Practice at “Guesstimating” Can Speed Up Math Ability

A person’s math ability can range from simple arithmetic to calculus and abstract set theory. But there’s one math skill we all share: A primitive ability to estimate and compare quantities without counting, like when choosing a checkout line at the grocery store. Practicing this kind of estimating may actually improve our ability to do the kinds of symbolic math we learn in school, according to new research published in Psychological Science, a journal of the Association for Psychological Science.
Previous studies have suggested a connection between the approximate number system, involved in estimating, and mathematical ability. Psychological scientists Elizabeth Brannon and Joonkoo Park of Duke University devised a series of experiments to test this association.
The researchers enrolled 26 adult volunteers and had them complete 10 training sessions designed to hone their approximate number skills. On each of these training sessions, the participants practiced adding and subtracting large quantities of dots without counting them.
They were briefly shown two arrays of 9 to 36 dots on a computer screen and then asked whether a third set of dots was larger or smaller than the sum of the first two sets, or whether it matched the sum.
“It’s not about counting, it’s about rough estimates,” explains Park, a postdoctoral researcher at Duke.
As participants improved at the game, the automated sessions became more difficult by making the quantities they had to judge closer to each other.
Before the first training session and after the last one, their symbolic math ability was tested with a set of two- and three-digit addition and subtraction problems, sort of like a third-grader’s homework. They solved as many of these problems as they could in 10 minutes. Another group of control participants took the math tests without the approximate number training.
Those who had received the 10 training sessions on approximate arithmetic showed more improvement in their math test scores compared to the control group.
In a second set of experiments, participants were divided into three groups to isolate whether there had been some sort of placebo effect in the first experiment that made the approximate arithmetic group perform better. One group added and subtracted quantities as before, a second performed a repetitive and fast-paced rank-ordering with Arabic digits, and the third answered multiple choice questions that tapped their general knowledge (e.g., “which city is the capital of France?”)
Again, the people who were given the approximate arithmetic training showed significantly more improvement in the math test compared to either control group.
“We are conducting additional studies to try and figure out what’s driving the effect, and we are particularly excited about the possibility that games designed to hone approximate number sense in preschoolers might facilitate math learning,” Park said.
Park and Brannon can’t yet isolate the mechanism behind their effect, but the research does suggest that there is an important causal link between approximate number sense and symbolic math ability.
“We think this might be the seeds — the building blocks — of mathematical thinking,” Brannon said.

Press release available on the APS website.
This research was supported by a James McDonnell Scholar Award, a grant from the Eunice Kennedy Shriver National Institute of Child Health and Human Development, and a Duke Fundamental and Translational Neuroscience Postdoctoral Fellowship.
For more information about this study, please contact: Joonkoo Park at
The APS journal Psychological Science is the highest ranked empirical journal in psychology. For a copy of the article “Training the Approximate Number System Improves Math Proficiency” and access to other Psychological Science research findings, please contact Anna Mikulak at 202-293-9300 or

Acceleration or Enrichment

Acceleration or enrichment: Report of a seminar held at the Royal Society
on 22 May 2000, The De Morgan Journal, 2 no. 2 (2012), 97-125.

Full title of the paper:

Acceleration or Enrichment?
Serving the needs of the top 10% in school mathematics.
Exploring the relative strengths and weaknesses of “acceleration” and “enrichment”.
Report of a seminar held at the Royal Society on 22 May 2000.

The report includes contributions from Tim Gowers, Gerry Leversha, Ian Porteous, John Smith, and Hugh Taylor.


This report was originally published in 2000 by the UK Mathematics Foundation (ISBN 0 7044 21828). It was widely red, and was surprising influential. However, it appeared only in printed form. Various moves made by the present administration have drawn attention once more to this early synthesis— which remains surprisingly fresh and relevant. Many of the issues raised tentatively at that time can now be seen to be more central. Hence it seems timely to make the report available electronically so that its lessons are accessible to those who come to the debate afresh.

While the thrust of the report’s argument remains relevant today, its peculiar context needs to be understood in order to make sense of its apparent preoccupations. These were determined by the gifted and talented policy’ adopted by the incoming administration in 1997, and certain details need to be interpreted in this context. There are indications throughout that many of those involved would probably have preferred the underlying principles to be applied more generally than simply to “the top 10%”, and to address the wider question of how best to nurture those aged 5–16 so as to generate larger numbers of able young mathematicians at age 16–18 and beyond. The focus in the report’s title and subtitle on “acceleration” and on “the top 10%” stemmed from the fact that those schools and Local Authorities who opted at that time to take part in the Gifted and Talented strand of the Excellence in Cities programme were obliged to make lists of their top 10% of pupils; and the only provision made for these pupils day-to-day was to encourage schools to “accelerate” them on to standard work designed for ordinary older pupils. The wider mathematics community was remarkably united in insisting that this was a bad move. This point was repeatedly and strongly put to Ministers and civil servants. But the advice was stubbornly resisted; (indeed, some of those responsible at that time are still busy pushing the same linez.

The present administration seems determined once more to make special efforts to nurture larger numbers of able young mathematicians, and faces the same problem of understanding the underlying issues. Since this report played a significant role in crystallising the views of many of our best mathematics teachers and educationists, it may be helpful to make it freely available—both as a historical document and as a contribution to current debate.

Read the whole paper. 

Tony Gardiner: Nurturing able young mathematicians

A. D. Gardiner, Nurturing able young mathematicians, The De Morgan Journal  2 no. 7 (2012), 87-96.


We summarise the developments of the last 20 years—highlighting the key underlying assumptions, and indicating certain unfortunate consequences. We show how official policy has been based on

  • persistent failure: (i) to develop and to implement a suitably challenging curriculum, and (ii) to provide ordinary teachers with good texts, suitable subject-specific professional development, and appropriate assessment targets;
  • a misconception of the curriculum as a one-dimensional ‘ladder’ (with each topic nominally the same for everyone, with uniform expectations for all pupils at a given ‘level’), up which pupils progress at their personal rate, and
  • associated accountability measures that have unintended consequences.

We then outline the alternative conception of a two-dimensional “*-curriculum”, in which each theme in the standard curriculum sequence is explored (and where necessary, assessed) to different depths, and where those who manage to dig deeper and to lay stronger foundations emerge naturally as the ones who are well-placed to subsequently progress further. In such a model, able pupils in Years 5 and 6 would not be pushed ahead to achieve a premature and superficial mastery of ‘Level 6’ material, but would spend time exploring harder problems at ‘Level 4’ and ‘Level 5’ (so-called 4* and 5* material). Similarly, able students in Years 10 and 11 would not be entered early for an accessible but superficial GCSE, but would instead be expected to master core GCSE material more deeply, so as to make the subsequent transition to A level in Year 12 straightforward.

Read the rest of the paper. 

Olympic Legacy

One of the justifications for the Olympic budget is the pious hope that people will be inspired to participate in sport more than hitherto.  No previous Olympic games achieved this, and it’s easy to see why.  The Olympics offer a model of sporting activity that is unavailable to most and unattractive to almost everybody.  Running 150 miles each week is not an option or an aspiration for all but a handful of talents.  If the powers that be really want to raise levels of participation, they should offer the models suited to the mass of the population, with facilities to match (proper cycle lanes, school sports fields, local swimming pools,etc.).  There are rewards that come from participating in sport at a very low level, but you’d never know it from watching the Olympics.

This matters to the DMJ because the same point applies to mental activity.  Tales of geniuses making astounding breakthroughs will not encourage kids into mathematics any more than Olympic gold will inspire sedentary Britons to take moderate exercise.  What we need are images of middling intellects getting something valuable out of mathematics.  This is especially important because in our assessment-driven system, children know from early on where they stand in the intellectual league tables.  The great majority know themselves to be middling intellects long before they make decisions about what to study.  We need stories about mathematics and illustrations of its value that speak to children thus informed.

Educating the Highly Able

The Sutton Trust  (see their Press Release) published today a report by Alan Smithers and Pamela Robinson, Educating the Highly Able. From the Executive Summary of the report:

Policy and provision for the highly able in England is in a mess. […]

When compared to other countries the consequences are stark. In the 2009 PISA tests only just over half as many achieved the highest level in maths as the average of 3.1% for OECD countries. England’s 1.7 per cent has to be seen against the 8.7 per cent in Flemish Belgium and 7.8 per cent in Switzerland. On a world scale, the picture is even more concerning – 26.6 per cent achieved the highest level in Shanghai, 15.6 per cent in Singapore and 10.8 per cent in Hong Kong. In reading, where the test seems to favour English-speaking countries, England is at the OECD average, but only a third get to the highest level compared with New Zealand and only half compared with Australia. The few top performers in England are in independent and grammar schools and almost no pupils in the general run of maintained reach the highest levels.

The root of the problem is that “gifted and talented‟ is too broad a construct to be the basis of sensible policy. As it has morphed from “intelligence‟ to “gifted‟ to “gifted and talented‟, it has become ever more diffuse. It is not just the conflating of “gifted‟ and “talented‟; it is that “gifts‟ and “talents‟ are often specific. A gift for mathematics and a gift for creative writing are rarely found in the same person. Few top footballers are also top artists. Continue reading