Timeline for Most harmful heuristic?
Current License: CC BY-SA 2.5
12 events
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| Oct 27, 2018 at 1:55 | comment | added | mc-lunar | @ToddTrimble I remember doing the same thing in high school - I was given something like (x-1)(x-5) = 0 and asked for the roots, my solution was to multiply them together and use the quadratic formula 😅 | |
| Aug 10, 2017 at 11:21 | comment | added | Tobias Kildetoft | @DouglasZare As long as they stick to square ones. I have seen so many students "compute" the determinant of non-square matrices (in many very weird ways). | |
| Oct 25, 2016 at 20:31 | comment | added | Douglas Zare | The reflex to multiply out any pair of binomials is similar to the reflex to take the determinant of any matrix. Some students are so used to being asked to compute determinants that they replace any small square matrix with a scalar, including when asked to solve $A\vec{x}=\vec{b}$. | |
| Aug 19, 2014 at 6:44 | comment | added | Benjamin Dickman | @PedroTeixeira Surely this is the way to proceed! Let $y = x - 2.5$; then $x = y + 2.5$, and the equation becomes: $0 = (y + 1.5)(y + 0.5)(y - 0.5)(y - 1.5) = (y^2 - 2.25)(y^2 - 0.25)$, which holds when $y = \sqrt{2.25}$ or $y = \sqrt{0.25}$. For each square root we obtain two possible $y$-values; add back the $2.5$ to each to get the four possible $x$-values. A similar approach can be found by observing $(x-1)(x-4) = (x-2)(x-3) - 2$; now denote the LHS by $y$ so that the original equation becomes $y(y+2) = 0$; solve for $y$ using the quadratic equation, etc. | |
| Apr 17, 2013 at 21:35 | comment | added | Pedro Teixeira | Todd: Ask students on an exam to solve an equation such as $(x-1)(x-2)(x-3)(x-4)=0$. I've done this a couple of times. A very common attempt of solution was to expand things out (often making mistakes along the way), contemplate the new, messy equation, and declare, "It can't be factored!". Sad... | |
| Aug 25, 2012 at 20:43 | comment | added | Todd Trimble | "teaching a human being to behave like a computer" -- or like a dog. Show students an expression like $(x-1)(x-3)$, and many will have a Pavlovian response "FOIL!" even if it doesn't do them any good. | |
| Jun 25, 2010 at 11:05 | comment | added | Pietro Majer | I partially disagree, because for the sake of computations algorithms and notations actually do not have to show "what is really going on": the more non-relevant information they suppress, the better to efficiency. Nevertheless, sooner or later a student should be told what's behind the sum of fractions, or the multiplication of polynomials. | |
| Apr 25, 2010 at 21:23 | comment | added | Blue | As a high school teacher, I usually encountered students after their first exposure to FOIL, so I made a point to revisit the process and introduce "Super-FOILing" (which, of course, was just applying the distributive property to two polynomials of any length). Yes, yes: I hammered proper terminology and all the conceptual stuff, too, but starting off with "Ah, so you can FOIL ... but can you SuperFOIL?" really made the ears perk right up! In a way, prior exposure to FOIL was helpful to me, providing an accessible object lesson that math is always "bigger" than any of us are ever taught. | |
| Apr 25, 2010 at 21:18 | comment | added | Willie Wong | As much as I dislike teaching mathematics "algorithmically", there is a reason why FOIL is taught as such: by forcing the user to adopted an algorithm, you can minimize mistakes. Doing things "in order" is a good habit, which should be encouraged. It is unfortunate the trend where "educators" take good practices, and distil from it something all but recognizable... | |
| Apr 25, 2010 at 18:43 | comment | added | Matt | I can't stand FOIL! It seems to indicate to students that order matters here. I don't see what FOIL adds, but it certainly detracts from the idea of just multiplying all the pairs and adding. Instead of teaching the idea (which they'll never forget), they now have something memorized (easy to forget). And I once had a student erase their correct work because they accidentally did FLOI or something and rewrite the same thing in a different order. | |
| Apr 25, 2010 at 18:14 | comment | added | LSpice | I am usually the first in line to bash blind application of algorithms, but, interestingly, I've never had the problem that others do with FOILed students. My calculus students insist on referring to multiplying any two (non-monomial) expressions as “FOILing it out”, but they seem—even the ones who also think that $x^3 - x = x^2$—perfectly able to multiply, say, two trinomials correctly, even though the acronym ‘FOIL’ makes no sense in that context. | |
| Nov 1, 2009 at 16:54 | history | answered | CJP | CC BY-SA 2.5 |