The difficulty, the need and the rules

What is the difficulty we need to teach to our students? Sometimes we teach things in a hard way because our teachers, in the past, taught us this way. But perhaps we could think about why we need to teach that way. In general, we could teach one thing more easyly if we don’t need it afterwards and so, we don’t need rules. The rules should be simplifications or generalizations after several rutine repetitions1. I’m going to explain it with an example: how can we teach how to calculate \((x+3)^2\)?2

  • We could do applying the formula \((a+b)^2 = a^2 + 2ab + b^2\):

    But we don’t need it. We surely have got our students confused. And we would introduce a mathematical arsenal that we will not need.
  • We could do as a product: \((x+3)^2 = (x+3) \cdot (x+3)\). So we could apply distributive property.

    This reminds our students the “rule” \(3(x+3) = 3x + 9\) seen before, but surely most of them will have a mess after their application. And surely most of them would not understand what would be doing.
  • Why not teach simply as we can: \((x+3)^2 = (x+3) \cdot (x+3)\). So let’s multiply it.

    This reduces the unexpected thing to familiar and known thing: algorithm for multiplying integers.

When does we need to know the formula \((a+b)^2 = a^2 + 2ab + b^2\)? In my opinion, only when our students need to expand expressions like \((x+3)^2\) many times or if they needed in order to solve a major problem (in which it is embeded) and they need not to spend their time in calculations (like they do in third way).

Note: For making this article, I used SimpleScreenRecorder [2] for recording an specific area of my desktop while running LibreOffice Writer formula editor [4] and Geogebra [3]. For conversion from webm to animated gif I used ffmpeg [1]

Update: Fred G. Harwood (@HarMath) suggested we could calculate it also visually. It gets better understanding to students and avoid typical errors.


[1] Albino, B. How to do i convert an webm (video) to a (animated) gif on the command line? 2014.

[2] Baert, M. SimpleScreenRecorder. maarten baert’s website.

[3] International GeoGebra Institute. Geogebra.

[4] The Document Foundation. Home. libreoffice.

  1. But sometimes (many more than it should be), the rules are taught before they are needed.

  2. I need it to teach it because I want it to pass from the form \(y=4(x+3)^2 -5\) to the form \(y=4x^2 + 24x + 31\) for treating quadratic functions. In particular for representing that in cartesian plane. Yes I know we could represent directly (and more easyly) quadratic functions in the form \(y=A(x-B)^2 + C\).