It's hoped the new NAPLAN format will help teachers better tailor lessons and support for students.
But the education union fears underprivileged schools, and kids who speak a second language, are at a disadvantage.
NAPLAN first began a decade ago, testing students from years 3, 5, 7 and 9, using the traditional tools of paper and pencils.
But over the next two to three years, the assessments will be conducted entirely online.
Nearly 4,000 schools nationwide are currently in the middle of a month-long trial.
The Australian Curriculum and Reporting Authority believe it will result in better assessments for individual schools and students.
CEO Robert Randall says the trials are trying determine if schools are capable of going completely online.
"School readiness testing is having a look at computers in the school, networks in the school, making sure teachers and students have the confidence to sit the test. The improvements that we're gonna realise through moving it online will address some concerns that people have previously had: about the time it takes to get the tests back, about one test for all children, how we can make it adaptive."
But the education union wants the new format scrapped.
Deputy federal president Maurie Mulheron says it has received hundreds of complaints from teachers.
"NAPLAN online is a logistical and ethical disaster. Particularly in areas of social disadvantage, their kids are just not on a level playing field. We've also got a problem with the students from a language background other than English, where the technology is, again, not as accessible."
Mr Mulheron claims remote schools with poor internet and limited use of computers will be also be adversely affected.
"At the moment schools are not yet ready for it. The system has not got the capacity to roll it out, and so we will need to ensure that the whole thing is put on hold until all the problems are worked out."
It will take 2 to 3 years before NAPLAN's digital transition is complete.
Mr Randall says schools not yet ready to go online should use that time to continue the trial, so they can identify and resolve all issues.
"If they have a doubt about their students' confidence or something, I'd encourage them not to take that step next year. Do another year or two of preparation. When we trial our items, we trial them with a whole range of young people. So we're confident with the English speakers, who only have English, and we're confident we trial them with those for whom English is a second or third language."
The paperless practice runs are mostly being conducted in New South Wales, which has never done such a digital trial.
Nearly 3,000 schools in the state are taking part.
At Edensor Park Public, Year 5 student Dylan Luu says he enjoyed doing the tests on his laptop.
"Yeah very easy on the computer, and much more better than on book and pencil. So I'm more used to computers, and I like computers."
His classmate, Marilyn Karkeas, agrees.
"It was pretty good. it was pretty easy. More efficient and possible."
But the education union says many students would not be as happy about the tests.
