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Primary Magazine - Issue 88: Digging Deeper


Created on 19 May 2016 by ncetm_administrator
Updated on 07 June 2016 by ncetm_administrator

 

Primary Magazine Issue 88'Digging' by Suzette (adapted), some rights reserved
 

Digging Deeper

Learning Multiplication Facts

The aims of the National Curriculum for Mathematics include a focus on fluency. There is a danger that fluency can be interpreted in a very narrow way, equated with speed and memorisation without an understanding of relationships, but this is not the intention, as the aim clearly states “…that pupils develop conceptual understanding and the ability to recall and apply knowledge…” (National Curriculum 2014).

Russell (2000) suggests that fluency “rests on a well-built mathematical foundation with three parts:

  • an understanding of the meaning of the operations and their relationships to each other - for example, the inverse relationship between multiplication and division;
  • the knowledge of a large repertoire of number relationships, including the addition and multiplication "facts" as well as other relationships, such as how 4 × 5 is related to 4 × 50;
  • a thorough understanding of the base ten number system, how numbers are structured in this system, and how the place value system of numbers behaves in different operations - for example, that 24 + 10 = 34 or 24 × 10 = 240.”

This focus on understanding relationships is important to consider when thinking about the learning of multiplication (and related division) facts; pupils will benefit most if the learning of multiplication facts is embedded within learning about multiplication. This will involve exploring multiplication in relevant contexts, representing the situations with mathematical images which expose structure and relationships, describing these using mathematical language and connecting all of these to a symbolic representation.

There are various strategies and approaches that can be used; the important starting point in all cases is to identify what the children already know and understand and build on this. The following questions are intended as prompts to support thinking about how to approach the teaching of multiplication facts as part of the teaching of multiplicative reasoning:

  • Do the children understand what ‘× 1’ means and can they use this understanding to multiply any number by one? This can be modelled using a Cuisenaire rod (or the bar model) alongside the language of multiplication and the symbols of multiplication, which will allow for generalisation.
    Cuisenaire rod
    For example: “I have one of the yellow rods. I have the yellow rod one time. Yellow rod multiplied by one equals yellow rod. If the yellow rod is eight, I have eight one time. Eight multiplied by one equals eight. If the rod is… ” matched to ‘Yellow × 1 = Yellow rod’ and ‘8 × 1 = 8’
    Contexts for multiplying by one could be linked to single or unique events or situations. For example: linking 5 × 1 = 5 to ‘in my purse all I have is one five pound note’; linking 80 × 1 = 80 to ‘tea bags come in boxes of eighty and we have one box’; and linking 250 × 1 = 250 to ‘bags of raisins weigh 250g and there is one bag.’
  • Do the children understand what '× 0' means and can they use this understanding to multiply any number by 0?
  • Do the children understand that doubling is the same as multiplying by two (× 2)? Explore doubling in different contexts and model using Cuisenaire rods (or the bar model).
    two Cuisenaire rods
    For example: “If I double a purple rod I have two purple rods or I have a brown rod. If the purple rod is ten, then the brown rod is double this, it is twenty; 10 × 2 = 20”. Cuisenaire rods and the bar model can be used to show how repeated addition and scaling are both part of multiplication.
  • Do the children understand how repeated addition can be represented as a multiplication? This can be modelled by making a visual link between counting in steps the children are familiar with and the building up of an array using rows of dots (rather than individual counters). For example, building up the array below, row by row saying “There are five dots in a row. We have five dots one time, two times, three times, four times. We have five dots, four times, five multiplied by four (5 × 4). We have five, ten, fifteen, twenty dots. Five multiplied by four equals twenty.” Contexts for building up arrays include planting vegetables in rows, setting out chairs in the hall and chocolates in boxes.
    five rows of five dots
  • Do the children understand the commutative property of multiplication? This can be modelled and explored with rectangular arrays, cut from squared paper, or in a context; it is important the children can explain how both multiplications are represented by the same array. For example, for the array below, discuss how this shows 5 × 4 - ‘There are five children in each row and there are four rows’ - and 4 × 5 - ‘The children are lined up behind each other. There are four children in each line and there are five lines.’
    four rows of five figurines
  • Which of the multiplication facts for twos, fives and tens do the children know/recall? From the points above, they should know that they know: 2 × 0, 2 × 1, 2 × 2, 5 × 0, 5 × 1, 5 × 2, 10 × 0, 10 × 1, 10 × 2. Support them to generalise about tens by focusing on the pattern of the calculation linked to an image (for example base ten or Numicon ten plates).
  • Do the children know how to use what they know to work out other facts, including:
    • doubling and halving – connecting the five and ten times tables, connecting the two, four and eight times tables, connecting the three and six times tables, connecting multiplying by ten with multiplying by five, connecting multiplying by two with multiplying by four and eight and connecting multiplying by three with multiplying by six
    • adjusting (one more or one less) – connecting the ten times table with the nine times table (use the array to model) and other known multiplication facts with nearby facts; for example using 6 × 6 for 6 × 7.
  • Special cases - are the children aware of different ways other children recall any of the facts including special cases, such as remembering 7 × 8 by thinking ‘5, 6, 7, 8…56 = 7 × 8’.

One teacher who has done some in-depth work in this area, as part of a Master’s degree, is Katie Crozier, a primary mathematics teaching for mastery specialist teacher in the Cambridge Maths Hub. She says:

When asked about strategies used to recall times table facts I found two responses were common: a) I just know it; and b) I count up in ___s. With further questioning, I started to realise that many children didn’t appear to have a strong visual representation of the structure of multiplication in its form of repeated addition. The times table facts were isolated points lined up in a sea of fog where the structure and connections lay hidden. I spent time devising a representation that clearly exposed the three parts to multiplication...

Read more about Katie’s research and her representation to support learning of multiplication facts, and you can also view the NCETM video on teaching multiplication.

Next time in Digging Deeper we'll explore journal-writing in mathematics.

Image credit
Page header by Suzette (adapted), some rights reserved

 

 
 
 
 
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Comments

 


16 October 2016 09:13
The associative, commutative and distributive laws in relation to the four operations are an absolute must for every primary school math's teacher and it is such a pity that new maths textbooks totally ignore this, especially the Abacus Evolve which my class uses at present. Unfortunately textbooks are about a mile wide in things to cover but about inch in depth when it comes to conceptual understanding.
By mizuno
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29 September 2016 11:38
Love the mathsticks site! Thanks johdu.
09 June 2016 12:00
I have started to use a multiplication grid which shows arrays as arrangements of squares. It can be found on the math-aids website in the multiplication section. They call it a 'sized' chart. Still trialling it to see if it helps the children to 'see' their times tables.
By G00fey
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07 June 2016 14:24
Thank you, this is very helpful. I've discussed similar methods to assist children developing fluency as part of CPD activities, and here: http://mathsticks.com

John
By johdu
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