Do you teach to the test or for understanding?

Last week I gave my students a copy of the article by Richard Skemp. It is a rather old article but unfortunately Mathematics education doesn't seem to have moved on.

He discuses the two different types of learning in Mathematics, relational and instrumental.

So what are the main differences about these two different types of understanding?



Instrumental understanding is learning associated rules without any reason. It is is simply learning a rule and applying it to the same type of question over and over again. For example, the area if a rectangle. The pupil will know the formula, base X height and will be able to apply it to low level questions, usually with quite a high success rate. However, when the question changes slightly, for example, the area of a rectangle where the base is 50cm and the height is 1m. The pupil would just recall the rule and wouldn't think about having to deal with the two different units. This is extremely low level understanding that requires lots of examples and chalk and talk. It is very easy to teach but does it have a high success rate? I have seen many different type of teacher who teaches rules rather than understanding and their exam results are sometimes usually impressive. This really annoys me, the pupils have not appreciated what maths can do and are unable to solve simple problems, unless the teacher has gone through every possible type of question associated with that topic (I am not sure that this is possible). The pupils become obsessed with just applying a rule and finding the answer rather than thinking for themselves. These types of teachers are TEACHING TO THE TEST and are not helping pupils develop their problem solving skills so they are able to use maths later on in life.
This type of traditional method of teaching is what Mr Gove is aspiring towards, even though he contradicts himself all the time by suggesting that he wants pupils to be equipped for life, which usually means they are able to use the skills they have learnt in school in real life and most importantly in their job.


The other type of understanding is relational which looks at a deeper aspect of understanding. Relational understanding allows the pupil to understand why and how. It gives them time to ask questions to further their understanding and because of this they are more likely to be able to transfer their skills to another problem or topic. They are more adaptable and they strive to do their own research.
This type of understanding can be difficult and time consuming for many pupils but if a pupil actually understands why they are more likely to be able to reuse it.

So here are the positives for instrumental.

• Easy to remember rules

• Easy to teach 
• Allows pupils to become motivated by a page of correct answers (pupils like to get the answer correct). 
• Quicker to teach. We are able to get through the curriculum and on to past papers quicker, ready for the examinations. 
• Restores success with low ability pupils as it's easier for them to remember a rule rather than actually understand why. 
• Less thinking is involved so pupils will think it is easier. 

Positives for relational. 

• More adaptable to new tasks. Pupils should find it easier to transfer their skills to other topics and subjects. 
• Easier to remember. This is because they can usually derive their own formulas.

• It can motivate pupils to understand why and how which makes the teachers job a lot easier. 
• Pupils should actively seek out new ideas and do their own research. This will make pupils become independent learners. 

So looking just at the positives which type of teacher would you want to be?

I am concerned about the lack of mathematicians in England, with such a short uptake post 16 we are going to be desperate for decent graduates in a few years time. The focus on examinations can also be  difficult. I know that all teachers have got to keep a close eye on the league tables, especially since   performance related pay but why are we so obsessed with rushing through the curriculum? I know that relational understanding can take more time, time that we don't always have. But know Liz Truss has announced that Maths will become double weighted hopefully schools will now give us more time at KS4 to teach relationally. I do also understand that relational understanding can be very difficult and lots of students will find it difficult but this is something that needs addressing early on in their education so that they are able to deal with failure when learning. The other major issue is time, teachers barley have enough time to plan a normal lesson, never mind one that allows for relational understanding. Teachers in Canada are given one hour planning for every hour they teach, why is this not the case here in England?

And finally, the pressure from other staff to teach instrumentally because they have a quiet classroom where all students are on task and they all get a decent result at GCSE. You need to take a stand, children need inspirational teachers who want to provide the tools needed to become independent problem solving learners who are NOT afraid to fail. Teaching relationally won't be easy, the pupils will find it hard at first and so will you, but the benefits are far greater than just teaching them rules without reasons!!!