Post Made Community Wiki by François G. Dorais
2 added a bit abut marking

Students must surely get used to the language of set theory, but are in difficulties because of a lack of a training in grammmar.

I found it necessary in an analysis course to go through a proof of a particular statement of the form $A \subseteq B$ by asking: "What is the first line of the proof? and getting them to see it has to be: "Let $a \in A$." and then lower down the board ask for the last line, i.e. "Thus $a \in B$." After doing this on a number of occasions they get the idea.

I have also used the teaching method of "reverse chaining" (see also "backward chaining" in wikipedia) to teach proof structure. You write out a proof say that the limit of a product is the product of the limits, for which there is little chance they could do from scratch; then you blank out bits, so there are still lots of clues, and the exercise is to fill in the bits, using the clues from the rest of the proof. This can be analogous to one of us writing out a sketch proof and then trying to filling in the details. This is also very easy to mark!

See also wikipedia on "errorless learning". This, like reverse chaining, is based on the idea that you learn from success, which is standard in animal training, and for children, and is also true for grown up humans! (Surprise, surprise! But actually this was explained to me by an excellent psychologist in the 1960s on observing me teaching my handicapped child a simple sorting task, and saying: "You are very good and clear about saying "No" and less clear about saying "Yes". You should do it exactly the other way round." I thought: "That is a good lesson for a mathematician!" Also she suggested making the cards bigger and clearer!)

Another problem is that the language we use today in mathematics has a certain artificiality. When we say $2 + 2 = 4$ we don't mean the left had side is the same as the right hand side, but that the operation of adding gives the right hand side. So what ever language you use, a student has to get used to it, and used to expressing things in that language. John Baez gave the example that the picture

$$\matrix{|| & ||| \cr ||& |||}$$

is much clearer that the expression $2 \times (2 +3)= 2 \times 2 + 2 \times 3$, which uses all sorts of conventions.

The above is a bit rambling but I hope has some useful points!

1

Students must surely get used to the language of set theory, but are in difficulties because of a lack of a training in grammmar.

I found it necessary in an analysis course to go through a proof of a particular statement of the form $A \subseteq B$ by asking: "What is the first line of the proof? and getting them to see it has to be: "Let $a \in A$." and then lower down the board ask for the last line, i.e. "Thus $a \in B$." After doing this on a number of occasions they get the idea.

I have also used the teaching method of "reverse chaining" (see also "backward chaining" in wikipedia) to teach proof structure. You write out a proof say that the limit of a product of the limits, for which there is little chance they could do from scratch; then you blank out bits, so there are still lots of clues, and the exercise is to fill in the bits, using the clues from the rest of the proof. This can be analogous to one of us writing out a sketch proof and then trying to filling in the details.

See also wikipedia on "errorless learning". This, like reverse chaining, is based on the idea that you learn from success, which is standard in animal training, and for children, and is also true for grown up humans! (Surprise, surprise! But actually this was explained to me by an excellent psychologist in the 1960s on observing me teaching my handicapped child a simple sorting task, and saying: "You are very good and clear about saying "No" and less clear about saying "Yes". You should do it exactly the other way round." I thought: "That is a good lesson for a mathematician!" Also she suggested making the cards bigger and clearer!)

Another problem is that the language we use today in mathematics has a certain artificiality. When we say $2 + 2 = 4$ we don't mean the left had side is the same as the right hand side, but that the operation of adding gives the right hand side. So what ever language you use, a student has to get used to it, and used to expressing things in that language. John Baez gave the example that the picture

$$\matrix{|| & ||| \cr ||& |||}$$

is much clearer that the expression $2 \times (2 +3)= 2 \times 2 + 2 \times 3$, which uses all sorts of conventions.

The above is a bit rambling but I hope has some useful points!