Imagine a world in which tennis wielded enormous analogical power. In this world, by making the balls, racquets and nets represent different factors, players are able to predict the value of an investment or work out how much concrete is needed to build a swimming pool.

Tennis coaches would constantly be told that they had to make tennis relevant. Instead of merely teaching their students how to manipulate racquet and ball in a meaningless and abstract fashion, they would be urged to constantly relate tennis back to mundane everyday applications.

Imagine a student of tennis who is struggling with his serve and, as a result, has become demotivated. Instead of working on the serve until the student starts to feel a sense of success, the coach is urged to make the process of serving more relevant to the student’s everyday life, perhaps by relating it to popular music or a trip to the local mall.

Imagine this were true for all sports. Imagine, before playing a game of cricket, all the players had to agree on what the wickets represent, or before a game of football, they needed to decide on the meaning of the ball and the goals.

This is the role of maths in our world. It is a victim of its own predictive success. Maths is rarely left alone to just be itself, it always has to represent something else. Experts line up to condemn the abstract performance of mere procedures. If students are struggling at maths then this is because their teachers haven’t made it relevant enough.

I say we set maths free to just be maths. Children are actually pretty good at understanding abstractions when it comes to sport or music – but what do the notes actually mean? – and they are equally good at understanding abstraction in maths. What they sometimes struggle with is the maths. Let’s focus on that.

I would point out one important difference here though: tennis is not a compulsory subject at upper secondary level.

The need for a high level of abstraction if maths is to be mastered properly is clear, but I do think that in some cases it’s worth pausing now and then to point out the practical applications of, say, differential calculus (of which there are very many, of course). Only now and then. But there are some teachers who never do this at all – I don’t think this is ideal.

Do you have an example of a practical application of differential calculus? That is a full example of a problem that people actually solve in practice using high school calculus.

My point is that these are hard to come by so the value of high school calculus must be something other than you will end up using it in practice.

Well, it’s not my field so I’m not the best person to give you an example, but isn’t DC used (for instance) to determine the amount of material needed for various types of construction? That’s what we were always told when I was at school, anyway.

I take your point that the examples might end up being even more abstruse and confusing than the abstract mathematics involved, and I’m very much in Greg’s corner about the “tyranny of relevance”, but I think that at the same time it’s helpful if the kids are given at least a hint of the practical applications of such things, *especially* in a core discipline such as maths.

It is very hard to justify most of the math curriculum based on the need to be able to do the math in actual jobs. Since the advent of high end handheld calculators in the 1980’s the need to do much of the math is rare. Take working out integrals, there anyone who needed it in a job for the last 200 years would have a reference book. Today CAD tools will take care of any calculation of materials required.

If someone is telling math teachers to motivate by relevance to stuff students will use they had also better tell them to teach trig integrals well before anyone introduces critical thinking.

What happens is often tortured examples that are neither real or interesting. Motivating by application should take the opposite approach. What is a problem students might actually be interested in that requires this math? I recommend Randall Munroe’s (the XKCD guy) What If for a pile of fun examples. He doesn’t show the math where it involves calculus but if it is not already answered on math.stackexchange.com or physics.stackexchange.com someone will help out.

I think that for the majority of students a deeper understanding of a smaller range of skills and knowledge would improve their support for the subject. it is not just about relevance to everyday live, but adding the value to what they are doing to their lives. I teach in England and cannot for the world of me understand why students trying to get an average grade 4, need to know the exact trig values for 30 and 60 degrees. Surely having a sense of where and why trigonometry was developed would give them a far better insight into its value. I see relevance and context as more of making the maths real in the historical sense, e.g. how did the Egyptians make the pyramids so square?, well by using trig of course. the maths develops out of the need to do, make or explain something. It is not there already. That is what makes the subject come to life.

I teach math for life sciences at a selective public university in the US. Every year, we have some students who mainly need help with computational work, but there is also a population of students who do OK on the calculations but have enormous difficulties relating math to any kind of verbal description of a situation. This seems to be a skill separate from calculating and we are figuring out how to address it. But yes, it does need to be explicitly taught.

I would point out one important difference here though: tennis is not a compulsory subject at upper secondary level.

The need for a high level of abstraction if maths is to be mastered properly is clear, but I do think that in some cases it’s worth pausing now and then to point out the practical applications of, say, differential calculus (of which there are very many, of course). Only now and then. But there are some teachers who never do this at all – I don’t think this is ideal.

Do you have an example of a practical application of differential calculus? That is a full example of a problem that people actually solve in practice using high school calculus.

My point is that these are hard to come by so the value of high school calculus must be something other than you will end up using it in practice.

Well, it’s not my field so I’m not the best person to give you an example, but isn’t DC used (for instance) to determine the amount of material needed for various types of construction? That’s what we were always told when I was at school, anyway.

I take your point that the examples might end up being even more abstruse and confusing than the abstract mathematics involved, and I’m very much in Greg’s corner about the “tyranny of relevance”, but I think that at the same time it’s helpful if the kids are given at least a hint of the practical applications of such things, *especially* in a core discipline such as maths.

It is very hard to justify most of the math curriculum based on the need to be able to do the math in actual jobs. Since the advent of high end handheld calculators in the 1980’s the need to do much of the math is rare. Take working out integrals, there anyone who needed it in a job for the last 200 years would have a reference book. Today CAD tools will take care of any calculation of materials required.

If someone is telling math teachers to motivate by relevance to stuff students will use they had also better tell them to teach trig integrals well before anyone introduces critical thinking.

What happens is often tortured examples that are neither real or interesting. Motivating by application should take the opposite approach. What is a problem students might actually be interested in that requires this math? I recommend Randall Munroe’s (the XKCD guy) What If for a pile of fun examples. He doesn’t show the math where it involves calculus but if it is not already answered on math.stackexchange.com or physics.stackexchange.com someone will help out.

For an example:

https://what-if.xkcd.com/71/

Optimization is an obvious one.

I think that for the majority of students a deeper understanding of a smaller range of skills and knowledge would improve their support for the subject. it is not just about relevance to everyday live, but adding the value to what they are doing to their lives. I teach in England and cannot for the world of me understand why students trying to get an average grade 4, need to know the exact trig values for 30 and 60 degrees. Surely having a sense of where and why trigonometry was developed would give them a far better insight into its value. I see relevance and context as more of making the maths real in the historical sense, e.g. how did the Egyptians make the pyramids so square?, well by using trig of course. the maths develops out of the need to do, make or explain something. It is not there already. That is what makes the subject come to life.

I teach math for life sciences at a selective public university in the US. Every year, we have some students who mainly need help with computational work, but there is also a population of students who do OK on the calculations but have enormous difficulties relating math to any kind of verbal description of a situation. This seems to be a skill separate from calculating and we are figuring out how to address it. But yes, it does need to be explicitly taught.