The story you are looking for is "Brain in a Pocket" by Rob Chilson, published in Analog, May 1986.
The particular story I'm looking for involves a group of educators that are discussing student use of these machines, with much the same arguments that were used against student use of calculators once they'd dropped in price to the point they were virtually free.
Paul stirred, set his cup down reluctantly. Seven pairs of eyes looked back from around the table, and he seated himself, touched the papers.
"Five of these are the sort of thing we've been seeing all our academic lives," he said. "The other six—"
The other six were different.
"The other six are the products of computer-enhanced minds," he said. "Students so brilliant they verge on genius might well produce such papers; no others but computer-enhanced ones could. Not even brain pills could do this.
"And we're supposed to grant a degree for this?" Hank Allis said.
"According to the rules of the University—" John West began.
"We can't take refuge behind that one, John," said Rothery. "The purpose of this committee is to determine if the University rules should be changed, and if so, how. We went through all this with memory-enhancing pills."
Our protagonist suggests that they all wear the devices for a week to see how things go.
"This is hardly a fair proceeding. Only one of us has used a pocket brain extensively, and only a couple of others have even tried them." He looked around at them. "I suggest we all try using pocket brains for a week before we pass on their use."
[I]t was a tiny device about the size of a modern iPod with a lanyard that looped around your neck - this was also an antenna that allowed the computer to access your brain.
In some alarm, Paul Carson picked up his. It was about the size of a small pocket watch but slimmer. A safety clip at the top held it in the pocket. Beside the safety clip was a snap at the end of the neuro-antenna, which was wound up inside. West handed out instruction books.
He pulled on the snap and a long thin line came out, looking like a nylon monofilament. He looped this around his neck and snapped the clip over the line. This was the neuro-antenna. It hung in a loose loop around his neck, and he dropped the brain into his inside jacket pocket, not bothering with the security clip.
At least one model of this device is the 'aardvark cerebi', a pun on the Cerebus The Aardvark series.
If the pocket brain was accepted, students would be buying-probably already were buying-Pomegranates and Aardvark Cerebi.
At one point he talks to a woman who was illiterate until she put on the computer. Suddenly, she can read as the computer instantly translates for her. She is so delighted, she exclaims 'I can read!'
"[...]Professor, that ain't all! I can read!"
"With your pocket brain."
"Yes, sir, it's only an old Medulla, I got it second hand, but first thing I did was learn how to read!" Tears streamed down her face. "Professor, lots of people can't remember when they couldn't read. They don't know! Every day I get up, I say, I can read! I can read! I can read! [...]
Later on, he goes home and notices that the Chinese language plaque on the wall does, in fact, say 'welcome friends' but there are also many more meanings to the phrase that he now instantly understands.
He tapped the plaque again, frowning. But it no longer bothered him that the four Chinese characters were side by side rather than vertical.
They were fan rong fang xiong, and in this usage horizontal was right. Of course they didn't mean "Bless this House" or "Fortune's Favor" as they'd explained to many guests.
Yet they did.
Paul frowned. The general thrust was "Prosperity," but this was a complex statement.—
Then at the climax the protag and his friends decide to listen to some music since our hero is a big music aficionado and has a huge collection. Turning on a bit of classic music, for the first time he truly understands how all the notes and themes relate to one another and he appreciates music on a vastly deeper level than he used to. He's crying and re-iterates the woman's words from earlier 'I can read!'
Partly in avoidance of this uncomfortable praise, and partly because of a lifelong irritation, Paul's attention turned to the music. He was about ready to go and look at the glowing letters.
"I have a mental image of a society of mental cripples hobbling around on crutches," Morea was continuing, unaware that he had lost his audience.
But the moment Paul's attention was turned consciously to it, he knew, in the way that he knew the time and the tide. For the pocket brain had all this time been recording the music as heard through his ears, and there were no defects in its short-term memory. In a mere moment-it was ultravolant-the Pocket Peach had searched through its store of music and made a match.
"Musikalisches Opfer," he said involuntarily. "The 'Musical Offering.' Bach again."
"Why, yes." Professor Morea dismissed Bach with a wave. "I was saying—"
Paul stared at him. "You don't understand, Professor. I recognized it!" He had an abrupt understanding of how Georgie's mother had felt. I can read! I can read! "Morea, I can follow the music!"
"I can follow the music! My God, I was never able to follow—oh, my God! Music!"
Music indeed: Professor Morea had cut directly to the six-part fugue and to Paul's unbelieving ear, worlds were opening. For the first time he comprehended the beauty of music, witty and inventive as words arrayed as poetry of Shakespearean height. Six voices, each singing the Royal Theme, entering like the voices in a canon but with vastly greater complexity, delayed, backwards, inverted, all playing off each other with a rich harmonic accompaniment... words could not express the almost visual complexity that burst upon Paul all at once. It wanted a shout, a cry, a song.