My Math Education Blog

"There is no one way"

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.

Thursday, February 8, 2018

Mind Maps

Alison Blank makes good points in her interesting presentation: "Math is not linear", where she encourages us to make connections, go on tangents, preview future topics and review past ones. In short, we should not be trapped in the inflexible sequence suggested by textbooks and school culture. In a recent blog post, Jim Tanton makes the related point that presenting the content of a course to students on a "mind map" should help students see that math content is rich with interconnections, and should not be seen as a straightforward linear march through topics as suggested by a table of contents, or a syllabus. I very much agree with both of them.

However, as a teacher and curriculum developer, I cannot limit myself to think only about content. I prefer a more complicated framework,  based on "themes, tools, concepts" (TTC) which I summarize in this figure:
Traditional pedagogy is near the top. It need not be thrown away, but it can only be effective if it is based on a foundation of themes (contexts), multiple representations, and learning tools. I explained this in more detail in a 2014 post and in various articles over the years. Later in 2014, I sketched a mind map using the TTC framework for the concept "proportional relationships". Check it out. And way back in the 1990's, I made a TTC mind map for the theme "area". Here it is:

(If the type is too small, see the full-scale map here —scroll down to page 23)

Compare with the corresponding map in too many math programs:


For anyone planning a course, or preparing a unit, I strongly recommend starting with brainstorming a TTC mind-map for what needs to be taught. This can be done alone, but it is sure to be more productive as a departmental project. The teacher's familiarity with the TTC landscape and connections for that course or unit is sure to provide a stronger foundation for teaching or writing curriculum. (See related ideas about "forward design" in my article on The Assessment Trap.)

That said, Bill McCallum points out that while math is indeed not linear, time is linear, and curricula and lesson plans must be sequenced. Organizing things in a strictly logical sequence is almost always a mistake: such sequencing may make sense to a mathematically sophisticated person, but it is not necessarily best from a pedagogical standpoint. Doing things in a certain order because tradition requires it is another loser, as traditional sequencing is often terrible. And looking at lists of standards is not particularly helpful.

Effective sequencing, like everything else in teaching, requires paying attention to what happens with actual students. I did this for 30+ years, and came up with some ideas you may find useful. See my article on the Common Core for a discussion of realistic sequencing across the four years of high school. For sequencing within a course and within a unit, see my article "Nothing Works". But whatever sequencing you end up with, consider it provisional, and don't lose sight of the reality that if it doesn't work, you can and should change it. There is no God-given sequence for teaching math. Still, I suggest sequencing guidelines in my next post.


PS: I will present a Visual Algebra workshop  for math teachers in Silicon Valley this August, largely based on the TTC framework. More info.