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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44
votes
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 ...
13
votes
4answers
751 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 ...
52
votes
24answers
45k views

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 ...
32
votes
17answers
8k 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 ...
29
votes
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 ...
19
votes
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. ...
25
votes
9answers
7k 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 ...
22
votes
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-...
19
votes
8answers
946 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 ...
26
votes
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 ...
21
votes
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 ...
16
votes
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 ...
14
votes
4answers
454 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 ...
19
votes
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 ...
11
votes
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 ...
32
votes
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 ...
22
votes
16answers
1k 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 ...
23
votes
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 ...
15
votes
6answers
561 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 ...
17
votes
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" ...
13
votes
6answers
410 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 ...
8
votes
8answers
647 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. ...
4
votes
2answers
695 views

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 \...
15
votes
13answers
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 ...
11
votes
3answers
6k 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 ...
11
votes
3answers
310 views

How can you explain the importance of $e$ to those who have not taken calculus?

The number $e$ has many interesting and important properties, many of which are related to calculus. How can I explain what $e$ is and why it is important to those who have not had calculus (or even, ...
10
votes
7answers
3k views

Direct applications and motivation of trig substitution for beginning calculus students

Motivating what is often called "Calculus 2" can be hard, which is probably why there are multiple other attempts at motivating it here. I have just begun teaching such a course, beginning with the ...
6
votes
2answers
146 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 ...
4
votes
1answer
691 views

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 ...