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Why do students have problems with showing that something is well-defined? How can this be improved?
Maybe, your students have a belief problem. They will rarely (maybe never) have encountered problems where something was not well-defined.
If you have never been in trouble since everything you were shown was well-defined, then you don't even understand the problem! (Even harder: after proving that something is well-defined, the world looks right like it ...
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I think it is a good thing to talk about how there are some concepts where there are choices for where you start when definining them. It happens in linear algebra too, with the definition of linear dependence. You need to talk about how there is this web of connected properties, and it depends on what you're trying to achieve as to where you start. I do ...
16
I think there are serious pedagogical problems with such an approach. Here is a good general rule for explaining any kind of math:
Skipping over the motivation doesn't make something easier to understand.
As math educators, our instinct is always to simplify the math that we are presenting as much as possible. We always want students to understand the ...
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It's funny you should ask this now, because I just taught a Math Ed graduate class on this topic the other day, so a lot of these thoughts are fresh in my mind.
The pedagogical sequence we conventionally follow with respect to complex numbers -- first introducing them in a "just pretend" way, and only later (for some students) providing a geometric ...
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Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such:
It appears that even many serious professional mathematicians do not understand the difference between a "definitional" iff and an "assertive" iff. This is entirely parallel to an assignment equality versus an assertive ...
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Many high school geometry textbooks define an angle as simply
the union of two rays with a common endpoint
The advantage of this definition is its simplicity. Among its disadvantages:
It does not serve well for capturing the idea of a "direction": That is, there is no way to distinguish between a clockwise and a counterclockwise rotation.
It more or ...
15
A linear function is not necessarily a first degree polynomial function: zero function is also linear.
In France the terminology is more appropriate than the traditional English one: a linear function is a function of the form $f(x) = ax$, while a function of the form $f(x) = ax + b$ is called an affine function.
So, strictly speaking, constant functions ...
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Why do students have problems with showing that something is well-defined? How can this be improved?
I've never tried this in a classroom, but I suspect a lot of the trouble with functions is that students haven't been taught the vocabulary to deal with things that are weaker than functions. For example, they are trying to define a function $f: A \to B$. What they have written down probably defines SOMETHING: Perhaps a subset $R$ of $A \times B'$ for some $...
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I think a lot depends on context -- what course you are teaching and what the characteristics of that course are. If by "Freshman college calculus" you mean what I think you mean -- namely, a non-rigorous course that treats a lot of things using intuitive and heuristic arguments -- then I wonder if "define" is even the right verb to use. Students come into ...
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This is a very difficult question to answer; I recommend as a first place to look:
Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. The ideas of algebra, K-12, 8, 19. Link (no paywall).
As Usiskin writes (emphasis in original):
My thesis is that the purposes we have for teaching algebra, the conceptions we have of the subject, ...
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The situation does indeed change your interpretation of the terminology.
Volume is a general concept of the amount of 3D space something takes up.
For a solid object like a brick, or for a liquid there is no ambiguity and you can simply do a calculation (like $V = \pi r^2 h$ for a solid cylinder), or possibly compare its mass to the mass of a known volume ...
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I think this is a great question because it is really about the ambiguous/confusing way we use the word polynomial. So I want to use the terms "polynomial" and "polynomial function" separately here.
A polynomial is an algebraic expression. It consists of variables and coefficients and only uses the operations of addition, subtraction, multiplication, and ...
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While I haven't done a systematic survey, my impression is that the overwhelming majority of pre-calculus and calculus texts define a linear function to be one of the form $f(x) = mx + b$ with no stipulation that $m \neq 0$. Thus, if you define linear to be a polynomial of degree 1 you are likely to be contradicting whatever textbook you are using.
From a ...
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You could reply something like this:
Yes, associating each subset of the integers with its size would be an example of a function. And, in fact this does not just work for the integers but for all the subsets of any set. But note that to do it really for all sets would cause some problems as "the set of all sets" is a problematic notion. However, this ...
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Why do students have problems with showing that something is well-defined? How can this be improved?
I think that you hit the nail on the head, when you said some are not even aware of what well-defined means. As Anschewski suggests the problem may be that students have not encountered enough non-well-defined operations to fully appreciate the problem.
This Spring I was teaching freshman algebra, and while explaining equivalence relations (prior to getting ...
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Turning my comment into an answer as per request:
I think the graphical approach gets the idea across. You could represent $3\times 1$ by a row of three dots; likewise, $3\times 2$ is two rows of three dots; etc. If you asked them to guess what $3\times 0$ is, some would probably tell you you'd have no rows so no dots, so the answer is zero. Humans are very ...
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At the secondary level, students have not yet mastered formal mathematics and most will need to continue learning concepts before definitions in many cases. The van Hieles (the Dutch educators who developed the Van Hiele Model of how students learn geometry) insisted that definitions should never come first. They claimed that students should first have ...
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Have a look at the paper written by Nunez et all:
EMBODIED COGNITION AS GROUNDING FOR SITUATEDNESS AND CONTEXT IN MATHEMATICS EDUCATION.
In essence, they argue that it is better to be causious if you want to "motivate the formal definition of continuity starting from the intuition" you have suggested in your question. In the following passage, "natural ...
8
When I teach calculus I, my initial working definition for $e$ is that it is the real number implicitly defined by:
$$ \lim_{h \rightarrow 0} \frac{e^h-1}{h}=1 $$
Naturally, this allows me to derive that $\frac{d}{dx} e^x = e^x$ as
$$ \lim_{h \rightarrow 0} \frac{e^{x}e^h-e^x}{h}=e^x\lim_{h \rightarrow 0} \frac{e^{h}-1}{h}=e^x. $$
Of course, we know this. At ...
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I have two intuitions to offer:
A sequence $(a_n)$ may have cluster points (these are points such that every neighborhood contains infinitely many elements of the sequence, or, more precisely, for every neighborhood and every index $N$, there is an element $a_n$ with an index $n>N$ which is in this neighborhood). The $\limsup$ is the largest of these ...
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One of my most vivid memories of graduate school was working on a problem concerning the Gelfand-Kirillov dimension of certain rings. I had been puzzling over the definition (which was rather new to me) and had reached the conclusion that for a certain class of rings the GK-dimension was greater than 1. My advisor told me that I could not possibly be ...
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To answer the literal question of the title: yes, there are some disadvantages in portrayal of the complex numbers as "purely geometric" objects", although perhaps these disadvantages are more philosophical than mathematical. E.g., it is unclear whether or not we "have license" to declare that "the plane" is "numbers"... although we might have reasons to ...
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Why do students have problems with showing that something is well-defined? How can this be improved?
In my opinion this problem arises due to the fact that students are never told that, when introducing a new symbol which isn’t a variable (here I’m not using variable in the formal system sense of the word) and which does not depend on previously defined symbols, then one must prove that there exists exactly one object with the given property.
As an example ...
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The prototypical way for a function to not be continuous is that of a jump discontinuity. Imagine a jump discontinuity on the order of a few micrometers, like the width of a hair. If you are tracing the graph of the function with an everyday pencil, you would slide right across the discontinuity without even noticing its presence. However, if you shrunk ...
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A somewhat analogous point of view is the one of "continuous extensions". What looks weird in saying $\frac 1 x$ is a continuous function is, of course, what is happening around $0$.
What is happening around $0$ can be summarized by saying:
there exists no continuous function on the whole $\mathbb R$ extending $\frac 1 x$
The whole idea of removable ...
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I teach people (informally) how to make iOS apps. A lot of the people I teach are not people who were good math students. Of course in programming variables are important and anyone with a basic understanding of algebra picks up the concept pretty quickly. The term "variable" is pretty scary for even for people who did ok in algebra because it's a math term ...
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A textbook I have used first defines $$\ln x =\int_1^x \frac 1 t \, \mathrm{d}t,$$ and then defines the number $e$ as in your Definition a).
Next, the exponential function $e^x$ is defined as the inverse of the function $\ln x$. (They show that this definition of $e^x$ coincides with the usual one from repeated multiplication when $x$ is an integer.)
...
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Your second definition for the Cartesian product is the definition of a product of objects in a category (as I am sure that you are aware). So the question basically (as I am hearing it) becomes:
Is teaching category theory useful and when can you start introducing it?
I think category theory is extremely helpful for a mathematics student who is trying ...
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Firstly, I don't think it entirely makes sense to ask why a property is defined to be local rather than global. Being local or global emerges from the definition itself. Continuity asks about whether a function is well-behaved in a particular way, which is determined locally. We use the word 'continuous' to mean 'well-behaved everywhere', but we also have ...
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