This article is a composite of a number of blog posts, fragments

you may have seen elsewhere on this site, and new material.

### Table of Contents

## 1. Introduction: Heterogeneous Classes

It is of course true that students learn math at different rates. Everyone knows this. Can something be done about it? The answer depends on how you explain the discrepancies. Too often, I hear things like: "They should not have been allowed in this class. They don't work. They are not abstract thinkers. They can't remember the most basic things." And so on. If you think it's just that some students are incapable of doing math, well, then there's little you can do about it, and you're off the hook. *But what if some of those gaps are caused or increased by what we, the educators, do?*

### Tracking

- Schools group students in various ways:
- - by age
- - by prerequisites taken
- - by student (and/or parent) choice
- - by schedule constraints
- - by presumed ability

The last one is called tracking. It is standard to track students in one or more subjects, and in smaller schools this leads to de facto tracking in other subjects, because of a domino effect in scheduling.

- While tracking makes things easier for teachers, it does have negative consequences for students:
- - students can be mis-tracked
- - once in a lower track, it’s nearly impossible to move up
- - expectations can be too low in the low tracks
- - the absence of strong students debilitates the low tracks
- - in fact, the image of parallel tracks is misleading: the tracks diverge, and the gaps increase
- Moreover, tracking does not solve the challenge of teaching the extremes, who still stick out in their classes, no matter how tracked.

- Here are the key advantages of heterogeneous classes:
- - no ceiling on average and weaker student achievement
- - opportunities for students to learn from and teach each other
- - greater diversity of backgrounds and experiences, which benefits all

- On the other hand, untracked classes have their own challenges, mostly revolving around the fact that they are difficult to teach:
- - the teacher needs a deeper grasp of the subject matter
- - it is easier to aim for the middle -- the extremes are challenging
- - concern for weaker students being left behind
- - weaker students’ perceived need: less depth, less breadth
- - stronger students’ perceived need: cover more, not deeper
- - there are few models on how to do this (but see below!)

### Detracking

Yes, there are exceptionally gifted and extremely challenged students, but if taught properly, the vast majority between these extremes are capable of learning the essential ideas of K-12 math. This fact has been masked by different expectations, which led to inequities in what math is offered to different populations of students, and by whom. (Paradoxically, low tracks are often assigned to inexperienced teachers!) Students learning less in the low track is in fact a self-fulfilling prophecy.

The National Council of Teachers of Math (NCTM) recently addressed this issue in *Catalyzing Change in High School Mathematics*. This document challenges many commonly held beliefs about high school math, and it is intended to trigger conversations about the structures and culture of math education in grades 9-12. It is a wide-ranging research-based document, which I encourage you to read and discuss with your colleagues.

In there, NCTM argues that tracking reinforces inequities, and recommends its elimination. I agree, and I will not repeat those arguments here. They suggest that all students would take the same core classes for two and a half to three years, and then spread out among various electives. No high school math class should be a mathematical dead end. A detracked program keeps doors open for all students.

Of all the recommendations in *Catalyzing Change*, detracking is sure to meet the greatest resistance. It is widely believed by students, parents, teachers, and administrators that tracking is in the best interest of all students. Thus, there is tremendous political and societal pressure to maintain tracking. Some of that pressure comes from parents and educators who fear that heterogeneous classes will be watered down, thus lowering the mathematical level for the stronger students, and possibly endangering their college admission prospects. Some of the pressure comes from the fear that currently low-track students are incapable of "keeping up", and their grades will suffer. In the absence of adequate structures and strategies, these are valid concerns.

Having taught in an untracked high school for 32 years, I will share some ways to address these worries. My experience is that most students who would have been low-tracked can thrive in the same classes as students who are bound for MIT. Every year, I saw students whose middle school teachers deemed "low" ready for calculus by 12th grade, or certainly by the end of 12th grade. My suggestions will be especially relevant to anyone engaged in the discussion or implementation of detracking. But in fact they are relevant to anyone teaching math in middle or high school, because tracked or not, *all classes are heterogeneous.*

### Alternatives

In addition to tracking, schools use two main strategies to address differences among students: acceleration and differentiation.

*Acceleration* refers to the idea that students who are ready to learn specific material earlier in their schooling, are taught this material a year earlier than their cohort, or even earlier than that. NCTM supports well-founded acceleration, and I agree: a little acceleration can be a good thing. At my school, we had a two-year age spread in most of our classes. Instead of offering a watered-down version of a given course, and an honors version, all students took the same course, but some took it a year earlier. However, beware of hyper-acceleration, which is terribly counterproductive.

As its name indicates, *differentiation* aims to address differences among students in the same class. It is interpreted in various ways. An extreme version of it is a completely individualized program, where students work at their own pace, possibly on different assignments. This undermines the classroom as a community of learners, as it makes it difficult or impossible for students to work cooperatively, or to participate in an all-class discussion. Teacher help in that scheme is strictly one-on-one, and thus not as efficient. (Combining class discussion with group work is a powerful way to leverage the role of both teachers and students.) Another problem with this sort of differentiation is that it requires more prep work from the teacher, which is not a recipe for success in a country where teachers are overworked, underpaid, and not given enough time for preparation or collaboration.

Another approach to differentiation is to offer extra assignments to the stronger students. Such policies are often resented by those students, who don't see why they should have to do more work than anyone else. One version of this I have used with some success is to have much harder problems at the end of assignments, which I don't expect all students to complete, genuinely challenging bonus problems on tests and quizzes, open questions in graded take-home assignments, and so on.

But mostly, I like to differentiate not in content, but in timing: all students learn essentially the same math, but they learn it at different rates. Read on.

## 2. Underlying Principles

Before sharing specific techniques for time-based differentiation, I will list some general principles for working with heterogeneous classes. While my practical suggestions are powerful and relatively easy to implement, they may need some tweaking to work in any particular situation, because of differences in school and department cultures, and in teachers' experience and personalities. Teaching is complex, and nothing works everywhere. Still, there are overarching truths about teaching heterogeneous classes which need to be observed if one expects to reach the full range.

### Strong Students

It is of course necessary to support the students who need more help, both in class, and outside of class. That is obvious. What is not as obvious is the crucial importance of building an alliance with the strongest students. They are the key to our success in a heterogeneous class. This may be the most crucial component of a winning approach, and is probably the most difficult to understand. It is predicated on understanding that *in a student-centered classroom, the top students are the engine that drives the class*. If we do not keep the course challenging and interesting to them, we lose their respect and their cooperation, and thus we lose our key classroom allies. Moreover:

**Politically**: if the class does not challenge the strongest students, their parents will not remain silent. They will be in the principal's office demanding a return to tracking, or a different teacher for their kid, or who knows what. We can disapprove of this behavior all we want, but their kids are their priority, and they are not interested in our theories.**Ethically**: it is not true that the strongest students "will take care of themselves." They are children, and we are in charge of teaching them.**Philosophically**: our responsibility as educators is not merely to have our students pass the class. This could be accomplished easily by watering down its content and slowing its pace. We have a greater responsibility: passing on the mathematical heritage of our civilization to the next generation. Among our students are the future scientists, engineers, mathematicians, and math and science teachers. We all have a stake in their education.**Pedagogically**: if the strong students enjoy our classes, and feel they are learning, they will not resent group work, or the review of old material. Quite the opposite, they will help teach their classmates, and they will fulfill their leadership role in the classroom.

When I was a beginning high school teacher, I was given a terrible piece of advice: "teach to the middle". That is a betrayal of all but a few students. Instead we need to follow the Goldilocks strategy: every day, aim to do something too difficult, something too easy, something just right, and everything in between. Or think of your lesson as an elevator: it needs to stop on all the floors. This multilayered approach is the way to avoid lowering our expectations, which would be in no one's interest.

### Better Teaching

Teaching the whole range of students entails many ingredients. If we rely on a lot of memorization with little or no understanding, we are favoring students who are more docile, and have better memory. The gap between them and those who are less eager to please or more forgetful will increase. Likewise, if our lessons are boring, once again, we favor obedience over genuine engagement, and the gaps will increase. Here are some answers:

- A curriculum with many rich problems and activities, sometimes known as "group-worthy", or "low threshold, high ceiling". Much of the curriculum I share on this Web site strives to fit that description.
*Classroom routines*that support genuine discourse, and give students opportunities to think, agree, disagree, reflect, and discuss.*Classroom discussions*led by the teacher towards important insights and techniques.- A tool-rich pedagogy, including manipulatives, technology, and paper-pencil conceptual tools makes it possible to represent topics in multiple ways, to make preview and review interesting, and to generate deeper understanding. One consequence of this is that one should teach fewer topics, to make room for multiple representations of the most important ones. If omitting less crucial topics is not an option because of outside mandates, then give less time to them.
*Student collaboration*, also known as group work. Various models are available on how to organize this. I like Peter Liljedahl's "thinking classroom" approach, and summarized some of its highlights here and here. I also like some of the ideas known as "complex instruction", and recommended some books on that approach here. I share some of my own strategies here.

Making progress on all those fronts is important, but it is challenging, and it takes time to grow in all these directions. Also, it is not sufficient: even with great teaching that incorporates all these ideas, some students just need more time. I know: there's a lot to cover, and if you had to wait until everyone got it, you'd never get anywhere, and moreover, who wants to penalize the stronger students for the benefit of the others, who may not get it no matter what we do? We've got to keep up the forward motion!

### Forward Motion

I don't disagree, but I contend that you can have plenty of forward motion, while at the same time giving enough time to students who need that. The way to do it is to use some strategies to extend student exposure to ideas. This may be the easiest ingredient to incorporate into your practice, and it may well be the one providing the most bang for your effort. Yet it is not widely done. In the rest of this article, I will share some techniques to achieve *extended exposure*.

The idea is to strive simultaneously for *constant forward motion, and eternal review*. To achieve forward motion, it is necessary to start on a new topic before the previous one is grasped by everyone. Eternal review is of course impossible, but much review can be built into a course without harming the forward motion. This is aspirational, as it is sometimes necessary to pause the forward motion, and there may not be sufficient time or resources for eternal review. Still, it is a great thing to aspire to. Forward motion is essential to keep a course interesting, especially to our strongest students. Review is essential for all students if we want ideas and techniques to stick.

### Extending Exposure

I have devised some ways to reorganize things so as to achieve both forward motion and eternal review, and to extend student exposure to concepts and techniques. There are two main reasons why this is important. For one thing, retention is drastically improved if students see important ideas more than once. Also, it is hard to learn when under time pressure, whether facing the tyranny of the clock or the tyranny of the calendar. "You must solve this by 10:15! You must understand this by Friday!" For many students, far from being motivational, artificial deadlines are anxiety-provoking and paralyzing.

As it turns out, it is not difficult to extend exposure without taking more time, by rearranging the time you are already using. Just implement these strategies: lagging homework, lagging assessments, separating related topics, pursuing two units simultaneously.

Before I flesh this out, I should mention that it is important to have some form of support available outside of class for students who need it. How to do this will depend on the specifics of your school. As department chair, I asked all members of the department to take turns staffing "Math Café", a classroom where students could bring their lunch and their questions, and get help from a teacher (and sometimes from peers.)

## 3. Lagging Homework

One concept which has been extremely popular every time I've offered my workshops is *lagging homework*. The post about it is also by far the most visited page on my blog. The idea is to completely separate tonight's homework from today's class work, and instead to assign homework about ideas that were introduced about a week ago (more or less.) Here are some arguments in favor of this practice:

- Students do not rush through classwork in order to "get to" the homework and try to do as much of it as possible in class. As a result, they are more intellectually present, and more available for reflection, discussion, and collaboration.
- Teachers do not rush through the introduction of a new topic. They are not under as much pressure, because they don't need to reach all students before the end of the period. If today's lesson does not go well, there is always tomorrow.
- Lagging homework extends students' exposure to the ideas: what could (with perfect students) be done in one week now takes two weeks, which gives the students who need it more time to absorb the ideas. And this without harming the students who don't need the extra time. In fact, it gives them an opportunity to later review ideas that they may have absorbed too fast.
- This policy is also helpful to your stronger students in another way: it allows forward motion to a new topic before every single last student is "ready to move on." The student who is not quite ready knows that they will have another chance to grapple with the idea in next week's homework. It is a kind of differentiation that does not require the teacher to come up with a different program for different students.

But, you ask, **are students confused by this practice**? It is probably different from what they've done before, yes, but it does not take long for them to buy into this system, for all the reasons given above.

**Don't students need to practice a new idea soon after they hear about it?** Well, yes, but that need not be done at home. Giving them a chance to do their first practice in class means that they are doing it in the presence of classmates who can help, and of course a teacher.

**How does it work at the beginning and end of the term? **At the beginning of the term, you might give homework on a topic from last term, instead of spending precious class time on it. At the end, you might schedule a buffer week for review. But in any case, this is a general guideline, which obviously need not and cannot be implemented in a rigid manner.

**What if you like to assign homework that prepares the students for the next day’s lesson? **I love that idea. It it is not at all in conflict with lagging homework. My main point is that on most days, you should not assign homework based on the day’s lesson. A week’s delay, more or less, provides many advantages, the main one being extended exposure to each topic. This in no way precludes homework that sets up the next day’s lesson, as long as it is not usually based on the day’s lesson. Such homework can reveal a key idea, generate curiosity, or in some cases dispose of a necessary digression up front. In general, such activities (whether assigned as homework, or as warm-ups) are essentially long-lagged work, based on ideas that were introduced the previous semester, or the previous year, or whenever. Such long lags can also be used for review, as suggested above.

On the other hand, if preparing for tomorrow’s lesson requires completing homework about today’s lesson, and this needs to happen frequently, then I strongly discourage that as the collateral damage on some of your students is substantial.

### No Homework?

**How about abolishing homework altogether? **I sympathize with this question. Most learning happens in class, and one should not overdo homework. Too much homework only antagonizes kids and in most cases, it does not help their learning.

On the other hand, a small amount of homework is a good thing:

- It is a form of differentiation, as it allows kids to take different amounts of time to do the same assignment. (In a successful cooperative learning culture racing is discouraged, and kids work more or less at the same pace, with the faster students slowing down to help others when needed. This is not just altruistic, as doing this helps deepen their understanding.)
- It gets the message across that it takes work to learn anything substantial, and that while in class we work in groups, the ultimate goal is to understand the material well enough to deal with it on your own.
- It's a place to do the often necessary but often boring work of basic drill and review, thereby saving class time for more interesting and substantial engagement.
- Homework helps you sort out what you understand from what you are still struggling with, which will make it easier for you to get exactly the help you need when going over the homework with your group.

The message is: you need to know how to do this on your own, but you can get as much help as you need. You are not being rushed or pressured. My goal as the teacher is not to separate those who can from those who can't: it is to get to where everyone can.

In my class, a powerful argument in favor of homework came from the students themselves. In course evaluations, it was common for students to tell me that what helped them learn the most was going over the homework with their classmates. I usually allowed about 15 minutes for that at the beginning of class. During that time, I walked around and recorded a 0 (did not do it), 2 (great job), or 1 (somewhere in between). Students helping each other is far more efficient than me explaining things to the class, because it allows different groups to focus on the parts of the assignment they each need to focus on. True, there's a risk that all the students in a given group are doing something incorrectly, but that doesn't happen very often. Besides, after a quick homework check, I'm walking around, ready to intervene if I see this.

A teacher explanation at the board is sometimes needed, particularly if there is a similar question in more than one group. However in general it is not a good idea as it is boring for students who did the work correctly at home, and often insufficient for students who really don't get it. It is also a waste of precious class time. Asking students to write solutions on the board does allow others to focus on the ones they need help with, but live conversation is more effective than silent copying of ill-understood techniques, and again, it is a waste of time for students who did the homework correctly.

Still, you may be in a school where students do not or cannot do homework. Fair enough: homework is not a matter of principle for me. It is a component in a comprehensive approach, and the same goals can be accomplished in other ways.

If a student cannot do homework because of their home situation, the school should offer study hall time so that "homework" can be done at school. If the school cannot or will not do that, and many students are unable to do homework, then some class time can be allotted for that: time where you work on your own, after plenty of time working with peers, and prior to being quizzed. Of course, giving up class time for this will reduce coverage, but it may be a price worth paying.

However, I should say that I have heard from more than one teacher that homework completion increased dramatically once homework was lagged. It turned out that the reason many students weren't doing the homework was because they didn't know how to do it. Lagging gave them time to learn the concepts, and made homework possible. So before abolishing homework, I would try both lagging it, and offering in-school time for it. More students will learn more math than with no homework at all.

## 4. Quiz and Test Corrections

The quiz or test should not be the end of the road on a given topic. It helps students know what they know and don't know, and they should get a chance to work on test corrections for credit, with the knowledge that any topic (particularly the hardest ones) is likely reappear in a later assessment. One way to manage this is to have short, focused quizzes, but to make sure that more substantial tests are at least somewhat cumulative. I tell students that anything which stumped many of them on a test is sure to appear on a future test. It is not sufficient (let alone effective or fair) to only offer a cumulative assessment in a final at the very end of the semester or course.

I require quiz and test corrections from all students and give them a week to complete them as homework. Everyone knows that doing well the first time is better, but if learning is the goal, what difference does it make if the learning occurs a week later? In my version of this, I told students they could get half-way to a perfect score by turning in high-quality corrections. I allowed my students to get help from anyone, but all writing must be their own.

This assumes a standard of explanation that is higher than on the test itself. The reason is that the student will not learn much by just copying correct answers, or even by having those explained to them, and then writing them down. Writing detailed explanations in one's own words helps with both understanding and retention.

**Why require corrections from all students?** Why not? It does not add a lot to the grading burden: grading the corrections from the student who got 95% right on the quiz does not take hardly any time. It also makes it easier to ask tough "bonus" or "extra credit" questions on the quiz.

**I tried it, and some (or many, or most) students didn't take corrections seriously.** Fair enough, but maybe you gave up too quickly. This is like everything else: students need to learn how to do it in a useful way. That may require some changes in departmental or school culture, and that does not happen fast. Until then, give points for well-written corrections!

**But if you give points, it will change the grades**! If that reflects learning, it is a good thing! Cheating is not an issue since students are allowed to get help. And no credit should be given for anything not written in the student's own words, or for answers without explanations.

**Why not, instead, offer opportunities for students to retake tests?** Because it is logistically challenging (finding a time and place for that, supervising). It requires creating alternate versions of tests. It takes longer to grade (because you're already familiar with the test that is being corrected.) Finally it continues to favor students who work well under time pressure over students who take time to do a great job.

### Lagging Assessments

Exposure can be extended even more by scheduling the quiz a week after homework was assigned, and yet another week before quiz corrections are due:

(*Recycle* is the word we used to mean "quiz corrections".)

The message to the student is: I trust that you can learn this, given enough time. In my experience, some students need all that extra time, and only finally "get it" at that last stage, when writing their quiz corrections. If some students do need that much time, why not give it to them? You are not taking anything away from kids who learn fast. Quite the opposite: using all these lagging strategies allows you to keep moving on to new topics at a good clip!

You may be worried that students will forget everything by the time the quiz comes around. My experience is the opposite: giving more time for students to develop their understanding makes for longer-term retention. When I announce a quiz, my students usually want to make sure it's not on something recent, that they haven't yet had time to fully grasp. (Instead of worrying about remembering, they're worrying about understanding!)

But really, if your students can't hang on to the ideas for an extra week, then why teach those ideas at all?

At most schools, math is mostly assessed by way of quizzes and tests. Unfortunately, those are usually conducted under tremendous time pressure. But why does it have to be so? What if your quizzes were shorter? What if you gave students as much extra time as they need? What difference does it make if they answer a question at 10:20 and not at 10:15?

Many students do not work well under time pressure, so overemphasis on quizzes and tests can be unfair. Thus, I give substantial weight to test corrections, plus a couple of major at-home assignments per trimester (projects, reports, and/or very tough problem sets.)

## 5. Topic Sequencing

This "lagging" approach has some implications for topic sequencing. For one thing, difficult and important topics should not be introduced in May! By then both students and teachers are tired, and there is little chance of success. Teach those topics as early as possible. Starting early will leave sufficient time for the needed "eternal review".

I have written about general principles of topic sequencing here. In this article, I will focus on two strategies which contribute to extending exposure.

### Separating Related Topics

Separating related topics is a great way to extend exposure, which yields the same benefits as lagging homework and assessments, for the same reason. Even though it is not widely used (yet!), this is a powerful idea, and it is not too difficult to implement.

Most of us teach related topics consecutively, hoping that if the previous topic is still fresh in the student's mind, it will make the next one easier to grasp. Examples are teaching factoring right after teaching the distributive rule, teaching sine, cosine and tangent in one fell swoop, teaching series immediately after sequences, and so on.

- Well, most of us are wrong. It is far more effective to separate related topics. Here are some examples:
- - factoring, then some time later, the distributive law
- - the tangent ratio, then some time later, sine and cosine
- - quadratic functions, then some time later, quadratic equations
- - sequences, then some time later, series
- - exponential functions, then some time later, logs

If you separate related topics, when you get to the second topic, you have a built-in review of the first one, perhaps in homework. And thus you've created forward motion (moving on to something else after introducing the first topic), and extended exposure (revisiting the first topic when working on the second.) Another advantage of this approach is that it communicates to the student that they are learning for the long run: any important topic may and will come back.

The gap can be anything from a few weeks to a whole semester.

### Two Units

I had the good fortune of teaching in long periods for my whole career, in what is sometimes known as a block schedule. One way I used that structure was to focus on two units at any one time. For example, here is the outline of trimester 2 in a "Math 2" class I taught before my retirement:

(My point is not to recommend this exact sequence, which depends on many department-specific assumptions, but to use it as an example of what is possible.)

Here are some of the advantages of that approach:

- Any one day or week is more varied, which is helpful in keeping students interested and alert. Note in particular that we tried to match topics that are as unlike as possible.
- It takes roughly twice as many days to complete a unit. This is good for students who need that extra time.
- This takes nothing from students who pick up ideas quickly. In fact, they appreciate the variety.
- It makes it easier to balance challenging and accessible work: if you hit a difficult patch in one topic, you can ease up on the other one. More generally, you gain a lot of flexibility in your lesson planning.
- If your work on one topic hits a snag, you can emphasize the other topic while figuring out what to do.
- Perhaps most importantly, it carries a message to the students: you still need to know this when we’re working on something else.

In the long period, it is possible to hit both units in every class period, for example by introducing new ideas on one topic during the longer part of the period, and applying already-introduced ideas on the other topic in the remaining time. (Homework is typically on one or the other topic, not both.)

**Can this approach, or a version of it, be used in traditional 50-minute classes? **I don’t know. I am guessing that the answer is yes, but I have not tried it. It might involve, for example, focusing on each topic on alternate days, while the homework is on the other topic.

But, you ask, **is this not confusing to students? Don’t they prefer focusing on one single topic? **If they do, that is only because that is what they’re used to. In teaching, the biggest obstacles to making changes are the cultural ones: the expectations of students, colleagues, parents. administrators, and of course one’s own deep-seated habits. At my school, working on two units, separating related units, lagging homework, all those were department-wide policies. Once they’re used to it, students do not question any of them.

## 6. Conclusion: Spiraling Made Easy and Effective

In most math curricula, students work on a single topic at a time. The idea is that is that by focusing on the topic, you are helping students really learn it, before you move on to the next unit. Unfortunately, that is not how retention happens. It is much more effective, when learning a new concept, to see it again a few weeks later, and again some time after that. Thus the concept of spiraling. The algebra textbook I coauthored in the 1990’s is spiraled throughout: not just in the homework, but in the makeup of each chapter and many lessons.

- What I have learned since then is that spiraling can be overdone. Over-spiraling can in some cases result in learning that is more shallow, and almost always in materials that are difficult for the teacher to organize. (See my blog post: Spiraling Out of Control?) The strategies outlined in this article constitute an approach to spiraling which:
- - is unit-based, and allows for going in depth into each topic
- - is easy to implement and does not make unrealistic demands on the teacher
- - is transparent and does not hide from the teacher what lessons are about

Implementing these policies does not require more prep time, or more classroom time, and it creates a non-artificial, organic way to implement *constant forward motion, eternal review*. It helps all students with the benefits of spiraling, but without the possible disadvantages. Cumulatively, these techniques make a *growth mindset* possible, as students are getting so many opportunities to learn each concept. This sort of approach is far more effective than "growth mindset" bulletin boards or slide shows. It is not enough to tell students they can grow: *we need to make that growth possible*.

What kills me is that many people who hear about these ideas do not try them. Such is the force of habit and tradition. "There's no time" is a common objection to making changes to one's teaching routine, but it doesn't apply to this! Lagging homework and assessments, separating related topics, etc. does not take any more time: it's just rearranging the time you were already planning to take.

Of course, if all your students learn math at the same rate, you should ignore these suggestions!