Ask HN: Exploratory ways to teach math up to high school?
I am interested in interesting ways of exploring math concepts non-algebraically, and I was hoping the HN community could point me to interesting games, simulations, physical experiments etc.
Most adults that I come across cannot define a logarithm, why π shows up in so many places. They are mystified by complex numbers, calculus etc. I would like some interesting approaches that work with adults (as remedial education) and with 7th grade kids and higher.
I have had a modest degree of success teaching exploration using Scratch. For example, the kids were taught how to measure areas of arbitrary shapes by scanning from top to bottom, left to right and counting pixels. This not only gives an intuition for integral calculus (without saying it is so), but also the relief that there is an alternate way to get the job done without having to remember a formula. Whatever reduces the resistance to math, I say.
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e: Hot/cold stuff passively reaching ambient temperature (why does it get colder quicker right at the beginning but so slowly at the end??) as well as the time constant for capacitors (which you could analogize with how long it would take for water to drain out of a broken dam: quickly at first, slowly as the water level decreases)
integrals: My brother designed a boat that he would build. He wanted to know how much volume it would displace until water would spill inside. The walls were not straight nor right-angled. Using the slicing method and estimates of "this is somewhere between an oval and rectangle, so the area of this slice will be between those shapes", he was able to figure out the ballpark volume.
derivatives, and this one takes a LOT of care to avoid confusion... position, velocity, acceleration. I would start with straight distance examples, but it's also cool to note: On a Merry Go Round, the rider's acceleration would only be toward the center if the rotating speed stays the same. It has to! If it were even a little backwards, you'd slow down! If the Merry Go Round were wider, the acceleration would be not so strong. If the Merry Go Round were so wide that it's impossible to tell you are spinning, there would be nearly no acceleration.
When you drop that ball from a window, gravity is its acceleration. Gravity never changes its direction or gets stronger. That is intuitive. The velocity will keep getting larger, but at the same rate for a while (until real life interferes aka drag). The position, you can plot that and see that every second it is a farther interval. That also makes sense: A ball thrown upward doesn't bounce off the air at the top and come down right away at the same speed. It hangs for a brief moment. But it doesn't hang anywhere else after its peak because the position is the second derivative of acceleration, and that second derivative only has one point where it stays nearly the same for a brief moment.
It might not stick for everyone, but you can explain that 6 dB is always "twice as loud". Thus, 12 dB is "2x2 times as loud" and 18 dB is "2x2x2, or 8 times, as loud". The dB amount is always addition, but the "as loud" part is multiplied.
Similarly, every 10 to 30 years (usually about 20) the value of money drops in half due to inflation. You're adding years but multiplying or dividing value. That lets you know that there's a power/logarithmic relationship.
One other concept is more useless but fun for me... There's product-over-sum. It shows up in "how long does it take N lawnmower pushers of varying efficiency to trim so-and-so acres of lawn?" (If Joe does a lawn in 20 and Jake in 30, they do it in 12 together.) but also in things like parallel resistance. In this case, I tend to think of the electrons as the blades of grass in the lawn and the resistors as lawnmower pushers. The electrons don't care who "trims" them, and whichever resistor becomes available to process an electron (at its own given efficiency e.g. conductance/anti-resistance) will do it.
I am a big fan of the Street Fighting Math series from MIT, but I feel like there were much better resources for explaining the concepts (e.g. videos, lectures) than I can find today. The book is not as approachable as the concepts themselves when explained at an audience-appropriate level. The only downside it its lack of general practicality, but it does move some advanced concepts into the napkin math domain.
In one part, Dr. Mahajan shows that because the derivative of the LN function is very close to the derivative of x at 1 (that is, 1), you can approximate powers close to 1 by subtracting 1 to get them into the log domain and multiplying by their exponent. This probably doesn't make sense, so I'll solve an example from the book...
"If 5% of bacteria are mutated during a radi...
I love Street Fighting Mathematics too, but it's tough even for me. Meaning, it makes sense as I see the explanation, but these techniques of estimation have to be practiced over and over again to feel natural.
As for the gravity example, a small game using Scratch was enough to get them to appreciate acceleration, by associating velocity with audio. You intuitively know what a vehicle sounds like when it is maintaining constant speed, vs when it is accelerating or decelerating (negative acceleration)
From there it was a small delta to showing horizontal launch velocity but no acceleration, and how you can deal with the x and y components independently, and how a ball automatically traces a quadratic curve.