Do mnemonics help or hinder? I don’t really know that the answer to that question (and I
am sure that the context in which the mnemonic is employed is relevant to
answering it) but I do know that mnemonics can certainly mask student
understanding, well,  … lack of understanding.

In beginning algebra students are often taught to FOIL expressions of the
sort( x + 2)( x + 6) . And many students who will happily (?) foil away all sorts of
expressions of this ilk.

But happiness – faux or otherwise – will often dissipate with presented with
expressions of the ilk ( x + 2) ( x + 6)( x + 9) or ( x + y + 2)( x + w + z + 2 + a) . Where
is FOIL now?


Suppose a student does find a means to be playful and creative in these little tasks and decides to LIFO instead? Is he wrong? Is he right but should be marked wrong because the system is “first, outer, inner, last” in that order, for the sake of uniformity in mathematics? Does the mathematics care in which order one expands brackets?

Working on understanding instead is more powerful and more joyful.  Here is how I introduce the principle of expanding brackets to students. The mathematics doesn’t care one whit about the order in which one expands brackets (nor does any mathematician!). 

Download in pdf format
Curriculum Essay #3


    QUADRATICS, Permutations and Combinations, EXPLODING DOTS, and more!

    Written for educators - and their students too! - this website, slowly growing, takes all the content Tanton has developed in his books, videos, and workshops, and organizes it into short, self-contained, and complete, curriculum units proving that mathematics, at all points of the school curriculum, can be joyous, fresh, innovative, rich, deep-thinking, and devoid of any rote doing! Let's teach generations of students to be self-reliant thinkers, willing to flail and to use their common sense to "nut their way" through challenges, to assess and judge results, and to adjust actions to find success. (Great life skills!)


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