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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33
votes
13answers
9k 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-...
11
votes
2answers
266 views

How to motivate 'Theory of Computation'?

I am teaching Theory of Computation for the first time soon, and need a short introduction and justification to motivate students who have only Introductory Programming behind them. I am torn between: ...
16
votes
6answers
607 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 ...
23
votes
6answers
756 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 ...
10
votes
5answers
237 views

Lesson-planning: Teaching probability concepts via geometry

I am intending to teach a lesson covering some topic related to "Probability via Geometry" and, if possible, I would appreciate references or materials (or good ideas) that can help me. The target ...
9
votes
3answers
221 views

Teaching and motivating the use of Eigenvectors

I would like to know how to better demonstrate Eigenvectors. The texts that I have display the properties and methods to calculate them. There are plenty of great elementary examples to follow through ...
6
votes
1answer
196 views

Resources for Pell's equation

What is the best way to introduce Pell’s equation on a first elementary number theory course? Are there any practical applications of Pell’s equation? What are the really interesting questions about ...
4
votes
1answer
175 views

How to explain the topic of Fourier transform interactively?

This is a soft question . In the walk-in for the lectureship, I have decided to give demo lecture on the topic of Fourier transform. The principal of the institution ask me to take lecture ...
5
votes
2answers
225 views

Examples where roots are necessary for the solution

I currently write an article where I want to introduce roots. Thus I need to motivate them. Here I said, they can be used to find solutions of equations like $x^n=a$. Now I want to make some examples, ...
8
votes
2answers
297 views

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 ...
6
votes
5answers
1k views

Analogies for mathematical induction

What are the most successful analogies that are used to teach the concept of mathematical induction? To clarify, I am not looking for a formal explanation of the principle of mathematical induction, ...
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 ...
8
votes
3answers
214 views

How important is building up intuition for a theorem before trying to prove it?

For example, consider trying to prove that: If $A$ is a set and $F \subset P(A)$, then the relation $R := \{(a, b) \in A \times A $ such that for every $X \subset A - \{a, b\}$, if $X \cup \{a\} ...
28
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 ...
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 ...
11
votes
5answers
531 views

An intuitive derivation of Taylor polynomial coefficients

I'd like to introduce Taylor polynomials by generalizing the linear approximation of a function $f(x)$ to a quadratic approximation. The linear approximation formula $L(x)=f(a)+f'(a)(x-a)$ of $f(x)$ ...
11
votes
3answers
551 views

Good ways of explaining the idea of epsilon-delta limits to bio & chem majors?

I am cross-posting from MSE. The students in my calc classes tend to be primarily bio/chem majors, and not very much math / physics / engineering. I feel like there are pretty good ways to talk ...
5
votes
2answers
377 views

Show me the money

Apologies for the crass title. However, there are incentives for many children in grades (comparable to the U.S.) 3-12 to learn to do arithmetic well. (I recall Number Sense and related contests ...
10
votes
2answers
501 views

How to differentiate between mathematical skills and understanding of mathematical concepts?

How would my Colleagues here on Math Educators differentiate between mathematical skills and understanding of mathematical concepts? I'm a community college instructor and high school instructor ...
7
votes
4answers
4k views

How Can I Motivate Geometric Constructions?

When starting compass and straightedge geometric constructions in my grade 8-9 maths classes, I usually begin by mentioning a little about Euclid and the fact that constructions have been done for ...
19
votes
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) ...
20
votes
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.
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" ...
33
votes
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 ...
14
votes
4answers
415 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 ...
11
votes
3answers
355 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 ...
14
votes
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 ...
12
votes
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, ...
6
votes
1answer
142 views

Intuitive explantion: What is a Finsler metric?

Neither of the two most evident sources, MathWorld: "Finsler Metric." Wikipedia: "Finsler Manifolds." seems to provide me with the high-level intuition that I could convey to students in ~10 minutes....
17
votes
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 ...
10
votes
7answers
4k 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
393 views

Examples of Applications of Basic Mathematics to Computing

Let me start by saying that there isn't necessarily a very good selection of answers to this question. I am teaching some very basic mathematics (see below for module content) to a group of ...
10
votes
3answers
398 views

Passage from Descriptive to Inferential Statistics - analogies with other Math-fields?

I am a guest here, having responded to a general invitation extended to the Cross Validated community, to possibly contribute answers whenever some question related to Statistics comes up in this site....
17
votes
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 ...
10
votes
5answers
4k views

Why do we need perfect numbers?

Why are perfect numbers important? What is the best way of introducing these numbers to a first course on number theory? I could not find any application apart from the relation to Mersenne primes. ...
11
votes
6answers
751 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 ...
16
votes
4answers
708 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 ...
13
votes
4answers
797 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 ...
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 ...
11
votes
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 $\tan$...
46
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 ...
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
4answers
1k views

Reasons to teach Thales' theorem

In a classical course on Euclidean, compass-and-ruler geometry, Thales' theorem has always had a prominent place. However, as the Wikipedia article says, It is equivalent to the theorem about ...
25
votes
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 ...
11
votes
1answer
246 views

A more natural motivation for the appearance of generalized eigenvectors in linear system with repeated eigenvalue

When I teach constant coefficient linear differential equations, the usual guess of an exponential can be motivated because it is "approximately" a fixed point for the differentiation operator. The ...
16
votes
4answers
1k 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 ...
54
votes
24answers
58k 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 ...
11
votes
4answers
8k 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 ...
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 ...
24
votes
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 ...