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What are some good answers to questions e.g. "why do we need to study square roots"?

Of course the answer depends highly on who is asking. For the scope of my question, I have a student in mind, who is - given their interests for further education - likely never going to use square roots in adult life, but still has to learn them. (Of course my question is not about square roots specifically. It can be any topic which, realistically, will not be used "in real life" by the student.)

I have some answers, but I feel that they all fall short of being a satisfactory answer:

  1. "Because you have to know it for your grades, so later you will have more choice of what you want to do in life" - this is my honest answer, but I fear it might be a source of resentment for the student against the education system, which I don't want.

  2. Bring some forced example from "real life", e.g., "John has a son whose age is the square root of his age. John is 49 years old. How old is his son?" - I want to avoid these at any cost, as any student will see through these and will be enforced in his view, that "math is nonsense". Even if the example is better and I think more relevant "in real life", if it fails to resonate deeply with the student, then it will backfire as an example.

  3. "It teaches you to think abstractly" - I am deeply skeptical of some generic problem solving ability. I think we are good at solving problems that we practice solving, and it's not obvious how one transfers to another problem. And even if generic problem solving activity is a thing, it is hard to see why practicing square roots is the best thing to improve it, instead of something else that the student is more interested in. I think, hearing this answer, most students will nod along and still think to themselves that math is nonsense.

  4. "Distract from the question". The "why learn this" question usually arises from a thought process, that is just distracting the student. He finds square roots hard to understand, so he is looking for an excuse to think about something else. He never seems to ask the "why do this" question for activities that he otherwise just enjoys, e.g., reading a novel, playing music, drawing etc. and if asked "why", he will rationalize why it is a good thing to do. So anything that distracts from the "why question" and manages to bring focus back to thinking about the problem itself is good. (If you have any concrete strategies for this, I appreciate if you share it.) However: I still think, that the "why learn this" question is important and I would not like to ignore it completely.

  5. I am an applied mathematician myself, so I can bring real life examples where math (beyond basic arithmetic) is useful. But the student will likely not have the knowledge to fully appreciate it, so while he might be impressed momentarily, I don't think I can expect any consistent success. It certainly makes sense to try connecting the applications of math with school curriculum every once in a while, but honestly these are often very far away from each other.

Is there any good strategy to answer to the "why learn this" question, that I missed?

Did I make a conclusion above, which is absolutely wrong?

If possible, I am looking for answers which are based on at least some empirical evidence, but I am also interested in concrete examples where something has worked really well.

EDIT: thanks for all the contributions. Some further clarification about my question:

  1. I look for direct answers and not analogies. Math is not sports, art, or music. To me, such answers are essentially distraction (see point 4 above).

  2. Several answers are in the direction of "it's part of general knowledge". I see this as appeal to authority and as such, the nicer version of appealing to grades (point 1 above).

  3. Some answers are quite specific, e.g., lots of suggestions are about finance (which btw I personally find really boring). Some topics in math connect better with finance than others and having one "go-to example" isn't a really useful mindset when dealing with a concrete student and a concrete topic.

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    $\begingroup$ I think it's important to remember that one size doesn't fit all here. Different students will relate to different answers. $\endgroup$ – Amy B Nov 19 '20 at 10:18
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    $\begingroup$ I don't have a good answer but I find it naive to assume one would never need to be able to recognize square root relation between two quantities. Recognizing basic functional relationships like proporitonality, inverse proportionality, squares, roots, exponents is a huge perk in pretty much any nontrivial real life profession, be it art, management or advertising. $\endgroup$ – Džuris Nov 19 '20 at 14:21
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    $\begingroup$ Re, "they all fall short of...satisfactory..." But you have five of them. And maybe your question will elicit more from the audience. Don't underestimate the value of quantity. Sometimes a preponderance of "meh" reasons can be, in and of itself, a good reason. $\endgroup$ – Solomon Slow Nov 19 '20 at 17:40
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    $\begingroup$ If the question is why do "we" (meaning society) need math then the answer is pretty obvious. If the answer is why do "I" need math then the most honest answer (which I've given students, more or less) is "How do I know what you'll need in the future? The vast majority of people will never [insert mathematical procedure] so you most likely don't need to learn it. But maybe you will, and here we are in math class, so..." $\endgroup$ – Thierry Nov 19 '20 at 23:42
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    $\begingroup$ You need roots when doing such simple things as converting the quantities in a cooking recipe from one cake diameter to the other. Basic math is everywhere. $\endgroup$ – Polygnome Nov 20 '20 at 16:57

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As you said, it depends on the type of student you're dealing with. Some of them will have an interest in x, and you just have to make a connection to x. On the other hand, some students won't have enough perspective/experience to appreciate most answers you might give. I've had younger students argue with me about learning fractions and decimals. You'd think that bringing up money would quickly end the conversation, but you'd be surprised.

Sometimes the question is driven by psychology instead of curiosity/a need for motivation. You have to figure out where the question is coming from. Challenging the student with undirected open play will get you nowhere if the student is looking for a distraction as you wrote in #4. You don't always have an ideal student.

Anyway, there are lots of things that we learn that can't be appreciated in the moment. That has to be the perspective you should communicate. Why do you learn x, y, z in this other class? Is there a purpose to that? You're not always going to get instant gratification from everything that you learn.

Ultimately the student has to be ok with accepting something on trust. You can tell them "it teaches you think abstractly," but if they're already critical of the process, then is that going to be convincing?

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  • $\begingroup$ Thanks, I find this answer insightful. Especially the need for frequently connecting math to other areas of life, and using such questions as opportunities, "teaching moments" for patience and keeping up even without instant gratification. But it should be explained, how eventually, the expected future gratification is positive, no? $\endgroup$ – BKE Nov 20 '20 at 11:45
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    $\begingroup$ @BKE I agree, it should be explained. But there's only so much that we can explain. Education is a two-way street. The student should also have enough trust that the process will ultimately benefit them. Maybe I'm just going off the rails here. $\endgroup$ – Alex Gramatikov Nov 21 '20 at 2:38
  • $\begingroup$ I accepted this answer as I think it comes closest. There are many other insightful answers in this thread, however, unfortunately not among the highest votes ones. $\endgroup$ – BKE Jun 7 at 10:48
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I found that my former students (low achieving ninth graders in the U.S.) always responded best when I answered with this:

You enjoy watching sports right? Whether it's Football, Basketball, the Olympics, or whatever. Or maybe not sports, but you like watching a rock concert or orchestra. You pay money to see the famous athletes or famous musicians play in the championship or to play their concert. You don't pay money to see them lift weights in the gym or to practice their scales and arpeggios. But we all recognize how important practicing those things are: Practicing scales and arpeggios helps your ability to sight-read music because you pick up on similar patterns. Practicing sports plays over and over helps you recognize when it will be useful against what kind of defense.

Well, all of us sitting in the room solving for $x$ or graphing a linear equation, this is our version of lifting weights, practicing sports plays or arpeggios. This is not the useful stuff. We won't get paid money for doing this. But this skill we are doing will help us exercise our mind so that it can perform any future tasks on the fly. We are lifting weights in the weight room right now, so no one is trying to pay us for this. But this will help prepare you for different unexpected situations in the future.

I do also mention what Alex Gramatikov brought up: Why learn anything else in any other class? Each class has a set of key concepts or overall general learning goals that apply to life on the larger scale: History is about cause and effect and compromise, composition/literature is about writing and argumentation, science is about looking for evidence and testing hypotheses, etc. Each class has something useful to learn on the larger scale, but they use incredibly specific things (Like reading The Scarlet Letter, or learning about the structure of DNA ,or learning about the War of 1812) to convey those lessons.

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    $\begingroup$ Hi, thanks for the answer. I probably should have stated explicitly, that I am looking for direct answers and not analogies. Some of this also be US specific, eg. sports achievement doesn't matter as much elsewhere. No wonder, that students who are excited about eg. sports have a good reaction when, in math class, finally you talk about sports, and not math, but I wonder how much of the message really gets through. $\endgroup$ – BKE Nov 20 '20 at 10:02
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    $\begingroup$ @BKE "sports achievement doesn't matter as much elsewhere" [citation needed]. $\endgroup$ – Celos Nov 20 '20 at 10:20
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    $\begingroup$ In Europe for example, there's nothing like US secondary school and college athletics. Students enjoy doing and watching sports, but it's mostly extracurricular. $\endgroup$ – BKE Nov 20 '20 at 15:53
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    $\begingroup$ I see this answer as a mixture of points #1 (appeal to general culture as authority) #3 (appeal to some general problem solving ability) and #4 (distracting from the question with analogies). So it might be useful as an example for others to draw on, personally I find it very unconvicing and obviously to present it to others, one has to believe it in the first place... $\endgroup$ – BKE Nov 20 '20 at 15:59
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    $\begingroup$ @BKE If you need to believe the analogy to be able to use it yourself, then I guess you should start with why you think we should learn math. What do you believe? The problem is that the reason you think we should learn it probably won't resonate with all your students, so you need to convey multiple arguments, even if you don't personally believe some of them. $\endgroup$ – ruferd Nov 22 '20 at 7:22
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The closest I came to getting fired for something I said to a student. The student asked "When will I ever use this math in the future?" I responded, "Well, you won't, but the smart kids might."

In my opinion, the right answer, even at the high school level, is to ask what they might wish to do in the future. The well rounded education keeps all their options open. When I graduated High School, I chose electrical engineering, and that was my major as a college freshman. Now, I've seen curriculum that allows a student to not declare until junior year in college.

For those that are certain they'll not go into a STEM career and are willing to make that decision. I'd go down the path that there is a mathematical equivalent of illiteracy, called innumeracy. Similar to how one needs to read, directions on a medicine bottle, a simple passage in a magazine, there's a level of mathematical understanding that helps us make sound investment decisions. Helps us adjust a recipe when we have 75% or 150% of the required ingredients. I've made a habit that when a student says they hate math and don't get it, to talk money. Who doesn't like money? And who would be happy to get ripped off due to their own bad math? Granted, they may not need to use square roots, but we are graduating kids who can't calculate compound interest or balance a checkbook.

The made up word problems (your sec 2) usually seem contrived and uninteresting, students are eager to get problems they are likely to actually face in real life.

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    $\begingroup$ did you almost get fired for quoting smbc? $\endgroup$ – pplat Nov 19 '20 at 14:21
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    $\begingroup$ Well, I delivered the line as though it was spontaneous.... $\endgroup$ – JTP - Apologise to Monica Nov 19 '20 at 14:24
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    $\begingroup$ You know, that comic was specifically titled "Why I could never be a math teacher": smbc-comics.com/comic/why-i-couldn39t-be-a-math-teacher $\endgroup$ – Daniel R. Collins Nov 19 '20 at 16:09
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    $\begingroup$ 'who would be happy to get ripped off due to their own bad math? Granted, they may not need to use square roots' Well, there's working out the total (compound) interest that accrues during a six-month payment holiday, on a loan whose interest rate is expressed as an APR. That involves raising (1+interest rate) to the power $1/2$, i.e. a square root. $\endgroup$ – Daniel Hatton Nov 20 '20 at 9:17
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    $\begingroup$ @Discretelizard - on reflection, I regret writing that first paragraph. If I knew the comments referencing it would be removed as well, I'd happily edit it out. "Important" to me is a level of understanding that rises above innumeracy. To be clear, jokes aside, I'd expect a person to to be able to follow the directions on a medicine bottle to not poison their child, etc. Same as a reading level sufficient to fill out a job application. I am not from the "every student needs AP calculus in HS" camp. $\endgroup$ – JTP - Apologise to Monica Nov 20 '20 at 15:04
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Money

Most people will not need to compute the trajectory of a ballistic projectile, but everyone will need to deal with money to live in any advanced society. Furthermore, while many people can get by without doing any advanced math, it is pretty easy to introduce real-world examples which get hairy very quickly. Compound interest on credit cards are perhaps a simple example. Computing simple interest on a mortgage is also relevant to many people. I would go so far as to assume that the student will be sufficiently wealthy at some point that they will want to invest in the stock market, at which point you can introduce any number of concepts related to expected returns on various investments, options, etc.

Make it Personal

Just to drive home the point that being numerically literate is important, you could offer the student a small financial incentive. Tell them:

I have $5 (or a dollar or whatever) that I'll give you, if you can solve this math problem for me right here, with just pencil and paper, in 10 minutes or less. I promise it is a problem you will likely face one day.

If the student agrees, then you follow up with:

Suppose you are buying a house. A mortgage lender offers you a mortgage at 5% interest. You need $200,000 to buy the house. At the end of a 30 year mortgage, how much money will you have paid out?

Obviously, this is a pretty easy problem, mathematically speaking, but likely well beyond the skill of the typical "Why do I have to learn this?" student. Hopefully, they will get the point that math is pretty important. If they say: "I don't need to know that. There's calculators for that" then you can say:

How do you know you are using the calculator correctly? Most calculators allow you to solve the problem from multiple angles, by changing the interest rate, the monthly payment, the total cost, etc. How will you know the answer you are getting is even in the right ballpark?

Gamify

There's lots of directions you can take this, including estimating taxes on wages, payments for a car, etc. If you want to engage the whole class, then you can make a game where they have to compute some interest or such to get the best loan for a car, etc. Give them multiple loans for high-ticket items, multiple jobs that pay different amounts on different schedules (weekly, biweekly, monthly) and have different tax rates, and challenge them to "spend" their allocated "working hours" maximizing their income and buying the most things given your mini-market. Hopefully they will see that it's far more than a game, and the answers have real-world implications.

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  • $\begingroup$ How do you know you're not messing up with you own calculation? If you happen to have very different results than the calculator, which result would a "Why do I have to learn this?" student have in their own results vs the ones from a calculator? Up to a certain level, lots of things you learn in school are just general culture which might or not be useful in the future. $\endgroup$ – Laurent S. Nov 19 '20 at 12:58
  • $\begingroup$ @LaurentS. Ultimately the value of something depends on how you use it. Obviously, many people get by living incurious and sheltered lives. The point is to show that a curious mind will be able to better take advantage of opportunity. Being able to perform Fermi estimation is generally useful across every area of knowledge. But you have to solve real problems a few times to first convince yourself that it works as advertised. $\endgroup$ – Lawnmower Man Nov 19 '20 at 21:32
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    $\begingroup$ The point about calculators cannot be understated. I still remember how my peers in high-school were absolutely dumbfounded that I could do 2x2 matrix multiplication involving only single-digit integers in my head faster than they could on a calculator... $\endgroup$ – Austin Hemmelgarn Nov 20 '20 at 2:56
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    $\begingroup$ Calculator depends on question you ask. MLM 'works' as advertised unless when you consider than after 10 iterations we run out of the people on earth. I would add importance of statistics in politics/decision making. People lie with statistics and no calculator will help with difference between P(X|Y) vs. P(Y|X). And I don't think any calculator will help with Simpson's paradox or similar things. [There is also rule that something doubles every 72 / x years where x is growth in % which is accurate enough for many back of napkin calculations but much faster than any calculator] $\endgroup$ – Maciej Piechotka Nov 20 '20 at 10:56
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I think there is also a case to be made for "passive" uses. For any problem you encounter that you know is a math problem, you can always research and figure out how to do it. But there are situations that you won't necessarily realize are math related, but knowing math might help you. Getting scammed in pyramid and similar schemes is one such instance. Spotting lies is another.

For example, the movie Bloodsport is purportedly based on real events in the life of Frank Dux who claims, among other things, to have 56 consecutive knockouts in a single-elimination bracketed tournament. To someone without "advanced" math skills like square roots and exponents, this may seem plausible. To those with such skills, it quickly becomes evident why such a claim is, to put it extremely mildly, a bit unlikely

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    $\begingroup$ 56 consecutive knockouts in a single-elimination bracketed tournament --- Are you sure the intended meaning was that all 56 knockouts were in the same tournament? This sounds to me a case of slightly careless wording, where omitting the "a" and adding an 's' at the end of "tournament" would fix it. I see editing oversights like this all the time in math texts (unless written by Halmos). For example, Theorem 4.2.4 on p. 164 of Metric Spaces by Shirali/Vasudeva states: Let $O$ be a nonempty subset of $({\bf R},d)$. (continued) $\endgroup$ – Dave L Renfro Nov 19 '20 at 17:56
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    $\begingroup$ Then $O$ is a union of an at most countable family of open intervals, no two of which have common points. As worded, this allows for the possibility that some two of them, maybe every two of them, have exactly one point in common. The authors should have written "have a common point", not "have common points". $\endgroup$ – Dave L Renfro Nov 19 '20 at 18:00
  • $\begingroup$ David, since the tournament supposedly takes place once every 5 years, even if you spread it out over 4 tournaments spanning 15 years AND were willing to believe a fighter could stay in their prime long enough to win by knockout all their matches in all four of those tournaments, you'd still need 16,384 participants per tournament $\endgroup$ – Kevin Nov 19 '20 at 18:13
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    $\begingroup$ takes place once every 5 years --- I didn't know that. Although I've seen this movie at least twice (probably the first time when it came out at the theatres), it's been a while since I last saw it, probably over 20 years (via Blockbuster rental). $\endgroup$ – Dave L Renfro Nov 19 '20 at 18:34
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    $\begingroup$ +1 for using Bloodsport as an example. To this day I still chant "Kumite, Kumite, Kumite" when doing mundane tasks like chopping vegetables. $\endgroup$ – Thierry Nov 21 '20 at 2:09
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(This answer is specific to square roots, not to the broader question except as an example.)

I occasionally use square roots to make sense of the news. For example, suppose a fire has been reported to have burned 100,000 acres. How big an area is that, in terms that make sense to me? There are 640 acres per square mile, so divide by 640 and take the square root -- 100,000 acres is the area of a square 12.5 miles on a side. Now that's a description that gives me a sense of scale in terms of the world I live in.

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  • $\begingroup$ Thanks, I find this to be a useful idea in general. Understanding square roots is definitely one tool to help "see" the world. One thing is to look at a number that's written down, and another is to see what it means. $\endgroup$ – BKE Nov 20 '20 at 10:20
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Find questions which are actually interesting to the student. Here is just one idea:

Ask the students to make several rectangles with an area of 50. They might find $1 \times 50$, $2 \times 25$, $5 \times 10$, etc. Then challenge them to come up with some examples with non-integer side lengths. What are the perimeters of all of these rectangles? Can they figure out which one has the smallest perimeter? You might not be able to find it exactly, but see if you can make the side lengths as close as possible. What kind of rectangle is it?

I won't say that every student will find this interesting, but it is relatively undirected open play, and students can work with it at a variety of different levels of abstraction. Some students will write down algebraic expressions to help them, and some will just play numerically. At the end of the day, you will have many students with many different approximations for $\sqrt{50}$. Then you can introduce the concept to them, when it already has a natural application.

Of course, you can take any natural instance of the square root and try to devise a similar investigation. Another example would be the fact that you only need to check up to $\lfloor \sqrt{N} \rfloor$ numbers when checking if $N$ is prime via trial division (this is related, of course, to the first example I gave).

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    $\begingroup$ I agree, that raising interest is a good strategy. "Interesting" and "useful" are not necessarily overlapping and if we manage to convey one of them is already a success. On your concrete example, I don't think it would be particularly interesting or useful for someone who is not both already interested and knowledgeable in abstract math. "Undirected open play" as a tool can fail spectacularly at both motivation and learning. $\endgroup$ – BKE Nov 19 '20 at 9:14
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    $\begingroup$ @BKE I disagree with you. My experience is that even people who are "bad at math" can enjoy hunting for patterns, especially if they have the tools to start doing computational work immediately. Students who only have a basic understanding of multiplication and division can start working on this immediately by playing with their calculators. $\endgroup$ – Steven Gubkin Nov 19 '20 at 13:01
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    $\begingroup$ I don't think there is enough space to debate this here in the comments, and it's slightly off-topic as well. If you are interested in what I think, I would direct you to read section 3.3 of this book, which pretty much sums it up. In short, I don't think in general it would be the right approach, at least not at the exact time when such "why do we learn this" questions usually come up. $\endgroup$ – BKE Nov 19 '20 at 14:48
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    $\begingroup$ You want to grow 50 acres of wheat. Which shape lets you spend the least money on fences? $\endgroup$ – user253751 Nov 20 '20 at 17:02
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    $\begingroup$ @user253751 Indeed. Although in my real life, I want to grow a 50 acre perennial food forest, not wheat. $\endgroup$ – Steven Gubkin Nov 20 '20 at 17:08
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This is among the oldest questions about learning that has been asked.

When a young person started learning geometry with Euclid and asked him why he should learn geometry, Euclid replied, "Give him threepence, since he must make a gain out of what he learns."

I think you are overlooking some aspects related to your answer #3 about abstract thinking. Abstract thinking is not about developing problem solving skills per se in the sense of George Polya. It is about developing and maintaining plasticity of the brain, in order to reason with new information in unfamiliar contexts. Practicing square roots does not develop abstract thinking. But learning about them, becoming familiar with them, and then moving on to a fresh unfamiliar context such as inverse functions, perhaps eventually to the notion branch cuts exemplified by Riemann surfaces for $w=\sqrt{z}$ or $w=\ln{z}$, is a reward of abstract thinking--not the threepence Euclid gave to his student.

There is another reason why we have a requirement that all high school students will have an opportunity to learn some mathematics. In the USA, compulsory education became the norm in the late 19th century, with Mississippi becoming the last state to enact a compulsory attendance law in 1918. I think there are many good reasons for this expectation, including equity, equal access, and societal need. It is a scary thought exercise to imagine what would happen if we decided to make mathematics in high school be an elective on the par of personal finance or woodworking. If we allowed high school students to opt out of learning mathematics and science, we might not have enough people in educational pipelines to sustain societal need for STEM. Maintaining such pipelines is a reason why society invests so much in public education. Of course such a reason is not likely to satisfy a reluctant learner. But having to teach reluctant learners is a bargain we make with compulsory education policy.

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    $\begingroup$ You know, I wonder how interest in STEM education actually pans out in countries where education is not compulsory. $\endgroup$ – James S. Cook Nov 19 '20 at 4:56
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    $\begingroup$ I didn't know they had pennies in Ancient Greece. :) $\endgroup$ – Llewellyn Nov 19 '20 at 18:56
  • $\begingroup$ @LLewellyn: this is an amusing translation issue for sure. In ancient Greece, jurors (Dikastes) were paid "three obloli" or triobolon. Maybe there is hidden humor in Euclid's suggested payment: mapping (teacher,student) to (accused,juror). $\endgroup$ – user52817 Nov 19 '20 at 19:18
  • $\begingroup$ @JamesS.Cook Are there such countries? $\endgroup$ – pipe Nov 20 '20 at 16:07
  • $\begingroup$ @pipe: There are a few third-world countries. But another way to look at this is by NAEP data for the USA. Only about 60% of grade 12 students are at or above NAEP-basic for math, in spite of compulsory education. nationsreportcard.gov/mathematics/nation/achievement/?grade=12 This is not enough to meet STEM demand, as evidenced by the need to increase the quota for H-1B visas, and the persistence of NSF programs to improve STEM education. $\endgroup$ – user52817 Nov 20 '20 at 18:45
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I disagree with the idea that any student can be assumed to “never need” square roots, at the point where they are learning them. At that point the students are simply too young for us to make assumptions about their career paths.

And while we can go back and forth about the particular utility of square roots, they’re part of a basic canon of math. It’s the kind of math you will always be assumed to know when studying anything at all advanced, and therefore has utility if you use math in any major capacity in your life. (And in order to teach someone a subject at all you have to believe there is some use in it...)

I’m not sure how exactly how to package that for a student, but perhaps that is a start towards justifying it.

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    $\begingroup$ It's almost like the whole posture of certain modern students towards education is wrong. Where does this posture come from ? Why is it not realized that education is a luxury of our modern existence ? Why don't more students adopt a posture of thankful curiosity rather than entitled ignorance ? These are questions we ought to ask... $\endgroup$ – James S. Cook Nov 19 '20 at 6:35
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    $\begingroup$ @JamesS.Cook I do vaguely remember some slightly pre-Christian rambling by one philosopher or other about this exact issue :) It's a (non-)problem as old as humanity. I'm pretty sure there was a hunter-gatherer who complained about having to learn to toss a spear buried under the sands of time somewhere. $\endgroup$ – mishan Nov 19 '20 at 8:07
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    $\begingroup$ "students are simply too young for us to make assumptions about their career paths." - Exactly, I made this point in my answer as well. Why limit your future career choices while still in HS? $\endgroup$ – JTP - Apologise to Monica Nov 19 '20 at 11:24
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    $\begingroup$ @JTP-ApologisetoMonica By not goofing off and drawing they are limiting their career choices as an artist, by not skipping class and going to the gym they are limiting their career choices in sports, etc. Every choice you make limits your career choices, and one cannot do everything. $\endgroup$ – Steven Gubkin Nov 19 '20 at 13:34
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    $\begingroup$ @JTP-ApologisetoMonica I also find the number of answers which focus on money and the economy to be deeply disturbing. Framing mathematics as primarily a tool in service to economic growth is gross to me. I want it to be a tool of intellectual liberation. $\endgroup$ – Steven Gubkin Nov 19 '20 at 14:39
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Things taught at school can be divided into three rough categories:

  1. Key skills. Examples: reading, basic arithmetic.
  2. Knowledge that should leave a trace, causing someone to ask questions they wouldn’t otherwise think of. Examples: “If I’m thirsty during my camping trip, will I be able to drink water from the river? I vaguely remember something about microorganisms, I better look this up.” “My shoelaces constantly come undone, I think someone told me it matters which loop goes on top?” “Can I get pregnant before marriage?”
  3. Specialized knowledge that’s really best acquired at places other than secondary school, and only by those who are interested in such topics. Examples: accurate names of organs of a cockroach, exact dates of historical events, coding skills.

Square roots in particular are somewhere on the boundary of 1 and 2. It’s easy to imagine quite practical questions which require this knowledge, e. g. what is the longest rod that will fit into this box, or the relationship between the weight of a pizza and its diameter etc.

However, I have a feeling that if you’re confronted with this question, you’ve already lost and your students already consider mathematics a grudge, looking for excuses to avoid working on it as much as possible. Therefore, the best way to address the question is to prevent it from arising. Great amounts of literature have been written on how to teach mathematics in a way that it feels like playing and not like performing tedious calculations—not that many schools seem to implement any of it.

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  • $\begingroup$ Thanks, I think you have a really good point here. I agree, that preventing it is important, and a good way to achieve this is to present math in a way that is frequently connected with other areas of life. Would you mind sharing some starting points to the literature you mention? $\endgroup$ – BKE Nov 20 '20 at 9:54
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Following on from Lawnmower Man's answer, I think money is a very good example: one that people are familiar with and understand the importance of.  But you don't need to go as far as mortgages to see the need for powers and roots.  Perhaps an example that students are likely to encounter much earlier is credit card interest.

For example: Credit card A charges 50% APR, while credit card B charges 4%/month.  Which is more expensive per month?

(A naïve calculation might be: ‘50 / 12 ≈ 4.1667 > 4, so A is more expensive.’  Which is of course wrong.  To get the right answer, you need to understand percentages well enough to be thinking of factors of 1.04 and 1.50 — and then understand exponentiation well enough to take the twelfth root of the latter.  Giving it a monthly interest of ~3.4%.  This may come as a surprise to people unfamiliar with compound interest…)

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  • $\begingroup$ deleting my comment.... no evidence of you testing whether we were paying attention. :) $\endgroup$ – JTP - Apologise to Monica Nov 19 '20 at 14:23
  • $\begingroup$ That "naive calculation" is the leading-order binomial approximation to the exact calculation involving the twelfth root, but I'm not sure whether we could make any pedagogic use of that fact. $\endgroup$ – Daniel Hatton Nov 21 '20 at 23:54
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    $\begingroup$ Seems like one could solve this without roots: take the B value (1.04), raise it to the 12th power, and compare to the A value. Whichever one has the higher annual rate likewise has the higher monthly rate. $\endgroup$ – Daniel R. Collins Dec 14 '20 at 3:55
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Many/most of the comments and responses have assumed mathematics to be entirely in the field of hard sciences. I think that there is an argument to be had that mathematics as a field is also an art.

Mathematics is beautiful. It is frequently located in the Arts and Sciences departments of colleges and universities. Other specific art subjects (e.g., painting, sculpting, sketching) are hardly ever criticized with the phrase "when will I ever use this?". There are elements of every part of life that are more tedious and banal. Then there are parts that are invigorating and exciting to explore. These feelings are largely subjective to each person.

I think that part of being a human being is being able to find the beauty and enjoyment in the tasks that are otherwise tedious and banal. There is beauty in square roots in the same way there is beauty in paintings. Whether they're displayed in a museum or on your refrigerator.

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    $\begingroup$ While I agree with the sentiment generally, I remember vividly at art class, wondering why I have to draw ugly apples, and at music class, why I have to learn some songs that I dislike. $\endgroup$ – BKE Nov 20 '20 at 10:10
  • $\begingroup$ Yeah, this is just not correct. Any class that students are required to take will produce complaints of "when will I use this?" And if your argument is some poetry about the human soul, then their complaints are unanswerable. As you say, these feelings are subjective. The reason why math is required of all students is because it's indispensable in the hard sciences, which are indispensable to modern society. That's it. $\endgroup$ – Kevin Arlin Dec 14 '20 at 21:21
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I, my kids, and my grand children have all had toys where shaped blocks had to be placed inside a box with various shaped holes. None of the blocks would fit through a hole of a different size.
Block Shape Toy W also had large colored buttons with holes in them. And colored string to feed through the holes.
Colored Buttons

AS an adult, I can say that I have NEVER had the need to perform either of these tasks as an adult. Not have my children. Why, then do these toys exist?

Because they help develop basic skills that we will need later in our life. Babies have no idea why it is important to learn to place a square block in a square hole, and not a round hole. They cannot possibly understand the value of this activity. Their elders do.
Math is another form of skill development. It is a mental skill, not a physical skill.
Driving an automobile requires math skills, even though we don't use a calculator. We still have to estimate distance, speed, etc. Knowing if the cashier at the store is ripping you off is a math skill. After I pay my bills, how much money do I have left. If I invest money at x% interest, how much will I have .

That is why we learn math -- all type of math, in school.
$0.02

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  • $\begingroup$ Hi, thanks for the answer. I probably should have made it explicit in the question: I am looking for direct answers to the question, and not analogies. $\endgroup$ – BKE Nov 20 '20 at 7:57
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    $\begingroup$ @BKE The answer to the student implied in this answer to you is: "Because of reasons that I cannot articulate to you due to our large difference in experience. <present analogy with toddler toys>. You can see how those skills are useful in a way the toddler cannot understand even if someone describes the reason to them, so I ask you to extrapolate the situation with your understanding of the toddler's tasks' value, to my understanding of your tasks' value, even though I cannot describe it to you". $\endgroup$ – Luke Griffiths Nov 20 '20 at 23:20
  • $\begingroup$ But the analogy falls apart easily: toddlers play with such games instinctively, without being instructed or taught. This is unlike how most people approach math. And why make up analogies, only to say, essentially, that "just take my word for it?". $\endgroup$ – BKE Nov 20 '20 at 23:39
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    $\begingroup$ All analogies break down at some point. To be honest, there is no answer to the question. When a person asks that question, they have already decided that they do not want to do it. All you can do is try to add healthy dose of logic and reason to the "mud" that has formed in their brain. Maybe, it will take hold. Maybe it won't. $\endgroup$ – Scottie H Nov 21 '20 at 1:06
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You have 2 packs of 34 paving slabs. You want to lay a square patio. How many slabs wide and long can your patio be?

You are building a shed. The wall is 6 feet tall, and 8 feet long. If you need a diagonal brace for reinforcement, how long does it need to be?

You have a cupboard that is 42" wide, and 25" tall. Can you fit a 48" TV in it?

Do you have enough ribbon to tie around that present? Is your ladder tall enough to clean the gutters? Will that new wardrobe fit up the stairs to your bedroom? Will the party bunting reach between the ceiling beams? What diameter of cake tin do I need, to ensure that it isn't too thick or too thin?

Moving away from just square roots: If 1 18" pizza costs the same as 2 12" pizzas, which gives you more pizza? How much tax do you need to pay, on purchases or income? Which savings account or mortgage is better value?

Which equipment or attacks are better in their video games? Fixed or percentage buffs? Big hits, or Damage-over-time? Guaranteed small numbers, or a chance for a big number? Should they increase Critical Chance, or Critical Damage?

Finally: there are a lot of people and companies out there who make a lot of money out of other people being bad at maths: do your students want to be their suckers and victims?

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    $\begingroup$ Thanks. These are good examples (better than what my forced examples in the question). However, also a bit desperate. In real life, eg. you most likely just go and measure the diagonal of the cupboard etc. without any calculation. I am just wary of overusing them. There are many reasons NOT to use them 1. when it's artificial 2. when students aren't yet familiar with the abstraction, decoding and solving in one task can be too much to ask 3. students get defensive about yet another "real life example" that only the teacher finds interesting and practical. $\endgroup$ – BKE Nov 20 '20 at 16:34
  • $\begingroup$ I was about to give a very similar answer. I feel the natural motivation to study squares or square roots comes from its role in the computation of areas. And of course, the Pythagoras theorem. $\endgroup$ – Divakaran Divakaran Dec 14 '20 at 18:22
  • $\begingroup$ Upvoting for the pizza example, which I use all the time, along with "If a 12" pizza serves 2 hungry teenagers, how many pizzas do I need to order to serve the 27 of you in this classroom?" (Convert inches to cm as appropriate, i.e. basically everywhere in the world except the U.S.) $\endgroup$ – shoover Dec 16 '20 at 2:36
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Something I heard a few years ago. Add this to your arsenal.

"Will I ever need to use algebra?" You may not go into STEM. You may still get a white-collar job, as an office worker. (Most people never become an NBA player or a movie star.) The office may use spreadsheets to keep track of things.

Each office has one person who knows enough algebra to program their spreadsheets. The others just enter the numbers, and the spreadsheet computes everything automatically. So it may be true that the workers in that office do not need to know algebra.

Now imagine they have a job opening. After interviews, they end up with some qualified candidates. They all have similar credentials and experience, except that one can program spreadsheets. Which one will they hire?

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Because many of the cool jobs need maths.

Want to build robots? You need maths.

Want to become a demolition expert? You need maths.

Want to design fast cars or jet planes? You need maths.

Want to design rockets that will take us to Mars? You need maths.

Want to build computer games? You need maths.

Want to make lots of money on the stock exchange or sports betting? You need maths.

Want to research black holes and supernovas? You need maths.

Want to design buildings and bridges? You need maths.

Want to make your own AI-based human-to-animal face filter for Tiktok? You need maths.

Any answer to this question has to be exciting and inspirational. You're not going to motivate anyone with reasons like 'to allow you to get a job as a spreadsheet drone', even if that's the most likely actual reason for needing to learn maths. High-school kids haven't yet cottoned on to the fact that they'll most likely live a mediocre existence as adults. At that age, they're absolutely sure they'll become a rock star, the next Picasso, a professional athlete or the Prime Minister. Arguments towards pragmatism won't stick.

It's important to give kids heroes of a sort. Not just the intellectual giants of the past, but people who, with just a little bit of maths and a lot of gumption were able to do something cool. Introduce youtube channels like SmarterEveryDay and games like Kerbal Space Program and Poly Bridge if they're not already aware of them.

There are plenty of really cool vocations out there that need maths that these kids will not even be aware of. It's also important to note that they don't need to be Mega Maths Geniuses to have even a chance of getting these jobs, that even a little bit of basic maths knowledge will help them out down the line.

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  • $\begingroup$ Thanks. I also think the "infotainment" sources you linked can be useful. I think, appealing to "fear of missing out" on the cool stuff can be a powerful motivator. But I also think it's a double edged sword. There is plenty of opportunity for people to be involved in cool tech in your examples, without excelling in math. So need to be careful not to actually give the impression, that they have no chance in eg. the gaming industry and not paint an unrealistic picture about the possible future opportunities, in our efforts to get students get engaged with our subject. $\endgroup$ – BKE Nov 20 '20 at 11:17
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Although there are many good answers, I want to respond specifically to point 4 that you made there, that you believe the question "why learn this?" is distracting the student from actually learning.

I want to argue that learning "why learn this?" is actually more important than learning the topic.

I was always terrible at maths when I was young. I got a U in my Mathematics GCSE (that is, I did so badly I was "unmarked", I basically sat an exam and was so bad at it, I might as well have handed in a blank paper).

It was only when I went to University and really knuckled down to learn maths for my dream career (programming) that I really started to understand it. I was learning things that were so much harder than I started with, because I wanted desperately to learn it. Sadly, I was still as good at math as a sack of potatoes.

It was only when I begged my professor to sit down and explain to me why I needed to know these things that I finally started to learn. I started learning from the top down, not the bottom up. My professor started by explaining that finding the Hypotenuse of a Triangle is why you cut across the grass of a park rather than walk down one side of it and then turn. How knowing the square root of a number lets you write up a grid for that thing with equal rows and columns. How I can use maths to draw shapes on a computer. He explained to me in real terms what it means to use this particular parts of mathematics, then I understood why my answers were wrong, instead of just being told my numbers weren't right and to try harder next time. Thankfully, programming gave me context on how things could be used. Programmers use a lot of maths. Other people's lives won't need that.

All of that is to say: the "why learn this?" question can come from at least two places. It can be an excuse to not put in the effort as a "I won't need this anyway" response, or it can be the exact motivator to learn the skill in the first place. In the last 8 years since I started University, I haven't once learnt a mathematical technique I didn't understand the "why" of. Not one single time.

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  • $\begingroup$ Thanks. What I take away from this, is that if a student expresses the "why learn it" question, it's better to try to make this into asset, and build on that, for the long term. $\endgroup$ – BKE Nov 20 '20 at 14:22
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Most answers here (with the exception of @AndrewSanfratello's) focus on finding a practical purpose of mathematics (money, brain training, problem resolution, finding a job, keeping all doors open, etc.).

These are all a posteriori justifications of mathematics. Mathematics have not been "created" for money or to find a good job. Mathematics, history, arts, physics, philosophy, etc. is what make mankind.
It could be a definition of the difference between people and animals: We do practice them because we want, because we're curious, because we ask questions, because we imagine, because we observe, because we don't understand, because we have something to voice. What's the purpose of learning "History"? "Latin"? "Music"? "Physics"? At the end of the scholarship, only a very few part of everything you learned is "needed" in your work and day-to-day life.

You could live without knowing anything about the history of mankind, or without any hint of imaginary numbers, sure. The point is to show that this would be a poor living, and there's something bigger that can give more meaning to our lives, because we can think and create.
Obviously, no one will be an expert of all fields, and most will forget almost everything about imaginary numbers, or the art of Dostoevsky, or the replication of a human cell. But anyone will know that it exists, anyone can focus on a given topic if they want. Anyone could freely think of it later in their life. Anyone has thoughts in common with the rest of the mankind.

The truth is: we (try to) teach children with basics of everything important for mankind. We try to educate them as intelligent people, not as animals who just try to fill their instinctive needs. We want to show them the beauty of the thought, and the greatness of eminent people who came before us.
My suggestion is: you could simply explain this bigger-than-us truth instead of looking for low-brow explanations that will give them ground to easily counter-argue.

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    $\begingroup$ Thanks, I sympathize with this mindset. However, effort spent on one thing is effort not spent on another. So this argument isn't particularly helpful to justify effort spent on any particular topic, as it is equally valid for all topics. However, it is a really good argument to increase willingness to learn in general. $\endgroup$ – BKE Nov 20 '20 at 14:31
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    $\begingroup$ If the question from the student is "Why would I learn maths while history/music/biology is much more interesting", I'm afraid you can't say too much, unless you're ready for an argument with the history/music/biology teacher! This argument would however be a good starting point for another question somewhere on SE (probably not on ME if you want objective opinions): (why) are mathematics superior to other school subjects?! :) More seriously: while I highly prefer STEM subjects over arts, I cannot say musicians or philosophers understand our world even less, or don't contribute to mankind. $\endgroup$ – Evariste Nov 20 '20 at 16:50
  • $\begingroup$ Yeah, I also find a lot very sympathetic here, but I think in the end it's irrelevant to the question, which is about why students should be required to learn fundamentals of math. The beauty of the fundamental theorem of algebra, or whatever, is irrelevant here, just as "just look at the achievements of Beethoven!" is not much of an answer to a student wondering why they need to learn to play Hot Cross Buns on a plastic recorder. This is much better as an attempt at explaining what the goal of a secondary education ought to be--but that's at a higher level. $\endgroup$ – Kevin Arlin Dec 15 '20 at 5:46
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You learn math to build on top of those before you

We only get so many years in our life. The best of us barely scrape together a hundred of them. So there is something to be said for building on top of the work of those before us.

If you want to dance, you can always create your own dance. Indeed, some would argue that is the truest way. But typically we build on those before us -- those people who built and shaped ballet, tap dance, ballroom dancing, hip hop, and more. Every one of these has a language associated with it, and every one will point out that it's not reasonable to get good at their art without practicing so that you understand what is meant when people use terms like a pirouette.

If you want to play a sport, you can always pioneer your own way. Indeed, the games have been improved by this thinking. But typically we build on those before us -- those people who built strategies and practice regimes. We turn to a good coach. Every sport has a language associated with it, and every one will point out that it's not not reasonable to get good at their sport without practicing first so that you understand what people mean when they use terms like zone defense.

Likewise, there are many things where, if you want to get better at them, you turn to a physicist or an engineer. We have our own languages, and they happen to be heavily steeped in mathematics. If you want to get good at things they do, or leverage their words to make your life better, you will need to practice their art, just as one would practice a sport or dance.

So if a direct answer is what you seek, the direct value of learning mathematics stems from one's ability to tap into the work done before us which is written in that language. What work you find of value is really a person-by-person thing. Some might find value in seeing the equations behind mortgages rather than just signing the papers for one. Another might appreciate the ability work on the undercarriage of their car safely because they understand geometry. But the common thread in all of them is the need to be able to take advantage of the works of literally thousands of years of thinkers.

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There has alreadby been several good and thoughtful answers, but I think what it boils down to is that maths is always useful, and most maths up to highschool level has many practical, everyday uses that others have already mentioned.

It is also intensely boring to a teenager (unless you are a nerd, like me). At that age you are always questioning authority, and to make it worse, you have already had years of boredom imposed on you in the form of learning how to spell, add, multiply and so on, so it is no wonder that you are tired of it; but I don't think it is likely that you would find any teenager, who regrets being able to read and write (thus being able to play computer games, for example).

The trick to learn boring things is always the same: keep at it, don't think about why, pace yourself with small, achievable targets in a daily routine, and celebrate once you done your duty for the day. It's like washing dishes: Not a bundle of joy, but you get through it.

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You'll need square roots for high school chemistry and physics. If you never take those classes, yes you dont need the roots. If you do, you will. This answer is pretty generalizable. Math is a tool of science and engineering. You can't even be a nurse without some of this stuff.

And by tool, I mean a tool of the classes, not the practice. I worked as a mechanical engineer and it was all drafting and selecting pumps and valves out of catalogs on the job. But the courses required math.

If you are going to do a trade, then you don't need that stuff. And dont need about half of high school. Let alone college. Then again our society way overemphasizes school. Yale or jail fallacy.

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You'll probably only use ten percent of the things you study, so of course it would be much more efficient to concentrate on just that ten percent.

Pick now and limit your options forever.

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