Dan Meyer writes a lot about how to set up students so that they really feel the need for the next thing they're about to learn. "Give them a headache before you offer them the aspirin."

https://blog.mrmeyer.com/2015/if-proof-is-aspirin-then-how-d...

One of my best memories from high school math was our teacher taking us outside to measure the height of our building after teaching us trigonometry.

We took a triangular ruler outside, leveled the base and lined it up by eye with the top of the building. Then we measured the distance to the point where we were standing and the distance from the ground to the ruler.

I really wish education reduced the amount of rote learning involved and focused more on letting students explore and make mistakes with the educator as a guide.

My high school took this concept one step further. We were tasked with designing a solar hot dog cooker. The physics class specified the reflective area needed to gather enough sunlight to cook the hot dog. They tossed that over the wall to the algebra class, who determined the parabola shape and focal length from the dog. From there it was sent to drafting class, where the plans were drawn up. Those went to the kids in wood shop, to build the thing from the plans.

Then one day we all went outside with our solar cookers and hot dogs and had a cookout. IIRC, there was a lot of variation among the cookers. Different reflectors were used (piece of aluminum sheet, bent and held by end brackets. Mylar film laminated to a wooden parabola "bed". One kid got adventurous and designed his with a vertical dog holder, and a circular parabola that was actually several different profiles stacked to "smear" the focal point along the vertical axis. I don't remember if it worked well or not)

I think we were meant to learn some sort of process engineering in addition to our respective subjects? Or just "teamwork", I guess.

I love this. That kind of genuine curiosity and passion is what’s needed to really teach people to embrace maths.

I hated exercises like that. It always seemed like massive waste of time to me.

I suppose that this differs at how mathematically mature the student is. For undergrads and high school students it is, as you say, probably a good idea to give a lot of motivation and applied examples for the theory being taught. I suppose this article is more geared towards teaching university students, and then it can well be that examples and applications become more abstract. Of course, introductory topics such as Linear algebra and Calculus would lend themselves very well for a wide range of application examples. On the other hand, in more advanced and abstract university courses the 'realistic problems' should already be reasonably clear for the student when starting the course.

I'm making an assumption here wrt OP, but I think that a lot of people are taught that an integral is a set of fancy tricks that you use to simplify down equations until you get ... an answer.

In my experience, students are expected to memorize all sorts of equations without ever really being expected to understand why you're calculating the integral in the first place (even at decent universities). Obviously the risk with that is that you won't recognize any problem can be solved with integrals, no matter how concrete. Once I left the fancy tricks behind I felt like a fog lifted and it was actually really fun to see how integrals fit into both applied and abstract examples.

Most classes actually teaches why integrals are useful, the problem isn't the material but that student's quickly forget that and then only remembers that they did a lot of memorizations of formulas and solving algebra. Then once you have worked a lot with integrals you've built the intuition to actually understand what it does and then it seems like a simple explanation taught you everything, although most likely you already heard that simple explanation a long time ago and forgot about it since you didn't understand it at the time.

In this case you have to bring them back to reality once in a while (like: "This is what we are actually doing here"). Ofc there are gifted math geniuses which have no problem with everything being abstract, but for most people connecting what they learn to the world they live in is a prerequisite for maintaining at least a minimum level of interest for the topic. At school students are used to learning useless stuff without questioning it. They won't tell you if what you teach them is useless stuff they don't understand.

Going back to that simple explanation once people are a bit more experienced can make things click for a lot of people. Remaining people what they are dealing with and why can also help.

Fair point. Though I do think the focus on integration shortcuts was a huge distraction to the main content, and arguably irrelevant to the actual concepts.

My class consisted of hour after hour of solving integrals of different types using all the different shortcuts, and the exams were largely the same. I feel like that time and effort could have been much better spent.

> One important thing is: create realistic problems that need certain math.

Aren't people who try to do this a running joke?

'Adam has five apples and seven buckets and must drive to Rapid City...'

People hate that stuff.

My 9 year old has to do reflex math (fact memorization); she wants to get to 100% by the end of the school year. I created a spreadsheet, and started entering her data. I added a linear regression line & explained how it worked. She was immediately suspicious, and began questioning how all the lines worked, and why it’d take so long, etc. I ended up teaching her the intuition for linear regressions, functions, etc. Very enjoyable for both of us.

Why should any of your students give a damn about Adam and his five apples? You have to make it about the students and their interest. And I don't mean that every math problem you give them has to be wrapped in a story, but but the first one for each topic should be, so they understand why this new thing is useful for them.

You can do a ton of cool stuff with maths, including calculating the movements for figures in a computer game, extrapolating the distance to the moon, etc.

And before someone claims I am beeing unrealistic: in my youth I worked with teens from socially complicated backgrounds and helped them to learn for their maths exams, exactly like this. These were kids that in bad moments would pull a knife on you or throw furniture out of the window. This was literally the only way to get them to start caring about the result of a mathematical formula and then they demanded to know more..