3. Arrays & The Grid Method (Mike Thain)
Wow! One blog a day was pretty ambitious of me! I wasn’t able to keep that up along with planning, data, marking oh & a little bit of sleep as well. So here it is after a short delay my third blog about MathsCon from last weekend, if you haven’t read my first two you can find them here & here.
So my third session in Bristol was on Grid Method & Arrays hosted by Mike Thain. If you’re a maths teacher you are probably quite familiar with the concept of multiplication known as The Grid Method, some of you may know it as The Area Model a point I mentioned to Mike at the beginning of the session.
As a bit of a starter (or Do Now!) Mike posed the following question:
So the main ‘go to’ methods you thought of were most probably:
1.The Column Method (Long Multiplication)
3.Napier’s Bones (Lattice Method)
WHY WOULD YOU?!
Why so many methods? Why do students use what could be considered a method that’s ripe for misconceptions & errors (long multiplication)? Especially when you consider the student’s journey through primary school using multiplication.
Key Stage 1
Mike began by outlining how a student first sees the idea of multiples & multiplication. Arrays seem to be the order of business in the early stages. Students look at counting on in numbers larger than 1, they link these ideas to arrays & multiplications, using Concrete manipulatives.
In year 2 this concept is extended into grouping & sharing, again the concepts are explored through the notion of arrays & the use of manipulatives.
The use of arrays is further explored through the times tables & fact families in year 2.
Key Stage 2
In year 3 the use of arrays is EVEN FURTHER explored & linked to times tables, multiples again using concrete tools to manipulate them.
In years 4 & 5 a more formalised written method is introduced to students & we begin to see a move away from arrays toward the column method (which is essentially an abstract concept with little ability to be represented by concrete manipulatives), a few weeks ago Mike posed the following question to KS1 & KS2 teachers:
Notice the shift from KS1 (33% using arrays all the time) to KS2 (16% using arrays all the time). Does this mean that the natural progression of areas represented in arrays toward areas represented in grids is subverted by the KS2 curriculum or is it taught along side the formal column method? Well, a bit of both really, since students are also taught to use the ‘bus stop method’ in order to develop a written method for dividing two & three digit numbers by a one digit number.
One problem there though. The ‘bus stop method’ is actually a reverse Grid Method! So this natural progression is explored to an extent but there is a necessity to teach the abstract column method.
Year 5 is also the year where students first explore the distributative law which is wonderfully illustrated by the grid method.
So students have a greater understanding of grid method than most KS3 teachers might think, in years 7, 8 & 9 when secondary teachers ask students to take on board the grid there is no real reason why they shouldn’t be able to adapt to the concept of the grid method. The formal methods required to be taught in year 5 could potentially undo a lot very valuable work completed in earlier years. So we should definitely push students toward fully embracing it as they have all the pre-knowledge & concepts buried in their memory. It is a very adaptable method & should definitely forge ahead with teaching it as it can be used to complete:
- multiplication/division of large numbers
- linear algebraic expansion/factorisation
- polynomial expansion/factorisation
- matrix multiplications (apparently)
It would appear that most KS3/4/5 teachers do, in fact, embrace the grid/area model as Mike also posed the same question to KS3/4/5 teachers. Perhaps if we make the links to arrays in our teaching we could rekindle our students’ love for multiplication again.
At the beginning of the blog I mentioned that grid can be seen as The Area Model, Mike touched upon this during his presentation.
Well that’s your lot for tonight, tomorrow, Jo Morgan’s session on converting units.