Mathematics is in a state of constant flux. Our ways of knowing and understanding mathematical concepts is entwined with the tools and technologies available to us. At a time when the smartphone you carry in your pocket possesses more processing power than the machines that sent man to the moon, we must reflect on the kind of mathematics the students of today will need to lead tomorrow’s world.
One facet of mathematical thinking that is gaining currency among maths educators is number sense. Two words, three syllables and a trigger for a whole host of questions: what is number sense and is it anything new? Does this mean we ditch times tables? Can number sense be developed? What are the most effective strategies for teaching number sense?
Answers – and a few more questions – follow.
What is number sense?
To appreciate what number sense is, and why it matters, first consider its opposite: procedural fixedness. There are countless instances of students carrying out mathematical procedures with speed and accuracy, only to come unstuck when asked to apply the same knowledge in less familiar contexts. In a famous study by Kurt Reusser, three quarters of students offer a numerical answer to the following question:
There are 125 shepherds and 5 dogs in a flock. How old is the shepherd?
Of course, the shepherd’s age is not discernible from the information provided. Guess what answers the students proffered? 120, 130, 25 and 625 were among the selections – answers easily obtained by combining the two numbers given using standard operations. My nephew even suggested 30 remainder 2 when I posed the question to him, though I’ve yet to determine how he reached his exotic figure.
Number sense is the antidote to this kind of mindless computation. To have number sense is to use numbers flexibly and fluidly, and to hold strong and reliable intuitions towards their behaviours. In other words, number sense is about having a good feel for numbers. You may consider the notion too holistic for a subject that supposedly possesses cold objectivity, yet number sense is a defining marker of the present-day mathematician. The days of ‘human computers’ are long gone; we no longer employ humans for their calculational prowess alone because we have, well, calculators for that. Where humans can excel is in checking that those calculations make sense within the context that they’re being applied to. For example, number sense is what will alert you to suspicious transactions that may otherwise slip under the radar in the age of contactless. You’ll know you’ve been overcharged for dinner if you have a good sense of what the bill should be, even if the precise amount escapes your scrutiny.
Developing number sense
The good news is that humans are endowed with a natural number sense. Babies have demonstrated an affinity for basic quantities: the tiny tots can discriminate between 2 and 3 and by six months can match the number of sound beats to the same number of objects. Even as adults, however, our eye for precision, however, is restricted to around four objects. Beyond that, we rely on a deeply embedded sense of approximation. It is only through the formal development of mathematical systems such as counting and place value that we were able to get a precise handle on larger quantities. The maths we learn at school is based on these systems that emerged over centuries. When learning and teaching those systems, we must not neglect the sense of number that comes naturally to us.
Number sense does not give us license to discard times tables, or mathematical procedures. Rather, it is an opportunity to engage those constructs in ways that reinforce our intuitions around number. If multiplication is reduced to a ritual of memorising isolated facts, we end up with the senseless shepherd calculations. But when factual and procedural knowledge paired with an equal emphasis on understanding, we get the best of all worlds: fluency with recall and flexibility to apply the same facts of procedures flexibly.
Achieving this fine balance is aided by several instructional techniques.
Seek powerful representations that convey the underlying patterns of number (here is a selection of compelling representations around times tables). Representations such be varied so that students interact with numbers in a range of contexts. It also helps to make deliberate use of language that seeks to enlighten students on where mathematical ideas derive from. For instance, did you know that the word ‘calculation’ derives from the word ‘calculus’, which is Latin for pebble? Why pebbles, you ask? Just think back to the original counting system employed by our ancestors – pebbles were the concrete representations of their time. Students must now tame today’s calculating machines; it is not an either/or as much as a question of to what extent? You do not want students reaching for the calculator to compute the most basic sums, but it is also futile to have them labour through really large calculations when a computer can offer a precise answer in a heartbeat. Graphing calculators can also serve as powerful representations of what is being calculated. Lastly, it is crucial to expand students’ mathematical worldview beyond binary questions that have a single right or wrong answer. Precision and accuracy have their place, of course, but so too does developing a sense of what might constitute a reasonable guess.
With those goals in mind, here is a selection of four activities for fostering number sense:
• Fermi problems: named after the physicist Enrico Fermi, this category of problems calls on your powers of reasonable estimation by posing a question with a large answer that is virtually impossible to obtain. Examples include: “how many toothbrushes are there in the UK right now?” or “how many pennies would fill your classroom?” The answer matters far less than the approach: to arrive at a reasonable estimate, you need to set up a model, make some assumptions and, yes, carry out some calculations. Then you can interrogate the reasonableness of your final answer against your intuitions or competing solutions.
• Countdown: the numbers game of the classic gameshow remains a potent way of embedding fluent arithmetic skills within an engaging context. The game offers variety and can be extended to multiple formats (have you ever tried Countdown Fractions?). Students cannot escape the importance of applying number facts and procedures flexibly.
• Number Talks: popularised by Cathy Humphrys, a number talk starts with a simple calculation – such as 5×18 – and invites students to compare their calculational methods. Some will use partitioning (5×10 + 5×8), others will employ a variant (5×20 – 5×2) and others still will draw on elegant symmetry techniques (5×18 = 10×9). The aim of a Number Talk is to visualise and discuss each method to show the plurality with which arithmetic unfolds.
• Exploration: there is more depth to number than even a professional mathematician can bear. Expose students to the perpetual mystery of prime numbers through the joyful exploration of Prime Climb, a simple board game premised on the fundamental role that primes play in multiplication.
Number sense is the mathematician’s weapon of choice in the twenty first century. It is not a new or even novel concept, but its importance has never been felt so sharply. The curriculum abounds with opportunities to instil number sense alongside the usual focus on mathematical facts and procedures. These goals can be mutually reinforcing, so long you make a deliberate effort to combine them in your instruction. We hope this article is a useful summary and would love to hear how you are making number sense a cornerstone of your teaching.