How Do We Know That We Know?


You may or may not have seen articles in national newspapers recently discussing the new ‘End of Year 6 Assessments’ which will focus on times tables.

Debate has been rolling as to whether these new assessments will be introduced nationally in the summer of 2017, with some parts of the Mathematic sector screaming that this is just another way of increasing the stress levels of students in Year 6.

At Bede’s, we are not particularly worried if these assessments do come along as we not only take learning multiplication facts seriously but we also celebrate the inverse, learning division facts too, well before the National Curriculum tells us that we should.

Indeed, we teach multiplication and division side by side, and aim to have all facts covered by the end of Year 4. And how they are taught - and why our method works - brings me to my next blog topic.


Richard Skemp (1919 – 1995) was a British pioneer who developed theories by intertwining Mathematics and Psychology. His main theory was based upon how children learn in Mathematics, either through Instrumental Understanding or Relational Understanding.

Instrumental Understanding is when a child learns through fixed ideas, procedures and end points, such as always concentrating on the answer to a calculation or problem.

This results in children building small ‘Schemas’ or mental plans to complete individual tasks in individual situations.

Relational Understanding, on the other hand, is when children learn conceptual understanding and build a larger ‘Schema’ or mental plan that can be built upon and used for a variety of smaller or larger calculations or problems.


I believe both Instrumental and Relational have their uses within Mathematics teaching, but strongly advocate the teaching of Relational Understanding first before moving onto Instrumental Understanding – and a perfect example of how I feel we get it right is how we teach division and multiplication.

If you squint, you may be able to see in the examples below the sorts of ways that children in Year 4 at Bede's Prep are taught how to multiply by grouping concrete objects first, before viewing a calculation with the ‘x’ operations sign.


When Bede's Prep mathematicians move onto division they are purposefully taught how to group objects equally and place objects into equal groups. The children are specifically taught that multiplication can be used to aid division, and are then taught how to complete the inverse of this.

I believe this leads to children making clear conceptual links between division and multiplication, which they then apply to solving worded problems.


In the two examples above, both Student A and B may show excellent division and multiplication knowledge on the surface, but there are some subtle differences conceptually.

Student B has misunderstood the concept of putting items into ‘equal groups’ rather than ‘groups of’ for question 2, and this conceptual misunderstanding has also crept into question 3 where student B has misunderstood question C also by creating groups of 7 rather than 7 equal groups.

The more students investigate and discuss these concepts the more flexible their approach is to problem solving, and this reduces the likelihood that they will make this sort of mistake.

Indeed, once this concept has been explored and understood the children find completing the ‘Superhero’ multiplication and division challenges we run at Bede’s much easier and more meaningful.


This side of our curriculum ensures that the pupils’ Instrumental Understanding is still being provided, but in a less stressful manner to reciting times tables orally and not applying them to problem solving.

Another example of relational understanding being used to aid learning, rather than memorising, is when children in Year 7 at Bede’s Prep have to find the area of a trapezium.

If taught instrumentally, children might be taught the formula of A = (a+b) / 2 x h and then spend a couple of lessons finding out the area of different trapeziums only to find out that three weeks down the line they have forgotten the method taught.


At Bede’s we offer the children plenty of time to build a conceptual understanding of how to find the area of different shapes, starting in Year 4, and this conceptual transition through the year enables children to build upon Schemas from an early age.

There are probably a million different ways to find the area of a trapezium, but it works best if the children find the method that works for them. The teacher can skilfully show them how the formal method is created, but even if the children cannot remember it, they at least have built a schema that if they do not recognise a shape, they could cut it into smaller shapes or think of other alternatives to succeed.


As can be seen from the picture above, with a little persistence and encouragement from a teacher, this student will have many more ‘Schemas’ to use in the future when coming across different shapes, rather than knowing the one formula which only works for one shape, without knowing why.

I believe that with this style of teaching, and having teachers with expert knowledge in mathematical pedagogy and psychology, the children leaving Bede’s Prep will be much better equipped at tackling the new GCSE syllabus and moving on to A Level Mathematics, while also enjoying the subject.

Finally, below is a small diagram which might help to make clearer what happens if a teacher can teach relational understanding and the students can also learn relationally.


It also serves as a food for thought about if we as a country are taking this research seriously when implementing the new Mathematics curriculum…


Read more from Mr Fasciolo-Barnes’ Maths Blog >

Learn more about the Bede’s Prep School Maths department >