Questions tagged [concept-motivation]
For questions how to motivate a mathematical concept (i.e., the motivation and examples of definitions, theorems, etc.) or general concepts of mathematics. Please use the [student-motivation] tag for questions about how to motivate students in general.
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Is this motivation for the concept of a limit a good one?
tldr: There is a simple intuitive definition of a limit for monotone sequences, and I suggest that it can be used to motivate the (more complicated) standard definition. I am asking for feedback on my ...
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Does induction really avoid proving an infinite number of claims?
I am teaching calculus $1$ this semester, and I saw the following motivation for using induction by another teacher:
Since we can't go over "manually proving" all claims $1,2,\ldots$ and ...
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Favorite linear programming (not integer) examples?
I am wondering what examples you like to give when introducing linear programming, where the examples are not clearly better suited as integer linear programs. I would like a few examples where we can ...
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How well can students learn abstract concepts through concrete examples?
In my own personal experience in teaching linear algebra, where many students encounter abstract ideas for the first time, I find that most students have trouble consolidating observations from ...
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Introducing direct substitution in an intro calculus course
I'm revisiting the materials I've put together for students taking a non-proof-based intro to calculus, and my goal is for them to have a clear but rough sense of a limit as a bound (basically enough ...
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Probability — analytical results instead of simulations
After students learn how to use probabilistic simulations, what strategies can one use to encourage them to understand analytical results anyway? For example, I'm struggling to find a compelling ...
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How to explain loss of significance in numerical analysis?
As I have myself struggled a bit with this concept, I would like to present my own explanation of it.
Context:
Loss of significance is a loss of precision, not necessarily accuracy.
And for a long ...
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An introductory example for Taylor series (12th grade)
I (a student) am doing a presentation on Taylor series in my class (12th grade, in Germany if this is relevant). I am looking for a good example where you can see when Taylor series might be useful. ...
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How to teach the Pythagorean theorem in a satisfying way to high school students?
I've been pretty dissatisfied with the way the Pythagorean theorem is usually taught, mainly for two reasons:
The chosen proof feels like magic and I don't feel like I have a better understanding of ...
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Replacement for the Pac-Man grid analogy
To most people, a torus is a donut-like shape. Topologists like to describe the torus differently: you start with a square, and "identify opposite sides". We can imagine gluing together one ...
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What is the motivation for teaching Factoring by Grouping?
This seems like such a niche trick to teach students when factoring polynomials. Like, the polynomials I've seen textbooks ask students to factor by grouping seem so cherry picked that I can't imagine ...
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Math websites/apps for high school students
I am undergraduate math student who is interested in being a high school math teacher. I have been given an assignment to present to my class (for a total of about 20 minutes) a teaching tool or a ...
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What's the point of learning equivalence relations?
I teach an introductory discrete mathematics course at a community college to math and computing majors, usually in their sophomore year. As is common, it's partly used as the first foray into formal ...
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Are there direct practical applications of differentiating natural logarithms?
The textbook I am using to teach Calculus I includes in the exercises of most chapters a number of interesting real-world applications of the concepts from that chapter. However, the chapter on the ...
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How can I introduce the idea of eigenvectors and matrix decompositions to a general audience in an engaging manner?
So I'm doing a freelance writing job, writing a script for a YouTube video about eigenvectors/values. It took me a while to decide what the focus was going to be, but I finally settled on focusing on ...
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"Feynman effect" in teaching mathematics
In his book "Surely you're joking Mr. Feynman", Richard Feynman relates the following story. As he was supervising a group of calculators for Manhattan project, he at some point gave them a lecture on ...
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Do you teach different proofs or calculations of same question?
Recently I asked a question on math.stackechange about the most ways to differentiate the same function and it didn't seem to generate any interest - rather, the reason why I'd ask such a question was ...
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How to intuitively understand how the trig ratios are calculated
I've asked a question on Math Stack Exchange, but it was suggested it might be a better idea to post it on this Educators instead.
Here's the question link:
https://math.stackexchange.com/questions/...
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Is there a game or analogy for searching for a number $R$ that is multiple of some number $A$ but not multiple of another $B$?
$\DeclareMathOperator{\lcm}{lcm}$There are many applications for finding the GCD or LCM of two numbers. I'm now interested in finding anything that could be used to illustrate the following ...
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How do I convince my teachers that a book on maths must focus on conceptual understanding?
I am a senior teacher at this school. We have to select the textbooks for the upcoming session. I am proposing that we have to select books (in maths) that focus more on conceptual understanding and ...
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Should the limits of one system of elementary set theory be the limits of a student's mathematical world? [closed]
In teaching elementary set theory, suppose we refrain from emphasizing historical decisions that were made in theory construction.
Is there a danger that students may see the mathematical language ...
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Applied ODEs for Numerical Methods
I am looking for a list of ODEs to use as examples in the teaching of a numerical methods course for engineers.
I am looking for first and second order examples - the more applied (to engineering) ...
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Rhombuses, kites etc
As a high school teacher, I sometimes wonder about the usefulness of certain topics. Some topics seem to be in the textbook because they have always been there, not because they lead anywhere ...
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Practical applications of integration by substitution where integrand is unknown
I posted this question on the Mathematics Stack Exchange a while ago, and got no responses, so I thought I would ask it here. I'm looking for any real-life applications of integration by substitution ...
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Complex numbers and encourage justification
In remedial algebra, we learn that the graph of $y=(\sqrt x)^2$ is only in the first quadrant. We know this is the correct graph for the equation. This is because we know $y=x$ and $x \ge 0$.
However,...
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Studies about group tutoring sessions
I’m not sure if this question belongs here, so I apologize if it doesn’t.
I work in a tutoring center at my university where we tutor every subject. Mathematics is in high demand, and occasionally my ...
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Why are we even studying cyclotomic polynomials?
My students found an exercise about cyclotomic polynomials in the AOPS precalculus text. They asked me why this construction exists in the first place and what it's good for... I am looking to give ...
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Making modular arithmetic interesting for school kids
This is a pattern even school kids could discover (when gently pointed to). I never did conciously, and cannot remember to have been pointed to explicitly, neither at school nor later:
$$\color{red}{\...
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Justifying the multi-variable chain rule to students
Suppose that $f(x,y,z) = x + 2xy^2 - yz$, and that $\gamma(u,v) = \langle uv, u\sin(v), u\cos(v)\rangle$. Use the chain rule to calculate $\partial(f \circ \gamma)/\partial u$.
This is an exercise ...
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Motivation for uniform continuity
What are some problems or theorems that motivate the distinction between continuity and uniform continuity? In particular, I would like:
a) A useful, appealing theorem that applies to uniformly ...
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teach that $\frac10$ not defined properly
there're some students, who belive that $$\frac10 = \infty $$
I need to teach them that this is not true and $\frac10 $ is undefined, mathematically and give a good picture (for their minds)
what is ...
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Motivation for Fibonacci: Bees
I want to talk about the Fibonacci sequence in my Linear Algebra class. So I tried to look online for examples where the sequence appears naturally.
One of the most often mentioned is that of the ...
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Explaining genus to students
I need to do a presentation on my thesis, which is in arithmetic geometry. This presentation is meant for all students of mathematics, but I will assume some knowledge of abstract algebra (i.e. groups,...
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Activities for biology undergraduates taking integral calculus
After searching for applications of calculus for biology students, I've found that many of the results are all either contrived exercises, or are way over the heads of students that are seeing ...
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Is there a simple real-world problem I can use to motivate a formula for $\displaystyle \sum_{i=1}^n i $?
I would like to know if there is a simple real-world problem which requires knowing a closed form for $\displaystyle \sum_{i=1}^n i$ and/or the sum of the first $n$ even/odd numbers. The only ...
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How to introduce Wilson's Theorem?
What is the most motivating way to introduce Wilson’s Theorem?
Why is Wilson’s theorem useful? With Fermat’s little Theorem we can say that working with residue 1 modulo prime p makes life easier ...
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What is most motivating way to introduce Fermat's Little Theorem
What is the best way to introduce Fermat’s Little Theorem (F$l$T) to students?
What can I use as an opening paragraph which will motivate and have an impact on why students should learn this theorem ...
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Moving from discrete probability distributions to continuous ones
I'm teaching an introductory statistics class at a community college, and we've just finished a unit on discrete probability. At the moment, the students' conception of the probability of an event A ...
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Finding the correct mathmatics based on application as opposed to the other way around?
I just tried to pitch a Math Recommendations site, and it was shot down because it is too broad and that Stack Exchange sites operate on a separation of concerns principle; which makes sense to me but ...
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Is there a toy example of an axiomatically defined system/ structure?
Day 1 in my "Into to Pure Maths" class...
I'd like to have a very simple set of axioms defining something, not necessarily a useful thing, but a system that is suitable for making short deductions.
...
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Why do we care about multiple proofs of the same theorem?
I am teaching a math appreciation course to high school students who are approximately 17 years old, in their last year of high school, and who do not believe they will choose a STEM major in ...
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Good way to explain fundamental theorem of arithmetic?
Some students understand how this works, like they know what the theorem means, but, say, imagine some student asks why, not how. Not really proving the theorem, rather why does it exist or why is it ...
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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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Teaching intuition for the universal property of the product (category theory)
In category theory, there's the idea of the product as an object satisfying a particular universal property.
Can you suggest ways to make the concept of the product intuitive? (So far, my attempt ...
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Why do we study ordinary differential equations?
What is a good answer to the question: Why should one study ordinary differential equations?
I would give the answer: ODEs are used in many models to determine how the state of this model is changing ...
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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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Why should we study continuity?
This question is related to How can I motivate the formal definition of continuity? Imagine a student asks the question why it is worth it to study continuity. What is a good response to this question?...
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Do "overview" sections increase learning outcome?
I write a math textbook for which I want to make an overview chapter for each part of the textbook. In those overview chapters I want to motivate and introduce the new mathematical concepts. I also ...
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Sketching paraboloids on paper
I have to teach sketching paraboloids on paper by looking at it's equation. Last year when I taught this topic no one was interested in learning this particular thing. They felt the topic difficult ...
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How can one motivate the adjugate matrix?
The adjugate matrix of an $n \times n$ matrix $A$ is defined by $(\mathrm{adj}\ A)_{k\ell} = (-1)^{k+\ell}\,\det M(\ell,k)$, where $M(\ell,k)$ is the minor matrix obtained from $A$ by deleting row $\...
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