Decomposition - 2025

1. Probe description

The Decomposition probe assesses students’ understanding of part-whole relationships with numbers. Students are shown a total quantity with some hidden, and must identify how many are hidden. Students select their answer from 4 multiple-choice options (numbers).

In 2025 there are two main forms used in the Foundation screening:

  • Decomposition to 5 (DMT5) – Term 1
  • Decomposition to 10 (DMT10) – Term 1
  • Decomposition to 5 V2 (DMT5V2) – Term 3 / Term 4
  • Decomposition to 10 V2 (DMT10V2) – Term 3 / Term 4
Note

Decomposition is not a timed probe, so there is no fluency analysis. This probe was completed only by Foundation students.

Changes from Term 1 (BOY) to Term 3 (MOY) / Term 4 (EOY)

The task presentation and quantity of items changed between BOY and MOY/EOY assessments:

Form Term 1 (BOY) Term 3 (MOY) / Term 4 (EOY)
Decomposition to 5 5 blocks are in a cup and some fall out (10 items) Up to 5 counters in a line, given the total identify the hidden quantity (6 items)
Decomposition to 10 10 blocks are in a cup and some fall out (10 items) Up to 10 counters in a line, given the total identify the hidden quantity (6 items)

Term 1 examples

Term 3 / Term 4 example

Download Foundation student data (wide format)

2. Overview of test results

Term Test name Test ID Students Median items attempted Median accuracy (%)* Median avg RT/item (sec)
1 Decomposing Measure to 10 DMT10_2025 1197 10/10 30.0 15.10
1 Decomposing Measure to 5 DMT5_2025 1288 10/10 61.3 11.50
3 Decomposition Measure 10 DECOMP-MEASURE-10 647 6/7 60.0 13.83
3 Decomposition Measure 5 DECOMP-MEASURE-5 785 7/8 71.4 9.71
4 Decomposition Measure 10 DECOMP-MEASURE-10 525 7/7 66.7 12.57
4 Decomposition Measure 5 DECOMP-MEASURE-5 566 7/8 85.7 10.14
* Accuracy is calculated after removing non-attempted items.

3. Accuracy distribution

Accuracy distribution

Term Students 25th percentile Median 75th percentile
Term 1 1288 30 61.2 90

Bottom quartile vs rest

Term Group Students 25th percentile Median 75th percentile
Term 1 Bottom 25% 356 10.0 20 22.2
Term 1 Rest 932 57.1 80 100.0

Accuracy distribution

Term Students 25th percentile Median 75th percentile
Term 3 785 33.3 71.4 100

Bottom quartile vs rest

Term Group Students 25th percentile Median 75th percentile
Term 3 Bottom 25% 201 14.3 20.0 28.6
Term 3 Rest 584 66.7 85.7 100.0

Accuracy distribution

Term Students 25th percentile Median 75th percentile
Term 4 566 57.1 85.7 100

Bottom quartile vs rest

Term Group Students 25th percentile Median 75th percentile
Term 4 Bottom 25% 167 22.5 40 50
Term 4 Rest 399 85.7 100 100

Accuracy distribution

Term Students 25th percentile Median 75th percentile
Term 1 1197 20 30 60

Bottom quartile vs rest

Term Group Students 25th percentile Median 75th percentile
Term 1 Bottom 25% 432 10 11.1 20
Term 1 Rest 765 30 50.0 80

Accuracy distribution

Term Students 25th percentile Median 75th percentile
Term 3 647 33.3 60 83.3

Bottom quartile vs rest

Term Group Students 25th percentile Median 75th percentile
Term 3 Bottom 25% 204 16.7 20.0 33.3
Term 3 Rest 443 50.0 66.7 100.0

Accuracy distribution

Term Students 25th percentile Median 75th percentile
Term 4 525 42.9 66.7 85.7

Bottom quartile vs rest

Term Group Students 25th percentile Median 75th percentile
Term 4 Bottom 25% 161 16.7 28.6 42.9
Term 4 Rest 364 60.0 85.7 100.0

4. Response time distribution

5. Distractor selected

Tip

Click on any plot to view an enlarged version in a lightbox popup.

No distractor data (dictionary coverage not available for these Term 3 items).

No distractor data (dictionary coverage not available for these Term 3 items).

No distractor data (dictionary coverage not available for these Term 3 items).

No distractor data (dictionary coverage not available for these Term 3 items).

6. Item statistics

NoteItem discrimination

Item discrimination (point-biserial) measures how well an item separates higher-ability from lower-ability students. Values above ~0.3 are typically “good”, below ~0.2 “weak”. In timed tests, late items can look artificially strong because only fast/able students reach them.

Download item stats (CSV)

Correct response time (sec)
Question Task Target (btn) No. response % Correct Most common error Median RT 95th RT Item discrim.
1 5-4 1 (btn-1) 1257 64.6 1 (btn-1) 14 46.00 0.58
2 5-5 0 (btn-3) 1182 63.8 3 (btn-2) 13 49.00 0.53
3 5-1 4 (btn-4) 1212 55.4 1 (btn-3) 8 31.00 0.68
4 5-3 2 (btn-2) 1218 57.3 3 (btn-4) 9 35.15 0.63
5 5-0 5 (btn-3) 1133 72.2 1 (btn-1) 9 40.00 0.42
6 5-2 3 (btn-1) 1214 50.7 2 (btn-3) 10 33.00 0.59
7 5-4 1 (btn-4) 1180 55.8 5 (btn-3) 9 32.10 0.63
8 5-0 5 (btn-2) 1119 68.8 1 (btn-1) 7 24.00 0.47
9 5-3 2 (btn-3) 1169 68.7 4 (btn-4) 7 26.90 0.48
10 5-2 3 (btn-4) 1187 55.4 2 (btn-1) 8 23.00 0.66
Correct response time (sec)
Item ID Task Target (btn) No. response % Correct Most common error Median RT 95th RT Item discrim.
DMT5-1-hiding – – 722 60.9 – 7 22.00 0.59
DMT3-1-hiding – – 753 66.0 – 6 20.20 0.62
DMT3-3-hiding – – 704 67.6 – 8 29.25 0.49
DMT4-2-hiding – – 726 73.8 – 8 28.25 0.40
DMT4-3-hiding – – 752 61.7 – 10 37.00 0.50
DMT5-2-hiding – – 726 65.2 – 8 25.00 0.67
DMT5-4-hiding – – 739 71.6 – 9 30.60 0.40
Correct response time (sec)
Item ID Task Target (btn) No. response % Correct Most common error Median RT 95th RT Item discrim.
DMT3-2-hiding – – 556 79.1 – 22 74.0 0.32
DMT3-1-hiding – – 539 73.5 – 5 16.0 0.67
DMT3-3-hiding – – 533 76.7 – 7 26.0 0.49
DMT4-2-hiding – – 542 79.3 – 6 19.0 0.44
DMT4-3-hiding – – 557 72.4 – 7 30.0 0.63
DMT5-2-hiding – – 545 74.3 – 6 19.0 0.65
DMT5-4-hiding – – 551 77.5 – 7 22.7 0.42
Correct response time (sec)
Question Task Target (btn) No. response % Correct Most common error Median RT 95th RT Item discrim.
1 10-9 1 (btn-2) 1153 31.0 9 (btn-1) 24 64.40 0.44
2 10-7 3 (btn-3) 1069 39.8 6 (btn-2) 24 72.60 0.38
3 10-6 4 (btn-1) 1076 38.9 7 (btn-4) 17 57.00 0.48
4 10-4 6 (btn-3) 1100 40.7 5 (btn-1) 14 49.00 0.52
5 10-2 8 (btn-2) 1079 43.0 1 (btn-4) 11 42.00 0.56
6 10-1 9 (btn-1) 1062 42.0 3 (btn-2) 9 36.50 0.63
7 10-3 7 (btn-4) 1084 39.7 1 (btn-3) 11 40.00 0.57
8 10-8 2 (btn-1) 1106 34.2 8 (btn-3) 13 42.15 0.56
9 10-5 5 (btn-2) 1094 57.9 6 (btn-3) 8 28.00 0.18
10 10-0 10 (btn-4) 1090 38.6 8 (btn-1) 11 32.00 0.47
Correct response time (sec)
Item ID Task Target (btn) No. response % Correct Most common error Median RT 95th RT Item discrim.
DMT5-2-hiding – – 604 55.8 – 6 19.0 0.47
DMT5-4-hiding – – 619 77.2 – 8 28.0 0.39
DMT10-3-hiding – – 596 46.1 – 13 32.0 0.40
DMT10-6-hiding – – 624 48.6 – 11 37.8 0.45
DMT8-5-hiding – – 619 48.9 – 24 65.9 0.34
DMT8-7-hiding – – 613 70.0 – 14 46.0 0.36
Correct response time (sec)
Item ID Task Target (btn) No. response % Correct Most common error Median RT 95th RT Item discrim.
DMT5-2-hiding – – 473 66.8 – 6 20.0 0.53
DMT5-4-hiding – – 496 80.0 – 7 20.0 0.43
DMT3-2-hiding – – 510 76.3 – 15 52.6 0.17
DMT10-3-hiding – – 475 52.2 – 13 36.0 0.49
DMT10-6-hiding – – 498 52.2 – 10 36.0 0.53
DMT8-5-hiding – – 512 51.2 – 17 47.9 0.38
DMT8-7-hiding – – 503 73.8 – 11 41.0 0.46

7. Item correct response time

8. Correlation between target number and performance

This section shows how item difficulty varies with the target number (the hidden amount students must identify, i.e., Correct Answer in the dictionary).

The size of each dot represents the number of attempted responses for that item. The regression line is weighted to give more influence to items with higher response counts.

Note

Correlation plots are only available for items with dictionary coverage (Term 1 items). Term 3 items without dictionary entries show a message.

`geom_smooth()` using formula = 'y ~ x'

No correlation data (dictionary coverage not available for these Term 3 items).

No correlation data (dictionary coverage not available for these Term 3 items).

`geom_smooth()` using formula = 'y ~ x'

No correlation data (dictionary coverage not available for these Term 3 items).

No correlation data (dictionary coverage not available for these Term 3 items).

7. Age and Response Accuracy Analysis

This section examines how student age relates to response accuracy patterns on decomposition tasks. The analysis addresses two research questions:

  1. Are older children less likely to choose the visible quantity when it’s incorrect?
  2. Are older children more likely to give exact answers vs. being off by 1 or 2?

Data Processing

For each student response, we calculate:

  • Age at testing: Student’s age in years and months at the time of their test (using attempted_at timestamp)
  • Response accuracy categories (mutually exclusive):
    • Exact: Response matches correct answer (difference = 0)
    • Off by 1: Response differs by exactly 1 from correct answer
    • Off by 2: Response differs by exactly 2 from correct answer
    • Off by 3+: Response differs by 3 or more from correct answer
  • Chose visible flag: TRUE if student selected the visible quantity when it was incorrect

These metrics are aggregated to the student × term level, showing the percentage of items in each accuracy category.

Processing decomposition responses for age analysis...

       exact     off_by_1     off_by_2 off_by_3plus         <NA> 
       39071         8690         6170         7257        13466 
Student × term records with valid age data: 4943

7.1 Age Distribution Overview

Age distribution by testing term
Term N Students Age Range Mean Age
Term 1 (BOY) 2454 4 yrs 8 mo - 8 yrs 11 mo 5 yrs 6 mo
Term 3 (MOY) 1409 5 yrs 0 mo - 9 yrs 4 mo 5 yrs 10 mo
Term 4 (EOY) 1080 5 yrs 4 mo - 9 yrs 0 mo 6 yrs 2 mo

7.2 Age vs. Exact Accuracy

Research Question: Are older children more likely to give exact correct answers?

`geom_smooth()` using formula = 'y ~ x'

Effect Sizes and Practical Significance

Statistical significance (p-values) tells us whether an effect exists, but effect sizes tell us how large and meaningful that effect is. Below we report multiple effect size metrics to provide a comprehensive picture of the relationship between age and accuracy.

Effect Size Interpretation Guidelines:

  • Correlation (r):
    • Small: 0.10 - 0.29
    • Medium: 0.30 - 0.49
    • Large: 0.50+
  • R² (Variance Explained):
    • Small: 1% - 9%
    • Medium: 9% - 25%
    • Large: 25%+
Enhanced correlation: Age and % exact correct responses
Term N Correlation (r) 95% CI R² Effect Size p-value Sig.
Term 1 (BOY) 2454 0.219 [0.181, 0.256] 0.048 Small 0 ***
Term 3 (MOY) 1409 0.116 [0.064, 0.167] 0.013 Small 0 ***
Term 4 (EOY) 1080 0.176 [0.118, 0.233] 0.031 Small 0 ***
Note:
*** p < 0.001, ** p < 0.01, * p < 0.05
Linear regression: Change in % exact per year of age
Term Slope (pp/year) R² Adjusted R²
Term 1 (BOY) 17.41 [14.34, 20.48] 0.048 0.048
Term 3 (MOY) 5.60 [3.09, 8.11] 0.013 0.013
Term 4 (EOY) 9.01 [6.00, 12.02] 0.031 0.030
Note:
Slope shows percentage point increase in exact accuracy per additional year of age. 95% confidence intervals in brackets.

Interpretation: The correlation coefficients show the strength of the relationship between age and exact accuracy. R² indicates the proportion of variation in accuracy that is explained by age. The slope from linear regression shows the practical effect: for each additional year of age, exact accuracy increases by the indicated number of percentage points.

Beta Regression Analysis

Why use beta regression? Accuracy percentages are bounded between 0% and 100%, but standard linear regression assumes outcomes can range from negative infinity to positive infinity. Beta regression is specifically designed for percentage data and provides more accurate estimates when many students score at or near 100% (ceiling effects).

How to read the table below:

  • Estimate: The effect of age on the log-odds scale (technical detail for statisticians)
  • Marginal Effect: For every additional year of age, accuracy increases by this many percentage points on average
  • 95% CI: We’re 95% confident the true effect falls within this range
  • Pseudo-R²: Proportion of variation in accuracy explained by age (similar to R² in linear regression)
Beta regression: Age predicting % exact correct responses
Term Estimate (95% CI) Marginal Effect (pp/year) Pseudo-R² p-value Sig.
Term 1 (BOY) 0.704 [0.578, 0.830] 14.94 0.039 0 ***
Term 3 (MOY) 0.358 [0.216, 0.501] 5.23 0.010 0 ***
Term 4 (EOY) 0.446 [0.270, 0.622] 5.34 0.022 0 ***
Note:
*** p < 0.001, ** p < 0.01, * p < 0.05. Marginal effect calculated at mean age.

Key Finding: Beta regression confirms the correlation results - older children have significantly higher exact accuracy rates. The marginal effects show the practical significance: each additional year of age is associated with a meaningful increase in exact accuracy across all terms.

Comparing Linear and Beta Regression

Both approaches confirm that age positively predicts accuracy, but beta regression provides more appropriate estimates for bounded percentage data.

Comparison of Linear and Beta Regression Results
Linear Regression
Beta Regression
Term Linear Slope (pp/year) Linear R² Beta Marginal Effect (pp/year) Beta Pseudo-R²
Term 1 (BOY) 17.41 [14.34, 20.48] 0.048 14.94 0.039
Term 3 (MOY) 5.60 [3.09, 8.11] 0.013 5.23 0.010
Term 4 (EOY) 9.01 [6.00, 12.02] 0.031 5.34 0.022
Note:
Both methods show similar effect sizes and lead to the same conclusion: older children perform better.

Key Differences:

  • Beta regression typically shows slightly different marginal effects because it better handles the bounded nature of percentage data
  • Confidence intervals from beta regression are more reliable for percentage outcomes with ceiling effects
  • Both methods lead to the same substantive conclusion: older children have higher exact accuracy rates
  • The pseudo-R² from beta regression is comparable to R² from linear regression

7.3 Age vs. Choosing Visible Quantity

Research Question: Are older children less likely to choose the visible quantity when it’s incorrect?

`geom_smooth()` using formula = 'y ~ x'

Effect Sizes and Practical Significance

Warning: There were 2 warnings in `summarise()`.
The first warning was:
ℹ In argument: `r = cor(age_decimal, pct_chose_visible, use = "complete.obs")`.
ℹ In group 2: `term_label = "Term 3 (MOY)"`.
Caused by warning in `cor()`:
! the standard deviation is zero
ℹ Run `dplyr::last_dplyr_warnings()` to see the 1 remaining warning.
Warning: There were 2 warnings in `summarise()`.
The first warning was:
ℹ In argument: `p_value = cor.test(age_decimal, pct_chose_visible)$p.value`.
ℹ In group 2: `term_label = "Term 3 (MOY)"`.
Caused by warning in `cor()`:
! the standard deviation is zero
ℹ Run `dplyr::last_dplyr_warnings()` to see the 1 remaining warning.
Enhanced correlation: Age and % choosing visible quantity
Term N Correlation (r) 95% CI R² Effect Size p-value Sig.
Term 1 (BOY) 2454 -0.128 [-0.167, -0.089] 0.016 Small 0 ***
Term 3 (MOY) 1409 NA [NA, NA] NA Large NA
Term 4 (EOY) 1080 NA [NA, NA] NA Large NA
Note:
*** p < 0.001, ** p < 0.01, * p < 0.05
Linear regression: Change in % choosing visible per year of age
Term Slope (pp/year) R² Adjusted R²
Term 1 (BOY) -2.28 [-2.98, -1.58] 0.016 0.016
Term 3 (MOY) 0.00 [0.00, 0.00] NaN NaN
Term 4 (EOY) -0.00 [0.00, 0.00] NaN NaN
Note:
Negative slopes indicate older children are less likely to choose visible quantity. 95% confidence intervals in brackets.

Interpretation: The correlations show that age is negatively associated with choosing the visible quantity - older children are less likely to make this error. However, note the floor effects at MOY and EOY (nearly all students avoid this error regardless of age), making BOY the most informative timepoint for this analysis.

Beta Regression Analysis (BOY Only)

Beta regression for choosing visible quantity is only meaningful at BOY, where students show variation in this behaviour. At MOY and EOY, nearly all students (regardless of age) have learnt not to choose the visible quantity distractor, resulting in a “floor effect” with insufficient variation to model reliably.

Beta regression: Age predicting % choosing visible quantity (BOY only)
Term Estimate (95% CI) Marginal Effect (pp/year) Pseudo-R² p-value Sig.
Term 1 (BOY) -0.198 [-0.301, -0.095] -0.72 0.013 2e-04 ***
Note:
*** p < 0.001, ** p < 0.01, * p < 0.05. Negative marginal effect indicates older children are less likely to choose visible quantity. MOY and EOY not modelled due to floor effects.

Key Finding: At BOY, each additional year of age is associated with a decrease in choosing the visible quantity when incorrect. This confirms that older children are less susceptible to this misconception at the start of the school year. By MOY and EOY, this error has largely been eliminated across all age groups through instruction.

7.4 Accuracy Category Distribution by Age Group

This analysis shows how the distribution of accuracy categories changes across age groups.

Mean accuracy percentages by age group and term
Term Age Group N % Exact % Off by 1 % Off by 2 % Off by 3+
Term 1 (BOY) 4.5-5.0 264 41.1 29.6 19.5 9.9
Term 1 (BOY) 5.0-5.5 997 51.4 24.5 16.3 7.9
Term 1 (BOY) 5.5-6.0 974 57.1 21.4 14.3 7.2
Term 1 (BOY) 6.0-6.5 198 65.4 17.0 12.5 5.0
Term 1 (BOY) 6.5-7.0 14 70.1 16.0 11.1 2.8
Term 1 (BOY) 7.0-7.5 5 72.7 10.8 14.0 2.5
Term 1 (BOY) 8.0-8.5 1 100.0 0.0 0.0 0.0
Term 1 (BOY) 8.5-9.0 1 100.0 0.0 0.0 0.0
Term 3 (MOY) 4.5-5.0 3 62.1 32.1 3.0 2.8
Term 3 (MOY) 5.0-5.5 309 71.8 15.0 8.8 4.3
Term 3 (MOY) 5.5-6.0 674 76.7 12.0 7.9 3.4
Term 3 (MOY) 6.0-6.5 398 79.5 10.7 7.4 2.4
Term 3 (MOY) 6.5-7.0 19 81.3 8.9 6.9 2.8
Term 3 (MOY) 7.0-7.5 2 66.2 24.7 4.5 4.5
Term 3 (MOY) 8.0-8.5 2 58.3 18.3 18.3 5.0
Term 4 (EOY) 5.0-5.5 55 70.8 14.0 10.8 4.4
Term 4 (EOY) 5.5-6.0 380 77.4 11.2 8.0 3.4
Term 4 (EOY) 6.0-6.5 497 83.4 8.1 6.3 2.2
Term 4 (EOY) 6.5-7.0 139 84.5 7.2 5.9 2.3
Term 4 (EOY) 7.0-7.5 7 80.8 6.2 11.6 1.4
Term 4 (EOY) 7.5-8.0 1 81.8 9.1 9.1 0.0
Term 4 (EOY) 8.5-9.0 1 100.0 0.0 0.0 0.0

7.5 Downloadable Student-Level Data

The table below contains student-level aggregated statistics for age and response accuracy. Each row represents one student in one term.

Note: Accuracy percentages (Exact + Off by 1 + Off by 2 + Off by 3+) sum to 100%. The “Chose Visible” percentage is independent and may overlap with the accuracy categories.

Student-level age and accuracy data (first 20 rows)
student_id term exam_group age_years_at_test age_months_at_test n_items_attempted pct_exact pct_off_by_1 pct_off_by_2 pct_off_by_3plus pct_chose_visible
00107783-010a-f011-bcb9-8fb474f3e22e 1 Foundation A 2025 5 4 11 18.2 36.4 36.4 9.1 27.3
007a8bfd-a207-f011-bcb9-dcac624517bd 1 Foundation A 2025 5 6 11 72.7 0.0 18.2 9.1 0.0
0095b32f-d60f-f011-a972-9ebec6053e0e 1 Foundation B 2025 5 2 11 18.2 36.4 36.4 9.1 0.0
00b14cb6-a10a-f011-bcbb-d1bac03748a2 1 Foundation A 2025 5 8 11 27.3 54.5 18.2 0.0 18.2
00b44cb6-a10a-f011-bcbb-d1bac03748a2 1 Foundation B 2025 5 1 2 50.0 50.0 0.0 0.0 0.0
00bc4cb6-a10a-f011-bcbb-d1bac03748a2 1 Foundation B 2025 5 4 11 27.3 54.5 18.2 0.0 9.1
00c3afda-f00e-f011-a972-97d85066398c 1 Foundation B 2025 5 3 10 20.0 10.0 40.0 30.0 0.0
00c42e4a-a007-f011-bcbb-e211f1a75204 1 Foundation B 2025 5 1 3 66.7 0.0 33.3 0.0 0.0
01231dd9-f213-f011-a972-c3fdec2d1746 1 Foundation A 2025 5 6 10 100.0 0.0 0.0 0.0 0.0
01233a51-a007-f011-bcbb-e211f1a75204 1 Foundation A 2025 5 0 11 81.8 9.1 0.0 9.1 0.0
012e3a51-a007-f011-bcbb-e211f1a75204 1 Foundation A 2025 5 9 11 54.5 27.3 0.0 18.2 18.2
014993f7-a207-f011-bcb9-dcac624517bd 1 Foundation B 2025 5 10 10 80.0 0.0 10.0 10.0 0.0
01597283-110c-f011-a972-a0475d3d08ed 1 Foundation B 2025 6 1 3 100.0 0.0 0.0 0.0 0.0
015b00f3-a50e-f011-a973-c418bf5f696c 1 Foundation A 2025 5 6 10 20.0 60.0 10.0 10.0 0.0
0181e1ee-b90a-f011-bcb9-8b81c0530db1 1 Foundation B 2025 5 8 11 72.7 0.0 27.3 0.0 0.0
01b00da4-650f-f011-a972-bc284488ed1c 1 Foundation A 2025 5 7 9 100.0 0.0 0.0 0.0 0.0
01b84cb6-a10a-f011-bcbb-d1bac03748a2 1 Foundation B 2025 5 10 11 81.8 9.1 9.1 0.0 0.0
021c4d4f-1c14-f011-a972-b33d457dfc0c 1 Foundation A 2025 5 6 11 100.0 0.0 0.0 0.0 0.0
025d7283-110c-f011-a972-a0475d3d08ed 1 Foundation B 2025 4 11 10 60.0 0.0 30.0 10.0 0.0
0265be5b-5f0c-f011-a973-ed10459e87e9 1 Foundation A 2025 5 6 11 100.0 0.0 0.0 0.0 0.0

Download Full Dataset (CSV)

**Dataset Summary:**
- Total records: 4943 
- Unique students: 2617 
- Terms: 1, 3, 4