Electron balance for biomass synthesis: is my professor right?
The terms are different.
Oxidation number is the difference in number of electrons (not pairs) from the elemental form (which for many elements exists only on paper) of an atom.
Degree of reduction is a more uncommon term that is basically the number of electrons that atoms in a molecule are donating per atoms of a given element. It is calculated formally by summing the number of electrons donated to reach a full shell (H=1, C=4, N=-3, O=-2, P=5, S=6 etc.) divided by the number of a given element, but that is more confusing than anything.
Carbon can be oxidised four times, e.g. methane > methanol > formaldehyde > formate > carbon dioxide (not biological route, just an example), while oxygen can be reduced twice: to hydrogen peroxide and then to water. As proof, most O2 dependent enzymes produce hydrogen peroxide (e.g. FADH2 enzymes). That is a cheat way to work out where those numbers come from.
I would recommend forgetting about the formal calculations of oxidation numbers —in biology oxidation numbers of carbon are generally not discussed.
I will open the brief parenthesis of the oxidation states of carbon just to point out they are not those: -1 for a bond with hydrogen, +1 with oxygen and 0 for a C-C bond. Carbon monoxide therefore has an oxidation state of +4, while a C in glucose (on average) has a -1. Molecular oxygen has two identical atoms therefore is 0. Elemental carbon has 6 electrons so hypothetically you cannot strip carbon more than +6. 
Here what is important is figuring out how many times a carbon in a molecule can be oxidised, that is how many electron pairs a molecule can give away as every time that happens an NAD+ (catabolism) or NADP+ (anabolism) is reduced, which is key to the cell. A dehydratase removes a hydroxyl group leaving an alkene, but it does not carry out a redox reaction: the geometry is no longer sp3, but sp2, and the oxidation state has decrease by 1 in that carbon, yet carbon kept its electrons —the other carbon increased by 1 as it deprotonated, so no net change and NADH is not generated.
Knowing how many electron pairs are given away going from glucose to carbon dioxide is important because cells cannot make them disappear and a terminal electron acceptor is needed either in the form of respiration or fermentation. There is a further concept, that of reduction potential. Different forms of terminal electron acceptors are employed in Nature: oxygen, Fe(III), sulfate etc. and these have different "energy" levels at which they accept the electrons (reduction potential, measured in volts), which translated to different amounts of energy that can be syphoned off during the reaction.