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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How can we motivate that Newton's method is useful?

If you teach Newton's method for finding roots of real functions on the high school (or freshmen) level, I think some students may reason like a variant of the following: Why do I need learn such a &...
Julia's user avatar
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9 votes
0 answers
155 views

Course materials for developing a mathematical theory from "natural questions to ask"

Educational setting. I'm teaching math courses - typically consisting of lectures, weekly homework sheets, and an exercise class where the homework questions are discussed - for undergraduate and ...
Jochen Glueck's user avatar
11 votes
7 answers
3k views

How can we best motivate the study of polynomials to high-school students?

We all know how important and ubiquitous polynomials are in mathematics. However, when faced with a (not so much in love with the subject) 14-year-old asking us why they should care about these things,...
Federico's user avatar
  • 219
12 votes
8 answers
937 views

Any meaning/interpretation for $\frac{1}{0!}+\frac{1}{1!}+\frac{1}{2!}+\dots (= \mathrm e)$ (sum of reciprocals of factorials)?

One common way to introduce Euler's number $\mathrm e$ is $$\mathrm e = \lim_{n\to \infty} \left(1+\frac{1}{n}\right)^n,$$ where the right-hand expression has an "interest rate interpretation&...
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3 votes
6 answers
1k views

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 ...
Asaf Shachar's user avatar
12 votes
7 answers
3k views

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 ...
Asaf Shachar's user avatar
3 votes
1 answer
157 views

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 ...
usul's user avatar
  • 179
9 votes
1 answer
301 views

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 ...
Bilbo's user avatar
  • 271
1 vote
1 answer
204 views

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 ...
Rax Adaam's user avatar
  • 229
4 votes
3 answers
284 views

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 ...
Paula's user avatar
  • 251
5 votes
4 answers
467 views

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 ...
ArminAshrafi's user avatar
15 votes
7 answers
3k views

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. ...
jng224's user avatar
  • 261
14 votes
3 answers
1k views

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 ...
IssaRice's user avatar
  • 259
14 votes
8 answers
3k views

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 ...
Misha Lavrov's user avatar
2 votes
3 answers
987 views

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 ...
Mike Pierce's user avatar
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2 votes
0 answers
92 views

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 ...
AfronPie's user avatar
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29 votes
15 answers
4k views

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 ...
Daniel R. Collins's user avatar
15 votes
6 answers
3k views

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 ...
Amos Hunt's user avatar
  • 351
0 votes
0 answers
157 views

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 ...
Mikayla Eckel Cifrese's user avatar
9 votes
2 answers
1k views

"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 ...
Kostya_I's user avatar
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8 votes
2 answers
218 views

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 ...
user avatar
5 votes
5 answers
891 views

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/...
user523384's user avatar
1 vote
0 answers
96 views

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 ...
user724963's user avatar
6 votes
3 answers
435 views

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 ...
gpuguy's user avatar
  • 281
-2 votes
2 answers
136 views

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 ...
ELM's user avatar
  • 352
3 votes
1 answer
121 views

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) ...
JP McCarthy's user avatar
6 votes
8 answers
863 views

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 ...
Dan Monroe's user avatar
0 votes
2 answers
648 views

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 ...
1123581321's user avatar
7 votes
3 answers
340 views

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,...
user9054364's user avatar
6 votes
2 answers
175 views

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 ...
Thomas Davis's user avatar
4 votes
2 answers
296 views

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 ...
Mason's user avatar
  • 313
4 votes
1 answer
152 views

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}{\...
Hans-Peter Stricker's user avatar
6 votes
3 answers
432 views

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 ...
Mike Pierce's user avatar
  • 4,845
11 votes
1 answer
544 views

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 ...
benblumsmith's user avatar
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10 votes
6 answers
819 views

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 ...
Bad English's user avatar
5 votes
2 answers
310 views

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 ...
Martin Argerami's user avatar
7 votes
1 answer
219 views

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,...
Krijn's user avatar
  • 173
10 votes
5 answers
2k views

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 ...
Mike Pierce's user avatar
  • 4,845
13 votes
6 answers
470 views

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 ...
Ovi's user avatar
  • 725
6 votes
2 answers
227 views

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 ...
matqkks's user avatar
  • 1,243
11 votes
4 answers
696 views

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 ...
matqkks's user avatar
  • 1,243
12 votes
5 answers
958 views

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 ...
Jared's user avatar
  • 2,223
2 votes
0 answers
86 views

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 ...
leeand00's user avatar
  • 173
9 votes
8 answers
880 views

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. ...
Gordon Royle's user avatar
25 votes
7 answers
4k views

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 ...
Amanda's user avatar
  • 553
5 votes
2 answers
922 views

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 ...
Buffer Over Read's user avatar
19 votes
5 answers
557 views

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, ...
Joseph O'Rourke's user avatar
13 votes
3 answers
742 views

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 ...
Patrick Stevens's user avatar
12 votes
9 answers
12k views

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 ...
Stephan Kulla's user avatar
9 votes
8 answers
921 views

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, ...
Jessie's user avatar
  • 131