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I am writing a paper directed at a physics-education journal and I want to briefly refer to the phenomenon of students “overassociating” (in lack of a better term) mathematical concepts with each other which have similar notations, names or properties, but are not similar in all respects.

For example, the multiplication of real numbers, scalar multiplication and the cross product are all called multiplication and partially share their symbols, because they have similar properties. I am interested in the phenomenon that a student treats, e.g., a scalar multiplication like a normal multiplication where this is not allowed and divides by a vector.

Another example is the misassumption of universal linearity.

As giving the above explanation or something similar would be too much, I am looking for one of the following:

  • An established term for such an association going too far, if one exists.
  • A general reference from mathematics-education literature about this phenomenon.
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Wrzlprmft
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    Does the term conflating not suffice? – Benjamin Dickman Aug 01 '14 at 01:13
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    is this like when the students don't know an asymptote from a hole in the graph ? – James S. Cook Aug 01 '14 at 03:12
  • @BenjaminDickman: It is certainly better than what I have for now, thanks. – Wrzlprmft Aug 01 '14 at 06:43
  • @JamesS.Cook: It’s not a problem I have ever encountered, but it goes into that direction. I have added another example. – Wrzlprmft Aug 01 '14 at 06:46
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    @James S. Cook's comment is a bit tongue-in-cheek; it's a play on the phrase "to know one's ass from a hole in the ground" http://www.oxforddictionaries.com/us/definition/american_english/not-know-one's-ass-from-a-hole-in-the-ground-or-from-one's-elbow – Brendan W. Sullivan Aug 01 '14 at 15:34
  • @brendansullivan07 joke aside, the problem of not distinguishing new concepts from old (such as confusing scalar multipication and the dot-product) suggests to me such students do not spend much effort on the context in which knowledge applies. Generality, domain, examples, counter-examples, any or all of the above discourage this (very real) phenomenon of the careless student. – James S. Cook Aug 01 '14 at 16:31
  • @JamesS.Cook: Good point about the importance of context! (I was just pointing out the joke because it might be lost on non-native English speakers.) – Brendan W. Sullivan Aug 01 '14 at 16:33
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    Transfer is a technical term in education for correctly applying a technique or solution to a new problem. This sounds like "poor transfer" which is common in novice learners. Would that term fit? – Adrienne Aug 01 '14 at 17:30
  • @Adrienne: Except for not being specific to this problem, it sounds good. – Wrzlprmft Aug 02 '14 at 18:40
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    Some possible sources to look through (especially if you want a general reference from which to cite the error as "common") can be found in my earlier answer here: http://matheducators.stackexchange.com/a/1616/262 What you discuss might be found, e.g., in Radatz (1979) as an error due to incorrect associations as discussed even earlier elsewhere (Pippig, 1975). – Benjamin Dickman Aug 03 '14 at 06:15
  • Hey @Wrzlprmft, checking in to see what happened with this paper! – Brendan W. Sullivan Mar 27 '17 at 22:11
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    @brendansullivan07: It is long published. As there was no answer suited for my purposes, I figured I might as well accept yours. – Wrzlprmft Mar 28 '17 at 11:31

2 Answers2

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I am not aware of standard terminology for this. Perhaps you have a chance to introduce some!

I propose "concept overload".

This suggests the underlying principle, that students have one concept in mind but are piling too many "things" onto it. This could be because they truly are similar concepts (like a limit of a function vs. a limit of a sequence), or because they share a name or notation (like the "multiplication" examples you mention).

In your article, you could briefly explain this idea early on, define the terminology (whichever you choose), and continue to use it throughout.

I also think this is a good idea because of the similarity to "function overloading" in programming:

[It] allows creating several methods with the same name which differ from each other in the type of the input and the output of the function. It is simply defined as the ability of one function to perform different tasks.

Brendan W. Sullivan
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    This is a nice term +1. I would add, my experience, any time I cite operator overloading as I reuse the same letter with different meaning, the computer science majors get happy. That term takes them away from calculus and back to their happy place. That said, it seems to me, to understand this overloading, we also need some concept to quantify or describe the inability of students to carefully understand definitions and terms. I call it "fuzzy thinking", some of my colleagues are less generous... Is there a method to quantify the extent of the fuzzyness? – James S. Cook Aug 01 '14 at 16:35
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    +1, but unfortunately, as this is only a minor aspect of my paper, introducing a new term would be out of place. (Would somebody be so kind as to quickly publish a paper on this, so I can cite it? -ommorow would be fine, as I will probably not find the time to submit today.) – Wrzlprmft Aug 01 '14 at 16:51
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    just cite bendansullivan07's answer here. Problem solved. – James S. Cook Aug 01 '14 at 20:35
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When a student draws improper analogies between two concepts after wrongly associating one with the other, that would be "false association."

The concept is similar to "false friends" in foreign languages, where two similar- sounding/looking words actually have different meanings.

Tom Au
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  • The only problem is that it’s perfectly fine to associate, e.g., scalar multiplication and regular multiplication with each other – they are both called multiplication for a reason, after all. The problem arises when association turns into identification. – Wrzlprmft Aug 02 '14 at 18:43
  • @Wrzlprmft: The first would be "true" association, and the second, "false association." – Tom Au Aug 10 '14 at 14:22