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Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students

Roles Conceptualization, Investigation, Writing – original draft, Writing – review & editing

* E-mail: [email protected]

Affiliations Department of Biological Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada, Department of Ecology and Evolution, University of Toronto, Toronto, Ontario, Canada

Affiliation Department of Ecology and Evolution, University of Toronto, Toronto, Ontario, Canada

Affiliation Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario, Canada

Affiliation Department of Biology, Memorial University of Newfoundland, St John’s, Newfoundland, Canada

• Korryn Bodner, 
• Chris Brimacombe, 
• Emily S. Chenery, 
• Ariel Greiner, 
• Anne M. McLeod, 
• Stephanie R. Penk, 
• Juan S. Vargas Soto

Published: January 14, 2021

• https://doi.org/10.1371/journal.pcbi.1008539
• Reader Comments

Citation: Bodner K, Brimacombe C, Chenery ES, Greiner A, McLeod AM, Penk SR, et al. (2021) Ten simple rules for tackling your first mathematical models: A guide for graduate students by graduate students. PLoS Comput Biol 17(1): e1008539. https://doi.org/10.1371/journal.pcbi.1008539

Editor: Scott Markel, Dassault Systemes BIOVIA, UNITED STATES

Copyright: © 2021 Bodner et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: The authors received no specific funding for this work.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Biologists spend their time studying the natural world, seeking to understand its various patterns and the processes that give rise to them. One way of furthering our understanding of natural phenomena is through laboratory or field experiments, examining the effects of changing one, or several, variables on a measured response. Alternatively, one may conduct an observational study, collecting field data and comparing a measured response along natural gradients. A third and complementary way of understanding natural phenomena is through mathematical models. In the life sciences, more scientists are incorporating these quantitative methods into their research. Given the vast utility of mathematical models, ranging from providing qualitative predictions to helping disentangle multiple causation (see Hurford [ 1 ] for a more complete list), their increased adoption is unsurprising. However, getting started with mathematical models may be quite daunting for those with traditional biological training, as in addition to understanding new terminology (e.g., “Jacobian matrix,” “Markov chain”), one may also have to adopt a different way of thinking and master a new set of skills.

Here, we present 10 simple rules for tackling your first mathematical models. While many of these rules are applicable to basic scientific research, our discussion relates explicitly to the process of model-building within ecological and epidemiological contexts using dynamical models. However, many of the suggestions outlined below generalize beyond these disciplines and are applicable to nondynamic models such as statistical models and machine-learning algorithms. As graduate students ourselves, we have created rules we wish we had internalized before beginning our model-building journey—a guide by graduate students, for graduate students—and we hope they prove insightful for anyone seeking to begin their own adventures in mathematical modelling.

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Boxes represent susceptible, infected, and recovered compartments, and directed arrows represent the flow of individuals between these compartments with the rate of flow being controlled by the contact rate, c , the probability of infection, γ , and the recovery rate, θ .

https://doi.org/10.1371/journal.pcbi.1008539.g001

Rule 1: Know your question

“All models are wrong, some are useful” is a common aphorism, generally attributed to statistician George Box, but determining which models are useful is dependent upon the question being asked. The practice of clearly defining a research question is often drilled into aspiring researchers in the context of selecting an appropriate research design, interpreting statistical results, or when outlining a research paper. Similarly, the practice of defining a clear research question is important for mathematical models as their results are only as interesting as the questions that motivate them [ 5 ]. The question defines the model’s main purpose and, in all cases, should extend past the goal of merely building a model for a system (the question can even answer whether a model is even necessary). Ultimately, the model should provide an answer to the research question that has been proposed.

When the research question is used to inform the purpose of the model, it also informs the model’s structure. Given that models can be modified in countless ways, providing a purpose to the model can highlight why certain aspects of reality were included in the structure while others were ignored [ 6 ]. For example, when deciding whether we should adopt a more realistic model (i.e., add more complexity), we can ask whether we are trying to inform general theory or whether we are trying to model a response in a specific system. For example, perhaps we are trying to predict how fast an epidemic will grow based on different age-dependent mixing patterns. In this case, we may wish to adapt our basic SIR model to have age-structured compartments if we suspect this factor is important for the disease dynamics. However, if we are exploring a different question, such as how stochasticity influences general SIR dynamics, the age-structured approach would likely be unnecessary. We suggest that one of the first steps in any modelling journey is to choose the processes most relevant to your question (i.e., your hypothesis) and the direct and indirect causal relationships among them: Are the relationships linear, nonlinear, additive, or multiplicative? This challenge can be aided with a good literature review. Depending on your model purpose, you may also need to spend extra time getting to know your system and/or the data before progressing forward. Indeed, the more background knowledge acquired when forming your research question, the more informed your decision-making when selecting the structure, parameters, and data for your model.

Rule 2: Define multiple appropriate models

Natural phenomena are complicated to study and often impossible to model in their entirety. We are often unsure about the variables or processes required to fully answer our research question(s). For example, we may not know how the possibility of reinfection influences the dynamics of a disease system. In cases such as these, our advice is to produce and sketch out a set of candidate models that consider alternative terms/variables which may be relevant for the phenomena under investigation. As in Fig 2 , we construct 2 models, one that includes the ability for recovered individuals to become infected again, and one that does not. When creating multiple models, our general objective may be to explore how different processes, inputs, or drivers affect an outcome of interest or it may be to find a model or models that best explain a given set of data for an outcome of interest. In our example, if the objective is to determine whether reinfection plays an important role in explaining the patterns of a disease, we can test our SIR candidate models using incidence data to determine which model receives the most empirical support. Here we consider our candidate models to be alternative hypotheses, where the candidate model with the least support is discarded. While our perspective of models as hypotheses is a view shared by researchers such as Hilborn and Mangel [ 7 ], and Penk and colleagues [ 8 ], please note that others such as Oreskes and colleagues [ 9 ] believe that models are not subject to proof and hence disagree with this notion. We encourage modellers who are interested in this debate to read the provided citations.

(A) A susceptible/infected/recovered model where individuals remain immune (gold) and (B) a susceptible/infected/recovered model where individuals can become susceptible again (blue). Arrows indicate the direction of movement between compartments, c is the contact rate, γ is the infection rate given contact, and θ is the recovery rate. The text below each conceptual model are the hypotheses ( H1 and H2 ) that represent the differences between these 2 SIR models.

https://doi.org/10.1371/journal.pcbi.1008539.g002

Finally, we recognize that time and resource constraints may limit the ability to build multiple models simultaneously; however, even writing down alternative models on paper can be helpful as you can always revisit them if your primary model does not perform as expected. Of course, some candidate models may not be feasible or relevant for your system, but by engaging in the activity of creating multiple models, you will likely have a broader perspective of the potential factors and processes that fundamentally shape your system.

Rule 3: Determine the skills you will need (and how to get them)

Equipping yourself with the necessary analytical tools that form the basis of all quantitative techniques is essential. As Darwin said, those that have knowledge of mathematics seem to be endowed with an extra sense [ 10 ], and having a background in calculus, linear algebra, and statistics can go a long way. Thus, make it a habit to set time for yourself to learn these mathematical skills, and do not treat all your methods like a black box. For instance, if you plan to use ODEs, consider brushing up on your calculus, e.g., using Stewart [ 11 ]. If you are working with a system of ODEs, also read up on linear algebra, e.g., using Poole [ 12 ]. Some universities also offer specialized math biology courses that combine topics from different math courses to teach the essentials of mathematical modelling. Taking these courses can help save time, and if they are not available, their syllabi can help focus your studying. Also note that while narrowing down a useful skillset in the early stages of model-building will likely spare you from some future headaches, as you progress in your project, it is inevitable that new skills will be required. Therefore, we advise you to check in at different stages of your modelling journey to assess the skills that would be most relevant for your next steps and how best to acquire them. Hopefully, these decisions can also be made with the help of your supervisor and/or a modelling mentor. Building these extra skills can at first seem daunting but think of it as an investment that will pay dividends in improving your future modelling work.

When first attempting to tackle a specific problem, find relevant research that accomplishes the same tasks and determine if you understand the processes and techniques that are used in that study. If you do, then you can implement similar techniques and methods, and perhaps introduce new methods. If not, then determine which tools you need to add to your toolbox. For instance, if the problem involves a system of ODEs (e.g., SIR models, see above), can you use existing symbolic software (e.g., Maple, Matlab, Mathematica) to determine the systems dynamics via a general solution, or is the complexity too great that you will need to create simulations to infer the dynamics? Figuring out questions like these is key to understanding what skills you will need to work with the model you develop. While there is a time and a place for involving collaborators to help facilitate methods that are beyond your current reach, we strongly advocate that you approach any potential collaborator only after you have gained some knowledge of the methods first. Understanding the methodology, or at least its foundation, is not only crucial for making a fruitful collaboration, but also important for your development as a scientist.

Rule 4: Do not reinvent the wheel

While we encourage a thorough understanding of the methods researchers employ, we simultaneously discourage unnecessary effort redoing work that has already been done. Preventing duplication can be ensured by a thorough review of the literature (but note that reproducing original model results can advance your knowledge of how a model functions and lead to new insights in the system). Often, we are working from established theory that provides an existing framework that can be applied to different systems. Adapting these frameworks can help advance your own research while also saving precious time. When digging through articles, bear in mind that most modelling frameworks are not system-specific. Do not be discouraged if you cannot immediately find a model in your field, as the perfect model for your question may have been applied in a different system or be published only as a conceptual model. These models are still useful! Also, do not be shy about reaching out to authors of models that you think may be applicable to your system. Finally, remember that you can be critical of what you find, as some models can be deceptively simple or involve assumptions that you are not comfortable making. You should not reinvent the wheel, but you can always strive to build a better one.

Rule 5: Study and apply good coding practices

The modelling process will inevitably require some degree of programming, and this can quickly become a challenge for some biologists. However, learning to program in languages commonly adopted by the scientific community (e.g., R, Python) can increase the transparency, accessibility, and reproducibility of your models. Even if you only wish to adopt preprogrammed models, you will likely still need to create code of your own that reads in data, applies functions from a collection of packages to analyze the data, and creates some visual output. Programming can be highly rewarding—you are creating something after all—but it can also be one of the most frustrating parts of your research. What follows are 3 suggestions to avoid some of the frustration.

Organization is key, both in your workflow and your written code. Take advantage of existing software and tools that facilitate keeping things organized. For example, computational notebooks like Jupyter notebooks or R-Markdown documents allow you to combine text, commands, and outputs in an easily readable and shareable format. Version control software like Git makes it simple to both keep track of changes as well as to safely explore different model variants via branches without worrying that the original model has been altered. Additionally, integrating with hosting services such as Github allows you to keep your changes safely stored in the cloud. For more details on learning to program, creating reproducible research, programming with Jupyter notebooks, and using Git and Github, see the 10 simple rules by Carey and Papin [ 13 ], Sandve and colleagues [ 14 ], Rule and colleagues [ 15 ], and Perez-Riverol and colleagues [ 16 ], respectively.

Comment your code and comment it well (see Fig 3 ). These comments can be the pseudocode you have written on paper prior to coding. Assume that when you revisit your code weeks, months, or years later, you will have forgotten most of what you did and why you did it. Good commenting can also help others read and use your code, making it a critical part of increasing scientific transparency. It is always good practice to write your comments before you write the code, explaining what the code should do. When coding a function, include a description of its inputs and outputs. We also encourage you to publish your commented model code in repositories such that they are easily accessible to others—not only to get useful feedback for yourself but to provide the modelling foundation for others to build on.

Two functionally identical codes in R [ 17 ] can look very different without comments (left) and with descriptive comments (right). Writing detailed comments will help you and others understand, adapt, and use your code.

https://doi.org/10.1371/journal.pcbi.1008539.g003

When writing long code, test portions of it separately. If you are writing code that will require a lot of processing power or memory to run, use a simple example first, both to estimate how long the project will take, and to avoid waiting 12 hours to see if it works. Additionally, when writing code, try to avoid too many packages and “tricks” as it can make your code more difficult to understand. Do not be afraid of writing 2 separate functions if it will make your code more intuitive. As with writing, your skill as a writer is not dependent on your ability to use big words, but instead about making sure your reader understands what you are trying to communicate.

Rule 6: Sweat the “right” small stuff

By “sweat the ‘right’ small stuff,” we mean considering the details and assumptions that can potentially make or break a mathematical model. A good start would be to ensure your model follows the rules of mass and energy conservation. In a closed system, mass and energy cannot be created nor destroyed, and thus, the left side of the mathematical equation must equal the right under all circumstances. For example, in Eq 2 , if the number of susceptible individuals decreases due to infection, we must include a negative term in this equation (− cγIS ) to indicate that loss and its conjugate (+ cγIS ) to the infected individuals equation, Eq 3 , to represent that gain. Similarly, units of all terms must also be balanced on both sides of the equation. For example, if we wish to add or subtract 2 values, we must ensure their units are equivalent (e.g., cannot add day −1 and year −1 ). Simple oversights in units can lead to major setbacks and create bizarre dynamics, so it is worth taking the time to ensure the units match up.

Modellers should also consider the fundamental boundary conditions of each parameter to determine if there are some values that are illogical. Logical constraints and boundaries can be developed for each parameter using prior knowledge and assumptions (e.g., Huntley [ 18 ]). For example, when considering an SIR model, there are 2 parameters that comprise the transmission rate—the contact rate, c , and the probability of infection given contact, γ . Using our intuition, we can establish some basic rules: (1) the contact rate cannot be negative; (2) the number of susceptible, infected, and recovered individuals cannot be below 0; and (3) the probability of infection given contact must fall between 0 and 1. Keeping these in mind as you test your model’s dynamics can alert you to problems in your model’s structure. Finally, simulating your model is an excellent method to obtain more reasonable bounds for inputs and parameters and ensure behavior is as expected. See Otto and Day [ 5 ] for more information on the “basic ingredients” of model-building.

Rule 7: Simulate, simulate, simulate

Even though there is a lot to be learned from analyzing simple models and their general solutions, modelling a complex world sometimes requires complex equations. Unfortunately, the cost of this complexity is often the loss of general solutions [ 19 ]. Instead, many biologists must calculate a numerical solution, an approximate solution, and simulate the dynamics of these models [ 20 ]. Simulations allow us to explore model behavior, given different structures, initial conditions, and parameters ( Fig 4 ). Importantly, they allow us to understand the dynamics of complex systems that may otherwise not be ethical, feasible, or economically viable to explore in natural systems [ 21 ].

Gold lines represent the SIR structure ( Fig 2A ) where lifelong immunity of individuals is inferred after infection, and blue lines represent an SIRS structure ( Fig 2B ) where immunity is lost over time. The solid lines represent model dynamics assuming a recovery rate ( θ ) of 0.05, while dotted lines represent dynamics assuming a recovery rate of 0.1. All model runs assume a transmission rate, cγ , of 0.2 and an immunity loss rate, ψ , of 0.01. By using simulations, we can explore how different processes and rates change the system’s dynamics and furthermore determine at what point in time these differences are detectable. SIR, Susceptible-Infected-Recovered; SIRS, Susceptible-Infected-Recovered-Susceptible.

https://doi.org/10.1371/journal.pcbi.1008539.g004

One common method of exploring the dynamics of complex systems is through sensitivity analysis (SA). We can use this simulation-based technique to ascertain how changes in parameters and initial conditions will influence the behavior of a system. For example, if simulated model outputs remain relatively similar despite large changes in a parameter value, we can expect the natural system represented by that model to be robust to similar perturbations. If instead, simulations are very sensitive to parameter values, we can expect the natural system to be sensitive to its variation. Here in Fig 4 , we can see that both SIR models are very sensitive to the recovery rate parameter ( θ ) suggesting that the natural system would also be sensitive to individuals’ recovery rates. We can therefore use SA to help inform which parameters are most important and to determine which are distinguishable (i.e., identifiable). Additionally, if observed system data are available, we can use SA to help us establish what are the reasonable boundaries for our initial conditions and parameters. When adopting SA, we can either vary parameters or initial conditions one at a time (local sensitivity) or preferably, vary multiple of them in tandem (global sensitivity). We recognize this topic may be overwhelming to those new to modelling so we recommend reading Marino and colleagues [ 22 ] and Saltelli and colleagues [ 23 ] for details on implementing different SA methods.

Simulations are also a useful tool for testing how accurately different model fitting approaches (e.g., Maximum Likelihood Estimation versus Bayesian Estimation) can recover parameters. Given that we know the parameter values for simulated model outputs (i.e., simulated data), we can properly evaluate the fitting procedures of methods when used on that simulated data. If your fitting approach cannot even recover simulated data with known parameters, it is highly unlikely your procedure will be successful given real, noisy data. If a procedure performs well under these conditions, try refitting your model to simulated data that more closely resembles your own dataset (i.e., imperfect data). If you know that there was limited sampling and/or imprecise tools used to collect your data, consider adding noise, reducing sample sizes, and adding temporal and spatial gaps to see if the fitting procedure continues to return reasonably correct estimates. Remember, even if your fitting procedures continue to perform well given these additional complexities, issues may still arise when fitting to empirical data. Models are approximations and consequently their simulations are imperfect representations of your measured outcome of interest. However, by evaluating procedures on perfectly known imperfect data, we are one step closer to having a fitting procedure that works for us even when it seems like our data are against us.

Rule 8: Expect model fitting to be a lengthy, arduous but creative task

Model fitting requires an understanding of both the assumptions and limitations of your model, as well as the specifics of the data to be used in the fitting. The latter can be challenging, particularly if you did not collect the data yourself, as there may be additional uncertainties regarding the sampling procedure, or the variables being measured. For example, the incidence data commonly adopted to fit SIR models often contain biases related to underreporting, selective reporting, and reporting delays [ 24 ]. Taking the time to understand the nuances of the data is critical to prevent mismatches between the model and the data. In a bad case, a mismatch may lead to a poor-fitting model. In the worst case, a model may appear well-fit, but will lead to incorrect inferences and predictions.

Model fitting, like all aspects of modelling, is easier with the appropriate set of skills (see Rule 2). In particular, being proficient at constructing and analyzing mathematical models does not mean you are prepared to fit them. Fitting models typically requires additional in-depth statistical knowledge related to the characteristics of probability distributions, deriving statistical moments, and selecting appropriate optimization procedures. Luckily, a substantial portion of this knowledge can be gleaned from textbooks and methods-based research articles. These resources can range from covering basic model fitting, such as determining an appropriate distribution for your data and constructing a likelihood for that distribution (e.g., Hilborn and Mangel [ 7 ]), to more advanced topics, such as accounting for uncertainties in parameters, inputs, and structures during model fitting (e.g., Dietze [ 25 ]). We find these sources among others (e.g., Hobbs and Hooten [ 26 ] for Bayesian methods; e.g., Adams and colleagues [ 27 ] for fitting noisy and sparse datasets; e.g., Sirén and colleagues [ 28 ] for fitting individual-based models; and Williams and Kendall [ 29 ] for multiobject optimization—to name a few) are not only useful when starting to fit your first models, but are also useful when switching from one technique or model to another.

After you have learned about your data and brushed up on your statistical knowledge, you may still run into issues when model fitting. If you are like us, you will have incomplete data, small sample sizes, and strange data idiosyncrasies that do not seem to be replicated anywhere else. At this point, we suggest you be explorative in the resources you use and accept that you may have to combine multiple techniques and/or data sources before it is feasible to achieve an adequate model fit (see Rosenbaum and colleagues [ 30 ] for parameter estimation with multiple datasets). Evaluating the strength of different techniques can be aided by using simulated data to test these techniques, while SA can be used to identify insensitive parameters which can often be ignored in the fitting process (see Rule 7).

Model accuracy is an important metric but “good” models are also precise (i.e., reliable). During model fitting, to make models more reliable, the uncertainties in their inputs, drivers, parameters, and structures, arising due to natural variability (i.e., aleatory uncertainty) or imperfect knowledge (i.e., epistemic uncertainty), should be identified, accounted for, and reduced where feasible [ 31 ]. Accounting for uncertainty may entail measurements of uncertainties being propagated through a model (a simple example being a confidence interval), while reducing uncertainty may require building new models or acquiring additional data that minimize the prioritized uncertainties (see Dietze [ 25 ] and Tsigkinopoulou and colleagues [ 32 ] for a more thorough review on the topic). Just remember that although the steps outlined in this rule may take a while to complete, when you do achieve a well-fitted reliable model, it is truly something to be celebrated.

Rule 9: Give yourself time (and then add more)

Experienced modellers know that it often takes considerable time to build a model and that even more time may be required when fitting to real data. However, the pervasive caricature of modelling as being “a few lines of code here and there” or “a couple of equations” can lead graduate students to hold unrealistic expectations of how long finishing a model may take (or when to consider a model “finished”). Given the multiple considerations that go into selecting and implementing models (see previous rules), it should be unsurprising that the modelling process may take weeks, months, or even years. Remembering that a published model is the final product of long and hard work may help reduce some of your time-based anxieties. In reality, the finished product is just the tip of the iceberg and often unseen is the set of failed or alternative models providing its foundation. Note that taking time early on to establish what is “good enough” given your objective, and to instill good modelling practices, such as developing multiple models, simulating your models, and creating well-documented code, can save you considerable time and stress.

Rule 10: Care about the process, not just the endpoint

As a graduate student, hours of labor coupled with relative inexperience may lead to an unwillingness to change to a new model later down the line. But being married to one model can restrict its efficacy, or worse, lead to incorrect conclusions. Early planning may mitigate some modelling problems, but many issues will only become apparent as time goes on. For example, perhaps model parameters cannot be estimated as you previously thought, or assumptions made during model formulation have since proven false. Modelling is a dynamic process, and some steps will need to be revisited many times as you correct, refine, and improve your model. It is also important to bear in mind that the process of model-building is worth the effort. The process of translating biological dynamics into mathematical equations typically forces us to question our assumptions, while a misspecified model often leads to novel insights. While we may wish we had the option to skip to a final finished product, in the words of Drake, “sometimes it’s the journey that teaches you a lot about your destination”.

There is no such thing as a failed model. With every new error message or wonky output, we learn something useful about modelling (mostly begrudgingly) and, if we are lucky, perhaps also about the study system. It is easy to cave in to the ever-present pressure to perform, but as graduate students, we are still learning. Luckily, you are likely surrounded by other graduate students, often facing similar challenges who can be an invaluable resource for learning and support. Finally, remember that it does not matter if this was your first or your 100th mathematical model, challenges will always present themselves. However, with practice and determination, you will become more skilled at overcoming them, allowing you to grow and take on even greater challenges.

Acknowledgments

We thank Marie-Josée Fortin, Martin Krkošek, Péter K. Molnár, Shawn Leroux, Carina Rauen Firkowski, Cole Brookson, Gracie F.Z. Wild, Cedric B. Hunter, and Philip E. Bourne for their helpful input on the manuscript.

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• Published: 31 October 2023

The sub-dimensions of metacognition and their influence on modeling competency

• Riyan Hidayat 1 , 2 ,
• Hermandra 3 &
• Sharon Tie Ding Ying 4  

Humanities and Social Sciences Communications volume  10 , Article number:  763 ( 2023 ) Cite this article

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Mathematical modeling is indeed a versatile skill that goes beyond solving real-world problems. Numerous studies show that many students struggle with the intricacies of mathematical modeling and find it a challenging and complex task. One important factor related to mathematical modeling is metacognition which can significantly impact expert and student success in a modeling task. However, a notable gap of research has been identified specifically in relation to the influence of metacognition in mathematical modeling. The study’s main goal was to assess whether the different sub-dimensions of metacognition can predict the sub-constructs of a student’s modeling competence: horizontal and vertical mathematization. The study used a correlational research design and involved 538 participants who were university students studying mathematics education in Riau Province, Indonesia. We employed structural equation modeling (SEM) using AMOS version 18.0 to evaluate the proposed model. The measurement model used to assess metacognition and modeling ability showed a satisfactory fit to the data. The study found that the direct influence of awareness on horizontal mathematization was insignificant. However, the use of cognitive strategies, planning, and self-checking had a significant positive effect on horizontal mathematization. Concerning vertical mathematization, the direct effect of cognitive strategy, planning, and awareness was insignificant, but self-checking was positively related to this type of mathematization. The results suggest that metacognition, i.e., awareness and control over a person’s thinking processes, plays an important role in modeling proficiency. The research implies valuable insights into metacognitive processes in mathematical modeling, which could inform teaching approaches and strategies for improving mathematical modeling. Further studies can build on these findings to deepen our understanding of how cognitive strategies, planning, self-assessment, and awareness influence mathematical modeling in both horizontal and vertical contexts.

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Changing curriculum content and instructional styles in teaching and learning processes for regular mathematics classes is critical to promote more meaningful engagement with mathematics (Schoenfeld, 2016 ). A shift to searching for solutions, exploring patterns, and formulating conjectures rather than simply memorizing procedures and formulas or completing exercises can lead to deeper understanding and more versatile problem-solving skills. Incorporating mathematical modeling into classroom activities by engaging students in authentic problem-solving within complex systems and interdisciplinary contexts can help develop the competencies to tackle increasingly complex problems. Mathematical modeling can strengthen problem-solving skills and connect mathematics to real-world situations, making it relevant to students’ current and future lives (Hidayat and Wardat, 2023 ). The importance of mathematical modeling is further underscored by its inclusion as a primary component in the mathematics assessment of the Program for International Student Assessment (PISA) (Niss, 2015 ). Students can tackle non-routine real-life challenges by engaging in modeling activities and working collaboratively on realistic and authentic mathematical tasks. However, traditional instructional methods for assessing student modeling proficiency are inadequate. This information underscores the need for improved methods of evaluation that capture the full range of students’ modeling abilities and the development of their problem-solving skills. Educators should consider incorporating alternative assessment methods such as project-based assessments, performance tasks, or reflective journals to better assess student modeling skills. In addition, professional development opportunities for teachers to learn effective strategies for integrating mathematical modeling into their instruction can contribute to more successful implementation and assessment of these skills.

Mathematical modeling is a multifaceted skill beyond solving real-world problems (Mohd Saad et al., 2023 ; Niss et al., 2007 ). As Minarni and Napitupulu ( 2020 ) point out, students can apply modeling abilities to describe context problems mathematically, organize tools, discover relationships, transfer between real-world and mathematical problems, and visualize problems in various ways. In modeling real-world problems, students activate other competencies, such as representing mathematical objects, arguing, and justifying (National Council of Teachers of Mathematics, 1989 ). Engaging in mathematical modeling in the classroom helps students clarify and interpret phenomena, solve problems, and develop social competencies necessary for effective teamwork and collaborative knowledge building. Mathematical modeling instruction aims to improve students’ mathematical knowledge, promote critical and creative thinking, and foster positive attitudes toward mathematics (Blum, 2002 ). Cognitive modeling combined with task orientation is more effective in increasing the likelihood of success. In high school curricula, students can connect mathematical modeling to different courses, reinforcing the importance of this skill in different contexts (Hernández et al., 2016 ). Integrating mathematical modeling into different subject areas can help students develop a comprehensive understanding of the relevance and applicability of mathematics in real-world situations, ultimately leading to better problem-solving abilities and an appreciation for the power of mathematical thinking.

Numerous studies have shown that mathematical modeling is challenging for many students (Anhalt et al., 2018 ; Corum and Garofalo, 2019 ; Czocher, 2017 ; Kannadass et al., 2023 ). Metacognitive competencies improve students’ modeling abilities (Galbraith, 2017 ; Vorhölter, 2019 ; Wendt et al., 2020 ). Metacognition, the ability to reflect on and regulate one’s thinking, can significantly impact expert and student success in problem-solving (Schoenfeld, 1983 , 2007 ). Productive metacognitive behaviors can help students better understand the given problem, search for and distinguish relevant and irrelevant information, and focus on the overall structure of the problem (Kramarski et al., 2002 ). These behaviors can lead to improved understanding and problem-solving abilities. Although the benefits of metacognition to learning are widely recognized, there is limited research on the specific types of metacognitive strategies that are most effective in helping students (Wilson and Clarke, 2004 ). Future research should focus on identifying these strategies and understanding how they can best be used in educational settings to improve students’ mathematical modeling and problem-solving abilities. This research could include exploring the most effective methods for teaching metacognitive skills, examining how metacognition can be tailored to individual student needs, and examining the impact of metacognitive interventions on student modeling performance. Thus, this study aimed to investigate how the sub-dimensions of metacognition can predict modeling performance. The study questions are as follows: (a) Do the sub-constructs of metacognition (awareness, cognitive strategy, planning, and self-checking) predict horizontal mathematization? (b) Do the sub-constructs of metacognition (awareness, cognitive strategy, planning, and self-checking) predict vertical mathematization?

Theoretical perspective

Models and modeling perspective (mmp).

The term ‘model’ is a collection of elements, connections between elements, and actions that describe or explain how the elements interact (English, 2007 ; Lesh and Doerr, 2003 ). Modeling exercises allow students to reveal their multiple forms of reasoning, create conceptual frameworks, and develop effective ways to represent the structural features of the topic (Carreira and Baioa, 2018 ). Models and Modeling Perspective (MMP), also known as contextual modeling (Kaiser and Sriraman, 2006 ), is considered a method to understand real-life situations and develop formal mathematical knowledge based on students’ understanding (Csapó and Funke, 2017 ; Lesh and Doerr, 2003 ). Students must move from a real-world situation to a mathematical world using their previously learned mathematical concepts as a modeling tool that goes beyond calculational prescriptions (Sevinc, 2022 ) and learning theories (Abassian et al., 2019 ). Moreover, MMP considers the mathematical model as a conceptual tool of a mathematical system that emerges from a specific real-world situation (Lesh and Lehrer, 2003 ). In brief, MMP is a new concept that incorporates real-world context into the teaching and learning of mathematical problem-solving because MMP prepares students to be mentally active in modeling. An important feature of MMP is the recognition that problem-solving typically involves numerous modeling cycles in which descriptions, explanations, and predictions are continuously refined. In contrast, solutions are modified or discarded depending on their interpretation of the world.

Students will use their internal conceptual systems to organize, understand, and make connections between events, experiences, or issues (Erbas et al., 2014 ) to adapt to MMP. Student learning through the use of MMP will also facilitate communication between peers and teachers through project-based learning or problem-based learning (Ärlebäck, 2017 ) as they practice solving authentic problem situations by engaging in mathematical thinking that involves interpreting situations, describing and explaining, computing through procedures, and deductive reasoning (English et al., 2008 ). MMP summarizes a cycle of activities that, in the first step, requires students to understand the real-world situation, followed by structuring the situation model, mathematizing to develop a mathematical model, and collaborating mathematical models to develop results that are considered and validated within the real-world situation, and finally presenting a solution to a real-world situation.

Mathematical modeling and mathematization

Modeling is also known as organizing representative descriptions in which symbolic representations and formal model structures develop (Hidayat et al., 2018 ; Niss, 2015 ). According to the South African Department of Basic Education (2011), mathematical modeling is an important curriculum focus, and real-world situations should be included in all areas, such as economics, health, social services, and others. Mathematical modeling is a process of mathematization or mathematization in which students can discover relevant issues or assumptions in a given real-world scenario by mathematizing, interpreting, and evaluating solutions to resulting mathematical problems related to the given circumstance (Leong and Tan, 2020 ). The mathematization method can be applied as a series of activities directed toward the activity system object, with the goal of the modeling project serving as the activity object itself (Araújo and Lima, 2020 ). Students with mathematical skills can acquire mathematical knowledge through logical reasoning using problem-solving. Formal mathematical information is obtained during the mathematization process by referring to informal knowledge, including components of actual problem situations (Freudenthal, 2002 ). Mathematical modeling can be divided into many tasks: simplifying, mathematizing, computing, interpreting, and validating. When students are proficient in the modeling process, they can independently and insightfully perform all components of a mathematical modeling process (Hankeln et al., 2019 ), with the focus of the competencies being on identifying specific fundamental capabilities.

A mathematical model is created using mathematization (Yilmaz and Dede, 2016 ). The concept of mathematization involves using mathematical methods to organize and examine various aspects of reality. The idea of the mathematization of actual reality is formulated in two forms of mathematization (Treffers, 1978 ; Treffers and Goffree, 1985 ), namely horizontal and vertical mathematization. Horizontal and vertical mathematization are complementary processes in mathematical modeling and problem-solving (Freudenthal, 1991 ). The process of horizontal mathematization begins with understanding the problem and extends to problem-solving (Galbraith, 2017 ). Horizontal mathematization involves translating real-world problems into mathematical representations, while vertical mathematics involves working within mathematics to solve the problem. Both processes are important for students to develop a comprehensive understanding of mathematics and its applications in real-world situations. Horizontal mathematization refers to translating a real-world problem into a mathematical problem or representation. Students identify relevant mathematical structures, concepts, and relationships related to the given problem in this phase. They may simplify the problem by making assumptions, recognizing patterns, or constructing a model. Horizontal mathematization aims to create a mathematical representation that captures the essence of the real-world situation and can be analyzed using mathematical tools. Simplification is about understanding the core problem and using mathematics to construct a model based on reality (Kaiser and Schwarz, 2006 ). Students must be able to clarify the essential elements of the situation, formulate the problem, and create a simplified version that can be analyzed mathematically. A further step is to identify relevant mathematical concepts, variables, and relationships that capture the essence of the real situation (mathematization). Students must be able to translate the problem into mathematical language using appropriate notations or visual representations (Kaiser and Stender, 2013 ). This study defines horizontal mathematization as simplifying assumptions, clarifying the objective, formulating the problem, assigning variables, establishing parameters and constants, formulating mathematical expressions, and selecting a model (Yilmaz and Dede, 2016 ).

Vertical mathematization occurs after the problem has been translated into a mathematical representation through horizontal mathematization. In this phase, students work within the domain of mathematics to solve the problem by using mathematical techniques, calculations, proofs, or manipulations. Vertical mathematization is about delving deeper into mathematical concepts, exploring connections, and gaining new insights. The focus here is on applying mathematical knowledge and reasoning to find a solution to the problem. Vertical mathematization refers to exploring the realm of formal symbols (Selter and Walter, 2019 ). Vertical mathematization also refers to the mathematical processing and improvement of real-world problems transformed into mathematics (Treffers and Goffree, 1985 ). Learners apply their mathematical knowledge or intuitive procedures to solve the problem within the framework of the mathematical model (Maaß, 2006 ). This model may involve calculations, manipulations, or proof to derive a mathematical solution. Once a mathematical solution is found, students must interpret the results in the context of the original problem (Garfunkel and Montgomery, 2016 ). To do this interpretation, they must understand the relationship between the mathematical solution and the real-world situation and place the solution in terms of the problem’s context. The final step is to review the solution for accuracy and critically evaluate the assumptions made, the model used, and the overall process (Kaiser and Stender, 2013 ). Students must determine if their solution is reasonable and sensible and if improvements or changes can be made to the model or assumptions. This paper defines vertical mathematization as interpreting, validating, and relating the result to a real-world context.

Metacognition

Metacognition encompasses two aspects: the capacity to recognize and understand one’s cognitive processes (referred to as metacognitive knowledge) and the ability to manage and adapt these processes (known as metacognitive control) (Fleur et al., 2021 ). This study must consider metacognition because modeling issues are typically worked on in small groups (Biccard and Wessels, 2011 ). Metacognition includes students’ understanding of their cognitive processes and their capability to regulate and manipulate them (Kwarikunda et al., 2022 ). Metacognition is the knowledge or cognitive activity that targets or controls any component of a cognitive effort (Flavell, 1979 ); for example, students use metacognition to solve issues while studying. Students must manage their cognitive processes during learning so that their learning achievement be measured afterward (Bedel, 2012 ). Metacognition is often divided into two parts: metacognitive knowledge and techniques, which are often complemented by an affective-motivational aspect (Efklides, 2008 ; Veenman et al., 2006 ). Planning cognitive activities, monitoring progress toward goals, selecting methods to solve difficulties, and reflecting on past performance to improve future outcomes are all examples of metacognitive techniques (Kim and Lim, 2019 ). Furthermore, O’Neil and Abedi ( 1996 ) operationalize students’ metacognitive inventory as a construct that includes planning, self-checking, cognitive strategy, and awareness. Metacognition is understanding how individuals gain information and manage the process (Schraw and Dennison, 1994 ).

Metacognitive abilities have a significant impact on student learning and performance. They enable students to identify areas of difficulty and select appropriate learning strategies to understand new concepts. Metacognition has been found to improve students’ problem-solving abilities (García et al., 2016 ). However, metacognitive skills differ among students with varying levels of modeling competence, with some putting little effort into organizing or expressing knowledge differences (García et al., 2016 ). Students with high levels of modeling competence tend to pay more attention to time management, which may contribute to their success in problem-solving tasks. Interestingly, metacognitive training is particularly beneficial for lower-performing students because it allows them to improve while working on the same tasks as their peers (Karaali, 2015 ). This finding suggests that metacognitive instruction can help level the playing field for students with different abilities and allow all learners to develop their problem-solving skills more effectively. In summary, metacognition is critical in mathematics and affects students’ abilities differently. Educators should integrate metacognitive training into their instructional practices to support all learners and help them develop self-awareness, reflection, and regulation skills to benefit their mathematical problem-solving efforts.

Relationship between metacognition and modeling competency

Metacognition can help with goal-oriented modeling and overcoming various challenges (Stillman, 2004 ), depending on students’ knowledge and experience. The success of metacognitive activity can be attributed to students’ responses to specific problem-solving scenarios that can activate metacognition (Vorhölter, 2021 ). Metacognition is an essential method associated with mathematical proficiency and problem-solving skills. Teachers can help students develop appropriate individual techniques for dealing with modeling challenges and various metacognitive activities, such as mathematizing across different circumstances and environments (Blum, 2011 ). Mathematizing is a horizontally sequential process of translating parts of the real world into the language of symbols and abstracting in a vertical direction (Freudenthal, 2002 ). The mathematization process is horizontal mathematization because it requires the learner to transform real life into mathematical symbols. Horizontal mathematization leads to results based on different problem-solving strategies and the concrete problem case (Gravemeijer, 2008 ). The process of horizontal mathematization focuses primarily on organizing, schematizing, and constructing a model of reality so that it can be treated mathematically (Piñero Charlo, 2020 ). Horizontal mathematization is highlighted as a learning difficulty in an instructional strategy where teachers do not recognize horizontal mathematization as a learning problem (Yvain-Prébiski and Chesnais, 2019 ), and students also have difficulty discovering connections and transferring real-world problems to known mathematical models. Changing models, merging and defining a connection in a formula, and improving and integrating models are challenges of vertical mathematization (Suaebah et al., 2020 ). Real-world modeling activities that promote the horizontal mathematization process can help students experience mathematics as a value by strengthening their understanding and tangible connection between mathematics and the effort expended, i.e., by improving their metacognition skills (Suh et al., 2017 ).

Awareness of metacognition is critical in developing and improving students’ problem-solving skills. Studies have shown a significant positive correlation between metacognition awareness and problem-solving abilities (Sevgi and Karakaya, 2020 ). Effective mathematical problem-solving is also associated with planning and revision techniques (García et al., 2019 ). Students can improve their problem-solving skills through self-reflection on planning, monitoring, and evaluating their thinking processes (Herawaty et al., 2018 ). This finding highlights the link between metacognition and modeling abilities such as awareness, self-checking, planning, and cognitive strategy. By using planning techniques, students can improve their problem-solving abilities, for example, through verbalization (Zhang et al., 2019 ). Although the transfer of metacognitive knowledge to mathematical modeling is modest, using planning and revision procedures still contributes positively to student success. The sub-dimension of monitoring can predict a student’s engagement in a discussion (Akman and Alagöz, 2018 ). Using cognitive strategies during the formulation phase of the modeling process provides a sense of guidance (Krüger et al., 2020 ). Awareness of metacognition and using metacognitive strategies such as planning, monitoring, and revising are essential to improve students’ problem-solving and mathematical modeling abilities. Educators should aim to incorporate metacognitive strategies into their teaching methods to support the development of these skills in students.

Metacognition has been recognized as critical for solving complicated tasks, such as modeling tasks (Wilson and Clarke, 2004 ). Individuals can cultivate a more methodical and comprehensive approach to horizontal mathematization by integrating the sub-constructs of metacognition (awareness, planning, self-checking, and cognitive strategies). For example, horizontal mathematization is enhanced by providing students with useful tools and tactics for planning, analyzing, and solving modeling tasks through awareness, planning, self-checking, and cognitive strategies. Students can recognize mathematical patterns and structures within a modeling task when they know the relevance and use of mathematics in everyday situations. Creating a plan allows students to break difficult tasks into manageable parts. Students can be disciplined and avoid errors or omissions by setting goals, outlining necessary mathematical operations, and choosing a sequence of tasks. Cognitive techniques enable effective information processing, allow students to connect different mathematical ideas, and promote creative thinking when solving modeling tasks. Finally, self-checking promotes error detection and correction, leading to a better understanding of mathematical ideas. At the same time, the sub-constructs of metacognition (awareness, planning, self-checking, and cognitive strategies) would help enhance vertical mathematization skills. For example, students can identify the relevant mathematical relationships and structures needed to build a mathematical model by improving their awareness. To fulfill this aim, they must recognize the mathematical concepts and principles that apply to the current real-world problem. Again, the objectives are set in the planning phase, variables and parameters are selected, and the mathematical operations and transformations are described. The problem is analyzed using cognitive techniques, and the mathematical solution is found through reasoning, pattern recognition, and visualization. Finally, self-validation assures that the mathematical model is accurate and reliable. Students can locate any errors or inconsistencies and correct them by examining and checking the model frequently.

The hypotheses of the research are as follows:

Significant relationships will occur between awareness and horizontal mathematization.

Significant relationships will occur between cognitive strategy and horizontal mathematization.

Significant relationships will occur between planning and horizontal mathematization.

Significant relationships will occur between self-checking and horizontal mathematization.

Significant relationships will occur between awareness and vertical mathematization.

Significant relationships will occur between cognitive strategy and vertical mathematization.

Significant relationships will occur between planning and vertical mathematization.

Significant relationships will occur between self-checking and vertical mathematization.

Methodology

Participants and design.

This study used a correlational research design (Creswell, 2012 ; Shanmugam and Hidayat, 2022 ), which explores the level of interrelation between metacognition and mathematical modeling using structural equation modeling (SEM). The current study sample consisted of college students studying mathematics education in Riau Province, Indonesia, with similar modeling experiences. These students were prospective mathematics teachers who were prepared to teach mathematics at the secondary level. First-year (133 or 24.7%), second-year (223 or 41.4%), and third-year (182 or 33.8%) students participated in the study, with a total of 538 samples. The fourth-year study samples were not included due to practical exercises. All participants were selected using cluster random sampling from universities with similar characteristics such as location and modeling experience. We used this type of sampling because this research focused on groups rather than individuals, which resulted in students coming from selected universities to take the test. Although the current research found that the percentage of gender resulted in more female (483 or 89.8%) than male (55 or 10.2%) samples, we did not use gender as a moderator or covariate for analyzing the data. The Department of Investment and Integrated One Stop Services, Indonesia, approved the study. Subsequently, all selected samples received written informed consent. We explained the study’s objectives and the voluntary nature of participation before the test was administered. All students from the selected universities took 60 min to complete the metacognitive inventory instrument and the mathematical modeling test.

To measure mathematical modeling competence, we developed and used the Modeling Test (Haines and Crouch, 2001 ), which we divided into two sub-constructs: horizontal and vertical mathematization. The items were assessed by multiple-choice questions with a three-level scoring (0=wrong answer, 1=partially correct answer, and 2=true answer). The modeling test had 22 questions and a final score of 44. Moreover, the test is also suitable for this study because the study included a large sample (Lingefjärd and Holmquist, 2005 ). Figure 1 shows one of the examples of measuring horizontal mathematization.

The examples of horizontal mathematization test.

Reliability scores for modeling competence followed the sub-construct: horizontal mathematization (18 items, α  = 0.861) and vertical mathematization (4 items, α  = 0.740). These overall reliability values were acceptable ( α  > 0.70) (Tavakol and Dennick, 2011 ). The internal consistency of the mathematical modeling test was good, with composite reliability values (CR) ranging from 0.775 to 0.925 (> 0.6). The value of the Average Variance Extracted (AVE) ranged from 0.500 to 0.501 ( > 0.5), indicating good discriminant validity. At the same time, the square roots of all AVE values were larger than the associations suggested among them or to the left of them, which underlined the discriminant validity of the mathematical modeling test. All these values were consistent with the recommendations of researchers (Fornell and Larcker, 1981 ; Hair et al., 2010 ; Nunnally and Bernstein, 1994 ), which were satisfactory.

The metacognitive inventory (O’Neil and Abed, 1996 ) was adopted for measuring metacognition, which comprised four sub-scales: awareness (5 items), cognitive strategy (5 items), planning (5 items), and self-checking (5 items). The example of the item for each sub-contract provided (awareness; I am always aware of my thoughts in modeling task ), (cognitive strategy; I am trying to find the main idea in the modeling task ), (planning; I am trying to understand the purpose of the modeling task before attempting to solve it ) and (self-checking; If I notice any mistakes while working on the modeling task, I always correct them ). Reliability scores for metacognition followed the sub-constructs of awareness ( α  = 0.825), cognitive strategy ( α  = 0.853), planning ( α  = 0.842), and self-checking ( α  = 0.828). These overall reliability values were acceptable ( α  > 0.70) (Tavakol and Dennick, 2011 ). The internal consistency of the metacognitive inventory was high, with composite reliability values (CR) ranging from 0.775 to 0.925 (>0.6). The value of the Average Variance Extracted (AVE) ranged from 0.500 to 0.526 (>0.5), indicating good discriminant validity. The square roots of all AVE values were higher than the associations suggested among them or to the left of them, underlining the discriminant validity of the metacognition scale. These values were consistent with what researchers proposed (Fornell and Larcker, 1981 ; Hair et al., 2010 ; Nunnally and Bernstein, 1994 ), which were satisfactory.

Strategy of data analyses

In the first analysis, we used descriptive statistics for all sub-constructs with missing data, outliers (boxplots), means, standard deviations, skewness, and kurtosis. At the same time, the relationships between latent variables were calculated using Pearson correlations to determine multicollinearity. According to Kline ( 2005 ), the relationship between the latent variables should be less than 0.900 for the observed variables to be free from multicollinearity. For the cut-off value of univariate normality, we used skewness (±2.0) (Tabachnick and Fidell, 2013 ) and kurtosis (±8.0) (Kline, 2005 ) in this paper. Then, SEM (AMOS version 18.0) was used to evaluate the hypothesized model. First, we calculated a measurement model (Confirmatory Factor Analyzes—CFA) for each variable to test whether or not the dimensional structures of the instruments could be confirmed for the sample in the present study. For the construct of metacognition, we assessed awareness models, cognitive strategy, planning, and self-checking sequentially. The following measurement model assessed two-dimensional modeling competence (horizontal and vertical mathematization). Next, we set up the hypothetical model to test the effect of the sub-dimensions of metacognition on mathematical modeling (horizontal and vertical mathematization). Model fit was assessed using the standardized root mean residual (SRMR) (<0.080), chi-square values ( P  > 0.05), comparative fit index (CFI) (>0.950), Tucker-Lewis index (TLI) (>0.950), the root mean square error of approximation (RMSEA) (<0.080) (Bandalos and Finney, 2018 ; Dash and Paul, 2021 ), and the goodness-of-fit index (>0.900) (Dash and Paul, 2021 ). SRMR was determined by taking the average of the residuals from the comparison of the observed and implied matrices (Bandalos and Finney, 2018 ). The chi-square test assessed the discrepancy between the observed sample data and the covariance matrices within the model. CFI and TLI compare the goodness of fit of a model to that of a null or independent model. Finally, to assess the discriminant validity, reliability, and convergent validity of the measures, we used the composite reliability (CR) (>0.60), Cronbach’s alpha values (0.60–0.70), and average variance extracted (AVE) (>0.50).

Descriptive results

Table 1 shows the descriptive results and correlation matrix for the sub-construct of metacognition (awareness, cognitive strategy, planning, and self-checking) and the sub-construct of modeling competency (horizontal and vertical mathematization).

As indicated in Table 1 , the highest relationship was between awareness and cognitive strategy ( r  = 0.677), while horizontal and vertical mathematization ( r  = 0.342) were the lowest correlated. Again, the students’ awareness, cognitive strategy, planning, and self-checking were moderate ( M  = 3.940, M  = 3.737, M  = 3.951, M  = 3.910, respectively). The skewness score ranged between −0.658 and −0.124 ( ± 2.0), while the kurtosis values ranged between 0.087 and 2.343 ( ± 8.0). The outputs indicated that no values exceeded the cut-off score for all of the four sub-constructs (Kline, 2005 ; Tabachnick and Fidell, 2013 ), which was normally distributed. At the same time, the students’ horizontal and vertical mathematization were also moderate ( M  = 0.914, M  = 0.848, respectively). The skewness score ranged between 0.095 and 0.195 ( ± 2.0), while the kurtosis scores ranged between −0.670 and 0.032 ( ± 8.0). The results showed that no scores exceeded the cut-off score for the two sub-constructs (Kline, 2005 ; Tabachnick and Fidell, 2013 ), which was normally distributed.

Measurement models

The measurement model was employed to confirm that observed variables reflected unobserved variables before evaluating the hypothetical structural model. We employed CFA to measure the fitness of the latent variables of metacognition (20 indicators) and mathematical modeling competency (22 indicators). The outputs of maximum likelihood estimation revealed that the measurement model of metacognition for the four sub-constructs indicated an acceptable match; χ 2  = 325.454, χ 2 / df  = 1.984, RMSEA = 0.043, SRMR = 0.036, CFI = 0.965, GFI = 0.955, TLI = 0.959 (Table 2 ). Moreover, the measurement model of mathematical modeling competency also revealed that two sub-constructs indicated an adequate fit of the model to the data; χ 2  = 261.077, χ 2 / df  = 1.305, RMSEA = 0.024, SRMR = 0.041, CFI = 0.975, GFI = 0.958, TLI = 0.971. Despite the significance of the chi-square result, χ²/ df , RMSEA, SRMR, CFI, GFI, and TLI recommended that the a priori model had an adequate factor structure.

Factor loading and coefficient of SEM regression are shown in Table 3 . All factor loadings from sub-constructs of horizontal mathematization (around 0.617–0.837), vertical mathematization (from 0.660 to 0.703), awareness (around 0.662–0.738), cognitive strategy (from 0.770 to 0.758), planning (around 0.660–0.757) and self-checking (from 0.662 to 0.760), were significant. Each item within every sub-construct exhibited statistically significant factor loadings ( P  < 0.001), affirming the correlation among items for each sub-construct. The standardized estimate for factor loading indicated that all items had factor loadings greater than 0.50, which surpassed the desired criteria (Hair et al., 2010 ).

Testing the hypothesized models

Similar to the examining measurement model, some cut-off scores were also applied for each measurement to evaluate the hypothesized model; χ 2 / df  < 5.00, RMSEA < 0.080, SRMR < 0.080, CFI > 0.950, GFI > 0.900, TLI > 0.950. The results of SEM indicated a highly satisfactory fit to data, χ 2  = 1163.570, χ 2 / df  = 1.460, RMSEA = 0.029, SRMR = 0.043, CFI = 0.950, GFI = 0.908, TLI = 0.950 (see Fig. 2 ). The hypothesized model shown in Fig. 2 was the final structural model that indicated the relationship between the sub-construct of metacognition and mathematical modeling competency. The parameter estimates for whole structural paths in the hypothesized model were statistically significant.

The final model.

Next, Table 4 shows detailed statistics on the final model (e.g., standardized estimate, unstandardized estimate, standard errors, CR, and P value).

As appeared in Table 4 , the direct path coefficient was significant: (a) cognitive strategy → horizontal mathematization [ β  = 0.26, P  < 0.05, t  = 2.535], (b) planning → horizontal mathematization [ β  = 0.23, P  < 0.05, t  = 2.369], (c) self-checking → horizontal mathematization [ β  = 0.23, P  < 0.05, t  = 2.470]. The hypothesis was fully accepted. Students who used cognitive strategy, planning, and self-checking accomplished well in horizontal mathematization. Conversely, the direct path coefficient of awareness to horizontal mathematization was insignificant [ β  = 0.17, P  > 0.05, t  = 1.685]. Thus, the hypothesis was not fully supported. It implied that awareness alone might not strongly predict success in horizontal mathematization. At the same time, the direct path coefficient was not significant: (a) cognitive strategy → vertical mathematization [ β  = 0.24, P  > 0.05, t  = 1.763], (b) planning → vertical mathematization [ β  = 0.15, P  > 0.05, t  = 1.180], (c) awareness → vertical mathematization [ β  = 0.08, P  > 0.05, t  = 0.635]. It showed that awareness, cognitive strategy, and planning alone may not strongly predict success in vertical mathematization. The direct path coefficient of self-checking → vertical mathematization was significant [ β  = 0.27, P  < 0.05, t  = 2.138]. Students who used self-checking accomplished well in vertical mathematization. In conclusion, cognitive strategy (26%), planning (23%), and self-checking (23%) accounted for a variance for horizontal mathematization; at the same time, self-checking (27%) accounted for a variance for vertical mathematization.

Integrating mathematical modeling across subject areas can give students a more meaningful and context-rich understanding of mathematics. Numerous studies have shown that many students find mathematical modeling difficult and complex (Anhalt et al., 2018 ; Corum and Garofalo, 2019 ; Czocher, 2017 ). For example, some students have difficulty translating real-world problems into mathematical terms, while others have difficulty finding appropriate mathematical models to represent complex systems and phenomena. This study aimed to examine whether the different sub-dimensions of metacognition could be used to predict a student’s level of competency in modeling.

We found no significant or positive relationship between awareness and horizontal or vertical mathematization. Despite numerous studies that do not support the finding of a significant and positive relationship between these variables (Kreibich et al., 2022 ; Sevgi and Karakaya, 2020 ; Toraman et al., 2020 ), previous research has primarily focused on metacognitive awareness rather than the sub-domain of awareness within metacognition. Indeed, much of the research in mathematics education has focused on problem-solving and not specifically on the context of mathematical modeling. This focus on problem-solving has led to valuable insights into how students learn, think, and apply mathematical concepts. However, certain aspects of mathematical modeling may have been less explored or understood in the process. One possible explanation could be insufficient mathematical knowledge in mathematical modeling. Leong ( 2014 ) indicated that incorporating mathematical modeling into the curriculum may face challenges, including teacher readiness, time constraints, and educator dispositions. The extent of a student’s mathematical understanding can influence the connection between awareness and horizontal or vertical mathematization. Students who do not have the requisite mathematical foundations may have difficulty making connections or applying problem-solving techniques, regardless of their level of awareness. For example, increased awareness can help students identify relevant information, recognize patterns and relationships, develop appropriate assumptions, select mathematical tools, and reflect on their modeling process.

Our results show a positive and significant correlation between cognitive strategy and horizontal mathematization; however, no significant relationship was found between cognitive strategy and vertical mathematization. This result confirms previous research in this area (Hidayat et al., 2020 , 2022 ; Krüger et al., 2020 ). This observation can be attributed to the complexity of the tasks. Horizontal mathematization involves translating real-world problems into mathematical representations, whereas vertical mathematization involves working within the domain of mathematics to solve problems. Cognitive strategies, such as organizing information, recognizing patterns, and selecting appropriate tools, may be more applicable to horizontal mathematization. This result is consistent with Krüger et al.‘s ( 2020 ) view that using cognitive strategies provides direction in the formulation phase of the modeling process. Conversely, in vertical mathematization, tasks may be more complex or abstract and require higher mathematical knowledge or skills. Vertical mathematization involves going deeper into the mathematical domain, working with more abstract concepts, and using advanced problem-solving techniques. Cognitive strategies typically focus on organizing, planning, and selecting tools that may not be as influential in this more abstract and complex domain. Consequently, cognitive strategies alone may not be sufficient to influence vertical mathematization. Another possible explanation is that students’ different cognitive styles may lead to different approaches to mathematization processes. Students with different cognitive styles may lead different approaches to mathematization processes (Mariani and Hendikawati, 2017 ).

This research’s results indicate a significant and positive relationship between planning and horizontal mathematization, but no significant correlation was found between planning and vertical mathematization. This result is consistent with previous research (García et al., 2019 ; Herawaty et al., 2018 ; Zhang et al., 2019 ). In a horizontal mathematization context, verbalization can potentially explain this observation. Zhang et al. ( 2019 ) indicated that students can improve their problem-solving skills through planning strategies such as verbalization. Verbalization, i.e., talking about the problem and their thought processes, can also help students clarify their thinking and identify possible errors or inconsistencies in their reasoning. By breaking down complex problems into smaller, more manageable steps, students can more easily understand the problem and develop an action plan for solving it. In horizontal mathematization, students must be able to analyze the problem, identify the most important variables and relationships, and develop a plan to solve the problem using mathematical concepts and procedures. However, the sub-domain of planning is not used effectively in vertical mathematization. Vertical mathematization requires students to engage in a more analytical and abstract form of thinking, which can be more challenging than the more concrete and tangible aspects of horizontal mathematics. In addition, vertical mathematization often involves multiple mathematical concepts and procedures, making it more challenging to plan a clear and effective problem-solving strategy. Students may rely on trial-and-error methods or intuitive problem-solving approaches rather than explicit planning.

Our study shows a significant positive correlation between self-checking and horizontal and vertical mathematization. This result is consistent with previous studies conducted on this topic, such as those by Akman and Alagöz ( 2018 ), García et al. ( 2019 ), and Herawaty et al. ( 2018 ). This consistency of results between studies highlights the importance of self-checking or monitoring in mathematical modeling. One possible explanation for this consistent finding is that self-checking is beneficial for students to identify errors, ensure accuracy, and build confidence in their mathematical abilities. Using self-checking techniques, students monitor their understanding and advancement as they work through the problem. This monitoring can help them identify errors or misunderstandings early on and correct their thought processes or methods accordingly. Self-checking can also help students stay organized and focused as they solve the problem, reducing the chance of making mistakes or overlooking important details. For example, modelers correctly identified the relevant variables and relationships in the problem. Similarly, monitoring strategies can improve vertical mathematization by helping students stay organized and focused, reflecting on their problem-solving approaches, and interpreting the outcomes of their solutions. For example, monitoring or self-checking can help students interpret the results of their problem-solving efforts in the context of the original problem. By reflecting on the meaning of the solution and its relation to the real world, students can develop a deeper understanding of mathematical concepts and their applications. In addition, monitoring can help students stay organized and focused as they work through a problem, reducing the likelihood of making mistakes or missing important details. Research has shown that the sub-dimension of monitoring can predict student engagement in classroom discussions (Akman and Alagöz, 2018 ).

Mathematical modeling involves applying mathematical concepts and techniques to real-world situations and requires students to think critically, creatively, and systematically about problems. Students need opportunities to engage in various tasks that require applying their mathematical knowledge to real-world situations and sufficient time to gain experience and develop their skills. Metacognition plays an important role in mathematical modeling by helping students become more aware of their thinking processes, monitor their understanding, and decide when to seek help or additional support. According to this research, awareness alone did not significantly impact horizontal mathematization. However, using cognitive techniques, making intelligent plans, and self-checking significantly improved horizontal mathematization. To improve learners’ horizontal mathematics skills, it is important to motivate them to use proper cognitive methods, acquire efficient planning techniques, and develop the habit of self-checking. In addition, the results pave the way for further research on the exact cognitive strategies, planning methods, and self-checking procedures that support effective horizontal mathematization. By analyzing how these variables interact and influence student performance, insights can be gained into instructional strategies and interventions that support successful mathematical modeling. Finally, these discoveries improve our understanding of the intricate connection between metacognition and mathematical modeling. Awareness may not directly affect horizontal mathematization, but cognitive techniques, planning, and self-checking are critical. The unique processes and techniques associated with different types of mathematical modeling must also be considered, as demonstrated by the differential effects on vertical mathematization. These findings extend our theoretical understanding of the relationship between mastery of mathematical modeling, metacognitive processes, and specific cognitive skills.

Limitations and suggestions

It is common for research studies to have limitations, and the current study is no exception. Acknowledging and considering the study’s limitations in future research is essential. Firstly, some hypotheses are fully supported by the research findings, while others are not. It is possible that other factors, such as students’ prior mathematical knowledge and experience, their motivation and engagement in mathematical modeling, and the quality of instruction, play a more important role in promoting horizontal and vertical mathematization. Further research is needed to fully understand the complex interplay of factors contributing to horizontal and vertical mathematization and to identify effective strategies for promoting mathematization in students. Secondly, although the current study found correlations among variables, it is important to note that correlational studies cannot prove causality. Future research may therefore benefit from using experimental designs or other methods to establish causal relationships among variables. These methods may involve interventions or manipulations designed to directly change the independent variable and observe its effects on the dependent variable. Such methods allow researchers to understand the causal relationships between variables better and draw more meaningful conclusions about the effects of various factors on the outcome of interest. Finally, a potential limitation of the current study is that it relied on self-reported measures of variables that could be susceptible to bias or error. Future research could benefit from using objective measurements or multiple data sources to increase the validity of the results. Objective measurements may include direct observation or physiological measurements, providing more accurate and reliable data. In addition, using multiple data sources can contribute to a more comprehensive understanding of the phenomenon under study, as different data sources may capture different aspects of the measured construct. Using such methods, researchers can increase the validity and reliability of their findings and draw more meaningful conclusions about the relationship between different variables.

Data availability

All relevant data can be found in the manuscript and its accompanying supplementary files.

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Hidayat, R., Hermandra & Ying, S.T.D. The sub-dimensions of metacognition and their influence on modeling competency. Humanit Soc Sci Commun 10 , 763 (2023). https://doi.org/10.1057/s41599-023-02290-w

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• PMCID: PMC6491984
• DOI: 10.1002/jrsm.1333

Mathematical modeling studies are increasingly recognised as an important tool for evidence synthesis and to inform clinical and public health decision-making, particularly when data from systematic reviews of primary studies do not adequately answer a research question. However, systematic reviewers and guideline developers may struggle with using the results of modeling studies, because, at least in part, of the lack of a common understanding of concepts and terminology between evidence synthesis experts and mathematical modellers. The use of a common terminology for modeling studies across different clinical and epidemiological research fields that span infectious and non-communicable diseases will help systematic reviewers and guideline developers with the understanding, characterisation, comparison, and use of mathematical modeling studies. This glossary explains key terms used in mathematical modeling studies that are particularly salient to evidence synthesis and knowledge translation in clinical medicine and public health.

Keywords: evidence synthesis; glossary; guidelines; knowledge translation; mathematical modeling studies.

© 2018 The Authors. Research Synthesis Methods Published by John Wiley & Sons Ltd.

• Calibration
• Computer Simulation
• Decision Making
• Evidence-Based Medicine*
• Extensively Drug-Resistant Tuberculosis / prevention & control
• Extensively Drug-Resistant Tuberculosis / therapy
• Guidelines as Topic*
• Markov Chains
• Models, Statistical
• Models, Theoretical*
• Monte Carlo Method
• Public Health
• Research Design / standards*
• Stochastic Processes
• Translational Research, Biomedical
• World Health Organization

Grants and funding

• 001/WHO_/World Health Organization/International
• JA Simpson's research is supported by two NHMRC Centres of Research Excellence (Victoria Centre for Biostatistics, ViCBiostat; and Policy relevant infectious disease simulation and mathematical modelling, PRISM).
• Special Programme for Research and Training in Tropical Diseases (TDR)
• 1104975/JA Simpson is funded by a NHMRC Senior Research Fellowship

Mathematical Modeling Doctor of Philosophy (Ph.D.) Degree

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The mathematical modeling Ph.D. enables you to develop mathematical models to investigate, analyze, predict, and solve the behaviors of a range of fields from medicine, engineering, and business to physics and science.

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Overview for Mathematical Modeling Ph.D.

Mathematical modeling is the process of developing mathematical descriptions, or models, of real-world systems. These models can be linear or nonlinear, discrete or continuous, deterministic or stochastic, and static or dynamic, and they enable investigating, analyzing, and predicting the behavior of systems in a wide variety of fields. Through extensive study and research, graduates of the mathematical modeling Ph.D. will have the expertise not only to use the tools of mathematical modeling in various application settings, but also to contribute in creative and innovative ways to the solution of complex interdisciplinary problems and to communicate effectively with domain experts in various fields.

Plan of Study

The degree requires at least 60 credit hours of course work and research. The curriculum consists of three required core courses, three required concentration foundation courses, a course in scientific computing and high-performance computing (HPC), three elective courses focused on the student’s chosen research concentration, and a doctoral dissertation. Elective courses are available from within the School of Mathematics and Statistics as well as from other graduate programs at RIT, which can provide application-specific courses of interest for particular research projects. A minimum of 30 credits hours of course work is required. In addition to courses, at least 30 credit hours of research, including the Graduate Research Seminar, and an interdisciplinary internship outside of RIT are required.

Students develop a plan of study in consultation with an application domain advisory committee. This committee consists of the program director, one of the concentration leads, and an expert from an application domain related to the student’s research interest. The committee ensures that all students have a roadmap for completing their degree based on their background and research interests. The plan of study may be revised as needed. Learn more about our mathematical modeling doctoral students and view a selection of mathematical modeling seminars hosted by the department.

Qualifying Examinations

All students must pass two qualifying examinations to determine whether they have sufficient knowledge of modeling principles, mathematics, and computational methods to conduct doctoral research. Students must pass the examinations in order to continue in the Ph.D. program.

The first exam is based on the Numerical Analysis I (MATH-602) and Mathematical Modeling I, II (MATH-622, 722). The second exam is based on the student's concentration foundation courses and additional material deemed appropriate by the committee and consists of a short research project.

Dissertation Research Advisor and Committee

A dissertation research advisor is selected from the program faculty based on the student's research interests, faculty research interest, and discussions with the program director. Once a student has chosen a dissertation advisor, the student, in consultation with the advisor, forms a dissertation committee consisting of at least four members, including the dissertation advisor. The committee includes the dissertation advisor, one other member of the mathematical modeling program faculty, and an external chair appointed by the dean of graduate education. The external chair must be a tenured member of the RIT faculty who is not a current member of the mathematical modeling program faculty. The fourth committee member must not be a member of the RIT faculty and may be a professional affiliated with industry or with another institution; the program director must approve this committee member.

The main duties of the dissertation committee are administering both the candidacy exam and final dissertation defense. In addition, the dissertation committee assists students in planning and conducting their dissertation research and provides guidance during the writing of the dissertation.

Admission to Candidacy

When a student has developed an in-depth understanding of their dissertation research topic, the dissertation committee administers an examination to determine if the student will be admitted to candidacy for the doctoral degree. The purpose of the examination is to ensure that the student has the necessary background knowledge, command of the problem, and intellectual maturity to carry out the specific doctoral-level research project. The examination may include a review of the literature, preliminary research results, and proposed research directions for the completed dissertation. Requirements for the candidacy exam include both a written dissertation proposal and the presentation of an oral defense of the proposal. This examination must be completed at least one year before the student can graduate.

Dissertation Defense and Final Examination

The dissertation defense and final examination may be scheduled after the dissertation has been written and distributed to the dissertation committee and the committee has consented to administer the final examination. Copies of the dissertation must be distributed to all members of the dissertation committee at least four weeks prior to the final examination. The dissertation defense consists of an oral presentation of the dissertation research, which is open to the public. This public presentation must be scheduled and publicly advertised at least four weeks prior to the examination. After the presentation, questions will be fielded from the attending audience and the final examination, which consists of a private questioning of the candidate by the dissertation committee, will ensue. After the questioning, the dissertation committee immediately deliberates and thereafter notifies the candidate and the mathematical modeling graduate director of the result of the examination.

All students in the program must spend at least two consecutive semesters (summer excluded) as resident full-time students to be eligible to receive the doctoral degree.

Maximum Time Limitations

University policy requires that doctoral programs be completed within seven years of the date of the student passing the qualifying exam. All candidates must maintain continuous enrollment during the research phase of the program. Such enrollment is not limited by the maximum number of research credits that apply to the degree.

National Labs Career Fair

Hosted by RIT’s Office of Career Services and Cooperative Education, the National Labs Career Fair is an annual event that brings representatives to campus from the United States’ federally funded research and development labs. These national labs focus on scientific discovery, clean energy development, national security, technology advancements, and more. Students are invited to attend the career fair to network with lab professionals, learn about opportunities, and interview for co-ops, internships, research positions, and full-time employment.

Students are also interested in: Applied and Computational Mathematics MS

The College of Science consistently receives research grant awards from organizations that include the National Science Foundation , National Institutes of Health , and NASA , which provide you with unique opportunities to conduct cutting-edge research with our faculty members.

Faculty in the School of Mathematics and Statistics conducts research on a broad variety of topics including:

• applied inverse problems and optimization
• applied statistics and data analytics
• biomedical mathematics
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• dynamical systems and fluid dynamics
• geometry, relativity, and gravitation
• mathematics of earth and environment systems
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Learn more by exploring the school’s mathematics research areas .

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Curriculum for 2023-2024 for Mathematical Modeling Ph.D.

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Mathematical Modeling, Ph.D. degree, typical course sequence

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Full-time study is 9+ semester credit hours. International students requiring a visa to study at the RIT Rochester campus must study full‑time.

Application Details

To be considered for admission to the Mathematical Modeling Ph.D. program, candidates must fulfill the following requirements:

• Complete an online graduate application .
• Submit copies of official transcript(s) (in English) of all previously completed undergraduate and graduate course work, including any transfer credit earned.
• Hold a baccalaureate degree (or US equivalent) from an accredited university or college.
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• Submit a current resume or curriculum vitae.
• Submit a statement of purpose for research which will allow the Admissions Committee to learn the most about you as a prospective researcher.
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English Language Test Scores

International applicants whose native language is not English must submit one of the following official English language test scores. Some international applicants may be considered for an English test requirement waiver .

International students below the minimum requirement may be considered for conditional admission. Each program requires balanced sub-scores when determining an applicant’s need for additional English language courses.

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Cost and Financial Aid

An RIT graduate degree is an investment with lifelong returns. Ph.D. students typically receive full tuition and an RIT Graduate Assistantship that will consist of a research assistantship (stipend) or a teaching assistantship (salary).

Additional Information

Foundation courses.

Mathematical modeling encompasses a wide variety of scientific disciplines, and candidates from diverse backgrounds are encouraged to apply. If applicants have not taken the expected foundational course work, the program director may require the student to successfully complete foundational courses prior to matriculating into the Ph.D. program. Typical foundation course work includes calculus through multivariable and vector calculus, differential equations, linear algebra, probability and statistics, one course in computer programming, and at least one course in real analysis, numerical analysis, or upper-level discrete mathematics.

Development in Mathematical Modeling

• First Online: 02 June 2021

Cite this chapter

• Corey Brady 5 &
• Richard Lesh 6  

Part of the book series: Early Mathematics Learning and Development ((EMLD))

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The chapters in this section represent noteworthy steps in a research agenda with implications far beyond traditional conceptions of models and modeling, addressing key questions such as: How does the K-12 mathematics curriculum need to adapt to prepare students for the rapidly changing nature of “mathematical thinking” outside of school? What does it mean to “understand” the most important “big ideas” in elementary (K-16) mathematics? How do these ideas and understandings develop? How can these developments be documented and assessed, in their earliest manifestations? How can assessments of students’ most important conceptual achievements be based on operational definitions that do not simply reduce them to checklists of factual and procedural knowledge? Our goal here is to briefly describe how these chapters are situated within this larger context.

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Brady, C., Lesh, R. (2021). Development in Mathematical Modeling. In: Suh, J.M., Wickstrom, M.H., English, L.D. (eds) Exploring Mathematical Modeling with Young Learners. Early Mathematics Learning and Development. Springer, Cham. https://doi.org/10.1007/978-3-030-63900-6_5

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• Advances in Educational Technology and Psychology
• Vol 7, Issue 15, 2023

Research on Integrating Mathematical Modeling Thinking into Large, Medium and Small School Teaching

DOI: 10.23977/aetp.2023.071507 | Downloads: 31 | Views: 360

Zhiqiang Hu 1 , Xiaodong Zhao 1 , Zhongjin Guo 1 , Xiaoqian Li 1 , Shan Jiang 1

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1 School of Mathematics and Statistics, Taishan University, Tai'an, Shandong, 271000, China

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This study explores how mathematical modeling thinking can be integrated into large, medium, and small school teaching to enhance students' mathematical abilities and interdisciplinary thinking. Using various research methods, including literature review, surveys, teaching experiments, statistical analysis, and expert interviews, we aim to establish a localized model for integrating mathematical modeling, thinking into large, medium, and small school teaching to optimize the quality of mathematics education. Research both in China and internationally has shown that mathematical modeling thinking has garnered significant attention in the field of education and holds promise as an effective approach to improving students' mathematical thinking skills and overall quality of education. The results of this research are expected to offer new insights and methods for mathematical education within the context of large, medium, and small school integration and provide scientifically sound assessment standards for education. This study is not only academically significant but also expected to support educational reform and practice in schools and educational institutions, contributing to the development of innovative and practical talents.

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Zhiqiang Hu, Xiaodong Zhao, Zhongjin Guo, Xiaoqian Li, Shan Jiang, Research on Integrating Mathematical Modeling Thinking into Large, Medium and Small School Teaching. Advances in Educational Technology and Psychology (2023) Vol. 7: 63-68. DOI: http://dx.doi.org/10.23977/aetp.2023.071507.

[1] Cai L., Li X., Han X. Exploring the Teaching of Mathematical Modeling Core Literacy Levels to Enhance Mathematical Thinking Quality. Teaching and Management, 2022, 901(36): 99-103. [2] Wang H., Wang J. Research on the Integration of Mathematical Thinking and Modeling in Comprehensive Education from Primary to High School. Modern Educational Technology, 2021, 10(2): 100-105. [3] Ma J. A Study of Strategies for Integrating Mathematical Thinking and Modeling in Comprehensive Education from Primary to High School. Mathematics Education, 2020, 32(7): 18-23. [4] Yu J. The Application of Mathematical Thinking and Modeling in STEM Education from Primary to High School. Educational Research and Experiment, 2020, 9(3): 36-40. [5] Liu Y., Chen W. Research on Strategies for Teaching Mathematical Thinking and Modeling in Comprehensive Education from Primary to High School. Journal of Mathematics Education, 2021, 40(2): 1-8. [6] Lesh R., Caylor B. Introduction to the special issue: Modeling as application versus modeling as a way to create mathematics. International Journal of computers for mathematical Learning, 2007, 12: 173-194. [7] Kertil M., Gurel C. Mathematical Modeling: A Bridge to STEM Education. International Journal of Education in mathematics, science and Technology, 2016, 4(1): 44-55. [8] Sturgill R. Mathematical modeling: Issues and challenges in mathematics education and teaching. Editorial Team, 2019, 11(5): 71. [9] Brady C., Lesh R. Development in mathematical modeling. Exploring Mathematical Modeling with Young Learners, 2021: 95-110. [10] Michelsen C. Mathematical modeling is also physics—interdisciplinary teaching between mathematics and physics in Danish upper secondary education. Physics Education, 2015, 50(4): 489. [11] Tan L., An K. A school-based professional development programme for teachers of mathematical modelling in Singapore. Journal of Mathematics Teacher Education, 2016, 19: 399-432. [12] Liu C., Wu C., Wong W. Scientific modeling with mobile devices in high school physics labs. Computers & Education, 2017, 105: 44-56. [13] Leung A. Exploring STEM pedagogy in the mathematics classroom: A tool-based experiment lesson on estimation. International Journal of Science and Mathematics Education, 2019, 17(7): 1339-1358. [14] Armutcu Y, Bal A P. The Effect of Mathematical Modeling Activities Based on STEM Approach on Mathematics Literacy of Middle School Students. International Journal of Educational Studies in Mathematics, 2022, 9(4): 233-253. [15] Yaman B. Preservice Mathematics Teachers' Achievement and Evaluation of Mathematical Modelling. Acta Didactica Napocensia, 2022, 15(2): 164-184.  [16] Zhang C., Liu X.. An Analysis on the Learning Rules of the Skip-Gram Model. 2019 International Joint Conference on Neural Networks, 2019, pp. 1-8. [17] Memon Christoph. Sound Field Optimization of Construction Machinery Cab Structure based on Ergonomics and Mathematical Modeling. Kinetic Mechanical Engineering (2021), Vol. 2, Issue 3: 20-29.

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Advanced Mathematical Modeling Research Using Machine Learning Gets Started at Idaho State University

April 25, 2024

Building accurate models relating to climate, machines, or the cosmos requires sophisticated mathematics. That's where folks like Idaho State University Professor Yury Gryazin and his students are stepping in.

Recently, Gryazin, an expert in numerical analysis and scientific computation, and his collaborators at the Berkeley Lab in California and Rice University in Texas started researching novel machine learning algorithms related to simulations and computer modeling. To support this effort, the U.S. Department of Energy awarded the team a $4 million grant under a newly established "Scientific Machine Learning for Complex Systems" program. Digital models are used across the STEM disciplines and simulate everything from Earth's weather and the movement of seismic waves to the operation of particle accelerators and nuclear reactors.

“Practically every new development is first modeled using advanced numerical approaches,” said Gryazin. “Recently, major advances in machine learning algorithms have opened new directions for developing faster and more accurate computational methods for such simulations.”

Gryazin and his students are focusing on the uncertainty quantification part of simulation problems. Typical uncertainty quantification problems of interest include certification, prediction, model and software verification and validation, parameter estimation, data assimilation, etc. Uncertainty quantification helps researchers determine how reliable the model's predictions are.

"You've probably seen a survey or poll with a margin of error of plus or minus a given percent," Gryazin said. "Uncertainty quantification is similar to that, except it is for models and simulations."

Because the development of many models requires data from sometimes millions—or more—data points, finding the margin of error for these scientific models is exponentially more complex than the average survey. Using advanced methods of applied mathematics and statistics, Gryazin and the students will create and train new neural network algorithms—a type of machine learning that is structured like a human brain—on already known data and will subsequently test their approach. During testing, the algorithm's success or failure will hinge on how well it can predict the correct results when given a new set of real-world data that was not present in the initial training set. Specifically, they'll work with algorithms used for subsurface imaging systems like those used to detect landmines, air pockets, pollutants, and more. Three students from multiple disciplines at Idaho State will be working on the project. 

“This new opportunity to collaborate with experts from esteemed institutions like Berkeley Lab offers an invaluable chance for learning,” said Adil Ahmed, a junior majoring in mechanical engineering. “Also, working alongside leading researchers in the field makes this project especially exciting to be a part of.”

The group hopes to publish its initial results by the end of 2024, when their first new subsurface neural network algorithms will be developed and tested.

“Students working on this project will gain invaluable experience solving important problems in exciting new areas of research in collaboration with world-class researchers from the country's top scientific centers,” Gryazin said. 

For more information on the Department of Mathematics and Statistics, visit isu.edu/math .

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The Role of Mathematical Modeling in Medical Research: “Research Without Patients?”

Computer controlled mathematical models of medical outcomes are commonly found in the current medical literature. What is less common is an understanding of the methods used to construct such models, leaving the consumers of medical research to accept the interpretations as presented. A basic knowledge of the concepts used to generate models will provide the clinician with the insight needed to critically evaluate medical literature based on mathematical models.

The development of computerized mathematical models used to simulate medical outcomes is a growing area of specialization (1–6) . A current MEDLINE search of articles using mathematical models yielded 43,764 articles dating from 1966. The majority (97%) of the manuscripts including mathematical models were published only since 1990. Since 1999, 9219 articles were published. That is 21% of the medical manuscripts using mathematical modeling over the last 35 years published in only this last year.

Clinicians and administrators are accepting the conclusions drawn from modeling, often without realizing the data are simulated. I have often been asked to comment on a journal article only to realize well into the critique that my clinical colleague did not know that the tables, charts, and figures were referring to computer generated cases. The surprise was best phrased by the question, “Do you mean that we can do research without patients?” The answer is, “Yes and no.”

The “yes” part of the answer hinges on the soundness of the methodologies employed. Regression methods, the most commonly seen in modeling, use some variation of the classical linear model, y=mx+b, according to a transformation or derivation that plots a math function closely describing the data ( 1 , 7 , 8 ). This is not new to biometrics, but the historical use is to compare two groups by the parameters of their lines from measured observations. Using the derived regression to predict outcomes in individuals from the same population has always been an accepted application of regression. Making the jump to using mathematical modeling to generate simulated patient populations, and even model their outcomes for therapies of the future, is a more difficult stretch.

The “no” part of the answer is rooted in the skepticism to believe something that was not only not seen and measured by the reader, but was also not seen or measured by the math modelers. Stopping here, however, can deprive the reader of the benefits of mathematical modeling. Some problems simply cannot be solved with a single math function or formula (8) . One solution is to repeat trial and error tests, possibly over many lifetimes. Another is to simulate the process in a computer model. The keys to the validity of modeling are the known dependent probabilities, associated variances, and coefficients determining the relative significance of each factor to the model ( 1 , 7 , 8 ). This means that a model must be based on sound research, with actual data that are widely accepted as valid by the medical science community.

Mathematical modeling is presented by various names like predictive modeling, simulation, or decision analysis. By far the most common methodology is the Markov Chain Monte Carlo simulation. The two parts of this method each have their own Mesh headings on MEDLINE, and together they have evolved into the acronym MCMC (pronounced “mac-mac”). Understanding the process of a MCMC simulation can go far in making one a better consumer of mathematical modeling because it contains the elements basic to modeling by any other name (8) .

Markov Chain

A Markov Chain, first used in the 1940s to model nuclear reactions, is simply a series of conditional probabilities in a fixed, dependent order (1) . Used by physicists for this limited application, this technique went unknown to the statistical community until the 1970s when it was generalized to any application for which one could not derive a single probability function (1) . The first practical applications appeared in the 1980s in the fields of neuroscience (1) and economics (7) .

The classical example used to teach Markov Chain theory is the random draw of one of two balls from a bag with replacement. We begin with unpainted balls. When an unpainted ball is drawn, a coin is flipped to decide to paint it red or black. The ball is painted and put back in the bag. When a red ball is drawn, it is painted black; when a black ball is drawn, it is painted red. Because the same individual ball can be drawn sequentially, it is not possible to derive a probability function to predict the probability of drawing a red ball from the bag at any given trial. Since there are only two balls, there are three possible probabilities of drawing a red ball at any one draw. One could be drawing from two black, two red, or one of each color at any given draw. The possibility at each draw depends upon the entire sequence of events, from the first draw to the draw in question. Every time the experiment is repeated, the n th trial can present a different possibility. There is the additional possibility of sequentially drawing the same individual ball in runs of varying length during the experiment. Convergence is achieved when the model looses perceivable dependence on the starting point (9) . The time prior to convergence is referred to as a “burn in” period. Runs of sequential draws of the same individual ball have more effect on the chain of events during the burn in period.

A computer simulation was written to illustrate this example (Table 1) . In the first Markov Chain, the probability of drawing a red ball on the draw 4 is 0.50 because after draw 3 there was one red and one black ball in the bag. In the third Markov Chain, the probability of drawing a red ball on the draw 4 is 1.00, and in the fifth Markov-chain the same request has a probability of 0.00.

The probability of drawing a red ball for draws 1-5, 10, 100, and 1000 for the first 10 individual simulations of the Markov Chain example. The values used to describe the possible outcomes in the text are in boldface

The problem of calculating event probabilities in this classical example is used because only modeling can solve it; a global math function or formula is not possible, although processes that can be solved by a global math function can also be modeled. Usually, to solve a math problem, one would prefer to solve the single function, but to test a process or the effect of sequential occurrences at the extremes of the known variances, it is often useful to develop a model. This is where regression methods are employed. While regression functions from least-squares methods are used, it is more common to see Bayesian methods. Bayesian-derived coefficients are encountered when outcomes are expressed as probabilities, such as logistic or probit regression (8) .

Monte Carlo

Monte Carlo simulation came into useful application in the same era as Markov-Chain processes ( 1 , 7 ). Monte Carlo simulation is a series of random draws, simulating an event within the known parameters of the probability distribution of the event ( 1 , 7 ). The name originated with early developers of the method who used a roulette wheel to generate random numbers. The wheel generated a gambling atmosphere as well, inspiring remembrance of the famous Monaco city, Monte Carlo (10) .

To illustrate the analytical synergy of these methods, and why the combination is more common than the individual parts alone, let's expand the random ball draw example into a Monte Carlo simulation. The same software that ran the individual experiments for the previous example was modified to loop 10,000 times and write the resulting probabilities of drawing a red ball on draws 1-5, 10, 100, and 1000. Since the three possible probabilities are 0.0, 0.5, and 1.0, we know that the average probability will migrate toward 0.5, and that the error term will reach equilibrium. We do not know, however, at which draws in the model the mean will reach 0.5 or the error term will stabilize.

As shown in Table 2 , the mean probability reached 0.5 by the tenth draw, and the standard deviation (used for the error term) stabilized at 0.36. Further analysis will be insightful because there are two extremes by which these parameters can be reached. First, most values can be very near the mean, minimizing the error term. Second, half of the values can equal the minimum value and half the maximum value. This gives an error term of maximum magnitude, or a “worst case scenario” error term. Comparing the frequencies of the possible probabilities between various depths into the chain can reveal when the model stabilizes. For this evaluation, two nonparametric analyses were run. A sign test, to pin down when the possible outcomes stabilized symmetrically around 0.50, and the Kolmogorov-Smirnov Test, a non-parametric analysis of variance (11) . Frequency table analysis using either the Pearson or Cochran X 2 tests is possible, and would yield confirming results, but the presentation is clearer with the chosen tests.

Simple descriptive statistics of the probability of drawing a red ball for draws 1-5, 10, 100, 1000 for the Monte Carlo example of n=10,000

As shown in Table 3 , frequencies of possible outcomes between draws 2-5 and every other draw are statistically dissimilar. Draws 10, 100, and 10,000 are statistically similar. That is, the model has reached convergence at the tenth draw in the chain, as proven by the stability of the model through the 1000 th draw in the simulation. Once convergence is achieved, the model is assumed to be stable and can be used for the intended purpose ( 8 , 9 ).

Statistical comparison of frequencies of possible outcomes between draws. P-values greater than 0.1 lead to the conclusion that the frequencies of possible outcomes are not different between these draws

Putting It All Together

A simple medical model will put all of the concepts together. This is a simplified example to illustrate the concepts; modeling a specific medical process or outcome requires the inclusion of many cofactors that make interpretation and presentation more difficult. First, a theoretical rare disease, D, is always fatal if untreated within a short time after onset (this allows us to skip time-dependent covariates). The first known treatment, A, has a success rate of 0.40 from a study of 200 subjects. Treatment B was recently tested against A in a trial of the same size that confirmed the success rate of A and established the success rate of B at 0.50. Figure 1 shows the outcomes of the two groups and the significant p-value of 0.04. Is this enough information to decide to make treatment B the treatment of choice? Remember, disease D is a rare event and these studies took years and big budgets to coordinate nationwide data collection. No additional outcome studies will be around any time soon. Furthermore, from only 200 subjects the success rates of 0.40 and 0.50 have 95% confidence intervals (0.33-0.47 and 0.43-0.57, respectively). The 95% confidence interval is the range in which we expect to find the success rates for 95 of the next 100 studies of the same size (11) . The observed overlap may lead one to suspect that null studies are certainly possible as well as studies reaching the opposite conclusion.

A comparison of hypothetical treatments, of sample size 200, for a hypothetical disease for the medical model example. The chi-square test ( X 2 ) =4.04 with 1 degree of freedom yields a p-value of 0.04. The risk ratio of treatment failure in treatment A relative to B is 1.2 with a 95% confidence interval of 1.03 to 1.39. The statistical inference of this single study would be that treatment B was superior to treatment A

The one clinical trial can be modeled into many trials and give estimates of the range of outcomes (which we suspect when we observe the confidence intervals) and the likelihood of reaching the same conclusion in successive clinical trials. The simulation can assign the outcome four different ways by two different divisions. The first division is the object of the simulation, which can be either individuals or populations. A model based on individuals is a more lifelike simulation and will yield more variation. Population-based models might be better suited for public health or community medicine applications. From our example, treatment group A can have a success rate of 0.40, but any patient in treatment group A cannot have a success rate of 0.40. Each patient's success rate must be either 0 or 1. The assignment of the 0 or 1 is the next division of strategy. If the confidence interval or other measure of variance is not known, each unit can have a random number between 0 and 1 generated and tested against the known rate (0.40 for group A in our example). If the random number for a unit is below 0.40, the unit is a success. If the random number is above 0.40, the unit is a failure. Since the error term is not known, this method is said to be a parameter estimation method and is called Gibbs sampling ( 8 , 12 ). (The name comes from the first use of this strategy in pixel imaging where the Gibbs probability distribution is used [8].) In assigning a binary outcome, this method yields a larger error term, but in applications where the error term is already at an extreme of small or large, this is not a concern. It is the only choice when the error term is unknown or a parameter such as sample size or a highly disputed denominator (as occurs in national databases or national surveillance) is encountered. In our example, the error terms are known, so, rather than compare the generated random number to 0.40, it can be compared to a rate randomly selected from the range of the confidence interval. In this most realistic simulation, we are comparing a randomly generated number between 0 and 1 to a randomly generated number between the lower and upper bounds of the 95% confidence interval to decide whether the case is a success or a failure. In the case of population-based simulation, the population's success rate is drawn from the range of the confidence interval.

The results of our MCMC simulation are summarized in Table 4 . Of 100 simulations of the known clinical trial, 91 observed that treatment B had a better success rate than treatment A. Of the simulations with observed outcomes confirming the clinical trial, only 50 (55%) had significant p-values (mean = 0.21, standard deviation = 0.29). Comparing the studies demonstrating the superiority of treatment B with studies refuting that finding for statistical significance with a Fisher's exact test gives a p-value of 0.0013 (Figure 2) . The interpretation is that a study refuting the superiority of treatment B is less likely to be statistically significant. The model gives us reason to not be so confident about the finding in the clinical trial. While I might personally prefer to be on treatment B if I had disease D, I can predict that if the clinical trial were repeated 100 times nine research centers out of the 100 would not want to switch from treatment A at all.

Status of statistical significance between simulations with confirming results to those refuting the superiority of treatment B. The p-value of the Fisher's exact tes t of this comparison is 0.0013. The statistical inference is that statistical significance is dependent on (more likely) confirming the superiority of B

Simple descriptive statistics of the medical model example comparing hypothetical treatments in a simulation of n=100 clinical trials

Modeling is increasingly common in the medical literature and is more often becoming the basis for managed care policy (3–6) . It is most commonly encountered for predicting outcomes, as in the simple example used in this paper (13–15) . Predicting the way an intervention or a technology might drive demand on a medical system is another variant of this use (16) . This is helpful when limitations like a rare event prohibit repeating actual studies or expanding research on actual patients.

A novel application for mathematical modeling is the determination of sample size requirement (17) . Estimates of the population parameters can direct a simulation that increases one patient at a time until a statistically significant difference is detected between the experimental groups. A series of such simulations can give investigators a range and midpoint of a sample size that should satisfy the test of their hypothesis. This is most useful in experimental designs with categorical, nonparametric, or otherwise non-normally distributed data. In some of these circumstances there are no functions to determine sample size, and where math functions are established the simulation method usually predicts a much more reasonable sample size requirement.

Another innovative use of MCMC is estimation of missing data points (18) . Most strategies to replace missing values use a point of central tendency like the mean or median. Such strategies usually have cutoff criteria for the minimum allowable proportion of missing fields to allow “filling in.” Usually more than 50% of the data for the case and the variable (throughout all cases) must be present. Such homogenous value replacement effectively reduces the variance. MCMC estimated values preserve the actual variance.

In all of the applications, math models in medicine are a reality to be faced. The impact of this, or any other, research methodology will only be to the betterment of patient care if the medical community at large undertakes a basic understanding, guiding the application of findings based on reasonable confidence after critical review. Modeling can answer otherwise unanswerable questions and greatly expand our knowledge base from actual study data. It can do all of this efficiently and risk-free, without real patients. However, all logical and mathematical components of a model must be based on valid research of actual patients.

Richard Chambers is a Biostatistician in the Outcomes Assessment Department of the Alton Ochsner Medical Foundation Division of Research and the Statistical Consultant for The Ochsner Journal

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Artificial intelligence can develop treatments to prevent 'superbugs'

Researchers used reinforcement learning to design antibiotic regimens to prevent treatment resistance.

Cleveland Clinic researchers developed an artficial intelligence (AI) model that can determine the best combination and timeline to use when prescribing drugs to treat a bacterial infection, based solely on how quickly the bacteria grow given certain perturbations. A team led by Jacob Scott, MD, PhD, and his lab in the Theory Division of Translational Hematology and Oncology, recently published their findings in PNAS .

Antibiotics are credited with increasing the average US lifespan by almost ten years. Treatment lowered fatality rates for health issues we now consider minor -- like some cuts and injuries. But antibiotics aren't working as well as they used to, in part because of widespread use.

"Health agencies worldwide agree that we're entering a post-antibiotic era," explains Dr. Scott. "If we don't change how we go after bacteria, more people will die from antibiotic-resistant infections than from cancer by 2050."

Bacteria replicate quickly, producing mutant offspring. Overusing antibiotics gives bacteria a chance to practice making mutations that resist treatment. Over time, the antibiotics kill all the susceptible bacteria, leaving behind only the stronger mutants that the antibiotics can't kill.

One strategy physicians are using to modernize the way we treat bacterial infections is antibiotic cycling. Healthcare providers rotate between different antibiotics over specific time periods. Changing between different drugs gives bacteria less time to evolve resistance to any one class of antibiotic. Cycling can even make bacteria more susceptible to other antibiotics.

"Drug cycling shows a lot of promise in effectively treating diseases," says study first author and medical student Davis Weaver, PhD. "The problem is that we don't know the best way to do it. Nothing's standardized between hospitals for which antibiotic to give, for how long and in what order."

Study co-author Jeff Maltas, PhD, a postdoctoral fellow at Cleveland Clinic, uses computer models to predict how a bacterium's resistance to one antibiotic will make it weaker to another. He teamed up with Dr. Weaver to see if data-driven models could predict drug cycling regimens that minimize antibiotic resistance and maximize antibiotic susceptibility, despite the random nature of how bacteria evolve.

Dr. Weaver led the charge to apply reinforcement learning to the drug cycling model, which teaches a computer to learn from its mistakes and successes to determine the best strategy to complete a task. This study is among the first to apply reinforcement learning to antibiotic cycling regiments, Drs. Weaver and Maltas say.

"Reinforcement learning is an ideal approach because you just need to know how quickly the bacteria are growing, which is relatively easy to determine," explains Dr. Weaver. "There's also room for human variations and errors. You don't need to measure the growth rates perfectly down to the exact millisecond every time."

The research team's AI was able to figure out the most efficient antibiotic cycling plans to treat multiple strains of E. coli and prevent drug resistance. The study shows that AI can support complex decision-making like calculating antibiotic treatment schedules, Dr. Maltas says.

Dr. Weaver explains that in addition to managing an individual patient's infection, the team's AI model can inform how hospitals treat infections across the board. He and his research team are also working to expand their work beyond bacterial infections into other deadly diseases.

"This idea isn't limited to bacteria, it can be applied to anything that can evolve treatment resistance," he says. "In the future we believe these types of AI can be used to to manage drug-resistant cancers, too."

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• Davis T. Weaver, Eshan S. King, Jeff Maltas, Jacob G. Scott. Reinforcement learning informs optimal treatment strategies to limit antibiotic resistance . Proceedings of the National Academy of Sciences , 2024; 121 (16) DOI: 10.1073/pnas.2303165121

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Production Technology for Deep Reservoirs

Complex fracture description and quasi-elastic energy development mathematical model for shale oil reservoirs Provisionally Accepted

• 1 Research Institute of Petroleum Exploration and Development (RIPED), China
• 2 Artificial Intelligence Technology R&D Center for Exploration and Development, CNPC, Beijing, China, China

The final, formatted version of the article will be published soon.

Reservoir numerical simulation is an important tool and method for the reasonable and efficient development of shale reservoirs. Accurate description of three-dimensional fractures in shale reservoir development is a necessary and sufficient condition to improve the accuracy and robustness of shale reservoir numerical simulation. This paper achieves precise characterization of complex fracture shapes and oil, gas and water flow by establishing an embedded discrete fracture model based on a non-structural network, which has advantages in the fine characterization of complex morphological fractures in the reservoir and the grid division of the reservoir. In the large matrix solution method, the Newton-Raphson method is used to linearize the nonlinear equations, the Jacobian matrix is constructed, the ILU method is used for preprocessing, the conjugate gradient method is used to solve the linear equations, and the shale oil quasi-elasticity is established A fully implicit solution method for mathematical models of energy development.shale oil, quasi-elastic energy development, complex fracture modeling, fully implicit solution, numerical simulation 1 Introduction China's continental shale formations are rich in huge shale oil resource potential, with diverse lithofacies, frequent phase changes, developed bedding fractures, and strong heterogeneity. In view of the characteristics of shale oil reservoirs with low permeability and low porosity, which are difficult to be exploited through conventional development methods, hydraulic fracturing is usually used to reform the reservoir first and then develop it. Therefore, for shale oil reservoirs, whether they are artificial fractures or natural fractures, the development of fractures has become a key technical issue. Naturally, in recent years, domestic and foreign scholars have focused their research on the laws and mechanisms of crack development and expansion (Meng, X. B., et al., 2024). Hou Bing et al. conducted an indoor true triaxial indoor fracturing physical simulation experiment and used concrete to wrap a full-diameter downhole core to test the initiation and vertical extension of hydraulic fractures in a

Keywords: Shale oil, quasi-elastic energy development, complex fracture modeling, Fully implicit solution, numerical simulation

Received: 26 Mar 2024; Accepted: 26 Apr 2024.

Copyright: © 2024 Wang, Lei and Jia. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY) . The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mx. Han Jia, Research Institute of Petroleum Exploration and Development (RIPED), Beijing, China

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