Establish Mathematics Goals to Focus Learning

In this episode of Room to Grow, Joanie and Curtis continue the season 5 series on the Mathematics Teaching Practices from NCTM’s Principles to Actions, celebrating its 10th anniversary. This month’s practice is “Establish Mathematics Goals to Focus Learning.” This is defined as follows:

Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions.

In today’s conversation, our hosts unpack the key components of this principle. First, they discuss how learning goals, focused on important mathematical understandings, differ from procedural, process goals, which may include skills and procedures that are not directly connected to the underlying mathematical concepts. Next, the discussion turns to situating goals within a learning progression, which helps teachers stay focused on what is relevant to their grade level or course, and provides a venue for students to be active in their progress toward learning. Finally, effective mathematics goals guide instructional decisions, helping educators know which tangents to explore and which are distractions from the intended learning. We hope you enjoy the conversation, and that it extends your thinking on mathematics goals for learning.

Additional referenced content includes:

·       NCTM’s Principles to Actions

·       NCTM’s Taking Action series for grades K-5, grades 6-8, and grades 9-12

·       NCTM’s myNCTM forums (membership required).

·       How learning goals serve as a guide – NCTM Teaching Children Mathematics blog post

·       Rachel Harrington’s appearance on the Math Learning Center podcast/blog discussing mathematical goals

Did you enjoy this episode of Room to Grow? Please leave a review and share the episode with others. Share your feedback, comments, and suggestions for future episode topics by emailing roomtogrowmath@gmail.com . Be sure to connect with your hosts on X and Instagram: @JoanieFun and @cbmathguy. 

Transcript

Season 5 Episode 5:

Establish Mathematics Goals to Focused Learning

00:00:00:00 - 00:00:02:00

Opening music

00;00;02;15 - 00;00;21;27

Joanie tease clip

A learning goal is not going to be skill based. Now that being said, the skill is a part of it, right? Like certainly being able to recognize that I need to find common denominators and know how to find common denominators is important. But that's not the underlying mathematics. 

00;00;22;29 - 00;00;57;10

Joanie intro

In today's episode, Curtis and I unpack the importance of establishing mathematics goals to focus learning. We speak carefully about how a mathematics learning goal, as described in NCTM’ s Principles to Actions, is more than just the restating of a standard or a statement about what skill students will master. Instead, these goals focus on big, important mathematical ideas and drive all other instructional decisions we make as teachers, as well as how students participate in their learning.

This is a meaty topic, so let's get growing.

00;01;00;13 - 00;01;26;11

Curtis

Well, hey, Joanie, I am super excited to be joining you once again this month. We're chatting on the Room to Grow podcast this month, we're recording our fifth episode of season five of the Room to Grow podcast, and we are on episode 49 overall, which I'm really excited for. That's very fun. So for our 50th episode, we got to do something cool for that.

00;01;26;11 - 00;01;52;19

Curtis & Joanie

I don't know, we do. We got it. Let's start planning around that. Yeah, right. Let's start planning. We're recording it next week, by the way. I'm just kidding. Just kidding, just kidding. So, hey, look, we're going to be talking today, about effective teaching of mathematics. We are continuing our progression through the Principles to Actions book and the, mathematics teaching practices.

00;01;52;21 - 00;03;08;20

Curtis

And today, we're talking about this idea of effective teaching and mathematics establishes clear goals for the mathematics that students are learning. It situates goals within learning progressions and uses the goals to guide instructional decisions. And this idea of establishing clear goals or establishing goals is, is an important one. I think it's a really…it's kind of a different guiding principle, maybe, than some of the other, things that we, we've talked about, here because I think this one spent we spend a lot of time in our planning, space, which I know is a passion for you. And I'm sure we'll have an opportunity to think a little bit about that in this next 30 minutes or so that we get to get a chance to talk about this here. But, I've got a question for you. Yeah. I want to maybe kick this thing off with a very broad question that we can take all kinds of different rabbit trails from. And that is why, why is this a standard for math, mathematical practice of teaching? Why is this important for teachers to do? Why is this a practice teachers should be doing?

00;03;09;19 - 00;03;29;18

Joanie

Well, that's an excellent question. And what a great set up for me to jump right in and say, this is like the most important one. And you know, since principals, two actions came out, we've seen lots of different interpretations of it and lots of folks embedding it in other, you know, situating it in other professional learning situations and one of the things I've seen and used in recent years is kind of like a visual diagram that shows the relationship between the eight mathematics teaching practices outlined in principles, two actions, and that we've been focusing on during this season of Room to Grow. And this one, the established mathematical…I got to read the right language because I'm doesn't establish mathematical goals to focus learning. That was not coming to me. Sorry. Is it's at the top of any diagram that you might see, because without this, it's really hard to select the right task. It's really hard to know what are the right questions to ask. It's really hard to know. How do I think about uncovering and using different representations? It's really hard to know what types of student thinking I want to focus in on. So having this establishing the mathematics goal for the lesson or for the week, or for the unit or whatever your time frame is, is, is like the thing you have to do first, because that's the thing that drives all of the other decisions that we make as an educator. And I want to take some time, Curtis, before we get too far into our conversation to to really call out, like, this one is so important. And it also sort of feels like, well, duh. Like I establish goals every day, like my principal requires me to write on the board every day. Students will be able to write like so. Of course I'm going to write goals. But we're not just we're not talking about general goals when we're talking about this foundational mathematics teaching practice. There are lots of kinds of goals that teachers think about and write and utilize in instruction. But this is a very specific kind of goal. And, you know, you read those words. I want to go back and read them a little more slowly establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional decisions. So we're not talking about like, students will be able to add two fractions with unlike denominators. That's not a mathematics goal. That's a performance goal that might be, you know, something we're trying to accomplish by the end of a lesson. But that's not the mathematics that that we're focused on. So this mathematics goal that establishes learning is, is deeper than that and it's about what is the heart of the mathematical understand that the lesson that we're working on, or the unit that we're working on is built. 

00;06;16;24 - 00;09;00;27

Curtis

I think that’s an important distinction and I almost, well one  of the things that came to mind and you can, guide this as, as it needs to be. But one of the things that came to mind for me was maybe the difference between a goal and a skill. And, and when you said, okay, we're going to be able to add fractions with unlike denominators. And there's, there's some skills students will be able to write. There are some procedures and skills that are important there, to be able to do. Right. But I think the larger part of that is the concept that, hey, these are in two different units of measure. These are right. So we have, we've we've got to kind of get ourselves on a level playing field. And that, that concept of, of what does it mean to put two units, two things together that are maybe represented with different kinds of, denominators, which means we have different denominations of, thing that we're counting by, right? We might have thirds in one, we might have, you know, eighths in another. And we need to be able to think about, what those units are that we're dealing with, and then we can start. Oh, well, where do they have a common unit? I you see what I'm saying? Like this, this idea of, of being able to perform the perform the task, be able to perform the skill. There are things that I need to do. We need to come up with a common denominator. We need to, you know, figure out how that that works in terms of, of my, my current fraction, my one third and my 3/8 or whatever it is that I had. Right. So I can work with those together and come up with things where I'm now beginning to count together. But the establishing of a mathematics goal might be that my students understand. Yeah, what it means when I have common denominators. Right. That that the denomination of each of those two things. Now we're actually talking in the same units and I need to get them to the same units. That's different than the skills that it takes. Oh, I need to make sure that I now take those thirds and I break them up into eighths together. So that makes 24th. And oh, I got to take those eighths and I got to make thirds out of that. So now I've got 24’s  there. Now I can start kind of working together when I'm talking in terms of 24th and 24th, is that does that making sense? What I'm getting out of it is yeah, I'm talking about skill. When we talk about this idea of establishing clear goals.

00;09;01;22 - 00;09;17;22

Joanie

A learning goal is not going to be skill based. Now, that being said, a skill, the skill is a part of it, right? Like certainly being able to recognize that I need to find common denominators and know how to find common denominators is important. But that's not the underlying mathematics.

00;09;17;22 - 00;09;37;16

Joanie & Curtis

The underlying mathematics is understanding the concept of. I mean, there's a lot of ways we could go with this, and maybe I could have picked a better example than, than adding fractions. But this is kind of a fun one. No it looks good. Right. So everybody realizes that yeah, it's it's like a better understanding of what do we even mean by adding these things.

00;09;37;16 - 00;12;29;27

Joanie

And what do we each of these individual, you know, fraction terms mean. What do they represent. How right. You know, how they they're, they're part of the same whole but they're different part. You know what I mean? Like how do we it's thinking through all of those things. So, you know, maybe a more immediate, because we love to talk about the kids in our lives. You know, the more immediate work that I've been doing with my nephew has been around factoring quadratics, right, and thinking about factoring polynomials. And I even remember saying to him the last time he was here for doing math together, like, okay, here's how we factor. I may have mentioned on a previous podcast, like, this kid just gets factoring, it just makes sense to him. And so he's catching on really easily and I'm taking advantage of that. You know, the skill is coming easily to him to get deeper with him and say, what are we actually even doing here? Like, why are we factoring in? He's like, I don't know, because this is my homework. This is what the teacher told me to do. So, so being able to factor a trinomial into, you know, factor, quadratic trinomial into its linear factors is maybe a performance goal. That's a skill. But the mathematics behind that is understanding that there are equivalent forms, but that we can write polynomial functions in. And there's a purpose for different forms. And so there's something there's something that factoring into linear factors tells us about the polynomial that we don't necessarily immediately recognize about that polynomial function when it's not in factored form, right? So that there is a, another example that gives a nuance between, well sure, the, the skill of how do I get those linear factors, you know, what are the what are the processes that I go through? What do I have to think about that certainly a part of it. But the bigger and more important understanding is what why am I even doing this and why is it important mathematically? And how does this fit into the mathematics that I've already learned and the mathematics that I'm going to be learning? Right. So how how is what I'm working on right now part of a bigger picture of mathematics? And I would argue that the danger in focusing on performance goals, rather than focusing on the bigger learning goal, is we can have students and Curtis, I would argue for decades, our U.S. system has created students that might be perfectly successful with skills, but don't understand how those skills fit into the bigger picture of the mathematics. They don't understand the bigger picture of the mathematics, even if they can successfully complete the skill.

00;12;29;27 - 00;12;39;00

Music break

End of Segment 1

Start of Segment 2

00;12;39;00 - 00;15;44;11

Curtis

Speaking as a student who is recovering from that, very diagnosis that you just made, I mean, yeah, right. I graduated high school, went to college, got a degree in engineering. Began teaching mathematics to students. And it probably really wasn't until I was in my teaching career several years. And then maybe even afterwards, when I began to realize that I was good at skills. But I had a long way to go in understanding. Yeah, mathematics and making that connection between things. Yeah. Within mathematics. And I think, I think that's actually maybe the more important piece of this puzzle, at least for, for me, as it relates to and I know a second topic here was supposed to be how is this important for students? And I'm jumping the gun a little bit, but for students, I think we, we can get caught up in, hey, you know, this is my homework and I'm supposed to get this done and I'm supposed to get this executed, and, and that's what homework sometimes is. Yeah, we're need to learn practice. We need to we need to get a skill accomplished. You can't ask a student, when a ball is going to hit the ground that's been thrown up in the air and do some kind of a projectile motion type problem without being able to find the zeros. Right? Of a quadratic. And the way you find the zeros of a quadratic often is if it factors reasonably. At least sometimes we set things up so that they do that. I wonder if we really intend to trust test the concept or if we're testing the skill. But. Right. But you know, I the point is, I need to know that the idea of the ball hits the ground, and it's modeled well by a quadratic, and the ball hits the ground whenever that quadratic crosses the x axis. And the x axis is where we happen to have the roots of this quadratic, which happens to be the zeros which happen to be the solutions to linear function times linear function equals zero, right and, and so I can realize that I need this factor and I can execute the factoring. But until I really have an understanding of why that even matters or what's really even going on here, oh I have this linear factor. That's you know, some value. But when x is equal to that, this other linear factor happens to go to zero.

And so that means that at that time this overall quadratic is going to have a value zero. That that idea is is really what we're driving at when we're saying, hey we need to establish a learning goal.

00;15;44;20 - 00;18;23;00

Joanie

Exactly, exactly. And I think I, I love that you took us to the student perspective right now because I think, you know, again, I tried to do this with my own kids. I know from the stories you shared of your work with your own kids, my work with my nephew, certainly letting students in on the secret here. Right. Like, here's what the big learning that you're trying to get at provides them an opportunity to actually actively partake in their own learning, like if they understand where we're going, if, if, if my student gives an answer like my nephew did of you know, why?

Why are you factoring these trinomials? And he says, because that's what my homework is. Well, then his his own ability to track whether or not he's on track with his learning just becomes, did I get it done or not? Right. But but if we can if, if instead we can help him understand that, hey, you're learning to factor as the bigger part of understanding quadratic functions and all of these things about them.

And I'm just thinking, like I named one, you know, possible learning goal around factoring quadratic functions, which would be, you know, understanding equivalent forms of functions and why we would care to represent them in different ways. But even what you just talked about under understanding, you know, the zero product property and why setting one of the linear factors equal to zero gives us an x intercept. Like there is another like that's a that's a bigger learning as well than just the single skill of. Can you find the x intercepts. Right. So there's lots of bigger ideas that we could be working towards. And knowing which one is important is helpful for me as a teacher. Like, how do I you know, what what paths do I want to take the student down and which do I want to avoid right now? But it's also so great for the student to know what's important and what's noise. Right? And yeah, what's nuance and what's solid. And it just helps build the, as you said, the coherence and the connections. That's at the heart of learning. We know science tells us that we don't learn things in isolation. We learn things by building on previous knowledge. So helping students know, like, how does this fit into the bigger picture and what's important about what you're working on right now? It just brings in this active participation in their own learning that we don't get if we don't have those learning goals. 

00;18;23;00-00;19;52;07

Curtis

No joke. As a, as a math nerd, in my head just now as you, I'm not even being able to let go of this, this whole idea of the linear terms and what even in my thinking about I should have known started mean you shouldn't have done that because, because in my head, here's what's going on as we're talking about it. Because I've got it here, I've got a graph, I have a I have I have the graph of each of the linear terms. Right. In my head, I have these two linear functions that I've got going on in my head. And I'm taking the outputs and multiplying them and I'm plotting that. Yeah. And I'm creating this, this little…really it's a composition of functions. I've got y1 times y2 is equal to y3 right. So I've got this and I've got this I mean this is so ridiculous about what's going on in my head right now. But as I thought about like, oh that's where that zero is going to happen, is where one of these guys crosses the x axis.

I mean, what a beautiful visualization. Yeah. Of what we're what we're talking about there. And a fun way to, to, introduce that idea as the zero product property. Yeah. It's like, hey, this this quadratic is sort of created by multiplying these two linear functions together. How fun, how fun is that? As a different way to think about, quadratics.

00;19;52;07 - 00;20;09;15

Joanie

Oh, sure. One thing, though. No, I just can I just say really quickly because what you what you just did on kind of going off on what's running around in your head is, is exactly what I was saying before. Like, when you have this mathematical goal established as an educator, now you can make all sorts of decisions, right? Yeah. Like, how am I going to present this and what are other ways I could help them think about making sense of it? And you know, just what you just described, like plotting two different linear functions and, evaluating them for different values of the independent variable and noticing that when the dependent variable is zero, that's when we get, where it costs. And, you know, there's so much there and all of that, you put in the context of, I want students to understand this mathematical idea. So once you have that really solid, then how you navigate students through that learning becomes much, much, much more clear. 

00;20;51;24 - 00;23;30;00

Curtis

No joke, no joke. And I think the other piece of this puzzle, that this helps.

Also, you know, we were talking before we started the podcast that the idea of coherence, that this really is sort of the way we establish coherence, across what we're doing. And it also, I think, allows me as the teacher to, To, to have students who can think outside the box or outside of the skill set that we're focused on on a particular day. I promise you, not once when I was teaching algebra two, did I ever graph two linear functions, make a table, make a tea table with, you know, multiple, columns and say I've got y1, y2 and y3 and y3 equals y1 times y2 and mult do these things here. And then notice that y3 also is how that quadratic that multiplies. Right. The, the expanded. If I, if I multiply those two binomials together, I expand them and do that, I'm thinking to myself, why didn't I do that? Because that's, that's a cool. That's a cool way to kind of investigate and establish and create like a real understanding for my students of like, this is what's going on. When we factor this, we are establishing these two things. Now I am thinking about I think I've seen a, an activity out there on the, on the, the hub there on our website, where this actually is explored that way, but I, I totally am just visualizing in my head how fun would that be? But what I really am excited about here is that this gives me flexibility.

Because if I've established what our what our learning goal, the clear learning goal is, then there's lots of things that can fit in underneath of that goal, and it establishes flexibility and creativity. In my classroom, my students can think differently about how they solve a particular problem. It gives value. And I know that's an important piece of the puzzle, right? We, we recognize all the thinking that students do as valuable. Right. And so being able to connect their thing, their thinking, and creative way of, of attacking a problem, I think that's, that's really, a value piece.

00;23;30;27 - 00;23;42;12

Joanie

I 100%, 100%, because that ties back. And I know that opportunity for creativity and individuality and student learning is a huge passion of yours. And I think that's spot on when we know what the mathematics learning goal is. And students know what the mathematics learning goal is, it opens the door up for more opportunities around that creative thinking and exploration. And even, you know, I'm kind of connecting back to some of the ideas I jotted down. I read the Taking Action Series book, the high school 912 version of Taking Action. In preparation for this conversation and one of the little notes I jotted down to myself, is learning goals promote mathematical reasoning and problem solving. And that's exactly what you're talking about right there. Right? Is like how, how we how the goal is stated and communicated to students gives them the guardrails around what they should be reasoning about, right, and focus for where they should be, you know, engaging in productive struggle and and trying to figure out, you know, what this means and where the connections exist. So I love that you talked through that example like that.

00;24;47;16 - 00;24;57;10

Music break

End of Segment 2

Start of Segment 3

00;24;57;12 - 00;25;29;07

Curtis

You know, another thing that we, have thought a little bit about, and I've made mention of it, is the coherence, but I think there's more to be unpacked there. This idea of coherence and, and the linking together and the logical structure and the and the this follows that follows that follows that. Right. Ideas that we have in mathematics, both within unit within a school year. 

And then also just within a progression of topics that we have as an overarching math program at our school or within our district. And I think this also kind of you made mention of this before, this idea of establishing goals, mathematical goals. You can't do it in a silo, right? You can't do it by yourself. And I think this this opens the doors, even forces the conversation of planning, not just within my own classroom, but then also vertically, across the grades and in across the topic, levels.

00;26;10;15 - 00;28;04;04

Joanie

Yeah. You're making me think about. So, my first year teaching AP statistics, I found out kind of very late in the summer that I was going to be teaching AP statistics, and I had not taken a statistics course as part of my math degree. I had the choice of probability or statistics at my university, and I took probability.

So I was woefully unprepared to be an AP statistics teacher. But I did everything I could. But what I lacked was that coherence of the whole trajectory of the course. Right. And I think, you know, my poor students, I did a much better job year or two, year or three in year four than I did year one, because I didn't have that bigger understanding of the whole trajectory. And the other thing that I didn't have that I think, certainly is an issue in for for some of our listeners, but hopefully many of them is it's not an issue. Many of them have access to other teachers. And again, you know, my soapbox issue is around teacher planning and preparation. And I think establishing learning goals is collaborative work.

It is what we should be doing with other teachers. Like whether it's I'm in my algebra one PLC and talking with the five other algebra one teachers in my school about, okay, here's this upcoming unit. And you know, what is the idea here? But it's also that vertical articulation with, you know, the middle school teachers. What did they do in seventh and eighth grade that I'm building on, that I want to point out intentionally to my students about how this is connecting to what they've previously learned.

And do I also understand where it's going? Am I talking to my algebra two teachers, to my pre-calculus teachers, to my quantitative reasoning teachers, to to really know what the where the key ideas are going so that I can set students up to be ready for those. And like you said, it's very, very hard to do that in isolation. And we're we're always going to do it much better if we can do it in collaboration with others. 

00;28;04;04 - 00;30;10;04

Curtis

Yeah, I think that's I think that's super, super important, realizing that this is a commutative thing that I can I can do this in a community. And, and you know, of course, in my brain, I was also then thinking of the teacher who's in a rural area that they are the whole math department. Right? So this communication, this conversation has to happen for them, vertically with themselves, but hopefully they can hopefully there's an opportunity in, in to seek out. Hey, is there a regional, group that I can get a good. Yeah, I'm sort of plugged in to, to be able to have some of these conversations and say, hey, here's what I'm dealing with. Here's what I'm thinking. Because part of this, it isn't just, the, you know, the reason the algebra one team, the five people get together is, is because each one of those five people, brings a different perspective, brings that thought process, brings a different skill set. That's how I think about things. Our brains think differently. And, you know, when I'm sitting here visualizing in my head, this plot of two, linear functions and then this set of coordinate points, coordinate pairs that I've multiplied off out of those two and created this quadratic. I mean, I'm all excited and nerdy about that thing. Yeah. And that was an idea I had, but it wouldn't have come if you hadn't prompted that, by some of the things that you said and set up. And so, you know, doing things in collaboration. And this goal, this, this teaching practice, this, this practice definitely.

00;30;10;10 - 00;31;11;20

Joanie

And I just, I think about those, educators that are in those rural situations and, you know, certainly I've never worked in that experience. So, you know, I don't want to pretend I have all the answers there, but certainly finding ways to connect with others and, you know, find that community if it's not already built into your system. And I just want to put a plug in one of the great resources that you and I tap into all the time for these podcast conversations is the “My NCTM” forums. And, you know, the questions that get asked. They're always get these long strings of responses from other teachers. So if there is somebody listening who's like, I don't even know where to look for other people to support me, it's just me. That that might be a starting place. If people aren't aware of the my NTM platform and the ability to go in and read through threads or ask a question and, and get responses and, support that way. I just think NTM is such a great learning community. 

00;31;11;20 - 00;31;11;20

Joanie

Yeah, I really like that. That learning board, it's a, it's a great digest of things and the variety of topics is always, good in the variety of, of outlooks and in input and, a frankly a variety of opinions.

00;31;11;20 - 00;32;02;16 

Curtis

Right? So there's always a good, lively discussion, on topics. And so it's, it's, it's a good place to go to, to learn and to and to grow a little bit in that space. So, this is really been a I, I'm enjoying this conversation. This is this whole progress, of working our way, through the, the book, Principles to Actions has been, a whole lot of fun. This, this, this year Joanie. I feel like I'm learning a lot, and I'm growing, a bunch. 

00;32;02;16 - 00;32;02;16

Joanie

I, I agree, I agree, Curtis, and I like that, you know, we, we kind of made a conscious decision in January not to go through the mathematics teaching practices in order. And I feel okay about that decision. If I had it to do again, though, I might have started here because I think, at the for the future conversations we have planned around those mathematics teaching practices. Certainly coming back to this one and how this one is relevant to the others is going to be an important part of our conversation. I, I would love to maybe try and wrap us up just circling back to where we started, because I want to acknowledge that there's a lot of different kinds of goals that we write as teachers and that are important for our work. And I don't want to diminish because we've spent, you know, 20 and 25 minutes really focused on mathematics, learning goals that are about big, important mathematical ideas and that, you know, guide us through our instruction or decisions. I don't want to suggest that we never do another students will be able to or that we never say, hey, the focus of today's lesson is to add two fractions with unlike denominators. Those goals have their place. I guess I just kind of want to end with my punchline being don't stop there. Like, be sure that the overarching mathematics learning goal is the target, and know that those other performance goals, those skill based goals, those individual daily lesson, you know, progress types of goals are are part of that bigger picture, not the end.

00;33;40;18 - 00;33;58;22

Joanie Outro

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!

00;33;58;22 - 00;34;04;00

Music out