The teaching and learning of mathematics

'To do mathematics is to engage in an act of discovery and conjecture, intuition and inspiration: to be in a state of confusion - not because it makes no sense to you, but because you gave it sense  and you still don't understand what your creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost powerful beauty...'  (Paul Lockhart, 2002)

 'Pattern sniffing'
As toddlers, we find and create opportunities to unwittingly develop our natural 'mathematical powers' (Mason, 1998), which come about through touching and feeling, observing, exploring and classifying - pattern sniffing. This uninhibited curiosity allows children to notice 'maths' in daily life without ever giving it a name!  Soon after children start school, they are introduced to the formality of maths (procedures and memorisation), which has been responsible for leaving many children feeling scared and confused by the subject

"... [T]hey have been sent on the wrong path, from an early age, of trying to memorise methods..."  (Boaler, 2015)

Maths has the potential to be a very abstract subject. For this reason, it is essential that maths is taught in creative ways to support understanding and that connections are made to further support learning.  At HPS, we support mathematical understanding through concrete, visual and abstract representations during our lessons and for all learners. The resources we tend to use most are as follows, but don't be surprised if a Mars Bar, banana or slice of bread, cut up into 100 pieces, ends up being used to support thinking in the classroom.

Using actual resources

Using images

Imagining/ Expressing

Counting sticks
Cuisenaire rods
Base-Ten apparatus
Place value counters

Number lines
Bar models
Place value grids
Ten frames
Hundred squares
'Cherry trees'


Mastery (teaching for understanding)
We teach for mastery (depth of understanding), so that all children have daily opportunities to 'master' the mathematical content being taught - improving skills, building confidence and developing understanding. Contrary to popular opinion, 'mastery' isn't a special type of curriculum: it's a way of teaching maths that encourages children to make connections across the subject matter in ways that further thinking. Maths becomes all about exploration and discovery - opportunities to develop fluency with number, reason mathematically ('prove it/ explain your thinking') and apply this to problem solving questions.

To ensure that children are demonstrating their growing mastery (depth of understanding) of mathematical concepts, we actively encourage the following during lessons:

1. Representing a problem in multiple ways (pictorial and concrete)
2. Describing a concept to someone else... so that the other person understands it
3. Making up own examples of a problem
4. Specialising and generalising - starting with one set of numbers and extending this to find mathematical proof
5. Finding exceptions to the rule
5. Applying a known concept when working with the unknown (e.g. If I know that 8x3=24, I also know 2400 divided by 8).

Mastery isn't about introducing accelerated content or providing larger numbers to support harder work, but ensuring that children can extend their thinking more laterally. For example, a child in year 6 with a good grasp of the concept might be given the following problem to consider:  Three numbers have a difference of 3.22.  The first number is 8.091. What could the other two be? How many combinations can you think of?

Workshops for parents
In February 2017, we ran a workshop for parents, to explain the way in which we are now teaching maths at HPS, involving plenty of opportunities to explore the resources and use these to solve maths problems! Please let us know if you would like us to deliver this workshop again.

How can parents support their children with maths, at home?
1. Encourage your child to play maths puzzles and games - anything with dice will work!  
e.g Snakes and ladders, Yahtzee, Monopoly, maths games on websites and apps.
2. Find ways to use all aspects of measurement (time, mass, distance, capacity)
e.g. measuring ingredients for cooking, duration of tv programs/ movies, bus/train timetables, weighing produce at the supermarket
3. Encourage your child to use the methods that they are learning at school (e.g. bar models, the cherry-tree, fraction bars)
- these methods enable children to develop a deeper understanding of maths, which will eventually support the more abstract, rules-based approach that they will be introduced to at secondary school
4. Never associate maths with speed - it has the potential to create anxiety
e.g Don't use timed tests, speed drills or flashcards to test automatic recall
5. Rather than focus on a wrong answer, try focusing on the logic involved in the calculation.
6. Encourage number sense - the size of numbers and how they can be flexibily separated and recombined to make calculations more user-friendly
e.g. When working out 38 + 42, add to 38 to make 40 + take two away from 42 to make 40 
e.g. If I don't know 6 x 4, I know 3 x 4, so I can do that, then double it
6. Never share the idea that you were bad at maths at school!
We believe that everyone has the potential to be good at maths, through hard work and believing in themselves  With rich learning opportunities in the classroom, we support all children to become confident and capable mathematicians.
(Jo Boaler, Stanford University)