One of the features that makes mathematical cognition such an interesting field is its philosophical dimension. In this post, I will discuss one such issue: the relationship between number, time, and space.
In
2003, a researcher named Walsh developed a theory called A Theory of Magnitude (ATOM;
Walsh, 2003). According to
Walsh, space, time, and number are processed in the brain by a single,
domain-general magnitude system. This system processes magnitudes such
as length, area, volume, quantity, duration, sound intensity, light intensity,
and so forth. It processes any domain that we experience in terms of “more
than” or “less than” and that can be represented along a continuous scale of
increasing or decreasing magnitude or intensity.
When a person or an animal, such as a monkey, reaches toward a pile of nuts, the perceived size of the pile and its perceived distance determine how the arm is extended. Many bodily actions are influenced by the perception and processing of magnitudes. The magnitude system is probably located in the parietal cortex, in an area called the intraparietal sulcus, or IPS.
Brain-imaging studies have found
that the IPS is active when people process spatial magnitudes or numbers. The
IPS is also associated with the perception of time: when a task requires
attention to temporal intervals, the IPS is active. Brain stimulation
that disrupts activity in the parietal cortex impairs the processing of
magnitudes in space, time, and number. In macaque monkeys as well, regions of
the parietal lobe are active when the monkey processes spatial information,
durations, or quantities.
If
there is a single magnitude system, people should associate “more” in one
domain with “more” in another domain. Indeed, studies have found that people
perceive stimuli with greater magnitudes, such as stimuli containing more dots,
larger squares, a more intensely illuminated circle, or a digit with a higher
numerical value, as lasting longer than stimuli with smaller magnitudes (Xuan,
Zhang, He, & Chen, 2007).
Therefore,
if people with dyscalculia also have difficulties perceiving and processing
time, this would provide evidence that time and number are processed by the
same brain mechanism. This is not easy to test because when people are asked to
estimate how long a stimulus lasted, they often count silently. In other words,
they use numerical processing to judge time.
For
this reason, Tobia, Rinaldi, and Marzocchi (2018) decided to examine time
processing in preschool children who had not yet fully acquired the symbolic
number system. Some preschool children may be able to count while performing a
time-estimation task, but the researchers attempted to determine whether the
children were actually doing so and concluded that the participants in their
study were not.
The
study included 196 children with a mean age of four and a half years. The
children completed screening tests for dyscalculia. These tests identified 30
children who were considered at risk for dyscalculia, and each of these
children was matched with a typically functioning child for the control group.
Both
groups were then given additional tests of time perception and quantity
comparison. Parents and preschool teachers also completed questionnaires
assessing the children’s sense of time. Finally, the children performed visual
and verbal working-memory tasks.
I
will first review the tests that were used to screen for possible dyscalculia.
This is an opportunity to learn which tests can be used for this purpose.
Although some of these tests are not yet available to us, developing a thorough
understanding of the subject provides a foundation for future work with them.
Quantity
comparison
The children were asked to choose which of two baskets contained more fruit. The number of pieces of fruit ranged from 3 to 20. This is similar to the Panamath test.
Nonsymbolic
estimation
In
this task, the children were shown two chefs, each carrying a tray containing
biscuits. They were then shown two plates of biscuits and had to choose the
plate containing the total number of biscuits carried by the two chefs
together.
The
quantities to be added ranged from 1 to 7, with a maximum total of 12. The
children were asked not to count, but to estimate the total on the basis of
perceptual cues related to quantity. To prevent the use of counting strategies,
the stimuli were displayed for only two seconds.
Digit
comparison
The
children were asked to choose the larger of two digits displayed together on a
sheet of paper.
Knowledge
of digits
The
children were asked to identify digits named by the examiner, read digits
aloud, and match digits with quantities.
As
noted above, the 30 children whose performance on these tests was particularly
low were classified as being at risk for dyscalculia. These children and the
children in the control group then completed the following tests.
Time
reproduction
The
children saw a light that was illuminated for half a second, one second, three
seconds, or five seconds. After the light went out, they were asked to turn it
on again for the same length of time.
Duration
discrimination
The
children listened to two tones presented one after the other. While each tone
was played, an image of an animal wearing headphones was displayed. Two
different animals were used to help the children distinguish between the tones.
After hearing both tones, the child was asked to select the animal that had
heard the longer tone.
Assessment
of the children’s sense of time
The
children’s sense of time was assessed using a questionnaire completed by
parents and teachers. The questionnaire included the following statements:
The
child can complete a time-limited activity before the allotted time runs out.
The
child talks about past events appropriately, referring to them as events that
occurred in the past rather than as events occurring in the present or future.
The
child knows what to expect during the daily routine, for example, preparing in
the morning to leave for preschool.
The
child independently recognizes when a routine daily event is approaching, such
as lunchtime or going outside for an activity.
The
child asks, “What time is it?” or spontaneously refers to times during the day.
The
child understands terms such as “yesterday” and “tomorrow.”
The
child correctly understands and uses concepts such as “yesterday” and
“tomorrow,” and “before” and “after.”
The
children also completed a counting task and a task assessing number sense
through quantity comparison. In this task, the children saw two sets of dots
and had to identify, quickly and without counting, which set contained more
dots. Each set contained between 1 and 9 dots.
The
researchers found that both groups of children were able to reproduce durations
of half a second, one second, and three seconds. The children at risk for dyscalculia had greater
difficulty than the control group reproducing a duration of five seconds and
discriminating between durations.
A correlation was found between
quantity-comparison skills and duration-discrimination skills. The better the
children performed on the quantity-comparison task, the better they also
performed on the duration-discrimination task.
Parents and teachers of children
at risk for dyscalculia reported that these children had a weaker sense of time
than the children in the control group.
These
findings should be interpreted with caution because the children at risk for
dyscalculia performed less well than the control group on tests of visual and
verbal working memory. Reproducing a duration of five seconds, discriminating
between durations, and comparing quantities may all be influenced by working
memory.
The
researchers argue, however, that even after controlling for the effects of
working memory, the children at risk for dyscalculia still experienced greater
difficulty than the control group on tasks assessing time perception and
quantity perception.
This
study therefore supports ATOM, according to which there is a single brain
mechanism for processing magnitudes, including durations, quantities, physical
sizes, stimulus intensities, and other dimensions.
Tobia,
V., Rinaldi, L., & Marzocchi, G. M. (2018). Time processing impairments in
preschoolers at risk of developing difficulties in mathematics. Developmental
Science, 21(2), e12526.
Walsh,
V. (2003). A theory of magnitude: Common cortical metrics of time, space and
quantity. Trends in Cognitive Sciences, 7(11), 483–488.
Winter,
B., Marghetis, T., & Matlock, T. (2015). Of magnitudes and metaphors:
Explaining cognitive interactions between space, time, and number. Cortex,
64, 209–224.
Xuan,
B., Zhang, D., He, S., & Chen, X. (2007). Larger stimuli are judged to last
longer. Journal of Vision, 7(10), 2.
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