The Limpopo High Court has reserved judgment in the leave to appeal application in the Michael Komape case.
Legal advocacy group Section 27 represented the family on Friday in the High Court sitting in Polokwane, in appealing the dismissal of the family's R3m damages claim.
The five-year-old boy drowned in a pit toilet in 2014.
In April, Judge Gerrit Muller rejected the claim for constitutional and general damages. Instead, he ordered that two of Komape's siblings be paid R6 000 each for general medical expenses.
On Friday, Muller also heard the application for appeal.
The lobby group also asked him to grant the family leave to appeal, and that it be heard in the Supreme Court of Appeal.
Equal Education Law Centre (EELC) and the State also presented arguments.
Muller initially stated that Section 27's lodging of its last papers for leave to appeal was two days late, and therefore "inadmissible".
However, senior counsel, advocate Vincent Maleka, argued that notice was not late, as an earlier notice had been stamped on May 16.
Maleka said the second last page of the appeal showed that it was received by all attorneys on time.
"Preliminary remark is that at this stage of litigation we are not here to persuade My Lordship of being wrong, simply that that the reasonable chance that another court may come to a different conclusion," Maleka said during proceedings.
The judge then accepted Maleka's explanation.
Judgment will be handed down on Tuesday.