# How Should I Teach from This Month Onward? A State-Space Model That Helps Drive Whole Classes to Achieve End-of-Year National Standardized Test Learning Targets

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Work

^{2}adjusted is 0.637. However, all students attempted to solve a similar set of items. One of the challenges is that teachers prefer to have the flexibility to adapt the exercises to their own experience. The authors did not consider the situation where teachers select or design their own set of exercises. Ref. [7] also compared, month by month, their linear predictors without a pretest with baselines. They found an increasing performance as they gathered process information of more months.

## 3. Theoretical Framework

**Research Question 1:**To what extent can teacher-designed questions (low- or no-stake quizzes) help predict students’ long-term learning as measured by end-of-year standardized state tests?

**Research Question 2:**To what extent can a hidden Markovian state of the student’s accumulated knowledge up to the current month be estimated so that, along with the probable deliberate practice on the next months, it is sufficient to predict with good accuracy the end-of-year standardized state tests?

**Research Question 3:**To what extent can a state-space model help teachers to visualize the trade-off between asking students to perform exercises more carefully or perform more exercises, and thus help drive whole classes to achieve long-term learning targets?

## 4. Materials and Methods

- Numbers and operations;
- Patterns and algebra;
- Geometry;
- Measurement;
- Data and probabilities.

#### 4.1. The Sequence of Linear Regression Models

- ${p}_{i}$: pretest of the student i (${p}_{i}\in \mathbb{R}$, normalized).
- ${\Theta}_{j}^{\left(1\right)}$: class j average for “My behavior is a problem for the math teacher”.
- ${\Theta}_{i}^{\left(2\right)}$: student i response to “Math is easy for me”.
- ${a}_{i,j,k,t}$: percentage of exercises solved on the first attempt for a student i, on the k strand, at month t (normalized for each strand month).
- ${d}_{i,j,k,t}$: the difference between exercises solved on the first attempt and those that took more than one for a student i, on the k strand, at month t (normalized for each strand month).

#### 4.2. The State-Space Model

- $x\left(t\right)$ is the hidden state that represents the long-term knowledge of a student.
- $y\left(t\right)$ are the observed measurement as a grade or score.
- $u\left(t\right)$ has historical variables at $t=0$ and dynamic ones for $t>0$.

## 5. Results

#### 5.1. Prediction Errors of Both Models

#### 5.2. Optimal Values of the Parameters

#### 5.3. Contribution of the Different Variables to the Prediction

#### 5.4. The Optimal Control Problem: Ask Students to Do Exercises More Carefully or Do More Exercises

## 6. Discussion

## 7. Conclusions

#### 7.1. Theoretical Implications

#### 7.2. Managerial Implications

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GPA | Grade Point Average |

LRMOFR | Linear Regression for the Month with an Optimal Forgetting Rate |

OLS | Ordinary Least Squares |

RMSE | Root Mean Squared Error |

SD | Standard Deviation |

SIMCE | Sistema de Medición de la Calidad de la Educación (Mesurement System of the Quality of Education) |

SS | State Space |

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**Figure 1.**Root mean square error (RMSE) of the two models with best RMSE. The graph shows the RMSE in the test classes for each month of the school year. LRMOFR: linear regression for each month with an optimal forgetting rate model. SS: state-space model.

**Figure 6.**Pretest effect. This is the contribution on the SIMCE prediction of one standard deviation increase of the pre-test.

**Figure 7.**Effect of the class average of the degree of agreement with “My behavior is a problem for the teacher”. This is the contribution to the SIMCE prediction of an increase of one standard deviation of the student belief.

**Figure 8.**Effect of the “Math is easy for me” subjective belief. This is the contribution to the SIMCE prediction of an increase of one standard deviation of the student belief.

**Figure 9.**Effect of an accuracy one standard deviation above the mean on every month. This is the contribution to the SIMCE prediction of an increase of one standard deviation of the student accuracy.

**Figure 10.**Effect of a difference one standard deviation above the mean on every month. This is the contribution to the SIMCE prediction of the difference between number of exercises solved correctly in the first attempt and the number solved correctly on other attempts.

**Figure 11.**Iso-effect curves for the number and operations strand, in September, where the last month with student data is August.

**Figure 12.**Effect gradients and iso-effect curves for the Number and Operations strand, in September, where the last month with student data is August.

**Figure 13.**Iso-effort, effect gradients, and iso-effect curves for the number and operations strand in September, where the last month with student data is August.

**Figure 14.**Iso-effort curves and effect gradients for the data and probability strand in September, where the last month with student data is August.

Strand | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Average |
---|---|---|---|---|---|---|---|---|---|

Numbers | 6 | 120 | 123 | 69 | 56 | 122 | 119 | 93 | 89 |

Patterns | 0 | 2 | 4 | 18 | 7 | 1 | 14 | 27 | 9 |

Geometry | 0 | 0 | 11 | 22 | 7 | 24 | 17 | 64 | 18 |

Measurement | 0 | 0 | 5 | 35 | 11 | 17 | 11 | 28 | 13 |

Data | 0 | 1 | 2 | 3 | 5 | 3 | 7 | 30 | 6 |

Total | 6 | 124 | 146 | 147 | 86 | 168 | 168 | 243 | 136 |

**Table 2.**Percentage of times in which the model had the lowest RMSE within the 24 models in the testing classes.

Model | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Average |
---|---|---|---|---|---|---|---|---|---|

Linear regression | 20.0 | 19.0 | 4.5 | 4.0 | 1.0 | 5.5 | 28.0 | 40.5 | 15.3 |

State-Space | 19.5 | 24.0 | 61.0 | 62.5 | 69.0 | 69.0 | 35.5 | 32.0 | 46.6 |

**Table 3.**Estimated parameters for the linear regression for each month with optimal forgetting rate model obtained with 200 cross validations.

Parameter | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct |
---|---|---|---|---|---|---|---|---|

$\rho $ | 0.71 (0.05) | 0.71 (0.05) | 0.71 (0.05) | 0.71 (0.05) | 0.71 (0.05) | 0.71 (0.05) | 0.71 (0.05) | 0.71 (0.05) |

${\gamma}_{\tau}$ | 33.16 (2.22) | 23.78 (2.24) | 18.77 (2.14) | 17.17 (2.16) | 14.56 (2.17) | 17.38 (1.91) | 15.87 (1.70) | 15.34 (1.59) |

${\beta}_{0}$ | 240.04 (3.45) | 240.04 (3.45) | 240.04 (3.45) | 240.04 (3.45) | 240.04 (3.45) | 240.04 (3.45) | 240.04 (3.45) | 240.04 (3.45) |

${\beta}_{1,\tau}$ | −11.80 (1.75) | −11.49 (2.04) | −11.38 (1.94) | −9.24 (1.73) | −9.26 (1.72) | −10.27 (1.76) | −8.96 (1.71) | −8.62 (1.65) |

${\beta}_{2,\tau}$ | 5.00 (1.12) | 4.83 (1.05) | 4.60 (0.95) | 3.98 (0.95) | 4.23 (0.83) | 4.01 (0.93) | 3.61 (0.95) | 3.00 (0.89) |

${\delta}_{1,\tau}$ | −1.00 (2.03) | 1.73 (3.29) | 7.93 (2.97) | 5.57 (2.99) | 6.75 (2.68) | 7.22 (3.24) | 12.21 (2.80) | 12.01 (2.49) |

${\delta}_{2,\tau}$ | −0.38 (1.73) | 3.58 (1.63) | 5.23 (1.71) | 3.98 (2.25) | 10.57 (1.85) | 7.08 (2.08) | 9.30 (1.90) | 6.70 (2.23) |

${\delta}_{3,\tau}$ | −5.69 (1.63) | −2.30 (2.12) | 0.52 (2.06) | 5.88 (2.31) | 7.18 (2.71) | 7.03 (2.69) | 6.86 (2.74) | 4.61 (2.40) |

${\delta}_{4,\tau}$ | −2.08 (1.89) | −0.66 (2.12) | 4.14 (2.24) | 8.00 (2.67) | 4.55 (2.90) | 2.61 (2.66) | 4.45 (2.35) | 3.18 (2.94) |

${\delta}_{5,\tau}$ | 1.83 (1.64) | 5.19 (1.92) | 2.28 (2.51) | 7.76 (1.48) | 13.82 (2.35) | 5.16 (2.54) | 8.07 (2.56) | 11.09 (2.30) |

${\eta}_{1,\tau}$ | 4.70 (1.69) | 14.64 (2.61) | 13.34 (2.22) | 15.76 (2.33) | 14.85 (2.07) | 9.85 (2.59) | 5.28 (2.54) | 3.03 (2.19) |

${\eta}_{2,\tau}$ | - | 1.94 (1.21) | 2.57 (0.99) | 0.71 (1.52) | −0.82 (1.87) | 1.82 (2.90) | 2.13 (2.56) | 7.64 (2.52) |

${\eta}_{3,\tau}$ | 4.10 (1.90) | 1.20 (2.74) | 1.86 (1.67) | −3.29 (3.02) | −6.90 (3.66) | −4.86 (3.17) | −3.59 (3.13) | 0.50 (3.52) |

${\eta}_{4,\tau}$ | - | 1.21 (1.58) | 2.59 (1.75) | 1.59 (2.05) | 3.49 (2.03) | 10.44 (2.89) | 5.24 (2.10) | 3.55 (2.02) |

${\eta}_{5,\tau}$ | - | −2.62 (1.83) | 1.26 (2.46) | −2.72 (1.55) | −6.25 (2.42) | −0.79 (2.08) | 1.09 (3.20) | 1.23 (2.54) |

Parameter | Mean | SD |
---|---|---|

${x}_{0}$ | 258.3901 | 3.4421 |

A | 1.0010 | 0.0006 |

K | 0.0835 | 0.0085 |

${Q}^{\frac{1}{2}}$ | 0.8982 | 0.1080 |

${R}^{\frac{1}{2}}$ | 10.4133 | 0.5388 |

${B}_{1}$ | 0.3787 | 0.2285 |

${B}_{2}$ | 0.5888 | 0.1422 |

${B}_{3}$ | 0.0770 | 0.1739 |

${B}_{4}$ | 0.1984 | 0.1855 |

${B}_{5}$ | 0.6834 | 0.1631 |

${B}_{6}$ | 1.1153 | 0.2337 |

${B}_{7}$ | 0.3170 | 0.1014 |

${B}_{8}$ | −0.0979 | 0.1365 |

${B}_{9}$ | 0.4526 | 0.1363 |

${B}_{10}$ | −0.0675 | 0.1544 |

${B}_{11}$ | 24.6766 | 2.0893 |

${B}_{12}$ | −14.0852 | 2.0651 |

${B}_{13}$ | 6.143 | 1.2942 |

Effect | LRMOFR | SS |
---|---|---|

Intercept | ${\beta}_{0}$ | $\begin{array}{c}{\scriptstyle {A}^{T-\tau -1}{(A-K)}^{\tau}\widehat{x}\left(0\right)}{\scriptstyle +{A}^{T-\tau -1}{\displaystyle \sum _{t=1}^{\tau}}{(A-K)}^{\tau -t}Ky\left(t\right)}\end{array}$ |

Pretest | ${\gamma}_{\tau}{p}_{i}$ | ${\scriptstyle {B}_{11}\left(\right)open="("\; close=")">{A}^{T-\tau -1}{(A-K)}^{\tau}{u}_{11}\left(0\right)}$ |

‘My behavior is a problem…’ | ${\beta}_{1,\tau}{\Theta}_{j}^{\left(1\right)}$ | ${\scriptstyle {B}_{12}\left(\right)open="("\; close=")">{A}^{T-\tau -1}{(A-K)}^{\tau}{u}_{12}\left(0\right)}$ |

‘Math is easy for me’ | ${\beta}_{2,\tau}{\Theta}_{i}^{\left(2\right)}$ | ${\scriptstyle {B}_{13}\left(\right)open="("\; close=")">{A}^{T-\tau -1}{(A-K)}^{\tau}{u}_{12}\left(0\right)}$ |

Accuracy | $\frac{{\displaystyle \sum _{t=1}^{\tau}}{\displaystyle \sum _{k=1}^{5}}{\rho}^{\tau -t}{\delta}_{k,\tau}{a}_{i,j,k,t}}{{\displaystyle \sum _{t=1}^{\tau}}{\rho}^{\tau -t}}$ | $\begin{array}{c}{\scriptstyle {\displaystyle \sum _{k=1}^{5}}{B}_{k}\left(\right)open="("\; close>{A}^{T-\tau -1}{\displaystyle \sum _{t=1}^{\tau}}{(A-K)}^{\tau -t}{u}_{k}\left(t\right)}\end{array}$ |

Difference | $\frac{{\displaystyle \sum _{t=1}^{\tau}}{\displaystyle \sum _{k=1}^{5}}{\rho}^{\tau -t}{\eta}_{k,\tau}{d}_{i,j,k,t}}{{\displaystyle \sum _{t=1}^{\tau}}{\rho}^{\tau -t}}$ | $\begin{array}{c}{\scriptstyle {\displaystyle \sum _{k=6}^{10}}{B}_{k}\left(\right)open="("\; close>{A}^{T-\tau -1}{\displaystyle \sum _{t=1}^{\tau}}{(A-K)}^{\tau -t}{u}_{k}\left(t\right)}\end{array}$ |

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**MDPI and ACS Style**

Ulloa, O.; Araya, R.
How Should I Teach from This Month Onward? A State-Space Model That Helps Drive Whole Classes to Achieve End-of-Year National Standardized Test Learning Targets. *Systems* **2022**, *10*, 167.
https://doi.org/10.3390/systems10050167

**AMA Style**

Ulloa O, Araya R.
How Should I Teach from This Month Onward? A State-Space Model That Helps Drive Whole Classes to Achieve End-of-Year National Standardized Test Learning Targets. *Systems*. 2022; 10(5):167.
https://doi.org/10.3390/systems10050167

**Chicago/Turabian Style**

Ulloa, Obed, and Roberto Araya.
2022. "How Should I Teach from This Month Onward? A State-Space Model That Helps Drive Whole Classes to Achieve End-of-Year National Standardized Test Learning Targets" *Systems* 10, no. 5: 167.
https://doi.org/10.3390/systems10050167