The future of math education will begin with the divorce of a very long-term marriage.

Ever since Massachusetts set up the nation’s first system of public schools, math education has been inextricably tied to what students should know based on their date of birth: 3rd graders are taught multiplication and division, 5th graders are taught to coordinate planes, 8th graders are taught to graph linear equations, and so forth.

But this marriage between math and grade-level instruction has always had an inherent conflict at its root. Here I explore what it will take to dissolve the marriage and finally end this conflict.

On one side, math is cumulative. The skills students are taught in one year are foundational to what’s taught in the next year. Learning about ratios in the 6th grade, for example, requires the ability to multiply with fractions and decimals (5th grade) and to compare fractions with different denominators (4th grade). Without those foundational skills, students have a far lower chance of mastering ratios.

On the other side, grade-level instruction is sequential. Students are enrolled in grade levels based on their age, and each state determines the specific math standards that students are taught in each grade.

The marriage works fine as long as students never fall too far behind. But when they do—which happens often—the conflict between what a student is ready to learn based on their knowledge and what they are supposed to learn based on their age causes the marriage to fail. The winner of the dispute is always the grade level since that’s what’s in the curriculum, that’s what the teacher has been trained on, and that’s what will be on the year-end summative assessment.

The result is that students matriculate from one grade to the next without a full understanding of all they should know. Those gaps make it harder for them to learn the next year’s concepts, which then make the subsequent years even more difficult. Many students get discouraged as they rightly surmise they may never catch back up.

For students who never fall behind, the marriage may seem to work fine. But for many others, grade levels place an invisible ceiling on their progress. Without the constraints of rigid scope and sequences tied to their date of birth, these students could often accelerate further. Instead, the marriage requires they spend precious instructional on topics they already grasp.

This is the harm that math’s dysfunctional marriage is having on its children.

Big Little Lies

I was about to start my first year as a teacher when I asked my principal what I feared was a dumb question. The performance data for my incoming 5th-grade class showed some students were performing at a 2nd-grade level, some at an 8th-grade level, and the rest everywhere in between. My job was to use the 5th-grade textbooks to teach the 5th-grade standards, which would ultimately be assessed on the 5th-grade end-of-year test.

I asked him how I was supposed to reconcile these two realities, and that’s when he let me in on education’s dirty little secret.

“Just do your best, Mr. Rose. That’s what we all do.”

Thirty years later, the reality hasn’t changed much. Just prior to the pandemic, the average 5th-grade classroom had students performing at seven different grade levels. That span has only widened since.

Nonetheless, policymakers, systems leaders, and advocates have been assuming our nation’s math woes could be solved by promoting a series of policies and reforms aimed at just about everything except how a teacher is supposed to differentiate instruction: raise standards, improve curricula, address teacher quality, adopt new assessments, instill accountability, lower class size, and increase teacher pay.

While each of these initiatives has been an important step forward for the sector, it’s hard to say that any one of them (much less all of them) has resulted in a material difference in student outcomes. In 2005, 30 percent of 8th-grade students in the United States were proficient in math on the National Assessment for Education Progress (NAEP). That number grew to 35 percent in 2011 but has since declined to 28 percent as of 2024.

Proponents of these reforms will often point to progress on different tests, with different grade spans, and at different periods of time. But squinting to find signs of progress masks the larger truth: Even with the incremental changes these tests sometimes reflect, it would still take at least another century before the vast majority of students leave high school with math skills that would allow them to succeed in college or a career.

If math outcomes are ever going to improve substantially, something more fundamental must change that can make it possible to differentiate teaching.

Beginning to Stray

In the early 2000s, as the Internet was reshaping nearly every sector in our economy, a friend suggested we meet at his office before venturing off to lunch together. I’d been working for a school management company, and he’d been running an adult learning center, training a generation of would-be technicians and programmers on the skills they needed to land a new job. As I sat waiting for him to come out, I saw a sign hanging in the waiting room that I couldn’t stop staring at: Choose Your Modality: Live, Online, or Blended.

My friend explained to me that a modality was simply a way of delivering instruction on a particular concept. All students within a course followed a standard curriculum, but the center scheduled in-person instruction on different modules at different times. It also provided students with asynchronous instruction on each of the modules, allowing them to learn everything at home or to come in periodically to engage more directly with the teacher. Common assessments helped to measure progress along the way.

I thought about Rolando, a former student of mine, and how lost he looked when I was teaching my class how to multiply decimals—because he didn’t yet understand what a decimal was. His time would have been much better spent learning about decimals instead of sitting through a lesson on how to multiply them. But since I couldn’t teach both lessons at the same time, wouldn’t Rolando have been better off spending that time block learning about decimals from a modality that wasn’t me?

On the other end was Eliana. She began the year well beyond grade level and was easily able to master what I was teaching. Should she have been learning something far more advanced, even if it were through a modality that wasn’t me?

For both students, even if the other modalities were not as good as what I could have done as a teacher, any loss in instructional quality might have been offset by the fact that each would have been working on the right skill at the right time.

Trying Something New

After visiting the adult learning center, I began to wonder what a multimodal math classroom would look like, and I drafted a proposal for how it might work. I called it School of One and shared the draft with Joel Klein, the New York City Public Schools chancellor, for whom I was by then working. He took to the idea and helped to raise the initial funding to get it off the ground.

Here is how it worked:

At the beginning of the school year, each participating student took a diagnostic assessment that generated a personalized set of math skills (including procedural, conceptual, and applied understanding) to work on over the course of the school year. That list could include a combination of pre-grade, on-grade, and post-grade skills, depending on each student’s baseline.

When it was time for math, students would enter the math center (often several combined classrooms set up with multiple learning stations) and view an airport-like television monitor that displayed their name and which station they should report to. The stations were designed to support different learning modalities, including teacher-led instruction, collaborative learning, real-world tasks, and independent learning.

Each day, students would work on a specific skill through two different modalities before taking an online “exit slip” assessment and then transitioning to their next subject. The data from the exit slip fed a sophisticated scheduling engine, which in turn generated recommended plans and student groupings for the next day based on what each student now understood.

In effect, it was an approach to math education oriented around personalized competency-based learning (PCBL) instead of grade-levels.

It was also a hit. Time named School of One a “Best Invention of 2009,” and, with the support of several national funders, we later spun the idea out of the New York City Department of Education into a new nonprofit (New Classrooms) and renamed the program Teach to One so we could begin to expand to other schools and districts across the country.

Over the next eight years, the program expanded to serve more than 100 schools across 20 states. National and local press wrote extensively about the program, and hundreds of visitors from all around the world came to see it in action. Most importantly, a third-party study showed students learning 23 percent more than the national average and 53 percent more in schools where the conditions allowed for higher levels of implementation fidelity.

Taking it Slow

While many systems leaders and advocates found Teach to One fascinating, schools weren’t partnering with us as fast as we had initially hoped. District administrators had other priorities designed to improve outcomes: the adoption of new standards, assessments, curricula, and teacher training initiatives—all of which kept math’s marriage to grade levels intact.

There were also practical limitations that stood in the way of adoption. Implementing Teach to One initially required knocking down walls (we eventually figured out a single-classroom version), retraining teachers to teach collaboratively and across multiple grade levels, paying several times more for the program than what a textbook costs, and challenging the wisdom of accountability systems that incentivized teaching only the grade-level standards.

As one superintendent shared, “Teach to One has to be the future of math. But we just aren’t ready for it. It’s too big of a leap from where we are now.”

He was right. So, in 2020, we took what we learned from our initial model and launched Teach to One Roadmaps, an all-digital, asynchronous version that supplements a school’s core curriculum. In less than three years, it became more widely adopted than the original Teach to One program ever was.

Why has this new approach become so much more popular? Teach to One Roadmaps supplements the marriage rather than challenging it. It also solves an acute problem teachers have today: the lack of tools to both diagnose and address foundational learning gaps in ways that still connect to their core curriculum. Roadmaps allows them to spend time during a math intervention block focusing on the skills they both don’t yet understand and that are foundational to what’s being taught in their regular math class.

At the same time, Roadmaps previews for schools what’s possible when the focus of math instruction is on academic skill building instead of grade-level curriculum coverage. That in itself raises new questions.

As one principal recently observed, “I see that in their supplemental time, our students are learning two new math skills each week. I bet that’s at least 2x what they are learning in their core math block because of the lack of foundational skills. Why can’t we use an approach rooted in PCBL as our core?”

For that to happen, the marriage must end.

The Barriers to Breaking Up

What keeps more schools and districts from embracing PCBL in math? Standing in the way are a series of misconceptions and a set of practical challenges that require creative solutions to overcome.

Confronting Misconceptions

Remediation. “Meeting students where they are” echoes the language of justification for remedial education, where lower-performing students are shuffled into classes that focus on material from prior grade levels but never actually catch up.

There is also ample evidence that remediation disproportionately affects students from marginalized communities who may never access grade-level material because their teachers simply don’t expect enough of them. Expectations matter, and it’s true that students can’t learn what they haven’t been taught.

At the same time, it is mastery, not access, that drives social mobility. For many students, a focus on access may undermine mastery, because precious instructional hours are spent teaching content students are unlikely to master.

PCBL offers a third approach: meet students where they are and provide them with a viable pathway to get to where they need to be.

Skills at the expense of abstract thinking. Another objection comes from educators who argue that a systemic, skills-based approach to math education is anathema to the abstract thinking and problem solving that are at the heart of a quality math education. The critique hearkens back to the math wars of the 1990s, when debates raged about whether deeper levels of conceptual and applied understanding should be incorporated into math education. It resurfaced again recently the new California math framework.

A shift to PCBL does not imply a return to rote teaching. The fact that some students need to focus on pre-grade skills does not mean that those skills should be taught in ways devoid of high-quality pedagogy. Nor does it suggest that conceptual and applied understanding should not be part of mastery. On the contrary, a shift to PCBL recognizes that better understanding of foundational concepts is a prerequisite for deeper levels of understanding.

Risks of incoherence. Opponents fear that a math classroom organized around PCBL will be less instructionally coherent and thus less effective—especially if students are jumping around to different mathematical domains or are using materials from a variety of sources that teach concepts in different ways.

Dismissing PCBL because of concerns about coherence not only disregards the science of learning and the need for foundational understanding. It also mistakenly presumes that the only way of achieving instructional coherence is by strict adherence to a fixed pedagogical approach on a predetermined sequence of connected skills applied to a whole class. Thoughtfully designed PCBL programs can be both instructionally coherent and effective for each student.

If the only way to drive instructional coherence is with a cohort-based instructional program, then any gaps students develop all but guarantee they will have an incoherent understanding of math—with significant implications for their futures.

Overcoming Practical Challenges

A shift in mindset. School board members, district and school administrators, teachers, and parents have all grown up in a world where math instruction was based on a student’s enrolled grade. The century-old predominance of this notion has created a set of expectations, built over generations, that will not be dislodged easily. It’s simpler and more familiar to keep using a grade-level textbook, following a standard scope and sequence, and testing students relative to grade-level standards.

Some teachers who hold onto the idea that they should be the sole source of instruction, that they should initiate and design the full student experience, or that it’s their job to cover every skill included in the grade-level standards may especially struggle with the transition to PCBL. Such entrenched thinking is incompatible with an embrace of PCBL, and deprogramming must be at the heart of professional learning.

Grade-level-only assessments. Federal law requires each state to assess all students in grades 3 through 8 (and once in high school) in reading and math, to report on how students performed relative to grade-level standards, and to implement a statewide accountability system that incorporates those results. Until the law changes, the federal focus on grade-level assessments raises the stakes for the school and district leaders looking to move away from an exclusive focus on teaching grade-level material for fear that near-term scores might go down.

Time. Most middle and high schools have either a single block (45 to 60 minutes) or double block (80 to 90 minutes) allocated for their core math instruction each day. They are not yet looking to jettison their core math program in favor of comprehensive PCBL models like the original Teach to One, whether because of the current accountability system, their recent investments in high-quality curriculum, or a belief that it would simply be too much change.

For those with double blocks or separate intervention periods, PCBL tools like Teach to One Roadmaps are now being used during the last 20 to 30 minutes of a core instructional block. This allows schools to provide both grade-level curriculum and PCBL. Other schools have scheduled intervention periods or afterschool sessions for struggling students to work on their PCBL. Some may even use PCBL as the basis for personalized homework.

Till Death Do Us Part?

Math is cumulative based on skills. Grade levels are sequential based on age. Those realities inherently conflict, no matter how strong the teacher, how good the curriculum, or how stringent the accountability system.

Historically, the age-grade paradigm has always prevailed. It is ingrained within our schools’ operating systems, our collective mindsets, and our nation’s policies. Yet efforts to move past incremental gains and meaningfully improve math education within this paradigm have never failed to disappoint—because it can’t be done.

The tools and capabilities to move beyond an age-grade mindset are now within our grasp. Thoughtfully deployed, they can enable true differentiated learning, with multiple modalities unlocking the ability to differentiate learning pace in ways that far exceed what happens in most classrooms today.

Divorce can be sad, but sometimes it’s necessary. The courage to persevere through it can open up new opportunities for all involved. On the other side of this divorce is a new world where math classrooms are designed to meet each student’s needs, where teachers have a more sustainable and fulfilling role, and where families can become true partners in their child’s success.

If we have the will to make this transition, the future of math will be known to a new generation of students as just math.

Joel Rose is the CEO and Co-Founder of New Classrooms Innovation Partners and a contributor to School Rethink 2.0.