rate your own
data.
• Interpret data from some tasks and what they suggest about
numerical abilities.
• Overview of Assignment 2 – more to be discussed in Lab 4.
Recent research suggest that humans have two core
systems for representing magnitude / number:
• A precise small number system (n ≤ 4)
• An approximate (large) number system
Evidence from:
• Preverbal infants
• Animal studies
• Guppies show quantity discrimination similar to
humans, implying that the building blocks of uniquely
human mathematical abilities may be evolutionarily
ancient (Agrillo et. al., 2012).
Core Number Abilities
Alex the parrot:
https://www.youtube.com/watch?v=P3w6OYsKJCc
Differences in Core Number Abilities
Recent research suggests children (and adults)
have different response time (RT) signatures
when performing tasks that assess core number
abilities.
• These differences may distinguish between
normal and atypical number development.
• Differences in how we represent and process
core number information may affect our acquired
(cultural) number development.
Current approaches to assessing young
children’s numerical development:
• Focus on understanding very basic processes underlying
numerically relevant computations.
• Use tests of capacity besides tests of attainment
Øe.g., core number abilities versus standardized math test
• Separate number understanding from calculation
Øunderstanding concept of numerosity versus operations on
numbers
• Moving beyond normative, age-as-a-proxy-for-
development analytic approaches
Øaggregated level versus individual level
Various tests are commonly used to assess
different aspects of numerical development:
• Counting tasks
• Dot Enumeration task
• Magnitude Comparison task
• Reading Numbers
• Single digit Addition
• and so on….
Assessing Numerical Development
Counting Tasks
• “start counting out loud and keep going until I
say, Stop!”
• “give me” n items
• Ask children to count items in an array.
Ø different arrays/arrangements
Ø different starting points
Counting Tasks
Keeping track….
Dot Enumeration Task
• A measure of the precise small number system
• Identify the total number of dots in an array
• Instruction: “I am going to show you some
pictures of dots. Tell me as fast as you can,
without making mistakes, how many dots you
see.”
• Record: Child’s numeric response and (laptop)
response time
Dot Enumeration
Dot Enumeration
Dot Enumeration
Dot Enumeration
Task Participation
• Now we will participate in the Dot Enumeration
task. Go to http://go.unimelb.edu.au/ix8r
• As fast and as accurate as you can, say the
number of dots on the screen and press the
arrow. We are interested in your RT so move
onto the next page as soon as you answer.
• Once you begin the task, you cannot stop until the end so
get ready before you begin. http://go.unimelb.edu.au/ix8r
• You will get a series of response times at the end of the
task. Do not close until you copied the results.
• To copy your data, highlight all the cells and press Ctrl+C.
• Paste the results into a Google sheet, and we will plot the
results (under Raw Data sheet):
https://docs.google.com/spreadsheets/d/1mA4nZpwyREd-
IwAavmH_vlbSQQ8w_tTXVB5pXQ1Rlsc/edit#gid=1996590192
• Click the first blank cell on the left, and press Ctrl+V to
paste ALL your results. Click “undo”, never delete!
Let’s look at our data!
https://docs.google.com/spreadsh
eets/d/1mA4nZpwyREd-
IwAavmH_vlbSQQ8w_tTXVB5pX
Q1Rlsc/edit?usp=sharing
Look under Combined Data sheet.
Display Size
R
T
in
m
se
cs
0
1000
2000
3000
4000
5000
6000
7000
1 2 3 4 5 6 7 8
Mean reaction time as a function of set size for
the Dot Enumeration task in children aged 5.5-
6.5 years (Reeve et al., 2012)
Data from Dot Enumeration task
What does this plot tell us?
• What’s the subitizing range?
• What’s the counting range?
• What do the error bars
suggest?
• Did you use strategies to
enumerate the dots?
• How might your (adult’s)
performance differ from
children’s?
• Measures ability to process numerical
magnitudes approximately
• Compare two numbers/quantities and indicate
which one is larger.
• Non-symbolic vs Symbolic formats
• Symbolic: children compare two numbers
and identify the one larger in magnitude
• Non-symbolic: children compare two sets of
dots/ squares/ other items, and identify the
set with a larger number of quantity.
Magnitude Comparison
• 72 trials: all combinations except
for doubles (2:2) of n = 1-9 (e.g.,
1:2, 3:5, 8:4 etc).
• Children are asked to decide which
side has “more squares” or “larger
number”.
• Responses are recorded by
accuracy and response time.
• Why are the squares of different
sizes for non-symbolic?
Ø Controls for total surface area of
the squares, so that the number of
squares is not confounded with
the amount of blue-ness
Magnitude Comparison
8 2
• Stimuli are coded according to
the ratio of the number of
squares.
• 5:6 = 0.83 = falls under ratio 8
• 3:2 = 0.67 = falls under ratio 6
E.g., ratios 0.1 to 0.19 falls under
ratio 1 and 0.2 to 0.29 falls under
ratio 2 and so forth…
Ø Smaller ratios are easier
Ø Bigger ratios are harder
Magnitude Comparison
Example Instructions for Non-symbolic Comparison:
• “In a moment you will see a picture of squares.
One side of the picture will always have more
squares than the other.
• If you think this side [indicate left side] has more,
press the yellow dot [touch left shift key].
• If you think that side [indicate right side] has more,
press the red dot [touch right shift key].”
Magnitude Comparison





8 2
6 9
4 1
8 5
1 3
Figure 1. Non-symbolic Number Comparison response
times as a function of ratio size and year level.
Non-Symbolic versus Symbolic
• How are the two response time graphs “similar” or “different”?
• Slight increase in RTs for larger ratios in the symbolic comparison vs.
Larger increases in RTs for larger ratios in the non-symbolic.
• How might your (adult’s) performance differ from children’s?
Figure 2. Symbolic Number Comparison response
times as a function of ratio size and year level.
Acquired (Cultural) Number Knowledge
(Third Number System)
Number knowledge that is learned over time
(c.f. core number knowledge)
For example:
• reading numbers
• writing numbers
• simple arithmetic (single-digit addition)
Reading Numbers
• 36 trials, six of each string length: 1-digit, “teen”, 2-,
3- 4, and 5-digits.
• Error types
105
1963,201
79,86528
Single-digit Addition
• Instruction: “I’m going to show you some adding
sums. Please tell me the answer. [When child
responds, ask] How did you work that out?”
• Record:
• Child‘s answer
• Counting support (use of fingers, counting words)
• Problem-solving strategy (counting-all, counting-
on, decomposition, retrieval)
• Response time
Single-digit Addition
• 30 trials: all combinations (except for doubles) of 2 to 7
4 + 3 = 2 + 7 =
Other Perspectives
• Some researchers still argue that differences in numerical
competence arise from differences in general cognitive abilities…
• Therefore, arguing differences in core number abilities would be
underpinned by general cognitive abilities
• How can we answer this question?
Ø Use standardised measures of cognitive abilities such as
working memory, and processing speed.
Ø In the assignment, we used:
- Ravens Coloured Progressive Matrices measuring non-verbal IQ
- Basic Reaction Time measuring processing speed
• If core number ability is a domain specific capability for
numerical competence, what would we expect?
Ø That differences in core number ability would not be associated
with differences in general cognitive abilities.
Ravens Coloured Progressive Matrices
• Children are shown a series of
36 patterns each with a piece
missing
• For each pattern, children are
presented with 6 patterned
pieces, and are asked to identify
the piece that completes the
pattern.
• What is the RCPM testing?
(scored using age norms)
Summary
• Evidence suggests that we are predisposed to
represent magnitude / number in two basic ways:
• precise small number (n ≤ 4)
• approximate large number
• Tests of numerical development can assess
numerical capacity (versus. attainment)
• We can use assessments of general cognitive
ability to test whether core number knowledge is
domain specific.
Assignment 2 Overview
• 1,000 words Discussion section based on the findings from a model Results
section
Ø All materials are already available in Assignment Information on LMS.
• You do not need to include the Intro/Method/Results into your submission.
Just begin with the Discussion. The title is: Discussion.
• Discussion must be in APA 7th format
Ø Guides on lab report writing are referenced in the Assignment Information,
additional resources are available on LMS and Lab 4 post-lab video will also
cover an overview of lab report writing, particularly the Discussion section.
• Questions for Assignment 2 can be:
• raised in your lab class with your tutor,
• posted in the Discussion board and they will be collated and answered in
Assignment 2 Q&A workshop on Monday 11 September @ 2-3PM PAR-Redmond
Barry-200 (Rivett Theatre Lvl 2) – also on Zoom and audio-recorded
• Assignment 2 FAQ will also be available on LMS.
Next lab class
• Understanding variability in numerical competence
• Differences, Deficit versus Delay
• Discuss and unpack the results for your
Discussion (Assignment 2)
• More information regarding Assignment 2
• How to write a Discussion (post lab video)