Achievement in Mathematics
Running Head:
Achievement in Mathematics
A Determinate Model of Achievement in Mathematics
1
Achievement in Mathematics
Abstract
An expolinear function based on psychometric findings is derived to explain student
achievement in mathematics over time. The function’s two parameters, Cm and k, are set
by Spearman’s g, as measured by student mental age. Using data from large longitudinal
samples, estimates of the function were made on 11 performance levels. Nonlinear
regression analyses unexpectedly resulted in a significant R2 = 0.99. The disparity
between previously reported results (R2 > .50) and these results imply the presence of a
hidden variable in parameter, Cm. The hidden variable may be teacher effect.
Keywords: asymptote, determinate growth, expolinear function, integration,
logistic function, longitudinal achievement, mental age, mental development, rational
function, Spearman’s g, teacher effect
2
Mathematics Achievement 3
A Determinate Model of Achievement in Mathematics
There have been a great number of mathematical functions specified to explain
student achievement in the forty years since the 1966 Coleman Report. Most of these
functions explained achievement and attempted to measure effectiveness of schools or
teachers by either of two statistical methods. Achievement has been analyzed primarily
by using ANCOVA or multiple regression to find whether relationships existed between
the myriad of factors thought to affect the learning of mathematics. The first model relied
on averages, which has made finding small differences in achievement hard to detect.
The second model included factors that were not under the control of schools
(McCaffrey, Lockwood, Louis, Hamilton, 2004). A model that accurately explains
achievement growth in mathematics is still needed. The purpose of this paper is to
propose and test a functional model that relates student achievement in mathematics to
time (development).
Two studies support the idea that achievement is functionally related to
intelligence. (Gedye, 1981) found by factoring teacher grading of elementary aged
students (including math grades) that about half of the variance (r = 0.75, R2 = .56) could
be explained by the Stanford-Binet test of intelligence. In addition, using a structural
model, (Brodnick, 1995) found 46.4% of college student achievement, as measured by
grades in mathematics and English, was causally linked with intelligence test scores.
Both studies are consistent with Jensen’s conclusion that Spearman’s g is strongly
correlated with student achievement (Jensen, 1998). Spearman’s g is a general mental
ability that enters into every kind of activity requiring mental effort. Never observed
directly, it underlies observed or measured variables and is called a latent variable. Since
Mathematics Achievement 4
Spearman’s g cannot be directly measured, the best measure of it appropriate for
education is the IQ test (Jensen, 1998).
Since strong cases (Jensen, 1980, 1991, 1993, 1998; Matarazzo, 1972; Snow &
Yalow, 1982) have been made that achievement is related to intelligence, it is paramount
to know how students mature mentally. Mental growth of students is analogous to human
physical growth from early childhood to maturity. It follows a determinate curve in time.
Bloom (1964) thought systemic environmental influences could alter mental growth
curves. He went on to assert, “We do not subscribe to the thesis that intelligence is a
physical or neurological growth function analogous to height growth and that it must
have a definite terminal growth point” (Bloom, 1964, p.81). First, we now know
environmental influences on intelligence are not systemic, such as social and historical
forces. Rather, the influences are small and idiosyncratic, like childhood diseases (Jensen,
1998, Chapter 7). Moreover, intelligence is indeed, a growth function like height growth.
It has been demonstrated by studies of monozygotic twins (Jensen, 1998) that mental age
grows in a determinate, biologic pattern over time set by inheritance. Moffitt (1993)
found, while there were short run improvements in IQ for 90% of individual children,
those improvements were not systematically associated with environmental changes. Of
the remaining 10% with large improvements and plateaus, “this change is variable in its
timing, idiosyncratic in its source, and temporary/transient in its course” (Moffitt, 1993,
p.498). Overall, it was found IQ is “elastic, rather than plastic” (Moffitt, 1993, p.496).
This implies stability in mental development as a student matures, and suggests that deep
biological forces driven by heredity navigate the course for academic achievement.
Mathematics Achievement 5
The key is to focus on developmental influences of intelligence. Since
achievement is a function of intelligence, then the rise in student intelligence during
maturation must accelerate the rate of achievement. Since intelligence is not a precise
concept, it is replaced in the following discussion by the operational construct of mental
age, because it is the measurable construct used to compare students on a relative basis to
one another. The relationship between development (time) and student mental age is
based on research demonstrating that mental age increases at a decreasing rate over time
(Bayley, 1949, 1955; Heinis, 1924; Thorndike, 1927; Thurston, 1928). Also, Moffitt
(1993) found, mental age changes in late adolescence were negligible. Bloom (1964)
plotted five studies as shown in Figure 1 with time as the independent variable and
percent of mature mental age as the dependent variable. The authors plotted the logistic
curve over Bloom’s data for reference and discussed throughout the paper. They found
percent of mature mental age ranged between 0% and 100% (one) at maturity. It has been
found that infants can perform some mathematical operations without instruction shortly
after birth (Wynn, 1992; Wynn, 1995; Slaughter, Kamppi & Paynter, 2006). Hence, 0%
is set to conception, not birth.
Mathematics Achievement 6
100
90
Percent of Mature Mental Age
80
70
60
50
40
Bayley Pt.
Heinis
30
Thorndike
Thurstone
Bayley Grp.
20
Logistic
10
0
0
2
4
6
8
10
12
14
16
18
Age Since Conception
Figure 1. Plot of Five Studies
The exact curve of mental age growth has not been determined by research,
although Bloom characterized this curve as parabolic (Bloom, 1964). This is consistent
with his notion that there is no terminal limit to the growth of intelligence, since a
parabola has no horizontal asymptote. Data from many studies demonstrating the mental
age curve has a horizontal asymptote is summarized by Bayley (1955) and supported by
Moffitt (1993). Instead of a parabola, it is likely a logistic curve, but not the curve of a
rational function. Although both curves may be constructed, as in Bloom’s chart, with a
zero intercept and a horizontal asymptote at 1, rational functions are unacceptable. This is
because rational functions possess coefficients that can have no biological interpretation
Mathematics Achievement 7
(Yin, 2003). The continuous changes inherent in biological growth are compatible with
natural exponential growth found in logistic functions (Yin, 2003). Furthermore, logistic
growth curves are based on the constraint of “carrying capacity,” a concept frequently
found in biology (Weisstein, 2003). This constraint respects the finding by psychometry
of mental maturity as the “carrying capacity” of intelligence. There may be a single
logistic growth curve of mental age for all students. This is based on observations found
in (Bayley, 1949, 1955; Heinis, 1924; Thorndike, 1927; Thurstone, 1928). It has not been
determined by research to what degree the student logistic curves vary among one
another.
In summary, since mental age grows during students’ education, and mental age is
highly correlated with achievement in mathematics, then there should be an impact from
this monotonic increase. The model entitled, The Determinate Model of Mathematics
Achievement uses time as the independent variable, and student achievement as the
dependent variable.
Model Description
Because contemporary public, legislative, and academic communities are
concerned with student achievement in mathematics, this investigation was designed to
explain the learning of mathematics through a conceptual model. The determinate model
is proposed herein as an alternative explanation of mathematics achievement. The model
is based upon the most probable accounting of previously reported statistical findings
explained and supported by the most current literature (Scllitz, 1966). The model is
considered to be dynamic and thereby open to revision as new hypotheses are formulated
Mathematics Achievement 8
and empirically tested. The following paragraphs present assumptions and considerations
based upon epistemological and mathematical reasoning supported by the literature.
The first postulate of the determinate model is student achievement in
mathematics is produced by student intelligence. When students begin learning
mathematics, they are not mentally mature. If they were mature, they could learn
mathematics at a constant rate. In this model, the rate of learning starts at a slow pace and
increases until it peaks at maturity. As discussed in the introduction, the authors postulate
the rate of learning to be directly related to the maturation of student intelligence. The
process of maturation in student intelligence can be tracked by mental age. Mental age is
developmental and can be defined by using the following equation:
(1)
The term g in the equation is any student’s mental age now. M is a mature mental
age, and ƒ is the percent of mature mental age.
In concurrence with Bloom (Bloom, 1964) the ƒ term in the above equation grows
over time with a definite shape. Research on the development of mental age verified the
shape to be a logistic curve for all students. See Chart One for a scatter plot of data points
taken from Bloom (1964) with a logistic curve applied for reference. The equation in (2)
is a logistic function that relates how the percent increases with any student’s age.
Mathematics Achievement 9
(2)
The k term is a decimal exponent value, which is less than one and is called the
coefficient of growth, and t is the term for time expressed in years since conception. This
growth function adapted from biology (Goudriaan, 1990; Lee & Goudriaan, 2003) has a
predictable path that begins at zero and increases to a horizontal asymptote; therefore, a
basic assumption in this model is that mental development starts at 0 with conception and
increases to 100% at maturity in late adolescence.
The model claims the rate of learning is set by the mental age of students. The
greater the mental age of a student, the faster new math content and skill can be learned.
This functional relationship is in equation (3).
(3)
The term
is the rate of learning of mathematics, whereas A is achievement, t is
time, g is student mental age, and a is a parameter and the coefficient of proportionality
between g and
. In addition, a is expressed as units of math per year per mental age.
Since student mental age grows to a maximum at maturity, then the maximum rate of
growth in achievement can be represented by the following equation.
Mathematics Achievement 10
(4)
The term
is the maximum rate of growth in achievement at maturity
represented in units of mathematics per year and M is student mature mental age. As
stated earlier, current student mental age is a percent ƒ of mature mental age. Then
student achievement growth at any time is the product of the percent ƒ and the maximum
rate of growth in achievement. Therefore, the equation becomes:
(5)
This means achievement growth is set by the percent of mature mental age, ƒ, and
the maximum rate of growth in student achievement. The determinate model states that
this percent grows along a path that is logistic and common to all students. Therefore, one
equation builds upon another equation when the logistic expression from the second
equation is substituted in for ƒ:
(6)
Mathematics Achievement 11
From this equation, we can predict student achievement levels. This is done by
summing all of the achievement growth as students progress through math education.
Student achievement growth changes each year because student effort becomes more
productive due to maturation of their mental age. This continues until a maximum is
reached at maturity. The summation of achievement growth is done by the mathematical
operation of integration. To do this, equation (6) must be re-arranged in proper form. This
is shown in equation (7) where the term dt is moved to the other side of the equation.
(7)
This function allows the summation of all achievement growth between
conception and any time thereafter. Since Cm is constant, it is not summed. Now, the
mathematical process of integration can occur as indicated in the following equation.
(8)
Mathematics Achievement 12
The integration process creates a function. Summing up all of the changes in
achievement and all of the increases to student productive efforts over time by integration
yields the final equation.
(9)
This model of mathematics achievement elucidates the assumption to be tested:
Mathematics Achievement is a joint product of the maximum rate of mathematics
achievement, Cm and the anti-derivative of the logistic mental maturity function. This
function is more than just a summing of all the mathematics achievement since students
become more mathematically productive as their mental age grows. The function in
equation (9) is shown as a curve in Figure 2. It has two phases to it. The exponential
phase, where the rate of learning begins at zero and rapidly rises as mental age grows in
early childhood. In the second phase, the student’s mental development rate slows,
causing the student’s rate of learning to grow at a constant or linear rate. Because of these
two phases equation (9) is called an expolinear function.
Mathematics Achievement 13
180
160
Linear Phase
Math Achievement Score
140
120
100
80
Asymptote
60
40
Exponential
Phase
20
0
0
2
4
6
8
10
12
14
16
Time
Figure 2. Expolinear Growth Curve
Testing the Determinate Model of Mathematics Achievement
To test whether mathematics achievement is functionally related to a student’s
age the expolinear function must be stable over different student achievement levels. To
test the model longitudinal data was used with a uniform measure of mathematics
achievement. The dataset used is from Choi, K. S., M. (2005). It is only available as CDROMs through UCLA, CREEST. The data points used in this study begin at Zero
equaling conception. Whereas, the remaining sources for the data-points beginning at
0.75 and 1.75 years of age are from Wynn, K. (1992) and Slaughter, V. (2006). The
dataset for kindergarten through eighth grade is called Early Childhood Longitudinal
Study (ECLS). It is in three parts: Rathbun, A. W., West, J. (2004), and the second is
found in Princiotta, D., Flanagan, K.D. & Germino Hausken, T. (2006) and the third is
Mathematics Achievement 14
found in Walston, J., Rathbun, A., and Germino Hausken, E. (2008). The dataset for
seventh through tenth grade is from the Longitudinal Study of American Youth (LSAY).
It is found in Miller, J. D., Kimmel, L., Hoffer, T.B. & Nelson, C. (2000). All
achievement data from the two longitudinal studies was converted from grade levels to
mean age. The mean age at the beginning of the school year for kindergarten is 5.25, 6.25
for first grade, 7.25 for second grade and on in the same manner for each school year.
The ECLS and the LSAY are compatible datasets. Both used item response theory
methodology and NCES math standards arriving at identical sample distributions and
difficulty levels. Though both studies were done at different times, the study periods are
comparable. This is true because there are no practical differences in student achievement
levels between the eras of each study on the National Assessment of Educational
Progress (Perie, M., 2005).
Adjustment for Differences in Relative Math Scales
The adjustment for the differences in the relative scales for achievement on the
LSAY and ECLS was made by the use of overlapping scores from the two datasets. The
LSAY has a fall test score for eighth grade and ninth grades, while the ECLS has an eight
grade spring test score that lies halfway between those two. To compare the scales for the
ECLS and the LSAY, a mean was taken for the two LSAY scores and a ratio was found
between the mean for the LSAY and the spring ECLS scores. This ratio was then used to
adjust the LSAY to the ECLS scale.
Statistical analyses of dataset and model parameters
Regression coefficients for the two parameters, k and Cm, in equation (9) were
estimated by minimizing residual sum of squares (least squares method). The
Mathematics Achievement 15
independent variable is time and the dependent variable is math achievement with mental
age latent. This was done for the eleven decile and quartile levels of the dependent
variable by using the non-linear fitting procedure of the SPSS package. Eleven estimates
for the parameters k and Cm resulted from these regressions. Statistical significance tests
were then performed on the estimated parameters for hypothesis testing.
Results of the Statistics on the Model
Table 1
Summary of Non-linear Regression Analysis of Expolinear Function and Estimates of
Parameters
Performance
C
k
R
10
10.953
0.234
0.979
20
11.843
0.257
0.984
25
12.287
0.267
0.986
30
12.495
0.272
0.987
40
13.073
0.285
0.988
50
13.603
0.295
0.988
60
14.137
0.306
0.990
70
14.728
0.317
0.991
Percentile
Mathematics Achievement 16
Table 1
Summary of Non-linear Regression Analysis of Expolinear Function and Estimates of
Parameters
Performance
C
k
R
75
15.052
0.323
0.991
80
15.409
0.329
0.991
90
16.367
0.344
0.992
Percentile
The following tests were performed on the three hypotheses:
1. The achievement function (equation 9) was tested for eleven regressions. This
function was estimated for all deciles and quartiles ranging from 90% to 10% (See
figure 4). The F-statistics were significant, ranging successively from 1745 (90%) to
466 (10%) with p < 0.001. The R2 ranged successively from 0.992 (90%) to 0.979
(10%) with six regressions above 0.990. The results for the estimates of the two
parameters, Cm and k, are given in Table Two. The standard error of the estimate for
Cm ranged between 0.1526 and 0.09851 from the 10% level to the 90% level,
respectively. All estimates of Cm were significant at p < 0.001.
2. The growth parameter, k, for the logistic function of mental age varied between 0.234
and 0.344. The standard error of the estimate varied between 0.0084 and 0.0073 from
the 10% level to the 90% level, respectively. The estimates were found significant at
p < 0.001.
Mathematics Achievement 17
3. The logistic function’s growth parameter, k, was tested for homogeneity between
decile and quartile levels using ANCOVA. The mean value for k, 0.294, was
compared to the eleven estimated values for k. The covariates were the predicted
achievement values. The null hypothesis was not rejected with F = 0.170, p = .680.
240
220
90%
200
Math Achievement Scores
180
50%
160
140
120
10%
100
80
60
40
20
0
0
2
4
6
8
10
12
14
16
Time
Figure 3. Three Expolinear Curves, 90%, 50%, 10%
Discussion
The results are consistent with the mathematical model. Sample curves are shown
2
in figure 2. The achievement function has significant predictive capacity. The R statistics
average about 0.988 for the eleven regressions (refer to Table 1). These are far higher
than expected based upon past literature. The maximum rate of growth at maturity, Cm,
was found to be stable and changes consistently with the level of achievement. It was
found that the logistic function’s growth parameter, k, is stable and statistically
significant. This is also consistent with psychometric studies of the relative stability of
Mathematics Achievement 18
mental development. The results indicated that there may be minuscule variations in k.
Furthermore, based on differences among achievement levels, the results from ANCOVA
support Bloom’s theory that mental maturation is both law-like and uniform among
students. There is statistical support, then, for a single logistic function operating for all
students by achievement level.
The dissonant finding that emerged in the study is the very high explanatory
power of the achievement function. This is not consistent with past studies of the link
between achievement and mental age. The variance explained was expected to be 50%.
Instead, almost all of the variation in achievement is explained by the function. Since
mature mental age, M cannot explain achievement by itself, the only possibility is a
specification error somewhere with a hidden variable. Logically, the source of the
unexpected explanatory power in the function rests in its components - the logistic
function’s anti-derivative or Cm. There is nothing about the anti-derivative that would
allow for the presence of a powerful underlying variable. It is strictly a function of time
with two constraints. The likely source of the specification error, then, is in Cm. In
equation (4), parameter a was hypothesized to directly link mature mental age, M with
Cm. Parameter a may be confounding the existence of two things: an expected constant
of proportionality and the activity of an underlying variable. If so, that underlying
variable would have to be random and normally distributed like the other two variables. It
would also have to be highly correlated with the dependent variable in equation (4) Cm.
A likely candidate for that missing variable is the teacher input to achievement.
To capture the input from teachers is most problematic. Teacher effect involves many
variables and resists being represented by a single input number. To overcome this, some
Mathematics Achievement 19
researchers have operationally defined it as a productive output. This single number is
teacher effect. There are two suggestive studies on teacher effect. (Rivers, 1999) found in
her large study of two school districts in Tennessee that teacher effect explains 49.6% of
the variance in achievement. If true, then Rivers’ finding that 50% explained variation in
achievement by teacher effect is of the right magnitude. It complements the demonstrated
50% explanatory power of mental age. Additionally, Hanushek (1992) found that teacher
effect has a range from 0.5 grade level equivalents to 1.5 grade level equivalents and was
a random variable with a normal distribution and mean at 1.0. These facts make teacher
effect a good candidate for the missing variable in equation (4). It is also reasonable from
a functional standpoint. If the unit of analysis is the classroom, the inputs to achievement
come from the students and the teacher. It is not known, however, where the mean of
teacher effect is located relative to the mean of math achievement. Aside from the
existence of bias, both the standard deviation of teacher effect and its correlation with
mental age are unknown.
The implication of this study is that a functional relationship probably exists
between mental age, teacher effect and achievement in mathematics for students. Further
study should be conducted to test for these relationships.
Mathematics Achievement 20
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