The beautiful array

    Published: 03 September 2019

    Imagine this situation (adapted from Hackenberg, 2013)

    In a classroom, there are four rows of desks with 6 seats in each row. The teacher then added three more rows.

    Hopefully this has triggered a visualisation, the four original rows of desks and the new three additional rows. We can see both dimensions – the number of rows and the number of seats in each row, and we understand how they relate to each other. For example, increase the number of rows, the number of seats will increase in a proportional pattern.

    But what do your pupils think has changed? The total number of seats has changed from 4 x 6 but to what? 7 x 6? 4 x 9? Pupils know that there is ‘a three more’ involved somewhere – but where exactly? They may be unable to translate multiplicative situations into arrays and back again with any level of confidence

    Research that the Herts for Learning primary mathematics team are currently undertaking, reveals that pupils' understanding of arrays is very fragile. Few of our LKS2 pupils were able to construct an array confidently at the beginning of the project.

    Even when they were able to construct arrays accurately, probing a little deeper revealed that they don’t really understand what they built, beyond putting the correct number of dots in the correct number of rows. Many can't adjust the arrays to match a changing situation nor link the arrays to multiplicative stories. These children aren’t able to coordinate the two different dimensions, instead they over focus on one dimension, either the number of rows or the number of items in a row (or number of columns). The problem is that this lack of understanding can stay hidden on our classrooms until about mid-year 4, then many pupils start to hit the multiplicative rocks.

    I would argue that pupils can be compliant in our classrooms – drawing the requested arrays, but they are unable to describe, play with or explore them.  Therefore the key of why we use arrays is missed. If they are not fully understanding the arrays that they are constructing, then how can they possibly visualise more complex situations which require them to compact the models into composite units? Many pupils became confused when they were asked;

    ‘f you know 5 x 7 is 35, how can you use this to solve 6 x 7?


    Is it one more group of 7, one more group of 6 or just one more?  Understanding this will help them cope with questions such as;

    SATs question
    KS2 SATs mathematics paper 3

     

    Consider the calculation 4 x 6.  A child who is still developing understanding may see this as four groups of the unit 6 (or a unit of 4 six times).

    array

    They will, with guidance from teachers, represent this as an array.

    array

    But what do they really see? Many pupils would still solve the question of how many dots through either skip counting (6, 12, 18, 24) or a form of hybrid counting (6,12,13,14,15, 16,17, 18,19, 20,21,22,23, 24). Which, whilst a fine way to solve this problem, doesn’t necessarily;

    • support pupils in developing strategies which they can use to help solve other calculations that go beyond counting efficiency
    • develop understanding of scale/ratios/multiplicative magnitude
    • develop what is different between additive and multiplicative situations

    Beware the fact barkers

    A pupil who is fluent in their times tables would be able to suggest that  4 x 6  is 24 – they are not having to use up cognitive space to track counting or repeated addition which is brilliant and so helpful, however through the course of our research we are starting to stumble upon ‘fact barkers’. These are pupils who are able to recall multiplication facts within the 12 x 12 Multiplication tables check requested range – but they have no conceptual understanding of what it really means and therefore cannot apply or adapt their knowledge in any way, without support these pupils can stall and struggle in UKS2 and beyond.

    It is not enough that they will pass the MTC.

    Strategy Development.

    This blog is limited to considering how understanding an array can support strategy development.

    A pupil who understand arrays, may not be able to INSTANTLY recall the correct answer to 4 x 6 but they can reconstruct it. They can also generalise so that they can apply these strategies to solve previously unencountered answers; they can adapt and interpret (like in the original problem at the top of the blog) and they translate from multiplicative situations into language/stories and vice versa.

    When presented with 4 x 6 a pupil who is developing deeper understanding of the array might be able to see;

    Commutativity

    deeper array

    Conservation of area

     

    conserving the area

     

    Related KS2 SAT examples from past papers

    Sats questions

     

    Distributive law (there are so many possibilities, I've limited the selection to include the more obvious)

    distributive arrays

     

    Related KS2 SAT examples from past papers

    SATs questions distributive

     

    Associative law using factorisation

    associative arrays

     

    Related KS2 SAT examples from past papers

    Sats questions

    Considerations

    How much time do your pupils spend just describing and playing with arrays?

    What do learners really see when you ask them to draw an array – is it just a collection of dots which they can use to help them count?

    Do they see the relationship between the two dimensions and how they are proportionally linked or are they stuck in some kind of additive reasoning loop?

    Personally, I would prefer a roomful of these curious tinkerers at this stage, some of whom may not be able to get to the correct response within the MTC time limit but have great visualisation and manipulation skills, rather than those who have spent two years being drilled to recall times table facts but have nothing of the structure.  Children with a great structural understanding of multiplication are far better prepared for learning to come in UKS2 and beyond than those without - whether they can bark the facts or not. 

    It is perfectly possible to have both speed and understanding - let's given them both.


    If you would like to explore the development of the array in greater detail, then we would love to continue the conversation with you via;

    Joining the ‘Making sense of x – building multiplicative fluency in LKS2'. This is a genuine classroom based research project explore how pupils perceive multiplicative situations (including but not limited to the array) and what we can do to improve their understanding.

    Attend one of our fantastically engaging central courses:
    Mental maths: the secrets to success KS1
    Mental maths: the secrets to success LKS2
    Mental maths: the secrets to success UKS2


    References:

    Hackenberg, A. J. (2013). The fractional knowledge and algebraic reasoning of students with the first multiplicative concept. Journal of Mathematical Behavior, 32(3), 538–563. doi:10.1016/j. jmathb.2013.06.007

    DfE Maths SAT papers 2016, 2017, 2018 and 2019.

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