Professor Barbara Oakley caused a stir recently with her New York Times Op-Ed, in which she advocated for more rote practice in maths. Oakley’s advice was specifically targeted at girls, though she does conclude with a recognition that boys too would benefit from additional maths practice. The article has prompted a familiar debate around whether rote learning has a place in mathematics.
There is some merit to Oakley’s thesis. Her push toward rote practice is premised on the cognitive benefits of being able to rapidly recall basic mathematical facts and procedures. Our working memories are only able to manipulate around four objects at a time; when dealing with more advanced concepts, we can ill afford to devote our limited ‘thinking space’ to retrieving those basic details.
Oakley does not glamorise rote practice; she encourages parents to embed maths practice in their child’s learning “even if she finds it painful.” It is a sentiment that may resonate with athletes, who endure painful, intense workouts in order to maximise their overall performance. Perhaps rote practice is simply the price students must pay for their mathematical development.
Yet pain seems too high a price to pay for mathematics. When so many students – and adults – are afflicted with maths anxiety, an ailment known only to this subject, we must surely challenge ourselves to adopt instructional techniques that instil both love and understanding of mathematics.
First, we must ensure that rote practice never undermines students’ understanding of mathematical concepts. There is always a danger when learning mathematics of over-relying on memorisation. Imagine you were asked to commit the following few lines to memory:
Charlie David lives on George Avenue
Charlie George lives on Albert Zoe Avenue
George Ernie lives on Albert Bruno Avenue
Charlie David works on Albert Bruno Avenue
Charlie George works on Bruno Albert Avenue
George Ernie works on Charlie Ernie Avenue
The task above is actually just arithmetic in disguise. Each word has been substituted in for a number or operation: Charlie = 3, David = 4, George = 7, Albert = 1, Zoe = 0, Ernie = 5, Bruno = 2, ‘lives on’ = +, ‘works on’ = x.
The first three lines then correspond to addition facts:
Charlie David lives on George Avenue -> 3 + 4 = 7
Charlie George lives on Albert Zoe Avenue -> 3 + 7 = 10
George Ernie lives on Albert Bruno Avenue -> 7 + 5 = 12
The next three lines correspond to multiplication facts:
Charlie David works on Albert Bruno Avenue -> 3 x 4 = 12
Charlie George works on Bruno Albert Avenue -> 3 x 7 = 21
George Ernie works on Charlie Ernie Avenue -> 7 x 5 = 35
The example is taken from Stanislan Dehaene’s excellent book, The Number Sense. It vividly illustrates how arbitrary and confusing learning can become when memorisation is emphasised at the expense of deeper engagement and understanding.
Even accepting the virtues of rapidly recalling maths facts, memorisation is rarely the most efficient or effective learning method. While Oakley does acknowledge that “the foundational patterns must be ingrained before you can begin to be creative”, she fails to realise that patterns are to be understood and explored, and that probing them significantly reduces the need for memorisation.
Oakley’s advice neglects the mutual dependency between procedural knowledge and conceptual understanding. Rote learning may have its place, but its goals are undermined when equal emphasis is not placed on why those techniques work. The more we immerse ourselves in the wondrous landscape of numbers, the stronger our mental representations, and the more those fact will fall out as a natural by-product.
Consider the child who effortlessly names the players in their favoured football team. Have they studiously committed those names to memory? Of course not – those facts are instead a by-product of supporting that team, knowing their style and formation, and following their matches from one week to the next. All of these touch-points, some of them quite immersive, leave the child so acquainted with their team that memorising players’ names is redundant.
This should be the aspiration for all maths educators – to help their students develop an intimate relationship with numbers (and indeed other mathematical objects) that renders memorisation almost moot. In learning times tables, for example, when students appreciate why the commutativity holds (the order of multiplication does not change the outcome), the daunting task of learning 144 multiplication facts collapses down to just 72. Times tables will further yield to your probing, as mathematician Eugenia Cheng brilliantly illustrates – she can recall and quickly derive number facts from her understanding of their relationships and inner workings.
Oakley remarks that your daughter will thank you “in the long run” for rote practice. Let’s instead offer students a new bargain: mathematics for joy and understanding, in the here and now. It is a subject replete with patterns and structure that awaits our exploration.