On the Origin of the Matthew Effect: Insights from a Quantitative Theoretical Model

https://osf.io/vs9hp/

Matthew T. McBee, Matthew C. Makel, & Natasha L. Godkin

4/15/2018

Open materials!

This slideshow

http://rpubs.com/mmcbee/origin

Preprint of the paper

https://psyarxiv.com/8a5qs

OSF project page with paper, code, slides, and supporting materials

https://osf.io/vs9hp/

The Matthew Effect

As one Nobel interviewee stated, “The world… tends to give the credit to already famous people” (Merton, 1988, p. 607).
“For whoever has will be given more, and they will have an abundance. Whoever does not have, even what they have will be taken from them.” Matthew 25:29. NIV
Further widening the gap between those who had from those who had not

Excellence and Achievement gaps

Matthew Effects are particularly relevant for gifted education due to pervasive excellence gaps. Excellence gaps represent performance differences between sub-groups at the high end of achievement (Plucker and Peters, 2016)

Ceci and Papierno (2005)

Matthew Effects have been found in numerous domains
Those who had an initial advantage or accomplishment, reaped not just rewards, but outsized rewards
Relevant for policy decision
Who gets the interventions?

ZPD

Origin of the Matthew Effect

Origin of the Matthew effect. X-axis: time. Y-axis: achievement. Students are observed at a time point denoted by the vertical dashed line. The process creating the discrepancies in achievement at time t will continue unless interrupted or constrained, resulting in growing inequality in achievement over time.

ZPDGrowthTrajectories model

Uses a quantitative theoretical model to understand the consequences of various interventions on achievement and excellence gaps.
Model generates synthetic acheivement growth trajectories.
Implemented as an R package https://github.com/mcbeem/ZPDGrowthTrajectories
Model is based on four underlying variables:
- The child’s basic learning rate,
- The educational quality of the child’s home environment
- How quickly the child forgets things learned
- Educational experiences to which the child is exposed in school

Model equations

\[\begin{equation} \begin{split} \textit{Learning}_{ti} = & (\textit{LearnRate}_i) \, \times \, \\ & \big[ ( \textit{dosage}_{ti}) \int_{0}^{\infty} \textit{ZPD}_{ti}(x) * S_t(x) \, dx + \\ & (1-\textit{dosage}_{ti}) (\textit{HomeEnv}_i) \int_{0}^{\infty} \textit{ZPD}_{ti}(x) * H_t(x) \, dx \big] \end{split} \end{equation}\]

where * denotes the pointwise product, \(LearnRate_i\) the \(i\)th student’s learning rate, \(HomeEnv_i\) the \(i\)th student’s home environment, dosage a constant between 0 and 1 that describes the relative exposure to school versus home during academic years, \(x\) denotes achievement, \(ZPD_it(x)\) denotes the function describing the \(i\)th student’s zone of proximal development at time \(t\), \(S_t(x)\) denotes the function describing the home curriculum at time \(t\), and \(H_t(x)\) the function describing the home curriculum at time \(t\).

Model equations

Achievement is updated for time interval \(t\) by adding the amount of learning during that interval to the student’s prior achievement and then multiplying the result by a value slightly smaller than 1.0 to model decay due to forgetting.

\[\begin{equation} \textit{Achievement}_{i,t}=(\textit{Achievement}_{i,t-1} + \textit{Learning}_{i,t})(1 - \textit{Decay}_i) \end{equation}\]

NWEA MAP national norms by subject and grade.

Representation of current achievement and the ZPD

The student’s current level of achievement in a particular subject is represented by the vertical line. The student’s ZPD is the shaded curve. Instruction is effective at producing academic growth when it aligns with this range.

Interaction of the ZPD with the school curriculum function

The growth rate during interval t is proportional to the intersection area of the ZPD curve (blue) with the school curriculum function (red).

Interaction of the ZPD with the home curriculum function

The growth rate during interval t is proportional to the intersection area of the ZPD curve (blue) with the home curriculum function (red).

Real versus simulated piecewise model results

Level	Component	Empirical	Simulated
Fixed effects	Intercept	193.72 (.084)	190.29 (.053)
	High-achieving	25.24 (.244)	25.79 (.313)
	School year	1.54 (.024)	1.58 (.009)
	School year * High-ach	-1.28 (.025)	-1.32 (.052)
	Summer	-0.12 (.033)	-.083 (.015)
	Summer * High-ach	0.24 (.036)	0.504 (.090)
	School year squared	-0.02 (.001)	-.021 (.003)
	School year squared * High-ach	0.03 (.001)	.032 (.002)
	Summer squared	.001 (.002)	-0.004 (.001)
	Summer squared * High-ach	-0.001 (.002)	-.025 (.007)
Random effects	Level-1 eij	29.890	28.896
	Intercept	30.601	30.761
	School year growth	0.100	0.286
	Summer growth	-	-
	School year growth squared	0.000	0.000
	Summer growth squared	-	-

Individual-differences parameters for the three simulated students

Student	Learning rate	Home environment	Decay rate
1	0.145	0.09	0.010
2	0.175	0.21	0.008
3	0.190	0.34	0.001

Model 1: Business-as-usual

Synthetic trajectories and gaps for three students. Condition: no intervention; standard multilevel curriculum (e.g., typical, remedial, and advanced). Panel 1: Achievement trajectories for birth through grade five in reading. Panel 2: Gaps (achievement differential). The ‘achievement gap’ is difference between student three and student one; the ‘excellence gap’ is the difference between student three and student two.

Model 2: Intervention to increase learning rates

Condition: intervention with additive impact on learning rates; standard multilevel curriculum (e.g., typical, remedial, and advanced). Panel 1: Achievement trajectories. Panel 2: Gaps. Dotted reference lines display the no-intervention results for comparison.

Model 3: Intervention to improve the home environment

Condition: intervention with additive impact on home environment; standard multilevel curriculum (e.g., typical, remedial, and advanced). Panel 1: Achievement trajectories. Panel 2: Gaps. Dotted reference lines display the no-intervention results for comparison.

Model 4: Increase dose (lengthen school day)

Condition: intervention to increase educational dose; standard multilevel curriculum (e.g., typical, remedial, and advanced). Panel 1: Achievement trajectories. Panel 2: Gaps. Dotted reference lines display the no-intervention results for comparison.

Model 5: Completely individualized instruction

Condition: Individualized curriculum.

Descriptive statistics of achievement by model and student

Achieved MAP score at the beginning of each grade by model and student.
Model	Grade	1	2	3
1	GradeK	137.5	162.6	183.0
	Grade1	153.5	171.1	187.5
	Grade2	168.2	188.7	199.1
	Grade3	181.4	204.9	214.1
	Grade4	192.6	216.7	223.0
	Grade5	202.1	224.6	229.5
	Grade6	209.4	229.1	233.8
2	GradeK	143.2	170.6	191.6
	Grade1	161.5	176.6	196.1
	Grade2	181.2	194.6	204.2
	Grade3	198.5	212.3	218.2
	Grade4	212.9	221.9	226.1
	Grade5	222.7	228.4	232.1
	Grade6	228.0	232.2	236.1
3	GradeK	146.4	169.4	188.0
	Grade1	160.7	175.3	192.5
	Grade2	176.0	191.6	201.6
	Grade3	189.3	207.7	215.5
	Grade4	201.0	218.3	223.8
	Grade5	210.8	225.5	230.1
	Grade6	218.2	229.8	234.3
4	GradeK	137.5	162.6	183.0
	Grade1	155.8	171.6	187.1
	Grade2	173.4	191.0	199.7
	Grade3	189.0	209.1	215.7
	Grade4	202.8	220.0	224.5
	Grade5	214.3	227.0	230.8
	Grade6	222.3	231.1	234.9
5	GradeK	137.5	162.6	183.0
	Grade1	155.3	184.7	207.6
	Grade2	172.3	205.3	230.6
	Grade3	188.5	224.8	252.8
	Grade4	204.2	243.6	274.5
	Grade5	219.4	261.9	295.9
	Grade6	234.2	279.7	317.1

Excellence gaps by grade and model

The excellence gap is defined as the achievement differential between the top two students. Excellence gaps at the beginning of each grade by model.
	Grade	Model1	Model2	Model3	Model4	Model5
1	GradeK	20.4	21.0	18.7	20.4	20.4
3	Grade1	16.4	19.5	17.2	15.6	22.9
5	Grade2	10.4	9.6	10.0	8.7	25.4
7	Grade3	9.2	5.9	7.8	6.6	28.0
9	Grade4	6.3	4.2	5.5	4.5	30.8
11	Grade5	5.0	3.7	4.6	3.8	34.0
13	Grade6	4.6	3.9	4.4	3.8	37.4

Achievement gaps by grade and model

Achievement gaps at the beginning of each grade by model.
Grade	Model1	Model2	Model3	Model4	Model5
GradeK	45.5	48.4	41.6	45.5	45.5
Grade1	34.1	34.6	31.8	31.3	52.3
Grade2	30.8	23.0	25.6	26.3	58.4
Grade3	32.7	19.7	26.2	26.7	64.3
Grade4	30.4	13.2	22.8	21.7	70.3
Grade5	27.5	9.4	19.3	16.5	76.4
Grade6	24.3	8.1	16.0	12.5	82.9

Why does this all matter?

Huge implications for Educators, Policy makers and Quantitative Researchers
Could help make more accurate predictions regarding changes in Excellent Gaps.

Closing

Those who had an initial advantage or accomplishment, reaped not just rewards, but outsized rewards

This slideshow

http://rpubs.com/mmcbee/origin

Preprint of the paper

https://psyarxiv.com/8a5qs

OSF project page with paper, code, slides, and supporting materials

https://osf.io/vs9hp/