Summary of the mechanism of reification

Question

The concept of "reification" in mathematics education is interesting. Roughly, if I understand this correctly, one "reifies" processes into mathematical objects. Very recently, it occurred to me that although this reification seems like a clear thing mathematicians do…perhaps without thinking about it…I don't have a very clear picture of the impetus for reification. What leads us to "reify"? Is it that we perform a process often enough that it becomes a recognizable pattern and thus is given a name and becomes more of a noun? This would make sense, but I'm not sure it is the only reason for reification…there may be deeper reasons connected to the compressive function of mathematical thought, similar to the use of highly leveraged notation.

Sorry for the blathering. I am writing to ask mathematics educators for a summary of the current understanding of the mechanism for reification, understood as the impetus for reification and the process by which it takes place.

I am aware of some controversy surrounding this, but a summary statement may help me understand my students' resistance to thinking about objects rather than processes. I imagine what is needed may be similar to the transfer from the mindset of classical procedure-based programming to object oriented programming. I apologize that this question is somewhat vague, but I'd appreciate any answer to it that may help me understand the various mechanisms...

Answer 1

I think that you might find useful Harel & Kaput's chapter in Tall's Advanced Mathematical Thinking (1991). You can find it for free download at http://www.math.ucsd.edu/~harel/publications/Downloadable/Conceptual%20Entities.pdf.

To summarize, they identify three roles for "conceptual entities" (including reified objects):

1. Alleviating working memory or processing load when concepts involve multiple constituent elements.
2. Facilitating comprehension of complex concepts: the cases of "uniform" operators, "point-wise" operators, and "object-valued" operators.
3. Assisting with the focus of attention on appropriate structure in problem solving.

If you have not read this chapter before, you may find it helpful.

Answer 2

Not a high-quality answer, since I can only refer to some decades' observation, rather than systematic/deliberate study of the phenomenon:

Having observed maybe 1,000 pretty-darn-good grad students, some of them very good, I find that it is not typical among mathematically-interested, mathematically-talented people (at least 20-30 yrs.) to have a good instinct for abstraction, which is surely a part of reification. I see this throughout basic grad courses: to write Fourier transform as a linear mapping is vaguely disturbing... as is declaring "integral" a continuous linear mapping (rather than a limit of Riemann sums, or something described by the fundamental theorem of calculus).

That is, in my observation (of others and myself), people become very attached to particulars, to specifics of a landscape, and it is jarring to find them declared irrelevant or illusory. This attitude to "progress/abstraction" can be manifest in particular to objection to processes becoming objects, since "object" does substantially connote inertness (though it's not so clear why it should). Actors versus acted-upon? (Stress over the second dual of a topological vector space having an imbedded copy of the original?!?)

All that leads me to believe that reification (and various other processes) are not widely natural even among "mathematically talented" people, whatever that may say about our specific psychology.

Some people successfully broaden or enhance their world-view to embrace this idea, while others balk. It is possible to survive while rejecting (to a significant degree) various abstraction-methodologies, in my observation, though, to my taste, much of this seems like just killing time. Tastes vary.

In summary, I think that reification and other comparable abstraction processes are not immediately natural even for people who have sympathy for and talent for mathematics. Hence the resistance and difficulties.

Answer 3

The literature lacks a clear mechanism, that is why theories on the process-object-duality are criticized sometimes. Anna Sfard's reification, Dubinsky's APOS and Tall's procept may help describing the situation, but hardly give specific advice what to do. (Note that Anna Sfard had suddenly stopped her work on reification and moved to commognition, a socio-constructive view on learning.) The clear strenght of these theories is, however, that they can well explain, why difficulties with seemingly simple problems occur, which build on object views. Think of power-sets or indexed variables which use the same concept twice (set of sets, variable variable) thus enforcing to think on object level.

My personal feeling from the literature and my teaching and learning experience would give you two advices: First, do not think of a clear dichotomy of process or object view. Students often can handle something more or less like an object, maybe depending on the context. Second, give reification or encapsulation a chance. You cannot force it, but try to find contexts in which an object view would make the situation much easier.

Take the example of natural numbers. (Here, a missing object-view is often associated to discalculia.) Children start counting, this a process obviously. It is rather like saying a poem while pointing on physical objects. If they manage to not point on two objects while saying "se-ven", you may let them count two objects on each step like 2, 4, 6, ... or even three objects. Still, it is a process but the role of the numbers slightly change. You may also want to look at the reverse process like counting 5, 4, 3, ... and so on. Then you treat numbers as if they were objects, asking what is 5+2? The process-view says it is counting on two more stepts, starting from 5. Then ask, what is 2+5? As a process, it is something completely different from 5+2, but identifying the results may help students recognizing that they just coordinate counting processes which are somewhat "the same". You could go on asking for patterns (like what is 9+1, 8+2, 7+3, ...) so you essentially treat the numbers as objects but the students may still start working with a proces view. Slightly, you may move to questions like "how many numbers are there between 4 and 8?" treating numbers as objects while still students might start the counting process in answering.

You can do very similar things on higher content like in undergraduate programs. Dubinsky et al. had great success in making their students programming on computers. A function, for example, needs to be distinguished from its values and treated like an object. Similarly to kindergarten, you may just play around, chaining functions, inversing functions (if possible, discussing the impossibility if not), restricting functions, extending functions, making functions become input or/and output of functions (operators!) and so on.

You will still find the "resistance to thinking about objects rather than processes" mentioned in your question. It is natural and abstraction will always take time. But without challenging this view in demanding tasks, reification/encapsulation is very unlikely to occur.