Challenges in Math #5 - Right answer Wrong work

[@fightocondria]

Linear algebra to solve simple circuits. Got marked 0 until I explained to the teacher how I did it.

[@AGDinCA]

On two separate occasions, I've had a student challenge the standard method for solving a particular problem. In both cases, the student had the correct answer, but incorrect methodology. I asked each student to prove to me that they could replicate their results (i.e. obtain the correct answer) using their method for a similar problem. In one case, the student never found success with their method, but that conversation still opened up a path to discuss WHY it didn't work in all cases. We both learned a little something that day. He learned the proper methodology and why it was necessary. I learned a little bit more about how students think and how they approach problems. And the more you know about how students think, the better teacher you can become! My other student, on the other hand, blew my mind. Somehow she had come up with a completely different way of looking at something that I had never seen before. When I took some time to think about how she approached the problem it actually made quite a bit of sense even if it was not a method I had ever been introduced to. In the end, her method still would not stand up to complete scrutiny because there were certain limitations that would have precluded her from getting the correct answer every single time. But she definitely opened my eyes to see new possibilities and we got to explore just how far she could take that other method that she had come up with. The limitations aside, it was extremely eye opening and humbling. And I have never been so proud to be humbled in my whole life.

[@TheTaylorSeries]

Isn't that the best thing ever? That's awesome. :)

[@keithplayzstuff2424]

What was the mind blowing method, you can't just not tell us! :)

[@lostwizard]

I remember a physics class in high school where I had something like that happen. The teacher had handed out a stack of problems where the point was the working out so the answers were also provided. The teacher was working through a few problems in class and half the time or more, he didn't get the same answer. Usually, it was only off in the third decimal or something like that, but one time, it was off by an entire order of magnitude. I had, however, gotten the answer on the problem sheet. The next day, after he had "graded" our work (it wasn't actually for grades), he asked me to explain what I did and why, which I was happy to do. See, where the teacher was going wrong is that he would start with the various equations needed to solve the problems and plug the numbers in right away. After each calculation, he would round to the number of significant figures required in the answer plus one. In most problems, the error accumulated due to this was minimal due to a small number of steps required to work through the various equations. This one problem, however, required using three different equations, one of them twice. I think it worked out to something like 10 or more actual arithmetic operations, all with rounding and their own contribution to the final error in the result. What he should have been doing was carrying more significant figures through the intermediate calculations (which is what I think the creators of the problem sheet expected). What I did instead was plug the equations together as required for the problem (with some variable renaming), did some algebra, and ended up with a single division. Granted, my way with the algebra was a bit confusing to follow, but it had the effect of carrying effectively infinite precision through the intermediate steps by virtue of eliminating most of them. This seemed obvious to me but it apparently wasn't to the teacher. We had a good conversation about it and he actually presented my solution to the class. It gave all of us a much better appreciation for how much of an error can accumulate through a chain of calculations. If only all my teachers had been like that. (I had an English teacher insist that "extant" wasn't a word one time. Even after being shown the word in the dictionary. That led to some low grade fireworks in the classroom since I was disinclined to put up with idiocy at that point in time.)

[@TheTaylorSeries]

Ah! That's an awesome experience. I am glad the teacher handled it well. I know what you're talking about; you generally want to plug in at the very end, or it's as you say: rounding errors will creep in and trash everything. I hope the teacher was ecstatic to extract exactly the extent of the extant nature of the word extant.

[@lostwizard]

​@@TheTaylorSeries Doing the algebra first just seemed obvious to me, even when it got complicated. I think it was because I hate doing actual numerical calculations. But I like doing algebra for the most part. :) The English teacher eventually grudgingly accepted I was right and dropped it. I almost wish she would have sent me to the "office" for disrupting class, though. It would have been fun to watch that play out (the administration of that school was slightly less knowledgeable than the English teacher so parents would have been called).

[@MultiJuice05]

Have you ever had the feeling where you can’t stop watching this channel?

[@TheTaylorSeries]

I just realized that I did the thing with the lighting reflecting off my glasses again. Whoops!

[@dvfantail]

Nice! Haven't checked this channel out in a minute. Looking good!

[@ikkuhishikawa7982]

Yay! He bacc

[@connorconnor2421]

No, he *_bacc_*

[@treyebillups8602]

B A C C

[@DiamondIceNS]

I had an experience similar to another commenter in my high school AP Physics course. We were studying the unit on 2D Newtonian projectiles (i.e. parabolic trajectories) and throughout the practice problem sets were many nested applications of sines, cosines, and tangents. One astute classmate, Mr. Peterson, was playing with some algebraic expressions and discovered that, for a specific set of initial conditions, a solution could be deduced by a highly simplified expression. Seriously, it would collapse maybe half a sheet of college-ruled notebook calculation steps into four or so. I brought this up to the teacher and he largely dismissed it, stating that it was a nice property, but he'd rather see us perform the full work steps on all assignments and exams. Undeterred, I decided to draft an explicit proof document. I clearly defined under what initial conditions the conjecture held, the algebraic steps used to derive the conjecture, and even drew a diagram for clarity. I turned it in to my professor after class hours under the title "Peterson's Property". He was very amused, and told me that I could declare its use in coursework. Come next week, on our unit exam, I indeed found a work problem where "Peterson's Property" applied. I pointed out the existence of the initial conditions, declared use of the property, and effectively sidestepped an entire math problem with ease. I received full credit for the work.

[@helloiamenergyman]

Second

[@TheTaylorSeries]

Was there a first? :)

[@dylutante]

The music makes it hard to listen to you. The video would be better without it, or at least much quieter.

[@connorconnor2421]

?noitammuS od ouy lliw nehW

[@TheTaylorSeries]

?wonk ot tnaw uoy od noitammus tuoba tahW

[@connorconnor2421]

.ti no edosipe na ekam tsuj ,gnihtoN