# 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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24answers
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### Optimization problems that today's students might actually encounter?

Our students are not fencing in farm fields, cutting wires and folding them, or designing windows, so they are often uninspired by the optimization problems we give them. They seem like something that ...
10answers
4k views

### Should we avoid indefinite integrals?

I am very uncomfortable with indefinite integrals, as I have a hard time giving them a precise sense that matches the way they are written and the usual meaning of other symbols. For example, when ...
17answers
9k views

### Dividing by zero

I was having a discussion with a friend and fellow mathematics teacher the other day when the topic of dividing by zero came up. She is the department head and had this in a questionnaire she gave to ...
18answers
2k views

### How to teach someone that $-3>-4$?

I am trying to teach a teenage person math, but he doesn't seem to be able to grasp the concept of negative numbers and $0$. Again and again he finds $-4$ greater than $-3$ because he has spent ...
11answers
4k views

### For calculus students, what should be the intuition or motivation behind series?

I've noticed that series are one of the most difficult portions of calculus for new students to learn. I think the level of abstraction has to do with this. Limits, derivatives, and integrals, as ...
13answers
8k views

### Should I be teaching point-slope formula to high school algebra students?

I'm student teaching this semester, and so far I'm loving it! Our next section in the book teaches point-slope formula, and my cooperating teacher (a 24-year veteran teacher) is convinced that point-...
14answers
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 ...
5answers
1k views

### Wonder as motivation

Like all mathematicians, I have a deep appreciation of the beauty of mathematics. Many theorems I find amazing even after I fully understand their proofs. (Example: Euler's formula, $V-E+F=2-2g$. That ...
9answers
8k views

### How to justify teaching students to rationalize denominators?

I'm teaching an "intermediate algebra" college course ($\approx$ junior high school or beginning high school algebra) and we have a bunch of problems on rationalizing denominators. How do I motivate ...
7answers
3k 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 ...
4answers
2k views

### What are some good examples to motivate the implicit function theorem?

I always had problems when teaching the implicite function theorem in advanced analysis courses. This result is motivated by later applications, but it would be great to be able to provide easily ...
6answers
733 views

### Too much motivation?

This is something that I felt like was difficult for me in some classes, especially lower division differential equations and linear algebra classes. I know professors want to motivate certain topics ...
11answers
2k views

### What is a good motivation/showcase for a student for the study of eigenvalues?

Courses about linear algebra make great demands on looking for eigenvalues and transforming matrices to diagonal matrices (or, at least, to Jordan normal form). This is somehow a technical, recipe-...
16answers
2k views

### How to motivate equivalence classes

Equivalence classs are very useful in mathematics, but many of the applications require further background, like quotient spaces in topology or quotient groups in algebra. One good example is residue ...
8answers
1k views

### Counterintuitive consequences of standard definitions

Let me motivate my question with the following situation. While teaching the concept of continuity, I usually start with motivating the concept. Then, when we see that there is an important and ...
7answers
881 views

### Is $e^{i\pi}+1=0$ a good motivation for introducing $e$ or $i$? Why (not)?

Most mathematicians would agree that $$e^{i\pi}+1=0$$ is one of the most impressive formulas. Imagine your students have just learned about the definition of $e$ or $i$ (just assume it's $e$, ...
12answers
9k views

### How to explain that we live in a three-dimensional world?

How does one explain, clearly and simply, that we live in a three-dimensional world? The explanation has to be understandable for a twelve year old child.
5answers
1k views

### Physical applications of higher terms of Taylor series

Depressingly many of the physical "applications" of Taylor series that I can find in textbooks and online are actually just applications of linear approximation, since they only take the constant and ...
7answers
1k views

### 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. ...
8answers
1k views

### What are some good mathematical applications to present in an abstract algebra course?

One of the main difficulties for a student learning abstract algebra is understanding the motivations behind concepts like groups, normal subgroups, rings , ideals etc. Also, many have difficulty ...
14answers
4k views

### How to teach binary numbers to 5th graders?

I already tried the direct approach, starting with "this is how it works". That turned out ok but took too long and was boring for all of us. My second attempt was using the twofingered alien. This ...
5answers
462 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, ...
6answers
2k views

### Motivating the study of matrices

In Brazil's curriculum students are taught matrices in high school. Here, however, there is no linear algebra or pre-calculus, therefore matrices end up being just tables with lots of "arbitrary" ...
2answers
2k views

### What is a good way to explain the Lebesgue integral to non-math majors?

A few days ago I had my last discussion session on probability theory as a TA. In the end I asked students to ask me questions as this is the last class. One of the student asked me about the (real) ...
3answers
1k views

### “Proof” meaning in maths and society

When we ask students to prove a particular result in a math class, students often reply with examples. For example, if I state: if a number is even its square will be even, and ask the students to ...
6answers
2k 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 ...
6answers
585 views

### How to get students in a under-graduate linear algebra course interested in determinants?

Before teaching the chapter on determinants in a linear-algebra course for beginning undergraduate students (mathematics and computer science, more specifically) I would like to give a small ...
8answers
2k views

### How to teach Mathematical Induction mathematically?

I am exhausted of teaching Mathematical induction to my little brother. I have given him many examples, Domino effect, aligned shops of hot dogs etc and every time he says that he got it but when I ...
4answers
535 views

### What are some good ways to motivate and introduce reasoning abstractly about abstract algebra?

I've found one of the hardest topics to introduce to students early on is abstract algebra. Even if they've already written proofs, it's hard for them to work directly from axioms. They seem to have ...
4answers
694 views

### Historically Motivating Concepts

I have been reading this site for a while, and was glad to find an entire tag devoted to "concept motivation," which is currently my area of interest. However, my particular focus has not been ...
4answers
984 views

### What is the motivation for characterizing second order linear PDEs as hyperbolic, elliptic, or parabolic?

I'm teaching an Intro to PDEs course (I'm an analyst, but PDEs are a bit outside my bailiwick) and I'm covering the basic examples: Heat, Wave, and Laplace. How should I move from these examples to ...
8answers
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 ...
5answers
536 views

### When is it a good idea to avoid talking about why something works?

I am teaching, among other things, a college algebra course this semester. In this course we do a lot of conceptual things but we also do some techniques to prepare students for calculus. One of the ...
4answers
408 views

### 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 ...
4answers
1k views

### Using joke / song / film / pop culture to exhibit a new mathematical concept

Questions: Do you have any examples from pop culture (say, a joke, or an episode of a show, or a song lyric, etc.) that utilize and/or effectively illustrate a mathematical concept? Have you used any ...
6answers
427 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 ...
6answers
3k views

### 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?...
4answers
774 views

### Wiggins' question #12

There's an interesting read: Conceptual Understanding in Mathematics by Grant Wiggins. In that text the author proposes "a test for conceptual understanding" which should be given "to 10th, 11th, and ...
6answers
711 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 ...
5answers
726 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 ...
5answers
9k views

### Real life examples to motivate the study of linear functions

Some years ago I used mobile phone or internet rates (for example, with basic fees and a given charge per minute or by data volume) to introduce and motivate the study of linear functions. However, ...
3answers
493 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 ...
2answers
206 views

### 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 ...
9answers
7k 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 ...
6answers
1k views

### How can I convince students that Fourier series are useful?

Main question: Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are ...
6answers
736 views

### is it possible to motivate square roots e.g. $\sqrt{2}$ in business math?

My class is mostly are in business and politics and most have made the choice never to look at math again. Is there still any chance I can motivate exact notions like $\sqrt{2}$ in a manner that is ...
5answers
6k views

### How to motivate the geometric definition of trigonometric functions on the unit circle

Suppose your students know already the geometric definition of $\sin$ $\cos$ and $\tan$ for angles between $0^{\circ}$ and $90^{\circ}$. How can I motivate the definition of $\sin$, $\cos$ and $\tan$...
4answers
7k views

### What are easy examples from daily life of constrained optimization?

A standard example of motivating constrained optimization are examples where the setup is described in a lot of lines, e.g., when you own a company and the company is making some products out of ...
3answers
292 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 ...
3answers
344 views

### How can one convincingly present the alternating group?

I am soon going to explain to my students what the alternating group is. The definition is subtle.... one must prove that the notation of an "even permutation" is well defined. There seem to be two ...