#now i realize it's intentional that harry takes the credit for it while LV gets no credit LOL

[slitheringghost]

Hermione As Teacher And Connections To Lily

An interesting narrative thread is how often Hermione's magic and teaching influences other characters' and how often it goes unrecognized, and particularly how it parallels a lot of Lily's own magic saving Harry from Voldemort. (Read on Ao3)

1. Triwizard Tournament

Summoning Charm

he did so badly at Summoning Charms in Professor Flitwick’s class that he was given extra homework — the only person to get any, apart from Neville. “It’s really not that difficult, Harry,” Hermione tried to reassure him [...] — she had been making objects zoom across the room to her all lesson, as though she were some sort of weird magnet [...] “You just weren’t concentrating properly —” (Ch 18, GoF)

Harry still hadn’t mastered Summoning Charms, he seemed to have developed something of a block about them, and Hermione insisted that learning the theory would help. They consequently spent a lot of time poring over books during their lunchtimes (Ch 19, GoF)

“Hermione, I need to learn how to do a Summoning Charm properly by tomorrow afternoon.” And so they practiced. They didn’t have lunch, but headed for a free classroom [...] At two o’clock in the morning, Harry stood near the fireplace, surrounded by heaps of objects [...] Only in the last hour had Harry really got the hang of the Summoning Charm. “That’s better, Harry, that’s loads better,” Hermione said, looking exhausted but very pleased. (Ch 20, GoF)

Flitwick then spends most of a lesson "talking to Harry about the perfect Summoning Charm Harry had used during the First Task".

1.2 Shield Charm

(Full analysis of the Shield Charm and how it's textually linked to Lily is in my meta When Lily Cast Her Life As A Shield)

He was still having trouble with the Shield Charm, though. This was supposed to cast a temporary, invisible wall around himself that deflected minor curses [...] “You’re still doing really well, though,” Hermione said encouragingly, looking down her list and crossing off those spells they had already learned. “Some of these are bound to come in handy.” [...] Come on, Harry,” she added briskly [...] “let’s try that Shield Charm again.” (Ch 31, GoF)

Which Harry then teaches the DA:

He was improving so fast it was quite unnerving and when Harry taught them the Shield Charm, [...] only Hermione mastered the charm faster than Neville. (Ch 25, OoTP)

Then, what Fred and George say makes the real money in the joke shop:

“We’ve just developed this more serious line,” said Fred. “Funny how it happened...” “You wouldn’t believe how many people, even people who work at the Ministry, can’t do a decent Shield Charm,” said George. “’Course, they didn’t have you teaching them, Harry.” “That’s right... Well, we thought Shield Hats were a bit of a laugh, you know, challenge your mate to jinx you while wearing it and watch his face when the jinx just bounces off. But the Ministry bought five hundred for all its support staff! And we’re still getting massive orders!” “So we’ve expanded into a range of Shield Cloaks, Shield Gloves...” “...I mean, they wouldn’t help much against the Unforgivable Curses, but for minor to moderate hexes or jinxes...” (Ch 6, HBP) Although Snape did not know it, Harry had taught at least half the class (everyone who had been a member of the D.A.) how to perform a Shield Charm the previous year. None of them had ever cast the charm without speaking, however. A reasonable amount of cheating ensued [...] Typically, ten minutes into the lesson Hermione managed to repel Neville’s muttered Jelly-Legs Jinx without uttering a single word (Ch 9, HBP)

I love that Harry is then the one to teach Hermione the Patronus Charm - which Remus tells Harry is "a kind of anti-dementor - a guardian that acts as a shield between you and the dementor".

1.3 Four-Point Spell

and the Four-Point Spell, a useful discovery of Hermione’s that would make his wand point due north, therefore enabling him to check whether he was going in the right direction within the maze. (Ch 31, GoF)

“Point Me,” he whispered to his wand, holding it flat in his palm. The wand spun around once and pointed toward his right, into solid hedge. (Ch 31, GoF)

It's reasonable to extrapolate that the spell is Hermione's invention, given that it's the only spell with an English incantation, and inventions are referred to as "discoveries" several times - Remus wrt the Wolfsbane Potion in PoA, Dumbledore's work ("the discovery of the twelve uses of dragon’s blood", "does Skeeter deny the brilliance that led to Dumbledore’s many magical discoveries?"), Harry has to "memorize the dates of magical discoveries and goblin rebellions" and mentions books titled "Important Modern Magical Discoveries and A Study of Recent Developments in Wizardry", Fred says they "spent six months developing" the Ton-Tongue Toffees, etc.

Hermione inventing this is particularly compelling given that it's similar to what Harry's wand does in DH (related to Lily's spellwork deflecting the Killing Curse), and Harry's conflict with her after she breaks his wand.

[...] his wand acted of its own accord. He felt it drag his hand around like some great magnet, saw a spurt of golden fire through his half-closed eyelids, heard a crack and a scream of fury. (Ch 4, DH) He knew exactly what Hermione would say if he expressed any of this: The wand is only as good as the wizard. But she was wrong, his case was different. She had not felt the wand spin like the needle of a compass and shoot golden flames at his enemy. (Ch 18, DH)

Also notable is Hermione solving Snape's Potions riddle, and Harry echoing this with solving the Sphinx riddle during the Third Task (Harry thinking “it was Hermione who was good at this sort of thing, not him” and then “amazed at his own brilliance” when he solves it).

2. Other examples

2.1 Impervius Charm

“I’ve got no chance with these on,” Harry said exasperatedly, waving his glasses. At that very moment, Hermione appeared at his shoulder; she was holding her cloak over her head and was, inexplicably, beaming. “I’ve had an idea, Harry! Give me your glasses, quick!” He handed them to her, and as the team watched in amazement, Hermione tapped them with her wand and said, “Impervius!” “There!” she said [...] “They’ll repel water!” Wood looked as though he could have kissed her. “Brilliant!” he called hoarsely after her [...] Hermione’s spell had done the trick. (Ch 9, PoA) "Harry, didn’t you do something to your glasses to stop the rain fogging them up when we played Hufflepuff in that storm?” “Hermione did it,” said Harry. He pulled out his wand, tapped his glasses and said, “Impervius!” “I think we all ought to try that,” said Angelina. “[...] all together, come on — Impervius!" (Ch 18, OoTP)

Additionally, when Padfoot comes to watch during that PoA match and dementors swarm the field, making Harry hear Lily's murder for the first time, Harry falls from his broom, his Nimbus crashes into the Whomping Willow, others say they thought he was dead and "Lucky the ground was so soft" - of course, it wasn't luck, it was Dumbledore; Hermione then repeats these actions in DH - during the waterfall in the Thief's Downfall (evokes the drowning feeling from dementors; and Hermione additionally uses the Shield Charm there) and immediately after they escape LV in Godric’s Hollow and Harry finally gets the full memory of Lily's death (Hermione’s eyes being emphasized):

“Dumbledore was really angry,” Hermione said in a quaking voice. “I’ve never seen him like that before. He ran onto the field as you fell, waved his wand, and you sort of slowed down before you hit the ground. Then he whirled his wand at the dementors. Shot silver stuff at them. They left the stadium right away [...]" “Then he magicked you onto a stretcher,” said Ron. “And walked up to school with you floating on it. Everyone thought you were...” (Ch 9, PoA) “Yes,” said Hermione. “I had to use a Hover Charm to get you into your bunk, I couldn't lift you [...]" There were purple shadows under her brown eyes and he noticed a small sponge in her hand: She had been wiping his face. (Ch 17, DH) Water filled Harry’s eyes and mouth: He could not see or breathe: [...] Harry heard the cart smash into pieces against the passage wall, heard Hermione shriek something, and felt himself glide back toward the ground as though weightless, landing painlessly on the rocky passage floor. “C-Cushioning Charm,” Hermione spluttered (Ch 26, DH)

2.2 Murtlap Essence

During that PoA practice, Fred and George were debating using Fever Fudge to get out of flying:

“— but you get these massive pus-filled boils too,” said George, “and we haven’t worked out how to get rid of them yet.” “I can’t see any boils,” said Ron, staring at the twins. “No, well, you wouldn’t,” said Fred darkly, “they’re not in a place we generally display to the public —” “— but they make sitting on a broom a right pain in the —” (Ch 18, OoTP)

The knowledge Hermione uses to heal the I must not tell lies scar on Harry's hand helps the twins finish the Snackboxes:

“Here,” she said anxiously, pushing a small bowl of yellow liquid toward him, “soak your hand in that, it’s a solution of strained and pickled murtlap tentacles, it should help.” Harry placed his bleeding, aching hand into the bowl and experienced a wonderful feeling of relief. (Ch 15, OoTP) When Harry next saw Lee, the back of his hand was bleeding rather badly. Harry recommended essence of murtlap. (Ch 25, OoTP) Hermione cast him a stern look. “You’ve got exams coming!” “Told you already, we’re not fussed about N.E.W.T.s,” said Fred. “The Snackboxes are ready to roll, we found out how to get rid of those boils, just a couple of drops of murtlap essence sorts them, Lee put us onto it...” (Ch 26, OoTP)

2.3 Magical Eavesdropping Methods

Harry frankly marveled at the fact that Hermione could research magical methods of eavesdropping as well as everything else they had to do. (Ch 28, GoF)

People have theorized that Hermione’s research while she was taking revenge against Rita Skeeter may have aided Fred and George develop the Extendable Ears, esp. since she was staying at 12GP that summer.

2.4 DA Galleons

Hermione's inspired by Voldemort's magic to invent the DA's communication method; Draco then gets the idea from that to carry out his Death Eater mission, enchanting his own coins to secretly communicate with Rosmerta, and also gets the idea to poison the mead from Hermione, having "heard her talking in the library about Filch not recognizing potions".

“You know what these remind me of?” “No, what’s that?” “The Death Eaters’ scars. Voldemort touches one of them, and all their scars burn, and they know they’ve got to join him.” “Well... yes,” said Hermione quietly. “That is where I got the idea... but you’ll notice I decided to engrave the date on bits of metal rather than on our members’ skin...” “Yeah... I prefer your way,” said Harry, grinning, as he slipped his Galleon into his pocket. (Ch 19, OoTP)

Notably, Lily's blood magic - unlike Hermione's and very like Voldemort's - does burn Voldemort's skin.

Similarly significant is Hermione's invention of the SNEAK curse - where the DA members all sign the binding contract in the Hog's Head, the same establishment where Trelawney gave the prophecy, and the binding contract incorporated in Lily's magic. Hermione enchants ~28 coins for the whole DA and creates the curse within the span of ~a month.

2.5 O.W.L. exam

Three rows to his right and four seats ahead, Hermione was already scribbling… He lowered his eyes to the first question: a) Give the incantation, and b) describe the wand movement required to make objects fly... Harry had a fleeting memory of a club soaring high into the air and landing loudly on the thick skull of a troll... Smiling slightly, he bent over the paper and began to write

On the whole Harry thought it went rather well; his Levitation Charm was certainly much better than Malfoy’s had been, though he wished he had not mixed up the incantations for Color-Change and Growth Charms (Ch 31, OoTP) the exam to which Harry was looking forward least and which he was sure would be the one that would be the downfall of his ambitions to become an Auror. Sure enough, he found the written exam difficult, though he thought he might have got full marks on the question about Polyjuice Potion: He could describe its effects extremely accurately, having taken it illegally in his second year. (Ch 31, OoTP)

Hermione is linked to Harry's Charms and Potions exams, the subjects most directly tied to Lily. Important to note that Professor Marchbanks praises Dumbledore for having “done things with a wand I’d never seen before” during his Charms and Transfiguration N.E.W.T.s - which Hermione didn't get to take 'cause of the war; and Harry and Neville are acknowledged as doing better in Potions without Snape's presence - clearly true of Hermione's potential too as her best accomplishment was brewing a N.E.W.T. level potion second year, outside Snape's supervision (which took a month, and then she spent 2 months in the hospital due to petrification/Polyjuice turning her into a cat).

2.6 Body Bind Curse

Harry turned to Hermione. “Do something,” he said desperately. [...] “Neville,” she said, “I’m really, really sorry about this.” “Petrificus Totalus!” she cried [...] Neville’s arms snapped to his sides. His legs sprang together. His whole body rigid, he swayed where he stood and then fell flat on his face, stiff as a board. [...] “What’ve you done to him?” Harry whispered. “It’s the full Body-Bind,” said Hermione miserably. (PS)

Interestingly, Harry instinctively copies what he saw of Sirius in SWM during the DoM sequence, using the full Body Bind for the first time (at least, that we see on screen). Hermione compliments Harry on it, calling back to her being the first one to show him first year - and her being cursed right after echoes the description of Sirius falling through the veil: "the second jet of light hit him squarely on the chest", his eyes widening in shock and the "look of mingled fear and surprise".

Sirius said, “Petrificus Totalus!” and Snape keeled over again at once, rigid as a board. “LEAVE HIM ALONE!” Lily shouted. She had her own wand out now. James and Sirius eyed it warily. (Ch 28, OoTP) “Petrificus Totalus!” shouted Harry, as the second Death Eater raised his wand [...] “Well done, Ha —” But the Death Eater Hermione had just struck dumb made a sudden slashing movement with his wand from which flew a streak of what looked like purple flame. It passed right across Hermione’s chest; she gave a tiny “oh!” as though of surprise and then crumpled onto the floor (Ch 35, OoTP) Harry seized his chance: “PETRIFICUS TOTALUS!” The spell hit Dolohov before he could block it, and he toppled forward across his comrade, both of them rigid as boards and unable to move an inch. (Ch 35, OoTP) Springing up, Harry yelled, “Petrificus Totalus!” Once again, Dolohov’s arms and legs snapped together and he keeled over backward, landing with a crash on his back. “Nice one!” shouted Sirius, forcing Harry’s head down [...] (Ch 35, OoTP)

2.7 Unbreakable Charm

Not something Hermione specifically taught, but also interesting is Hermione mirroring Barty Crouch Jr. during his Unforgivables lesson (and saying Harry gave her the idea when he mentioned "bugging").

Moody got heavily to his mismatched feet, opened his desk drawer, and took out a glass jar. Three large black spiders were scuttling around inside it. (Ch 14, GoF)

“Oh not electronic bugs [...] Rita Skeeter [...] is an unregistered Animagus. She can turn —” Hermione pulled a small sealed glass jar out of her bag. “— into a beetle.” [...] Inside were a few twigs and leaves and one large, fat beetle [...] Hermione took the glass jar back from Ron and smiled at the beetle, which buzzed angrily against the glass. [...] “I’ve put an Unbreakable Charm on the jar, you see, so she can’t transform. And I’ve told her she’s to keep her quill to herself for a whole year. See if she can’t break the habit of writing horrible lies about people.” (Ch 37, GoF)

Hermione imprisoning Rita Skeeter in a jar vaguely references Lily and Harry's actions in 1981 leading to an "imprisoned" LV in Albania; this also happens a few chapters after Priori Incantatem, and Harry and LV under the web of light evokes insects trapped underneath glass:

The golden thread connecting Harry and Voldemort splintered; though the wands remained connected, a thousand more beams arced high over Harry and Voldemort, crisscrossing all around them, until they were enclosed in a golden, dome-shaped web, a cage of light, beyond which the Death Eaters circled like jackals, their cries strangely muffled now... “Do nothing!” Voldemort shrieked to the Death Eaters, and Harry saw his red eyes wide with astonishment at what was happening, saw him fighting to break the thread of light [...] and the golden thread remained unbroken. “Do nothing unless I command you!” (Ch 34, GoF)

Then the phoenix song comes from "every thread of the light-spun web vibrating around Harry and Voldemort", and a voice saying Don't break the connection (elaborated here).

3.0 Blasting Curse

All this comes full circle in the Godric's Hollow graveyard in DH (a deeper analysis also for another post), where Hermione mirrors what Harry did because of Lily as a baby (and also Harry's actions at the start of DH while flying during the Battle of the Seven Potters): blows up the house (“Confringo") and enrages Voldemort as she and Harry fly out the window - fly from death - together.