Response to Students Who Say "This Is Not Important"

Question

Lately, my students keep telling me why what we are learning is not important. They ask me when will we use this in the real world?

I explain how math is important in gambling, cooking, finance, engineering, programming, electronics, philosophy, and logic. I state how we live in a digital world, and there is a ton of application out there. As a pure mathematician at heart, I point out how important proofs are in research and how difficult theorems are to come up with. I explain how pure mathematicians don't even need to find appication, and their work is incredibly important, e.g. Euclid and his discoveries in number theory and geometry. I explain how math is beautiful and similar to art, patterns, and design. I have played movies where mathematicians had huge roles and impacts on history. I explain the importance of education and study in order to find happiness and get out of poverty. I explain how education is important to get what you want in life.

Still... students comment how little math they use and others use generally in their daily lives. It's like if I am not using the math I am using tonight, then why do I bother learning it?

What is a good response to get students out of this mindset? How do you dare them to try to do these things? I am really into psychology and how to persuade students to change their mindset.

Answer 1

"Lately, my students keep telling me why what we are learning is not important. They ask me when will we use this in the real world?"

There's a quick reply to this that I think people won't like, and you would have to be very careful using, but actually makes a more serious than it first appears. One possible answer is: "You're quite right. You don't need maths to mop the toilets out for a living. Is that your plan?" Many people do end up in basic jobs. There's nothing wrong with that, if that's genuinely what you want out of life. And many people who do it are happy and fulfilled with their lot. It's their life. You just need to be sure it's an informed choice, that they're going into with their eyes open to the consequences.

But really you need to find out why they're asking. It could be that they genuinely find it hard to connect what they're doing to real world applications, or that they do see a need for maths in daily life but not the stuff you're teaching. (People need to do things like fill in tax returns, pay bills, plan finances, calculate interest for loans, run a business, calculate fuel usage, buy supplies in the right amounts, understand economics, etc. Maths lessons often don't teach any of that.) In this case, find out what they want to do with their lives, or what they do do, and then give examples built around their particular applications.

Or it might be that they don't enjoy it, feel they're not good at it, and are seeking some excuse not to do it, or a reason not to feel bad about not being able to do it. Dismissing it as "unimportant" is one way to do that. In which case, it's a waste of time giving reasons why it really is essential - that just makes them feel even worse. There you need to build their confidence, help them catch up.

Or it may be a cultural issue. Sometimes people belong to a cultural identity that has negative views about maths and studying. Girls may think it's a "boy's subject". Boys may think it's "unmanly" or "geeky/nerdy" to be studious/academic/clever (as opposed to being physically tough, good at sports, etc.). Teens are very sensitive to what characteristics are seen as attractive or unattractive to the opposite sex. Sometimes you get racial, religious, or class stereotypes, where studiousness is associated with particular groups that are negatively regarded, or where there are strong religiously-based expectations that oppose, for example, girls studying, or the adoption of Western values, or non-religious values generally. Where gangs and criminality are a problem, respecting authority and conforming to the expectations of mainstream society are often seen very negatively by their peers. Friends, families, and communities all apply social pressures. Cultural conflict is a much harder problem to solve. There may be a way to fit mathematics positively into their worldview, but it may be that the only way is to tackle it head on and change their worldview, and that obviously has some very serious ethical complications that need to be considered first. Is is right to destroy elements of their culture, even if it is "for their own good"?

Without knowing what their issue is, trying to come up with solutions is a very broad question. Ask more questions about what they want from life, what they value, and what they think they have to offer. And be open to the possibility that maybe they're right, and maths is not what they need.

Answer 2

Math is just as useless as almost any other subject

As a math tutor, I've thought about this a lot over the last 15 years or so. Aside from tutoring, I don't use my math education in "the real world". Here's a list of educational requirements that I have also never used in "the real world":

• American literature (Hemingway, Flannery O'Connor, Langston Hughes)
• British literature (Shakespeare, ummm... I'm sure I studied some other British authors)
• Gym class (volleyball, crab soccer, archery, etc.)
• Latin
• Biology
• Chemistry
• Physics
• 95% of History (e.g., "the age of explorers!") - besides most of what I learned is either completely false, presented poorly, or heavily biased in favor of the myth of American exceptionalism
• Finding library books using a card catalog and the Dewey Decimal system

For some reason, math seems to be the only subject students are forced to learn that causes them to ask "why do we have to learn this?" I don't know why. But outside of a very short list of actually applicable subjects (expository essays in English class, health class, driver's education, and shop class, and art (!!) for me), the vast majority of primary and secondary education is just as "useless" as mathematics.

So why do we teach any of this stuff?

There is one thing that I said to a student one time about why we learn math and he said it made sense to him. It was similar to this:

If you want to excel in certain sports, you might do some strength training. So you might do bench presses, curls, dead lifts, or other esoteric muscle isolation exercises. None of those actions, like curling a barbell, are thing you would actually do in a sport like baseball or football (American) or football (Soccer). But manly athletes do strength training to help them be more effective in their sport. The point is, the things that you do to strengthen muscles does not have to be the same thing you are doing when you use your stronger muscles. The muscles don't care how they got strong - they are strong and humans with strong muscles can do things that humans with weaker muscles can't.

School is like strength training for your brain muscles. We tell you to do all of these arbitrary mental tasks, like the mental equivalent of curling a barbell, not because you'll be doing mental barbell curls later on in life, but simply to strengthen that part of your brain, so you can do whatever you want with that part of your brain later in life. The same reasoning applies to learning British literature and Physics. Your brain doesn't care how it got strong. And humans with stronger brains can do things that humans with weaker brains can't.

---

Note: only after typing all of this up did I discover that this question is almost certainly a duplicate of a question where the highest voted answer is quite similar to my answer here:

How should I answer questions about the purpose of learning math?

Answer 3

I've never had success with giving a list of applications to such students - because, realistically, we don't use most of the math we teach. For example, I teach early equations in one of my classes, things like "solve the equation $5x = 15$". If asked why this question is "useful", I could give this answer:

• Well, suppose you were planning a barbeque with four friends, and you had 15 hamburger patties in the freezer, and you wanted to know how many hamburgers each person could have.

Except we don't use math like that. Nobody, no matter how steeped in math, would solve the above problem by defining a variable, setting up an equation, and solving that equation. Even I would just solve this problem by dividing 15 by 5.

I could, instead, give this answer:

• Solving equations like $5x = 15$ is the first step in learning to solve more complicated equations, which you'll need to do for higher-level math classes.

But this isn't terribly convincing either; if they aren't planning on taking later math classes, what's the point? And even if they are, it just pushes the problem back one step - why are those more complicated equations useful? If I'm being totally honest, even I hardly ever use an equation for anything that isn't math.

Instead, my response is always about the modes of thought. Solving an equation like $5x = 15$ is an exercise in precise thought; writing equations, defining variables, and so on are exercises in precise communication. And thinking and communicating precisely are crucial skills in literally every field. If you haven't developed the level of organized thinking required to handle the exponent laws, it's going to be very hard for you to make sense of all the complicated details of a loan; if you aren't able to connect the numbers in an expression to their underlying meaning, it's going to be hard to tell whether two for \$3 is a better or worse deal than one for \$1.

Answer 4

The American Mathematical Society provides posters promoting awareness of mathematics, its beauty, and applications. Here's a quote from the AMS Posters website:

"Students frequently ask when they will use the math we learn in real life, and your posters provide great visuals to support the answers to this question."

The AMS also have "Mathematical Moments," a "series of posters that promote appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture." These are available in different languages.

Answer 5

Trigger warning: math enthusiasts do not like this answer.

They ask me when will we use this in the real world?

When students ask when they will use something in "real life", they are rarely expecting an actual answer to how something is used.

They are telling you: "I don't understand this."

Rather than coming up with try-hard ways to directly answer their question, focus your energy on teaching effectively, so that they understand the math that is taught to them.

It is not to say that it is not important to highlight connections of math to other areas of life, when they arise naturally.

I recommend that you read "How I wish I'd taught maths" from Craig Barton. Chapter 2.3 discusses this topic at some length.

Answer 6

(not complete)

You should have mentioned which were some of the topics where your student's noise pollution rates increased drastically. We could have been able to tailor our responses specific to your curriculum.

Anyway, I remember one of my friends arguing why the chapter on polynomials was a silly chapter with no real-life application [ questions like $p\left(x\right)=4x^{3}+10x^{2}+5x\ +4$ find the zeroes, etc ]. I must confess I really believed in his argument that what we were learning was not useful. But then came high school-level physics and chemistry which was essentially piggybacking here and there based on the core principles of the chapter on polynomials, such as finding zeros. I would also argue that there could have been more applications on the chapter that I just can't recollect, but if I remember changing my mind afterwards then that chapter must have been really useful.

This may sound like a silly anectode, but math has always been like that. You learn something at an elementary level realises its application only later. This is true at a scientific level too. There are many times where abstract ideas in the field of mathematics were given real-world application (courtesy of physics) only several decades after the discovery.

I suggest maybe going to the maths and science history stack exchange and see if you can ask them for an example of mathematical abstract which was given use only many years later. The only example coming to my mind is the heavy reliance of the General theory of relativity on Topological space as opposed to Euclidean space (which to be frank must have had 0 application) and of course, the Euler number making its way into Electrical engineering and Quantum mechanics. Oh yes along with iota. If you can find any formulas that knock them out, then they may understand how even the square root of negative 1 can be useful (ill edit my answer better tomorrow as its late-night)

Now assuming you have found a great example, ask your students how much of them want to become scientists, engineers

In the words of Richard Feynmann

Mathematicians prepare abstract reasoning that’s ready ‘to be used’ even though they don’t know what it’s being used for

(I must also confess i am not a teacher but just a student who likes maths)

Answer 7

You do math to get better at logical thinking, not to get better at math.

Similar to how world-class athletes train in the gym by doing many exercises that you'd never see them stop and do mid-match, learning and doing math trains your brain to make you better at other every day tasks.

A lot of math (most of it, in my opinion) is learning how to analyze a problem and break it down into steps you know how to solve. As you advance in your education the problems get more complex and you learn to identify which variables are relevant and what they relate to from context. That's a skill that's applicable in nearly everyone's life, from choosing whether you want an electric vs gas car based on purchase price and operating cost over 5 years, to planning a barbecue with friends. They might not think that those are very complicated problems, but that's the point: thanks to their their mathematical foundations they can easily analyze the problem.

Answer 8

W.G.,

I think you have to engage based on the situation. If it is someone wanting an argument, or to disrupt the class, you can engage a little, but at a certain (quick) point, need to move on and just teach. It's not a winnable argument, per se.

That's not to say that no motivation should ever be given. It is good if you can do so at times along the way (without long discussions, just little asides like "this helped crack the Nazi's code, or this is important in biology, or whatever). And of course, this is not the only motivation. You should be motivating by modulating your voice, having energy, being excited, showing JOY when you solve a problem. In other words, the personal dynamic.

As far as pure math and beauty and all that: Even crusty old applied people like I were thrilled when Wiles solved FLT. So there is some thrill to pure discoveries. But really in a sense students get that since integration by parts is new TO THEM. But you've gotta realize 95%+ of the population sees math as a tool of science (and business and the like). So, mix in a little beauty at times. But if you think everyone is heading to being like you, thinking like you, you're ignoring reality.

Really, the major driver for kids to learn math is to be able to handle undergrad science/engineering classes. And it's not a stupid rationale. It's a very real thing in front of them. Calculus as a co/pre-req for entry level physics for instance.

If they want to be doctors, they need to take chem, physics, bio and organic chem. That's just to deal with the MCAT. You could abstractly argue about how much of that is needed for doctors, but we live in the real world. You want a big house and a trophy wife, gotta pass those MCATs. Let's deal with what we're actually facing.

Probably for your own good, it would be nice to learn at least a little more about how math IS used in college homework problems in chem, physics, engineering, econ, etc. You don't need to run off an get a masters. But you could learn a lot from skimming a few textbooks. And I say this with humility, but I've seen a huge pattern of pure types lacking even the start of awareness of these areas. There aware that there's some stuff called applications. But they have zero feel for it. Do things like citing an incredibly minor and high level diffyQ for one random Ph.D. thesis in biophysics modeling, but don't know that imaginary numbers are key in all of alternating current EE. But the good thing about being very ignorant, is that can very quickly move from zero to something. Large payoff for minor investment.

P.s. Based on your questions, I'm assuming you're dealing with college-bound HS students. But of course you should have told us this and told it to yourself. The audience is a key variable in all these pedagogy questions. 8th graders are different than 12th than college. Top track different from average. Etc.

Answer 9

Because the sum of all the subjects helps to make for a well rounded education. This keeps all opportunities in life open to you.

The math learned by sophomore year of high school may actually be enough math for the average person. But even though a lawyer may not ever use the law of cosines, the law school they want to get into expects a HS transcript with good grades across the board.

At another level, I was tasked to teach an MCAS (our local state standardized test for sophomore HS) class, teaching to the skills required. To that class, I asked if they were comfortable reading street signs, or the articles in a paper or magazine. They said they were. I told them there was a level of math understanding that was similar. Enough to not get ripped off in a store, to adjust recipes ingredients up or down, and so on.

Answer 10

A simple answer would be: "You will date more than one person in your life but you won't know in advance who is the one, unless you give them all a substantial try. If you never do that you'll miss an important part of a full life."

Answer 11

Give them practical examples

Engineering is a good start. But equally well, you could relate it to simple DIY.

The most obvious example is Pythagoras. You're building a covered area next to your house. The house wall is 3m before you get to the gutters; and you want the posts to be 2.5m tall. You've got 3m of patio. How long is the roof? because you need to make sure you buy enough tiles for it.

Frighteningly, I have had a conservatory fitted by someone who couldn't do this calculation to work out how big the roof needed to be. He needed to get someone to hold a tape measure for him to measure that distance. I told him beforehand what the answer would be, and it was. :) To be fair, he did a decent job of putting it together, but he was completely out of his depth on any kind of design work.

Or even more complex trigonometry. Last year I built raised flower beds on a square area, using wooden sleepers. To make it look nice and give me access, I decided to put paths along the diagonals. Then I decided that it'd look better to make the outside walls curve around. To make that look good, I had the path running at 45 degrees, then three sections of wall to make the curve. How long did they need to be, and what angle did I need to cut to make them join properly? And then since I was building with overlapping joints with 10cm of overlap to hold everything together, how long did each section have to be in total, and how many 2.4m sleepers did I need to buy to build it?

This isn't an imaginary example, by the way. I have physically built this, and I literally did draw the diagram and use the Sine Rule to work out lengths of each section. Sure, I could have chopped things roughly to fit and hacked it around a bit, but this way I could guarantee it would fit (within the bounds of final fine adjustment.)

Answer 12

Would it not be interesting to ask students who voice such opinions to substantiate their answers with examples of situations where one would think that mathematics would be of importance but is not?

One could ask a student to give a presentation of why number theory is of no importance in cryptography or why calculus is of no importance for optimisation problems. The goal should be that the student can substantiate his/her opinion but this will then require him/her to get enough of an understanding of mathematics to be able to explain convincingly why it is actually of no importance. And then the student will discover that his/her objection did not make sense.

Answer 13

Perhaps it's the mathematical way of thinking about and approaching problems that is useful, and this way of mathematical thinking can be carried over into many domains of study and work. So while there might not immediately be direct real world applications to parts of it, it's the way of mathematical thinking and approaching problems that is useful. That being said, mathematics can be such a rewarding and interesting subject of study. Perhaps there is room for improvement in this area. How can mathematics be approached to where students can genuinely gain more of an interest in it?

Answer 14

Give real world examples of how maths applies to everyday job functions, go beyond cash, tax and cost planning and apply it to physical tasks.

It is important in almost any real world building project to check for square and for setout. An understanding of triangulation, pythagoras and the use of 3/4/5 triangle rule in measurement is the only thing that will guarantee success when you are trying to construct either horizontal or vertical elements.

Painters, carpet layers, dress makers or anybody that buys things in liquid form or on rolls need to analyse and calculate coverage of that material to simplify the task and minimise wastage. How much paint/cloth one needs to coat/wrap a spherical object requires a mathematical solution. So is the amount of Helium required to inflate 10 party balloons, cook steaks on a BBQ grill from a propane tank etc.

Anybody who handles loose materials will need to calculate and consider bulk density, pile mass, slump angle, friction and surface area for storage and handling. How much space do you need for the dirt you are digging out of the hole and how far away should the spoil pile be so it doesn't fall back in on you? This requires an understanding of geometry, angle of repose and base shape. Show your students that some animals dig conical pits in sand to catch prey and how the hole created is on the verge of slumping until it's disturbed by the prey and collapses aka the Great Pit of Carkoon!

Answer 15

My kids have asked me this and i just responded "You are wrong. History, geography and most subjects i learnt at school were pretty much useless to me, and i have forgotten most of it, but maths is the only subject I have needed on a daily basis. I need it for all kinds of things." From time to time, i involve them in the maths i need. When i am working out materials i need for DIY. When i was calculating flow for draining the garden. Budgeting. Etc. My young kids now fully accept and understand that they need it.

Part of the problem is that many of the subjects ARE useless and i dont understand why they are taught. There is too much academia. Kids should learn gardening, DIY, electrics, money management and dealing with government offices, sports, diet, cooking, typing, web development, image processing, and a plethora of other topics far more important than the 18th century textbook based crap schools offer today. Perhaps if schools offered more topics where kids had to work and think for themselves then they would see when and why they need to calculate things and come to you for guidance with the context of real projects.

Answer 16

Here a lot of subjects at university require quite a bit of math. Students who didnt pay a lot of attention during high school have a hard time and a lot fail. And it is usually a surprise for the students when studying majors like psychology, or archeology that they have to pass one or two math modules (mostly aiming at statistics, but you need also a bit of analysis when talking about statistical functions)

Answer 17

Why do people go to the gym to work out or take spin classes? Is lifting heavy weights or spinning a wheel on a stationary bike a useful skill they need to practice to succeed in their ordinary life? For most people, no. They do it, because going through those motions helps train their body to endure physical stress without wearing out. That helps them everyday, whether it's making a shopping trip to the mall, playing with their kids, going on a fun mountain bike trek, or just climbing the stairs to get to their office for work. They do hard physical things to make all the other physical things in their life easier.

Math education is the same way. Most people don't need to know how to derive a function, or calculate the cosign of an angle, or even know what those things mean. I'm a software engineer by profession with a Computer Science degree, so I had to take a lot of math in college. I can pretty honestly say I have not done any hard math other than some light algebra since I graduated almost 20 years ago. I have forgotten far more than I remember from school. But I am ever so thankful for the math education I got, because I know its there every time I do my taxes, or try to figure out if that discount at the store is really a good deal, or have to solve a hard logic problem at work. I don't use most of what I learned in my math classes, but because I went through the hard work of learning it, my brain is better trained to handle the everyday problems I do encounter. It makes everything in my life easier.

Answer 18

It Depends, do they want a Good job?

Consider this list made by US News and World Report of the 100 Best Jobs. Almost all of the jobs on this list fall into one of two groups:

1. Medical/Health Careers, and ...
2. Jobs that require A Lot of Math Skills

Go ahead, go through the list: software developer, statistician, financial analyst, etc. What do they all have in common? That's right, those all require lots of math knowledge and skills. And it doesn't matter if you use a different source or list, they all pretty much look like this one. Lots of medicine and lots of math. Take your pick (but you're not going to get into med school with bad grades in math either).

Answer 19

To be honest, if we're speaking of secondary school mathematics, most people don't need them in real life. Most people won't ever find a situation, where they would need them.

Most of the occupations that need math need nothing that isn't in the elementary school syllabus. And still, elementary school syllabus has things that have no practical applications for the very majority of people. Who needs trigonometry? For sure, you have needed them to setup the trebuchet, but even then, you've just used the tables. And for finance analysis, you don't even need to understand the concept of function to make the calculations, or type the formula in the excel.

So don't pretend to find out abstract examples that show how useful the stuff is in the real life. Pupils are not stupid, they have a good filling if something is reality near or alienated. Concentrate on the real positives.

First of all, studies. Math is a requirement for technical studies. Can a secondary school pupil know for sure he/she will never want to study on the technical university? Most likely not. Not learning math will take that opportunity away from them. It doesn't matter if math is really needed. I've studied information sciences, I needed to master math to pass the entrance exams and to pass first 2 years, I've never ever used stuff I've learned there in the 'real' life, but I would never get to that 'real' life without that.

Second, brain training. Mathematics are just the training for the abstract thinking. It's a great training, probably one of the best training that you can do in school. It's very hard to overestimate how hardly we need it in the occupations that doesn't directly need higher mathematics.

So if you're asked, why do they need to learn that stuff, ask them, are they sure, they want to loose the opportunity to choose every uni they want in the future, and to train their brain now?

Answer 20

You need applied mathematics to make reasonable life choices.

Talk with them about compound interest and payday loans. (And make sure that the example also includes some fees.) Talk about the reproduction number of a virus and what increasing or decreasing it by a couple of percent means, some months down the line. Talk through different mobile phone contracts, comparing prepaid with fixed costs. How many minutes do you have to use the phone per day/week/month to make contract X better than contract Y? How much will it cost you? Talk to them about the meaning of pre-election polls, exit polls, and election results, and how one would expect them to differ.

You need pure mathematics to understand and practice applied mathematics.

Using real-world examples might enhance the learning experience, but each example takes a lot of time and effort. Also, too many real-world terms might obscure the underlying pattern. Both the loan example and the pandemic example are exponential growth. But a real-world loan also has fees, and a real-world pandemic happens in a finite population, so neither of them is easy to calculate.

So show them the basic formula, and ask them if it helps them to understand the practical application better.

Answer 21

There is an extra dimension, because they might not be using mathematics directly. I am a big fan of knowing how and why something works, and there are many applications of mathematics they use every day but don't think about.

• The Adds they see when theybrowse the net are based on statistical algorithms, using many factors
• The phones they use to communicate work internally using binary numbers, so if we didn't have some of the mathematical concepts we do, they would have to still use paper strips and cards to use them
• The production facilities producing their clothes and gadgets most likely use CAM and CAD to function - also in essence based on counting.
• the games and such they play use maths to calculate all sorts of aspects - view angle, effects, etc - there is a reason that graphics cards are being hoarded for bitcoin mining - they crunch a lot of numbers.

I would play some games - like "find the maths in your hobby" - or "go looking for the mathematical aspects to every piece in your bag", or if should be more direclty available "how close can you get to the sum in your head when you buy groceries" or "where can we get to with this amount of gas/level of charge on our battery" - and "can we get home again".

It could also be stuff like the ratio of the sides of Ax paper being sqrt(2)and the possibilities that gives - if mathematic wasn't involved - we'd have a lot of different not related sizes maybe...

Personally I am a big fan of Presh Talwaker and NumberPhile on Youtube, and i like to be aware of the potentials, but i rarely get to apply it. appart from recently when I was building CAD models of a woodworking project. parameterization of angle calculations and side lengths is directly transferable. The issue is of course,that the students don't necessarily have that sort of hobby - yet. And when they get there later in lfe I hope they remember some of what they did learn in school.

You could also ask them to try to come to class only bringing stuff that has NO relation to mathematics at all - challenge them to prove to you that mathematics isnt' everywhere.

TL:DR

use analogies and real world examples of how the students' life would be without mathematics.

Answer 22

Have actual applications in class

Some examples:

At ISCED level 1 (elementary school, barneskole, alakoulu), maybe you can have pupils plan and budget for some kind of party or trip. There will be plenty of training on addition, multiplication, what prices mean, being systematic, digital tools (maybe a spreadsheet), gathering data (who are coming, are there allergies or other concerns) and handling uncertainty (someone might not come after all and we might not know how many cookies everyone eats).

At ISCED level 3 (high school, lukio, gymnasium, videregående) you might ask whether it is useful to have solar panels on electric cars.

Some related theories

1. Olav Skovsmose has written about landscapes of investigation (undersøgelseslandskaber), where mathematics tasks are classified along two axes: first is whether the task is open or closed, and the second is whether it lies within pure mathematics (what kinds of patterns are there in Pascal's triangle), semi-reality (Kauko is 32 years elder than Teija and next year Kauko will be five times as old as Teija will; how old are they?) or reality (the examples above).
2. Etnomathematics is all about the mathematics that a given group (maybe an ethnicity, or maybe a subculture, a profession or a hobby group) actually uses; the mathematical content, the expressions used, and so on. Often the pupils might do very poorly at school but fluently handle the same calculations in some other context. Bridging the gap and showing how the pupils actually already have lots of competency and do use what can be abstracted as school mathematics in their everyday life might be helpful, and then the abstract mathematics might bleed back in and help in their everyday context, too.
3. Some theories of interdisciplinarity (my translation from Norwegian tverrfaglighet, might not be the jargon actually in use in English) classify things according to whether the non-mathematical discipline is just mentioned (a farmer has 100 meters of fence and wants to fence off as large an area as possible; a straight cliff edge already works as a fence...), up to actual integration of the subject matters, like in the examples above.

All of these have the shared idea of not just touching upon some real-life context, but actually using it in a way natural to it, and seeing the mathematics there.

Answer 23

I point them to GH Hardy's essay: A Mathematician's Apology. Aside from any "applications", I got into Math for the beauty of it, not because of any "applications", so I try to show that to my students. There are a lot of things we do in life that may not have a utility of purpose, but are nonetheless enjoyable or beautiful.

As Hardy said: "Exposition, criticism, appreciation, is work for second-rate minds. [...] It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done."

Answer 24

Response to Students Who Say "This Is Not Important". Lately, my students keep telling me why what we are learning is not important.

Why not go full on Lakatos-ian, ask them "What is it that you mean by 'important'? Can you try to give a clear cut definition?". Then you, collectively, try to see what things that are 'intuitively' thought of as 'important' do not fit into the definition, and, dualy, what 'non-important' things do, and refine it

After some rounds of refinement, try to explain to them that this activity you've been doing mirrors, to some extent, an important (!) part of mathematical activity, that of precisifying intuitive notions, capturing as many 'essential' examples and barring as many monsters as possible. Tell them we have to start somewhere, and we start with numbers and shapes because, believe it or not, such abstracted things are much less complicated than basically everything 'out there in "real life"/"the real world"'

This may be hopelessly naïve, impractical suggestion, that can't be brought to bear in most places, times and situations, but it's something I would really like to see