There was a viral post in the past week where a parent shared to social media their challenge to their child’s teacher for teaching that “1 divided by 0 is 0”. Eventually the teacher backed down, but explained away as “It was how I was taught in the 90’s”. What was interesting about the post was the huge divide in what those responding believed the solution to be (0 or 1) and often those with the ‘correct answer’ having their confidence in their interpretation knocked. 

Division is a core mathematical operation – one that is introduced within primary education and key concepts and applications are developed all through secondary and post 16 education, the aforementioned ‘division by 0’ embedding itself within elements of vectors, functions and calculus within post-16 education. Not having basic concepts in place creates a huge barrier for young people progressing through mathematics – and there are issues that I often see, even in high achieving students with regards the concept of division, which don’t manifest with the other mathematical operations.

The general structure of a division calculation is;

Dividend ÷ Divisor = Quotient

I will make use of these terms within this post – but to aid accessibility will only do so when no other alternative is appropriate. 

What is division?

One of the initial barriers with division is that there are two ‘schools of thought’ as to how to describe and conceptually model division. Both of these ideas stem from the same source, that division is the inverse process of multiplication – however how these manifest as cognitive processes differ. 

Division as Sharing: In my experience this is the general ‘go to’ of the lay-person with regards division. The “I have 10 cakes and share them between 5 people” approach to “10÷5”. Sharing of items is a concrete example that young people are aware of, this inversion of multiplication is also more in line with the multiplication calculation structure, can be tied ‘inverting the presentation’ of ‘times tables’, adapted to introduce simple fractions (e.g. 1 cake shared between 4 people) and can be used to support young people in an appreciation of rearrangement of mathematical calculations, an important pre-requisite for algebraic content that will follow in their schooling. Interestingly observation in studies such as Fischbein et.al (1987) and Mulligan & Mitchelmore (1997), have shown that as a ‘spontaneous concept’ the untrained approach to ‘division as sharing’ is often to treat the problem as a multiplication problem, using ‘repeated addition’ until the dividend is reached – students literally modelling the process of ‘sharing out items’ until the available resources are exhausted. How this method sits alongside multiplication can be seen below, as well as adaptation to simple fractions.

Division as repeated Subtraction: As stated, multiplication is essentially a ‘repeated addition’ (at least early conceptualisation, whilst working with integers). If division is the inverse process of multiplication, the inverse process of addition is subtraction. We can therefore conceptualise division as ‘repeated subtraction’. So for example “10÷5” can be approached by “How many times can 5 be subtracted from 10?”. Again this method can make use of previously taught concepts but it is arguably of greater benefit for fractional divisors and dividends at later point in study? Examples are provided below;

What does research say?

There will be some maths teachers and primary education specialists furious that I haven’t used the terms ‘Partitive Division’ and ‘Quotative Division’ – these are the terms as they tend to appear in curriculum and are the accepted technical definitions of the different approaches to division. These come about from the commutative nature of multiplication, that 5×2 is the same as 2×5. Speaking in generalities, division as sharing is ‘Partitive Division’ and division as repeated subtraction is ‘Quotative Division’.

The reason why I have avoided these terms in building the two division structures is that these structures often have more to do with how worded problems involving division are asked than the actual cognitive approach or visualisation that the student undertakes when solving the problem. For example, Fleischman et.al (1987) conducted one of the most famous studies into the ‘intuitive’/‘spontaneous’ approaches children have to division, and in that study the problem type was derived at creation due to the wording of the problem, not retrospectively based on the method or reasoning of the student. 

PARTITIVE EXAMPLE: In 8 boxes there are 96 bottles of mineral water. How many bottles are in each box?

QUOTATIVE EXAMPLE: The walls of a bathroom are 280cm high. How many rows of tile are needed to cover the walls if each row is 20cm?

(Example Problems from Fleischman et.al, 1987)

In the first problem the solution is going to be based on ‘things’ – the number of ‘bottles’ in each box. The 2nd solution is going to be based on the ‘amount – the ‘amount of tiles required’. However, how the student visualises and constructs their solution could be identical – both can be done using ‘sharing out’ principles, or both could be done using ‘repeated subtraction’ principles. Furthermore I’m more interested in the generalities, what if there was no context and a student merely had to calculate 96÷8? This is why I’ve avoided the Partitive and Quotative terminology prior to this point.

Primarily the research into the area of division is focussed on youngsters with limited (if any) formal teaching on the concepts – researchers looking into what ‘intuitive’/‘spontaneous’ concepts of division young people are able to construct. Fleischman et.al (1987) research led to a conjecture based on results that the Quotative method of division is not ‘intuitive’, ‘primitive’ or ‘spontaneous’ in its construction, and must instead by formally taught. In summing up their findings they also state that there are some evidence that the initial didactic models continue to subconsciously influence how children and adults think even after formal mathematical training on a concept – this could lead to the reasoning for ‘sharing’ being to ‘go-to’ empirically for the lay-person regarding the mathematical division concept.

The idea that ‘Quotative division’ is not ‘intuitive’ or ‘spontaneous’ is not one that is necessarily confirmed by future research. Correa, Nunez and Bryant (1998) found that whilst partitive performance is stronger and that the suggestion is that ‘sharing’ forms the initial development of a child’s knowledge of division, older students were able to begin to construct methods to solve quotative problems – even without direct instruction on the topic. Mulligan and Mitchelmore (1997) delve more into detail on the processes and justifications given by young people solving problems. Their research helps emphasise the point I made earlier on, that despite being a ‘Partitive division’ problem, students can often pursue a ‘Quotative division’ method of repeated subtraction. They used other classifications of method – although indicated that they eventually bundle into the same categories that Fleischman et.al described – the research again showing that ‘sharing’ methods were significantly more common – although a few students made use of ‘repeated subtraction’ as an intuitive method without direct instruction.

Other research such as studies conducted by Squire and Bryant (2002 and 2023) show that diagrammatic representations and how physical objects are grouped can be used as an intervention to support young people in solving types of problems. Students without prior formal teaching will improve in ‘quotative’ division problems if objects are grouped in such a manner that they can be directly allocated without adjustment. These interventions tended to have less of an effect as students grew older, again suggesting the eventual outcome of realising the two processes ‘are the same’.

My interest is more at this point, on “What happens after formal teaching?” – how do students and adults longterm continue to conceptualise division? I recently wrote about the ‘game of the classroom’, and how students are almost trained through the years of schooling on the idea that the metaphors, analogies and models provided by teachers aren’t necessarily a requirement in their reconstruction of that knowledge. This is a barrier when it comes to division, as realistically both ‘schools of thought’ are required for the comprehension of division, and in order to reconstruct knowledge of division and its application in a variety of different scenarios through mathematics – but once students realise that the calculations are the same – do they discount one of these conceptual models?

At lower levels a sole reliance on the sharing model means division involving fractions and/or decimals become incredibly difficult arithmetic concepts for students to deal with. Furthermore, it can cause problems with regards the transition to working in fractions (another aspect of higher level mathematics). “How do you visualise 3/4?” – is an eye opening question to ask young people. Whether they visualise 3/4 as an operation 3÷4, or as a number positioned on a number line is often very mixed, however asking the same of 0.75 generates a ‘100%’ response to the latter. If students literally don’t see fractions as numbers, instead seeing as the outcome of what happens in a division calculation, this means the barrier to transition goes significantly beyond ‘just getting used to’ using a new format/representation. Purely conjecture, I foresee a potential link between this and the ‘cake sharing’ metaphors that permeate through conceptualisation of fractions – which in turn stems from holding division as a product of ‘sharing’.

The recent viral situation is also evidence of what happens when only the sharing model of division is held – whilst ‘dividing by zero’ feels like an arbitrary and irrelevant concept to the everyday, the process permeates a lot of areas of the A Level Mathematics and Further Mathematics specifications, within the post-16 environment – and Maths/Physics/engineering disciplines beyond.

So what happens when we try to divide by zero?

The standard scientific calculator will show “syntax error”, Microsoft Excel a “#DIV/0!” error. A minor misconception that some students hold is that the solution is therefore an ‘error’. This is not the case, the printout of ‘error’ is coming from the computer programme that runs the spreadsheet or the calculator, not the mathematics itself. This error manifests as the result to the calculation of ‘anything divided by 0’ is undefined. Interestingly one of the questions I ask on my Further Mathematics diagnostic is for the students to “Discuss the possible solutions to 0/0” – I am looking to see how confident students are in articulating mathematical ideas.

A lot of students will know the calculation derives an error, some will know that this means the calculation is undefined – but as the viral post showed, this is not always the case, and even when it is, the reasoning is not always solidified (most conceptualisations again returned to sharing out to nobody, or placing into non-existent boxes’) or were merely a “It’s not possible” response – which as previously stated led to some with a ‘correct’ answer starting to feel swayed. So why is there such a problem with this?

This again stems from the cognitive model of division that is held by people. This screenshot of a post in the discussion is my absolute favourite – it has ‘internet hubris’, “it’s impossible to divide by zero” whilst providing a solution, and questionable use of ‘half”, but the reason why I love it is because it is a perfect representation of the limitation of the ‘sharing’ concept of division, and why an over reliance on that mental model can breed misconceptions later on.

The statement at the end “Take a pizza and cut it 0 times. How many pieces are left?”. On the surface this appears to make sense, yeah the pizza is still in 1 piece! But that’s not actually what the quotient of a division using the sharing method is. Returning to 10/5 from before, or to link to the pizza context “I cut a pizza into 10 pieces and share them between 5 people” – the solution to this isn’t “How many pieces are left” (still 10) it’s “How many pieces does each person get?”. Thinking of division exclusively in this way can yield misconceptions of this, not just with 0, but fractional and decimal values as well.

If instead we use the repeated subtraction cognitive model of division, we get a much clearer overview of the conundrum of dividing by zero;

Using repeated subtraction you can identify it is impossible to define how many times 0 can be subtracted by 1. Philosophically, It’s not even that you can’t necessarily literally do the calculation, it’s that if you do it, you cannot conclude on a solution. As previously stated this method of division has benefits for decimalised and fractional divisors – this method can also be used to transition towards ideas of limits and ‘infinite’ which make up factors of calculus in higher level mathematics (see the ending bonus).

Solidifying a diversity of cognitive concepts for division

Identifying the cognitive models that students use to undertake division, and encouraging diversity of these models are vital in allowing young people to build a robust understanding and ability to apply arithmetic, whilst also setting up learners for theoretical concepts that permeate STEM disciplines in the future. There are other areas of Mathematical study, particularly within secondary curriculums that allow us to embolden learners in their conceptualisation of division, namely straight line graphs.

Gradients are a prime opportunity to embed different cognitive models of conducting and visualising division into learners, as fundamentally the gradient is just a visual representation of a division calculation. It also again provides opportunities to distinguish the difference between 0 as a dividend and zero as a divisor, due to the very different geometric constructs of a horizontal or vertical line respectively – reinforcing conceptual ideas that should be formally taught earlier on.

As educators in mathematics it’s so easy for us to revert to exam board ‘method’ check-boxes when analysing a students working or thought process. An appropriate process, whatever that might look like, and a correct answer – have a nice tick. The problem is can yield blind spots in understanding that may not immediately manifest themselves to us and the youngsters – so diversity of approach and discussion is something that should be encouraged. We should be am to be in a position where the next generation of teachers don’t fall back onto “That’s what I was taught” or a “It’s not possible” response – we should have more respect for our discipline than that.

BONUS TIME – Can we ever divide by zero?

Without going too technical, if higher maths is not your specialism and you’ve braved it this far, you may have noticed during the repeated subtraction method of dividing by zero, you are in theory caught within an ‘infinite loop’ – hence the calculation issue of dividing by zero. What happens as a variable denominator/divisor approaches zero is a key part of calculus and limits discussed earlier on. Within this there is a method called ‘Differentiation via First Principles’, where the gradient of a curve at a point on a curve can be found by taking the gradient between two points on the curve, and reducing the horizontal distance between them to zero (infinitesimal change). Again in lay-terms, this is possible as we are able to bypass ‘actually dividing by zero’ within the algebraic formula, as the horizontal change representing the denominator of the gradient can be cancelled out of the function. So we are ‘dividing by zero’, but not actually dividing by zero.

Historically not everyone was happy about this, and I will sign off with the beautifully articulate (although not one that stands the test of time) rebuttal of Gottfried Leibniz and Isaac Newton’s works on calculus from philosopher, George Berkeley in 1734.

“And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?”

References and Wider Reading

Correa, C; Nunes, T; Bryant, P (1998). Young children’s understanding of division: The relationship between division terms in a noncomputational task. Journal of Educational Psychology. Vol.90(2).

Fischbein, E; Deri, M; Nello, MS; Marino, MS (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education. Vol.16(1).

Mulligan, JT; Mitchelmore, MC (1997). Young children’s intuitive models of multiplication and division. Journal for Research in Mathematics Education. Vol.28(3).

Squire, S; Bryant, P (2002). The influence of sharing on children’s initial concept of division. Journal of Experimental Child Psychology. Vol.81.

Squire, S; Bryant, P (2003). Children’s models of division. Cognitive Development. Vol.18.