# Difference in difficulty between removing and adding structures?

In Fantasy math, Peter Saveliev remarks:

Removing structures from consideration is hard; adding is much easier. Compare how you start with point-set topology -- by removing the geometry from the Euclidean space -- and how you start with inner product spaces -- by adding this structure to vector spaces.

Questions: Has research been carried out on the difference in difficulty, if any, between removing and adding structures when it comes to learning a mathematical concept or topic? If so, what were the conclusions? Is it inherently easier to think about additional structure on a familiar object, and inherently harder to let go of already familiar structures? Are any aspects of this special to mathematics or is it merely an example of a more general phenomenon of difficulty in unlearning what you have already learned?

(Mathematical examples and counterexamples are welcome.)

Note: while I am using the term "structure" somewhat loosely, it may be helpful to see the Wikipedia article on mathematical structure(s). I quote the opening sentences below:

In mathematics, a structure on a set is an additional mathematical object that, in some manner, attaches (or relates) to that set to endow it with some additional meaning or significance.

A partial list of possible structures are measures, algebraic structures (groups, fields, etc.), topologies, metric structures (geometries), orders, events, equivalence relations, differential structures, and categories.

• In the context of middle school mathematics education, your question reminds me, in spirit, of an article in MTMS (Nov 2015) related to rules that expire, and its references. – Benjamin Dickman Jan 9 '16 at 15:55
• This is very hard to do, as the "structures" you are talking about are wildly different, and discussing the removal requires varying levels of "mathematical maturity" and the result are a veritable zoo. – vonbrand Jan 9 '16 at 16:21
• I guess the thing that makes adding structures easier is that everything you know already holds, whereas when you remove a structure you have to remember not to use what you knew before. – Jessica B Jan 14 '16 at 10:28
• Another example we might consider here is the differing approaches to teaching abstract algebra. Some texts begin with rings and things then later go to groups. For example, the undergraduate text by Hungerford. It seems most other texts start with groups. I don't have much experience to form an opinion on which is best. I see virtue in both. – James S. Cook Jan 14 '16 at 19:34
• Have you tried looking up research on 'generalising and specialising'? I think that is what is going on when you remove/add structure. – Jessica B Jan 28 '16 at 12:33

There is a sense in which it is not fair to say that one "removes structure" when one considers a more general class of mathematical objects. Usually one tries to do analogous things with the more general class that one was able to do with the more specific class, and this requires new definitions and techniques.

Let's take the example of "removing the geometry from Euclidean space". Just forgetting about distance and angles etc. is easy to do. But if I want to actually do anything in this new geometry-less world, that's another story. Let's say, as in the question, that I've moved into point-set topology. First need to know what a topology is, and that takes a certain level of mathematical maturity to grasp. Once I understand what a topology is, I will probably learn that there are many different possible topologies on a set. I will also learn that the notion of "distance" I used to have in Euclidean geometry gives one such topology. Then I can check my understanding by convincing myself that some of these new notions I've learned about really do specialize to the things I used to do when I was doing geometry.

This is a common pattern in mathematics: generalize with an eye towards recovering old results or analogs of old results in the generalized setting. This almost invariably leads to a net gain in number of possible structures, and these possible structures are the things that wind up being studied.

On a more concrete level, Jessica B's comment also addresses a difficulty that can occur when removing structure: we can't use the rules we used to use. Consider a linear algebra student encountering matrices and matrix algebra for the first time. The student would like to do algebra with matrices just like with numbers. But now there are things that are not invertible, and even when a matrix is invertible the student can't "divide" both sides of an equation by a matrix because they don't have commutativity. So the student needs to adjust the way that they've always done algebra.

It is hard to remove the structures, usually knee-jerk manipulations, $x^n \leadsto nx^{n-1}$ type of thing, when teaching calculus at the college level. Another example is un-teaching them to use l-Hopital without checking the hypotheses first.

An example perhaps more along the lines of what the OP had in mind is the tendency to attribute certain characteristics to the word "number", for example assuming that the term can only refer to a finite quantity. Paradoxically, this is the opposite of what the greats of the past considered intuitive; thus, Leibniz and Euler naturally assumed that a number can be infinite, and moreover used such numbers in numerous articles.

• I don't think this is really what the OP is asking about. – Jessica B Jan 11 '16 at 17:24
• I think this might be an answer to a separate question, focusing not on structures the way I meant them, but on difficulty getting learners to replace suboptimal habits. That said, I appreciate the answer and perhaps I could clarify what I mean by "structure" (and presumably that is also what Saveliev meant). – J W Jan 11 '16 at 17:39