Why Moving Ahead in Math Isn’t Always the Right Move

As schools work to help students recover from pandemic-era learning loss in math, much of the focus has been on struggling students. But what about those who are already excelling?

“A lot of students, too many students, are not meeting our goals for math learning," Professor Jon Star says. "And they deserve our careful and thoughtful attention and maybe even the bulk of our attention … because we've been working so hard, and with limited success, at addressing the needs of struggling students, we see much fewer opportunities and organized efforts to address the needs of advanced students than we used to. I'm trying to make sure that stays on our agenda.”

For too long, the go-to for advanced students has been acceleration — sometimes moving them a full grade ahead. But as Star points out, while this seems obvious, it can create new problems: harder-to-manage classrooms, wider gaps between students, and the message that math is all about speed.

“If students start to perceive that mathematics and the curriculum is just a race, it's how fast you can get through everything, that essentially incentivizes superficial learning and learning that isn't the depth that we want,” he says. “It's just about being able to solve a few problems and move on to the next topic.”

Instead, Star proposes shifts in practice that provide new challenges and push advanced students to think more deeply. Typically, this issue arises in grades 3 to 5, when parents begin to notice their child is bored in math. In his new book, Diving Deeper with Upper Elementary Math: Low Prep, High Reward, Challenging Math Enrichment Activities, Grades 3–5, Star offers lessons and mini-units that are easy to use and help advanced students build deeper understanding. Through thoughtful questioning and purposeful extensions, teachers can encourage students to think critically, test their assumptions, and develop a richer grasp of mathematical concepts.

“There's so many interesting topics in mathematics that we can dive into that aren't accelerating, that are deepening their knowledge of the grade level content, that should be our preferred strategy for challenging our students,” Star says.

He also explains how these exercises illustrate what meaningful math enrichment looks like in practice, even for students who aren’t advanced.

In this episode, the EdCast explores why we need to think differently about advanced math learners and the simple ways teachers can shift practices to truly challenge them.

Transcript

JILL ANDERSON: I'm Jill Anderson. This is the Harvard EdCast. 

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Jon Star believes we need to think more carefully about how we challenge high-performing students in math, even though much of the conversation tends to focus on students who are struggling. He's a professor at the Harvard Graduate School of Education and also teaches middle school math.

He says moving high-performing students ahead to higher level material might seem like the obvious solution, but it can create new challenges for teachers and may not actually help these students. Instead, he argues, students benefit more when they're pushed to think deeply about the math they're already learning.

Given the recent results from the National Assessment of Educational Progress showing that math scores are still recovering from pandemic era decline, I asked Jon why we even need to pay attention to those students who are doing well.

JON STAR: I should start by acknowledging that we should be putting a lot of our focus on underperforming students. A lot of students, too many students, are not meeting our goals for math learning. And they deserve our careful and thoughtful attention and maybe even the bulk of our attention. 

But I think that my work is pushing us to think also about some of the students who are doing quite well in math and how they can be challenged. For a lot of reasons, our elementary schools are focusing a lot of their efforts on meeting the needs of struggling students. And this is important. And budgetary reasons aside, this is a good thing for elementary schools to do because the negative consequences of, let's say, a grade 2 student not mastering the math and getting behind when they go to grade 3 are probably much more detrimental to that student than the negative consequences of a grade 2 student being a little bored in math class because they already understand everything. And I understand that. Some districts, in the past, and even in the present, do have gifted and talented programs for thinking about the needs of their more advanced students. But on the whole, because we've been working so hard, and with limited success, at addressing the needs of struggling students, we see much fewer opportunities and organized efforts to address the needs of advanced students than we used to. I think I'm trying to make sure that stays on our agenda, even if it's not as critical that we do so as meeting the needs of our most challenged students.

JILL ANDERSON: And I try to imagine what it's like being an educator in a classroom where you are faced with this challenge of reaching both students who are underperforming, students who are performing above grade level. That must be a real reality for a lot of educators. Can you talk about the tension between these two things in practice, and what it might end up looking like for a teacher?

JON STAR: So, I think what we're talking about here is what I would call differentiation. And differentiation can be thought of as the need for teachers to adapt their instruction to meet the learning needs of all students in a class. And we know that differentiation is one of the core challenges of teaching. There will always be a range of student abilities in a class. And figuring out, as a teacher, how to appropriately challenge all students is really hard.

There's been a lot written about how to do so. Teachers are taught strategies such as modifying tasks or reteaching content and other strategies. But we have a hard time doing this well. So, differentiation is known to be hard for teachers. And it's something we're working on.

But when some students are challenged by accelerating them into higher math grades, let's say, then this makes the already difficult task of differentiation in the elementary school grades even harder. Let's say that I have a second-grade class where the range and ability levels and knowledge in math in that one class, let's say, range from first grade to third grade, even though it's a second-grade class. There's some kids who are more at the first-grade level, some at the third-grade level.

This is already a hard differentiation challenge for teachers. But when those students who are more advanced are accelerated, but still in that room, then perhaps they might make it to the fifth-grade level in that class. And this presents the teacher with a potentially insurmountable challenge in terms of differentiation.

Certainly, it makes sense that one way that you might try to differentiate and challenge those advanced students is just giving them some above grade level material. But that just compounds the challenge that the teacher faces in terms of differentiation.

I think it's also to note that there's a limit to what elementary schools can reasonably even provide to students who have been accelerated so far above their grade level, two or more grades above their age. As we know, elementary schools tend to be very small. And this is by design and a good thing. It's optimal for addressing the needs of our youngest students.

But as a result of being small, it may not be possible for elementary schools to offer extra classes, or extra teachers, or extra instructional time that's specifically geared toward advanced students in math. So, I think we are obligated, we really are caring deeply about challenging all students. And the most advanced students that we work with, it's a real question as to what's the best way to challenge them. 

And I think what I'm pushing on here is the most easy seeming strategy to challenge them is to accelerate them, and they're excited by that. If you have a third grader and you give them fifth grade material, they would love it. They're super excited to learn new stuff.

But that creates problems downstream for the student and for the teacher. And it also might not be the best for the student in terms of their mathematics learning. We want students to really develop this deep understanding of mathematics, all students.

And if students start to perceive that mathematics and the curriculum is just a race, it's how fast you can get through everything, that essentially incentivizes superficial learning and learning that isn't the depth that we want. It's just about being able to solve a few problems and move on to the next topic.

But we really want students to sink their teeth in the math and really deeply understand it, all of our students. And the gifted students, the advanced students, they're particularly capable of doing so. They can really wrestle with some fundamental deep issues, questions about mathematics. Why not engage them in those conversations, rather than just moving them on to the next topic?

JILL ANDERSON: You've already mentioned that acceleration might be like an obvious solution for a lot of educators. And you just hinted at creates its own set of problems. Beyond just this idea of they're going so quickly through the lessons and at a surface level, it feels like it almost becomes a checklist, and elementary kids are really excited about that, are there other issues with acceleration?

JON STAR: Well, we've talked about the challenges it poses for teachers, that the range of ability levels in their class is exacerbated. Let's say it's further increased, making differentiation even more harder, perhaps even impossible to do. We've talked about the ways in which the acceleration might advance a set of beliefs about mathematics that aren't productive, that it's just a race to get through it. It's just about solving problems correctly. The first one there wins.

It's not about our understanding goals, our deep understanding goals. These are the main concerns that I have about acceleration. Our goal for students in learning math is that they develop this mathematical understanding that we're after. And to what extent does acceleration serve those goals?

If we felt like there was no other way to challenge a student except accelerate them, then maybe we'd be having a different conversation. I would say we need to challenge students, and the only way to do so is accelerating them. But I don't think that's the case.

There's so many interesting topics in mathematics that we can dive into that aren't accelerating, that are deepening their knowledge of the grade level content, that that should be our preferred strategy for challenging our students. Not to suggest that's easy. It's not. But I think it has fewer downstream negative effects for teachers, and schools, and students than acceleration does. 

Now, one thing I should probably mention here, though, because I'm saying a lot about my concerns about acceleration, is I think it's important to note that there are some students for whom acceleration is exactly the right thing to be doing. There's a small, small sliver of students that we might say are prodigies. They are truly geniuses, truly prodigies in math, extraordinarily gifted. Now, I don't know whether we should think of these as the top 1%, or the top 0.1%, or 0.01%, but it's not 0. There are such students. 

Wherever we draw this line, for the students who are above the line, it might be completely appropriate for them to be allowed to opportunity to accelerate through the math curriculum as quickly as they're able. Often, these students end up relying on out-of-school experiences to support their acceleration. Or they go to different schools that are especially geared toward people who are extraordinarily gifted.

There are a lot of students out there who love math, who are very good at math, and who deserve to be challenged in math, but only a tiny, tiny part of those are truly prodigies. I'm not talking about those students when I'm down on acceleration.

I'm talking about all the rest of those students who are really good, who love math, who deserve to be challenged. And I'm arguing that for those students, going deeper into the grade level content is a preferable strategy for differentiation than acceleration.

JILL ANDERSON: Right, and so what does enrichment math look like?

JON STAR: Well, I've been talking about how we are interested in getting students to dive deeper into the math that they're learning, rather than moving on to above grade level math. And so, for me, that's what enrichment is, diving deeper into grade level math. 

So that might mean that once the student has mastered today's lesson, then I have some other extension tasks or questions that I can ask them that are about today's lesson, but are pushing them to think a little differently, a little more deeply, or to explain their thinking a little more thoroughly. And that's what I'm looking for.

Some teachers might say that enrichment for them means that when a student finishes their work for the day, the teacher pulls out a puzzle sheet, which is a common way that teachers don't accelerate and they aim for enrichment. So, oh, you finished today's work, here's a fun puzzle sheet you could do.

Kids do like those puzzle sheets. I'm not saying those are bad things. But I still feel that there's a missed opportunity there for the teacher to have some other kinds of questions or activities that the student could engage in, even independently, that push their thinking around the grade level topic.

For example, one of my favorite questions that I ask my students in my middle school class that I teach is once they've finished the task they're working on, I might ask them something like, is that always true? Or what if I change this feature of the problem to this other feature? Does your strategy still work, or is what you claim still true?

That pushes them to think about the generality of their strategy or of the conclusion they raised. And that is such a deeply mathematical way of thinking, to push them in that direction. I, in my class, feel like often, that I want all students to be asked that question. That's a great question for everyone to be asked. 

But there are some topics where I feel like it might be a little much for us to go too deep into that with everyone, but that's always a question that my more advanced students would benefit from hearing and trying to answer. Why is it true? Is it always true? Under what conditions is it true? 
What if I change the problem to look like this, is your method still true? What's a general principle you could articulate about the thing that we've been talking about? Those are great extension questions. And I hope this makes sense that that's not acceleration. I'm not pushing them into the next grade level. I'm just pushing them to deepen the way they're thinking about the thing that we learned today.

JILL ANDERSON: Because I have to imagine teachers doing their lesson in their math block and the work goes out, and then there's always probably one or two kids in the class who are done really quickly and sitting there. Is that stressful for teachers to turn around and see there's that one or two, maybe more, who are just sitting there? Like you said, maybe they get out a puzzle sheet or something that's fun for those kids to do, but you still have another 15 minutes of the math block left.

JON STAR: That is very stressful for teachers. And part of what I'm arguing is that even if a teacher or a school is pro-acceleration, and they've done a lot of acceleration, and they've accelerated a group of students in their class, so they're working on their own, and they don't have to deal with them at all, then what you just described is still going to be the case for the students that are left.

No matter how many students you decide are core to your class and you're not accelerating, there is a range of abilities in that group. And there's always going to be differences in how quickly they do the task. So, no matter what, you're always going to have to solve this very problem, always.

You're always going to think, well, what do I do with those students who finish early? And how do I give the support to the students who need a little more time? That's a core problem of teaching. And I don't think we can ever eliminate it, even if we accelerate.

So, what I'm proposing, then, is that this becomes a strategy for differentiation more generally. So, when I'm planning my lesson, I'm always going to be thinking about extensions. That's a built-in part of the curriculum I'm using, of the way I lesson plan. And I'm going to give you some special thought to varieties of extensions that might be appropriate for some of my more advanced students, such as some of the questions I just asked.

So again, this is a core challenge, no matter what. But I'm thinking of ways that I can address this challenge with my more advanced students by having additional extension tasks that I might have in my back pocket, if you will, that I can pull out, or questions I can pull out of my back pocket to ask that don't take a lot of my time to monitor as a teacher necessarily, because maybe I have two or three students who finish early.

And I group the three of them together, and I say, hey, you three, I have another question for you. What if I change this number to this? Then is what you just said still true? Can you talk about that amongst yourself? That's a way that I've pushed those three students who have finished early into some deeper thinking and working with each other, so that I can devote more attention to the students who are still struggling.

JILL ANDERSON: Jon, I'm thinking about how in your latest work, you're really looking at these high-performing students in third, fourth, and fifth grade particularly. And you have a lot of resources for educators to use in how to do this enrichment work. Let's talk a little bit about why those later elementary school years are really pivotal in math learning.

JON STAR: Yeah, on one level, I would say that every year is critical in a student's math journey. So, I'm not sure I can make a claim that grades 3 to 5 are more important than any other years. They are important. And they're important because this is when some of the elementary school arithmetic foundational skills are mastered and solidified. And we start to move toward the greater abstraction that students will be wrestling with in algebra. We start that journey in these grades.

So, every year is important. That's why grades 3 to 5 are particularly important. But from my perspective, grades 3 to 5 are especially important when we're thinking about this acceleration issue. Because in the very younger grades, in grade 1 and grade 2, some of our students show real interest in math, love of math, they're curious about it. That's wonderful. And the math world is their oyster. They're excited about everything.

Those grade 1 and grade 2 teachers, I don't think, feel, typically, that press for accelerating from students and their parents. Students are just loving it, the ones that are really-- they're there. They're in it. But I think in grades 3 to 5 is when we start to see some kids say they're getting bored, parents getting concerned about their being bored, and parents starting to press schools to do something about their kids being bored.

The parents might be pushing for leveling, let's say, so an advanced fourth grade math class and a regular fourth grade math class, which many elementary schools don't want to do, understandably. And yet parents are coming to the school saying, my kid loves math and they're not challenged. They know all this stuff already. What can you do for them? 

So, I think this is when this issue first rises to the need to deal with radar screen of teachers and administrators. And this is where teachers and administrators might first get the idea that they can accelerate as a way to answer this problem.

They may first say, oh, I'm getting all these concerns raised by parents about their kid being challenged. Let's advance those kids to the next grade level or two grade levels. That will solve the problem. I think this is why I feel like it's especially important for me to be talking to those teachers and administrators at those grade levels, to argue that this is precisely when you're going to be asked to do this, and this is precisely when you should resist doing this. And you should focus on deepening.

By the time students get to say, high school, they're in an environment in most high schools where this issue is, for better or worse, solved for them, because high schools offer many different levels of classes for students at all grades. And you have mixed grade classes. 

So, if you're in 10th grade, you can take calculus. You can take geometry. You can take algebra. That's all good. So, kids can choose their own adventure, essentially, by the course that they pick. But elementary schools are not designed to do that.

And neither are middle schools, for the most part. And so yeah, grades 3 to 5 are when this first comes as a really tough issue for schools to have to deal with. And it's when they need to be deliberate about how they're choosing to deal with it.

JILL ANDERSON: I'm curious what it's when you get a student who has been accelerated in your classroom in math. What is it like on that end to be the middle school teacher?

JON STAR: It's really hard. As I said, differentiation is a core challenge in teaching, but this makes it virtually impossible. Imagine, as in my case, sometimes, that I'm in a class where 50% to 75% of the class has already learned everything, I'm about to cover today, and tomorrow, and the next week, and for the next month. They know it all already.

And yet I have another group of students, a quarter of my class, let's say, who haven't. And they are sharp. They are motivated. They are talented students. They just haven't seen the materials that I'm teaching before. So, what do I do in that case?

It seems not responsible to take the 50% of students who've seen it before to just say, oh, you guys go off and do this puzzle sheet for the next month. I still am their math teacher. And I want them to think, and learn, and deeply reflect on mathematics.

I'm not going to be confident that they understand the math as well as I want them to be understanding it because they've seen it from someone else. I want to ensure that that's the case. I want to ask them questions. I want to keep them in the conversation. And that's really hard if they've seen everything before.

If I let them drive my pace I say, oh, you've all seen this before, let's go really fast through this, then how am I taking care of the students who haven't been accelerated, who are bright, who are motivated, who are talented, they just haven't had this acceleration as part of their math history?

JILL ANDERSON: It's really interesting, Jon, because hearing you say that I wonder how rare is it that that's happening in math classes across the country? I know you haven't been in every single math classroom, but I'm wondering if it's rare to see math being taught in such a way.

JON STAR: One thing I would say about that is that nothing that I just said about the way that I think of mathematical understanding is my own original idea and something that I'm saying for the first time. These are things that people before me have been saying for a long time, and that people have been working really hard to do for a long time.

And there are a lot of teachers who are doing all of these things out there in the field. We can use a lot more teachers doing these things as well. But teachers generally know that these are good things to do. The implementation of them is quite challenging.

So, I don't know how many teachers, in actuality, are doing all of these things for their classrooms. But I would venture to say that many teachers, maybe even most teachers, know that these are good things to be doing. And we just need to continue to be working with them on how to actually do them amidst all the challenges that teachers face in their classrooms, some of which we've talked about, but there are many others.

JILL ANDERSON: The reason I ask that is because I can only reflect on my own experience in math. And it was a long time ago. And math education, how kids learn math has changed quite a bit. But I don't ever remember any conversations happening in math class. I just explicitly remember there was a problem on the board, they show you how to do it. Maybe another kid gets up and does a couple of the problems. And you're just trying to get to the answer. That's what I remember from math. And so, the idea of being in a math classroom, where it's even led by critical thinking and conversation, to me, is so interesting in and of itself. It feels like it would be an entirely different experience.

JON STAR: I think over the past, let's say, 40 years or 50 years even, we, as in math education professionals in the US and abroad, have been pushing for these kinds of changes pretty hard and pretty steadily.

And as a result, I think that in many elementary school classrooms around the country, this vision is what you would see, more or less, depending on the teacher. And again, it's challenging to implement this well, but I think we do a great job in many of our elementary schools of pushing toward this vision. We have a lot of excellent curricula now that incorporate this vision into the way that they're designed. So, I think in elementary schools, you would see a lot of this. 

Now, I will say that maybe one measure of how successful we've been in elementary school at advocating this vision could be reflected in how prevalent it is for some parents to be unhappy with what goes on in elementary school.

Because if you're someone who experienced a math background like you described in your own elementary school, let's say, and you send your kid to elementary school and they're doing something completely different, that algorithm you learned for multiplication, they're not even taught that. Or they're taught at two years after you learned it. And they're doing it this long involved way. And you know a quicker way.

And they're not being taught that. And that's frustrating to you as a parent. And so, you're unhappy. So maybe the prevalence of that kind of unhappiness is a signal that these things we've been trying to change are starting to be changed.

Now, I think the process of changing middle schools and high schools is more complicated. And I'm not sure that the things we're after in elementary school are exactly the same things we should be after in high school and middle school. But I think here, we're talking about elementary school. And these kinds of things happen, I hope, more often than not in our elementary schools.

JILL ANDERSON: Jon Star is a professor at the Harvard Graduate School of Education and a middle school math teacher. He's the author of Diving Deeper with Upper Elementary Math: Low Prep, High Reward, Challenging Math Enrichment Activities. I'm Jill Anderson. This is the Harvard EdCast produced by the Harvard Graduate School of Education. Thanks for listening. 

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