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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28
votes
14answers
3k 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 ...
2
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0answers
65 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 ...
16
votes
4answers
489 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 ...
16
votes
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 ...
0
votes
0answers
127 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 ...
7
votes
2answers
188 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 ...
9
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2answers
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 ...
32
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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 ...
15
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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
4answers
993 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 ...
5
votes
5answers
303 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/...
44
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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 ...
1
vote
0answers
86 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 ...
0
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2answers
218 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 ...
6
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3answers
348 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 ...
-2
votes
2answers
106 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 ...
9
votes
6answers
580 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 there minds) what ...
3
votes
1answer
99 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) ...
10
votes
5answers
1k 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 ...
6
votes
2answers
148 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 ...
7
votes
3answers
259 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,...
11
votes
9answers
6k 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 ...
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" ...
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 ...
27
votes
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-...
3
votes
2answers
180 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 ...
4
votes
1answer
106 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}{\...
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-...
16
votes
4answers
858 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 ...
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. ...
6
votes
3answers
338 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 ...
19
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.
53
votes
24answers
46k 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 ...
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 ...
11
votes
4answers
624 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 ...
12
votes
5answers
635 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 ...
19
votes
8answers
960 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 ...
11
votes
1answer
276 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 ...
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 ...
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 ...
13
votes
6answers
412 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 ...
11
votes
2answers
263 views

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 $\...
5
votes
2answers
196 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 ...
6
votes
1answer
193 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,...
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 ...
23
votes
6answers
699 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 ...
6
votes
2answers
188 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 ...
10
votes
5answers
2k 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. ...
14
votes
4answers
396 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 ...
2
votes
0answers
81 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 ...