In the recent HfL blog ‘Get outdoors and do some maths’, Deborah Mulroney refers to the skill of estimation and how children can be reticent to use it because they ‘might get it wrong’. She mentions the premise that, as adults, we realise that it is worth pursuing as is it is a relevant life skill. The blog shows an example of estimating how much decking would be needed to make a wildlife viewing area before creating it exactly. This reminded me of my father, who was a builder and very adept at estimation as it was an integral part of the professional skills that he needed, not only in the process of construction but also in the cost of materials, the estimation of the time it would take and how much to charge to make a profit.
This led me to further investigation regarding my understanding of estimation as a mathematical concept. Looking at the content of the National Curriculum for Mathematics (2014), there are fourteen specific references to ‘estimate’ (and one to ‘approximate’) across the primary age range. This confirmed its importance and relevance and in exploring research articles, I discovered the idea of ‘Visual Estimation’ as a key skill which is defined as ‘the process of using your eyes and mind to come up with a value that is close enough to the correct answer, without actually counting or otherwise measuring an object.’ In ‘Estimating – Making sense of things’ (2017), Mark Chubb refers to ‘too many students being asked to DO something instead of them being asked to THINK about something’. I realised that this was my father’s key skill and he was fluent in ‘Visual Estimation’ which had been developed over many years of practise in thinking about real life situations. It never ceased to amaze me how he could tell me how many bricks were needed to build a wall or how many floor tiles would be needed to cover a floor just by looking at them. It also made me realise that perhaps we do not spend enough time exploring it when we teach maths.
For me, this follows the idea that children learn maths successfully through handling objects, representing them in drawings and using pictures to explore situations (a common theme running through ESSENTIALmaths) to gain approximate answers before carrying out exact calculations. In the blog: ‘Ever thought about what comes before counting? Pre-number learning in the early years’ and in the resource ‘Essential Foundations for Counting’, the development of number sense is paramount. Its relation to the understanding of number magnitude (the size of numbers in relation to zero and to each other) is discussed as being an integral part of developing effective mathematical understanding. Without this understanding, estimation would not be possible but it can also be augmented; allowing children to think more deeply by being involved in exploration, which can be done in both the home and school environment.
In the Early Years and Year One, use counters or cubes (or coins or any objects that are the same size as each other) and tens frames (which can be drawn). Lay the objects out initially and ask the children to estimate whether the amount looks greater than or less than ten and twenty. The correct answer can then be checked and rehearsed with different numbers. This can be used as a game where the person closest to the correct amount wins. This will not only enhance ‘number sense’ but will also improve the children’s understanding of their ‘reasonableness of answers’ where ‘wild’ guesses will be lessened.
As children learn their times tables in Years Two, Three and Four, this can be extended into arrays. As shown below, objects can be spread out first and the children could be asked, ‘What array can you make?’ They can then estimate the total by visualising arrays that could be made to make the approximated amount and then checking their answers. Ask questions such as, ‘Could you do this another way?’, ‘Can you make a different array to show the same amount?’ or ‘Could you make two arrays that when added together, show the approximate total?’ This can also be made into a game of ‘Estimation’ where the closest answer wins. It could include, for example, 26 counters being used as an array of 5 x 5 with one object on its own. This will improve the children’s skills of recognising times table within amounts.
In Year 3 and 4, the children explore times tables, multiplication and division further and Kate Kellner-Dilks introduced me to the use of pasta as an object to enhance these skills as it is usually readily available. You could take a pile of it and ask children to estimate the amount within a range e.g. between 10 and 20 pieces. This might be difficult initially but with practise, children will get used to the idea. The objects can then be put into approximate smaller piles in a line (a number line) to gain a clearer picture and the children can use repeated addition or multiplication to reach an estimation of the total amount. Remember that the idea is to find an approximate amount initially through visual estimation and a focus on the magnitude (size) of a number, relieving the ‘pressure’ of having to gain a perfectly correct answer. In the example below, an initial estimation of between 30 and 40 was realised as there are 6 groups of approximately 6. A key question to ask would be, ‘How else could this be done?’ This will further enhance the children’s use of ‘subitising’ (recognising the amount of objects or symbols in a group without counting them individually).
The example above could also be seen as being rectangular in shape and this could be explored by the children looking at how area is calculated using squares. Draw a rectangle (or a square) and one smaller square and ask the children to estimate how many squares would fit into the rectangle visually rather than drawing a completed grid and asking them to count or calculate the amount. This would allow an initial focus on visual estimation rather than calculation. You could then check how close the estimate was to the actual amount by drawing or cutting them out.
This can be practised with different sized rectangles or squares, with larger or smaller squares of paper to fit inside, to further explore number sense. It could also be extended into ‘compound shapes’ (shapes which are made up of two or more simple shapes) to understand the further complexities of calculating area.
Ask the children to estimate how many squares would fit inside the shapes. Would the total in each shape be the same or different? Why? How could we use the two parts to estimate the total area of the whole shape?
The idea of using squares and rectangles as visual representation can then be further explored by placing the pasta differently as shown below where the initial estimation of between 30 and 40 is represented visually as 6 groups of approximately 6 in the shape of a different rectangle.
You could also define the number of groups by using times tables that the children know and you could also refer to the inverse operation of division (36 ÷ 6 = 6). Again, a key question to ask would be ‘How else could this be done?’ This can be carried out many times with different amounts and you could also make more than one pile with different amounts and ask the children to compare by estimating the difference between the amounts. This could also be explored with different sized objects e.g. how much space would the same amount of Lego® pieces (of the same size) take up?
As children progress through Key Stage Two, they learn different ways of calculating and working with larger amounts including more complex multiplication. In the example below, the initial amount of coins might be more difficult to estimate in a large group but if the coins are put into rows and columns showing approximate amounts, the picture becomes clearer.
If the first column has approximately 25 and there are 6 columns of approximately the same amount then the estimate would be 25 x 6 = 150. This could also be calculated as five rows of 30 being equal to 150. The children might also think of other ways of estimating the total amount using their multiplication skills and show different combinations of how the amount of coins could be represented. They might be grouped in tens, for example. This could also be used to enhance their understanding of unitisation (where objects or symbols have a value other than ‘one’) by estimating the actual amount of money, which could involve multiplying the answer of 150 by 2 (as each coin is worth 2p) which would equal 300p or £3. The initial task then would be, ‘Estimate how much money is in the pile’.
Using the same idea, this could also be investigated through drawing pictures. The difference here is that the objects cannot be moved so the method of estimation is trickier. In this example, a grid has been drawn and then dots have been drawn into the grid. This allows children to be visually aware that the area has been divided into approximately equal sections (squares) but the amount in each section varies between 2, 3 and 4. Using a visual estimate, children might identify that each square contains roughly 3 dots. Using their knowledge of calculating area, the children could then identify the total number of squares by multiplying the number of squares in the length by the number of squares in the width (7 x 5 = 35). This would then be multiplied by the approximate amount of dots in each square and would result in 35 x 3 and an overall estimate of 105. This would also enhance the children’s understanding of unitisation as each square represents an amount which is different to one.
This could be further extended by drawing a crowd of people or other items or symbols and asking the children to draw a grid to help them with their visual estimation. This could also be carried out online with pictures of shoals of fish or crowds at a football match, for example. These pictures could either be printed or children could use software to draw grids on the online pictures.
All of the activities I have described can be repeated and adapted many times to allow the children to gain a conceptual understanding through visual estimation. Remember that the idea is not just to get to the right answer but to allow the children to value the concepts of estimation and approximation through ‘thinking’ about the amounts and how they could be represented before they ‘do’ the calculation. This will enhance their number sense and understanding of number magnitude, unitising, subitising and the reasonableness of their answers. They will be able to visually recognise instantly, for example, that the jar below contains approximately 120 jelly beans.
In my experience of teaching maths across the primary age range, some children can generally become aware of numbers and amounts without the realisation of ‘what they actually look like’ in context. I see the teaching of visual estimation as being an effective strategy in achieving that outcome. In ‘The Power of Estimation’ (2008), Francisca Rebullida explains that it is important to teach the students estimation strategies because it will help them understand the magnitude of small and large numbers. They will be able to take a guess with common sense when they fully grasp the concept. Thus, estimated numbers will be meaningful to them, and they should use the skills learned as often as possible in their everyday life to maintain them; just as my father did.
My research also led me to the book ‘Ish’ by Peter H. Reynolds, in which a boy learns to enjoy the process of drawing rather than always worrying about the end result. When he thinks his drawing of a vase doesn’t look right, his little sister says it looks “vase-ish,” or close enough to the real thing. My general aim in this blog is for children to appreciate the value of utilising ‘ish’ and become fluent at it to enhance both their mathematical and their life skills.
Other ideas to develop visual estimation can be found in ‘Maths Everywhere’ and on @Hertsmaths #finditfriday:
Reynolds, P. H. ‘Ish’ (2004) Somerville, Massachusetts: Candlewick Press
National Council of Teachers of Mathematics