9
$\begingroup$

During my studies I observed that while most lecturers try to explain theorems and their proofs, only very few of them try to explain definitions. However, in my opinion, definitions are the base of maths education. In principle, it is possible to take them as given and just work with them, but establishing a link between the content of a definition and the name of the object or property we define seems very important to me. However, while some people asked questions concerning theorems and proofs, I never saw anybody asking about the plausibilty of definitions.

Let me give you an example: The definition of linear (in)dependency. Quoting Wikipedia:

The vectors in a subset $S=\{\vec v_1,\vec v_2,\dots,\vec v_k\}$ of a vector space $V$ are said to be linearly dependent, if there exist scalars $a_1,a_2,\dots,a_k$, not all zero, such that: $$a_1\vec v_1+a_2\vec v_2+\cdots+a_k\vec v_k= \vec 0,$$ where $\vec 0$ denotes the zero vector.

Now, imagine a student asking you: Why do we actually call this property "linear dependency"? Please take a minute or two if you can't come up with an answer immediately before looking into the spoiler below. Hint: The answer "It's a definition and there is no point in asking why a definition is the way it is!" is not the one I had in mind.

A variable $y$ is called dependent on another variable $x$ if there is a function $f$ such that $y=f(x)$. This also works for several dependencies: A variable $y$ is called dependent on other variables $x_1, ..., x_n$, if there is a function $f$ such that $y=f(x_1, ..., x_n)$. This also works for a vector valued function which takes vectors as arguments: A vector variable $\vec y$ is dependent on other vector variables $\vec x_1, ..., \vec x_n$ if there is a function $f$ such that $\vec y=f(\vec x_1, ..., \vec x_n)$. If the function $f$ is linear, it can be written as $f(\vec x_1, ..., \vec x_n)=c_1\vec x_1 + ... + c_n \vec x_n$ for some coefficients $c_1, ..., c_n$. Then it is natural to say that $\vec y$ is linearly dependent on the vector variables $\vec x_1, ..., \vec x_n$, if there are coefficients $c_1, ..., c_n$ such that $\vec y=c_1\vec x_1 + ... + c_n \vec x_n$. Now, how does this relate to the definition quoted above? If there is a non-trivial solution of $a_1\vec v_1+a_2\vec v_2+\cdots+a_k\vec v_k= \vec 0$, there is an $a_i\neq 0$, so we can solve for the corresponding $\vec v_i$ and express it through the other $\vec v_j$, $j\neq i$ and get that $\vec v_i$ is linearly dependent on the other $\vec v_j$ in the sense as described above.

Now that I have written it down, the explanation looks trivial and lengthy to me, but I recall the time when I was wondering what the relation between linear dependency of vectors and dependency in the sense of a functional dependency of two variables is and had a tough time to really pinpoint it. Also, there are definitions (for example from algebra, say, the different kinds of field extensions) where I am still missing the link between the name and the content (hints to good literature for this topic are highly appreciated).

So I am wondering:

  1. Is it a common phenomenon that giving intuition for definitions, in particular, relating name and content is treated as an orphan, or is it just my subjective perception?
  2. Would it make sense to spend more time on giving intuition for definitions?
  3. What are possible reasons for the phenomenon?
$\endgroup$
  • 2
    $\begingroup$ My observations refer to university level education but in principle the same questions can be asked about school maths as well, though rigorous definitions appear much less frequently there. $\endgroup$ – Photon Jun 13 at 14:48
  • 1
    $\begingroup$ I know the example is just an example, but it sounds like a case of there having been a better term but not enough users adopting it early enough in the language: I present my intuitive way of thinking of the satisfaction of a non-trivial linear relation as 'linear redundancy'. (TBH the possibly fictional etymology in the yellow box confuses me; yes you can solve for one in terms of the rest as in the implicit function theorem, but I think of the vectors as fixed not varying. It's akin to the difference in connotations to me between an identity and an equation to be solved.) $\endgroup$ – Vandermonde Jun 13 at 20:50
  • 3
    $\begingroup$ To me, your explanation is more impenetrable than the definition. What is really needed is a motivating example, such as the set of vectors (1,0), (0,1), (1,1). $\endgroup$ – Ben Crowell Jun 14 at 13:14
  • 1
    $\begingroup$ The title question seemed to be quite general, and sparked my interest. But the body of the question was pretty narrow. I am curious why you felt the need to use the 'spoiler' html feature. $\endgroup$ – JTP - Apologise to Monica Jun 14 at 15:07
  • 1
    $\begingroup$ @JoeTaxpayer: Yes, you are right, the title was too broad, I should have put it differently. Concerning the spoiler, I wanted readers to think about the question before reading my proposal of an answer, I think that's how spoilers are meant to be used. :) $\endgroup$ – Photon Jun 14 at 16:23
7
$\begingroup$

(1). It's going to depend on level to a huge degree. In my experience, up into advanced undergraduate you could expect definitions to be named sensibly and an instructor to explain how a sensible name like linear dependence corresponds to the definitions content.

(3). Here would be my guess: the more advanced you get, definitions become more utilitarian and axiomatic, with the purpose of picking out formal objects that we don't have ordinary language to describe. For example, if I want to talk about a normal subgroup of a connected semisimple Lie group, I'm going for precision and brevity in my naming conventions over more English language explanatory power. We start giving definitions that we don't have short pithy English language referents for. So in higher math your question largely doesn't apply, since the naming conventions are trying to do something different.

Bringing it back, I wouldn't be surprised if mathematicians lecturing at lower levels can forget the importance of the expository power of a sensibly named definition. The attitude is much more "A definition separates out classes of objects that we would like to talk about. These classes must be justified by theoretical results, or a practical intuitive definition. Oh, an it's called a principle component for historical reasons seem less important than it's use."

(2). Do I think we could spend more time on it? Yes, absolutely, especially if the naming conventions are explanatory. Your example of linear dependence is a good one: It's a great place to teach/reinforce mathematical vocabulary like linear and dependence. Such an explanation will hopefully help students make connections between the content and the name, but also between the content of the given definition and other parts of math.

(Edited to get the number order correct)

$\endgroup$
  • $\begingroup$ Thanks for the reply! Definitions as sets of properties to separate a subclass of some bigger class are quite common, indeed. But there are such definitions which have telling names and also such which have names which are hard to make sense of. Say, an open set is a set without boundary which seems quite intuitive. But a normal subgroup, which you mentioned, gives me a hard time, why should a subgroup which is invariant under conjugation be called normal? $\endgroup$ – Photon Jun 14 at 16:39
  • $\begingroup$ In the case of a normal subgroup the normal is in analogy to orthogonal. For example, in $\mathbb{R}^n$ any $k$-dimensional subspace $V \cong \mathbb{R}^k$ decomposes $\mathbb{R}^n$ as $\mathbb{R}^n \cong V \oplus V'$, where is the group quotient $V' = \mathbb{R}^n/\mathbb{R}^k$. Note that $V$ is a group under addition. In general, a normal subgroup is one that allows you to form such quotients on a general group. But even if the quotient can be formed, you cant necessarily for the orthogonal decomposition $G = G'\oplus G/G'$, so normal is already a deviant usage because it's a new concept. $\endgroup$ – Nate Bade Jun 14 at 16:58
  • 1
    $\begingroup$ @Photon long ago the term for "normal subgroup" was "invariant subgroup". See an answer at math.stackexchange.com/questions/898977/… for other terms that were used for this concept once. $\endgroup$ – KCd Jun 18 at 16:25
4
$\begingroup$

I agree with Nate Bade about the helpfulness of making connections between the names of properties and how those properties are defined, particular when there are direct connections that can be articulated.

At the same time, there are terms in mathematics, with specific definitions, for which the choice of specific terms may best be explained by understanding the etymology of those terms (when those terms were first coined, and by whom.) I.e., there are historical factors that figure in to the choice of some mathematical terms/properties their definitions, that make sense w.r.t the time period at which the term/property was defined, but may not be so clear at this point in time.

I'd argue that explaining the etymology of such terms and properties, when introducing them to students, and their definitions, is also useful in helping students develop an understanding/intuition connecting terms/properties with their definitions.

Added

I empathize with and share Daniel R. Collins' reality-check on the idea of calling on teachers and profs, etc, to spend time when teaching to help students make connections between terms, theorems, etc.. Even teaching math at the secondary school level, responsible for covering a curriculum state-dictated curriculum, on which students will ultimately be tested, makes teaching math a sort of "triage" based, sometimes remediation, leaving teachers very limited opportunities for them to wonder off into things like the origin of terms, etc. When it can be done, great.

However, why presume it ought to be, from grade school through undergraduate years, that educators ought to give students such connections. Better yet, we need to encourage independent thinking, critical reading, and independent research skills of students, starting from the primary grades, through middle school. So, e.g., when a student wonders "I know I'm taking Algebra, and my textbook tells me so. But where did the name "Algebra" come from?" they take initiative to first find out, if they can. "Hmm, let me plug Algebra into a Google search bar ... Wow, lots of hits including Wikipedia which discusses the history of algebra, and here we go, I see 'The origins of Algebra'! ...ahhh, oh cool, I get it!"

The capacity to think independently, work and research independently and within groups, read critically, etc., are skills students need to develop from their first day in first grade, and all the way through the pre-college education. Hopefully, a student will then be able to better engage in their own education, and realize that their education is not merely defined by what teachers/profs lecture on, or the texts they read from, but also, perhaps more importantly, it's determined also by what the student does with what information.

This applies to much more than math education. Students, for too long, have been described as "consumers" of educational "products", which seems to imply the student/consumer is passive: "Give me (the student) the educational product you promote, and I'll consume it. And I can't be bothered to hunt or gather or otherwise find what I might also need to consume to supplement the delivered product."

We would all be better off if students thought of themselves as partners in their education.

$\endgroup$
  • 5
    $\begingroup$ As an example, there's this really annoying thing in physics where everything is named after a person instead of it's function. It makes it impossible to read physics papers without memorizing a bunch of names... But once you know it, you get this whole story about the history of the people in physics which help you track the timeline of discovery, experiment and understanding. Now, that story has been critiqued and maybe obfuscates the collective effort, but they etymology does add another way in which you can understand the theory. $\endgroup$ – Nate Bade Jun 15 at 3:22
1
$\begingroup$

I sympathize very much with this. Re: "What are possible reasons for the phenomenon?", I think the primary issue is simply time limitations. As a math/computing lecturer at a 2-year school, I'm wildly desperate for time in class, and already have to slash out half of any standard book section's content to try and get my students up and running on the core required skills. On the other hand, I also know of some faculty who do prefer a purely mechanical class presentation, and are simply disinterested in drawing connections to (even slightly) larger social or historical meaning.

I've made the point numerous times in the past that math/computing courses inherently run at a higher conceptual level than other courses. E.g.: Other faculty reference Bloom's taxonomy and bemoan why they're operating at a low level of base knowledge (testing on definitions of terms), and wishing they can escalate to higher levels (like applications and analysis). But I don't know any math faculty who test on definitions -- we don't have time, and all of the work is jumping instantly to applications all the time already.

There's a lot of times when I get a bit frustrated with students not picking up the "obvious" meaning of a certain name for a particular definition. E.g.: In a statistics course I speak of a distribution's "tail", and then on tests some students write this as "tale", having heard what I verbalized but not associating any meaning to it. Sometimes in response I wind up drawing a distribution with a cat's face on one end to emphasize the meaning. Is that really a good use of time? If I did that in a lecture, what percentage of students would be bored by the pedantry, and what other topic would be getting pushed aside by it? (Other examples: Say I'm in a discrete mathematics course presenting basic logic identities; in one table we have the Identity laws, Domination laws, Commutative laws... and about 20 others. Are these names all obvious? Should I spend time drawing out the English meaning of all these names?) That's the dilemma for me, and I suspect for many others.

If those naming backgrounds were in sidebar in the text, and my students could dependably read them (which they can't), then I would consider that ideal.

Alternatively, it might make for an interesting research question to survey students and see what proportion can identify/reason out the motivation for different naming traditions on their own, versus those that need explicit instruction (and so help to prioritize use of in-class time).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.