# Statistical Analysis of the Evolutive Effects of Language Development in the Resolution of Mathematical Problems in Primary School Education

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## Abstract

**:**

## 1. Introduction

- Theoretical teacher explanation.
- Discussion among the teacher and the students.
- Appropriate practical work.
- Consolidation and practice of basic skills and routines.
- Troubleshooting, including the application of mathematics to real-life situations.

- P1.
- In a vase there are three red flowers and five yellow flowers. How many flowers are there in the vase?
- P2.
- Mary has taken three flowers from a vase, and Peter five. How many flowers have they taken?
- P3.
- Peter took three wilted flowers from a bouquet of Mary. Now the bouquet has five flowers; how many flowers did the bouquet have at the beginning?

#### 1.1. Classification of Additive Problems

- Change. These problems are characterized by an action that produces a change (increase or decrease) in an initial amount. If the problem is based on the sum operation, $a+b=c$, it leads to three different types of problems depending on which variable is the unknown: a, b or c. In a similar way, when a subtraction operation is involved, $a-b=c$, there are three other different types of problems also depending on which variable is the unknown.
- Combination. In these problems, there are two amounts that can be considered isolated or being a part of a whole without any action between them. We distinguish two types of problems: when the question is about the set (the union or total) and when the question is about a subset.
- Comparison. These are problems where there is a comparative action between two different quantities. The problem could be either to compute the difference between them or to find an unknown quantity related to the other. When using the words “more” or “less” in the statement of the problem, two different problems (one with “more” and the other with “less”), asking either for the difference, the comparison or the reference quantity appear. This means six different problems for this case.
- Equality. This category involves elements from change and comparison problems. The statement of the problems includes an implicit action based on the comparison of two different quantities. As in previous cases, six different problems are obtained.

- To study the influence of mental representation in the resolution of additive problems for children from 6 to 12 years old.
- To find a model relating language skills and academic course with problem resolution (one-operation problems with either addition or subtraction).
- To quantify the influence of mental representation of problems separately from the cognitive level of the students according to their grade.
- To provide a cutoff in the mental representation test score as a tool for teachers to predict the ability of the students in solving additive problems correctly.

## 2. Materials and Methods

- Identifying non-redundant data.
- Identifying the question.
- Operating and expressing the solution correctly.

#### 2.1. Mathematical Background

#### 2.2. Influence of Mental-Representation Level in Problem-Solving

**M1:**- Simple linear regression model. Using all the data but not taking into account the existence of groups:$$Y=a+bX+\epsilon .$$
**M2:**- Multiple linear regression model including a categorical variable. A multiple linear regression model adding fictitious variables (${G}_{i}$) accounting for the effects of the different groups. This is the right model when the relationship between X and Y is identical for all the groups, but for a scalar difference from the mean response:$$Y=a+bX+{c}_{1}{G}_{1}+{c}_{2}{G}_{2}+\dots +{c}_{k-1}{G}_{k-1}+\epsilon ,$$
**M3:**- Simple linear regression models in each group. This is the appropriate model when the relationship between X and Y is different in each group beyond the intercept term, that is, when the coefficients ${b}_{i}$ are different:$${Y}_{{G}_{i}}={a}_{i}+{b}_{i}{X}_{{G}_{i}}+{\epsilon}_{{G}_{i}},i=1,\dots ,k.$$

**Step 1 (Compare M1 to M3):**The first test compares the variance explained by models**M1**(without groups) and**M3**(the most complex model that takes the groups into account) in order to determine whether the groups are significant or not in the relationship between the two quantitative variables. The null and alternative hypotheses of the test are, respectively:- ${H}_{0}$:
- The groups are non significant; that is, the gain of variability explained when considering different linear regressions in the groups is small.
- ${H}_{1}$:
- It is necessary to take the groups into account.

The F-statistic to conduct this test, ${F}^{1\to 3}$, is described in (4).**Step 2 (Select model)**If the test in**Step 1**is not significative, model**M1**should be selected, and finishing the groups is not needed. Otherwise we conclude that the existence of the groups is important for the variability, and since there are two models that take into account the groups (**M2**and**M3**), they should be compared in order to decide which one is the most convenient (**Step 3**).**Step 3 (Compare M2 versus M3):**In the case of rejecting the null hypothesis in Step 1, it should be checked whether the**M2**model is enough for fitting the data or if the individual linear models in**M3**are necessary. The test hypotheses are in this case as follows:- ${H}_{0}$:
- The relation between the quantitative variables Y and X is the same in every group; that is, the coefficient of X in the model, b, does not depend on the groups.
**M2**is the right model. - ${H}_{1}$:
- There is interaction between the groups and the regressor X; that is, the coefficient of X in the regression line varies with the groups. Thus, model
**M3**is the best one.

That is, the coefficient of X in the regression line varies with the groups. Thus, model**M3**is the best one.The F-statistic to conduct this test, ${F}^{2\to 3}$, is described in (4).**Step 4 (Select model):**If the test in Step 3 is significant, then model**M3**should be chosen; otherwise, model**M2**should be chosen.

- Both the dependent and the explanatory variables should be continuous.
- The grouping factor is composed of two or more categories of independent groups.
- The observations must be independent, for example, selecting different people.
- There should not be significant atypical values; this could have a negative effect on the validity of the results.
- For each category of the independent variable, the residuals should follow a normal distribution. This hypothesis may be violated in a certain way while the tests still provide valid results. In order to check normality, Shapiro–Wilk or Kolmogorov–Smirnov tests and P–P or Q–Q charts can be made.
- Homoscedasticity (similar variances of the dependent variable for the different groups) is assumed. This requirement can be checked, for instance, by the Levene test. Even when the hypothesis is violated, the above tests are still reliable provided that the groups sizes do not differ very much (none of the groups is twice the size of any other one).
- The relationship between X and Y should be linear. This assumption can be tested by a simple linear regression analysis between the covariate X and the response Y.

#### 2.3. Predicting Problem-Solving Ability

## 3. Results

#### 3.1. Models Relating PS with MR

- The dependent variable and covariate are continuous.
- The independent variable, MR, has two independent categories. The observations are independent since the groups are disjoint.
- There are no atypical values in the ${G}_{2}$ and ${G}_{3}$ groups.
- The residuals of linear models are approximately normal for groups ${G}_{2}$ and ${G}_{3}$.
- The Levene test for checking homocedasticity returns a p-value of 0.419, and thus the homogeneity of variances cannot be rejected.
- The assumption about the linear relationship between PS and MR is corroborated by a simple linear regression analysis.

**M1:**- Simple linear regression model.$$\mathrm{PS}=-0.982+0.798\mathrm{MR},$$
**M2:**- Multiple linear regression model including a categorical variable.$$\mathrm{PS}=-0.55+0.789\mathrm{MR}-0.995{G}_{2},$$
**M3:**- Simple linear regression model by group.A linear regression model in each one of the groups is computed.
- $\mathbf{M}{\mathbf{3}}_{{\mathbf{G}}_{\mathbf{2}}}$:
- Model 3 in ${G}_{2}$ group:$$\mathrm{PS}=-1.811+0.840\mathrm{MR},$$

with 38 degrees of freedom, $RSS=134.282$ and ${R}_{3,2}^{2}=0.581$.- $\mathbf{M}{\mathbf{3}}_{{\mathbf{G}}_{\mathbf{3}}}$:
- Model 3 in ${G}_{3}$ group:$$\mathrm{PS}=-0.402+0.762\mathrm{MR},$$

with 62 degrees of freedom, $RSS=166.673$ and ${R}_{3,3}^{2}=0.633$, totaling $g{l}_{3}=62+38=100$ degrees of freedom and $RS{S}_{3}=134.282+166.673=300.955$.

**Step**

**1**

**(Compare**

**M1**

**versus**

**M3):**Computing the test in (4)

**Step 3**).

**Step**

**3**

**(Compare**

**M2**

**versus**

**M3):**The statistic test (4)

#### 3.2. Classification Models for the Groups

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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PS | ${\mathit{G}}_{1}$ | ${\mathit{G}}_{2}$ | ${\mathit{G}}_{3}$ |
---|---|---|---|

Mean | 1.165 | 2.537 | 3.757 |

Standard Deviation | 1.962 | 2.865 | 2.686 |

Number of students | 74 | 40 | 64 |

Real Value | ||
---|---|---|

Prediction | Positive | Negative |

Positive | A | B |

Negative | C | D |

${\mathit{G}}_{2}$ | ${\mathit{G}}_{3}$ | |
---|---|---|

$E\left[X\right]$ | 2.537 | 3.757 |

s | 2.865 | 2.686 |

N | 40 | 64 |

Group | LR Model | AUC | MR Cutoff |
---|---|---|---|

${G}_{1}$ | $p\left(MR\right)$ = $\frac{1}{1+{exp}^{(18.2310-2.4040MR)}}$ | $99.11$ | 6.5 |

${G}_{2}$ | $p\left(MR\right)$ = $\frac{1}{1+{exp}^{(14.3667-1.8461MR)}}$ | $97.07$ | 6.5 |

${G}_{3}$ | $p\left(MR\right)$ = $\frac{1}{1+{exp}^{(7.4048-0.9952MR)}}$ | $90.06$ | 7.2 |

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Rodríguez-Hernández, M.M.; Pruneda, R.E.; Rodríguez-Díaz, J.M.
Statistical Analysis of the Evolutive Effects of Language Development in the Resolution of Mathematical Problems in Primary School Education. *Mathematics* **2021**, *9*, 1081.
https://doi.org/10.3390/math9101081

**AMA Style**

Rodríguez-Hernández MM, Pruneda RE, Rodríguez-Díaz JM.
Statistical Analysis of the Evolutive Effects of Language Development in the Resolution of Mathematical Problems in Primary School Education. *Mathematics*. 2021; 9(10):1081.
https://doi.org/10.3390/math9101081

**Chicago/Turabian Style**

Rodríguez-Hernández, M. M., R. E. Pruneda, and J. M. Rodríguez-Díaz.
2021. "Statistical Analysis of the Evolutive Effects of Language Development in the Resolution of Mathematical Problems in Primary School Education" *Mathematics* 9, no. 10: 1081.
https://doi.org/10.3390/math9101081