"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.

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