2.1: Models in Science and Engineering - Mathematics

Science is an endeavor to try to understand the world around us by discovering fundamental laws that describe how it works. Such laws include Newton’s law of motion, the ideal gas law, Ohm’s law in electrical circuits, the conservation law of energy, and so on, some of which you may have learned already. A typical cycle of scientiﬁc effort by which scientists discover these fundamental laws may look something like this:

1. Observe nature.
2. Develop a hypothesis that could explain your observations.
3. From your hypothesis, make some predictions that are testable through an experiment.
4. Carry out the experiment to see if your predictions are actually true.

• Yes→Your hypothesis is proven, congratulations. Uncork a champagne bottle and publish a paper.

• No→Your hypothesis was wrong, unfortunately. Go back to the labor the ﬁeld, get more data, and develop another hypothesis.

Many people think this is how science works. But there is at least one thing that is not quite right in the list above. What is it? Can you ﬁgure it out?

As some of you may know already, the problem exists in the last part, i.e., when the experiment produced a result that matched your predictions. Let’s do some logic to better understand what the problem really is. Assume that you observed a phenomenon P in nature and came up with a hypothesis H that can explain P. This means that a logical statement H → P is always true (because you chose H that way). To prove H, you also derived a prediction Q from H , i.e., another logical statement H → Q is always true, too. Then you conduct experiments to see if Q can be actually observed. What if Q is actually observed? Or, what if “not Q” is observed instead?

If “not Q” is observed, things are easy. Logically speaking, (H → Q) is equivalent to (not Q → not H) because they are contrapositions of each other, i.e., logically identical statements that can be converted from one to another by negating both the condition and the consequence and then ﬂipping their order. This means that, if not Q is true, then it logically proves that not H is also true, i.e., your hypothesis is wrong. This argument is clear, and there is no problem with it (aside from the fact that you will probably have to redo your hypothesis building and testing).

The real problem occurs when your experiment gives you the desired result, Q. Logically speaking, “(H → Q) and Qdoesn’t tell you anything about whether H is true or not! There are many ways your hypothesis could be wrong or insufﬁcient even if the predicted outcome was obtained in the experiment. For example, maybe another alternative hypothesis R could be the right one (R → P, R → Q), or maybe H would need an additional condition K to predict P and Q (H and K → P, H and K → Q) but you were not aware of the existence of K.

Let me give you a concrete example. One morning, you looked outside and found that your lawn was wet (observation P). You hypothesized that it must have rained while you were asleep (hypothesis H), which perfectly explains your observation (H → P). Then you predicted that, if it rained overnight, the driveway next door must also be wet (prediction Q that satisﬁes H → Q). You went out to look and, indeed, it was also wet (if not, H would be clearly wrong). Now, think about whether this new observation really proves your hypothesis that it rained overnight. If you think critically, you should be able to come up with other scenarios in which both your lawn and the driveway next door could be wet without having a rainy night. Maybe the humidity in the air was unusually high, so the condensation in the early morning made the ground wet everywhere. Or maybe a ﬁre hydrant by the street got hit by a car early that morning and it burst open, wetting the nearby area. There could be many other potential explanations for your observation.

In sum, obtaining supportive evidence from experiments doesn’t prove your hypothesis in a logical sense. It only means that you have failed to disprove your hypothesis. However, many people still believe that science can prove things in an absolute way. It can’t. There is no logical way for us to reach the ground truth of nature1.

This means that all the “laws of nature,” including those listed previously, are no more than well-tested hypotheses at best. Scientists have repeatedly failed to disprove them, so we give them more credibility than we do to other hypotheses. But there is absolutely no guarantee of their universal, permanent correctness. There is always room for other alternative theories to better explain nature.

In this sense, all science can do is just build models of nature. All of the laws of nature mentioned earlier are also models, not scientiﬁc facts, strictly speaking. This is something every single person working on scientiﬁc research should always keep in mind. I have used the word “model” many times already in this book without giving it a deﬁnition. So here is an informal deﬁnition:

A model is a simpliﬁed representation of a system. It can be conceptual, verbal, diagrammatic, physical, or formal (mathematical).

As a cognitive entity interacting with the external world, you are always creating a model of something in your mind. For example, at this very moment as you are reading this textbook, you are probably creating a model of what is written in this book. Modeling is a fundamental part of our daily cognition and decision making; it is not limited only to science.

With this understanding of models in mind, we can say that science is an endless effort to create models of nature, because, after all, modeling is the one and only rational approach to the unreachable reality. And similarly, engineering is an endless effort to control or inﬂuence nature to make something desirable happen, by creating and controlling its models. Therefore, modeling occupies the most essential part in any endeavor in science and engineering.

Exercise (PageIndex{1})

In the “wet lawn” scenario discussed above, come up with a few more alternative hypotheses that could explain both the wet lawn and the wet driveway without assuming that it rained. Then think of ways to ﬁnd out which hypothesis is most likely to be the real cause.

Exercise (PageIndex{2})

Name a couple of scientiﬁc models that are extensively used in today’s scientiﬁc/engineering ﬁelds. Then investigate the following:

• How were they developed?

• What made them more useful than earlier models?

• How could they possibly be wrong?

1This fact is deeply related to the impossibility of general system identiﬁcation, including the identiﬁcation of computational processes.

Successful K-12 STEM Education

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The final version of this book has not been published yet. You can pre-order a copy of the book and we will send it to you when it becomes available. We will not charge you for the book until it ships. Pricing for a pre-ordered book is estimated and subject to change. All backorders will be released at the final established price. As a courtesy, if the price increases by more than $3.00 we will notify you. If the price decreases, we will simply charge the lower price. Applicable discounts will be extended. Downloading and Using eBooks from NAP What is an eBook? An ebook is one of two file formats that are intended to be used with e-reader devices and apps such as Amazon Kindle or Apple iBooks. Why is an eBook better than a PDF? A PDF is a digital representation of the print book, so while it can be loaded into most e-reader programs, it doesn't allow for resizable text or advanced, interactive functionality. The eBook is optimized for e-reader devices and apps, which means that it offers a much better digital reading experience than a PDF, including resizable text and interactive features (when available). Where do I get eBook files? eBook files are now available for a large number of reports on the NAP.edu website. If an eBook is available, you'll see the option to purchase it on the book page. Types of Publications Consensus Study Report: Consensus Study Reports published by the National Academies of Sciences, Engineering, and Medicine document the evidence-based consensus on the study’s statement of task by an authoring committee of experts. Reports typically include findings, conclusions, and recommendations based on information gathered by the committee and the committee’s deliberations. Each report has been subjected to a rigorous and independent peer-review process and it represents the position of the National Academies on the statement of task. Contributors Description Science, technology, engineering, and mathematics (STEM) are cultural achievements that reflect our humanity, power our economy, and constitute fundamental aspects of our lives as citizens, consumers, parents, and members of the workforce. Providing all students with access to quality education in the STEM disciplines is important to our nation's competitiveness. However, it is challenging to identify the most successful schools and approaches in the STEM disciplines because success is defined in many ways and can occur in many different types of schools and settings. In addition, it is difficult to determine whether the success of a school's students is caused by actions the school takes or simply related to the population of students in the school. Successful K-12 STEM Education defines a framework for understanding "success" in K-12 STEM education. The book focuses its analysis on the science and mathematics parts of STEM and outlines criteria for identifying effective STEM schools and programs. Because a school's success should be defined by and measured relative to its goals, the book identifies three important goals that share certain elements, including learning STEM content and practices, developing positive dispositions toward STEM, and preparing students to be lifelong learners. A successful STEM program would increase the number of students who ultimately pursue advanced degrees and careers in STEM fields, enhance the STEM-capable workforce, and boost STEM literacy for all students. It is also critical to broaden the participation of women and minorities in STEM fields. Successful K-12 STEM Education examines the vast landscape of K-12 STEM education by considering different school models, highlighting research on effective STEM education practices, and identifying some conditions that promote and limit school- and student-level success in STEM. The book also looks at where further work is needed to develop appropriate data sources. The book will serve as a guide to policy makers decision makers at the school and district levels local, state, and federal government agencies curriculum developers educators and parent and education advocacy groups. Mathematical Modeling in Science and Engineering : An Axiomatic Approach Mathematical and computational modeling makes it possible to predict the behavior of a broad range of systems across a broad range of disciplines. This text guides students and professionals through the axiomatic approach, a powerful method that will enable them to easily master the principle types of mathematical and computational models used in engineering and science. Readers will discover that this axiomatic approach not only enables them to systematically construct effective models, it also enables them to apply these models to any macroscopic physical system. Mathematical Modeling in Science and Engineering focuses on models in which the processes to be modeled are expressed as systems of partial differential equations. It begins with an introductory discussion of the axiomatic formulation of basic models, setting the foundation for further topics such as: Mechanics of classical and non-classical continuous systems Solute transport by a free fluid Flow of a fluid in a porous medium Throughout the text, diagrams are provided to help readers visualize and better understand complex mathematical concepts. A set of exercises at the end of each chapter enables readers to put their new modeling skills into practice. There is also a bibliography in each chapter to facilitate further investigation of individual topics. Mathematical Modeling in Science and Engineering is ideal for both students and professionals across the many disciplines of science and engineering that depend on mathematical and computational modeling to predict and understand complex systems. Author Bios GEORGE F. PINDER, PhD, has a primary appointment as Professor of Engineering with secondary appointments as Professor of Mathematics and Statistics and Professor of Computer Science at the University of Vermont. He is the author, or co-author, of nine books on mathematical modeling, numerical mathematics, and flow and transport through porous media. He is a recipient of numerous national and international honors and is a member of the National Academy of Engineering. Methods to develop mathematical models: traditional statistical analysis Abstract Mathematical models for kinematics, kinetics, and muscles potentials activities from sEMG based on traditional statistical analysis are developed using different methods for data analysis, where each model is represented using a structure with a linear dynamic form, explicit and discrete, that can be verified as stochastic process and arising from empirical finding. In this chapter, Mathematical tools are studied with the objective of obtaining Mathematical Models from: traditional stochastic methods from probability and statistics as probability models, probability distributions, statistical inferences using statistical hypotheses testing parameters, z-tests, t-tests, paired t-tests, ANOVA. We apply them to: Linear equations, Regression methods, and Autoregressive equations. The different methods explained are applied to research Biomechanics examples to model and detect data behaviors, and this chapter is concluded with the development of a special software application of Mathematical Models for Analysis of Continuous Glucose Monitor (CGM) for Diabetic subjects. Note: Others Mathematical Models based on Domain/Conversion/Transform analysis, and Machine Learning Models Analysis are studied in the next chapters. Science, Technology, Engineering, and Math, including Computer Science In an ever-changing, increasingly complex world, it's more important than ever that our nation's youth are prepared to bring knowledge and skills to solve problems, make sense of information, and know how to gather and evaluate evidence to make decisions. These are the kinds of skills that students develop in science, technology, engineering, and math, including computer science—disciplines collectively known as STEM/CS. If we want a nation where our future leaders, neighbors, and workers can understand and solve some of the complex challenges of today and tomorrow, and to meet the demands of the dynamic and evolving workforce, building students' skills, content knowledge, and literacy in STEM fields is essential. We must also make sure that, no matter where children live, they have access to quality learning environments. A child's zip code should not determine their STEM literacy and educational options. Department Offices that Support STEM Office of Planning, Evaluation, and Policy Development (OPEPD) Office of Career, Adult, and Technical Education (OCTAE) Office of Elementary and Secondary Education (OESE) Office of Special Education and Rehabilitative Services (OSERS) Office of Postsecondary Education (OPE) Office of Non-Public Education (ONPE) Office of Educational Technology (OET) Office of English Language Acquisition (OELA) Institute of Educational Sciences (IES) White House Initiatives Federal Student Aid (FSA) Office of Communications and Outreach (OCO) Open ED Funding and Other Opportunities The Fiscal Year (FY) 2021 funding season officially kicked-off on October 1, 2020. The grants forecast is located here and you can find all open ED grants here. Education Innovation and Research (EIR): Mid-Phase & Expansion-Phase Grants The Department just released two notices inviting applications (NIAs) for Mid-Phase and Expansion-Phase EIR projects. The EIR program provides funding to create, develop, implement, replicate, or take to scale entrepreneurial, evidence-based, field-initiated innovations to improve student achievement and attainment for high-need students and rigorously evaluate such innovations. The EIR program is designed to generate and validate solutions to persistent education challenges and to support the expansion of those solutions to serve substantially larger numbers of students. The FY 2021 Mid-Phase grant competition includes four absolute priorities, one competitive preference priority, and two invitational priorities: • Absolute Priority 1--Moderate Evidence • Absolute Priority 2--Field-Initiated Innovations—General • Absolute Priority 3--Field-Initiated Innovations—Science, Technology, Engineering, or Mathematics (STEM), which includes a Competitive Preference Priority that focuses on expanding opportunities in computer science for underserved populations • Absolute Priority 4--Field-Initiated Innovations--Fostering Knowledge and Promoting the Development of Skills That Prepare Students to Be Informed, Thoughtful, and Productive Individuals and Citizens (social emotional learning or SEL) • Invitational Priority 1--Innovative Approaches to Addressing the Impact of COVID-19 on Underserved Students and Educators • Invitational Priority 2--Promoting Equity and Adequacy in Student Access to Educational Resources and Opportunities, including STEM courses and teacher certification. The FY 2021 Expansion-Phase grant competition includes two absolute priorities and two invitational priorities. Absolute priorities require strong evidence and field-initiated innovations. Invitational Priority 1 seeks innovative approaches to addressing the impact of COVID-19 on underserved students and educators. Invitational Priority 2 seeks projects promoting equity and adequacy in access to educational resources and opportunities, such as those addressing the Department's Civil Rights Data Collection STEM Course Taking Report, 2018. Total estimated funding for the EIR Early Phase (to be published later), Mid-Phase, and Expansion-Phase grants is$180 million. For Mid-Phase projects, the Department intends to award an estimated $32 million in funds for STEM projects and$32 million in funds for SEL projects. Applications are due July 7, 2021.

GEAR UP Funding
The Department issued a notice inviting application (NIA) for fiscal year 2021 (FY21) for the Gaining Early Awareness and Readiness for Undergraduate Programs (GEAR UP) Partnership Grants. GEAR UP is a discretionary grant designed to help eligible low-income students, including students with disabilities, in obtaining a secondary school diploma and prepare for postsecondary education. Activities must include postsecondary financial aid information, reduce remediation at the postsecondary level, and improve the number of students who obtain a secondary school diploma, complete applications, and enroll in postsecondary education. Activities may include mentoring tutoring dual or concurrent enrollment programs for students in science, technology, engineering, or mathematics (STEM) academic and career counseling financial and economic literacy education and exposure to college campuses. Applications are due June 28, 2021.

New to the Department's grantmaking process? The Department offers introductory resources about its grantmaking. The Department is always seeking experts in STEM education and other fields to serve as peer reviewers of grant applications. See sections below for more details.

Examples of the Department's discretionary grants that can support STEM

Below are investments made in FY 2020:

• $3.6 million for the Alaska Native Education Equity Program •$300,000 for Braille training (rehabilitation services demonstrations and training)
• $5.1 million for the College Assistance Migrant Program (CAMP) •$5 million for the Comprehensive Centers Program
• $185 million for the Education Innovation and Research Program (EIR) (awarded in early FY 2021) •$124.7 million for Gaining Early Awareness and Readiness for Undergraduate Programs (Partnership Grants) (GEAR-UP)
• $23 million for Graduate Assistance in Areas of National Need •$25 million for Innovative Approaches to Literacy
• $5.7 million for the Jacob K. Javits Gifted and Talented Students Education Program •$900,000 for Migrant Education Consortium Incentive Grants (CIG)
• $29 million for the Native Hawaiian Education Program •$12.6 million for the Minority Science and Engineering Improvement Program (MSEIP)
• $1.4 million for the Perkins Innovation & Modernization Grant Program •$300,000 for Strengthening Asian American and Native American Pacific Islander Serving Institutions (AANAPISI)
• $2.3 million for Strengthening Native American Nontribal Serving Institutions (NASNTI) •$1.5 million to provide special education programs in educational technology, media, and materials for students with disabilities via a cooperative agreement with the Center on Early STEM Learning for Young Children
• $9.3 million to provide special education programs educational technology, media, and materials for individuals with disabilities via Stepping Up •$151.2 million for Federal TRIO Programs
• $73.7 million for Supporting Effective Educator Development (SEED) •$49.4 million for the STEM Investment Summary FY2018-2020

You can search for open discretionary grant opportunities or reach out to the Department's STEM contacts noted below. The Forecast of Funding Opportunities lists virtually all Department discretionary grant programs for FY 2021.

Grant Applicant Resources

The Department published in spring 2020 two new grant applicant resources. These resources were developed to (1) provide an overview of the discretionary (or competitive) grants application process and (2) offer more details intended to be used by prospective applicants, including new potential grantees. These support one of the Secretary's new administrative priorities on New Potential Grantees that was published in March 2020. They can also be found under the "Other Grant Information" on the ED's Grants webpage.

Call for Peer Reviewers

The Department is seeking peer reviewers for our Fiscal Year 2021 competitive/discretionary grant season, including in the STEM/CS areas (among others). The Federal Register notice spotlights the specific needs of the Office of Elementary and Secondary Education (OESE), the Office of Postsecondary Education (OPE), and the Office of Special Education and Rehabilitative Services (OSERS). The How to Become a Peer Reviewer slide deck provides additional information and next steps.

America's Strategy for STEM Education

The STEM Education Strategic Plan, Charting a Course for Success: America's Strategy for STEM Education, published in December 2018, sets out a federal strategy for the next five years based on a vision for a future where all Americans will have lifelong access to high-quality STEM education and the United States will be the global leader in STEM literacy, innovation, and employment. It represents an urgent call to action for a nationwide collaboration with learners, families, educators, communities, and employers—a "North Star" for the STEM community as it collectively charts a course for the Nation's success. The Department is an active participant in each of the interagency working groups focused on implementation of the Plan.

In December 2020, the Office of Science and Technology Policy at the White House issued the Progress Report on the Implementation of the Federal STEM Education Strategic Plan. This progress report describes ongoing efforts and implementation practices across the Federal Government as it works to accomplish the goals and objectives of the Strategic Plan. This report also compiles budget information from all Federal agencies that have investments in STEM education during Fiscal Year (FY) 2019. Additionally, this document is meant to fulfill the requirements under the America COMPETES Reauthorization of 2010 that the Office of Science and Technology Policy (OSTP) must transmit a report annually to Congress at the time of the President's budget request providing an update on the STEM Education Federal portfolio performance and an inventory of Federal STEM education investments. The 2019 Progress Report was issued in October 2019

Secretary's STEM Priority

STEM is a centerpiece of the Department's comprehensive education agenda. The STEM priority has been used across the Departments' discretionary grant programs to further the Department's mission, which is "to promote student achievement and preparation for global competitiveness by fostering educational excellence and ensuring equal access."

U.S. Department of Education STEM Newsletter

In February 2020, the Department created the U.S. Department of Education STEM Newsletter. Please go to our newsletter subscription page to sign-up.

STEM Education Briefings

The STEM Education Briefings are live-streamed, close-captioned and archived for your convenience.

Upcoming STEM Briefings

POSTPONED 2021– Advanced Manufacturing: Industry of the Future, Register Here

Other Federal Agency STEM websites

The following are federal agencies that the Department collaborates with to support the aims of the STEM Education Strategic Plan (see above section for more details) and support the Department's stakeholders.

4. Models and Theory

An important question concerns the relation between models and theories. There is a full spectrum of positions ranging from models being subordinate to theories to models being independent of theories.

4.1 Models as subsidiaries to theory

To discuss the relation between models and theories in science it is helpful to briefly recapitulate the notions of a model and of a theory in logic. A theory is taken to be a (usually deductively closed) set of sentences in a formal language. A model is a structure (in the sense introduced in Section 2.3) that makes all sentences of a theory true when its symbols are interpreted as referring to objects, relations, or functions of a structure. The structure is a model of the theory in the sense that it is correctly described by the theory (see Bell and Machover 1977 or Hodges 1997 for details). Logical models are sometimes also referred to as &ldquomodels of theory&rdquo to indicate that they are interpretations of an abstract formal system.

Models in science sometimes carry over from logic the idea of being the interpretation of an abstract calculus (Hesse 1967). This is salient in physics, where general laws&mdashsuch as Newton&rsquos equation of motion&mdashlie at the heart of a theory. These laws are applied to a particular system&mdashe.g., a pendulum&mdashby choosing a special force function, making assumptions about the mass distribution of the pendulum etc. The resulting model then is an interpretation (or realization) of the general law.

It is important to keep the notions of a logical and a representational model separate (Thomson-Jones 2006): these are distinct concepts. Something can be a logical model without being a representational model, and vice versa. This, however, does not mean that something cannot be a model in both senses at once. In fact, as Hesse (1967) points out, many models in science are both logical and representational models. Newton&rsquos model of planetary motion is a case in point: the model, consisting of two homogeneous perfect spheres located in otherwise empty space that attract each other gravitationally, is simultaneously a logical model (because it makes the axioms of Newtonian mechanics true when they are interpreted as referring to the model) and a representational model (because it represents the real sun and earth).

There are two main conceptions of scientific theories, the so-called syntactic view of theories and the so-called semantic view of theories (see the entry on the structure of scientific theories). On both conceptions models play a subsidiary role to theories, albeit in very different ways. The syntactic view of theories (see entry section on the syntactic view) retains the logical notions of a model and a theory. It construes a theory as a set of sentences in an axiomatized logical system, and a model as an alternative interpretation of a certain calculus (Braithwaite 1953 Campbell 1920 [1957] Nagel 1961 Spector 1965). If, for instance, we take the mathematics used in the kinetic theory of gases and reinterpret the terms of this calculus in a way that makes them refer to billiard balls, the billiard balls are a model of the kinetic theory of gases in the sense that all sentences of the theory come out true. The model is meant to be something that we are familiar with, and it serves the purpose of making an abstract formal calculus more palpable. A given theory can have different models, and which model we choose depends both on our aims and our background knowledge. Proponents of the syntactic view disagree about the importance of models. Carnap and Hempel thought that models only serve a pedagogic or aesthetic purpose and are ultimately dispensable because all relevant information is contained in the theory (Carnap 1938 Hempel 1965 see also Bailer-Jones 1999). Nagel (1961) and Braithwaite (1953), on the other hand, emphasize the heuristic role of models, and Schaffner (1969) submits that theoretical terms get at least part of their meaning from models.

The semantic view of theories (see entry section on the semantic view) dispenses with sentences in an axiomatized logical system and construes a theory as a family of models. On this view, a theory literally is a class, cluster, or family of models&mdashmodels are the building blocks of which scientific theories are made up. Different versions of the semantic view work with different notions of a model, but, as noted in Section 2.3, in the semantic view models are mostly construed as set-theoretic structures. For a discussion of the different options, we refer the reader to the relevant entry in this encyclopedia (linked at the beginning of this paragraph).

4.2 Models as independent from theories

In both the syntactic and the semantic view of theories models are seen as subordinate to theory and as playing no role outside the context of a theory. This vision of models has been challenged in a number of ways, with authors pointing out that models enjoy various degrees of freedom from theory and function autonomously in many contexts. Independence can take many forms, and large parts of the literature on models are concerned with investigating various forms of independence.

Models as completely independent of theory. The most radical departure from a theory-centered analysis of models is the realization that there are models that are completely independent from any theory. An example of such a model is the Lotka&ndashVolterra model. The model describes the interaction of two populations: a population of predators and one of prey animals (Weisberg 2013). The model was constructed using only relatively commonsensical assumptions about predators and prey and the mathematics of differential equations. There was no appeal to a theory of predator&ndashprey interactions or a theory of population growth, and the model is independent of theories about its subject matter. If a model is constructed in a domain where no theory is available, then the model is sometimes referred to as a &ldquosubstitute model&rdquo (Groenewold 1961), because the model substitutes a theory.

Models as a means to explore theory. Models can also be used to explore theories (Morgan and Morrison 1999). An obvious way in which this can happen is when a model is a logical model of a theory (see Section 4.1). A logical model is a set of objects and properties that make a formal sentence true, and so one can see in the model how the axioms of the theory play out in a particular setting and what kinds of behavior they dictate. But not all models that are used to explore theories are logical models, and models can represent features of theories in other ways. As an example, consider chaos theory. The equations of non-linear systems, such as those describing the three-body problem, have solutions that are too complex to study with paper-and-pencil methods, and even computer simulations are limited in various ways. Abstract considerations about the qualitative behavior of solutions show that there is a mechanism that has been dubbed &ldquostretching and folding&rdquo (see the entry Chaos). To obtain an idea of the complexity of the dynamics exhibiting stretching and folding, Smale proposed to study a simple model of the flow&mdashnow known as the &ldquohorseshoe map&rdquo (Tabor 1989)&mdashwhich provides important insights into the nature of stretching and folding. Other examples of models of that kind are the Kac ring model that is used to study equilibrium properties of systems in statistical mechanics (Lavis 2008) and Norton&rsquos dome in Newtonian mechanics (Norton 2003).

Models as complements of theories. A theory may be incompletely specified in the sense that it only imposes certain general constraints but remains silent about the details of concrete situations, which are provided by a model (Redhead 1980). A special case of this situation is when a qualitative theory is known and the model introduces quantitative measures (Apostel 1961). Redhead&rsquos example of a theory that is underdetermined in this way is axiomatic quantum field theory, which only imposes certain general constraints on quantum fields but does not provide an account of particular fields. Harré (2004) notes that models can complement theories by providing mechanisms for processes that are left unspecified in the theory even though they are responsible for bringing about the observed phenomena.

Theories may be too complicated to handle. In such cases a model can complement a theory by providing a simplified version of the theoretical scenario that allows for a solution. Quantum chromodynamics, for instance, cannot easily be used to investigate the physics of an atomic nucleus even though it is the relevant fundamental theory. To get around this difficulty, physicists construct tractable phenomenological models (such as the MIT bag model) which effectively describe the relevant degrees of freedom of the system under consideration (Hartmann 1999, 2001). The advantage of these models is that they yield results where theories remain silent. Their drawback is that it is often not clear how to understand the relationship between the model and the theory, as the two are, strictly speaking, contradictory.

Models as preliminary theories. The notion of a model as a substitute for a theory is closely related to the notion of a developmental model. This term was coined by Leplin (1980), who pointed out how useful models were in the development of early quantum theory, and it is now used as an umbrella notion covering cases in which models are some sort of a preliminary exercise to theory.

Also closely related is the notion of a probing model (or &ldquostudy model&rdquo). Models of this kind do not perform a representational function and are not expected to instruct us about anything beyond the model itself. The purpose of these models is to test new theoretical tools that are used later on to build representational models. In field theory, for instance, the so-called &phi 4 -model was studied extensively, not because it was believed to represent anything real, but because it served several heuristic functions: the simplicity of the &phi 4 -model allowed physicists to &ldquoget a feeling&rdquo for what quantum field theories are like and to extract some general features that this simple model shared with more complicated ones. Physicists could study complicated techniques such as renormalization in a simple setting, and it was possible to get acquainted with important mechanisms&mdashin this case symmetry-breaking&mdashthat could later be used in different contexts (Hartmann 1995). This is true not only for physics. As Wimsatt (1987, 2007) points out, a false model in genetics can perform many useful functions, among them the following: the false model can help answering questions about more realistic models, provide an arena for answering questions about properties of more complex models, &ldquofactor out&rdquo phenomena that would not otherwise be seen, serve as a limiting case of a more general model (or two false models may define the extremes of a continuum of cases on which the real case is supposed to lie), or lead to the identification of relevant variables and the estimation of their values.

Interpretative models. Cartwright (1983, 1999) argues that models do not only aid the application of theories that are somehow incomplete she claims that models are also involved whenever a theory with an overarching mathematical structure is applied. The main theories in physics&mdashclassical mechanics, electrodynamics, quantum mechanics, and so on&mdashfall into this category. Theories of that kind are formulated in terms of abstract concepts that need to be concretized for the theory to provide a description of the target system, and concretizing the relevant concepts, idealized objects and processes are introduced. For instance, when applying classical mechanics, the abstract concept of force has to be replaced with a concrete force such as gravity. To obtain tractable equations, this procedure has to be applied to a simplified scenario, for instance that of two perfectly spherical and homogeneous planets in otherwise empty space, rather than to reality in its full complexity. The result is an interpretative model, which grounds the application of mathematical theories to real-world targets. Such models are independent from theory in that the theory does not determine their form, and yet they are necessary for the application of the theory to a concrete problem.

Models as mediators. The relation between models and theories can be complicated and disorderly. The contributors to a programmatic collection of essays edited by Morgan and Morrison (1999) rally around the idea that models are instruments that mediate between theories and the world. Models are &ldquoautonomous agents&rdquo in that they are independent from both theories and their target systems, and it is this independence that allows them to mediate between the two. Theories do not provide us with algorithms for the construction of a model they are not &ldquovending machines&rdquo into which one can insert a problem and a model pops out (Cartwright 1999). The construction of a model often requires detailed knowledge about materials, approximation schemes, and the setup, and these are not provided by the corresponding theory. Furthermore, the inner workings of a model are often driven by a number of different theories working cooperatively. In contemporary climate modeling, for instance, elements of different theories&mdashamong them fluid dynamics, thermodynamics, electromagnetism&mdashare put to work cooperatively. What delivers the results is not the stringent application of one theory, but the voices of different theories when put to use in chorus with each other in one model.

In complex cases like the study of a laser system or the global climate, models and theories can get so entangled that it becomes unclear where a line between the two should be drawn: where does the model end and the theory begin? This is not only a problem for philosophical analysis it also arises in scientific practice. Bailer-Jones (2002) interviewed a group of physicists about their understanding of models and their relation to theories, and reports widely diverging views: (i) there is no substantive difference between model and theory (ii) models become theories when their degree of confirmation increases (iii) models contain simplifications and omissions, while theories are accurate and complete (iv) theories are more general than models, and modeling is about applying general theories to specific cases. The first suggestion seems to be too radical to do justice to many aspects of practice, where a distinction between models and theories is clearly made. The second view is in line with common parlance, where the terms &ldquomodel&rdquo and &ldquotheory&rdquo are sometimes used to express someone&rsquos attitude towards a particular hypothesis. The phrase &ldquoit&rsquos just a model&rdquo indicates that the hypothesis at stake is asserted only tentatively or is even known to be false, while something is awarded the label &ldquotheory&rdquo if it has acquired some degree of general acceptance. However, this use of &ldquomodel&rdquo is different from the uses we have seen in Sections 1 to 3 and is therefore of no use if we aim to understand the relation between scientific models and theories (and, incidentally, one can equally dismiss speculative claims as being &ldquojust a theory&rdquo). The third proposal is correct in associating models with idealizations and simplifications, but it overshoots by restricting this to models in fact, also theories can contain idealizations and simplifications. The fourth view seems closely aligned with interpretative models and the idea that models are mediators, but being more general is a gradual notion and hence does not provide a clear-cut criterion to distinguish between theories and models.

Criticisms of STEM

Critics of STEM education believe the in-depth focus on science, technology, engineering, and math shortchanges students' learning and experiences with other subjects that are also important, such as art, music, literature, and writing. These non-STEM subjects contribute to brain development, critical reading skills, and communication skills.

Another criticism of STEM education is the belief—alleged to be mistaken—that it will fill a coming shortage of workers in fields related to those subjects. For careers in technology and many careers in engineering, this prediction may be true. However, careers in many scientific areas and in mathematics currently have a shortage of jobs available for the number of people seeking employment.

JOTS v25n2 - Models of Curriculum Integration

The notion of curriculum integration is not new. Dewey and Kilpatrick advocated forms of integration early in the century ( Vars, 1991 ). More recently, however, educational theorists have been advocating curriculum integration for a number of reasons. The challenge has been for those who attempt to put theory into practice. The purpose of this paper is to define curriculum integration, discuss selected research related to curriculum integration, present several curriculum models for integration, and discuss some of the implications curriculum integration will have on education.

Integrated Curriculum Defined

"The very notion of `integration' incorporates the idea of unity between forms of knowledge and the respective disciplines" ( Pring, 1973, p. 135 ). In practice this can take many forms. Those who consider astronomy, biology, chemistry, geology, and physics as distinct disciplines consider a general science course a step in the direction of integration. They use the metaphor of a marble cake versus a layer cake to signify different levels of integration. The layer cake means each of the sciences maintains an identity in a general science course while the marble cake is more problem based with the various sciences contributing to the solution of the problem. They argue that the layer cake is more of an interdisciplinary approach to curriculum because the boundaries among the disciplines are maintained. Therefore, if one is discussing curriculum integration with a science educator, one must first determine the context because integration could refer to integration within the sciences rather than integration among a wide range of disciplines so that the learner experiences a number of interconnections among disciplines.

An interdisciplinary curriculum can be closely related to an integrated curriculum. Most educators represent the view that knowledge in interdisciplinary studies is a repackaging and, perhaps, enhancement of disciplinebased knowledge ( Kain, 1993 ). In Jacobs' ( 1989 ) definition, interdisciplinary means conscientiously applying methodology and language from more than one discipline to a theme, topic, or problem.

Whether a curriculum is interdisciplinary or integrated is not the main issue. Rather, the focus should be on designing a curriculum that is relevant, standards based, and meaningful for students. At the same time, the curriculum should challenge students to solve real world problems.

Research Supporting Curriculum Integration

During this decade, cognitive scientists have been able to use advanced imaging technologies to study the operation of the brain.

Much of this research has yet to be directly translated into curriculum and pedagogy. This research is spawning a dynamic educational philosophy referred to as "constructivism" which refers to engaging students in constructing their own knowledge. "The single best way to grow a better brain is through challenging problem solving. This creates new dendritic connections that allow us to make even more connections" ( Jenson, 1998, p. 35 ).

And one of the best ways to promote problem solving is through an enriched environment that makes connections among several disciplines ( Wolf & Brandt, 1998 ).

Educational researchers have found that an integrated curriculum can result in greater intellectual curiosity, improved attitude towards schooling, enhanced problem-solving skills, and higher achievement in college ( Austin, Hirstein, & Walen, 1997 Kain, 1993 ). Barab and Landa ( 1997 ) indicated that when students focus on problems worth solving, motivation and learning increase.

Some schools have used an integrated curriculum as a way to make education relevant and thus a way to keep students interested in school ( Kain, 1993 ). In a traditional program, relevancy can be a problem. One of the most common questions in a mathematics class is, "Why are we learning this math?" And the common response is, "Because you will need to know it in your math class next year." This response seldom satisfies the learner. Schools report higher attendance rates when students are engaged in an integrated curriculum ( Meier & Dossey, unpublished manuscript ). Having the opportunity to utilize knowledge and skills from several disciplines does offer increased opportunities for making the curriculum relevant. A word of caution is in order, however. Just because a curriculum is integrated does not automatically mean that it is relevant.

A number of organizations support integrated learning. Project 2061's benchmarks for science literacy calls for an interdisciplinary, integrated development of knowledge organized around themes that cut across various science disciplines, mathematics, social studies, and technology ( American Association for the Advancement of Science, 1993 ). The National Science Education Standards ( National Research Council, 1996 ) and the Mathematics Standards ( National Council of Teachers of Mathematics, 1989 ) also promote integrated learning. The pending Technology Education Standards ( International Technology Education Association, 1998 ) actually include a major section on making "technological connections." This section refers to ways that technology education relates to other disciplines.

Another premise supporting the move towards integrated curricula is that the current system of discipline-based education is not as effective as it must be. The assumption is that most real world problems are multidisciplinary in nature and that the current curriculum is unable to engage students in real world situations. Thus, a discipline-based curriculum should be replaced with an integrated curriculum ( Kain, 1993 ).

Implications of Implementing an Integrated Curriculum

No matter which model is selected, there are several common factors that tend to emerge. First, teachers must shift their belief system from one that is primarily didactic in nature to one that has a foundation in constructivism. Rather than asking students to follow the steps of procedure, memorize facts, or verify given principles or laws, students work together to discover knowledge, applying their knowledge as they solve real world problems.

Second, an extensive amount of professional development is needed for teachers. This includes a significant intervention of two or three weeks of knowledge development in curriculum areas other than the one they are certified to teach. Also, this professional development must include extensive practice in the use of constructivist-oriented pedagogy.

Third, the teachers need to become members of learning communities. At one level this means working with one's peers to improve education. At another level teachers work with their students in solving problems that have multiple answers.

Fourth, teachers need to become skilled in facilitating small group learning. Research has shown that learning is a social process and that students learn a great deal by interacting with one another.

Fifth, teachers need to manage experiential-oriented instruction. This includes inventorying and storing materials the safe operation of instrumentation, machines, and equipment and leading students toward efficient progress.

Sixth, teachers need to learn to use authentic assessment strategies such as portfolios, performance exams, and rubrics to document student progress.

Seventh, administrators and school boards need to be oriented so the necessary resources and ongoing support can be provided to the teachers.

Eighth, public information strategies need to be implemented in order to inform the community and parents that a new paradigm of education is being used. The expectation is for education to be provided as it has always been, and unless the public is informed of changes to be made, there is likely to be resistance.

Finally, changing to an integrated curriculum requires systemic reform. This includes the way teachers are prepared, certified, and assessed. Attention must also be given to statewide assessment of students and the process whereby teacher credentials are renewed.

Conclusion

Given the implications listed above, the prospect for moving to the implementation of an integrated and/or interdisciplinary curriculum on a nationwide basis is bleak. On the other hand, research in the area of education as well as in cognitive science suggests that some form of an integrated curriculum is likely to promote more learning. This being true, the topic of integrated curriculum is destined to receive a lot of attention soon.

References

American Association for the Advancement of Science. (1993). Project 2061: Bench marks for science literacy, . New York: Oxford University Press.

Austin , J. D., Hirstein, J., & Walen, S. (1997). Integrated mathematics interfaced with science. School Science and Mathematics , 97(1), 4549.

Barab , S. A., & Landa, A. (1997). Designing effective interdisciplinary anchors. Educational Leadership , 54(6), 5258.

International Technology Education Association. (1998). Standards for technology education: Content for the study of technology . Blacksburg, VA: Author.

Jacobs , H. H. (Ed.). (1989). Interdisciplinary curriculum: Design and implementation . Alexandria, VA: Association for Supervision and Curriculum Development.

Jensen , E. (1998). Teaching with the brain in mind . Alexandria, VA: Association for Supervision and Curriculum Development.

Kain , D. L. (1993). Cabbages and kings: Research directions in integrated/interdisciplinary curriculum. The Journal of Educational Thought , 27(3), 312331.

LaPorte , J., & Sanders, M. (1996). Technology science mathematics . New York: Glenco/McGraw-Hill.

Meier , & Dossey, unpublished manuscript, Illinois State University.

National Research Council. (1996). National science education standards . Washington, DC: National Academy Press.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics . Reston, VA: Author.

Pring , R. (1973). Curriculum integration. In R. S. Peters (Ed.), The philosophy of education (pp. 123149). London: Oxford University Press.

Vars , G. F. (1991). Integrated curriculum in historical perspective. Educational Leadership , 49(2), 1415.

Wolf , P., & Brandt, R. (1998). What do we know from brain research? Educational Leadership , 56(3), 813.

Science and Engineering Practices

The practices describe behaviors that scientists engage in as they investigate and build models and theories about the natural world and the key set of engineering practices that engineers use as they design and build models and systems. The NRC uses the term practices instead of a term like “skills” to emphasize that engaging in scientific investigation requires not only skill but also knowledge that is specific to each practice. Part of the NRC’s intent is to better explain and extend what is meant by “inquiry” in science and the range of cognitive, social, and physical practices that it requires. Although engineering design is similar to scientific inquiry, there are significant differences. For example, scientific inquiry involves the formulation of a question that can be answered through investigation, while engineering design involves the formulation of a problem that can be solved through design. Strengthening the engineering aspects of the Next Generation Science Standards will clarify for students the relevance of science, technology, engineering and mathematics (the four STEM fields) to everyday life.

A practice of science is to ask and refine questions that lead to descriptions and explanations of how the natural and designed world works and which can be empirically tested.

Developing and Using Models

A practice of both science and engineering is to use and construct models as helpful tools for representing ideas and explanations. These tools include diagrams, drawings, physical replicas, mathematical representations, analogies, and computer simulations.

Planning and Carrying Out Investigations

Scientists and engineers plan and carry out investigations in the field or laboratory, working collaboratively as well as individually. Their investigations are systematic and require clarifying what counts as data and identifying variables or parameters.

Analyzing and Interpreting Data

Scientific investigations produce data that must be analyzed in order to derive meaning. Because data patterns and trends are not always obvious, scientists use a range of tools—including tabulation, graphical interpretation, visualization, and statistical analysis—to identify the significant features and patterns in the data. Scientists identify sources of error in the investigations and calculate the degree of certainty in the results. Modern technology makes the collection of large data sets much easier, providing secondary sources for analysis.

Using Mathematics and Computational Thinking

In both science and engineering, mathematics and computation are fundamental tools for representing physical variables and their relationships. They are used for a range of tasks such as constructing simulations statistically analyzing data and recognizing, expressing, and applying quantitative relationships.

Constructing Explanations and Designing Solutions

The products of science are explanations and the products of engineering are solutions.

Engaging in Argument from Evidence

Argumentation is the process by which explanations and solutions are reached.

Obtaining, Evaluating, and Communicating Information

Scientists and engineers must be able to communicate clearly and persuasively the ideas and methods they generate. Critiquing and communicating ideas individually and in groups is a critical professional activity.

2.1: Models in Science and Engineering - Mathematics

• Teachers should select instructional models for teaching the practices based on their commitments, preferences, and their local context.
• District staff and PD providers should highlight for teachers that multiple instructional models can be used to implement the practices although PD may focus on a specific one.
• School leaders should know what instructional models are used by their teachers and learn to recognize qualities of them within classroom.

What is the Issue?

The Next Gen Science Standards (NGSS) expect learners to engage in eight science and engineering practices in order to learn and apply conceptual ideas. People often assume that a particular instructional model is best for engaging students in the NGSS practices. In fact, there are multiple models that can be used effectively.

Authors:

PHILIP BELL AND ANDREW SHOUSE

• How satisfied are you with your current way of teaching science and engineering? How well does it engage students in the science and engineering practices?
• What instructional models do you currently use? Which additional ones might be suitable for your context?
• It is productive to take up small specific teaching practices that can be repeated and refined throughout your teaching. What teaching strategy might you want to focus on?

Things to Consider

• NGSS and the underlying NRC Framework do not say anywhere that there is only one instructional approach for engaging students in the practices. But specific curricula, instructional resources, and PD can reinforce this view by focusing on only one model at a time. There are actually multiple instructional models that can be productively used to implement the learning goals of NGSS.
• Explore the practice-focused instructional models listed in the table and select one(s) that fit your situation and personal preferences.
• Selecting an instructional model that fits a particular classroom should be based on local circumstances. This can involve supporting instruction that fits a teacher's personal history, goals, or commitments. Or it can be based on what instructional model is in use in the local curriculum. The district's or school's instructional strategy or a professional learning community may also shape teachers' orientation to an instructional model.
• Implementing an instructional model may require adaptation of available curriculum to engage students in the practices.
• Multiple instructional models can be integrated, but it is important for the learning experience to be coherent so that a rigorous and engaging classroom culture can be cultivated.

Attending to Equity

• Instructional strategies vary in terms of how they relate the science being learned to the lives and interests of the learners and the communities they are part of. Some instructional models—for example, culturally relevant instruction—actively connect to and build upon the life experiences and practices of learners.
• In order to make science teaching and learning as inclusive as possible, educators should select instructional models that engage students with the practices in different, locally relevant ways.

Recommended Actions You Can Take

• Learn which instructional models for science instruction are used in your district, school, or PD. Determine if it focuses on practices.
• Connect to others who use a desired instructional model in order to share materials and learn about the finer points of using a specific instructional model.