Development of mathematical competences in a foreign language

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IX Jornadas Internacionales de Innovación Universitaria Retos y oportunidades del desarrollo de los nuevos títulos en educación superior

DEVELOPMENT OF MATHEMATICAL COMPETENCES

IN A FOREIGN LANGUAGE

Vela-Pérez, María1, Tirado, Gregorio2

1: Departamento de Ciencias Escuela Politécnica Universidad Europea de Madrid

C/ Tajo, Villaviciosa de Odón, 28670, Madrid (Spain) e-mail: maria.vela@uem.es

2: Departamento de Estadística e Investigación Operativa Facultad de Ciencias Matemáticas

Universidad Complutense de Madrid Plaza de Ciencias 3, 28040, Madrid (Spain)

e-mail: gregoriotd@mat.ucm.es

Abstract.

The main objective of the paper is to evaluate the development of mathematical competences when a foreign language is employed for teaching. This study is focused, in particular, on understanding how English language skills affect the mathematical learning process, and here it is especially interesting for evaluating how the deep knowledge of a foreign language is decisive when considering mobility between different countries. Some preliminar results show that the English language skills do not seem to be a decisive factor in the development of mathematical competences. In the paper a complete statistical study of these results is presented and analyzed.

Keywords: Mathematical competences, assessment, language skills, correlation coefficients.

1. INTRODUCTION

This paper deals with evaluating the development of mathematical competences, in particular in the area of Statistics, when a foreign language (English in this experience) is employed for teaching. Due to the fact that we are in the framework of the European Space of Higher Education, it is particularly interesting for evaluating how the deep knowledge of a foreign language is decisive when considering mobility between different countries.

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IX Jornadas Internacionales de Innovación Universitaria

Retos y oportunidades del desarrollo de los nuevos títulos en educación superior

The rest of the paper is organized as follows. First, in Section 2 a complete description of the experience is provided. Section 3 reviews the main mathematical background that is used afterwards in Section 4, where different statistical analyses are performed on the available data and the main results obtained are commented, and finally Section 5 summarizes the main conclusions of this work.

2. DESCRIPTION OF THE EXPERIENCE

Since 2011, all students of the degree of “Mechanical Engineering” must take one module per semester studied in English; in the second semester of 2011/2012, the module studied in English is the one considered in this study, “Statistics for Engineering”, that corresponds to the third module studied in English during the whole degree.

The module “Statistics for Engineering” consists of 5 units: Descriptive Statistics (Uni and Bidimensional), Probability, Random Variables (Discrete and Continuous), Important Probability Models and Statistical Inference (Random Sampling, Point Estimation, Confidence Intervals and Hypotheses Testing).

The evaluation system is divided in several parts. There are two tests based on problems and exercises solved in the lectures (30% of the final grade each): one mid-term test where the first 3 units are evaluated and another second-term test where the last 2 units are evaluated. Besides, there are two team projects (15% of the final grade each) developed by groups with 3-5 students which include an oral presentation of the main results of the projects; the first one consists on making a statistical study of a given data base and explain the main results in class; the second team project, where students have to compile their own data (in this case, the time that it takes to go to the university and to come back home), consists on performing different ANOVA tests with their own data to obtain several conclusions about the mean time for students to arrive and come back from the university. Finally, general skills assessment (10% of the final grade) are evaluated based on the results of a questionnaire filled by the classmates during the oral presentations, a self-assessment questionnaire about the autonomous work developed and a troubleshooting assessment evaluated by the teacher.

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3. MATHEMATICAL BACKGROUND

For analyzing the results obtained from the experience described in the previous section, we have calculated the value of three coefficients that evaluate in different ways the correlation between two samples of data: the Pearson’s coefficient, the Kendall’s coefficient and the Spearman’s coefficient. In this section we will review how these coefficients are calculated and how their values should be interpreted (see, for example, Vélez & García, 1997). In what follows, for the definition of these coefficients let us denote the two data samples subject of study as



x1, , xn



and



y1, , yn



, respectively.

The sample Pearson’s coefficient measures the linear dependence between two given samples by estimating the covariance of the associated variables divided by the product of their standard deviations. It is usually denoted by r, and it is always a value between +1 and −1 inclusive. Values close to +1 indicate a positive correlation between the variables, while values close to -1 indicate a negative correlation and values close to 0 indicate no correlation. It is widely used in science as a measure of the strength of linear dependence between two variables and it is calculated as follows:

1 1 1

2 2

1 1 1 1

n n n

i i i i

i i i

n n n n

i i i i

n x y x y

r

n x x n y y

             _ _ _ _    



 



The Kendall’s correlation coefficient measures the rank correlation, that is based on the similarity of the orderings of the data when ranked by each of the attributes measured by each sample. It is usually denoted by τ, and it is always a value between +1 and −1 inclusive. This coefficient is usually used on a non-parametric hypothesis test for

statistical dependence and is calculated as 2( ) ( 1)

a b n n

  

 , where







 







# ( , ) 1, , , | _i _j and _i _j or _i _j and _i _j

a i j  _ n i j x x y  y x x y  y







 







# ( , ) 1, , , | _i _j and _i _j or _i _j and _i _j

b i j  _ n i j x x y  y x x y  y . If  1, the ranking of all elements in both samples are identical, and thus values close to +1 indicate again a positive correlation. Analogously, if   1 the rankings of the elements in both samples are reversed, and thus values close to -1 indicate again a negative correlation, while values close to 0 indicate no correlation.

Under a null hypothesis of the two variables being independent, the sampling distribution of τ (for sample sizes of n10) can be approximated by a normal distribution with mean zero and variance 2(2 5)

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n n n



 . Hence, we calculate what we will

call the Normalized Kendall’s coefficient (denoted by _norm) by dividing by the standard

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Spearman’s correlation coefficient measures the rank correlation by assessing how well the relationship between the two samples or variables considered can be described using a monotonic function. It is usually denoted by ρ, it is again a value between +1 and −1 and it is interpreted in the same way the other two introduced coefficients are: positive values indicate positive correlations and negative values negative correlations. Like Kendall’s coefficient, Spearman’s coefficient is used on non-parametric hypothesis tests for statistical dependence and is calculated as

2 1

2

6 ( )

1

( 1) n

i i

i

r s

n n

 

  





where ri and si are the ranks of the i-th element of each sample when they are sorted.

As it was also done with Kendall’s coefficient, under a null hypothesis of the two variables being independent, the sampling distribution of ρ (for sample sizes of n10) can be approximated by a normal distribution with mean zero and variance 1

1

n .

Hence, we calculate what we will call the Normalized Spearman’s coefficient (denoted by _norm) by dividing by the standard deviation, as _norm  n1.

4. ANALYSIS OF THE RESULTS

The main objective of this section is analyzing, for each group of students, if there is a correlation between the grades obtained in each of the tests of the module of Statistics and the previous knowledge of English language (measured by the UEM Lab Level). For this purpose, the values of the coefficients (r, τ and ρ) introduced in the previous section are calculated using the grades of Test 1 as one of the samples and the UEM levels as the other sample (see Table 1). These coefficients are first calculated for each group considered independently (see columns 2 to 4) and then considering all students together as one single class (see last column). For each column, the Pearson’s coefficient is given in row 2, Kendall’s coefficient in row 3 and Spearman’s coefficient in row 5. Besides, the normalized values of the last two coefficients are given in rows 4 and 6, respectively.

Test 1 Group 1 Group 2 Group 3 All students

r 0.263 0.000 0.470 0.123

τ 0.218 -0.067 0.283 0.091

τnorm 1.222 -0.413 1.531 0.958

ρ 0.409 0.002 0.485 0.211

ρnorm 1.637 0.007 1.880 1.524

Tabla 1. Results about Test 1

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coefficients obtained using the grades of Test 2 as one of the samples and the UEM levels as the other sample. Finally, the results obtained with the averages of both tests and considering all students together are provided in Table 3.

Test 2 Group 1 Group 2 Group 3 All students

r 0.254 -0.109 0.446 0.190

τ 0.253 -0.116 0.361 0.119

τnorm 1.418 -0.715 0.243 1.259

ρ 0.365 -0.101 0.494 0.218

ρnorm 1.461 -0.439 1.914 1.574

Tabla 2.Results about Test 2

r τ τnorm ρ ρnorm

0.263 0.218 1.222 0.409 1.637 Tabla 3. Results using final grades and considering all students together

Concerning the Pearson’s coefficient, it can be observed that all values are between -0.5 and 0.5, indicating that the correlation between the English level and the mathematical knowledge acquired is not strong. In fact, most values are very close to 0, showing that the correlation is indeed small, and all of them except the one obtained considering Test 2 of Group 2 are positive, indicating that the correlation, if any, would be positive, as expected. The values of r for Group 2 are 0 for Test 1 and -0.109 for Test 2, showing a behavior that is a bit different from the other groups but indicating clearly not correlation.

To interpret Kendall’s and Spearman’s coefficients we will use directly the normalized values given in Tables 1-3. For a significance level of 95%, the result of the hypothesis test would be to accept the null hypothesis (independent variables) if the absolute value of the normalized coefficient is less than or equal to zα=1.96. In

the tables it can be observed that all absolute values of τnorm and ρnorm are below this

value, from what follows that the analyzed data do not provide enough evidence to state that the English level and the mathematical knowledge acquired are correlated. If a smaller significance level is used, for instance 90% (already quite small), the corresponding critical value is zα=1.65, and again the result of the test is accepting

the hypothesis of independence in all but a couple of cases (Spearman’s test for Group 3 with both tests).

These results let us conclude that, in general, English level and mathematical knowledge acquired are not correlated, but the three groups do not behave completely identically: Group 2 shows the largest evidence of non-correlation, even providing positive and negative coefficients; however, all coefficients for Group 1, though not leading to reject the null hypothesis, are all positive, indicating that the correlation, if any, would be positive; lastly, the coefficients for Group 3 are always the largest, leading in a few cases to reject the null hypothesis and admitting that some positive correlation could exist.

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the performances of the students in Test 1 and Test 2 to try to find out if the groups also behaved differently in this context.

Figura 1. Linear regression of data concerning group 1

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Figura 4. Linear regression of data considering all students together

Figures 1-4 show the linear regression graphs considering independently groups 1, 2 and 3 and all students together, respectively. In each figure, the graph on the left is drawn using the grades of Test 1 in the horizontal axis and the grades of Test 2 in the vertical axis, while for the graph on the right the averages of both tests are used in the horizontal axis and the English level on the vertical axis. These graphs on the left illustrate the correlation between the grades obtained in both tests on each group, showing that in group 2 they are much less correlated (note that R2, the correlation coefficient, is notably smaller) than in the other groups and thus suggesting that this group is somehow different: if the grades of the students vary much from one test to the other, it is not surprising that they are not correlated at all with the English level. The graphs on the right illustrate the correlation between average grades on the module and the English level, confirming the results provided by the coefficients calculated previously and already commented: group 2 behaves differently, but in general it seems there is not a high dependence between mathematical skills learning and English background.

5. CONCLUSIONS

The statistical study presented in this paper provides a first approach to the evaluation of statistical competences when a foreign language, English in this case, is employed.

It is shown that there is not enough statistical evidence to state that there is any correlation between the student grades and their level of English. Depending on the groups, this lack of correlation is stronger.

As a final conclusion, we can state that within the context of the European Space of Higher Education, students can consider mobility between different countries despite of their own level of the correspondent foreign language. At least, this has been observed in degrees directly related with mathematics, but it seems to be transferable to scientific degrees in general. If the students work hard, they should not have any problem to acquire the required background and pass the subjects.

REFERENCIAS

Vélez, R. & García, A. (1997). Cálculo de probabilidades y estadística matemática.