Practically speaking yes, this will almost certainly be a step-down transformer, but I would agree with those other teachers: it can't really be concluded from the wire thickness. It's perfectly possible to build a transformer in which the secondary coil has more windings, but nevertheless use thicker wire.
As John Rennie explained, this doesn't normally make sense, because then there would be more AC current in the thinner primary wire than in the secondary, though that could cope with more. However, that's not necessarily the only current that's relevant: there are many applications where the voltage stepping is not even the reason to use a transformer but to decouple voltages and/or currents. In particular, you could have a high DC current or low-frequency AC flowing on the secondary side, and use the transformer to modulate a much weaker, high-frequency control signal on top of it. In this case, you will need a thicker wire on the secondary side regardless of the winding ratio.