Guest Blog Posts

Expertise Development in Young Mathematicians by Dylan Smith

Recently, Mark Chubb published an engaging blog post on The Matthew Effect, the social-scientific idea that those who enjoy early advantage over others are likely to see their advantage widen. The theme of Mark’s post was that Math readiness gaps in preschoolers can be expected to widen and become Math achievement gaps. Mark, an experienced instructional coach, wondered why and how this occurs, and what strategies might be adopted to level the field and promote engagement and Math proficiency for all students.

In this guest post, I spin off Mark’s post, presenting the view that for most children, readiness differences on entry to kindergarten are due to experienced-based differences in perceptual skill, broadly speaking, and that a better understanding of this cognitive landscape is likely to suggest home-and-school-friendly strategies for reducing early inequities in Math learners. Let’s begin with a historical anecdote.

Perceptual skills are “the hands and feet of genius”

In 1986, Dr. Benjamin Bloom — you know, the thinking skills fellow — wrote an article for Educational Leadership on the topic of automaticity in learning (Bloom, 1986). Bloom already enjoyed wide acclaim regarding talent development in children, including mathematical talent. In one passage of the article, Bloom quoted a well-known 19th century study on how telegraph operators, with repeated practice, developed remarkable levels of expertise:

“The learner must come to do with one stroke of attention what now requires half a dozen, and presently, in one still more inclusive stroke, what now requires thirty-six. He must systematize the work to be done and must acquire a system of automatic habits corresponding to the system of tasks. When he has done this, he is a master of the situation in his [occupational or professional] field. Finally, his whole array of habits is swiftly obedient to serve in the solution of new problems. Automatism is not genius, but it is the hands and feet of genius.” (Bryan & Harter, 1899)

We tend to think of perceptual abilities as ever ready to serve learning and development, yet rather fixed in how they operate throughout childhood. We also tend to notice the awkwardness of a youngster’s learning performances alongside those of adults. And we tend to believe a youngster’s schooling in Math, for example, is largely founded on the early learnings we might find in a curriculum document, such as counting to a benchmark, representing simple amounts or patterns, comparing dimensions of adjacent objects, etc. But while these popular beliefs are of course true to some extent, each of them profoundly underestimates the miracle of how a small child acquires proficiency as a learner.

The nature of “perceptual learning”

In the past few decades, cognitive science has firmly established that the perceptual abilities of young children allow them to rapidly adapt to new environments and learning situations. This cognitive process has been coined “perceptual learning” and may be defined as long-term performance enhancement that results from perceptual experience. There are a number of models that compete to describe how perceptual learning works and develops, but we may simply say that children improve their perceptual skills with practice.

We do know that as children explore their world, they learn to extract information that was perhaps previously overlooked, but now viewed with purpose. Task-relevant features of learning situations or tools are identified and configured in terms of one another, while task-irrelevant attributes are either ignored or unlearned. Importantly, sets of features previously thought of as tied may now distinguish themselves, either pre-consciously or to the child’s attention, while still other constellations of independent features are reconfigured for identification as a unit, as with facial recognition. With repeated exposures and practice, children learn to dovetail their processes for info-pickup with improved efficiency. In this way, over time, a child develops and elegantly refines long-lasting perceptual proficiencies which can, in favouring circumstances, accelerate and evolve into task automaticity, expertise, and confidence.

Examples will prove helpful. I am eager to first share what happened in my car only a few days ago. Each weekday morning, I drive my 15 year-old, an ardent car enthusiast, to a nearby bus stop from where he has a non-stop ride to school. On this particular morning, he all of a sudden announced the make, model, and year of a nearby car, some sort of Bentley. “Where is this car?” I asked, and he directed my gaze to an oncoming and snow-covered car that was in traffic and had to be at least 150 metres away on the other side of an intersection when he first spoke. When I questioned how he could possibly know all that vehicle info from a car as distant and obscured as it had been, he replied that when he reads something he’s interested in, it sticks. I didn’t challenge him on that, but quietly I knew his remarkable expertise was, in truth, founded on his experience-based ability to instantly perceive meaning in the relevant pattern of nuanced situational detail.

Other examples of expertise relating to visual discrimination have been well-documented, and include the recognition of abnormal chest x-rays (Myles-Worsley, 1988), reading, and the sorting of young chickens by gender (Goldstone, 1998; Watanabe, 2015), and how parents learn to tell infant twins apart (Johnson & Proctor, 2017). Examples of perceptual learning also exist for each of the other sensory modalities, including professional wine-tasting and piano tuning (Li, 2016), as well as for complex multisensory circumstances, such as landing an aircraft (Kellman & Massey, 2013).

It is also worth noting that Kellman and Massey (2013) examined the influence of perceptual learning in higher-order cognition, and paid special attention to intermediate and senior grades mathematics. The authors surveyed a number of studies attesting to the importance of pattern recognition and fluency in mathematics, citing benefits for graphing and equations, transformations, fractions, and proportional reasoning. Indeed, the authors concluded that the inability of conventional instruction to promote perceptual learning may be “disproportionately responsible for students’ difficulties in learning” (p.125).

Perceptual learning and math readiness in young children

But what about math readiness in very young children? Can cognitive science guide us in better preparing all youngsters for elementary Math programs? Can educators design “favoring circumstances” for the development of perceptual proficiencies in early childhood, proficiencies which would be, in the words of Bryan and Harter, “swiftly obedient to serve in the solution of new problems”? And if so, can strategies be designed and adopted, as Mark asked in his original post, to reduce early math inequities?

I am encouraged to say yes, and believe we have already left the starting line in important ways. Educators endorsing more “playful” instructional approaches are on a promising track. And surely the Maker movement has something to offer, as do Ontario Ministry of Education undertakings to coordinate parents at home, classroom teachers, and before- and after- and pre-school programs on early learning. Each of these initiatives can draw fresh vitality from the cognitive science of perceptual learning.

With all this in mind, I will close with a recommendation for my colleagues. We teachers, like the children we lead, develop tremendous expertise through our classroom explorations, and much more than we can ever know or articulate. In our efforts to further our own unrivalled expertise on K-12 classroom learning and achievement, we would do well to pay closer attention to the work of related fields. True, we might expect that if we were more open to evidence-informed contributions from other disciplines, such as cognitive psychology, our view of “playful,” for example, might require an adjustment with regards to the measure and manner we view it to be purposeful, or task-specific, or tied to particular materials, skills, contexts, fluencies, or developmental milestones. But wouldn’t such examples of deeper understanding equate to progress, and lead to sharper instructional outcomes? Nothing will be sitting on a plate for us as we look beyond our own field to pull our expertise forward. Work will be required. But applying ourselves to such work is bound to lead to gainful change.


Bloom, B. S. (1986). Automaticity: The Hands and Feet of Genius. Educational Leadership, 43(5), 70-77.

Bryan, W. L., & Harter, N. (1899). Studies on the telegraphic language: The acquisition of a hierarchy of habits. Psychological Review, 6(4), 345-375.

Goldstone, R. (1998). Perceptual learning. Annual Review of Psychology, 49, 585-612.

Kellman, P. J., & Massey, C. M. (2013). Perceptual learning, cognition, and expertise. Psychology of Learning and Motivation, 58, 117-165.

Li, W. (2016). Perceptual learning: Use-dependent cortical plasticity. Annual Review of Vision Science, 2, 109-130.

Myles-Worsley, M., & Johnston, W. A. (1988). The influence of expertise on X-ray image processing. Journal of Experimental Psychology: Learning, Memory, and Cognition, 14, 553-557.

Watanabe, T., & Sasaki, Y. (2015). Perceptual learning: Toward a comprehensive theory. Annual Review of Psychology, 66, 197-221.