The feel that the study of modern Mathematics has far too much focus on real-world applications and job-world applications (Relevance is subjective). I understand this may sound a little unorthadox, but please hear me out.

Me too! That’s the premise of this series. Engagement can arise not just from relevant / real experience (whatever that means) but from the cognitive conflicts that propelled the history of math itself.

The feel that the study of modern Mathematics has far too much focus on real-world applications and job-world applications (Relevance is subjective). I understand this may sound a little unorthadox, but please hear me out.

There is a beauty in Mathematics; learning for the sake of learning. It is through this beauty and aimless curiousity that we discover the unknown. We develop skills that appear to do not have an explicit use, that may (or may not) have a use in future.

To put it simply, studying Mathematics for life/job skills is setting the bar too low.

My wife has worked at the LHC in CERN, Switzerland during the search for the Higgs Boson. When people ask “What can we use it for when we properly understand it?” the answer is simply “We don’t know”. We dont know YET. Without the aimless curiousity, the search would never have begun.

Through teaching abstract Mathematics in my classroom I (along with yourself and may other teachers) am given the question “When will we ever use this?”. I tend to reply with the question “Why do boxers skip?”

Pupils will begin to say boxers skip to make them fitter and better at boxing. I say that I have NEVER seen a boxer skip in the ring nor have I seen a pair of skipping ropes anywhere near a ring during a boxing match, so boxers are wasting their time with skipping ropes. The pupils are very quick to defend, saying that they build muscles and agility. I can them summarise by saying that they are doing so to build a skill to apply to unfamiliar situations. Then usually finish with an incredibly cringeworthly line like “Mathematics is skipping for the brain”.

To summarise, the abstract, non-blatantly applicable elements of maths should not be seen as the enemy, but instead should be encouraged to enable our learners to become more broad, open-minded thinkers.

I really enjoyed this post and think and it as a lot of good/enjoyable stuff in it (and in the comments section) and I hope you have time to read/reflect on my (lenthy) reply.

* The first has a vertical asymptote
* The second has an x-intercept and a VA
* The third has an x-intercept and two vertical asymptotes

One question is: how do these intercepts and asymptotes interact? By this I mean, can we look at (say) graphs of the three addend functions and figure out how their sum will behave?

As Howard points out (comment 14), the answer is sensitive to the coefficients involved. If we multiply one of the three addend functions by a nonzero coefficient, this won’t affect its x-intercepts or vertical asymptotes, but may affect the x-intercepts and asymptotes of the sum.

This means that visually inspecting the addends can give us a hint of what to expect from the sum, but may not be conclusive. If we want to be sure about the behavior of the sum, going through the algebra seems indispensable.

I’ll admit this is still abstract–why are we interested in those graphs, or the sum of those three functions, in the first place?–but using algebra to check or clarify graph-based hunches is a lot more interesting than just doing algebra for its own sake.

Some related questions:
* For what numbers A, B, C will the function
f(x) = A/(x+1) + Bx/(x-6) + C(5x-2)/(x^2-5x-6) have exactly 1 vertical asymptote?
* For what numbers A, B, C will the function
f(x) = A/(x+1) + Bx/(x-6) + C(5x-2)/(x^2-5x-6) have no vertical asymptotes?
* Do the above questions get easier if we write 5x-2 as a linear combination of x+1 and x-6?

One could argue the actual skill of being able to do this simplification by hand is not necessary to teach. But this skill is just one piece in becoming proficient in algebraic manipulation. I think it’s a stretch to look for “cute” applications that make us feel as if it’d actually be used in some profession. It seems to me that someone taking this line of argument really means to argue that not *everyone* will need to be an algebra expert – and of course this is true. But in a world full of critical jobs (e.g. engineers, analysts, computational programming, bioinformatics researchers, etc.) demanding computational precision, it’s important that people in such positions be experts in employing algebraic manipulation and can confidently verify that their work is precise. Positions where creativity and higher order thinking are required are not positions where one can spend life “plugging in” to a computer, getting a mysterious output, and leaving it at that. For instance, a researcher in signal/image processing needs to apply higher order thinking to make novel progress – he/she must know what a Fast Fourier Transform does rather than just how to apply it to examples. And how can one expect to advance the vast modern applications of linear algebra without being an expert in elementary algebra? Though these people might not literally be simplifying rational functions by hand, they do need algebraic expertise for which simplification is foundational.

Presumably, we don’t want to track children into certain jobs early in life. So we need to be realistic that students not going into professions requiring such algebraic proficiency will end up having to spend time learning them along side those that do. We all have to learn skills in the other sciences that we don’t use; I had to learn how to analyze NMR spectra in organic chem and about various biological processes in high school but I ended up in mathematics and will never need them for my “real life.” Yet we understand that these are important because many important professions exist for which having such skills and foundational knowledge are necessary. Mathematics is a language which abstracts the real world and explores human thought and this abstract nature is distasteful to many people. In my opinion this is OK! Teachers don’t need to force all students to like it or believe it will be useful to them individually – because it won’t. But we should teach about the existence of important professions that require expertise in mathematics (particularly algebra) and hopefully fuel an appreciation from students that the physical sciences seem to earn so easily.

Obviously this is never going to be used in the real world. It’s an idealised problem designed to test a couple of specific concepts, in a way that can be marked consistently.

We are testing students with these problems to see if they have the right combination of mathematical ability and dedication to study. Those that do will get entry into those subjects at university.

Ordinary people, not intending to become engineers or scientists, should not be doing these in the first place. I advise students who don’t need this sort of thing just to avoid doing them — there are easier marks elsewhere that they are better to focus on. Even if you provide the “aspirin” that makes them want to do them, they will just get bogged down in technical errors.

I usually use something like this:

Our football team is doing a teeshirt fundraiser. We can get the shirts wholesale for $15 each. Getting our team logo printed on them is a fixed price of $75.

So how much is the cost per teeshirt?

The kids originally set up the linear equation–then I point out that this is total cost, not cost per tee. They usually work a few numbers and then realize it’s the linear equation divided by x. Then they run some numbers, realize that the more shirts bought, the less the price per tee. We talk about why–the $75 is fixed, so it’s allocated to each shirt. The more shirts, the lower each shirt cost–but of course, the greater the total cost. Also, no matter how many teeshirts, the cost can’t get below $15.

So then I show it as two graphs (15x + 75) and x. Then we graph the division.

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At that point, I usually just skip on to practice (like you’ve outlined above). But this year, I’m thinking of going deeper.

Say you’ve got multiple types of shirts. Pick one baseline–the boring shirt. Then you’ve got hoodies, performance tees, sweatshirts. Each one of those have a different per-tee price and maybe different printing costs. So you decide how many hoodies & others you’re going to buy in relationship to the baseline. twice as many, 15 more, whatever. So if hoodies were $25 each and printing of $200, and you were going to buy 15 less than the shirts, it’d be (25x+200)/(x-15)

Then you could say something like: well, we want people to buy two or three shirts, not just one. We know that our breakeven price changes based on how many shirts we buy. How ambitious should we be? How much can we maybe convince people to buy more shirts by pricing attractively? How will adding these functions help us find optimal pricing?