Questions tagged [definitions]
For questions related to the issue of concepts of definitions.
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Good exercises that force you to apply the definition of the derivative, without explicitly telling you to do so?
I'd like to ask my students whether some real function is differentiable at a certain $x_0$. I prefer not telling them that they have to use the definition of the derivative, but to instead present a ...
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Can you talk about (the rest of the) field axioms when the operations are not closed? [closed]
Note: Updated based on this.
In my course, my instructor posed the following exercise:
Let $S$ be the subset of $\mathbb R^n$, $S=\{(a_1,a_2,a_3...a_n) | a_2 = \pm a_1, a_3=...=a_n=0 \}$. Define ...
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Getting students to actually read definitions
I'm teaching a second year "Introduction to Theoretical Computer Science" course, and one of the skills/habits I've tried to instill in the students is to actually read definitions, take ...
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What can (and should) an educator do about ambiguous terms like "triangle", "square", etc?
The imagined students are in elementary school, say around 9-13 years old.
I want to use rather precise terminology when talking to my students. However, it seems like we typically use the same ...
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Why should or shouldn't we teach functions to 15 year olds?
Background
The students in my country are supposed to be able to work with and answer questions about functions at the age of around 15. This is asserted in the standard mathematics curriculum for ...
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Tension between the most intuitive definition vs. the most common definition of a concept
Many definitions in mathematics are "fully crystalized". Sometimes the form of these definitions might be somewhat baffling to the uninitiated.
For example, the definition of a relation ...
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Definition of Trapezoid
From one textbook we use in our High School -
Transcription:
A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases of the trapezoid.
And from ...
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Online Definition/Theorem Statement Quizzes
This fall I'll be teaching a standard "Introduction to Proofs" course and administration is advising us to be ready at any time to switch from in-person to fully online (or anything in ...
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Definitions of factors and terms
I have come across this question in a textbook
How many factors are there in the term $5ab(x+y)$? State what they are
It is being praised because it encourages thinking, which it does. However, I'm ...
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Why is a translated exponential function considered an exponential function?
I am tutoring a student preparing to take Calculus 1 at a university. This student hasn't taken precalculus for a year, so I have been drilling him on definitions, rules, and theorems from a college ...
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How to define "axes with the same scale" in Secondary/High School?
It's easy to recognize visually when an orthogonal coordinate system has its axes in the same scale. See, for instance, the following image. But I'm trying to write down a precise definition of it.
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Should we stop teaching "interchange $x$ and $y$" when finding the inverse function?
In one textbook I use for College Algebra, the author teaches that one should interchange $x$ and $y$ when looking for inverse functions. For example, the inverse function of $$y=2x+2$$ is $$y=0.5x-1.$...
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How is $\frac{a}{b}$ interpreted?
I was having a discussion with a colleague who is in the process of writing some curriculum, and we ended up having a discussion about what $\frac{a}{b}$ (with all the standard restrictions) meant. We ...
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Are degrees of polynomials illogically defined in elementary algebra, intermediate algebra and college algebra courses?
In most of books on elementary algebra, intermediate algebra and college algebra, the degree of the non-zero polynomial $$f(x)=a_nx^n+\cdots a_1x+a_0$$ with $a_n\neq 0$ is defined to be $n$.
But I ...
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Explaining why (or whether) zero and one are prime, composite or neither to younger children
There are lots of discussions out there about whether $1$ is a prime number (such as this one) and even about zero (such as this question, though note zero does generate a prime ideal in $\mathbb{Z}$ ...
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How important is making definitions plausible?
During my studies I observed that while most lecturers try to explain theorems and their proofs, only very few of them try to explain definitions. However, in my opinion, definitions are the base of ...
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Different Kinds of Variables
Students sometimes ask whether the $x$ in the expression
$$2x$$ the same kind of thing as the $x$ in the equation
$$2x = 4.$$
In the expression $2x, \;x$ can be any real value.
However, in the ...
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Difficulty in teaching the coordinates of a vector with respect to a basis $\{v_1,v_2,\ldots,v_n\}$
Let $V$ be a finite dimensional vector space and let $B=\{v_1,v_2,\cdots,v_n\}$ be a basis of $V$. If a vector $v$ can be written as
$$v=a_1v_1+a_2v_2+\cdots+a_nv_n,$$
we call $(a_1,a_2,\cdots,a_n)$...
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What is an intercept?
I have always taught my students that the $y$-intercept of a line is the $y$-coordinate of the point of intersection of a line with the $y$-axis, that is, for the line given by the equation $y=mx+y_0$,...
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When a geometrical figure a special case of another [closed]
Squares are special types of rectangles.
Are circles special types of ellipses/ovals?
Are cones special types of pyramids? I guess the answer is no because of the 2D basis: circles are not special ...
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Why define the names of quadrilaterals so that some categories (rhombus and rectangle) intersect and some (kite and trapezoid) are disjoint?
We're using Pearson's Geometry in my class.
As terms are defined there, Parallelograms include Rhombi (congruent sides), Rectangles (right angles), Squares (congruent sides and right angles, i.e. ...
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In what curricula are "rectangles" defined so as to exclude squares?
Most contemporary curricula define the word "rectangle" inclusively, so that all squares are automatically rectangles. Are there curricula in which this convention is not followed? That is,...
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Does solving crosswords help with recall of definitions?
I have an idea to help with self-study and recall of definitions.
It is to create a crossword where the clues are the definitions and the words to solve are the concepts being defined.
I haven't ...
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Differential equations - definitions
I am having a great deal of trouble with the definitions used throughout the book so far - i.e. linear, homogenous, non-homogenous, etc. I am not sure why exactly they are useful to know. I am having ...
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Against introducing precise definitions first
After introducing eight different ways of viewing the derivative of a function
(infinitesimal, symbolic, logical, geometric, rate, approximation, microscopic),
Thurston, in his famous essay,
...
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For purposes of teaching, should constant functions be considered "linear functions"?
I can see arguments both for and against classifying constant functions as linear functions.
Against: "Linear function" means "first-degree polynomial function", and constant functions are not first-...
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Definition of the term, equation
What is the definition of an equation (as a mathematical terminology)?
I have been using this term, equation, for a long time.
I don't even remember when and where I have learned this term (Possibly, ...
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In Polynomial Form, After Simplification (But Not Before!)
We know that $\dfrac{(x^2+1)x^3}{x^2+1}=x^3$ and $(\sqrt{|x|})^4=x^2$ for every $x\in \mathbb{R}$. Can $\dfrac{(x^2+1)x^3}{x^2+1}$ and $(\sqrt{|x|})^4$ be called polynomials?
Is there a general name ...
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Alternative limit for e
I have recently worked with some students motivating the development of $e^t$ and $e^{t i}$ as summing change over time, basically informally solving differential equations. My motivation for this is ...
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What is a variable?
There are two kinds of answers I'm looking for:
What do students think a variable is?
What do YOU, the teacher, think a variable is?
I'm also interested in why you think a variable is what you think ...
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Why is continuity only defined on its domain?
As mentioned in this question students sometimes struggle with the fact that continuity is only defined at points of the function's domain. For example the function $f:\mathbb R\setminus\{0\} \to \...
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Why is continuity defined as a local property?
The formal definition of continuity is a local property (the definition of continuity at a point is a property of the germ of the function at this point). Why is it a good decision to make the ...
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How can I motivate the formal definition of continuity?
In order to teach continuity of real valued functions $f:D\to\mathbb R$ one may start with the (in some sense wrong) intuition
$f$ is continuous when its graph can be drawn without lifting the pen.
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Defining vertical tangent lines
In looking at the definition of vertical tangent lines in some popular calculus texts,
I noticed that there are a few different definitions for this term, including the following:
A function $f$ ...
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What is the intuition behind the limit superior?
I want to write an article which explains the limit superior. I also want to present the intuition behind this concept. Currently I would describe the limit superior as the "least upper bound of a ...
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Rigorously defining the concept of an angle for high school students
Arriving at a rigorous definition of the concept of angle for high school students is not as easy as expected.
Google search provided me with many definition that are too technical or too vague IMO.
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Can students tell the difference between the "definition if" and the "theorem if"?
The word "if" is used in two meanings in mathematics:
Definition. A topological space is compact if every open cover has a finite subcover.
Theorem. A topological space is compact if it is ...
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What are the questions one should ask of oneself in trying to understand definitions and lemmas? [closed]
Definition (Finite series). Let $m,n$ be integers, and let $(a_i)_{i=m}^{n}$ be a finite sequence of real numbers, assigning a real nmber $a_i$ to each integer $i$ between $m$ and $n$ inclusive. Then ...
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Why is distribution prioritized over combining?
In every algebra (or basic analysis) book that I've seen, three properties of real numbers are taken as axiomatic: commutativity, association, and distribution of multiplication over addition [$a(b + ...
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Is multiplication by zero clear for and understood by K-3 students?
For K-3 students, perhaps it is not acceptable to introduce multiplication by zero as a property or definition. Instead, the child may think about multiplication as, e.g., repeated addition.
Examples ...
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how is volume different than capacity
I have in mind that volume is the amount of room or space a 3-d object takes up - its "outsideness" and that capacity with the amount of room or space a 3-d object can hold. Then I start thinking ...
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What is a good way to explain the slightly different kinds of continuity?
What is a good way to explain the slightly different kinds of continuity to students? I have in mind these kinds of continuity:
A function is continuous at a point. (This also has two sub-kinds: ...
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Explaining subjects whose justification requires demanding technical content
This is my first question and I hope it's appropriate.
Often in the process of teaching a subject I start with examples of a phenomenon, exhibiting similar properties between the examples and ...
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The definition of natural log and e
I'm asking this question from the point of view of an introductory non-rigorous calculus instructor. Calculus textbooks have different approaches about how to define $e$ and $\ln$. For example, my ...
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Are there disadvantages to teaching complex numbers as purely geometrical objects?
Complex numbers are, or at least were to me, generally introduced like this:
There's no number whose square is negative. That's a shame! Well, whatever - we'll make one up! Set $i^2=-1$ and declare ...
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Why do students have problems with showing that something is well-defined? How can this be improved?
I see a lot of students struggling when they have to show that something is well-defined. I have the feeling that this is often not understood.
Two examples:
When defining a sequence $x_n= g(x_{n-1}...
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Morphism-oriented definitions
For some objects there are alternate definitions, which are "morphism-oriented". To give some examples, there are two definitions of a prime number:
$p$ is prime if it is greater than $1$ and has no ...
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