My Math Education Blog

"There is no one way"

Monday, October 15, 2018

Spiraling Out of Control?

In most math curricula, students work on a single topic at a time. (When I taught elementary school, decades ago, I noticed that if we’re working on subtraction, it must be November! But the same applies at all grade levels.) The idea is that is that by really 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. Years ago, the Saxon books distributed homework on any one topic across the year, typically with one or two exercises per topic on any given day. Some more recent curricula do facilitate that sort of homework spiraling by including review homework in addition to homework on the current topic after each lesson. 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. This idea was so important to us, that there is an image of a spiral at the start of each chapter! (If you have the book, check that out! Or just look at it online.)

In this post, I want to argue that while I agree with the fundamental underlying idea of a spiraled curriculum,  there is such a thing as overdoing the spiral. I will end with specific recommendations for better spiraling.

Impact on Learning 

Too much spiraling can lead to atomized, shallow learning. If there is too much jumping around between topics in a given week, or in a given homework assignment, it is difficult to get into any of the topics in depth. Extreme spiraling makes more sense in a shallow curriculum that prioritizes remembering micro-techniques. In a program that prioritizes understanding, you need to dedicate a substantial amount of time to the most important topics. This means approaching them in multiple representations, using various learning tools, and applying them in different contexts. This cannot be done if one is constantly switching among multiple topics.

In particular, in homework or class work, it is often useful to assign nonrandom sets of exercises, which are related, and build on each other. For example, “Find the distance from (p, q) to (0, 0) where p and q are whole numbers between 0 and 10.” (This assignment is taken from my Geometry Labs.) At first sight, this is unreasonable: there are 121 such points. But as students work on this and enter their answers on a grid, they start seeing that symmetry cuts that number way down. In fact, the distances for points that lie on the same line through the origin can easily be obtained as they are all multiples of the same number. (For example, on the 45° line, they’re all multiples of the square root of two.) Nonrandom sets of problems can deepen understanding, but they are not possible in an overly spiraled homework system.

Impact on Teaching

The main problem with hyper-spiraling is the above-described impact on learning. But do not underestimate its  impact on the teacher. For example, some spiraling advocates suggest homework schemes such as “half the exercises on today’s material, one quarter on last week, one quarter on basics.” Frankly, it is not fair to make such demands on already-overworked teachers. Complicated schemes along these lines take too much time and energy to implement, and must be re-invented every time one makes a change in textbook or sequencing. Those sorts of systems are likely to be abandoned after a while, except by teachers who do not value sleep.
Another problem for teachers is that it makes using a hyper-spiraled curriculum difficult to use, because it is difficult to find where a given concept or technique is taught. (In the case of Algebra: Themes, Tools, Concepts we tried to compensate for that by offering an Index of Selected Topics and Tools. We also included notes in the margin of the Teachers’ Edition: “What this Lesson is About”. But even with all that, a hyper-spiraled approach makes extreme and unrealistic demands on teachers’ planning time. In fact, some hyper-spiraled curricula lack even those organizational features. Without them, a teacher needs to spend the whole summer working through the curriculum in order to be ready to teach it. This can be fun if the curriculum is well designed (e.g. the Exeter curriculum), but no one should feel guilty if they’re not up to that level of workaholism. 

Spiraling Made Easy and Effective

So, you ask, what do I suggest? In the decades following the publication of my overly-spiraled book, I developed an approach to spiraling that:
  • 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 what lessons are about (most of the time)
I have written a fair amount about this, under the heading extending exposure. The ingredients of this teacher-friendly approach are:
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. You really should try it!

-- Henri

Sunday, October 7, 2018

More on Extending Exposure

I have written several posts in which I argued that extending student exposure to mathematical concepts is one key to reaching the whole range of students. This is based on the simple observation that students learn math at different rates, and that extending exposure by making simple changes to our routines can benefit all students: those who pick up new ideas quickly, and those who need more time. If schools heed NCTM’s recommendation to eliminate tracking, making these simple changes becomes even more important.

On this blog, the most popular post on extending exposure is Lagging Homework, and it links to other posts where I describe additional strategies (separating related topics, lagging assessments, and more.) If you haven’t read those posts, you should. The problem is, it does involve a bit of clicking around. Thus, I decided to combine all that information in a single longer article on my Web site.

However, before I do that, I need to share a few more related tidbits which hadn’t made it into those posts. Writing about those here will help when I’m ready to put it all together.

Homework as Preparation

A number of people, over the years, have told me that they don’t agree with my lagging homework system because they like to assign homework that prepares the students for the next day’s lesson. That, my friends, is not a disagreement! I love that idea.

Lagging homework is not a rigid system that requires homework to be assigned exactly one week (or day, or month) after the corresponding class work. 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 which I described in my original post, 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. I described the characteristics of such preparatory problems in this post, based on Scott Farrand’s approach to in-class warm-ups.

In general, these 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 (better than taking precious class time for that). On the other hand, if preparing for the next day’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. (Unfortunately, the students who suffer from this policy will get the blame: they didn’t work hard enough, they’re not abstract thinkers, they don’t belong in this class, and so on.)

Two Units

I had the privilege of teaching in long periods for my whole career. One way I used what is sometimes known as a block schedule was to focus on two units at any one time. For example, here is the outline of semester 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, . The only way to tackle these obstacles is departmental collaboration, and a step-by-step approach: don’t make all the changes at once! 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.

Coming Soon

My next post will hopefully be the last in this series, and it may surprise you: Beware of Over-Spiraling! Once that is written, I’ll be ready to put all the pieces together in one article on teaching heterogeneous classes (in other words, any classes!)

-- Henri

Sunday, September 23, 2018

Understanding "understanding"

It is not uncommon to read articles about math education in the mainstream press, arguing that students must master basic skills before they can develop conceptual understanding. And moreover, that the road to such mastery is teacher explanation followed by repetitive drill. These essays frequently argue that it’s like learning to play the piano: you must practice scales before playing real music! When I mentioned this to a friend who is a piano teacher, he considered it to be an insult to his profession. He said that obviously these people are not piano teachers! Teaching piano is about music! Yes, students do need exercises, but if that’s all you had them do, you’d drive them away from music. The biggest motivator is the recital, when they play real music, not scales! 

My friend is right: the authors of these op eds are not piano teachers, but they’re often not math educators either! Still, it is important to address their ideas, because they reflect a broad cultural consensus among many parents, administrators, students, and teachers. Some proponents of the “skills first” approach equate teaching for understanding to what they call “fuzzy math”, a flaky anything-goes sort of teaching, with no specific learning goals, no accountability, just feel-good teachers who allow students to wallow in their ignorance.

It behooves those of us who disagree with this caricature to clarify what we mean by understanding. That is the main purpose of this post.

To get a straw man out of the way, my position is that understanding cannot be divorced from the acquisition of skills. Without understanding, it is hard to develop an interest in the skills, or to retain them; without skills, understanding is out of reach. Good teaching requires a skillful weaving of those two strands. It is, in fact, like learning to play the piano! Skills are important, but it’s all about the music.

But on to the main point: what is understanding? This is a difficult question, and the true fact that experienced math teachers can "recognize it when they see it" is not a sufficient answer. Here is an attempt at spelling it out. A student who understands a concept can:

  • Explain it. For example, can they give a reason why 2(x+3) = 2x+6? Responding “it’s the distributive rule” is evidence that the student knows the name of the rule, but a better explanation might include numerical examples, or a figure using the area model, or a manipulative or visual representation.  Therefore, we should routinely ask students to explain answers, verbally or in writing, even though many don't enjoy doing that. It is a way for us to gauge their understanding, and thus improve our teaching, and more importantly, it is a way for them to go deeper and guarantee the ideas stick.
  • Reverse processes associated with it. For example a student does not fully understand the distributive law if they cannot factor anything. More examples: can they create an equation whose multi-step solution is 4? Can they figure out an equation when given its graph? And so on. Reversibility is a both a test of understanding, and a way to improve understanding. 
  • Flexibly use alternative approaches. For example, for equation solving, in addition to the usual "do the same thing to both sides" for solving linear equations, students should be able to use the cover-up method, trial and error, graphs, tables, and technology. If they have this flexibility, they can decide on the best approach to solve a given equation, and moreover, they will have a better understanding of what equation solving actually is.
  • Navigate between multiple representations of it. Famously, functions can be represented symbolically, or in tables, or in graphs. Making the connections between these three is a hallmark of understanding. I have found that a fourth representation (function diagrams) can also help deepen understanding, and be used to assess it. Multiple representations on the one hand offer different entry points that emphasize different aspects of functions, but making the connections between the representations is part and parcel of a deeper understanding. 
  • Transfer it to different contexts. For example, ideas about equivalent fractions are relevant in many contexts, such as similar figures and direct variation.  Or, the Pythagorean theorem can be used to find the distance between two points, given their coordinates. If a student can only handle a concept in the form it was originally presented in class or in the textbook, then surely no one would claim they fully understand it. 
  • Know when it does not apply. When faced with an unfamiliar problem, students will tend to reach for familiar concepts, such as linear functions and proportional relationships. Sometimes, this makes sense, of course, but students need to be able to recognize situations where a given concept does not apply. 

Clearly aiming for all this is a high bar, and it is tempting to just have students memorize some facts and techniques, and then test them to see if they remember those a few weeks later. (This is often how the “skills first” approach plays out.) But what good would that do? It would just add to the vast numbers who got A’s and B’s in secondary school math, went to college, and now tell us “they’re not math people”.  Yes, teaching for understanding is ambitious, and it must be our goal for all the students. 

Alas, there are obstacles. For one thing, understanding cannot be easily conferred by explanations. (A naive traditionalist once suggested that it was easy to teach about variables: patiently explain to the students that variables behave just like numbers! Would that it were that easy.) Moreover, understanding is not always valued by students, parents, and administrators, many of whom believe that everything would be so much more straightforward if we could just have the students memorize facts and algorithms. Finally, understanding is difficult to assess. (Actually, the list above is one way to improve assessments: each item on the list suggests possible avenues for authentic assessment. To reduce complaints that it’s "not fair" to assess students that way, such assessments can be ungraded. As the current jargon would have it: consider them formative assessments.) Being able to reproduce a memorized set of steps is a good test of memory and obedience. To test understanding, non-rote assessments are the most revealing.

The above list is also a tool in forward design. When planning a unit, ask yourself how you can incorporate reverse questions, alternate approaches, multiple representations, varied contexts, and so on. A tool-rich pedagogy is helpful, as different manipulative, technological, and paper-pencil tools provide a way to do this and avoid boring repetition. Of course, the implication of such planning is that it takes more time to teach any given concept. Because of the enormous pressure of coverage at all costs, it is generally necessary to take less time on less important topics, and approach the most important topics in as many ways as possible. 

In any case, I hope you find this post helpful in your teaching, and also in your conversations with colleagues, administrators, parents, and students.

Good luck as you teach for understanding!

-- Henri

[This post includes part of my Nothing Works article, updated and expanded.]

Wednesday, August 15, 2018


I’m done with my summer workshops, and I hope to resume blogging when the inspiration strikes. Today, a brief post about catchphrases, one per paragraph. (And no, this is not because this word has six consecutive consonants, which may well be a record.)

I started thinking about this topic when I learned that Annie Fetter's catchphrase “What do you notice? What do you wonder?” had been trademarked by the alas now defunct Math Forum. The trademark is now owned by NCTM, which considers it community property, and has no intention of restricting its use by any and all math teachers. This is good, since in some situations, these two questions can jumpstart a worthwhile exploration, one which is initially driven by an actual expression of student understanding and interests. This is particularly important in a classroom that needs to move away from a rigid “listen to the teacher and then practice” model. Applied skillfully, these questions can honor student thinking and sense-making, and  launch a great lesson. Still, as I pointed out in this post, there are situations where this is not the best option. Like every other good idea about teaching, it cannot be expected to apply universally. Teaching is a complex enterprise, that does not lend itself to one-liner solutions.

And yet! I love coming up with my own catchphrases. For example, the point I made at the end of the previous paragraph is encapsulated in the catchphrase “Nothing works!”. I love this slogan because it is sure to get people’s attention, and it allows me to elaborate: nothing works for every class, every teacher, every lesson, etc. This is an important understanding in a culture of quick fixes and edu-fads. (Some links about this: “Nothing Works” article| blog post on being eclectic | “Art of Teaching”worksheet to help teachers and departments think about alternatives to current practices.)

A friend told me that “Nothing woris” is self-referential, since it too doesn’t work universally: she claims that some things I believe in do work. (Alas, I’m unable to remember what she was referring to!) A related catchphrase, the motto for my Web site and for this blog, is “There is no one way” (as the Zen Buddhist said to the traffic cop.)


To elaborate: teachers sometimes look for the “best” way to teach something. That is the wrong question: for anything important, we should know many ways to teach it so as to have the needed flexibility when reaching a wall in a given class or with a given student. Thus the importance of learning tools and multiple representations, which make that possible.

My favorite among my own catchphrases may be “Constant forward motion. Eternal review.” 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 if we want ideas and techniques to stick. Each requires the other to work well, and in combination, they make for extended exposure, a must for heterogeneous classes (i.e. all classes.) Striving for constant forward motion and eternal review is facilitated by such practices as lagging homework, separating related topics, test corrections, and no doubt other techniques. 

Incorporating those changes, or any changes, into one’s teaching (or into departmental practice) is a long term project. This is captured in my catchphrase “Fast is slow and slow is fast”.


What I’m trying to say with this cryptic formulation is that if you try to make too many changes in a hurry, you may find that in fact the changes are superficial, and the underlying classroom realities are not affected. Or you may conclude that “it didn’t work” and go back to your old ways. If on the other hand you pace yourself, and make incremental changes one step at a time, you will find that on the one hand you reap immediate benefits, and on the other hand the changes will take root and become the new normal. Be the tortoise, not the hare. Even better if you do this in ongoing dialogue and collaboration with your colleagues.

You’ll notice that my catchphrases are largely aimed at teachers, not students. This is because I’m in general agreement with “Nix the Tricks” a great compendium of how to avoid catchphrases in our teaching. (Download the book, which was put together by Tina Cardone and the MTBoS.) Shortcuts like “cross-multiply” or “FOIL” usually obscure the underlying mathematics. They often reflect a cynical attitude: the students will never understand these concepts, so I’ll give them an easy-to-remember shortcut

Nevertheless, I do occasionally use a catchphrase in my teaching. For example, in geometry I might be heard offering the hint: “When working with circles, you should listen to the radii”. It is a good hint, with substantial mathematical content, so it’s not really a trick that should be nixed. Overall, my stance is that formulas and tricks should encapsulate understanding, not substitute for it. That is the catchphrase I’ll end on.

Feel free to share catchphrases you love or hate in the comments!

Monday, April 2, 2018

April Travels, May Webinar, Summer Workshops

I'll be traveling a lot this month. Here's the plan, should you want to say hello.
New York City April 5, 4:30pm: I will present Geometric Puzzles at the Museum of Math Teachers’ Circle. Geometric puzzles are accessible to solvers of all ages, but they can also challenge even the most tenacious of solvers.  Join math education author and consultant Henri Picciotto in an exploration of hands-on polyomino puzzles that involve area, perimeter, symmetry, congruence, and scaling — you’ll even participate in some collaborative pentomino research!
New York City April 6, 6:30pm: I will present Playing with Pentominoes at the Museum of Math Family Friday. Pentominoes are simple to create — just join five equal-sized squares together — but provide a host of classic challenges in the world of recreational mathematics.  Discover them, play with them, and explore a variety of visual puzzles that span the whole range: from kindergarten to adult, from the most accessible to the most challenging, and from the meditative to the maddening.
Atlanta April 11-15: I will attend the Gathering for Gardner. I won't be presenting, but I look forward to seeing old and new friends. I contributed a thematic cryptic crossword to the conference book.
Washington, DC April 25 and 27 3:00pm: I will present Taxicab Geometry for the Math Teachers’ Circle sessions at the NCTM National Meeting (in the Networking Lounge). Many concepts depend on distance: the triangle inequality, the definition of a circle, the value of π, the properties of the perpendicular bisector, the geometry of the parabola, etc. In taxicab geometry, you can only move horizontally and vertically in the Cartesian plane, so distance is different from the usual "shortest path" definition. We will explore the implications of taxicab distance. There are no prerequisites, other than curiosity and a willingness to experiment on graph paper.
Washington, DC April 26, 2pm and April 27, 1pm, I will be at the Didax booth (#253 in the exhibit hall, NCTM National Meeting) for a 15-20 minute introduction to the Lab Gear. Participants will get a free sample!
Washington, DC April 27 8:00am: I will present “Quadratic Equations and Functions Use Manipulatives and Technology for More Access and More Depth” at the NCTM National Meeting. (Convention Center 209 ABC) Algebra manipulatives provide an environment where students can make sense of two ways to solve quadratic equations: factoring and completing the square. Graphing technology allows students to link those approaches to quadratic functions. Using these tools and connecting these concepts makes the algebra come to life for all students. [I will give away manipulatives to the first 72 attendees.] If you’re an experienced Lab Gear user, I would love it if you would assist me during that session. Get in touch!
And from the comfort of your own home:
Anywhere, May 17, 5:00pm Pacific Time: Reaching the Full Range, a webinar. As everyone knows, students learn math at different rates. What should we do about it? I propose a two-prong strategy based on alliance with the strongest students, and support for the weakest. On the one hand, relatively easy-to-implement ways to insure constant forward motion and eternal review. On the other hand, a tool-based pedagogy (using manipulatives and technology) that supports multiple representations, and increases both access and challenge. Click here.
Whether you attend these events or not, you can find handouts and links on my Talks page.

Summer Workshops

I'll be presenting two summer workshops for teachers, at Menlo School in Silicon Valley:
No Limits! (Algebra 2, Trig, and Precalculus, with Rachel Chou, Aug 1-3), and
Visual Algebra (grades 7-11, Aug 6-9.)
For more information about the workshops, visit my Web site.
Info about registration and logistics: Menlo School.

Friday, February 23, 2018


In my last post, I offered guidelines for sequencing math curriculum. The response I got on Twitter (and in one comment to the post) was quite positive. However, one point I made triggered some disagreement:

Start with definitions? No! Most students find it difficult to understand a definition for something they have no experience with. It is more effective to start with activities leading to concepts, and introduce vocabulary and notation when your students have some sense of what you’re talking about.

Michael Pershan wrote:

I also don't want to introduce vocab before kids are ready for it, and there are often times when I introduce vocab when it comes up in class. When that happens, I like to launch the next class with that vocab and give kids a chance to use it during that next lesson.

I don't disagree with what @hpicciotto says in this post either. I square the two by saying vocab probably shouldn't be the start of the unit, but it works nicely for me when it's the start of a lesson.

My heuristic is something like, I want the definition to be easy to understand, so I give the definition when it won't be hard to comprehend. And that often (not always) requires some prior instruction. But I do like introducing vocab at the start of a lesson.

I don’t disagree with Michael. None of the guidelines I gave in that post should be interpreted as rules one cannot deviate from. The essence of what I was trying to get across is to avoid definitions if students cannot understand them. Michael’s system does not violate that principle. Starting a particular lesson with a definition, if students are ready for it, is not a problem at all.

As always when thinking about teaching, beware of dogma! Be eclectic, because nothing works, not even what I say on this blog!

Mike Lawler gave a specific example:

Teaching via definitions: I found it useful to introduce a formal definition of division to help my younger son understand fraction division initially (this video is from 4 years ago) :

Then a few days later we did a more informal approach with snap cubes. Overall I thought this formal to informal approach was useful and helped him see fraction division in a few different ways:

You should definitely watch the videos. They provide a great example of starting with a definition, that worked. But let’s analyze this. The student, in this case, was indeed ready for a definition:

  • He already knew that fractions are (or represent?) numbers.
  • He already knew what a reciprocal is (not merely “flipping” the fraction, but the number by which you multiply to get 1, and from there he got to flipping)
  • He already knew, that a division can be represented by a fraction.

Many students who are told to “invert and multiply” know none of this, and for them defining division this way would not carry a lot of meaning. Moreover, students at this level usually have an idea of what division means, and it would be important to show that multiplying by the reciprocal is consistent with the meaning they already have in mind.

So my approach might be to start with something students know (for example, “a divided by b” can be said “b times what equals a?) and from there find a way to get to “multiply by the reciprocal”. (That is my approach in this document.) However it is not easy to do in this case, and defining first, and then getting to a familiar meaning may well be preferable.  In fact, that is very much the approach I use when defining complex number multiplication in high school. 

To conclude: yes, a lesson can start with a definition, as long as the students know what you’re talking about, and will not instantly turn off. This does not invalidate my point. To return to the example I gave in my last post, in a bit more detail, compare these two approaches to introducing the tangent ratio.

Standard approach: “Today we’re starting trigonometry. Please take notes. In a right triangle, the ratio of the side opposite the angle to the side adjacent to the angle is called the tangent. (etc.)” This approach is likely to lead to eyes glazing over, to some anxiety induced by the word “trigonometry”, and to a worry about remembering which ratio is which. The latter is allayed by the strange incantation “soh-cah-toa”, but alas that does not throw much light on the topic.

My suggested approach: “As you know, for every angle a line makes with the x-axis, there is a slope. For a given slope, there is an angle with the x-axis. [More than one, but no need to dwell on that right now.] We’ll use this idea, a ruler, and the 10cm circle to solve some real-world problems.” This allows students to right away put the tangent ratio to use, without knowing its name. (See Geometry Labs, chapter 11. Do Lab 11.2 after introducing the 10cm circle, but before making tables as suggested in Lab 11.1. Free download.) Once they’re comfortable with the concept, you can tell them there’s a word for this, a notation, and a key on the calculator. And yes, at that point, you can do all that at the beginning of a lesson. And a few months later, you can introduce the sine and cosine in a similar way.

-- Henri

PS: How to introduce trig, complex numbers, and matrices in Algebra 2 and Precalculus are among the topics Rachel Chou and I will address in our workshop No Limits! this summer. More info.

Thursday, February 15, 2018


In my last post, I argued that, as teachers and math education leaders in a school or district, we need to free ourselves from the sequencing preordained by the textbook, and instead pay attention to what actually works with our students. In this post, I will present some general guidelines for sequencing topics, and some specific suggestions. All these ideas are based on 3+ decades in the high school math classroom, with somewhat heterogeneous classes.

Is the topic age-appropriate? Mysteriously, not much attention is paid to this. For example, tradition requires completing the square and the quadratic formula as topics for Algebra 1, In my experience, it is much easier to teach this in Algebra 2, to students who have a little more maturity. This in turn frees up precious time in Algebra 1 to reinforce the sort of basics that we find so frustrating when they are missing later on.

Can the topic fit in a single unit? Some topics are important, but difficult for some students. It’s a good idea to spread those out over more than one unit, and sometimes more than one course. One example of that is linear, quadratic, and exponential functions, which can be approached in different ways at different levels, from Algebra 1 to Precalculus. Another example is trigonometry, which can be distributed among Geometry, Algebra 2, and Precalculus.

When in the school year? 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. There may also be some traditionally late topics that can be useful early on. For example, an exploration of inscribed angles has only very few prerequisites, and can be used to practice angle basics in an interesting context. I did this very early in my Geometry class. See my Geometry Labs (free download.)

Spiraling? There is much to be said for coming back to already-seen topics later in the school year. However spiraling can  be overdone, and result in an atomized curriculum consisting of chunks that are so small that students don’t get enough depth on each topic. One easy-to-implement policy, which provides the advantages of both spiraling and depth is to separate related topics. I have written about that in past blog posts (here and here).

Review? It is widely believed that one should start the year, and then each unit and even each lesson with review of relevant past material. Of course, I understand why that is a standard practice, but I believe it is counterproductive. Over time, it tells students “you don’t need to remember anything — I’ll make sure to remind you.” It is also profoundly boring for the students who don’t need the review, and takes away from the excitement of starting something new. It is far better to start  with a solid anchor activity, and use homework, subsequent class work, and if need be out-of-class support structures to do the review. It is especially catastrophic to start Algebra 1 with a review of arithmetic. The students who need it won’t get it, and those who don’t will be disappointed.

Anchor activity? If you can, start a unit with an interesting problem or activity (the anchor). It should be motivating and memorable, and it need not be easy. Examples of anchor activities are Geoboard Squares for the Pythagorean theorem (Lab 8.5 in Geometry Labs,) Rolling Dice for exponential functions, or Super-Scientific Notation for logarithms. A good anchor brings together key content with good practices, and generates curiosity and engagement. It is something you can refer to later on, to remind students of the basics of the unit.

Start with definitions? No! Most students find it difficult to understand a definition for something they have no experience with. It is more effective to start with activities leading to concepts, and introduce vocabulary and notation when your students have some sense of what you’re talking about. For example, see the Super-Scientific Notation lesson mentioned above. Another example: the tangent ratio can be introduced with the help of slope, without having to mention trigonometry or the calculator, instead using the 10-centimeter circle. Once students can use the concept to solve problems, you can name it and reveal that there is a key on the calculator for it. 

Concrete or abstract? Math is all about abstraction, but understanding is usually rooted in the concrete, so it is usually a good idea to start there. This can mean many things:

  • Discrete first, continuous later. Numerical examples first, generalization later. For example, work on the geoboard (both the standard 11 by 11 geoboard, and the circle geoboard) is strictly with specific examples based on the available pegs. But it lays the groundwork for a generalization using variables which would otherwise be impenetrable to many students.

  • Natural numbers to real numbers -- almost any new idea is more accessible if you start with whole number examples — as in NewImage.

  • Kinesthetics (link) and manipulatives (link) do not accomplish miracles, but they can improve classroom discourse and provide meaningful and memorable reference points. In particular, algebra manipulatives can provide both access and depth to an essentially abstract subject, by way of a visual / geometric interpretation.

  • Tables and graphs can help provide a concrete foundation to the study of functions. This is sometimes described as modeling: you start with a concrete situation, use tables and graphs to think about it, and generalize with equations. This is the approach I use a lot in Algebra: Themes, Tools, Concepts and in my Algebra 2 materials.

From easy to hard? Well, that is certainly implied in the previous segment. However, I will now challenge that assumption. (What can I say, sequencing curriculum doesn’t lend itself to simple choices.) In my view, it is a good idea to start with somewhat challenging material, then ease up, and keep alternating between hard and easy. Starting too easy can give the wrong impression, that the unit will not require work. In fact, most of the above guidelines are best implemented as a back and forth motion: for example, after introducing vocabulary and notation, one needs to re-introduce the concepts. Likewise for most of these guidelines.

This is all fine, but how does one deal with externally mandated sequencing? Alas, I have no experience with this, as most of my career was at a small private school, and moreover I chaired my department (with plenty of input from my colleagues.) I can only suggest discussing these ideas with colleagues and supervisors! Also, most of the suggestions in this post address sequencing within a unit, and thus may be implemented anywhere if there is any wiggle room at all in the mandated sequence.

-- Henri

PS: I’m offering two summer workshops (one on algebra, grades 7-11, and one with Rachel Chou on Algebra 2 / Precalculus. The workshops will include many of the bits of curriculum I linked to in this post. More info.


  • Much of this post is based on one section of my article with the cheerful title Nothing Works.

  • Some of it is from a previous post on sequencing (Mapping Out a Course), where I propose a step by step process for doing just that.