The development of arithmetical abilities – Butterworth (2005)
Variables:
Independent Variables: Innate number-specific capacities and general cognitive abilities
(reasoning, memory, spatial awareness).
Dependent Variables: Arithmetic abilities, including skills such as counting, addition, and
understanding numerosity.
Method: quantitative meta-analysis of participants ranging from infants to older children using test
measuring different parts of arithmetic.
Hypothesis: arithmetic skill development is influenced by both general cognitive abilities and innate
number-specific capacities.
Research question and results: Is the acquisition of arithmetic skills supported solely by general
cognitive abilities such as reasoning, memory, and spatial awareness, or are there innate, number-
specific capacities involved?
Infants can respond to numerical changes (e.g., three dolls becoming four) and perform
mental manipulations, suggesting that numerical understanding may not rely solely on
language or advanced reasoning.
Key counting principles such as the Stable Order Principle (counting words follow a fixed
order), One-to-One Principle, Cardinal Principle (final counting word represents the total
quantity), Abstractness (abstract sets like wishes can also be counted), and Order-Irrelevance
are fundamental to the acquisition of arithmetic skills. These principles are initially learned
through basic cognitive processes but also benefit from innate numerical capacities.
Different educational practice, e.g. learning commuted pairs (e.g., 3×5 = 5×3) and the using
the Principle of Complementarity (e.g., if 5 + 3 = 8, then 8 - 5 = 3), impact the development of
arithmetic skills. This can improve learning efficiency and accuracy.
Developmental dyscalculia (DD) is a condition reflecting an innate deficit in numerical
understanding, necessary for arithmetic. DD often co-occurs with other problems like literacy
and emphasizes the role of brain regions like the intraparietal sulcus in numerical processing.
Genetic factors, such as Turner’s Syndrome, are also relevant to understanding DD.
Conclusion: Arithmetic skills develop through a combination of innate numerical understanding and
general cognitive abilities. Early numerical skills, brain development, and specific genetic factors play
a role in arithmetic ability. Further research is needed to explore how these factors interact and
influence arithmetic learning.
Strengths:
Comprehensive review of existing literature and empirical studies.
Integration of cognitive, developmental, and neurological perspectives.
Highlights both innate and cognitive factors in arithmetic development.
Limitations:
Precise mechanisms behind developmental dyscalculia and its relationship with other
disabilities remain unclear.
Gaps in understanding the interactions between early numerical abilities, brain functions,
and educational practices.
, Flexibility and adaptivity in arithmetic strategy use: What
children know and what they show. – Hickendorff (2022)
Variables:
Practical vs. potential flexibility: using vs. knowing multiple strategies
Practical vs. potential task-based adaptivity: using vs. knowing the strategy most
appropriate for a task.
Practical vs. potential individual-based adaptivity: students using vs. knowing the strategy
that is optimal for themselves on a given problem (fastest strategy leading to a correct
answer).
Math performance, number knowledge, memory, and gender.
Method: Quantitative/ experimental. Arithmetic task for Dutch grade 3 kids with subtraction
problems up to 100.
Hypothesis: There's a gap between what students currently show and their potential in adaptive
expertise.
Research questions and results:
1. How well do students use and understand different strategies?
The mean potential flexibility was significantly higher than the mean practical
flexibility suggesting hidden potential in strategy knowledge
2. Do students use and know the best strategies for specific tasks?
The mean potential task-based adaptivity was significantly larger than the mean
practical adaptivity, indicating hidden potential in adaptivity to task characteristics.
3. Do students use and know the best strategies for their own needs?
The mean potential individual-based adaptivity did not differ from the mean practical
adaptivity.
4. How are students' strategy skills related to their math performance, number knowledge,
memory, and gender?
Potential flexibility was linked to all student factors except gender. Practical flexibility
and potential individual-based adaptivity were connected to math achievement. The
other three adaptive expertise factors—practical task-based adaptivity, potential
task-based adaptivity, and practical individual-based adaptivity—weren't linked to
any student factors studied.
Conclusion: Students’ strategy flexibility and task-based adaptivity is rather low. However, students
may not use certain strategies because they don’t work well for them, but they might still have the
ability to adapt strategies well if given the chance. Many students know more than they show
regarding flexibility and task-based adaptivity.
Strengths:
Free strategy choice: It assesses how students choose strategies in a real-world setting,
showing a wide range of strategies.
Performance Comparison: It compares students' own strategies with predefined ones,
showing that personal strategies can be more effective.
Ecological validity: It gives a clearer picture of students' adaptability by focusing on their
natural choices, revealing they are more adaptable than if only specific strategies are
considered.
Variables:
Independent Variables: Innate number-specific capacities and general cognitive abilities
(reasoning, memory, spatial awareness).
Dependent Variables: Arithmetic abilities, including skills such as counting, addition, and
understanding numerosity.
Method: quantitative meta-analysis of participants ranging from infants to older children using test
measuring different parts of arithmetic.
Hypothesis: arithmetic skill development is influenced by both general cognitive abilities and innate
number-specific capacities.
Research question and results: Is the acquisition of arithmetic skills supported solely by general
cognitive abilities such as reasoning, memory, and spatial awareness, or are there innate, number-
specific capacities involved?
Infants can respond to numerical changes (e.g., three dolls becoming four) and perform
mental manipulations, suggesting that numerical understanding may not rely solely on
language or advanced reasoning.
Key counting principles such as the Stable Order Principle (counting words follow a fixed
order), One-to-One Principle, Cardinal Principle (final counting word represents the total
quantity), Abstractness (abstract sets like wishes can also be counted), and Order-Irrelevance
are fundamental to the acquisition of arithmetic skills. These principles are initially learned
through basic cognitive processes but also benefit from innate numerical capacities.
Different educational practice, e.g. learning commuted pairs (e.g., 3×5 = 5×3) and the using
the Principle of Complementarity (e.g., if 5 + 3 = 8, then 8 - 5 = 3), impact the development of
arithmetic skills. This can improve learning efficiency and accuracy.
Developmental dyscalculia (DD) is a condition reflecting an innate deficit in numerical
understanding, necessary for arithmetic. DD often co-occurs with other problems like literacy
and emphasizes the role of brain regions like the intraparietal sulcus in numerical processing.
Genetic factors, such as Turner’s Syndrome, are also relevant to understanding DD.
Conclusion: Arithmetic skills develop through a combination of innate numerical understanding and
general cognitive abilities. Early numerical skills, brain development, and specific genetic factors play
a role in arithmetic ability. Further research is needed to explore how these factors interact and
influence arithmetic learning.
Strengths:
Comprehensive review of existing literature and empirical studies.
Integration of cognitive, developmental, and neurological perspectives.
Highlights both innate and cognitive factors in arithmetic development.
Limitations:
Precise mechanisms behind developmental dyscalculia and its relationship with other
disabilities remain unclear.
Gaps in understanding the interactions between early numerical abilities, brain functions,
and educational practices.
, Flexibility and adaptivity in arithmetic strategy use: What
children know and what they show. – Hickendorff (2022)
Variables:
Practical vs. potential flexibility: using vs. knowing multiple strategies
Practical vs. potential task-based adaptivity: using vs. knowing the strategy most
appropriate for a task.
Practical vs. potential individual-based adaptivity: students using vs. knowing the strategy
that is optimal for themselves on a given problem (fastest strategy leading to a correct
answer).
Math performance, number knowledge, memory, and gender.
Method: Quantitative/ experimental. Arithmetic task for Dutch grade 3 kids with subtraction
problems up to 100.
Hypothesis: There's a gap between what students currently show and their potential in adaptive
expertise.
Research questions and results:
1. How well do students use and understand different strategies?
The mean potential flexibility was significantly higher than the mean practical
flexibility suggesting hidden potential in strategy knowledge
2. Do students use and know the best strategies for specific tasks?
The mean potential task-based adaptivity was significantly larger than the mean
practical adaptivity, indicating hidden potential in adaptivity to task characteristics.
3. Do students use and know the best strategies for their own needs?
The mean potential individual-based adaptivity did not differ from the mean practical
adaptivity.
4. How are students' strategy skills related to their math performance, number knowledge,
memory, and gender?
Potential flexibility was linked to all student factors except gender. Practical flexibility
and potential individual-based adaptivity were connected to math achievement. The
other three adaptive expertise factors—practical task-based adaptivity, potential
task-based adaptivity, and practical individual-based adaptivity—weren't linked to
any student factors studied.
Conclusion: Students’ strategy flexibility and task-based adaptivity is rather low. However, students
may not use certain strategies because they don’t work well for them, but they might still have the
ability to adapt strategies well if given the chance. Many students know more than they show
regarding flexibility and task-based adaptivity.
Strengths:
Free strategy choice: It assesses how students choose strategies in a real-world setting,
showing a wide range of strategies.
Performance Comparison: It compares students' own strategies with predefined ones,
showing that personal strategies can be more effective.
Ecological validity: It gives a clearer picture of students' adaptability by focusing on their
natural choices, revealing they are more adaptable than if only specific strategies are
considered.
