# A Network Theory Approach to Curriculum Design

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Complexity and Curriculum

## 3. Mathematical Model

#### 3.1. Assumptions

#### 3.2. Network Construction

“One clear distinction that arises between the functions we have studied thus far is that while a linear function exhibits a constant rate of change, an exponential function exhibits a constant percent change”.

Introduced Topic | Directly Connected Topic |

Linear Functions | Rate of Change |

Linear Functions | Exponential Functions |

Exponential Functions | Percent Change |

Rate of Change | Percent Change |

- To start with, we make a list of all the topics in the chosen text under each chapter, section, and subsection. These topics are listed in a Table A1 in Appendix A. Each topic is given a numeric code. In the language of sets, $X=\{{k}_{i}:1<i<N\}$ pertains to the set of $N\in \mathbb{N}$ topics covered in the textbook, where ${k}_{i}$ refers to the i-th assigned code for each topics
- Based on the topics listed in first two columns of the Table A1, we then create a third column as shown in Appendix A, where the elements constitute the set $Y=\{{z}_{ij}:{z}_{ij}\in X,1<i<N,1<j<M,M\le N-1\}$: That is, topic ${k}_{i}$ may contain up to M direct connections outlined in the corresponding text being mapped, $M\in \mathbb{N}$. The elements of set Y therefore represent distinct topics in X which are related to ${k}_{i}$.

#### 3.3. Metrics Computed

- The
**Degree Distribution**(DD) helps us ascertain that the ‘textbook network’ does indeed display a power law profile and hence the metrics typically associated with the analyses of such networks are meaningful in this context. The power law nature of such a network reveals that there is a specific structure to curriculum which is not random. The degree distribution of the network is given by the probability function$$P\left(x\right)=c{x}^{-\alpha}$$ **Clustering Coefficient**(CC) tells us about the average number of connections for each node, giving us a glimpse into the variety of ways a particular topic in precalculus can be understood. A fundamental assumption of the constructivist model of mathematics is the potential to make meaning. Therefore the greater the clustering coefficient, the more diverse the ways in which a concept can be comprehended depending on the particular background and proclivity of the student. The local clustering coefficient, denoted ${C}_{i}$, is commonly given by the expression:$${C}_{i}=\frac{3(\mathrm{number}\phantom{\rule{4.pt}{0ex}}\mathrm{of}\phantom{\rule{4.pt}{0ex}}\mathrm{triangles})}{\mathrm{number}\mathrm{of}\mathrm{connected}\mathrm{triples}}$$**Average Path Length**(APL) tells us the average number of steps that must be taken to traverse between any two nodes. In the context of this study, the APL tells us about how efficiently one can move from one idea to another. It is particularly useful to strategize about how to resolve mathematical problems. A network possessing a low APL is preferable, since it makes explicit the links between concepts and provides a road-map to travel efficiently from one point to another. This, coupled with a high CC, makes for easy navigation between ideas and also increases the likelihood of exploring many possible ways to navigate between these ideas. The APL is given by the equation$$APL=\frac{1}{N(N-1)}\sum _{i\ne j}d({V}_{i},{V}_{j})$$**Hubs**(H) are nodes which have a large number of edges. The threshold number of edges to qualify to be a hub, in general, is determined by the nature of the problem itself. We use the minimum number of chapters from all the texts examined to decide a threshold to qualify for a hub. This number turns out to be 6 based on the book by Faires [30].

## 4. Results

#### 4.1. Power Law Distribution

#### 4.2. Comparing the Network Profiles

## 5. Network Resilience Analysis

#### 5.1. Stochastic Network Simulations

#### 5.2. Simulation Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Network Construction

**Table A1.**This table provides the codes and connected topics in the text by Stewart [33].

Topic Name | Code | Directly Connected Topics |
---|---|---|

Polynomial Functions | PF | RATZEROTHM, RF, REMAINALG, DIVALG, FACTTHM, GRAPHTECH, UPLOWBOUND, QUAD, LINEARFUNC, NONLIN, INEQ, F/R, DESCARTES, |

Rational Zero Theorem | RATZEROTHM | PF |

Rational Functions | RF | PF, GRAPH, MOD, PARTFRAC, LINEARFUNC, NONLIN, INEQ, F/R |

Transformations | TRANS | GRAPH |

Graphs | GRAPH | RF, TRANS, ASYM, MOD, EXP, LOG, EQN, PERIODICFUNC, LINEARFUNC, SYSLIN, LINPROG |

Asymptotes | ASYM | GRAPH |

Modeling | MOD | RF, GRAPH, EXP, LOG, EQN, PERIODICFUNC, LAWS, TRIGRAT, MATRIX, SYSLIN, HYPERBOLA, ELLIPSE, PARABOLA, SEQUENCES, BIN, RECURS |

Partial Fractions | PARTFRAC | RF |

Remainder Algorithm | REMAINALG | PF, SYNTH |

Synthetic Division | SYNTH | REMAINALG, DIVALG |

Division Algorithm | DIVALG | PF, SYNTH, LONGDIV |

Long Division | LONGDIV | DIVALG |

Factor Theorem | FACTTHM | PF |

Intermediate Value Theorem | IVT | GRAPHTECH |

Graphing with Technology | GRAPHTECH | PF, IVT |

Descartes’ Rule | DESCARTES | PF |

Upper and Lower Bounds | UPLOWBOUND | PF |

Quadratic Functions | QUAD | PF, QUADGRAPHING, OPTIM |

Graphing Quadratics | QUADGRAPHING | QUAD |

Optimization | OPTIM | QUAD |

Exponential Functions | EXP | GRAPH, MOD, LOG, EQN, NAT, F/R |

Logarithms | LOG | GRAPH, MOD, EXP, EQN, NAT, COM, LIQU, F/R |

Equation Representations | EQN | GRAPH, MOD, EXP, LOG, TRIG, IDENT, HYPERBOLA, ELLIPSE, PARABOLA, POLAR |

Natural Log | NAT | LOG, EXP |

Common Log | COM | LOG |

Application: Liquids | LIQU | LOG |

Trigonometry | TRIG | EXP, LOG |

Right Triangle Trig | RTTRI | LOG |

Unit Circle | UNITCIRCLE | LOG |

Trig Identities | IDENT | EQN, UNITCIRCLE |

Periodic Functions | PERIODICFUNC | MOD, RTTRI |

Inverses | INVERSE | RTTRI |

Trig Laws | LAWS | MOD, RTTRI |

Trig Ratios | TRIGRAT | MOD, RTTRI |

Matrices | MATRIX | MOD, METHODS, SYSOFEQNS |

Methods of Evaluating Systems | METHODS | MATRICES, DET, CRAMER, GAUSSJORD |

Systems of Equations | SYSOFEQNS | MATRIX, LINEARFUNC |

Determinants | DET | METHODS |

Cramer’s Rule | CRAMER | METHODS |

Gauss-Jordan | GAUSSJORD | METHODS |

Linear Functions | LINEARFUNC | PF, RF, GRAPH, SYSOFEQNS, F/R |

Non-Linear Functions | NONLIN | PF, RF, SYSLIN, F/R |

Systems of Non-Linear Eqns | SYSNLIN | GRAPH, MOD, NONLIN, INEQ |

Inequalities | INEQ | PF, RF, SYSLIN, F/R |

Linear Programming | LINPROG | GRAPH |

Conic | CONIC | HYPERBOLA, ELLIPSE, PARABOLA, POLAR, F/R |

Hyperbolas | HYPERBOLA | MOD, EQN, CONIC |

Ellipses | ELLIPSE | MOD, EQN, CONIC |

Parabolas | PARABOLA | MOD, EQN, CONIC |

Polar | POLAR | EQN, CONIC |

Sequences | SEQUENCES | MOD, INDUCTION, BIN, RECURS, GEO, ARITH, SERIES, F/R |

Proof by Induction | INDUCTION | PROOF, SEQUENCES |

Binomial Theorem | BIN | MOD, SEQUENCES |

Recursion | RECURS | MOD, SEQUENCES |

Geometric Sequences | GEO | SEQUENCES, PARTIALSUMS |

Arithmetic Sequences | ARITH | SEQUENCES, PARTIALSUMS |

Series | SERIES | SEQUENCES, PARTIALSUMS |

Partial Sums | PARTIALSUMS | GEO, ARITH, SERIES |

Proofs | PROOF | INDUCTION |

Functions and Relations | F/R | PF, RF, EXP, LOG, TRIG, LINEARFUNC, NONLIN, INEQ, CONIC, SEQUENCES |

## Appendix B. Codes for Hubs in All Books

Hub Name | Code |
---|---|

Functions and Relations | F/R |

Linear Functions | LF |

Polynomial Functions | PF |

Rational Functions | RF |

Trig | TRIG |

Conics | CONICS |

Sequences | SEQUENCES |

Limits | LIMITS |

Series | SERIES |

Graphing | GRAPH |

Modeling | MOD |

Inequalities | INEQ |

Equations | EQN |

Rate of Change | ROC |

Average Speed | AS |

Constant Rate of Change | CROC |

Quadratic | QUAD |

Composition | COMPOSITION |

Function Notation | FUNCTNOT |

Inverses | INV |

Domain and Range | D/R |

Exponential Function | EF |

Growth and Decay | GROW/DEC |

Transformations | TRANSF |

Roots and End Behavior | ROOTS/EB |

Circular Motion | CIRCMOT |

Angle Measure | ANGMES |

Cosine | COS |

Sine | SIN |

Right Triangle | RTTRI |

Non-Right Triangles | NRTTRI |

Average Rate of Change | AROC |

Tangent | TAN |

Change in Quantity | DELQ |

Covariation | COV |

Proportions | PROP |

Box Activity | BOX |

Percent Change | %DEL |

Logarithm | LOG |

Technology | TECH |

Real Number Line | REALLN |

X-Y Plane | XYPLANE |

Applications | APPS |

Distance Formula | DISTF |

Pythagorean Theorem | PYTHTHM |

Parabola | PARABOLA |

Symmetry | SYMM |

Calculus | CALC |

Complex Numbers | COMPLEX |

Periodic | PERIODIC |

Cotangent | COT |

Secant | SEC |

Cosecant | COSEC |

Trigonometric Identities | IDENTITIES |

Sum and Difference Formulas | SUMDIFF |

Hyperbola | HYPERBOLA |

Ellipse | ELLIPSE |

Circle | CIRC |

Reflection | REFLECT |

Quadratic Formula | QUADF |

Polar Form | POLAR |

Radians | RAD |

Reciprocal | RECIP |

Roots of Unity | ROOTUNITY |

DeMoivre’s Theorem | DEMOIVRE |

Secant Line | SECLINE |

Slope | SLOPE |

Euler’s Constant | E |

Factorial | FACTORIAL |

Tangent Line | TANLN |

Combinatorics | COMBINATORICS |

Mathematical Modeling | MATHMODEL |

Dependent Variable | DV |

Independent Variable | IV |

Applications: Free Falling Objects | FALLINGOBJECTS |

Oblique Triangles | OBLIQUE |

Vectors | VECTORS |

Differential and Difference Equations | DIFFEQ |

Set Representations | SETS |

Function Representations | REP |

Role of Numbers and Quantity | NUMBERS |

Permutations and Combinations | PERMUTCOMB |

Counting Principles | COUNTPRINC |

Probability | PROB |

Binomial Expansion | BINEXP |

Recursion | RECURS |

Difference Tables | DIFFTAB |

Tables | TAB |

Proof | PROOF |

Geometry | GEOM |

Analytic Geometry | ANALGEOM |

Coordinate Plane | COORDPL |

Exponent Value | EXPVAL |

Events | EVENTS |

Area Under Curve | AREAUNDCURVE |

Division Algorithm | DIVALG |

Operations | OPER |

## Appendix C. Hubs

**Table A3.**The hub topics for each textbook. Note that the threshold to qualify as a hub is based on the book by Faires which contains a minimum of six chapters.

Book | Hub Names |
---|---|

Abramson | RF, LF, PF, RF, TRIG, CONICS, SEQUENCES, LIMITS, SERIES |

Blitzer | GRAPH, MOD, INEQ, RF, PF, EQN, CONICS, SERIES, F/R |

Pathways | ROC, AS, CROC, QUAD, F/R, COMPOSITION, FUNCTNOT, INV, D/R, EF, GROW/DEC, PF, TRANSF, ROOTS/EB, RF, CIRCMOT, ANGMES, COS, SIN, RTTRI, NRTTRI |

Stewart | PF, RF, GRAPH, MOD, LOG, EF, SEQUENCES, EQN R/F |

Faires | R/F, LF, REALLN, XYPL, RF, APPLICATION, QUAD, D/R, GRAPH, TECH, PYTHTHM, TRANSF, PARABOLA, SYMM, CALC, INV, ROOTS, PF, COMPLEX, TRIG, SIN, COS, PERANG, ANG, TAN, COTAN, SEC, COSEC, IDENTITIES, SUMDIFF, RTTRI, EF, LOG, GROW/DEC, CONICS, ELLIPSE, HYPERBOLA, CIRC, REFLECT, QUADF, POLAR, DISTF, COMPOSITION |

CME | TAN, SIN, COS, PYTHTHM, GRAPH, ANG, RADIANS, TRIG, CIRC, EQN, RECIPROCAL, COMPLEX, DEMOIVRE, PF, IDENTITIES, SUM, RF, SECLINE, SLOPE, EF, E, FACTORIAL, TANLN, COMBINATORICS, PERMUTCOMB, R/F, PROB, BINEXP, RECURS, DIFFTAB, TAB, PROOF, GEOM, ANALGEOM, COORDPL, EXPVAL, EVENTS, CALC, AREAUNDCURVE, COUNTPRINC, ROOTUNITY |

Larson | DIVALG, PF, GRAPHS, MOD, EQNS, OPER |

COMAP | TRANSF, LINEAR, GEOMETRY, PF, F/R, GRAPHS, MATHMODEL, TABLES, EQN, DV, IV, EXP, LOG, INVERSE, MODELING, FALLINGOBJECTS, TECHNOLOGY, COMPLEX, PERIODIC, COS, SIN, RADIAN, TAN, RTTRI, OBLIQUE, VECTORS, POLAR, MATRIX, ANALGEO, PARABOLA, COUNTINGPRINC, DIFFEQ |

Rockswold | F/R, SETS, REP, GRAPHS, LINEAR, INEQ, NUMBERS, MODELS, ZERO, EQN, QUADRATIC, PF, DIVISION, RF |

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**Figure 1.**Schematic showing the complex nature of education and curricular design. While teaching and learning are recognized to proceed along an a-causal feedback loop, teaching at its intrinsic scale is feedback between the intended and enacted curriculum, each of which gets refined over time.

**Figure 2.**Network structure of the Pathways Combined curriculum. The various panels represent different books, namely: (

**a**) Abramson (

**b**) Blitzer, (

**c**) CME, (

**d**) COMAP, (

**e**) Faires, (

**f**) Larson, (

**g**) Pathways (combined), (

**h**) Rockswold and (

**i**) Stewart. In cases (

**d**,

**h**), we provide a zoomed image to showcase the details of the connectivity.

**Figure 3.**A Q-Q plot for the (

**a**) CME and (

**b**) Pathways books shows a linear profile indicating the strong possibility of a power law distribution.

**Figure 4.**This graph shows the degree distribution for the various textbooks along with the power law fit superimposed. The various panels represent different books, namely: (

**a**) Ambramson (

**b**) Blitzer, (

**c**) CME, (

**d**) COMAP, (

**e**) Faires, (

**f**) Larson, (

**g**) Pathways (combined), (

**h**) Rockswold and (

**i**) Stewart.

**Figure 5.**A visual display of the degree distribution behavior of two texts with outlier power law parameter values. (

**a**–

**d**) display the DD and all aforementioned Q-Q plots of Rockswold, while (

**e**–

**h**) display the same information for Larson.

**Figure 6.**A visual demonstration of the trends observed in the union of all sampled texts, where (

**a**) is the degree distribution fitted to a power law with relevant parameters, (

**b**) is the Q-Q plot of the union with respect to a power law fit, and (

**c**) with respect to an exponential distribution fit.

**Figure 7.**A visual display of both the variability of all fifty iterations for each value of n and the APL and CC metrics in (

**a**,

**b**), as well as a regression for the mean APL and CC across all iterations in (

**c**,

**d**).

Text (Reference) | First Author | Edition | Year | Publisher |
---|---|---|---|---|

[25] | Abramson | 1st | 2017 | OpenStax |

[26] | Blitzer | 5th | 2013 | Pearson |

[27] | Carlson | 8th | 2020 | MacMillan |

[28] | CME | - | 2013 | - |

[29] | COMAP | - | 2002 | W. H. Freeman |

[30] | Faires | 5th | 2011 | Cengage |

[31] | Larson | 3rd | 2000 | Houghton Mifflin |

[32] | Rockwold | 4th | 2010 | Pearson |

[33] | Stewart | 6th | 2012 | Cengage |

[34] | Sullivan | 11th | 2020 | Pearson |

**Table 2.**The results of the network analysis for all texts are summarized in this table. Quantities computed include the average path length, clustering coefficient, number of hubs, percentage of nodes that are hubs, number of edges and nodes for each text.

Book | APL | CC | % H | # H | Edges/Nodes | Mean Edges per Node | Power-Law Exponent | p-Value |
---|---|---|---|---|---|---|---|---|

Abramson | 19.0047 | 0.2662 | 11.54 | 9 | 116/78 | 1.487 | −5.851 | 0.14 |

Blitzer | 3.7652 | 0.0481 | 12.33 | 9 | 109/73 | 1.493 | −16.881 | 0.16 |

Carlson | 3.9001 | 0.2725 | 21.83 | 31 | 268/142 | 1.887 | −16.6774 | 0.07 |

Stewart | 3.361 | 0.0417 | 15.00 | 9 | 101/60 | 1.683 | −28.357 | 0.06 |

Faires | 3.2133 | 0.3414 | 35.25 | 43 | 327/122 | 2.680 | −38.478 | 0.84 |

CME | 3.4179 | 0.4152 | 32.03 | 41 | 314/128 | 2.453 | −29.716 | 0.06 |

Larson | 3.6915 | 0.156 | 14.63 | 6 | 66/41 | 1.610 | −68.315 | 0.051 |

COMAP | 3.6908 | 0.3397 | 21.19 | 32 | 302/151 | 2 | −22.996 | 0.22 |

Rockswold | 3.4128 | 0.2815 | 11.76 | 14 | 192/119 | 1.613 | −55.960 | 0.07 |

**Table 3.**The results of the stochastic network simulation analysis for the union of all texts are summarized in this table. Quantities computed include the average path length, clustering coefficient, and error metrics relative to the original union graph.

No. of Removed Nodes, n | n = 5 | n = 80 | n = 160 | n = 320 | n = 800 |
---|---|---|---|---|---|

Percent Equivalent of Total Edges | 0.3097 | 4.9566 | 9.9132 | 19.8265 | 49.5662 |

Absolute Percent Error in APL | 0.0666 | 11.7704 | 44.7580 | 66.4542 | 100.6488 |

Absolute Percent Error in CC | 0.8559 | 9.3076 | 10.4080 | 13.9279 | 16.2400 |

Mean APL | 5.1142 | 5.7123 | 7.3982 | 9.9291 | 12.9921 |

Standard Deviation of APL | 0.0029 | 0.2820 | 0.7283 | 0.7835 | 2.1169 |

Mean CC | 0.3138 | 0.2871 | 0.2836 | 0.2725 | 0.2651 |

Standard Deviation of CC | 0.0028 | 0.0210 | 0.0190 | 0.0132 | 0.0146 |

Rank | Network | Composite RMSD | Rank of Graph Size Based on # of Nodes |
---|---|---|---|

1 | Pathways | 1.2989 | 2 |

2 | COMAP | 1.7228 | 1 |

3 | CME | 4.8429 | 3 |

4 | Stewart | 5.1918 | 8 |

5 | Larson | 5.3651 | 9 |

6 | Blitzer | 6.6795 | 7 |

7 | Faires | 6.6892 | 4 |

8 | Rockswold | 7.0268 | 5 |

9 | Abramson | 10.7050 | 6 |

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O’Meara, J.; Vaidya, A.
A Network Theory Approach to Curriculum Design. *Entropy* **2021**, *23*, 1346.
https://doi.org/10.3390/e23101346

**AMA Style**

O’Meara J, Vaidya A.
A Network Theory Approach to Curriculum Design. *Entropy*. 2021; 23(10):1346.
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**Chicago/Turabian Style**

O’Meara, John, and Ashwin Vaidya.
2021. "A Network Theory Approach to Curriculum Design" *Entropy* 23, no. 10: 1346.
https://doi.org/10.3390/e23101346