Regression in high school: An empirical analysis of
Spanish textbooks
Marı́a Gea, Carmen Batanero, Pedro Arteaga, José Miguel Contreras,
Gustavo Cañadas
To cite this version:
Marı́a Gea, Carmen Batanero, Pedro Arteaga, José Miguel Contreras, Gustavo Cañadas. Regression in high school: An empirical analysis of Spanish textbooks. Konrad Krainer; Naďa
Vondrová. CERME 9 - Ninth Congress of the European Society for Research in Mathematics
Education, Feb 2015, Prague, Czech Republic. pp.658-664, Proceedings of the Ninth Congress
of the European Society for Research in Mathematics Education. <hal-01287065>
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Regression in high school: An empirical
analysis of Spanish textbooks
María M. Gea, Carmen Batanero, Pedro Arteaga, J. Miguel Contreras and Gustavo R. Cañadas
University of Granada, Faculty of Education, Granada, Spain, [email protected]
The aim of this study was to analyze the presentation of
regression in Spanish school textbooks aimed at Social
Sciences students. In a sample of eight textbooks we
analyzed the problem-fields and suggested procedures,
as well as the associated concepts, which are classified
according to the way they are introduced and the properties associated with them. Our results suggest that these
textbooks mostly reduce regression to lineal regression,
present many properties at an operational level without
deep discussion of their meaning, and there is a great
variety between the textbooks. Some suggestions to improve the presentation of this topic are included.
Keywords: Regression, textbooks, high school,
mathematical objects.
INTRODUCTION
Correlation and regression are fundamental statistical ideas, due to their usefulness to model phenomena
in different fields (Engel & Sedlmeier, 2011). Previous
research is mainly focused on students understanding of correlation and describes several misconceptions (Estepa & Batanero 1995; Estepa, 2008; Zieffler
& Garfield, 2009). There has not been, however, much
interest in the way the topic is taught or presented in
the textbooks, even although textbooks are important educational tools. From the official curricular
guidelines to the teaching implemented in the classroom, an important step is the written curriculum
reflected in textbooks (Herbel, 2007). The selected
textbook is an important part of teaching and learning mathematics at secondary and high school level,
since the presentation of the topics and the problems
proposed provide the main basis of why the topic is
taught (Shield & Dole, 2013). Moreover, in other mathematical topics, textbooks receive increasing attention
from the international community; see for example
Fan, and Zhu (2007).
CERME9 (2015) – TWG05
The aim of this research was to analyze the presentation of regression in high school Spanish textbooks
aimed at Social Sciences students. It is part of a wider
project where we compare the way in which correlation and regression are presented in the textbooks in
Spain. Due to length limitation we only present here
a part of our analysis; complementary results were
published in (Gea, Batanero, Cañadas, & Contreras,
2013), and (Gea et al., 2014).
THEORETICAL BACKGROUND
According to Rittle-Johnson, Siegler and Alibali (2001)
conceptual and procedural knowledge are the two
extremes in a continuous; conceptual knowledge is
more flexible and includes the implicit or explicit understanding of a domain and its interrelationships.
Sfard (1991) suggests that a concept is a construct
which corresponds to the mathematical universe
and distinguishes two types of definitions of the
same: structural (where concepts are introduced by
describing their essential conditions or properties)
and operational (when it is described by a formula)
definitions. In our analysis we will take into account
both types of definitions and will also describe in detail the properties associated with regression.
The division between conceptual and procedural
knowledge is analyzed in the onto-semiotic approach
to mathematics education (Godino & Batanero, 1998;
Godino, 2002) where mathematical activity plays a
central role. Knowledge in this framework is modelled
in terms of systems of practices, in which different
types of objects intervene when a subject faces the
solution of a given problem. “Object” is understood
in a broad sense for any entity which is involved in
mathematical practice and can be identified separately. More specifically in our analysis we will consider
the following types of mathematical objects: problem-situations; procedures; concepts and properties.
658
Regression in high school (María M. Gea, Carmen Batanero, Pedro Arteaga, J. Miguel Contreras and Gustavo R. Cañadas)
Previous research
In spite of the relevance of the topic, previous research
suggests poor results in people’s understanding of
correlation. For example, Chapman and Chapman
(1967) described illusory correlation where people
are often guided by their theories rather than by data
when estimating correlation. Estepa (1994) defined
several misconceptions including the deterministic
conception, where the student only considers whether
functional dependence exists. Sánchez Cobo (1999)
described students’ failures in ordering correlation
coefficients and in interpreting the relationship between the correlation coefficient and the slope of the
regression line. As regards research on textbooks,
Sánchez Cobo (1999) classified the definitions of concepts presented in 11 textbooks published in the period 1977–1990 as procedural, structural or a mixture of
both. Lavalle, Micheli and Rubio (2006) analyzed the
concepts and procedures included in 7 high school
textbooks from Argentina.
As stated in the introduction, this paper is part of a
more comprehensive project. In (Gea et al., 2013) we
analyze the problem-situations used to contextualize
correlation and regression in eight Spanish textbooks
and in (Gea et al., 2014) the symbolic, verbal and graphical language used in this topic. The current paper
complements these publications with the analysis of
concepts, properties and procedures included in the
study of regression in the same textbooks, which were
not included in these papers. In each of these types of
objects we analyzed all the categories that are relevant
for the teaching of the topic.
METHOD
The sample consisted of eight mathematics high
school textbooks aimed at Social Sciences students
and that were published just after the current curricular guidelines were introduced (MEC, 2007). These
particular textbooks have a wide diffusion in Spain
due to the publishers’ prestige and are still used in
the schools (See Appendix). We performed a content
analysis (Neuendorf, 2002) of the chapters devoted to
correlation and regression with the following steps: a)
following an inductive and cyclic procedure we first
categorized all the different mathematical objects
explicitly or implicitly included in the chapter (the
different types of problems, concepts, properties and
procedures); b) For each of them we analyzed the way
in which they are described or used in the textbooks.
In this paper we only describe the mathematical objects linked to regression; as the remaining results
have been published elsewhere (Gea et al., 2013; 2014).
PROBLEMS AND PROCEDURES
Anthony and Walshaw (2009) reported on the different types of tasks that have been analysed in mathematics education research, that include problems
focused on specific mathematical content; problems
that promote mathematical modelling; others that require discussion of aspects that vary; those that ask
students to interpret and criticise data and those that
prompt sense making and justification of thinking.
Gea and colleagues (2013) identified two main types
of problems in the study of regression in which one of
several of the above type of tasks may be combined: a)
Fitting a model to the data requires sense making and
criticism of data; mathematical modelling, discussion
of variation and justification of thinking, b) Using the
model to predict a value of the dependent variable is
a more computational type of task; however, much of
the time judgement of the goodness of fit involves criticism, decision making and justification. Both types of
problems appear with different frequency in all the
books; each of these types of problems range from
20% of all the problems posed in the chapter in [H4]
to 35% in [H6] 1. To solve these two types of problems,
the books introduce the following procedures:
P1. Fitting the least squares line. Lineal regression by
the least squares method is included in all the books,
with two different procedures to compute the line
of best fit: a) Equation of the line which includes the
gravity centre (point whose coordinates are the mean
for the two variables); and b) General expression for
the lineal function:
Regression line Y = f(X)
a) y − y = byx ⋅ (x − x)
b) y = a + byx ⋅ x
Regression line X = f(Y)
a) x − x = bxy ⋅ (y − y)
b) x = a’ + bxy ⋅ y
1
All the books also introduced the analysis of the sign and
strength of correlation between the variables as a previous
problem as well as problems related to graphical representation
of bivariate data. We are not including here these problems that
were analyzed in (Gea et al., 2013).
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Regression in high school (María M. Gea, Carmen Batanero, Pedro Arteaga, J. Miguel Contreras and Gustavo R. Cañadas)
P2. Fitting other regression models. Only [H8] includes
the regression towards the median developed by
Tukey, which is a robust method in the presence of
outliers as well as variable transformation to fit exponential and polynomial models.
P3. Computing the determination coefficient and interpreting the goodness of fit. Only a few textbooks introduce the determination coefficient r2 as a measure
of the goodness of fit; in all of them the coefficient is
previously defined.
P4. Prediction with the line of best fit and interpreting
the goodness of prediction. All the books propose some
tasks where the students have to estimate the response
variable Y given a value for the explanatory variable X.
Some of them ([H1], [H2]) suggest that when the correlation coefficient is strong it is possible to use the same
line to predict X from a value of Y. Since this property
is not general (as there are two different regression
lines) students could make an incorrect generalization. Some books qualitatively evaluate the goodness
of fit by comparing the estimated and observed values
for isolated data. This informal method is not reliable,
as the reliability of estimation depends on the goodness of fit (given by the determination coefficient) as
well as on the closeness of the estimated value to the
centre of the distribution of the explanatory variable.
In Table 1 we observe that all the books include the
least square line, as well as its use for prediction, while
only one includes the Tukey line and a few the evaluation of the goodness of fit. A missing point is the use
of informal “eye fitting” methods that are useful to
build the students’ intuitions and can be implemented
with applets (e.g., //docentes.educacion.navarra.es/
msadaall/geogebra/figuras/e3regresion.htm). Results
are better than those by Lavalle, Micheli and Rubio
(2006); where only half the books include the line of
best fit and 60% of the books include prediction activities.
CONCEPTS AND PROPERTIES
In our analysis we found the following concepts
linked to regression:
C1. Dependent (response) and independent (explanatory) variable: While correlation is symmetric, regression is asymmetric; for this reason we should discriminate the response and explanatory variables (Estepa,
1994). Only a few books make the distinction explicit
although in other books it is implicit when they introduce two different regression lines.
C2. Model of fit. The idea that the regression line is only
a model (and therefore does not exactly coincide with
all the data) is only implicitly introduced; a couple of
books make this idea explicit: “When there is strong
correlation between X and Y the analysis of regression
helps to find a mathematical function as a model to fit
the data. This function can be a straight line, a parabola,
exponential...“ ([H4], p.226).
C3. Line of best fit (linear model). All the textbooks include the definition and explanation of the minimum
squares method (in an informal way); however, only
a few of them justify the utility of the model to estimate values of Y in situations where the variable is
difficult to measure. Moreover, as in Sánchez Cobo
(1999), few texts highlight the predictive utility of the
regression line.
C4. Regression coefficients. Since there are two possible
lines of best fit (depending on which is the explanatory and response variable) there are two different
regression coefficients, but only a few books make
this explicit. They also include the interpretation of
these coefficients: “The line that minimizes the sum
of residuals Σd2i is given by the following expression:
y = y + σxy (x - x)/σx2. The slope σxy/σx2, is the regression
coefficient” ([H1], p. 230).
Procedures
H1
H2
H3
H4
H5
H6
H7
H8
P1. Fitting the least squares line
x
x
x
x
x
x
x
X
P2. Fitting other regression models
X
P3.Computing the determination coefficient and interpreting
the goodness of fit
P4. Prediction and interpretation (line of best fit)
x
x
x
x
x
x
x
x
x
X
Table 1: Regression analysis procedures included in the books
660
Regression in high school (María M. Gea, Carmen Batanero, Pedro Arteaga, J. Miguel Contreras and Gustavo R. Cañadas)
C5. Goodness of fit. Coefficient of determination. These
concepts help the student understand the meaning of
regression; however they are only included in [H5]
and [H6], where the accuracy of the model is identified
with the accuracy in the prediction for any particular
point. This is not true, as the accuracy is higher when
the point approaches the centre of the distribution of
the explanatory variable.
C6. Non lineal models of fit. Regression is a general
method for understanding relationships between variables (Moore, 2005), and therefore it is necessary to
introduce different models of fit; however, only a few
textbooks implicitly define non lineal models, where
[H5] and [H6] define them explicitly.
from the points to the line minimum. Usually the explanation is only visual (a formal deductive proof is
avoided).
P2. Two different regression lines. Most of the books implicitly remark that there are two different regression
lines and part of them warns the students of the danger of using an inadequate line to make a prediction.
Two books ([H2] y [H8]) do not remark on this property. This omission may reinforce the deterministic
conception of some students (Estepa, 1994), since in
deterministic dependence there is only one algebraic
expression (function) to express the dependence.
P3. Percentage of variance explained (r2). The determination coefficient measures the goodness of fit. Some
We summarize the definition of concepts in Table 2, textbooks also analyze its interpretation as the perwhere we observe the predominance of adding ex- centage of variance explained by the regression line:
amples to the definitions; usually the texts include “(r2x100)% is the percentage of variance of Y explained
scatter plots with the line of best fit added to show the
by the value of X”([H6], p.185).
residuals. The line of best fit is introduced in all the
books and is generally defined both in a structural P4. Estimation using the regression line. The regression
and operational way (Sfard, 1991). The presentation line serves to predict the value of response (Y) given
is very similar to that in Sánchez Cobo (1999). Other a value of the explanatory variable (X). The books
definitions (regression coefficients, dependent and implicitly indicate that these estimates are only apindependent variable, goodness of fit and non linear proximations. They insist that, contrary to functional
models) are missing in some textbooks or are only dependence, there are several values of Y for a given
defined in an operational way. The definitions are value of X, and the regression line provides the averintroduced in different orders; sometimes the exam- age of all these values: “These estimates are approximaples are followed by the definition and vice-versa. In tions and involve a probability; it is probable that when
the same way the order to introduce operational or x = x0 the value of y is approximately y(x0). ([H1], p.230).
structural definitions also varies.
P5. The regression line crosses the distribution centre of
Properties
gravity, a property only included in half the books in
The textbooks add different properties to the defini- the study by Sánchez Cobo (1999).
tion of the concepts or to relate different concepts, as
described below:
P6. Estimates are more accurate for values closer to the
centre of gravity. However some books only judge the
P1. Least squares property. Most textbooks explain reliability of estimates by the value of the correlation
that the regression lines make the sum of residuals coefficient.
Concepts
H1
H2
H3
H4
C1. Dependent and independent variable
H5
H6
H7
O
C2. Model of fit
C3. Line of best fit (linear models)
ESO
EOS
C4. Regression coefficients
O
SO
ES
SE
SO
SOE
SOE
H8
O
SOE
SOE
SO
O
C5. Goodness of fit.
SOE
OE
C6.Non linear models
SOE
SOE
E = Examples; O = Operational definition; S =Structural definition
Table 2: Concepts linked to regression and type of definition
661
Regression in high school (María M. Gea, Carmen Batanero, Pedro Arteaga, J. Miguel Contreras and Gustavo R. Cañadas)
P7. Reliability of estimates and sample size. It is included only in a few books: “The estimate accuracy
increases with the number of data; the regression line
computed with few data has little reliability even if r is
high” ([H5], p. 260).
P8. Strength of correlation and angle of regression lines:
This angle varies from perpendicular lines (independence) to only one line (perfect linear dependence).
P9. Regression line, covariance, correlation. Covariance
and correlation are interpreted as regards the closeness of the points to the regression line. Their sign is
related to the slope in the regression line: “Depending
on the position of (xi, yi) as regards (x, y), the product
(xi − x) ⋅ (yi − y) will be positive or negative. If many points
are close to a line with a positive slope, most of these products are positive and the covariance and correlation
are positive” ([H1], p. 228). These properties were not
found in the books analyzed by Sánchez Cobo (1999).
P10. Product of regression coefficients. Some books
suggest that the product of regression coefficients is
equal to the square of the correlation coefficient: r2.
This property was found in most books in Sánchez
Cobo’s research (1999) but is found in only two books
in our study.
PP11. Correlation and reliability of estimates. Most textbooks relate both concepts: “The higher the correlation
coefficient r, the higher the reliability of estimates: when
r is close to zero, there is not much sense in doing an
estimation; as r approaches to 1 or -1, the real values will
approach our estimates; when r = 1 or r = -1, real values
and estimates coincide” ([H4], p. 226).
In Table 3 we summarize the properties of regression
included in the books. There are great differences between textbooks, because while some of them ([H4],
[H8]) hardly describe any of the properties analyzed,
others ([H3]) include almost all of them. The most frequent property is the least squares, the existence of
two different regression lines, estimation with the
line, centre of gravity crossing the line, and relationship of reliability in the estimate, centre of gravity and
correlation coefficient. Globally the books introduce
a rich set of properties of regression. We remark that
some textbooks do not include the properties P8 and
P9; this omission may reinforce the students’ failures
in interpreting the relationship between the correlation coefficient and the slope of the regression line
described by Sánchez Cobo (1999).
DISCUSSION AND IMPLICATIONS
FOR TEACHING
Our results suggest little changes in the presentation
of regression in the high school textbooks, as regards
the analysis by Sánchez Cobo (1999), although the
books studied by this author were published between
1977 and 1990. Our analysis complement that research
and that by Lavalle, Micheli, & Rubio (2006), because
neither of these previous studies analyzed the properties linked to regression, in spite of the fact that Sfard
(1991) considered that the properties are an essential
part of the concepts. This study also complements our
previous studies: (Gea et al., 2013), where we analyzed
with more detail the problem-situations used to contextualize correlation and regression, and (Gea et al.,
2014) where we described the symbolic, verbal and
graphical language used in this theme.
H1
H2
H3
P1. Least squares property
x
x
x
P2.Two different regression lines
x
x
H4
x
P3. Percentage of variance explained (r2)
H6
H7
H8
x
x
x
x
x
x
x
x
x
x
x
x
P4. Estimation using the regression line
x
x
x
P5. Regression line and centre of gravity
x
x
x
x
x
x
x
P6. Reliability of estimates and centre of gravity
x
x
x
x
x
x
x
x
x
x
x
P7. Reliability of estimates and sample size
P8. Strength of correlation- angle of regression lines
x
x
P9. Regression line, correlation, covariance
x
x
P10. Product of regression coefficients
x
x
P11. Correlation coefficient – reliability of prediction
x
x
x
x
H5
x
x
x
x
x
x
x
Table 3: Properties of regression
662
Regression in high school (María M. Gea, Carmen Batanero, Pedro Arteaga, J. Miguel Contreras and Gustavo R. Cañadas)
Although all the textbooks introduce linear regression and propose methods to compute the line of
best fit and make predictions with the same, only a
few of them introduce procedures to compute the
determination coefficient and its interpretation as
a percentage of explained variance; however these
properties are introduced theoretically with no practical applications or procedures related to the same.
Similar to the study of Lavalle, Micheli, & Rubio (2006),
only a few textbooks introduce examples of non linear regression, even when a few propose tasks where
different models would be preferable. This coincides
with Sánchez Cobo’s research (1999) where some textbooks introduced examples of non linear regression
without discussing these models.
We found few definitions of the concepts linked to
regression, apart from the definition of the line of best
fit. This fact could be explained because these textbooks use most of the available space for this theme
in the study of correlation (computing correlation
coefficients and interpreting its sign and strength), as
we are shown in (Gea et al., 2013). Although correlation is no doubt an important concept, it doesn´t make
much sense that the books devote so much space to
its study if this study is not completed with the study
of regression; the need to fit a model to the data is the
reason to study correlation between variables; the
isolated study of correlation is useless. We therefore
recommend reinforcing the study of other concepts
linked to regression, such as model, centre of gravity,
regression coefficients, and parameters of the line of
best fit (slope; coordinate at the origin).
to study the reliability of prediction, with no connection, for example of the sign of correlation and the
slope of the regression line. In the same way there is
no discussion of examples where a low correlation
coefficient may be associated to a strong non linear dependence; for example a parabola. It is also important
to emphasize explicitly the existence of two different
regression lines; that may sometimes be very close
when r is close to +1 or -1, but may be very different
in the general case. Many times the difference is only
implicit.
When comparing the different books, [H5] and [H6]
are far more complete than the other books, as they
present all the procedures for the linear model; they
are the only books that define the goodness and non
linear models (although they do not include the procedures for these models) and introduce the majority
of properties analyzed.
ACKNOWLEDGEMENT
Project EDU2013-41141-P (MEC) and group FQM126
(Junta de Andalucía).
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