### More Constructivism

There's another aspect to my growing sense that Constructivism is a powerful lens through which to view learning: it has changed the way that I interpret misunderstandings. Whereas I was sometimes guilty in the past of just thinking "they haven't got it yet" when a pupil gave a wrong answer, I now try very hard to figure out where the wrong answer came from, in the sense that I am thinking "what model of this concept have they constructed that would lead them to give this answer?"

To give an example, I was working yesterday with a group of S1 pupils who were trying to master the ordering of decimals to two decimal places. To be honest, they were really just trying to grasp the concept of decimal fractions, and ordering was the aspect we happened to be looking at. I was using the decimal darts described here, and we were trying to compare 0.32 and 0.5

I asked them to write the bigger number on their show me boards. Joe [not his real name!] wrote 0.32

"Is that because 32 is bigger than 5?"

"Yes"

"But remember Joe, decimals don't work like that do they?" [a really poor response from me!] " Let's just look at the tenths - which is bigger three tenths or five tenths?" [holding up the two yellow .3 and .5 darts]

"three tenths" [this momentarily stumped me!"

"Oh - did you say that because you know that 5 is bigger than 3, and I told you decimals don't work like normal numbers?"

"Yes"

So at this point, Joe hadn't just "not got it". He had tentatively constructed a model of decimals that said something like "you put them in order like normal numbers, then just reverse the order". That was a perfectly reasonable model, based on the poor response he had received from me. If I had shot him down at that point, and tried to stuff the "right" model into his head, he would have received the message from me that his attempt to construct some sense out of the situation had been forlorn. I firmly believe that "kids that can't do maths" are often kids who have repeatedly experienced feedback which tells them that their intuitive construction of understanding in maths is flawed. In the face of such feedback, it is hardly surprising that they reach the conclusion that they can't do maths.

To give an example, I was working yesterday with a group of S1 pupils who were trying to master the ordering of decimals to two decimal places. To be honest, they were really just trying to grasp the concept of decimal fractions, and ordering was the aspect we happened to be looking at. I was using the decimal darts described here, and we were trying to compare 0.32 and 0.5

I asked them to write the bigger number on their show me boards. Joe [not his real name!] wrote 0.32

"Is that because 32 is bigger than 5?"

"Yes"

"But remember Joe, decimals don't work like that do they?" [a really poor response from me!] " Let's just look at the tenths - which is bigger three tenths or five tenths?" [holding up the two yellow .3 and .5 darts]

"three tenths" [this momentarily stumped me!"

"Oh - did you say that because you know that 5 is bigger than 3, and I told you decimals don't work like normal numbers?"

"Yes"

So at this point, Joe hadn't just "not got it". He had tentatively constructed a model of decimals that said something like "you put them in order like normal numbers, then just reverse the order". That was a perfectly reasonable model, based on the poor response he had received from me. If I had shot him down at that point, and tried to stuff the "right" model into his head, he would have received the message from me that his attempt to construct some sense out of the situation had been forlorn. I firmly believe that "kids that can't do maths" are often kids who have repeatedly experienced feedback which tells them that their intuitive construction of understanding in maths is flawed. In the face of such feedback, it is hardly surprising that they reach the conclusion that they can't do maths.

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