Since I noticed a lot of STEM undergrads struggle with the notation of basic analysis and algebra I've been thinking about alternative representations of both for middle school level teaching. I am pretty sure they struggle mostly because their calcified brains refuse to parse abstract notation. Since at least half the undergrads don't make it anyway I am not really concerned about fail rates, but having spent years with undergrads I have come to believe that a great deal of them have psychological problems on top of a skill issue.Inspired by feynman diagrams and the like I developed a rudimentary graphical notation for polynomials, derivatives and rational numbers. Addition is simply realized by putting two diagrams next to each other, while multiplication is achieved by having graphs touch/superimposed. Since derivatives, integrals and such for polynomials only need rational arithmetic I ended up with a system that could be taught in the first few years of middle school. I have a hunch that by framing the same problems as puzzles rather than math problems the kids would outperform undergrad crybabies who are primed to believe "math is hard :((".
>>17026558Draw an example of (x+1)(x-1) = x^2 - 1 so we see what you mean
>>170265651 is represented as a circle, x^n is a "pearl necklace" with n "pearls" (smaller circles). This way x^0 coincides with 1. -1 is a black circle. You can realize (...)(...) by putting the expressions in circles and have the circles "kiss". Integers are just asterisk-likes with n "points" for any n>1, negatives have a black circle in the center.
>>17026586[math]x\cdot x = x^2.[/math]You could also have the two just touch rather than be inside each other.If you wish to write, say, 2x, you just put a 2 inside the x.
0s are represented as dots.Fractions require a new symbol but that isn't difficult to come by. The division symbol is just a broken circle. You attach the denominator to the outside and the numerator to the inside.Picrel: 1/2.
>>17026586>>17026589I like it, it's intuitive and clear. Good job
I don't believe this notation is particularly groundbreaking, it mostly emphasizes counting numbers as much as it can. But my theory is that by breaking down abstract notations into something closer to finger counting, the same way an abacus represents numbers as beads on semantically distinct rods, might have an effect.>>17026600Thanks!
What the actual tittyfucking Christ?
>>17026642It's not as schizo as it looks, most of it is just replacing math symbols by arrangement and indices by tally marks.
>>17026558>I am pretty sure they struggle mostly because their calcified brains refuse to parse abstract notation.No, it's because they just don't care. Math is quite simple if you just put some effort into it. But hardly anyone in an undergrad math class actually cares about the topic (including the professors).
>>17026666The schizo thing is thinking this gobbledegook is somehow more readable than standard math notation
>precalculus midterm>question 1:
>>17026763Got a laugh outta me
>>17026666>It's not as schizo as it looksWhatever you say, Satan.
>>17026586What about x^100?Or x^x?x^2 + y^2 = 1?
>>17026957I thought about support for larger exponents, the most elegant solution would be to allow for numbers inside the small circle. I think multivariate calc is something I would at least for now avoid working on, because in my experience the people that manage univariate don't take long to get the generalization. You reminded me of something though, tensor train people have a concise notation for these sorta things, that would be a notation deeply rooted in linear algebra though.
>>17026755Remember that it is not meant to replace or compete with standard notation, it is meant to obfuscate things that people have strange emotional hangups over, in particular unknowns and indices. Nobody to whom standard notation is trivial would or should prefer it. But I sadly don't have access to school kids to test my ideas.
>>17026763I think I need to introduce proper notations for equations still, using "=" is unhelpful in this case. One consideration I had was to distinguish between a symbol to express exercises like yours and a distinct symbol for identities. I know this technique is used in elementary school math books.
>>17026586>>17026589interesting concept, bump
>>17027737How about this?
>>17027780yes, something of that sort, perfect.
The question is how far to willingly take this notation, it might be sane to extend it to complex numbers, exponential and trig functions.
>>17026558Kot
>>17026586This might work as an exercise for people who like math, but people who struggle with notation will just get more confused. It would be a good math olympiad exercise for middle schools.
>>17026586Googly eyes on tits with a cancerous lump?>>17026589An eyeball inside an eyeball? A head wearing goggles?>>17026597A magic wand!>>17026763Nets on a swimming pool?None of this shit is obvious. You need to establish some proper rules here. This just makes everything harder. I have a bachelor's in maths and this is nonsensical and doesn't make sense.
>>17030576>this just makes everything harderobviously I didn't post an in-depth introduction to it like I would prepare for a school setting, I can explain anything you wish, though the core of it is pretty straightforward. It gets clunky for large numbers (obviously because representations are effectively base 1)>I have a bachelor's in mathsThat's nice but have you ever actually taught it, especially to undergrads whose main subject wasn't math? It's a genuinely challenging task because what we find intuitive is often incomprehensible to them.>>17030562That might be, I am not confident in this being a working approach, the problem is that I know from experience the notation is a major roadblock for a lot of people without talent. Indices and unknowns seem to be major no-go
>>17030843I've taught it to undergrads in business. Will that suffice? Not only did they do well with my standards, they loved it. What's my secret? The ingredients you lack. It wasn't the formalism of symbolic algebra. It was the obsession on derivation and opaque rules. All they need is a definition, why it's used and many examples in thorough detail using rules they learned in middle school. Your technique doesn't do this.
>>17030903Since I never mentioned, my PhD is in theoretical physics, maybe it adds context.>The ingredients you lack.Lots of assumptions, anon. Let's keep this civil, I am not going to humor you trying to turn this conversation into some shit flinging. I have no interest throwing shade. Business is not STEM, though. The culture is different. It is also a matter of how far into the selection process the people you taugt are. Like I said, failing at least 10-20% of a first semester year is normal, even expected, in my neck of the woods. 2/3 of the people won't finish their Bachelor. It ties into>many examples in thorough detail using rules they learned in middle school.For example, did the sample of people under your care struggle with fractions, were errors such as 2/(2+3) = 1/(1+3) common? Did people resist the notion of complex numbers because the notation was too different from what they know? >All they need is a definition, Obviously.>why it's usedPersonally, if it were not for time constraints, I would encourage math to have a larger portion tied to discovery. People often resist a formalism if they see no sense in it.I get the feeling our experiences with students are way too different and you parse that as a skill issue on my end.
>>17030918>I get the feeling our experiences with students are way too different and you parse that as a skill issue on my endCorrect. My PhD is also in theoretical physics. My bachelor's is in maths. The other issue with your approach is the time waste arts and crafts would create on the board.
>>17031113>Correctalright, then you can go be smug somewhere else, you aren't here to engage with ideas but jerk yourself off.
Stop argueing with that jackass and show us more bubble numbers.
>>17031490Gladly, one thing that is driving me nuts is how to handle placeholders. Made progress with definite integral notation but not quite happy yet with that either. It would be natural to treat it as an inverse derivative operator [math]\partial^{-1}[/math] style but the caveat is obvious (+c)
i think this thread presupposes a lot regarding why people fail; I wonder who else here has taught a filter math course and looked into the problems of the people that are NGMI
>>17026586>>17026589>>17026600kek>>17026763I like it.
The first line would be a nice way to express[math]\frac1x = x^{-1}[/math]Assuming I use [math]\partial[/math] as the multiplicative operator for derivatives then the bottom line would show how the derivative looks in this notation, and the rightmost part is what happens when you allow the small circle to contain numbers themselves. Not sure if I like it. Definitely needs workshopping. I think the main point I am coming to realize is that perhaps one could involve an abacus type of device instead of changing notation..
>>17026558I noticed a lot of people are inferior to me. I've been thinking it would get me attention and adoration if I let everyone know how superior I am to most others. Anyone who disagrees that I'm superior is a crybaby.
