I remember reading a patent about an educational calculator that used a regular calculator chip, and had external circuitry to decide if an entered equation was correct. I bet that's what the 2nd PCB is for. I wonder if I can find it again.
Edit- it was actually Coleco Quiz Wiz's patent US4303398. One implementation was to use a calculator chip and have the inputs set up so that when you punched in a question #, you were entering an equation. When you put in your answer, that was subtracted from the answer the calculator chip got from the equation. Then that difference was used as a divisor, so if it was 0 (correct), a divide-by-zero error lit the green LED. Whew! Fig. 9 in that patent shows PLS, CHK and CLR keys, which are on the Lil Genius. The Quiz Wiz that I got used the other implementation, which was much more user-friendly.
And while I was searching for that, I hit patent US4051605 for an educational calculator from National Semiconductor using the MM57 chip. Quiz Kid Speller and Mattel Basketball use the NSC MM47. The patent has an instruction list and source code. It calls the chip a "Calculator Oriented Processor" or COP. I haven't found anything on the MM47, but there's quite a bit on the MM57: http://www.cpushack.com/2014/09/27/national-semiconductor-the-cop-before-the-cops/ Hopefully the MM47 is similar enough that this is enough info to figure it out.