders and that piece about not just substitute teachers but student teachers was really interesting to me because they are a part of your community once they enter your building so making sure that everybody is on the same page is so crucial.


And you were speaking to some of the core agreements and one of those being to use correct and consistent language, and along with equitable instruction, there's been a lot of talk about student identities. And the podcast here we talk a lot about building positive student identities and different experiences that can help with that. So, how does using correct and consistent language positively affect student identity?


Photograph of Dr. Sarah B. Bush smiling outside in front of a tree trunk
Dr. Sarah B. Bush

Sarah Bush 9:10

Thanks, Desiree! For this one, I think it might be important to back up a little bit and talk about teacher identity. When we think about the overall math pact and the mathematics whole school agreement, really at the core of the mathematics whole school agreement, it really has the potential to transform individual teachers' mathematics identities first. If you think about it as a team - a mathematics whole school agreement team is really working together, teachers start to build greater mathematical understanding themselves, more confidence that they've worked to implement new strategies that they're learning as they're team is working together, and there's really becoming this expectation that these strategies they're implementing and reflections from instructions will be shared among the team, and that's just going to foster their growth even more.


And so we believe with these positive shifts and teacher identity, the way in which the teachers cultivate their individual classroom cultures, their mathematics classroom cultures, this is going to cause some shifts to where students start to contribute in different ways and that will begin to foster their positive mathematics identities. So, it will kind of go from the teachers to the students through the mathematics whole school agreement.


Also, I think another thing that's important to point out, is that students as they're now moving from grade to grade, they're now getting a consistent message because of the mathematics whole school agreement. And so mathematics no longer appears to be this set of mysterious tips and tricks that are disconnected, and instead, students see as they're moving along their mathematics careers, they're building a true foundation of mathematical understanding. And so when students go from grade to grade, they see strategies and approaches they are familiar with now, and those - that knowledge can really serve as a launching point instead of everything always seeming new.


And so, this will contribute to both their confidence and their competence in mathematics and students will really start to see mathematics as something much more stable, and not something constantly changing every year, or even more often.


This, you know, I kind of think this phenomenon all around is something that could be easy to miss when we as teachers are only thinking about our individual classrooms, which is why a team mathematics whole school agreement approach is absolutely essential! So we don't miss this.


And, you mentioned the language component, but really all of these ideas apply to all the components of the mathematics whole school agreement. So we really feel that overall that a mathematics whole school agreement really nurtures both teachers' and students' positive mathematical identities through this work.


Desiree Harrison 12:07

Wow! I really like that idea about nurturing and you bring in that confidence because I find that some teachers that I work with- and I used to be one of them - just not having that confidence and just relying on the book. And if it wasn't in there, or just anything to help kids not struggle. But, you know, I've learned differently since then, since beginning, but it's important to think about where you've been and then where we can go together, too.


Another core agreement that you all have in this book is about evaluating rules that expire. So you started to touch upon it. So, how can a rule really expire, if we're talking about - that term expire, I think about food expiring and no longer being healthy for you to ingest, so, can you talk to us about how can a rule expire and how does this evaluation of these rules lead to more cohesion within and across grades.




Headshot of Dr. Barbara Dougherty
Dr. Barbara Dougherty

Barbara Dougherty 14:00

That is such a fabulous question because this work actually started in thinking about rules that expire. So, Karen, Sarah, and I, have three articles that first launched this kind of notion about rules that expire. And so, from that, we started thinking more broadly, like, across grade levels and within courses or across courses.


How does this effect students?


And so, when you think about something that expires, I like your food analogy where you just don't want it anymore right, and so what happens when students realize that rules have expired that they have held on to for so long - it's very unnerving for them because they trusted that. They think these rules are like the mathematics rules that if you break it -


I don't know if they expect like a mathematics policeman to come after them? But it's like, these are so set, that you can't mess with them.


So, much like using accurate math language consistently, it's really important to consider how the use of rules can effect student learning. And that in fact will also effect their student identity.


So here's a telling tale, from a study of over 5,000 high school students. We saw how persistent rules can be long after students leave the grade in which the rule was introduced or focused on.


Six or seven years after students were first exposed to multiplication, a large percentage of high school students still believe that multiplication makes bigger. And they see that this rule is something that even though they've had other math experiences and they've seen where that doesn't really apply, they hold on to that and it affects their thinking about working with other generalized quantities.


So, their maturity in thinking actually didn't even help them move past that. So, beginning to think about - "Okay, now I'm a high school teacher, how do I deal with this?"


To eliminate or unteach a rule is really tough because they become so embedded in student thinking. So the best way to eliminate is just to make sure that the rule that you give a student doesn't expire. And that's really important. And what that means is that the rule is:

•Long-lasting (it's not contradicted in later leaning). So, for example, the rule, "Multiplication makes bigger", that rule's going to fall apart when they're working with fraction multiplication, or when they're working with multiplication with a positive and a negative number. But actually, that rule wasn't a good one to start with because when they multiply by zero or one, it's already got a problem.


But, you kind of shove that one under the rug. But later, that rule is going to stick with them. So, instead of focusing on rules, we advocate in a Whole School Agreement to think about instead -


What generalizations are important for students to develop?


And those generalizations are going to stand that test of time and create that foundation. Basically a mathematical fabric that's going to weave together all the math concepts and skills and help to make that cohesive for students so that they're better able to see how things are connected across topics and within topics.


So, the generalizations are a much different way to go than rules. Because rules tend to be things like- you always have to do something this way. Where generalizations are these broader conceptual ideas that students form through well-designed tasks and they're able to see then how smaller ideas can be put together to create this much bigger idea. And it's where that stronger retention, and stronger retention leads to less re-teaching; less re-teaching means that we get to some significant math content in that grade level.


So, that evaluation of rules that expire, really forms that mathematical content discussion that's often needed within classrooms - across classroom teachers and within grade levels. Even across the whole school part. Hence, the Whole School Agreement.


Desiree Harrison 17:46

So as you were talking I was trying to find the chart in chapter 6, it's on page 92, where you all have a rule and generalizations sort. And I really liked taking that because it just helped me reflect on what I have been teaching and what I want to continue to teach to make sure that I am teaching generalizations and not those rules. And I also found it really helpful that you all have a multi-page chart in here about, not just the rules that expire, but you give a detailed explanation to help teachers understand for each specific one, so, that is extremely helpful, especially, as- like, you're getting a team together and you're thinking through this, you have some excellent talking points that you've given us.


Barbara Dougherty 18:35

Well, Desiree, I should add also, that those rules, actually came from much of our own classroom personal experience. These are rules that we started with, where we were like, "I've said that rule before". And so, I think it's just a good reminder for all of us that we've all done this in the light of trying to help students understand something or shortcut frustration, but in the end, it's not really helpful.


Desiree Harrison 19:02

Yeah, like the - I circled - "To factor, use a factor rainbow" - because I'm like, "Oh! I've done that!" I mean I've done lots of these, but that one was like, "Oh! I didn't even - that was like not on my radar -


Karen Karp 19:19

Well, that's how we know all these! We pulled all our knowledge, Desiree, to say, "Well, what did you teach that you regret!" [laughing]


And so, we're all learners! And I think everybody needs to know - as a matter of fact, we're using our mistakes to put together these ideas for others so that we can, if anything, shortcut their movement to get out of that.


Sarah Bush 19:41

I mean, we really kind have named our articles, "Rules We Regret Using and a Few Others", you know, because so many did come from us. And others we had heard about in our teaching experiences.


Karen Karp 19:53

Yeah, it could have been titled, "So Sorry Former Students". [laughing].


Barbara Dougherty 19:59

Right. because when we reflected on a lot of these rules as we were expanding our list of rules, and we realized how many of them we had actually used in our own teaching, we began to wonder what happened to those kids that we told that to? Where are they now? And, it's really an interesting thing when you reflect on your own practice. And one of my hopes as people read through the book is that it causes that self-reflection, right?


And even for us, writing it, we self-reflected.


So, I hope that it creates that opportunity for teachers and coaches to do that.


Desiree Harrison 20:37

Well, you definitely give us lots of opportunities because you have the "Try it out!" sections so we can write directly in the book, and some white space in some of the pages to make sure we can write some notes. So, I've definitely done a lot of reflecting, but I do definitely suggest that this is read in a group, so it's not just, read on your one and then come back together to start the agreement. But, read together and start the agreement together.


So, I'd like to come back to this idea of confidence and also expectations in terms of teachers. And how educators are always doing amazing things in the limited amount of time that is given, but one of the core ideas that you all have is that teacher lesson and unit planning time has to be protected. So, I just want you all to talk a little bit about why this planning time is a non-negotiable and how teachers and parents can advocate for this time.


Barbara Dougherty 21:43

We're going to approach this from a couple of different ways to answer that question. I think about when I'm working with teachers in their lesson and unit planning, that really an important part of that is being able to think very thoughtfully about what you are teaching. And, let's all be real, that there are times when you're like, "On my gosh! I need a quick lesson, so I'm going to do something that I did 3 years ago, but I really liked the lesson so I'm just going to throw it in because I didn't have time to plan."


In the elementary Catalyzing Change book from NCTM, published in 2020, there was a discussion about depth and coherence of what we teach. And those two characteristics are so important in so many ways. They're especially critical if we want students to understand and be prepared for the expectations of a math class.


The expectations also apply to teachers. It becomes expected that teachers will use learning progression. When you start to plan together, those learning progressions help to make the cohesion lesson to lesson to lesson be very clear and it's easier to establish the connections across topics and from lesson to lesson and also from topic to topic. So those connections and that coherence built on a really solid way of building students' mathematical understandings is important.


As you're working on a Whole School Agreement, being able to articulate within a grade level or across a school which learning progressions - because let's also understand that there are multiple learning progressions that might be picked from - but having a common one that teachers are working with provide that content development within a grade level and across grade levels. Also understanding that there are the Standards for Mathematical Practice. So, how are we going to use those within our lessons so that's also consistent.


And then emphasizing sense-making. I'm going to use a very particular example - one of the examples that we site in the book is with a charter school in Key West, Florida - [. ] Public Charter School. At that particular school, they began working on a Whole School Agreement and the best part of that- the teachers began to think about - what is it that we need to include in our planning time?


They already had planning time together - their principal had already structured that, so that was a good thing. But, you can have planning time together, but may not always be used effectively. So, one of the things that they started thinking about as we began working on the Whole School Agreement, is they really wanted to cut down at the beginning of the year on routine teaching. And they realized they all had different routines that they were teaching students. So they instantly know the first 20 days of class was going to be a wash, just trying to get routines established in a math class. So, that was one of the first things that they decided to focus on. They wanted students to know what to expect from year to year to year.


So, as they did that, the first step was to begin to plan together at grade level. So rather than taking whole school, individual grades started to plan together. They set up a timeline for their very first unit and started that planning by identifying the significant mathematical target learning that they wanted to come out from the unit.


That target learning included both generalizations and skills. Now, that sounds like that probably took an hour - oh no! That took a long time because that required them to unpack the mathematics.


When you start to create what generalizations you want, you have to unpack the math and say "How is this all connected together and what are the big ideas?"


Because sometimes our focus in on that particular skill- this is much bigger than that.


So, that took some time to do that. But after doing that, then they became more fine-grained and by components of what an individual lesson would look like. So they decided on starting with a warm-up, and having a homework discussion and the homework discussion was not about "what did you do on problem 3?"


They looked at students explorations and then they looked at how we end a lesson. So, having all of those components is great, but then they got more fine-tuned and they said -


What is in each of those individual components- what does a warm-up look like? What does an exploration look like? What does a homework discussion look like? How do you structure a reflection and summary?


And so, they began to identify all of that. Now the beauty of that was that they each now had a consistent language that they could talk to each other and to the principal and to the coach, all in the same way. So everybody talked the same language - if they said, "I'm doing a warm-up", everybody knew what that warm-up should like like.


The plus side to this is, the students knew what it looked like because they became much more transparent with students. And because the students knew, that started creating more

parent communication and conversations. So parents now had a common language.


So, when parents would come in for conferences, at each of the grading cycles, especially in the elementary grades, they could talk with parents about, you know, during our exploration time - and parents are like, "Oh! I know what that looks like! I know exactly what that looks like!"


And they can say, "Your child is really good with the discourse that is happening in the classroom."


The parents became very familiar with the same language. They became familiar with the student expectations, they knew now that if they're helping students with homework, it's not about showing your steps, it's about showing your thinking. So they became better able to also support students in creating better explanations of their math work. (27:28)


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