Explorative Study of Developing a Mathematical Model for Evaluating HOTS in the Mathematics Curriculum Operating in the KZN TVET Colleges
Abstract
:1. Introduction
2. Methodology
2.1. Formulation of the Data Collection Instrument
 ➢
 Working systematically through cases in an exhaustive way;
 ➢
 Interpret and extend solutions of problems;
 ➢
 Identifying possible applications of mathematics in the surroundings;
 ➢
 Translate a worded or graphically represented situation to relevant mathematical formalisms;
 ➢
 Use with reasonable skill the available tools for mathematical exploration.
2.2. Evaluation Model Development
2.2.1. Review of the SIR Model
 S(t): number of susceptible individuals;
 S′(t): rate of change in S;
 I(t): number of infected individuals;
 I′(t): rate of change in I;
 R(t): number of recovered individuals;
 R′(t): rate of change in R;
 β: disease transmission rate;
 γ: recovery rate.
 The duration of infection is the same for everyone;
 Once recovered, you are immune, and can no longer infect anyone;
 Only a fraction of contacts with the disease cause infection;
 The units of S, I, and R are persons;
 The units of time are days;
 The units of S′, I′, and R′ are persons per day, written person/day.
2.2.2. Development of the SVHIR Model
2.2.3. Determination of the SVHIR Model Parameters
2.2.4. Validation of the SVHIR Model
Development of the General Prediction Functions and Four Parameters (μ, θ, γ, and β)
Determination of the Constants of Integration (c_{2}, c_{3}, and c_{4})
Validation
2.2.5. Basic Reproductive Ratio
2.2.6. Association of the Actual Data with the SVHIR Model
 A score of less than or equal to 5% cannot be used to define the status of a student: it is a nil, given that this score is highly possible to be obtained by a person who guessed the answers without being exposed to the curriculum. Therefore, we equivalate this person as someone who never took the test; hence, this score is associated with the susceptible compartment;
 A student with a score between 5% and 50% counts as a failed; hence, this score is associated with the infection compartment;
 A student with a score at 50% and above counts as a pass; hence, this score is associated with the healthy or recovery compartment.
 A student who received nil in the first test and nil in the second test is considered susceptible. The first test shows symptoms of susceptibility (neither infected nor healthy but at risk of infection), and towards the end of the curriculum the second test confirms the symptoms remained the same, which means the student did not move to the vaccine compartment. Hence, the student stays in the susceptible compartment. Nonetheless, this does not mean the curriculum was not presented to the student but rather means it was presented and did not make any significant impact or sink in for the student. Therefore, the student is the same as the time of arrival, which happens at the susceptible stage.
 A student who received nil in the first test and failed in the second test is considered infected. The first test show symptoms of susceptibility, and towards the end of the curriculum the second test confirms that the student is infected. Hence the student will move from susceptible S(t), vaccinated V(t) and to infected I(t) compartment. In this case, the curriculum was presented and did make an impact to the student but not enough.
 A student who received nil in the first test and passed in the second test is considered recovered. The first test show symptoms of susceptibility, and towards the end of the curriculum the second test confirms the symptoms have improved. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the healthy I(t) compartment. When the curriculum is presented to this student, it is very impactful.
 A student who failed in the first test and received nil in the second test is considered infected. The first test shows symptoms of infection, and towards the end of the curriculum the second test confirms the symptoms of being at risk of infection. This student is considered infected. In the model, this student will move from the susceptible S(t) or vaccinated V(t) and to the infected I(t) compartment. In this case, the curriculum was presented and did make an impact on the student but not enough.
 A student who failed in the first test and failed in the second test is considered infected. The first test shows symptoms of infection, and towards the end of the curriculum the second test confirms the symptoms remained the same. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the infected I(t) compartment. In this case, the curriculum was presented and did make an impact on the student but not enough.
 A student who failed in the first test and passed in the second test is considered in recovery. The first test shows symptoms of infection, towards the end of the curriculum the second test confirms the symptoms have gotten better. Hence, the student will move from the susceptible S(t), vaccinated V(t) or infected I(t) and to the recovered R(t) compartment. In this case, the curriculum was presented and did make an impact on the student.
 A student who passed in the first test and received nil in the second test is considered infected. The first test shows symptoms of being healthy, and towards the end of the curriculum the second test confirms the symptoms of being susceptible. For a student to be moved from healthy to susceptible is the indication of degradation of the skill; and that can only happen when someone is infected. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the infected I(t) compartment. In this case, the curriculum was presented and did make an impact on the student but not enough.
 A student who passed in the first test and failed in the second test is considered infected. The first test shows symptoms of being healthy, and towards the end of the curriculum the second test confirms the symptoms of infection. For a student to be moved from healthy to susceptible is an indication of degradation of the skill; and that can only happen when someone is infected. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the infected I(t) compartment. In this case, the curriculum was presented and did make an impact on the student but not enough.
 A student who passed in the first test and passed in the second test is considered healthy. The first test shows symptoms of being healthy, and towards the end of the curriculum the second test confirms the symptoms have remained the same. Hence, the student will move from the susceptible S(t) or vaccinated V(t) and to the healthy H(t) compartment. This student is presumed to have arrived already equipped with HOTS; hence, when the curriculum is presented to them, it is very impactful.
 A student who received nil in the first test and did not get a chance to participate in the second test is excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the second test, the student could either be Outcome 1 or 2 or 3 in Table 3, which are three different compartments (susceptible, infected, and recovered) the student could possibly belong to, and the study is unable to conclude about the student’s compartment between the three in the absence of the second test score. Hence, the student is excluded.
 A student who failed in the first test and did not get a chance to participate in the second test is excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the second test, the student could either be Outcome 4 or 5 or 6 in Table 3, which are two different compartments (infected and recovered) the student could possibly belong to, and the study is unable to conclude about the student’s compartment between the two in the absence of the second test score in that case. Hence, the student is excluded.
 A student who passed in the first test and did not get a chance to participate in the second test is excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the second test, the student could either be Outcome 7 or 8 or 9 in Table 3, which are two different compartments (infected and healthy) the student could possibly belong to, and the study is unable to conclude about the student’s compartment between the two in the absence of the second test score in that case. Hence, the student is excluded.
 A student who did not participate in the first test and received nil in the second test is excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the first test, the student could either be Outcome 1 or 4 or 7 in Table 3, which are two different compartments (susceptible and infected) the student could possibly belong to, and the study is unable to conclude about the student’s compartment between the two in the absence of the first test score in that case. Hence, the student is excluded.
 A student who did not participate in the first test and failed in the second test is considered infected. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the first test, the student could either be Outcome 2 or 5 or 8 in Table 3, which are all the infected compartments.
 Lastly, this is a student who only participated in the second test and passed. This student will also be excluded in the current study. The reason is that with the three possible scores (nil, fail, and pass) the student could have obtained in the first test, the student could either be Outcome 3 or 6 or 9 in Table 3, which are two different compartments (recovered and healthy) the student could possibly belong to, and the study is unable to conclude about the student’s compartment between the two in the absence of the first test score in that case. Hence, the student is excluded.
2.2.7. Application of the SVHIR Model Instruction
2.3. Data Collection and Participants
3. Results of the Research
3.1. Partial Evaluation of the Content Delivery
3.2. Evaluation of HOTS
3.2.1. Validation of SVHIR Model
3.2.2. Application of SVHR Model
4. Results Discussion
5. Conclusions
6. Limitations
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. HOTS Questionnaire (PreAssessments)
 Simplify the following:
 (a)
 $\frac{{x}^{2}}{{y}^{2}}$
 (b)
 $\sqrt{\frac{{x}^{2}}{{y}^{2}}}$
 (c)
 $\frac{x}{y}$
 (d)
 $\sqrt{\frac{{x}^{4}}{{{x}^{2}y}^{4}}}$
 (e)
 None of these.
 2.
 The following equation has not more than two roots/solutions:
 2.1.
 Why the above equation has not more than two roots/solutions?
 (a)
 It is a cubic equation.
 (b)
 It has no solution.
 (c)
 It is a quadratic equation.
 (d)
 It has a constant number 1.
 2.2.
 Elaborate further what is meant by something being a root/solution of a particular equation?
 (a)
 It is any integer number.
 (b)
 It is a number that when substituted in a given equation satisfies it.
 (c)
 It is a number that when substituted in a given equation leaves the result undefined.
 (d)
 It is any constant number found in the equation.
 3.
 One chocolate and one apple cost a total amount of ZAR 50 while four chocolates and three apples cost a total amount of ZAR 190. How much is each chocolate and each apple?
 4.
 Write down the following sentences/statements in a form of mathematical equations.
 5.
 Given the following diagram
 5.1.
 Mention the method/s that can be used to find the distance AB.
 5.2.
 Use the abovementioned method/s to calculate the distance AB (e.g., if you mentioned two methods, find the distance of AB by the first method and after that use the second method).
Appendix B. HOTS Questionnaire (PostAssessments)
 Simplify the following:
 (a)
 $\frac{{x}^{2}}{{y}^{2}}$
 (b)
 $\sqrt{\frac{{x}^{2}}{{y}^{2}}}$
 (c)
 $\frac{x}{{y}^{2}}$
 (d)
 $\sqrt{\frac{{x}^{4}}{{{x}^{2}y}^{4}}}$
 2.
 The following equation has no more than two roots/solutions:
 2.1.
 Why does the above equation have no more than two roots/solutions?
 (a)
 It is a cubic equation.
 (b)
 It has no solution.
 (c)
 It is a quadratic equation.
 (d)
 It has a constant number 10.
 2.2.
 Elaborate further what is meant by something being a root/solution of a particular equation?
 (a)
 It is any integer number.
 (b)
 It is a number that when substituted into a given equation satisfies it.
 (c)
 It is a number that when substituted into a given equation leaves the result undefined.
 (d)
 It is any constant number found in the equation.
 3.
 One chocolate and one apple cost a total amount of ZAR 40, while four chocolates and three apples cost a total amount of ZAR 150. How much is each chocolate and each apple?
 4.
 Write down the following sentences/statements in a form of mathematical equations.
 5.
 Given the following diagram
 5.1.
 Mention the method/s that can be used to find the distance BC.
 5.2.
 Use the abovementioned method/s to calculate the distance AC (e.g., if you mentioned two methods, find the distance of AC by the first method and after that use the second method).
 6.
 Do you think all the above questions from 1–5 are familiar or relevant to what you have learnt from the N1 to N2 curriculum and class lessons?
 (a)
 YES
 (b)
 NO
Appendix C. Marks Scoring Grid
Question 1: Transfer—work systematically through cases in an exhaustive way. 

Total [1 mark]  
Question 2.1: Critical Thinking—interpret and extend solutions of problems. 

Total [1 mark]  
Question 2.2: Critical Thinking—interpret and extend solutions of problems. 

Total [1 mark]  
Question 3: Transfer—identify possible applications of mathematics in their surroundings. 

Total [4 marks]  
Question 4: Transfer—translate a worded or graphically represented situation to relevant mathematical formalisms. 

Total [3 marks]  
Question 5.1: Problem Solving—use with reasonable skill available tools for mathematical exploration. 

Total [1 mark]  
Question 5.2: Problem Solving—use with reasonable skill available tools for mathematical exploration. 

Total [2 marks] 
Appendix D. Pre and PostAssessment Students’ Scores for HOTS
Student Order ($\mathit{i}$)  PreAssessment Scores  PostAssessment Scores  SVHIR Model Compartment 
1  0  23  Infected 
2  69  0  Infected 
3  54  0  Infected 
4  53  77  Healthy 
5  53  15  Infected 
6  46  0  Infected 
7  38  46  Infected 
8  54  0  Infected 
9  38  0  Infected 
10  38  0  Infected 
11  38  0  Infected 
12  46  0  Infected 
13  38  0  Infected 
14  0  0  Susceptible 
15  15  0  Infected 
16  69  51  Healthy 
17  23  0  Susceptible 
18  23  8  Infected 
19  15  0  Infected 
20  15  31  Infected 
21  31  15  Infected 
22  38  0  Infected 
23  38  31  Infected 
24  46  31  Infected 
25  54  31  Infected 
26  23  31  Infected 
27  69  62  Healthy 
28  15  23  Infected 
29  84  100  Healthy 
30  69  92  Healthy 
31  15  31  Infected 
32  54  77  Healthy 
33  38  0  Infected 
34  53  0  Infected 
35  46  0  Infected 
36  54  23  Infected 
37  85  0  Infected 
38  31  0  Infected 
39  0  85  Recovered 
40  0  100  Recovered 
41  0  46  Infected 
42  0  23  Infected 
43  0  85  Recovered 
44  0  38  Infected 
45  0  38  Infected 
46  0  46  Infected 
47  0  46  Infected 
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Parameters  Description 

$\mu $  Vaccination rate 
$\theta $  Healthy individuals’ discovery rate 
$\beta $  Disease transmission rate 
$\gamma $  Recovery rate 
${t}_{0}$  Initial or 1st day ${t}_{0}=$ 1 
${t}_{f}$  Final or 180th day $\left({t}_{f}=180\right)$. 
${S}_{0}$  Susceptible individuals on the 1st day/ Initial susceptible individuals 
${S}_{f}$  Susceptible individuals on the 180th day/ Final susceptible individuals 
${I}_{0}$  Infected individuals on the 1st day/ Initial infected individuals 
${I}_{f}$  Infected individuals on the 180th day/ Final infected individuals 
${H}_{0}$  Healthy individuals on the 1st day/ Initial healthy individuals 
${H}_{f}$  Healthy individuals on the 180th day/ Final healthy individuals 
${R}_{0}$  Recovered individuals on the 1st day/ Initial recovered individuals 
${R}_{f}$  Recovered individuals on the 180th day/ Final recovered individuals 
${V}_{f}$  Vaccinated individuals on the 180th day/ Final vaccinated individuals 
${N}_{0}$  Total number of individuals on the 1st day/ Initial total number of individuals 
Order  Scores  Description  Compartment 

1.  $0\le {M}_{i}\le 5\%$  Nil  susceptible 
2.  $5\%<{M}_{i}<50\%$  fail  infection 
3.  ${M}_{i}\ge 50\%$  pass  healthy or recovery 
Outcome  Test 1 Marks (t = 0)  Test 2 Marks (t = 180)  Resultant Compartment (t = 180) 

1  $0\le {M}_{i}\le 5\%$  $0\le {M}_{i}\le 5\%$  Susceptible 
2  $0\le {M}_{i}\le 5\%$  $5\%<{M}_{i}<50\%$  Infected 
3  $0\le {M}_{i}\le 5\%$  ${M}_{i}\ge 50\%$  Recovered 
4  $5\%<{M}_{i}<50\%$  $0\le {M}_{i}\le 5\%$  Infected 
5  $5\%<{M}_{i}<50\%$  $5\%<{M}_{i}<50\%$  Infected 
6  $5\%<{M}_{i}<50\%$  ${M}_{i}\ge 50\%$  Recovered 
7  ${M}_{i}\ge 50\%$  $0\le {M}_{i}\le 5\%$  Infected 
8  ${M}_{i}\ge 50\%$  $5\%<{M}_{i}<50\%$  Infected 
9  ${M}_{i}\ge 50\%$  ${M}_{i}\ge 50\%$  Healthy 
10  $0\le {M}_{i}\le 5\%$  None  Excluded ${N}_{f}<{S}_{0}$ 
11  $5\%<{M}_{i}<50\%$  None  Excluded ${N}_{f}<{S}_{0}$ 
12  ${M}_{i}\ge 50\%$  None  Excluded ${N}_{f}<{S}_{0}$ 
13  None  $0\le {M}_{i}\le 5\%$  Excluded ${N}_{f}>{S}_{0}$ 
14  None  $5\%<{M}_{i}<50\%$  Infected ${N}_{f}>{S}_{0}$ 
15  None  ${M}_{i}\ge 50\%$  Excluded ${N}_{f}>{S}_{0}$ 
Students’ Response  Number of Students  Percentage 

Yes  26  55.6% 
No  2  4.3% 
No comment  19  40.4% 
Compartment Parameters  Actual Data Values 

Initial time (in days)  ${t}_{0}=1$ 
Final time (in days)  ${t}_{f}=180$ 
Initial susceptible individuals  ${S}_{0}=47$ 
Final susceptible individuals  ${S}_{f}=2$ 
Initial infected individuals  ${I}_{0}=0$ 
Final infected individuals  ${I}_{f}=36$ 
Initial healthy individuals  ${H}_{0}=0$ 
Final healthy individuals  ${H}_{f}=6$ 
Initial recovered individuals  ${R}_{0}=0$ 
Final recovered individuals  ${R}_{f}=3$ 
Final vaccinated individuals  ${V}_{f}={H}_{f}+{I}_{f}+{R}_{f}=45$ 
Initial total number of individuals  ${N}_{0}=47$ 
Final total number of individuals  ${N}_{f}=47$ 
Compartment Rate Names  Calculated Values 

Vaccination rate  $\mu =0.0180$ 
Healthy individuals’ discovery rate  $\theta =0.0007$ 
Disease transmission rate  $\gamma =0.0005$ 
Recovery rate  $\beta =0.0050$ 
Integration Constant  Predicted Compartment at ${\mathit{t}}_{\mathit{f}}$  Actual Compartment at ${\mathit{t}}_{\mathit{f}}$ 

  ${S}_{p}=1.84\approx 2$  ${S}_{f}=2$ 
${c}_{2}=7.21$  ${H}_{p}=6$  ${H}_{f}=6$ 
${c}_{3}=3.69$  ${R}_{p}=2.71\approx 3$  ${R}_{f}=3$ 
${c}_{4}=0.54$  ${I}_{p}=36$  ${I}_{f}=36$ 
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Mazibuko, G.N.; Maharaj, A. Explorative Study of Developing a Mathematical Model for Evaluating HOTS in the Mathematics Curriculum Operating in the KZN TVET Colleges. Educ. Sci. 2024, 14, 279. https://doi.org/10.3390/educsci14030279
Mazibuko GN, Maharaj A. Explorative Study of Developing a Mathematical Model for Evaluating HOTS in the Mathematics Curriculum Operating in the KZN TVET Colleges. Education Sciences. 2024; 14(3):279. https://doi.org/10.3390/educsci14030279
Chicago/Turabian StyleMazibuko, Godfrey Nkululeko, and Aneshkumar Maharaj. 2024. "Explorative Study of Developing a Mathematical Model for Evaluating HOTS in the Mathematics Curriculum Operating in the KZN TVET Colleges" Education Sciences 14, no. 3: 279. https://doi.org/10.3390/educsci14030279