Perception of Mathematical Equations Based on Prior Knowledge

This study explores the correlation between the visual perception of abstract algebraic notions, arithmetic equations and the prior mathematical knowledge in order to determine if the latter influences the perception of mathematical equations when presented differently to different sets of people. The study is based on Marghetis, Landy and Goldstone (2016) experiment of the correlation between the visual perception of abstract algebraic notions, arithmetic equations and the prior mathematical knowledge.

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Title: How perception of mathematical equations change based on prior knowledge
Introduction
This study based on Marghetis, Landy and Goldstone (2016) experiment of the
correlation between the visual perception of abstract algebraic notions, arithmetic equations
and the prior mathematical knowledge in order to determine if the latter influences the
perception of mathematical equations when presented differently to different sets of people.
As indicated by a multitude of studies, the notion of abstractness and perception of symbols
in humans have been historically evident since the days of the cave paintings (Goldstone et
al., 2017).
Landy and Goldstone (2007) have advocated for the high significance and relevance
of symbolic manipulation in the human cognition process in the cognitive research fields.
They have identified that Algebra is one of the best examples of pure symbolic manipulation
(landy & Goldstone, 2007). Sophian (2000) indicated that under experimental conditions
regarding spatial proportionality judgements, even children as young as 4 – 5 year old were
able to provide accurate responses; furthermore, were also able to retain the accuracy when
the experimental conditions were changed. Besides, Ottmar & Landy also indicated that the
aspect of algebraic abstractness learning is also dependent in children upon the perception of
concreteness and one of the key contributors to the process of learning has been attributed to
concreteness fading. When it comes to the visual perception effects and retention of
mathematical knowledge, Rivera and Garrigan (2016) also indicated that even with
administration of masking effects, the mental representations are retained, especially in the
form of visuospatial characteristics. These studies show that the concept of perceiving
mathematical notations and symbolic knowledge in humans is not as abstract and random but
there are possibilities of an underlying system of rule governed perception that is at play.

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The concept of mathematical reasoning in humans is also not an entirely random
effect as the effects of pre learned symbols and rules are identified to play a significant role in
the perception of mathematical expressions. The BODMAS rule for example, governs how
the alignment of the four operators in arithmetic equations play a hierarchical role when it
comes to perception of equations (Mukucha, 2010). This would indicate that expression
identification is primarily a structured method where prior knowledge of rules are significant
contributors to the process. Perceptual grouping is also another factor that seemingly plays a
significant role in the process of interpretation of mathematical expressions. The early 20th
century gestalt psychologists had identified a set of grouping principles that specified how
some constant image features led to between - element groupings while all other factors were
kept constant (Brooks, 2015)
In the current study designed after Marghetis, Landy and Goldstone (2016), the syntax
knowledge of participants regarding their perception of algebraic equations are studied. Using
a modified experimental setup from the original, the experiment has been able to describe
how the different conditions of spacing influences the differential perception in the
participants based on their prior knowledge of mathematical syntax. Thus, three hypotheses
were generated for this study as follows:
1. Mathematical syntax knowers show an advantage over syntax non knowers when
it comes to detecting colour changes within sub expression groupings than
between sub expression groupings.
2. When the sub expression groupings are supported by syntax consistent spacing,
the effect of mathematical syntax knowledge remains.
3. However, during sub expression groupings using inconsistent spacing, the effects
of syntax knowledge is eliminated.

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Methods
Participants
For this experiment a total of 642 first respondents were identified with 450 being
selected as the final sample size. The final sample size was divided into three groups for the
three conditions of the study. Each group had an equal 150 participants.
The participants were tested for their syntax knowledge beforehand and they were
grouped into two groups – Syntax Knowers and Syntax Non – Knowers respectively. Each
condition group had an equal number of participant subdivision with 75 belonging to the
syntax knowers group and 75 belonging to the Syntax Non – Knowers group. 74 % of the
participants were females (333/450) and all participants were aged between 17 and 62 years
with the mean age at 20.7 and a standard deviation of 5.71. Of the remaining participants, 101
were males, 6 belonged to other gender and 10 preferred not to answer.
Procedure
For the grouping task, a modified version of the Syntax Knowledge Measure
experiment that was used in Marghetis, Landy and Goldstone (2016) was used. The new
measure involved 25 math questions that was created to measure the participants’ knowledge
of mathematical syntax. A series of mathematical equations was presented on the screen for
the participants and the equations were designed with 4 numbers mixed with addition and
multiplication (e.g. 4 + 3 X 2 + 1). The equation was followed by four alternative responses.
One of them was correct when obtained by performing multiplication before addition, one
was correct when addition was performed before multiplication and the last one was correct
when obtained by performing the operation from left to right.
Each correct response received a score of 1. With the maximum score possible at 25,
participants who obtained a score of 85 % and higher were allotted to the Syntax Knowers

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group and the participants who received a score lower than 85 % were allotted to the Syntax
Non – Knowers group.
For the main task, the participants were randomly assigned into one of the three
experimental conditions as mentioned before. Each of the conditions had the same set of
tasks and instructions and each included four blocks of 88 trials building a total of 352 trials.
Each trial was started with a black text being presented for 3 seconds before either of the
following two changes happened – For the colour judgement task, the colour was changed in
two of the letters and for the algebraic equivalence task, another equation was presented on
the right of the = sign.
For both the tasks, the participants were required to respond as fast as possible. For
the colour judgement task, the participants were asked to respond if the two colour letters
were of the same or different colours. For the algebraic equivalence task, the participants
were asked to respond if both the equations on either side of the = were mathematically
equivalent.
For both the tasks, the equations stayed on the screen for 10 seconds within which the
participant had to enter a response to which either a ‘correct’ or a ‘wrong’ feedback was
given. Once 10 seconds had passed without any action, the screen displayed a ‘Too Slow’
message for 500ms before the next trial appeared.
Results
For each of the conditions, a two sample t-test was used to calculate the data obtained
from the study. The results for the first testing condition (neutral spacing) showed a mean of
0.36 for the non-knower group against a mean of 3.02 for the syntax knower group, with SDs

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of 1.99 and 1.94 respectively. The t value recorded was at 8.29 for 148 Df. The test indicated
a resulting p < .001. Therefore the condition is statistically significant.
Measure obs mean SE SD 95% CI
Non Knower 75 . 36 .22 1.99 - .09 .81
Knower 75 3.02 .22 1.94 2.57 3.47
Combined 150 1.69 .19 2.37 1.31 2.07
Diff 2.66 .32 3.30 2. 03
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For the second testing condition (consistent spacing), the mean for the non – knower
group was reported at - .28 and that for the knower group was at 4.46, with SDs of 2.66 and
1.81 respectively. The t value was recorded at 12.76 for 148 degrees of freedom. The test
indicated a resulting p < .001. Thus for this condition too, the result is statistically significant.
Measure obs mean SE SD 95% CI
Non Knower 75 -.28 .30 2.66 - .89 .33
Knower 75 4.46 .20 1.81 4.05 4.88
Combined 150 2.09 .26 3.28 1.56 2.62
Diff 4.74 .37 5.48 4.01
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For the third testing condition (inconsistent spacing), the mean for the non-knower
group was recorded at - .54 and that for the knower group was - .34, with SDs of 2.20 and

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2.13 respectively. The t value was recorded at 0.56 for 148 Df. The test result indicated a p =
0.57 (p > 0.5) indicating that the result for this condition is statistically not significant.
Measure obs mean SE SD 95% CI
Non Knower 75 - .54 .25 2.02 - 1.05 - .04
Knower 75 - .34 .24 2.13 - .83 .14
Combined 150 - .44 .17 2.16 - .79 - .09
Diff .2 .35 - .49 .89
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Discussion
The results of this study indicate that for the neutral condition and consistent spacing
condition, the relations were statistically significant. This indicates that for the first condition,
the hypothesis stands strong. In the first condition, the syntax knowers displayed an
advantage at detecting colour changes that occurred within sub-expressions than between
sub-expression groupings in algebraic expressions. For the second condition, the result
correlated that the hypothesis generated before stands true and that when the sub expression
groupings are supported by spacing that is syntax consistent, the syntax knowers have a
bigger advantage. However, the third condition was resultantly statistically not significant.
This indicated that the effects of prior mathematical knowledge does not persist in either of
the groups during inconsistent spacing condition.
Mathematical notation processing becomes a challenge for us given that the human
brain does not have any specialised region in order to process or perform mathematical
functions. Therefore, we have to resort to reusing or adapting the existing brain mechanisms
in order to process mathematical notations. Implementing rules like BODMAS can help,
however it is a slow and effortful process which is also prone to error if the rules are

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misremembered. Thus it becomes more efficient and the process is sped up if humans learn to
‘see’ the mathematical expressions in a way that is consistent with these rules. According
Gestalt psychology, the principles underpinning it are a set of rules that govern the ways we
group different visual features to form a holistic or unified pattern of wholeness instead of
simply observing an arbitrary series of nonsensical series of colours, lines and curves.
(Koffka, 2013)
Brooks (2015) defines perceptual grouping as the process of determining which
specific regions and parts of the visual scene belong together to form parts of higher order
perceptual units like patterns of concrete objects. In other words, perceptual grouping defines
how we group or perceptually organise visual information in order to determine what we see.
The principle of proximity allows us to perceive the relative closeness of elements and group
objects accordingly. In contrast, the principle of similarity uses our perception to group
elements that are physically similar. This happens even when the inter element distances are
the same.
Such patterning can indicate towards how we see mathematical expressions. The
perceptual features of mathematical expressions can support or interfere with the correct
implementation of the order of operation rules. Thus, people who have been trained to use the
BODMAS rule, perceive various mathematical expressions as containing different subgroups
or objects that are shaped by the order of the operations. Under that consideration the would
group ( X ) and ( ÷ ) together and ( + ) and ( - ) together. This further indicates that the ways
in which expressions are perceived by individuals are reliant on the knowledge they possess
about the rule based organisation of the expression. On the other hand, if the common pattern
of expression is somehow altered, the cognitive system has to redo the entire process in the
brain in order to justify and accommodate the changes. This is not possible in a short span of
time. The results of the tests can thus be justified saying that those with syntax knowledge are

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able to perceive symmetrical changes in patterning (consistent spacing) t with asymmetrical
changes (inconsistent spacing), the cognitive system fails to internalise it in one glance and
restarts the process of evaluation.
This experiment indicated how the special symbolic notations of algebra and
arithmetic equations are retained in the cognitive system of humans. This also indicates
towards the assertion made by Goldstone et al. (2017) where it has been indicated that the
human perception of abstract knowledge is an old phenomenon and has been persistent since
the ancient human history.
A gap in the study would be what Marghetis, Landy and Goldstone (2016) has
identified as, stating that the aspect of perception alone is not sufficient to account for
mathematical reasoning. However, the strength of the study lies in the fact that it has
successfully replicated, modified and evaluated differently, a crucial study in the field of
perception of abstractness and symbolic notations. By doing so, the study has been able to
provide future grounds for adaptation and replication of the process to account for further
possible variance in the data.
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References
Brooks, J. L. (2015). Traditional and new principles of perceptual grouping. Oxford
Handbook of Perceptual Organisations. Oxford university press.
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perception, action, and technology for mathematical reasoning. Current Directions in
Psychological Science, 26(5), 434-441.
Koffka, K. (2013). Principles of Gestalt psychology. Routledge.
Landy, D., & Goldstone, R. L. (2007). How abstract is symbolic thought?. Journal of
Experimental Psychology: Learning, Memory, and Cognition, 33(4), 720.
Marghetis, T., Landy, D., & Goldstone, R. L. (2016). Mastering algebra retrains the visual
system to perceive hierarchical structure in equations. Cognitive research: principles
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Mukucha, J. (2010). Mathematical reasoning in BODMAS. Mathematics: The Pulse of the
Nation, 178.
Ottmar, E., & Landy, D. (2017). Concreteness fading of algebraic instruction: Effects on
learning. Journal of the Learning Sciences, 26(1), 51-78.
Rivera, J., & Garrigan, P. (2016). Persistent perceptual grouping effects in the evaluation of
simple arithmetic expressions. Memory & cognition, 44(5), 750-761.
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ratios. Cognition, 75(2), 145-170.