Ensuring Every Student is Learning

One of the biggest challenges math teachers face is the variety of mathematical backgrounds students bring with them to math class.  Although state standards describe what students should know and be able to do at the end of the grade, more often than not, students move to the next grade or course with an incomplete mastery of pre-requisite content. In this episode of Room to Grow, Curtis and Joanie suggest that we reframe this reality: instead of thinking this is a problem to be solved, educators can embrace the diversity and leverage it to ensure that all students are progressing in their mathematical learning.

Our hosts start with an acknowledgement that all teachers are doing their best, and that all teachers want their students to be successful. From there, they discuss the importance of establishing strong relationships with student. Although it is cliché, it’s true that “students don’t care how much you know until they know how much you care.” When students know their teacher believes in them and is willing to do what it takes to help them understand math, they are often more willing to put forth the effort and work past obstacles to learning.

Curtis and Joanie complete this episode by sharing planning strategies that keep learning for all students front of mind, and suggest ways that math educators can make adjustments to their planning and instructional practice that help more learners engage. 

Be sure to check out these resources:

• Achieve the Core’s Coherence Map for tracing back pre-requisite math knowledge: https://achievethecore.org/coherence-map/  and this blog post that describes how the Coherence Map helps with unfinished learning: https://achievethecore.org/aligned/unfinished-learning-in-math-how-do-you-address-it/
• Research-backed strategies for supporting students: https://ies.ed.gov/ncee/wwc/Docs/PracticeGuide/rti_math_pg_042109.pdf 
• Article discussing Just-in-time vs. just-in-case intervention: https://pubs.nctm.org/view/journals/mtlt/113/10/article-p774.xml (*note: NCTM membership required)

Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com. Be sure to connect with your hosts on Twitter and Instagram: @JoanieFun and @cbmathguy. 

Transcript

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SPEAKER_01 0:00

Welcome to Room to Grow.

SPEAKER_00 0:01

I'm Curtis Brown. And I'm Joni Fenderberg.

SPEAKER_01 0:04

We work together at Texas Instruments, and we're glad you're here. We're looking to continually improve our practice, and we understand that you are too. We hope that you'll find this podcast as a room for you to grow along with us as we wrestle with and explore ideas about teaching math even better. So let's get growing. In this episode, Curtis and I dive right into one of the most challenging aspects of being a classroom teacher: managing the diverse mathematical backgrounds, experiences, and understanding that students bring with them to math class. In our conversation today, we consider some ideas for laying the foundation for this work with students through relationships and careful lesson planning.

SPEAKER_00 0:42

Ready?

SPEAKER_01 0:43

Let's grow.

SPEAKER_00 0:44

Well, Joni, it's great to be chatting with you again. Today we're going to be talking about this topic of ensuring students are learning. And really, when we talk about that, we really are talking about every student learning in our classrooms. And this is a this is a large conversation. I think there's a lot that goes into it. But maybe let's first of all just talk a little bit about maybe the what that means, the assumptions that we probably need to bring to the table or that we do bring to the table when we start talking about what we mean when every student is learning.

SPEAKER_01 1:21

Right. I I agree, Kurt. This is such a big topic. And this is something I think will speak to every math teacher. I know you and I have had this conversation before that this is something that for all the years that I was in the classroom and for you too, that, you know, being able to reach every single kid and be sure that every student in the room is learning is a huge challenge for teachers. So I think the the first assumption that I want to lay out is that we're all doing the very best we can for every single student. That every I've never met a teacher who didn't want every single student in their class to be successful. Even that kid that made you kind of crazy, or that kid that felt like they didn't care. Like they do care. Kids care, and you know, they'll they'll present their their you know safety when they're feeling uncomfortable in different ways. But um, I think uh coming from the assumption that we know teachers want their students to be successful, and I think that that is a a super important place for us to say that's where we're starting from.

SPEAKER_00 2:22

I think that's so true. Yeah, I've never met a teacher that didn't want every student sitting in every seat in their classroom uh to learn. And and really even the kids that are out in the hallway, right? I mean, the ones that aren't in your class. Teachers, we just have this, we just have this innate thing about um making sure that students are successful. And so, yeah, of course, um every teacher is doing all the things that they they know of to do um and the best that they know to do um to get those things done. A second one that kind of comes to mind um is this idea that um I I want that there's a that there's a silver silver bullet that there isn't one. No, uh I don't think silver bullet to students learning, right? Um and and just making sure that we understand that um students are individuals, that students have individual needs. I know that you and I uh are different and we each and have individual needs um on a, you know, when it comes to to us as learners. And so I think it's really important that we also remember that about our students, that we think about it um as them as individuals, and then and then look to find the things that work for each student. I think that's important.

SPEAKER_01 3:37

I think so too. And I'm I'm reminded of, you know, I heard Phil Darrow talk once and he said something that was so simple and so profound. He said, you know, we need to quit treating the diversity of learning experiences and backgrounds in our classroom as a problem to be solved. You know, he suggests, and I it certainly resonated with me, I taught for 20 years, and never once did I have every single student fully prepared for the for the greater course that I was teaching, right? Like that diversity in background and understanding is the norm. So he suggested instead of positioning that as a problem that we need to solve, let's just build on that as a strength. Like that should be uh something that we we take into consideration in our instruction and that we leverage to benefit all of the students in the room. And uh it was a really cool concept, and uh, we can tie back into some of that a little bit later in today's session. But I think understanding that, you know, everybody is an individual, but that doesn't mean as a teacher, I need to prepare 30 different lesson plans for the 30 different students in my classroom, right? There are still some commonalities and some ways of approaching mathematics that will be common across groups. So as long as I'm teaching in ways that provide opportunity for students to access learning in the way that makes sense for them and that is what they're ready for, um, I think that that that provides a reasonable strategy for educators to reach every student without feeling like I have to do one-on-one instruction with every kid. Now that we've established some assumptions, what do you think are the most important ideas for for teachers to think about as they begin planning around this idea?

SPEAKER_00 5:26

So as we've had this conversation, I think um probably one of the things that has come to my mind uh a little bit, and you and I have talked about this before, this idea of a teacher becoming an investigator, thinking uh a little bit about their students and and coming into this uh this room with you know looking at these students and trying to decide and figure out what their needs are and and how best to meet them as individuals and then as groups. Um you know, we've talked a little bit about this idea that that, yeah, you can't prepare um 30 lesson plans, right? I mean, that's a little bit um crazy to think about trying to do 30 lesson plans, but multiple strategies and multiple approaches and making sure that what we're doing is open enough and big enough for students to have uh multiple entry points, I think is is important um as we get started in some of these things.

SPEAKER_01 6:22

I I love that idea of being an investigator, right? And I think one of the key components to that is establishing really strong relationships with the students. And most importantly for me, especially for kids who struggle, is ensuring that they know that you believe in them. So, you know, I'm thinking back to in in my teaching career, there was a year where I taught um the classes that were the lowest performing seniors that were in this, you know, like very last chance for math credit. And these were kids who had been unsuccessful in math for years and years and years. And on the first day of class, I remember standing in front of this group of kids and saying, How many of you think you're bad at math? And every single one of them raised their hand. And I said, Well, here, I want you to know I don't think you're bad at math. I just don't think you've had a teacher who's been able to explain math in a way that makes sense to you. And I'm gonna do everything I can to make that different this year. And, you know, the the difference in their attitudes even from that first day and the willingness to work hard and and persevere just completely shifted because those kids knew I really believed in them. And then they could be part of that, you know, me being an investigator, like you said, like trying to figure out what worked for what worked for this kid who has, you know, traditionally found trouble and they then become a part of that and help me understand their thinking a little bit better. What are the go ahead.

SPEAKER_00 7:44

I think that's a really good point there. And and I think, you know, you you put something out there that's that's important, valuing these students and valuing their thinking. I think, you know, when you think about um students who are maybe shy to speak up, or someone who is a little unsure of where they are, if we are creating uh classroom environments where um student thinking, regardless of the preconception that they may uh bring up or expose, um, if we are able to value that thought process and and find the the good piece of of what they're doing and and begin to kind of capitalize on that and and use that as a as a way to kind of work into uh whatever the strategy is that we're trying to get across or the teaching uh topic that we're trying to get to. Um it opens the door for students to be a little bit more safe. I think, you know, we talk about creating safe environments, and the more we can create a safe environment, the more our brains are a little bit more open to learn, right? I think that's huge. Um and I also, you know, I have a friend of mine that that um maybe someday we'll get him onto the podcast. Um he talks a lot about um humor opening up uh the doors. And and the more we can use humor and and things that make class feel a bit more comfortable, um, I think all of those pieces kind of go into establishing this rapport with our students and these relationships and being able to be an investigator into what works for them. So um all of those things I think are important.

SPEAKER_01 9:30

Yeah, I'm I'm remembering too something I read years ago and trying to come up with good strategies for working with these students and uh valuing them, I it made me think about this idea of social and academic capital. Um, so this researcher was talking about how a teacher can really forward um the capital that students hold in the classroom. So by the teacher showing for, you know, there's you know, one or two or three kids in a class where most of the kids are successful and and the small handful aren't, the teacher ensuring that the whole class knows that the less successful students' responses and thinking is just as valuable goes so far towards their learning because it builds their social and academic capital within the classroom. So I agree, uh being really intentional about these things is super important.

SPEAKER_00 10:35

As we kind of think about maybe the next piece of this, um there there's this idea of, okay, so I've gotta figure out, I've gotta investigate all of my students, I've got to establish rapport with them, I've got to figure out what kinds of things work for them. Um but next we kind of think need to think a little bit about the planning and and that that piece and making sure that we as teachers are ready to go because um students are individuals and they require a lot of of uh preparation, if you will, right to to be prepared for each um each kind of group of students. So what are some things that maybe go into that?

SPEAKER_01 11:16

Yeah, I I think the planning is maybe the most important piece for ensuring that all students are learning, really educators taking the time to think very carefully about the content, about how the content fits into the bigger scope of students learning across the board, both what instruction have they had before, what learning have they had before, and then where is this going? Like what the thing that I'm gonna be working with them on this week or this month, where is that gonna go later in this course or later in their progression of learning? So really understanding the whole picture, I think is the first responsibility as an educator and to say, okay, I I can sort of see it's like having, it's like having an internal map of a city, right? Like the better you can understand all the roads and you know how things connect and how do I get from one place to another, that bigger picture is more helpful in helping somebody new to your town navigate, right? If you only know your little portion of the city that you live in, you're not gonna be able to help people that are there visiting in a broader scope. So I think really having a deep level of understanding and then being able to place the mathematical topic that I'm working on within that progression of learning. So thinking about it, you know, maybe one analogy is a ladder, right? Like if I want kids to reach the third rung of this ladder in today's lesson or this week's topic, what are the things that they need to have in place prior to that?

SPEAKER_00 12:46

I totally agree with you, Joni. Uh, one of the things when I got when I got started teaching, because I came into to uh teaching as uh say, I'll call it a second career, only this has really mostly been my career. But um I came in from engineering, from the engineering world. And and so math was just a thing that that made sense to me. And um I didn't know it at the time, but I hadn't established as myself as a math learner, um, all of the connections among the different topics. And I I am continually fascinated by the just these subtle connections between topics um in in mathematics that open doors for students to conceptualize things. Um, and so I think, you know, as a teacher, thinking about what are the play the ways that um this lesson or this this topic could go off the rails, um and and seeing those those potential um, we'll call them preconceptions creep in um that that will lead students to to understandings that just just simply aren't true. Um, you know, I I think those knowing those things going in, um being prepared for those kinds of things also is very, very important. So it's not just um and it's not just my understanding of maybe my my math's place in the sequence. I think that's really, really important. I think it's also a really good understanding of of the the topic itself, but then also what are the preconceptions that my students might come in that could cause some trouble in making making this connection or making this understanding uh as we go further in, right? So just kind of having this knowledge of we know what they've been taught, we know where we're headed, but then what are the little things that that maybe they've made a preconception on that that really isn't what should be knowledge, right? Or should be something that they've grabbed hold of and they've kind of latched onto that. What are the things I'm gonna have to potentially undo uh or or help them help them figure out? So I think all of the things that I've got to do.

SPEAKER_01 15:00

I love I love that you brought that up because I think first of all, I love the term preconception over misconception, right? Like it's not that they learned it wrong, it's just that they haven't learned it fully yet. So really coming again, that I know it's a simple tweak, but that strengths-based language instead of deficit-based language is so, so important. And it it ties back to what we've been saying about relationships and rapport. But I'm also thinking about this idea of obstacles and roadblocks, right? And I think there's there's some value, there's some really important components there because sometimes that obstacle or that that thing that we think is gonna misdirect and and have them gain uh an incorrect belief or understanding about the math, sometimes the smartest thing to do is to hit that obstacle head on so that you can really address it and move through it. There, and and and knowing when do I hit the obstacle head on, because this is actually gonna help enhance the learning of students in the classroom, versus what are the obstacles that are gonna segue the learning that are gonna actually take us off topic, that are gonna, you know, keep us from getting to the important nugget of mathematics that we're trying to learn in the lesson or unit. So being able to understand that, and I think back to when I was in early in my career, I was doing most, if not all, of my lesson planning by myself at my kitchen table on Saturday mornings. And, you know, I think there's so much value now later in my career when professional learning communities became a thing and our high school principal actually scheduled time for teachers of the same course to talk and plan together. And how much more value that brought to me to be able to talk with other teachers about, you know, what are the things that I I actually want to engage, the those sort of troublesome issues that I want to engage the kids in, and how do I help them navigate those in a way that's productive and gets them to learning in the end versus what are the things that are gonna, you know, throw us all off track and and make a mess? And and I don't know, as I'm saying that, there's there are certainly times where messy learning is perfectly wonderful. But I think it's it's easy to um it's easy to want to just be super directive as a teacher to totally control where the learning is going at all times. But I also think that that's not a good strategy for ensuring that every student learns. There has to be some room within my instruction, some flexibility to let kids go where it makes sense for them to go. But then that there's a whole lot more impetus on me as a teacher to know how to how to sort of funnel all of that to ensure that we get to the place that we're trying to get. So it's a really tricky, uh, important balance. And again, I think, you know, just the way our conversation is going shows how important this aspect of planning is. It's just huge.

SPEAKER_00 17:53

Yeah, I do I you uh you hit the nail on the head in that in that last little segment there, talking about um my knowledge of how do I get these students who, you know, if I'm really comfortable with a topic or a subject area, um it makes me much more confident to allow my students to go and explore uh the things that they know and and approach problems from different uh from a different direction than maybe I would or a maybe one of their students next to them. But being able to have a have a a good enough foundation in the topic itself to be able to circle the wagons a little bit and bring them back um into that learning, I think is is so so important. Um, so I think that there's there's a you know, we're doing this, we're recording this uh podcast still in the time of of the pandemic and and some more uh isolation. And there's a term that's kind of come up uh quite a bit in conversation and in articles and things. Um, this idea of just in time instruction or uh just in case instruction. And what does that mean? And what is that really talking about? I think that's something that maybe we should should address here. I think it makes sense.

SPEAKER_01 19:14

Yeah, I I've loved hearing this terminology come up. And we can link in the show notes. There's a great article from the NCTM journal back in, I believe it was October. My my good friend Kathy Martin, who's uh director in the Denver Public Schools, wrote about this idea of intervention and how we should be focused on just in time intervention rather than just in case intervention. So I really like this language because I know in my experience, we know that mathematics is cumulative. So the students, the the learning that students bring to the table helps contribute or deters from their learning of the grade level or course mathematics. But it doesn't mean that that past learning has to be fully fleshed out. You know, back to the Phil Darrow comment, right? This lots of varied backgrounds is the norm. It's not a problem to be solved. So instead of thinking like, okay, a ninth grade student is coming to Algebra One and doesn't know their multiplication facts, right? Doesn't don't, they don't know their times tables right off the top of their head. Well, we have to, instead of saying, well, they don't know their times tables, so we need to go back and instruct them from third grade content and fill in all the gaps uh all the way up to get them ready for ninth grade. That's just there's just not true. Like even research says that that's just not true. So that would be just in case intervention, right? Like filling everything, be sure they know everything that's prerequisite. Just in time intervention, though, is thinking about, okay, what's the mathematics of my lesson or unit? And then again, back to that prerequisite, understanding the bigger picture. What's what are the important mathematical ideas? Not the skills, not the, you know, can they get the right answer to, you know, six times eight, but what are what are the big understandings that need to be in place in order for them to access the important understandings of the lesson or unit? And those are the things that I'm actually gonna take the time to address prior to the learning. That's just in time intervention that uh attempts to give kids a more solid background foundation on the stuff that's actually relevant for what it is they're trying to learn today.

SPEAKER_00 21:30

So, Joni, I'm gonna try to give a concrete example uh of what you're talking about here, and you can tell me if I'm uh barking up the wrong tree a little bit. But let's, you know, if we think a little bit about, say, uh, a ninth grader in my uh algebra one class who's come in and uh it's it's readily apparent that this student is struggling with just knowing their their multiplication tables. The the the root memorization of uh uh you know, three times three and six times eight, and and this this being able to recall those right offhand, just not something that this student has totally grasped hold of and and struggles a little bit with remembering those uh those numbers. There are some things we can do to help them maybe over overcome that getting of the answer part um that that make it easy. Like we could give them a calculator, we could give them a multiplication table, we could allow them those tools that they'll have that they have access to to help them get those um I'll put answers in quotes here for just a second, right? But they can get those those answers. However, when I start in my um section on you know slope as rate of change and and we're beginning to explore this idea of linear relationships and looking at slope as as the the rate of of change of that graph my students if they only have an understanding of maybe multiplication as repeated addition. So they can remember three times three is I gotta get three groups of three and add them all up or they remember that you know that that is the case or even maybe the the point of of an array maybe in their mind this this idea of six times eight gives them six copies of eight rows or whatever. Like they can come up with those things but um the that is not necessarily super helpful when I start thinking about uh slope as rate or even maybe further in and multiplying three fourths of X uh plus seven like that just doesn't that isn't super helpful in understanding the way that that works right is that that that's kind of what you're addressing here.

SPEAKER_01 23:48

Exactly so what would be the mathematical concepts that you would actually take the time like it's not the it's not the getting them to spit out the answers faster. It's not flashcards kill and drill on their multiplication facts. It's what do they need to know about multiplication so that they can understand rate of change and what that means in terms of a a linear function or even a an exponential function.

SPEAKER_00 24:12

So where what are the big mathematical concepts that you would want to be sure they have a foundational understanding of to ensure they could access that rate of change idea so thinking about multiplication as scale thinking about um this idea of of multiplying three times um this this function this exponential function what is that actually doing across the entire um picture of my graph uh graphical representation how can I how can I approach that from a um coordinate points and what does that do to each coordinate point what does that do to the overall uh image of the graph those are the kinds of concepts that I want to kind of establish so this idea of uh multiplication uh as scale I think will make uh a difference right so I can go back in and and think and help them build that concept maybe before I get uh into this this idea of of slope as as rate right so so maybe maybe we establish that maybe we establish that and I'm thinking back to like for for our listeners that might be in a similar position that I was when I first started teaching and don't have a lot of other people to co-plan with sometimes knowing what is that what is that important prerequisite mass I just want to put up a quick plug in and again we can link this in the show notes too on achieve the core the coherence map such a great way to actually go in and just click on a standard click on the math topic that you're going to be working on and the coherence map will show you what are the prerequisite standards and you can see you can even click back into those prerequisite standards and get some tasks that you could then give to students to help you know assess where are they in terms of this prerequisite understanding and how do I find out what they already know about the prerequisite knowledge so that I can build on that as we're moving into this bigger concept.

SPEAKER_01 26:07

And it's great because you can track concepts like high school concepts, you can track all the way back just like we're describing, right? From you know an algebra one kind of idea, we can track that all the way back to the early, you know, the elementary school grades like late elementary school grades and and find some tasks that help us know what students understand and help bring them closer to the understanding they need to access the grade level content.

SPEAKER_00 26:30

Well I really like this idea of just in time instruction. I think that's a super important one uh to hold uh hold tight to and remember. Um so I just you know I really enjoyed this conversation today Joni I think um we've kind of covered a lot here most of it being um you know what can we as teachers do? Of course we acknowledge that there's responsibilities to students as well but there's always something more that we as teachers can do to inspire and and encourage uh our students and I think we hit on two really important ones um planning and and relationships today. So I'm really excited to continue this conversation though, um maybe in a future episode.

SPEAKER_01 27:15

Yeah I really see Kurt that this is going to be maybe even a thread that we take through you know each of the future episodes even just in this last planning piece I think we we laid a nice sort of foundation for our next session where we're going to talk about representations and exploring with mathematics I think we'll be able to make some connections back to this idea too. So great conversation and just you know my kind of last words for our our educator listeners out there is to you know be go easy on yourself. Set yourself up for success and you know make make progress each day rather than feeling like you know you're gonna listen to our 30 minute podcast and be able to change your behavior and suddenly all kids will learn it's about progression and moving them forward. So great great talking I'm looking forward to future conversations on this topic.

SPEAKER_00 28:01

All right we'll keep talking next time.

SPEAKER_01 28:04

Well that's it for this time. Be sure to check the show notes for the resources we mentioned and others you might want to explore. We would love to hear your feedback and your suggestions for future topics. And if you're enjoying learning with us consider leaving a review to help others find us and share the podcast with a fellow math educator. See you next time