These four visuals dispel the myths we’ve been fed about maths education
The case of Nancy and Liam
Every child is unique.
As parents, relatives and friends, we celebrate that fact. As well we should; our rich diversity is what makes us human. But as educators, we are often challenged by it, because it forces us to confront the individual needs and preferences of every learner. This is never truer than for mathematics.
Traditional schooling conspires to sidestep the challenge. Rather than embracing the individuality of learners, it imposes standardised curriculum and teaching practices that preclude all but the arbitrary few to excel as mathematical thinkers.
The assumptions of traditional schooling have persisted because so few viable alternatives have been put forward. That may be about to change with the advent of intelligent tutoring.
Intelligent maths tutors dispense with the devastating snapshot judgements of one-time testing. Instead they adopt a model of continuous, adaptive assessment that is more reliable and more thorough. And because students’ progress is tracked from moment to moment, intelligent tutors allow for sharper insights into the learning and teaching of mathematics.
I’ll explore some data I’ve encountered at Whizz — all real and anonymised, with permission granted to share these reports — to see how the myths of traditional schooling unravel. Views my own.
Myth #1: Students in a given class have the same overall maths ability
We’ll start with the distribution of overall maths ability in a Year 5 class. In the image below, each icon corresponds to an individual student. The scale plots their overall maths ability, calculated by the Maths-Whizz intelligent tutor and denoted by Maths Age. The Maths Age metric is based on Whizz’s international standards and has a natural interpretation: a child with a Maths Age of 9 has reached the overall level expected of a nine-year-old.
We would expect these students to have a Maths Age of around 10. Instead we see a gap of over four years behind the highest and lowest achieving students. Maths-Whizz serves thousands of schools across the world, and this multi-year gap is consistent wherever we go. There is some variation due to region, school type and grade level, but we always find that the spread of overall maths ability is larger than most teachers expect — even those fortunate to teach ‘gifted’ students in the most affluent schools.
Myth #2: Classes can be neatly divided into subgroups by ability
The class distrbution hints at the underlying challenge for maths educators — delivering an educational experience that meets each child where they are.
Ah, but we only have a few outliers to deal with at either extreme. So we’ll challenge the high achievers with some deeper problems, and remediate the laggards until they’re caught up. Those in the middle can continue as normal.
Not so fast…
It turns out that every one of these students has fundamental gaps in their learning. Let’s take Nancy, who sits in the central cluster. This was her learning profile two years ago when she first started on Maths-Whizz:
The red markers indicate where Nancy was placed in her initial assessment. Nancy was around seven and a half at the time and exhibited a jagged learning profile. In Place Value, her ability was close to a five-year-old; perhaps Nancy was absent the week her class covered Place Value, or maybe she did not grasp the concept of Tens and Ones. In any case, Nancy’s development as a mathematician will rest on a strong foundation in Place Value, so while her overall Maths Age may place her in the central cluster, she is in urgent need of intervention.
Myth #3: Students in a given class will share common gaps in their knowledge and understanding
Perhaps the whole class is weak on Place Value — it may not have been covered extensively in class, or may have been explained poorly. So we’ll single out Place Value for reinforcement and all students will benefit.
Before doing so, let’s take another student in the same class as Nancy, also within that cluster. He has the same teacher, receives the same support and instruction. His overall level is similar. So we might expect his learning profile to resemble Nancy’s.
And yet, here was Liam when he joined Maths-Whizz two years ago:
Go figure.
Place Value was among his strongest topics. His shaky foundations were in Fractions and Pencil/Paper methods. Here is the evidence that no two students are the same. Even in the same classroom environment, Nancy and Liam have disparate learning needs, which arise from the wildly uneven context of their individual lives.
In most schools, Nancy and Liam would simply be considered on grade. They would be assigned the same label — the same expectations and learning plan — based on their overall ability. Yet their learning profiles confirm that their needs were vastly different. One-size-fits-all won’t cut it, and neither will one-size-fit-many. Students need one-size-fits-one, which is where 1:1 tutoring finds its purpose.
Myth #4: Students’ maths abilities are fixed
The tragedy of school maths is that it imposes fixed beliefs around students’ maths abilities. Many students are left to believe they can not excel as mathematicians. These signals are often implicit; ability-grouping may be born from good intentions but is a sure-fire way of initiating a vicious cycle of low attainment, low confidence and low effort. It is an injustice that these students are often denied access to a complete maths curriculum, leaving them further behind their peers.
Perhaps it would be said of Nancy that she just can not grasp Place Value. She had her chance, and since students like Liam got along with it just fine, the fault was clearly hers (hey, maybe it’s genetic).
And yet, it’s amazing what can be achieved when students receive dedicated support that is targeted at their specific knowledge gaps. Nancy has interacted with the Maths-Whizz tutor for two years — the blue bars correspond to her improvement in each topic:
Still think Place Value was beyond Nancy? This visual embodies Carol Dweck’s idea of a growth mindset; the belief that our abilities are not fixed. Her research is clear on the positive impact that continuous, positive feedback (seeing those blue bars grow) can have on students’ confidence, effort and achievement — that’s a virtuous cycle.
The case of Nancy and Liam exposes the fatal flaw of traditional schooling. It is designed for the average student that doesn’t exist. It is why at Whizz we encourage parents and educators to interpret Maths Age at the level of the individual learner and topic — and to retain a sharp eye for deeper analysis even then.
In The End of Average, Todd Rose debunks the so-called averagarian approach to Education in both historical and mathematical terms. The principles of individuality he lists are evident in the case of Nancy and Liam: a) Talent is jagged, b) There is context behind data, c) Students should learn along individualised pathways. Most importantly, when we speak of equalising opportunity in education, we must recognise the need not just for equal access and quality, but also equal fit.
Every child is unique — it is time for educators to embrace it.
