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Using the Scientific Method to Solve Problems

How the scientific method and reasoning can help simplify processes and solve problems.

By the Mind Tools Content Team

The processes of problem-solving and decision-making can be complicated and drawn out. In this article we look at how the scientific method, along with deductive and inductive reasoning can help simplify these processes.

example of problem solving using scientific method

‘It is a capital mistake to theorize before one has information. Insensibly one begins to twist facts to suit our theories, instead of theories to suit facts.’ Sherlock Holmes

The Scientific Method

The scientific method is a process used to explore observations and answer questions. Originally used by scientists looking to prove new theories, its use has spread into many other areas, including that of problem-solving and decision-making.

The scientific method is designed to eliminate the influences of bias, prejudice and personal beliefs when testing a hypothesis or theory. It has developed alongside science itself, with origins going back to the 13th century. The scientific method is generally described as a series of steps.

observations/theory
explanation/conclusion

The first step is to develop a theory about the particular area of interest. A theory, in the context of logic or problem-solving, is a conjecture or speculation about something that is not necessarily fact, often based on a series of observations.

Once a theory has been devised, it can be questioned and refined into more specific hypotheses that can be tested. The hypotheses are potential explanations for the theory.

The testing, and subsequent analysis, of these hypotheses will eventually lead to a conclus ion which can prove or disprove the original theory.

Applying the Scientific Method to Problem-Solving

How can the scientific method be used to solve a problem, such as the color printer is not working?

1. Use observations to develop a theory.

In order to solve the problem, it must first be clear what the problem is. Observations made about the problem should be used to develop a theory. In this particular problem the theory might be that the color printer has run out of ink. This theory is developed as the result of observing the increasingly faded output from the printer.

2. Form a hypothesis.

Note down all the possible reasons for the problem. In this situation they might include:

The printer is set up as the default printer for all 40 people in the department and so is used more frequently than necessary.
There has been increased usage of the printer due to non-work related printing.
In an attempt to reduce costs, poor quality ink cartridges with limited amounts of ink in them have been purchased.
The printer is faulty.

All these possible reasons are hypotheses.

3. Test the hypothesis.

Once as many hypotheses (or reasons) as possible have been thought of, then each one can be tested to discern if it is the cause of the problem. An appropriate test needs to be devised for each hypothesis. For example, it is fairly quick to ask everyone to check the default settings of the printer on each PC, or to check if the cartridge supplier has changed.

4. Analyze the test results.

Once all the hypotheses have been tested, the results can be analyzed. The type and depth of analysis will be dependant on each individual problem, and the tests appropriate to it. In many cases the analysis will be a very quick thought process. In others, where considerable information has been collated, a more structured approach, such as the use of graphs, tables or spreadsheets, may be required.

5. Draw a conclusion.

Based on the results of the tests, a conclusion can then be drawn about exactly what is causing the problem. The appropriate remedial action can then be taken, such as asking everyone to amend their default print settings, or changing the cartridge supplier.

Inductive and Deductive Reasoning

The scientific method involves the use of two basic types of reasoning, inductive and deductive.

Inductive reasoning makes a conclusion based on a set of empirical results. Empirical results are the product of the collection of evidence from observations. For example:

‘Every time it rains the pavement gets wet, therefore rain must be water’.

There has been no scientific determination in the hypothesis that rain is water, it is purely based on observation. The formation of a hypothesis in this manner is sometimes referred to as an educated guess. An educated guess, whilst not based on hard facts, must still be plausible, and consistent with what we already know, in order to present a reasonable argument.

Deductive reasoning can be thought of most simply in terms of ‘If A and B, then C’. For example:

if the window is above the desk, and
the desk is above the floor, then
the window must be above the floor

It works by building on a series of conclusions, which results in one final answer.

Social Sciences and the Scientific Method

The scientific method can be used to address any situation or problem where a theory can be developed. Although more often associated with natural sciences, it can also be used to develop theories in social sciences (such as psychology, sociology and linguistics), using both quantitative and qualitative methods.

Quantitative information is information that can be measured, and tends to focus on numbers and frequencies. Typically quantitative information might be gathered by experiments, questionnaires or psychometric tests. Qualitative information, on the other hand, is based on information describing meaning, such as human behavior, and the reasons behind it. Qualitative information is gathered by way of interviews and case studies, which are possibly not as statistically accurate as quantitative methods, but provide a more in-depth and rich description.

The resultant information can then be used to prove, or disprove, a hypothesis. Using a mix of quantitative and qualitative information is more likely to produce a rounded result based on the factual, quantitative information enriched and backed up by actual experience and qualitative information.

In terms of problem-solving or decision-making, for example, the qualitative information is that gained by looking at the ‘how’ and ‘why’ , whereas quantitative information would come from the ‘where’, ‘what’ and ‘when’.

It may seem easy to come up with a brilliant idea, or to suspect what the cause of a problem may be. However things can get more complicated when the idea needs to be evaluated, or when there may be more than one potential cause of a problem. In these situations, the use of the scientific method, and its associated reasoning, can help the user come to a decision, or reach a solution, secure in the knowledge that all options have been considered.

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Sat / act prep online guides and tips, the 6 scientific method steps and how to use them.

General Education

When you’re faced with a scientific problem, solving it can seem like an impossible prospect. There are so many possible explanations for everything we see and experience—how can you possibly make sense of them all? Science has a simple answer: the scientific method.

The scientific method is a method of asking and answering questions about the world. These guiding principles give scientists a model to work through when trying to understand the world, but where did that model come from, and how does it work?

In this article, we’ll define the scientific method, discuss its long history, and cover each of the scientific method steps in detail.

What Is the Scientific Method?

At its most basic, the scientific method is a procedure for conducting scientific experiments. It’s a set model that scientists in a variety of fields can follow, going from initial observation to conclusion in a loose but concrete format.

The number of steps varies, but the process begins with an observation, progresses through an experiment, and concludes with analysis and sharing data. One of the most important pieces to the scientific method is skepticism —the goal is to find truth, not to confirm a particular thought. That requires reevaluation and repeated experimentation, as well as examining your thinking through rigorous study.

There are in fact multiple scientific methods, as the basic structure can be easily modified. The one we typically learn about in school is the basic method, based in logic and problem solving, typically used in “hard” science fields like biology, chemistry, and physics. It may vary in other fields, such as psychology, but the basic premise of making observations, testing, and continuing to improve a theory from the results remain the same.

The History of the Scientific Method

The scientific method as we know it today is based on thousands of years of scientific study. Its development goes all the way back to ancient Mesopotamia, Greece, and India.

The Ancient World

In ancient Greece, Aristotle devised an inductive-deductive process , which weighs broad generalizations from data against conclusions reached by narrowing down possibilities from a general statement. However, he favored deductive reasoning, as it identifies causes, which he saw as more important.

Aristotle wrote a great deal about logic and many of his ideas about reasoning echo those found in the modern scientific method, such as ignoring circular evidence and limiting the number of middle terms between the beginning of an experiment and the end. Though his model isn’t the one that we use today, the reliance on logic and thorough testing are still key parts of science today.

The Middle Ages

The next big step toward the development of the modern scientific method came in the Middle Ages, particularly in the Islamic world. Ibn al-Haytham, a physicist from what we now know as Iraq, developed a method of testing, observing, and deducing for his research on vision. al-Haytham was critical of Aristotle’s lack of inductive reasoning, which played an important role in his own research.

Other scientists, including Abū Rayhān al-Bīrūnī, Ibn Sina, and Robert Grosseteste also developed models of scientific reasoning to test their own theories. Though they frequently disagreed with one another and Aristotle, those disagreements and refinements of their methods led to the scientific method we have today.

Following those major developments, particularly Grosseteste’s work, Roger Bacon developed his own cycle of observation (seeing that something occurs), hypothesis (making a guess about why that thing occurs), experimentation (testing that the thing occurs), and verification (an outside person ensuring that the result of the experiment is consistent).

After joining the Franciscan Order, Bacon was granted a special commission to write about science; typically, Friars were not allowed to write books or pamphlets. With this commission, Bacon outlined important tenets of the scientific method, including causes of error, methods of knowledge, and the differences between speculative and experimental science. He also used his own principles to investigate the causes of a rainbow, demonstrating the method’s effectiveness.

Scientific Revolution

Throughout the Renaissance, more great thinkers became involved in devising a thorough, rigorous method of scientific study. Francis Bacon brought inductive reasoning further into the method, whereas Descartes argued that the laws of the universe meant that deductive reasoning was sufficient. Galileo’s research was also inductive reasoning-heavy, as he believed that researchers could not account for every possible variable; therefore, repetition was necessary to eliminate faulty hypotheses and experiments.

All of this led to the birth of the Scientific Revolution , which took place during the sixteenth and seventeenth centuries. In 1660, a group of philosophers and physicians joined together to work on scientific advancement. After approval from England’s crown , the group became known as the Royal Society, which helped create a thriving scientific community and an early academic journal to help introduce rigorous study and peer review.

Previous generations of scientists had touched on the importance of induction and deduction, but Sir Isaac Newton proposed that both were equally important. This contribution helped establish the importance of multiple kinds of reasoning, leading to more rigorous study.

As science began to splinter into separate areas of study, it became necessary to define different methods for different fields. Karl Popper was a leader in this area—he established that science could be subject to error, sometimes intentionally. This was particularly tricky for “soft” sciences like psychology and social sciences, which require different methods. Popper’s theories furthered the divide between sciences like psychology and “hard” sciences like chemistry or physics.

Paul Feyerabend argued that Popper’s methods were too restrictive for certain fields, and followed a less restrictive method hinged on “anything goes,” as great scientists had made discoveries without the Scientific Method. Feyerabend suggested that throughout history scientists had adapted their methods as necessary, and that sometimes it would be necessary to break the rules. This approach suited social and behavioral scientists particularly well, leading to a more diverse range of models for scientists in multiple fields to use.

The Scientific Method Steps

Though different fields may have variations on the model, the basic scientific method is as follows:

#1: Make Observations

Notice something, such as the air temperature during the winter, what happens when ice cream melts, or how your plants behave when you forget to water them.

#2: Ask a Question

Turn your observation into a question. Why is the temperature lower during the winter? Why does my ice cream melt? Why does my toast always fall butter-side down?

This step can also include doing some research. You may be able to find answers to these questions already, but you can still test them!

#3: Make a Hypothesis

A hypothesis is an educated guess of the answer to your question. Why does your toast always fall butter-side down? Maybe it’s because the butter makes that side of the bread heavier.

A good hypothesis leads to a prediction that you can test, phrased as an if/then statement. In this case, we can pick something like, “If toast is buttered, then it will hit the ground butter-first.”

#4: Experiment

Your experiment is designed to test whether your predication about what will happen is true. A good experiment will test one variable at a time —for example, we’re trying to test whether butter weighs down one side of toast, making it more likely to hit the ground first.

The unbuttered toast is our control variable. If we determine the chance that a slice of unbuttered toast, marked with a dot, will hit the ground on a particular side, we can compare those results to our buttered toast to see if there’s a correlation between the presence of butter and which way the toast falls.

If we decided not to toast the bread, that would be introducing a new question—whether or not toasting the bread has any impact on how it falls. Since that’s not part of our test, we’ll stick with determining whether the presence of butter has any impact on which side hits the ground first.

#5: Analyze Data

After our experiment, we discover that both buttered toast and unbuttered toast have a 50/50 chance of hitting the ground on the buttered or marked side when dropped from a consistent height, straight down. It looks like our hypothesis was incorrect—it’s not the butter that makes the toast hit the ground in a particular way, so it must be something else.

Since we didn’t get the desired result, it’s back to the drawing board. Our hypothesis wasn’t correct, so we’ll need to start fresh. Now that you think about it, your toast seems to hit the ground butter-first when it slides off your plate, not when you drop it from a consistent height. That can be the basis for your new experiment.

#6: Communicate Your Results

Good science needs verification. Your experiment should be replicable by other people, so you can put together a report about how you ran your experiment to see if other peoples’ findings are consistent with yours.

This may be useful for class or a science fair. Professional scientists may publish their findings in scientific journals, where other scientists can read and attempt their own versions of the same experiments. Being part of a scientific community helps your experiments be stronger because other people can see if there are flaws in your approach—such as if you tested with different kinds of bread, or sometimes used peanut butter instead of butter—that can lead you closer to a good answer.

A Scientific Method Example: Falling Toast

We’ve run through a quick recap of the scientific method steps, but let’s look a little deeper by trying again to figure out why toast so often falls butter side down.

#1: Make Observations

At the end of our last experiment, where we learned that butter doesn’t actually make toast more likely to hit the ground on that side, we remembered that the times when our toast hits the ground butter side first are usually when it’s falling off a plate.

The easiest question we can ask is, “Why is that?”

We can actually search this online and find a pretty detailed answer as to why this is true. But we’re budding scientists—we want to see it in action and verify it for ourselves! After all, good science should be replicable, and we have all the tools we need to test out what’s really going on.

Why do we think that buttered toast hits the ground butter-first? We know it’s not because it’s heavier, so we can strike that out. Maybe it’s because of the shape of our plate?

That’s something we can test. We’ll phrase our hypothesis as, “If my toast slides off my plate, then it will fall butter-side down.”

Just seeing that toast falls off a plate butter-side down isn’t enough for us. We want to know why, so we’re going to take things a step further—we’ll set up a slow-motion camera to capture what happens as the toast slides off the plate.

We’ll run the test ten times, each time tilting the same plate until the toast slides off. We’ll make note of each time the butter side lands first and see what’s happening on the video so we can see what’s going on.

When we review the footage, we’ll likely notice that the bread starts to flip when it slides off the edge, changing how it falls in a way that didn’t happen when we dropped it ourselves.

That answers our question, but it’s not the complete picture —how do other plates affect how often toast hits the ground butter-first? What if the toast is already butter-side down when it falls? These are things we can test in further experiments with new hypotheses!

Now that we have results, we can share them with others who can verify our results. As mentioned above, being part of the scientific community can lead to better results. If your results were wildly different from the established thinking about buttered toast, that might be cause for reevaluation. If they’re the same, they might lead others to make new discoveries about buttered toast. At the very least, you have a cool experiment you can share with your friends!

Key Scientific Method Tips

Though science can be complex, the benefit of the scientific method is that it gives you an easy-to-follow means of thinking about why and how things happen. To use it effectively, keep these things in mind!

Don’t Worry About Proving Your Hypothesis

One of the important things to remember about the scientific method is that it’s not necessarily meant to prove your hypothesis right. It’s great if you do manage to guess the reason for something right the first time, but the ultimate goal of an experiment is to find the true reason for your observation to occur, not to prove your hypothesis right.

Good science sometimes means that you’re wrong. That’s not a bad thing—a well-designed experiment with an unanticipated result can be just as revealing, if not more, than an experiment that confirms your hypothesis.

Be Prepared to Try Again

If the data from your experiment doesn’t match your hypothesis, that’s not a bad thing. You’ve eliminated one possible explanation, which brings you one step closer to discovering the truth.

The scientific method isn’t something you’re meant to do exactly once to prove a point. It’s meant to be repeated and adapted to bring you closer to a solution. Even if you can demonstrate truth in your hypothesis, a good scientist will run an experiment again to be sure that the results are replicable. You can even tweak a successful hypothesis to test another factor, such as if we redid our buttered toast experiment to find out whether different kinds of plates affect whether or not the toast falls butter-first. The more we test our hypothesis, the stronger it becomes!

What’s Next?

Want to learn more about the scientific method? These important high school science classes will no doubt cover it in a variety of different contexts.

Test your ability to follow the scientific method using these at-home science experiments for kids !

Need some proof that science is fun? Try making slime

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Melissa Brinks graduated from the University of Washington in 2014 with a Bachelor's in English with a creative writing emphasis. She has spent several years tutoring K-12 students in many subjects, including in SAT prep, to help them prepare for their college education.

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A Guide to Using the Scientific Method in Everyday Life

The scientific method —the process used by scientists to understand the natural world—has the merit of investigating natural phenomena in a rigorous manner. Working from hypotheses, scientists draw conclusions based on empirical data. These data are validated on large-scale numbers and take into consideration the intrinsic variability of the real world. For people unfamiliar with its intrinsic jargon and formalities, science may seem esoteric. And this is a huge problem: science invites criticism because it is not easily understood. So why is it important, then, that every person understand how science is done?

Because the scientific method is, first of all, a matter of logical reasoning and only afterwards, a procedure to be applied in a laboratory.

Individuals without training in logical reasoning are more easily victims of distorted perspectives about themselves and the world. An example is represented by the so-called “ cognitive biases ”—systematic mistakes that individuals make when they try to think rationally, and which lead to erroneous or inaccurate conclusions. People can easily overestimate the relevance of their own behaviors and choices. They can lack the ability to self-estimate the quality of their performances and thoughts . Unconsciously, they could even end up selecting only the arguments that support their hypothesis or beliefs . This is why the scientific framework should be conceived not only as a mechanism for understanding the natural world, but also as a framework for engaging in logical reasoning and discussion.

A brief history of the scientific method

The scientific method has its roots in the sixteenth and seventeenth centuries. Philosophers Francis Bacon and René Descartes are often credited with formalizing the scientific method because they contrasted the idea that research should be guided by metaphysical pre-conceived concepts of the nature of reality—a position that, at the time, was highly supported by their colleagues . In essence, Bacon thought that inductive reasoning based on empirical observation was critical to the formulation of hypotheses and the generation of new understanding : general or universal principles describing how nature works are derived only from observations of recurring phenomena and data recorded from them. The inductive method was used, for example, by the scientist Rudolf Virchow to formulate the third principle of the notorious cell theory , according to which every cell derives from a pre-existing one. The rationale behind this conclusion is that because all observations of cell behavior show that cells are only derived from other cells, this assertion must be always true.

Inductive reasoning, however, is not immune to mistakes and limitations. Referring back to cell theory, there may be rare occasions in which a cell does not arise from a pre-existing one, even though we haven’t observed it yet—our observations on cell behavior, although numerous, can still benefit from additional observations to either refute or support the conclusion that all cells arise from pre-existing ones. And this is where limited observations can lead to erroneous conclusions reasoned inductively. In another example, if one never has seen a swan that is not white, they might conclude that all swans are white, even when we know that black swans do exist, however rare they may be.

The universally accepted scientific method, as it is used in science laboratories today, is grounded in hypothetico-deductive reasoning . Research progresses via iterative empirical testing of formulated, testable hypotheses (formulated through inductive reasoning). A testable hypothesis is one that can be rejected (falsified) by empirical observations, a concept known as the principle of falsification . Initially, ideas and conjectures are formulated. Experiments are then performed to test them. If the body of evidence fails to reject the hypothesis, the hypothesis stands. It stands however until and unless another (even singular) empirical observation falsifies it. However, just as with inductive reasoning, hypothetico-deductive reasoning is not immune to pitfalls—assumptions built into hypotheses can be shown to be false, thereby nullifying previously unrejected hypotheses. The bottom line is that science does not work to prove anything about the natural world. Instead, it builds hypotheses that explain the natural world and then attempts to find the hole in the reasoning (i.e., it works to disprove things about the natural world).

How do scientists test hypotheses?

Controlled experiments

The word “experiment” can be misleading because it implies a lack of control over the process. Therefore, it is important to understand that science uses controlled experiments in order to test hypotheses and contribute new knowledge. So what exactly is a controlled experiment, then?

Let us take a practical example. Our starting hypothesis is the following: we have a novel drug that we think inhibits the division of cells, meaning that it prevents one cell from dividing into two cells (recall the description of cell theory above). To test this hypothesis, we could treat some cells with the drug on a plate that contains nutrients and fuel required for their survival and division (a standard cell biology assay). If the drug works as expected, the cells should stop dividing. This type of drug might be useful, for example, in treating cancers because slowing or stopping the division of cells would result in the slowing or stopping of tumor growth.

Although this experiment is relatively easy to do, the mere process of doing science means that several experimental variables (like temperature of the cells or drug, dosage, and so on) could play a major role in the experiment. This could result in a failed experiment when the drug actually does work, or it could give the appearance that the drug is working when it is not. Given that these variables cannot be eliminated, scientists always run control experiments in parallel to the real ones, so that the effects of these other variables can be determined. Control experiments are designed so that all variables, with the exception of the one under investigation, are kept constant. In simple terms, the conditions must be identical between the control and the actual experiment.

Coming back to our example, when a drug is administered it is not pure. Often, it is dissolved in a solvent like water or oil. Therefore, the perfect control to the actual experiment would be to administer pure solvent (without the added drug) at the same time and with the same tools, where all other experimental variables (like temperature, as mentioned above) are the same between the two (Figure 1). Any difference in effect on cell division in the actual experiment here can be attributed to an effect of the drug because the effects of the solvent were controlled.

In order to provide evidence of the quality of a single, specific experiment, it needs to be performed multiple times in the same experimental conditions. We call these multiple experiments “replicates” of the experiment (Figure 2). The more replicates of the same experiment, the more confident the scientist can be about the conclusions of that experiment under the given conditions. However, multiple replicates under the same experimental conditions are of no help when scientists aim at acquiring more empirical evidence to support their hypothesis. Instead, they need independent experiments (Figure 3), in their own lab and in other labs across the world, to validate their results.

Often times, especially when a given experiment has been repeated and its outcome is not fully clear, it is better to find alternative experimental assays to test the hypothesis.

Applying the scientific approach to everyday life

So, what can we take from the scientific approach to apply to our everyday lives?

A few weeks ago, I had an agitated conversation with a bunch of friends concerning the following question: What is the definition of intelligence?

Defining “intelligence” is not easy. At the beginning of the conversation, everybody had a different, “personal” conception of intelligence in mind, which – tacitly – implied that the conversation could have taken several different directions. We realized rather soon that someone thought that an intelligent person is whoever is able to adapt faster to new situations; someone else thought that an intelligent person is whoever is able to deal with other people and empathize with them. Personally, I thought that an intelligent person is whoever displays high cognitive skills, especially in abstract reasoning.

The scientific method has the merit of providing a reference system, with precise protocols and rules to follow. Remember: experiments must be reproducible, which means that an independent scientists in a different laboratory, when provided with the same equipment and protocols, should get comparable results. Fruitful conversations as well need precise language, a kind of reference vocabulary everybody should agree upon, in order to discuss about the same “content”. This is something we often forget, something that was somehow missing at the opening of the aforementioned conversation: even among friends, we should always agree on premises, and define them in a rigorous manner, so that they are the same for everybody. When speaking about “intelligence”, we must all make sure we understand meaning and context of the vocabulary adopted in the debate (Figure 4, point 1). This is the first step of “controlling” a conversation.

There is another downside that a discussion well-grounded in a scientific framework would avoid. The mistake is not structuring the debate so that all its elements, except for the one under investigation, are kept constant (Figure 4, point 2). This is particularly true when people aim at making comparisons between groups to support their claim. For example, they may try to define what intelligence is by comparing the achievements in life of different individuals: “Stephen Hawking is a brilliant example of intelligence because of his great contribution to the physics of black holes”. This statement does not help to define what intelligence is, simply because it compares Stephen Hawking, a famous and exceptional physicist, to any other person, who statistically speaking, knows nothing about physics. Hawking first went to the University of Oxford, then he moved to the University of Cambridge. He was in contact with the most influential physicists on Earth. Other people were not. All of this, of course, does not disprove Hawking’s intelligence; but from a logical and methodological point of view, given the multitude of variables included in this comparison, it cannot prove it. Thus, the sentence “Stephen Hawking is a brilliant example of intelligence because of his great contribution to the physics of black holes” is not a valid argument to describe what intelligence is. If we really intend to approximate a definition of intelligence, Steven Hawking should be compared to other physicists, even better if they were Hawking’s classmates at the time of college, and colleagues afterwards during years of academic research.

In simple terms, as scientists do in the lab, while debating we should try to compare groups of elements that display identical, or highly similar, features. As previously mentioned, all variables – except for the one under investigation – must be kept constant.

This insightful piece presents a detailed analysis of how and why science can help to develop critical thinking.

In a nutshell

Here is how to approach a daily conversation in a rigorous, scientific manner:

First discuss about the reference vocabulary, then discuss about the content of the discussion. Think about a researcher who is writing down an experimental protocol that will be used by thousands of other scientists in varying continents. If the protocol is rigorously written, all scientists using it should get comparable experimental outcomes. In science this means reproducible knowledge, in daily life this means fruitful conversations in which individuals are on the same page.
Adopt “controlled” arguments to support your claims. When making comparisons between groups, visualize two blank scenarios. As you start to add details to both of them, you have two options. If your aim is to hide a specific detail, the better is to design the two scenarios in a completely different manner—it is to increase the variables. But if your intention is to help the observer to isolate a specific detail, the better is to design identical scenarios, with the exception of the intended detail—it is therefore to keep most of the variables constant. This is precisely how scientists ideate adequate experiments to isolate new pieces of knowledge, and how individuals should orchestrate their thoughts in order to test them and facilitate their comprehension to others.

Not only the scientific method should offer individuals an elitist way to investigate reality, but also an accessible tool to properly reason and discuss about it.

Edited by Jason Organ, PhD, Indiana University School of Medicine.

Simone is a molecular biologist on the verge of obtaining a doctoral title at the University of Ulm, Germany. He is Vice-Director at Culturico (https://culturico.com/), where his writings span from Literature to Sociology, from Philosophy to Science. His writings recently appeared in Psychology Today, openDemocracy, Splice Today, Merion West, Uncommon Ground and The Society Pages. Follow Simone on Twitter: @simredaelli

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This has to be the best article I have ever read on Scientific Thinking. I am presently writing a treatise on how Scientific thinking can be adopted to entreat all situations.And how, a 4 year old child can be taught to adopt Scientific thinking, so that, the child can look at situations that bothers her and she could try to think about that situation by formulating the right questions. She may not have the tools to find right answers? But, forming questions by using right technique ? May just make her find a way to put her mind to rest even at that level. That is why, 4 year olds are often “eerily: (!)intelligent, I have iften been intimidated and plain embarrassed to see an intelligent and well spoken 4 year old deal with celibrity ! Of course, there are a lot of variables that have to be kept in mind in order to train children in such controlled thinking environment, as the screenplay of little Sheldon shows. Thanking the author with all my heart – #ershadspeak #wearescience #weareallscientists Ershad Khandker

Simone, thank you for this article. I have the idea that I want to apply what I learned in Biology to everyday life. You addressed this issue, and have given some basic steps in using the scientific method.

1.3: The Scientific Method - How Chemists Think

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Learning Objectives

Identify the components of the scientific method.

Scientists search for answers to questions and solutions to problems by using a procedure called the scientific method. This procedure consists of making observations, formulating hypotheses, and designing experiments; which leads to additional observations, hypotheses, and experiments in repeated cycles (Figure $\PageIndex{1}$).

Step 1: Make observations

Observations can be qualitative or quantitative. Qualitative observations describe properties or occurrences in ways that do not rely on numbers. Examples of qualitative observations include the following: "the outside air temperature is cooler during the winter season," "table salt is a crystalline solid," "sulfur crystals are yellow," and "dissolving a penny in dilute nitric acid forms a blue solution and a brown gas." Quantitative observations are measurements, which by definition consist of both a number and a unit. Examples of quantitative observations include the following: "the melting point of crystalline sulfur is 115.21° Celsius," and "35.9 grams of table salt—the chemical name of which is sodium chloride—dissolve in 100 grams of water at 20° Celsius." For the question of the dinosaurs’ extinction, the initial observation was quantitative: iridium concentrations in sediments dating to 66 million years ago were 20–160 times higher than normal.

Step 2: Formulate a hypothesis

After deciding to learn more about an observation or a set of observations, scientists generally begin an investigation by forming a hypothesis, a tentative explanation for the observation(s). The hypothesis may not be correct, but it puts the scientist’s understanding of the system being studied into a form that can be tested. For example, the observation that we experience alternating periods of light and darkness corresponding to observed movements of the sun, moon, clouds, and shadows is consistent with either one of two hypotheses:

Earth rotates on its axis every 24 hours, alternately exposing one side to the sun.
The sun revolves around Earth every 24 hours.

Suitable experiments can be designed to choose between these two alternatives. For the disappearance of the dinosaurs, the hypothesis was that the impact of a large extraterrestrial object caused their extinction. Unfortunately (or perhaps fortunately), this hypothesis does not lend itself to direct testing by any obvious experiment, but scientists can collect additional data that either support or refute it.

Step 3: Design and perform experiments

After a hypothesis has been formed, scientists conduct experiments to test its validity. Experiments are systematic observations or measurements, preferably made under controlled conditions—that is—under conditions in which a single variable changes.

Step 4: Accept or modify the hypothesis

A properly designed and executed experiment enables a scientist to determine whether or not the original hypothesis is valid. If the hypothesis is valid, the scientist can proceed to step 5. In other cases, experiments often demonstrate that the hypothesis is incorrect or that it must be modified and requires further experimentation.

Step 5: Development into a law and/or theory

More experimental data are then collected and analyzed, at which point a scientist may begin to think that the results are sufficiently reproducible (i.e., dependable) to merit being summarized in a law, a verbal or mathematical description of a phenomenon that allows for general predictions. A law simply states what happens; it does not address the question of why.

One example of a law, the law of definite proportions , which was discovered by the French scientist Joseph Proust (1754–1826), states that a chemical substance always contains the same proportions of elements by mass. Thus, sodium chloride (table salt) always contains the same proportion by mass of sodium to chlorine, in this case 39.34% sodium and 60.66% chlorine by mass, and sucrose (table sugar) is always 42.11% carbon, 6.48% hydrogen, and 51.41% oxygen by mass.

Whereas a law states only what happens, a theory attempts to explain why nature behaves as it does. Laws are unlikely to change greatly over time unless a major experimental error is discovered. In contrast, a theory, by definition, is incomplete and imperfect, evolving with time to explain new facts as they are discovered.

Because scientists can enter the cycle shown in Figure $\PageIndex{1}$ at any point, the actual application of the scientific method to different topics can take many different forms. For example, a scientist may start with a hypothesis formed by reading about work done by others in the field, rather than by making direct observations.

Example $\PageIndex{1}$

Classify each statement as a law, a theory, an experiment, a hypothesis, an observation.

Ice always floats on liquid water.
Birds evolved from dinosaurs.
Hot air is less dense than cold air, probably because the components of hot air are moving more rapidly.
When 10 g of ice were added to 100 mL of water at 25°C, the temperature of the water decreased to 15.5°C after the ice melted.
The ingredients of Ivory soap were analyzed to see whether it really is 99.44% pure, as advertised.
This is a general statement of a relationship between the properties of liquid and solid water, so it is a law.
This is a possible explanation for the origin of birds, so it is a hypothesis.
This is a statement that tries to explain the relationship between the temperature and the density of air based on fundamental principles, so it is a theory.
The temperature is measured before and after a change is made in a system, so these are observations.
This is an analysis designed to test a hypothesis (in this case, the manufacturer’s claim of purity), so it is an experiment.

Exercise $\PageIndex{1}$

Classify each statement as a law, a theory, an experiment, a hypothesis, a qualitative observation, or a quantitative observation.

Measured amounts of acid were added to a Rolaids tablet to see whether it really “consumes 47 times its weight in excess stomach acid.”
Heat always flows from hot objects to cooler ones, not in the opposite direction.
The universe was formed by a massive explosion that propelled matter into a vacuum.
Michael Jordan is the greatest pure shooter to ever play professional basketball.
Limestone is relatively insoluble in water, but dissolves readily in dilute acid with the evolution of a gas.

The scientific method is a method of investigation involving experimentation and observation to acquire new knowledge, solve problems, and answer questions. The key steps in the scientific method include the following:

Step 1: Make observations.
Step 2: Formulate a hypothesis.
Step 3: Test the hypothesis through experimentation.
Step 4: Accept or modify the hypothesis.
Step 5: Develop into a law and/or a theory.

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1.1: The Scientific Method

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Laci M. Gerhart-Barley
College of Biological Sciences - UC Davis

Biologists, and other scientists, study the world using a formal process referred to as the scientific method . The scientific method was first documented by Sir Francis Bacon (1561–1626) of England, and can be applied to almost all fields of study. The scientific method is founded upon observation, which then leads to a question and the development of a hypothesis which answers that question. The scientist can then design an experiment to test the proposed hypothesis, and makes a prediction for the outcome of the experiment, if the proposed hypothesis is true. In the following sections, we will use a simple example of the scientific method, based on a simple observation of the classroom being too warm.

Proposing a Hypothesis

A hypothesis is one possible answer to the question that arises from observations. In our example, the observation is that the classroom is too warm, and the question taht arises from that observation is why the classroom is too warm. One (of many) hypotheses is “The classroom is warm because no one turned on the air conditioning.” Another hypothesis could be “The classroom is warm because the heating is set too high."

Once a hypothesis has been developed, the scientist then makes a prediction, which is similar to a hypothesis, but generally follows the format of “If . . . then . . . .” In our example, a prediction arising from the first hypothesis might be, “ If the air-conditioning is turned on, then the classroom will no longer be too warm.” The initial steps of the scientific method (observation to prediction) are outlined in Figure 1.1.1.

Testing a Hypothesis

A valid hypothesis must be testable. It should also be falsifiable, meaning that it can be disproven by experimental results. Importantly, science does not claim to “prove” anything because scientific understandings are always subject to modification with further information. To test a hypothesis, a researcher will conduct one or more experiments designed to eliminate one or more of the hypotheses. Each experiment will have one or more variables and one or more controls. A variable is any part of the experiment that can vary or change during the experiment. The control group contains every feature of the experimental group except it is not given the manipulation that tests the hypothesis. Therefore, if the results of the experimental group differ from the control group, the difference must be due to the hypothesized manipulation, rather than some outside factor. Look for the variables and controls in the examples that follow. To test the first hypothesis, the student would find out if the air conditioning is on. If the air conditioning is turned on but does not work, then the hypothesis that the air conditioning was not turned on should be rejected. To test the second hypothesis, the student could check the settings of the classroom heating unit. If the heating unit is set at an appropriate temperature, then this hypothesis should also be rejected. Each hypothesis should be tested by carrying out appropriate experiments. Be aware that rejecting one hypothesis does not determine whether or not the other hypotheses can be accepted; it simply eliminates one hypothesis that is not valid. Using the scientific method, the hypotheses that are inconsistent with experimental data are rejected.

While this “warm classroom” example is based on observational results, other hypotheses and experiments might have clearer controls. For instance, a student might attend class on Monday and realize they had difficulty concentrating on the lecture. One observation to explain this occurrence might be, “When I eat breakfast before class, I am better able to pay attention.” The student could then design an experiment with a control to test this hypothesis.

Exercise $\PageIndex{1}$

In the example below, the scientific method is used to solve an everyday problem. Order the scientific method steps (numbered items) with the process of solving the everyday problem (lettered items). Based on the results of the experiment, is the hypothesis correct? If it is incorrect, propose some alternative hypotheses.

Observation
Hypothesis (answer)
The car battery is dead.
If the battery is dead, then the headlights also will not turn on.
My car won't start.
I turn on the headlights.
The headlights work.
Why does the car not start?

C, F, A, B, D, E

The scientific method may seem overly rigid and structured; however, there is flexibility. Often, the process of science is not as linear as the scientific method suggests and experimental results frequently inspire a new approach, highlight patterns or themes in the study system, or generate entirely new and different observations and questions. In our warm classroom example, testing the air conditioning hypothesis could, for example, unearth evidence of faulty wiring in the classroom. This observation could then inspire additional questions related to other classroom electrical concerns such as inconsistent wireless internet access, faulty audio/visual equipment functioning, non-functional power outlets, flickering lighting, etc. Notice, too, that the scientific method can be applied to solving problems that aren’t necessarily scientific in nature.

This section was adapted from OpenStax Chapter 1:2 The Process of Science

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Solving Everyday Problems with the Scientific Method: Thinking Like a Scientist (Second Edition)

This book describes how one can use The Scientific Method to solve everyday problems including medical ailments, health issues, money management, traveling, shopping, cooking, household chores, etc. It illustrates how to exploit the information collected from our five senses, how to solve problems when no information is available for the present problem situation, how to increase our chances of success by redefining a problem, and how to extrapolate our capabilities by seeing a relationship among heretofore unrelated concepts. One should formulate a hypothesis as early as possible in order to have a sense of direction regarding which path to follow. Occasionally, by making wild conjectures, creative solutions can transpire. However, hypotheses need to be well-tested. Through this way, The Scientific Method can help readers solve problems in both familiar and unfamiliar situations. Containing real-life examples of how various problems are solved — for instance, how some observant patients cure their own illnesses when medical experts have failed — this book will train readers to observe what others may have missed and conceive what others may not have contemplated. With practice, they will be able to solve more problems than they could previously imagine. In this second edition, the authors have added some more theories which they hope can help in solving everyday problems. At the same time, they have updated the book by including quite a few examples which they think are interesting. Readership: General public interested in self-help books; undergraduates majoring in education and behavioral psychology; graduates and researchers with research interests in problem solving, creativity and scientific research methodology.

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show/hide words to know

Biased: when someone presents only one viewpoint. Biased articles do not give all the facts and often mislead the reader.

Conclusion: what a person decides based on information they get through research including experiments.

Method: following a certain set of steps to make something, or find an answer to a question. Like baking a pie or fixing the tire on a bicycle.

Research: looking for answers to questions using tools like the scientific method.

What is the Scientific Method?

If you have ever seen something going on and wondered why or how it happened, you have started down the road to discovery. If you continue your journey, you are likely to guess at some of your own answers for your question. Even further along the road you might think of ways to find out if your answers are correct. At this point, whether you know it or not, you are following a path that scientists call the scientific method. If you do some experiments to see if your answer is correct and write down what you learn in a report, you have pretty much completed everything a scientist might do in a laboratory or out in the field when doing research. In fact, the scientific method works well for many things that don’t usually seem so scientific.

The Flashlight Mystery...

Like a crime detective, you can use the elements of the scientific method to find the answer to everyday problems. For example you pick up a flashlight and turn it on, but the light does not work. You have observed that the light does not work. You ask the question, Why doesn't it work? With what you already know about flashlights, you might guess (hypothesize) that the batteries are dead. You say to yourself, if I buy new batteries and replace the old ones in the flashlight, the light should work. To test this prediction you replace the old batteries with new ones from the store. You click the switch on. Does the flashlight work? No?

What else could be the answer? You go back and hypothesize that it might be a broken light bulb. Your new prediction is if you replace the broken light bulb the flashlight will work. It’s time to go back to the store and buy a new light bulb. Now you test this new hypothesis and prediction by replacing the bulb in the flashlight. You flip the switch again. The flashlight lights up. Success!

If this were a scientific project, you would also have written down the results of your tests and a conclusion of your experiments. The results of only the light bulb hypothesis stood up to the test, and we had to reject the battery hypothesis. You would also communicate what you learned to others with a published report, article, or scientific paper.

More to the Mystery...

Not all questions can be answered with only two experiments. It can often take a lot more work and tests to find an answer. Even when you find an answer it may not always be the only answer to the question. This is one reason that different scientists will work on the same question and do their own experiments.

In our flashlight example, you might never get the light to turn on. This probably means you haven’t made enough different guesses (hypotheses) to test the problem. Were the new batteries in the right way? Was the switch rusty, or maybe a wire is broken. Think of all the possible guesses you could test.

No matter what the question, you can use the scientific method to guide you towards an answer. Even those questions that do not seem to be scientific can be solved using this process. Like with the flashlight, you might need to repeat several of the elements of the scientific method to find an answer. No matter how complex the diagram, the scientific method will include the following pieces in order to be complete.

The elements of the scientific method can be used by anyone to help answer questions. Even though these elements can be used in an ordered manner, they do not have to follow the same order. It is better to think of the scientific method as fluid process that can take different paths depending on the situation. Just be sure to incorporate all of the elements when seeking unbiased answers. You may also need to go back a few steps (or a few times) to test several different hypotheses before you come to a conclusion. Click on the image to see other versions of the scientific method.

Observation – seeing, hearing, touching…
Asking a question – why or how?
Hypothesis – a fancy name for an educated guess about what causes something to happen.
Prediction – what you think will happen if…
Testing – this is where you get to experiment and be creative.
Conclusion – decide how your test results relate to your predictions.
Communicate – share your results so others can learn from your work.

Other Parts of the Scientific Method…

Now that you have an idea of how the scientific method works there are a few other things to learn so that you will be able test out your new skills and test your hypotheses.

Control - A group that is similar to other groups but is left alone so that it can be compared to see what happened to the other groups that are tested.
Data - the numbers and measurements you get from the test in a scientific experiment.
Independent variable - a variable that you change as part of your experiment. It is important to only change one independent variable for each experiment.
Dependent variable - a variable that changes when the independent variable is changed.
Controlled Variable - these are variables that you never change in your experiment.

Practicing Observations and Wondering How and Why...

It is really hard not to notice things around us and wonder about them. This is how the scientific method begins, by observing and wondering why and how. Why do leaves on trees in many parts of the world turn from green to red, orange, or yellow and fall to the ground when winter comes? How does a spider move around their web without getting stuck like its victims? Both of these questions start with observing something and asking questions. The next time you see something and ask yourself, “I wonder why that does that, or how can it do that?” try out your new detective skills, and see what answer you can find.

Try Out Your Detective Skills

Now that you have the basics of the scientific method, why not test your skills? The Science Detectives Training Room will test your problem solving ability. Step inside and see if you can escape the room. While you are there, look around and see what other interesting things might be waiting. We think you find this game a great way to learn the scientific method. In fact, we bet you will discover that you already use the scientific method and didn't even know it.

After you've learned the basics of being a detective, practice those skills in The Case of the Mystery Images . While you are there, pay attention to what's around you as you figure out just what is happening in the mystery photos that surround you.

Ready for your next challenge? Try Science Detectives: Case of the Mystery Images for even more mysteries to solve. Take your scientific abilities one step further by making observations and formulating hypothesis about the mysterious images you find within.

Acknowledgements:

We thank John Alcock for his feedback and suggestions on this article.

Science Detectives - Mystery Room Escape was produced in partnership with the Arizona Science Education Collaborative (ASEC) and funded by ASU Women & Philanthropy.

Flashlight image via Wikimedia Commons - The Oxygen Team

Chicago Manual of Style

CJ Kazilek and David Pearson. "Using the Scientific Method to Solve Mysteries ". ASU - Ask A Biologist. 08 October, 2009. https://askabiologist.asu.edu/explore/scientific-method

MLA 2017 Style

CJ Kazilek and David Pearson. "Using the Scientific Method to Solve Mysteries ". ASU - Ask A Biologist. 08 Oct 2009. ASU - Ask A Biologist, Web. 24 Mar 2024. https://askabiologist.asu.edu/explore/scientific-method

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Using the Scientific Method to Solve Mysteries

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15 Scientific Method Examples

scientific method examples and definition, explained below

The scientific method is a structured and systematic approach to investigating natural phenomena using empirical evidence .

The scientific method has been a lynchpin for rapid improvements in human development. It has been an invaluable procedure for testing and improving upon human ingenuity. It’s led to amazing scientific, technological, and medical breakthroughs.

Some common steps in a scientific approach would include:

Observation
Question formulation
Hypothesis development
Experimentation and collecting data
Analyzing results
Drawing conclusions

Definition of Scientific Method

The scientific method is a structured and systematic approach to investigating natural phenomena or events through empirical evidence.

Empirical evidence can be gathered from experimentation, observation, analysis, and interpretation of data that allows one to create generalizations about probable reasons behind those happenings.

As mentioned in the article published in the journal Nature,

“ As schoolchildren, we are taught that the scientific method involves a question and suggested explanation (hypothesis) based on observation, followed by the careful design and execution of controlled experiments, and finally validation, refinement or rejection of this hypothesis” (p. 237).

The use of scientific methods permits replication and validation of other people’s scientific analyses, leading toward improvement upon previous results, and solid empirical conclusions.

Voit (2019) adds that:

“…it not only prescribes the order and types of activities that give a scientific study validity and a stamp of approval but also has substantially shaped how we collectively think about the endeavor of investigating nature” (p. 1).

This method aims to minimize subjective biases while maximizing objectivity helping researchers gather factual data.

It follows set procedures and guidelines for testing hypotheses using controlled conditions, assuring optimum accuracy and relevance in concluding by assessing a range of aspects (Blystone & Blodgett, 2006).

Overall, the scientific method provides researchers with a structured way of inquiry that seeks insightful explanations regarding evidence-based investigation grounded in facts acquired from an array of fields.

15 Examples of Scientific Method

Medicine Delivery : Scientists use scientific method to determine the most effective way of delivering a medicine to its target location in the body. They perform experiments and gather data on the different methods of medicine delivery, monitoring factors such as dosage and time release.
Agricultural Research : Scientific method is frequently used in agricultural research to determine the most effective way to grow crops or raise livestock. This may involve testing different fertilizers, irrigation methods, or animal feed, measuring yield, and analyzing data.
Food Science and Nutrition : Nutritionists and food scientists use the scientific method to study the effects of different food types and diet on health. They design experiments to understand the impact of dietary changes on weight, disease risk, and overall health outcomes.
Environmental Studies : Researchers use scientific method to study natural ecosystems and how human activities impact them. They collect data on things like biodiversity, water quality, and pollution levels, analyzing changes over time.
Psychological Studies : Psychologists use the scientific method to understand human behavior and cognition. They conduct experiments under controlled conditions to test theories about learning, memory, social interaction, and more.
Climate Change Research : Climate scientists use the scientific method to study the Earth’s changing climate. They collect and analyze data on temperature, CO2 levels, and ice coverage to understand trends and make predictions about future changes.
Geology Exploration : Geologists use scientific method to analyze rock samples from deep in the earth’s crust and gather information about geological processes over millions of years. They evaluate data by studying patterns left behind by these processes.
Space Exploration : Scientists use scientific methods in designing space missions so that they can explore other planets or learn more about our solar system. They employ experiments like landing craft exploration missions as well as remote sensing techniques that allow them to examine far-off planets without having physically land on their surfaces.
Archaeology : Archaeologists use the scientific method to understand past human cultures. They formulate hypotheses about a site or artifact, conduct excavations or analyses, and then interpret the data to test their hypotheses.
Clinical Trials : Medical researchers use scientific method to test new treatments and therapies for various diseases. They design controlled studies that track patients’ outcomes while varying variables like dosage or treatment frequency.
Industrial Research & Development : Many companies use scientific methods in their R&D departments. For example, automakers may assess the effectiveness of anti-lock brakes before releasing them into the marketplace through tests with dummy targets.
Material Science Experiments : Engineers have extensively used scientific method experimentation efforts when designing new materials and testing which options could be flexible enough for certain applications. These experiments might include casting molten material into molds and then subjecting it to high heat to expose vulnerabilities
Chemical Engineering Investigations : Chemical engineers also abide by scientific method principles to create new chemical compounds & technologies designed to be valuable in the industry. They may experiment with different substances, changing materials’ concentration and heating conditions to ensure the final end-product safety and reliability of the material.
Biotechnology : Biotechnologists use the scientific method to develop new products or processes. For instance, they may experiment with genetic modification techniques to enhance crop resistance to pests or disease.
Physics Research : Scientists use scientific method in their work to study fundamental principles of the universe. They seek answers for how atoms and molecules are breaking down and related events that unfold naturally by running many simulations using computer models or designing sophisticated experiments to test hypotheses.

Origins of the Scientific Method

The scientific method can be traced back to ancient times when philosophers like Aristotle used observation and logic to understand the natural world.

These early philosophers were focused on understanding the world around them and sought explanations for natural phenomena through direct observation (Betz, 2010).

In the Middle Ages, Muslim scholars played a key role in developing scientific inquiry by emphasizing empirical observations.

Alhazen (a.k.a Ibn al-Haytham), for example, introduced experimental methods that helped establish optics as a modern science. He emphasized investigation through experimentation with controlled conditions (De Brouwer, 2021).

During the Scientific Revolution of the 17th century in Europe, scientists such as Francis Bacon and René Descartes began to develop what we now know as the scientific method observation (Betz, 2010).

Bacon argued that knowledge must be based on empirical evidence obtained through observation and experimentation rather than relying solely upon tradition or authority.

Descartes emphasized mathematical methods as tools in experimentation and rigorous thinking processes (Fukuyama, 2021).

These ideas later developed into systematic research designs , including hypothesis testing, controlled experiments, and statistical analysis – all of which are still fundamental aspects of modern-day scientific research.

Since then, technological advancements have allowed for more sophisticated instruments and measurements, yielding far more precise data sets scientists use today in fields ranging from Medicine & Chemistry to Astrophysics or Genetics.

So, while early Greek philosophers laid much groundwork toward an observational-based approach to explaining nature, Islam scholars furthered our understanding of logical reasoning techniques and gave rise to a more formalized methodology.

Steps in the Scientific Method

While there may be variations in the specific steps scientists follow, the general process has six key steps (Blystone & Blodgett, 2006).

Here is a brief overview of each of these steps:

1. Observation

The first step in the scientific method is to identify and observe a phenomenon that requires explanation.

This can involve asking open-ended questions, making detailed observations using our senses or tools, or exploring natural patterns, which are sources to develop hypotheses.

2. Formulation of a Hypothesis

A hypothesis is an educated guess or proposed explanation for the observed phenomenon based on previous observations & experiences or working assumptions derived from a valid literature review .

The hypothesis should be testable and falsifiable through experimentation and subsequent analysis.

3. Testing of the Hypothesis

In this step, scientists perform experiments to test their hypothesis while ensuring that all variables are controlled besides the one being observed.

The data collected in these experiments must be measurable, repeatable, and consistent.

4. Data Analysis

Researchers carefully scrutinize data gathered from experiments – typically using inferential statistics techniques to analyze whether results support their hypotheses or not.

This helps them gain important insights into what previously unknown mechanisms might exist based on statistical evidence gained about their system.

See: 15 Examples of Data Analysis

5. Drawing Conclusions

Based on their data analyses, scientists reach conclusions about whether their original hypotheses were supported by evidence obtained from testing.

If there is insufficient supporting evidence for their ideas – trying again with modified iterations of the initial idea sometimes happens.

6. Communicating Results

Once results have been analyzed and interpreted under accepted principles within the scientific community, scientists publish findings in respected peer-reviewed journals.

These publications help knowledge-driven communities establish trends within respective fields while indirectly subjecting papers reviews requests boosting research quality across the scientific discipline.

Importance of the Scientific Method

The scientific method is important because it helps us to collect reliable data and develop testable hypotheses that can be used to explain natural phenomena (Haig, 2018).

Here are some reasons why the scientific method is so essential:

Objectivity : The scientific method requires researchers to conduct unbiased experiments and analyses, which leads to more impartial conclusions. In this way, replication of findings by peers also ensures results can be relied upon as founded on sound principles allowing others confidence in building further knowledge on top of existing research.
Precision & Predictive Power : Scientific methods usually include techniques for obtaining highly precise measurements, ensuring that data collected is more meaningful with fewer uncertainties caused by limited measuring errors leading to statistically significant results having firm logical foundations. If predictions develop scientifically tested generalized defined conditions factored into the analysis, it helps in delivering realistic expectations
Validation : By following established scientific principles defined within the community – independent scholars can replicate observation data without being influenced by subjective biases or prejudices. It assures general acceptance among scientific communities who follow similar protocols when researching within respective fields.
Application & Innovation : Scientific concept advancements that occur based on correct hypothesis testing commonly lead scientists toward new discoveries, identifying potential breakthroughs in research. They pave the way for technological innovations often seen as game changers, like mapping human genome DNA onto creating novel therapies against genetic diseases or unlocking secrets of today’s universe through discoveries at LHC.
Impactful Decision-Making : Policymakers can draw from these scientific findings investing resources into informed decisions leading us toward a sustainable future. For example, research gathered about carbon pollution’s impact on climate change informs debate making policy action decisions about our planet’s environment, providing valuable knowledge-useful information benefiting societies (Haig, 2018).

The scientific method is an essential tool that has revolutionized our understanding of the natural world.

By emphasizing rigorous experimentation, objective measurement, and logical analysis- scientists can obtain more unbiased evidence with empirical validity .

Utilizing this methodology has led to groundbreaking discoveries & knowledge expansion that have shaped our modern world from medicine to technology.

The scientific method plays a crucial role in advancing research and our overall societal consensus on reliable information by providing reliable results, ensuring we can make more informed decisions toward a sustainable future.

As scientific advancements continue rapidly, ensuring we’re applying core principles of this process enables objectives to progress, paving new ways for interdisciplinary research across all fields, thereby fuelling ever-driving human curiosity.

Betz, F. (2010). Origin of scientific method. Managing Science , 21–41. https://doi.org/10.1007/978-1-4419-7488-4_2

Blystone, R. V., & Blodgett, K. (2006). WWW: The scientific method. CBE—Life Sciences Education , 5 (1), 7–11. https://doi.org/10.1187/cbe.05-12-0134

De Brouwer , P. J. S. (2021). The big r-book: From data science to learning machines and big data . John Wiley & Sons, Inc.

Defining the scientific method. (2009). Nature Methods , 6 (4), 237–237. https://doi.org/10.1038/nmeth0409-237

Fukuyama, F. (2012). The end of history and the last man . New York: Penguin.

Haig, B. D. (2018). The importance of scientific method for psychological science. Psychology, Crime & Law , 25 (6), 527–541. https://doi.org/10.1080/1068316x.2018.1557181

Voit, E. O. (2019). Perspective: Dimensions of the scientific method. PLOS Computational Biology , 15 (9), e1007279. https://doi.org/10.1371/journal.pcbi.1007279

Viktoriya Sus (MA)

Viktoriya Sus is an academic writer specializing mainly in economics and business from Ukraine. She holds a Master’s degree in International Business from Lviv National University and has more than 6 years of experience writing for different clients. Viktoriya is passionate about researching the latest trends in economics and business. However, she also loves to explore different topics such as psychology, philosophy, and more.

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Chris Drew (PhD)

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Solving sports problems with science

Download Student Worksheet

Directions for teachers:

Ask students to read the online Science News article “ Spiraling footballs wobble at one of two specific frequencies ” and discuss the first set of questions with a partner. A version of the article, “Why spiraling footballs sometimes miss the mark,” appears in the September 10, 2022 issue of Science News . Next, use the questions to run a class discussion about the scientific method. Have students answer the second set of questions individually and then share their answers with a partner. You may find that the topic of STEM careers in sports is either a good hook or a possible extension for some students. Check out the Science News Explores article “ Cool Jobs: Sports science ” to integrate it into your lesson.

Want to make it a virtual lesson? Post the online Science News article to your virtual classroom. Discuss the article and questions with your class on your virtual platform.

Step by step

1. What is a recent problem you solved? For example, did you fix something that didn’t work, mend tensions with a friend or family member or figure out how to get to the next level in a video game?

Student answers will vary.

2. Break down how you solved the problem into steps. How did you know there was a problem to solve? How did you decide to take the action you did, etc.? List the steps.

Student answers will vary. They should include at least some of the following steps: noticing something was off and wondering why, determining a way to try to resolve it, trying something to resolve it and deciding whether it worked or not.

3. The scientific method is a systematic way to solve problems and answer questions in science and engineering. List the steps of the scientific method. Use an external resource if necessary.

Make observations. Analyze your observations and develop a measurable, testable question, or a hypothesis. Develop a procedural method to test your hypothesis, while collecting appropriate data. Analyze your data to determine your results and new hypotheses.

4. Read the Science News article “ Spiraling footballs wobble at one of two specific frequencies .” Using the steps of the scientific method from your answer to the previous question, give an example of each step from the article.

Make observations: When a football is thrown it wobbles and veers away from its intended target. Analyze your observations and develop a measurable, testable question, or a hypothesis: What forces cause a football to wobble? Develop a procedural method to test your hypothesis, while collecting appropriate data: Create a computer simulation to determine wobble rates based on the football’s speed and spiral rate. Analyze your data to determine your results: Footballs wobble at rates of one or five times per second when the spinning momentum interacts with the twisting force. Develop new hypotheses or questions: How much does the wobble rate affect the football’s path?

Science in sports

1. What is your favorite sport? Do you like to play it? Do you watch it on TV? Have you ever watched it played professionally? Do you have a favorite professional team?

2. Choose a position in your favorite sport. What are some skills required to be successful in the position? For example, a soccer player that takes free kicks needs to be able to angle their foot correctly to put spin on the ball to get it over a wall of players.

3. Search for your sport in the Science News Explores archive and choose an article to read. What is the article about? What scientific question does it ask? If you can’t find an article about your chosen sport, check out these examples:

Why sports are becoming all about numbers – lots and lots of numbers

Let’s learn about the science of the Winter Olympics

These young researchers take aim at sports

4. Come up with a testable, measurable question you’d like to explore about the sport of your choice. When brainstorming a question, it might help to think about things related to your sport such as skills, health and physical abilities, performance, statistical averages etc.

Student answers will vary. For instance, questions could explore equipment use, like how the weight of a baseball bat relates to the distance the ball is hit; a skill, like whether the height of where a ball is hit affects the number of aces served in tennis; or statistics, such as whether a batter’s RBI (runs batted in) score is correlated to their place in the lineup in softball. Questions could also be about the health and physical abilities of athletes. For example, does the amount of sleep the night before a race impact a swimmer’s time?

5. Explain how you would attempt to answer the question using the steps of the scientific method. What would you do for each step?

Student answers will vary. As an extension, students could develop a full testing procedure and perform the experiment.

1-6. Label the parts of the following graph of the scientific method.

Possible Answers

1. What part of the scientific method does (1) represent?

2. What part of the scientific method does (2) represent?

3. What part of the scientific method does (3) represent?

4. What part of the scientific method does (4) represent?

5. What part of the scientific method does (5) represent?

6. to 10. In each of the following indicate which parts of the scientific method are present. Mark only those parts of the scientific method that are explicit in the problem description.

Parts of the Scientific Method

6. The U.S. Forest Service is given the task of managing Forest Service lands for multiple uses (e.g., lumber, recreation, conservation). As with any government agency, there is an elaborate set of regulations that must be followed in pursuing this charge, and of course, there is also an informal set of procedures that is and has been used to conduct this business. But this agency does not implement any procedure to consider what kind of job they do or how they might improve execution of those duties.

Answer: Goal and model are present. The goal is the task ostensibly assigned to them (manage the lands). The model is their set of procedures for implementing their duties. But they do not gather data on their performance nor evaluate whether their procedures are actually doing what is intended, so the latter 3 steps of the scientific method are absent. It is in fact typical of government agencies that no provisions are made for evaluating and improving their performance.

7. A car mechanic is given a 1966 Mustang to fix. The owner's complaint is that the engine runs roughly, especially when the car is accelerating. The mechanic's first guess as to the source of the problem is that the fuel mix is too rich. After adjusting the fuel mix valve, the problem persists, so the mechanic then guesses that the sparkplug advance mechanism is faulty. Replacing the sparkplug advance mechanism (the diaphragm attached to the distributor) fixes the problem, and the fixed car is returned to the owner.

Answer: All elements are present. The goal is clearly defined as fixing the engine trouble. The first model of the problem is the mechanic's first guess (fuel mix problems). Data are the observations made after adjusting the fuel mix, and evaluation is the mechanic's observation that the problem remains. Revision is the mechanic's second guess (sparkplug advance).

8. To determine the most effective sales strategy, consumer goods retailers sometimes do experiments. These experiments help the retailer understand how people decide what to purchase. For example, the retailer can change whether a product is displayed on a high or low shelf, or can change which products are displayed nearby. The retailer then analyzes these data, and changes their product displays so as to maximize sales.

9. To understand the AIDS epidemic and develop improved ways of controlling it, scientists use mathematical descriptions of the epidemic.

10. The typical university class is intended to convey specific information to the students enrolled. The usual format for conveying this material involves lectures and reading material which the students are expected to learn. The extent to which students actually succeed in learning the material is determined from exam scores.

Answer: The first four features of the scientific method are present. However, it is not commonplace for instructors to revise their teaching methods in response to poor class performance. Instead, a grading curve is implemented or future exams are made less difficult to increase the average score. In other words, evaluation serves to change the instructor's expectations but does not change the instructor's teaching methods. In fact, use of the scientific method in improving teaching is even more non-existent than is evident in this simple example. Instructors almost never evaluate student knowledge at the beginning of class, so they cannot evaluate what the student has learned, rather they can only evaluate what the student knows. The common assumption is that students don't already know the material being taught, but that assumption is obviously and often false.

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Original research article, learning scientific observation with worked examples in a digital learning environment.

1 Department Educational Sciences, Chair for Formal and Informal Learning, Technical University Munich School of Social Sciences and Technology, Munich, Germany
2 Aquatic Systems Biology Unit, TUM School of Life Sciences, Technical University of Munich, Freising, Germany

Science education often aims to increase learners’ acquisition of fundamental principles, such as learning the basic steps of scientific methods. Worked examples (WE) have proven particularly useful for supporting the development of such cognitive schemas and successive actions in order to avoid using up more cognitive resources than are necessary. Therefore, we investigated the extent to which heuristic WE are beneficial for supporting the acquisition of a basic scientific methodological skill—conducting scientific observation. The current study has a one-factorial, quasi-experimental, comparative research design and was conducted as a field experiment. Sixty two students of a German University learned about scientific observation steps during a course on applying a fluvial audit, in which several sections of a river were classified based on specific morphological characteristics. In the two experimental groups scientific observation was supported either via faded WE or via non-faded WE both presented as short videos. The control group did not receive support via WE. We assessed factual and applied knowledge acquisition regarding scientific observation, motivational aspects and cognitive load. The results suggest that WE promoted knowledge application: Learners from both experimental groups were able to perform the individual steps of scientific observation more accurately. Fading of WE did not show any additional advantage compared to the non-faded version in this regard. Furthermore, the descriptive results reveal higher motivation and reduced extraneous cognitive load within the experimental groups, but none of these differences were statistically significant. Our findings add to existing evidence that WE may be useful to establish scientific competences.

1 Introduction

Learning in science education frequently involves the acquisition of basic principles or generalities, whether of domain-specific topics (e.g., applying a mathematical multiplication rule) or of rather universal scientific methodologies (e.g., performing the steps of scientific observation) ( Lunetta et al., 2007 ). Previous research has shown that worked examples (WE) can be considered particularly useful for developing such cognitive schemata during learning to avoid using more cognitive resources than necessary for learning successive actions ( Renkl et al., 2004 ; Renkl, 2017 ). WE consist of the presentation of a problem, consecutive solution steps and the solution itself. This is especially advantageous in initial cognitive skill acquisition, i.e., for novice learners with low prior knowledge ( Kalyuga et al., 2001 ). With growing knowledge, fading WE can lead from example-based learning to independent problem-solving ( Renkl et al., 2002 ). Preliminary work has shown the advantage of WE in specific STEM domains like mathematics ( Booth et al., 2015 ; Barbieri et al., 2021 ), but less studies have investigated their impact on the acquisition of basic scientific competencies that involve heuristic problem-solving processes (scientific argumentation, Schworm and Renkl, 2007 ; Hefter et al., 2014 ; Koenen et al., 2017 ). In the realm of natural sciences, various basic scientific methodologies are employed to acquire knowledge, such as experimentation or scientific observation ( Wellnitz and Mayer, 2013 ). During the pursuit of knowledge through scientific inquiry activities, learners may encounter several challenges and difficulties. Similar to the hurdles faced in experimentation, where understanding the criteria for appropriate experimental design, including the development, measurement, and evaluation of results, is crucial ( Sirum and Humburg, 2011 ; Brownell et al., 2014 ; Dasgupta et al., 2014 ; Deane et al., 2014 ), scientific observation additionally presents its own set of issues. In scientific observation, e.g., the acquisition of new insights may be somewhat incidental due to spontaneous and uncoordinated observations ( Jensen, 2014 ). To address these challenges, it is crucial to provide instructional support, including the use of WE, particularly when observations are carried out in a more self-directed manner.

For this reason, the aim of the present study was to determine the usefulness of digitally presented WE to support the acquisition of a basic scientific methodological skill—conducting scientific observations—using a digital learning environment. In this regard, this study examined the effects of different forms of digitally presented WE (non-faded vs. faded) on students’ cognitive and motivational outcomes and compared them to a control group without WE. Furthermore, the combined perspective of factual and applied knowledge, as well as motivational and cognitive aspects, represent further value added to the study.

2 Theoretical background

2.1 worked examples.

WE have been commonly used in the fields of STEM education (science, technology, engineering, and mathematics) ( Booth et al., 2015 ). They consist of a problem statement, the steps to solve the problem, and the solution itself ( Atkinson et al., 2000 ; Renkl et al., 2002 ; Renkl, 2014 ). The success of WE can be explained by their impact on cognitive load (CL) during learning, based on assumptions from Cognitive Load Theory ( Sweller, 2006 ).

Learning with WE is considered time-efficient, effective, and superior to problem-based learning (presentation of the problem without demonstration of solution steps) when it comes to knowledge acquisition and transfer (WE-effect, Atkinson et al., 2000 ; Van Gog et al., 2011 ). Especially WE can help by reducing the extraneous load (presentation and design of the learning material) and, in turn, can lead to an increase in germane load (effort of the learner to understand the learning material) ( Paas et al., 2003 ; Renkl, 2014 ). With regard to intrinsic load (difficulty and complexity of the learning material), it is still controversially discussed if it can be altered by instructional design, e.g., WE ( Gerjets et al., 2004 ). WE have a positive effect on learning and knowledge transfer, especially for novices, as the step-by-step presentation of the solution requires less extraneous mental effort compared to problem-based learning ( Sweller et al., 1998 ; Atkinson et al., 2000 ; Bokosmaty et al., 2015 ). With growing knowledge, WE can lose their advantages (due to the expertise-reversal effect), and scaffolding learning via faded WE might be more successful for knowledge gain and transfer ( Renkl, 2014 ). Faded WE are similar to complete WE, but fade out solution steps as knowledge and competencies grow. Faded WE enhance near-knowledge transfer and reduce errors compared to non-faded WE ( Renkl et al., 2000 ).

In addition, the reduction of intrinsic and extraneous CL by WE also has an impact on learner motivation, such as interest ( Van Gog and Paas, 2006 ). Um et al. (2012) showed that there is a strong positive correlation between germane CL and the motivational aspects of learning, like satisfaction and emotion. Gupta (2019) mentions a positive correlation between CL and interest. Van Harsel et al. (2019) found that WE positively affect learning motivation, while no such effect was found for problem-solving. Furthermore, learning with WE increases the learners’ belief in their competence in completing a task. In addition, fading WE can lead to higher motivation for more experienced learners, while non-faded WE can be particularly motivating for learners without prior knowledge ( Paas et al., 2005 ). In general, fundamental motivational aspects during the learning process, such as situational interest ( Lewalter and Knogler, 2014 ) or motivation-relevant experiences, like basic needs, are influenced by learning environments. At the same time, their use also depends on motivational characteristics of the learning process, such as self-determined motivation ( Deci and Ryan, 2012 ). Therefore, we assume that learning with WE as a relevant component of a learning environment might also influence situational interest and basic needs.

2.1.1 Presentation of worked examples

WE are frequently used in digital learning scenarios ( Renkl, 2014 ). When designing WE, the application via digital learning media can be helpful, as their content can be presented in different ways (video, audio, text, and images), tailored to the needs of the learners, so that individual use is possible according to their own prior knowledge or learning pace ( Mayer, 2001 ). Also, digital media can present relevant information in a timely, motivating, appealing and individualized way and support learning in an effective and needs-oriented way ( Mayer, 2001 ). The advantages of using digital media in designing WE have already been shown in previous studies. Dart et al. (2020) presented WE as short videos (WEV). They report that the use of WEV leads to increased student satisfaction and more positive attitudes. Approximately 90% of the students indicated an active learning approach when learning with the WEV. Furthermore, the results show that students improved their content knowledge through WEV and that they found WEV useful for other courses as well.

Another study ( Kay and Edwards, 2012 ) presented WE as video podcasts. Here, the advantages of WE regarding self-determined learning in terms of learning location, learning time, and learning speed were shown. Learning performance improved significantly after use. The step-by-step, easy-to-understand explanations, the diagrams, and the ability to determine the learning pace by oneself were seen as beneficial.

Multimedia WE can also be enhanced with self-explanation prompts ( Berthold et al., 2009 ). Learning from WE with self-explanation prompts was shown to be superior to other learning methods, such as hypertext learning and observational learning.

In addition to presenting WE in different medial ways, WE can also comprise different content domains.

2.1.2 Content and context of worked examples

Regarding the content of WE, algorithmic and heuristic WE, as well as single-content and double-content WE, can be distinguished ( Reiss et al., 2008 ; Koenen et al., 2017 ; Renkl, 2017 ). Algorithmic WE are traditionally used in the very structured mathematical–physical field. Here, an algorithm with very specific solution steps is to learn, for example, in probability calculation ( Koenen et al., 2017 ). In this study, however, we focus on heuristic double-content WE. Heuristic WE in science education comprise fundamental scientific working methods, e.g., conducting experiments ( Koenen et al., 2017 ). Furthermore, double-content WE contain two learning domains that are relevant for the learning process: (1) the learning domain describes the primarily to be learned abstract process or concept, e.g., scientific methodologies like observation (see section 2.2), while (2) the exemplifying domain consists of the content that is necessary to teach this process or concept, e.g., mapping of river structure ( Renkl et al., 2009 ).

Depending on the WE content to be learned, it may be necessary for learning to take place in different settings. This can be in a formal or informal learning setting or a non-formal field setting. In this study, the focus is on learning scientific observation (learning domain) through river structure mapping (exemplary domain), which takes place with the support of digital media in a formal (university) setting, but in an informal context (nature).

2.2 Scientific observation

Scientific observation is fundamental to all scientific activities and disciplines ( Kohlhauf et al., 2011 ). Scientific observation must be clearly distinguished from everyday observation, where observation is purely a matter of noticing and describing specific characteristics ( Chinn and Malhotra, 2001 ). In contrast to this everyday observation, scientific observation as a method of knowledge acquisition can be described as a rather complex activity, defined as the theory-based, systematic and selective perception of concrete systems and processes without any fundamental manipulation ( Wellnitz and Mayer, 2013 ). Wellnitz and Mayer (2013) described the scientific observation process via six steps: (1) formulation of the research question (s), (2) deduction of the null hypothesis and the alternative hypothesis, (3) planning of the research design, (4) conducting the observation, (5) analyzing the data, and (6) answering the research question(s) on this basis. Only through reliable and qualified observation, valid data can be obtained that provide solid scientific evidence ( Wellnitz and Mayer, 2013 ).

Since observation activities are not trivial and learners often observe without generating new knowledge or connecting their observations to scientific explanations and thoughts, it is important to provide support at the related cognitive level, so that observation activities can be conducted in a structured way according to pre-defined criteria ( Ford, 2005 ; Eberbach and Crowley, 2009 ). Especially during field-learning experiences, scientific observation is often spontaneous and uncoordinated, whereby random discoveries result in knowledge gain ( Jensen, 2014 ).

To promote successful observing in rather unstructured settings like field trips, instructional support for the observation process seems useful. To guide observation activities, digitally presented WE seem to be an appropriate way to introduce learners to the individual steps of scientific observation using concrete examples.

2.3 Research questions and hypothesis

The present study investigates the effect of digitally presented double-content WE that supports the mapping of a small Bavarian river by demonstrating the steps of scientific observation. In this analysis, we focus on the learning domain of the WE and do not investigate the exemplifying domain in detail. Distinct ways of integrating WE in the digital learning environment (faded WE vs. non-faded WE) are compared with each other and with a control group (no WE). The aim is to examine to what extent differences between those conditions exist with regard to (RQ1) learners’ competence acquisition [acquisition of factual knowledge about the scientific observation method (quantitative data) and practical application of the scientific observation method (quantified qualitative data)], (RQ2) learners’ motivation (situational interest and basic needs), and (RQ3) CL. It is assumed that (Hypothesis 1), the integration of WE (faded and non-faded) leads to significantly higher competence acquisition (factual and applied knowledge), significantly higher motivation and significantly lower extraneous CL as well as higher germane CL during the learning process compared to a learning environment without WE. No differences between the conditions are expected regarding intrinsic CL. Furthermore, it is assumed (Hypothesis 2) that the integration of faded WE leads to significantly higher competence acquisition, significantly higher motivation, and lower extraneous CL as well as higher germane CL during the learning processes compared to non-faded WE. No differences between the conditions are expected with regard to intrinsic CL.

The study took place during the field trips of a university course on the application of a fluvial audit (FA) using the German working aid for mapping the morphology of rivers and their floodplains ( Bayerisches Landesamt für Umwelt, 2019 ). FA is the leading fluvial geomorphological tool for application to data collection contiguously along all watercourses of interest ( Walker et al., 2007 ). It is widely used because it is a key example of environmental conservation and monitoring that needs to be taught to students of selected study programs; thus, knowing about the most effective ways of learning is of high practical relevance.

3.1 Sample and design

3.1.1 sample.

The study was conducted with 62 science students and doctoral students of a German University (age M = 24.03 years; SD = 4.20; 36 females; 26 males). A total of 37 participants had already conducted a scientific observation and would rate their knowledge in this regard at a medium level ( M = 3.32 out of 5; SD = 0.88). Seven participants had already conducted an FA and would rate their knowledge in this regard at a medium level ( M = 3.14 out of 5; SD = 0.90). A total of 25 participants had no experience at all. Two participants had to be excluded from the sample afterward because no posttest results were available.

3.1.2 Design

The study has a 1-factorial quasi-experimental comparative research design and is conducted as a field experiment using a pre/posttest design. Participants were randomly assigned to one of three conditions: no WE ( n = 20), faded WE ( n = 20), and non-faded WE ( n = 20).

3.2 Implementation and material

3.2.1 implementation.

The study started with an online kick-off meeting where two lecturers informed all students within an hour about the basics regarding the assessment of the structural integrity of the study river and the course of the field trip days to conduct an FA. Afterward, within 2 weeks, students self-studied via Moodle the FA following the German standard method according to the scoresheets of Bayerisches Landesamt für Umwelt (2019) . This independent preparation using the online presented documents was a necessary prerequisite for participation in the field days and was checked in the pre-testing. The preparatory online documents included six short videos and four PDF files on the content, guidance on the German protocol of the FA, general information on river landscapes, information about anthropogenic changes in stream morphology and the scoresheets for applying the FA. In these sheets, the river and its floodplain are subdivided into sections of 100 m in length. Each of these sections is evaluated by assessing 21 habitat factors related to flow characteristics and structural variability. The findings are then transferred into a scoring system for the description of structural integrity from 1 (natural) to 7 (highly modified). Habitat factors have a decisive influence on the living conditions of animals and plants in and around rivers. They included, e.g., variability in water depth, stream width, substratum diversity, or diversity of flow velocities.

3.2.2 Materials

On the field trip days, participants were handed a tablet and a paper-based FA worksheet (last accessed 21st September 2022). 1 This four-page assessment sheet was accompanied by a digital learning environment presented on Moodle that instructed the participants on mapping the water body structure and guided the scientific observation method. All three Moodle courses were identical in structure and design; the only difference was the implementation of the WE. Below, the course without WE are described first. The other two courses have an identical structure, but contain additional WE in the form of learning videos.

3.2.3 No worked example

After a short welcome and introduction to the course navigation, the FA started with the description of a short hypothetical scenario: Participants should take the role of an employee of an urban planning office that assesses the ecomorphological status of a small river near a Bavarian city. The river was divided into five sections that had to be mapped separately. The course was structured accordingly. At the beginning of each section, participants had to formulate and write down a research question, and according to hypotheses regarding the ecomorphological status of the river’s section, they had to collect data in this regard via the mapping sheet and then evaluate their data and draw a conclusion. Since this course serves as a control group, no WE videos supporting the scientific observation method were integrated. The layout of the course is structured like a book, where it is not possible to scroll back. This is important insofar as the participants do not have the possibility to revisit information in order to keep the conditions comparable as well as distinguishable.

3.2.4 Non-faded worked example

In the course with no-faded WE, three instructional videos are shown for each of the five sections. In each of the three videos, two steps of the scientific observation method are presented so that, finally, all six steps of scientific observation are demonstrated. The mapping of the first section starts after the general introduction (as described above) with the instruction to work on the first two steps of scientific observation: the formulation of a research question and hypotheses. To support this, a video of about 4 min explains the features of scientific sound research questions and hypotheses. To this aim, a practical example, including explanations and tips, is given regarding the formulation of research questions and hypotheses for this section (e.g., “To what extent does the building development and the closeness of the path to the water body have an influence on the structure of the water body?” Alternative hypothesis: It is assumed that the housing development and the closeness of the path to the water body have a negative influence on the water body structure. Null hypothesis: It is assumed that the housing development and the closeness of the path to the watercourse have no negative influence on the watercourse structure.). Participants should now formulate their own research questions and hypotheses, write them down in a text field at the end of the page, and then skip to the next page. The next two steps of scientific observation, planning and conducting, are explained in a short 4-min video. To this aim, a practical example including explanations and tips is given regarding planning and conducting scientific for this section (e.g., “It’s best to go through each evaluation category carefully one by one that way you are sure not to forget anything!”). Now, participants were asked to collect data for the first section using their paper-based FA worksheet. Participants individually surveyed the river and reported their results in the mapping sheet by ticking the respective boxes in it. After collecting this data, they returned to the digital learning environment to learn how to use these data by studying the last two steps of scientific observation, evaluation, and conclusion. The third 4-min video explained how to evaluate and interpret collected data. For this purpose, a practical example with explanations and tips is given regarding evaluating and interpreting data for this section (e.g., “What were the individual points that led to the assessment? Have there been points that were weighted more than others? Remember the introduction video!”). At the end of the page, participants could answer their before-stated research questions and hypotheses by evaluating their collected data and drawing a conclusion. This brings participants to the end of the first mapping section. Afterward, the cycle begins again with the second section of the river that has to be mapped. Again, participants had to conduct the steps of scientific observation, guided by WE videos, explaining the steps in slightly different wording or with different examples. A total of five sections are mapped, in which the structure of the learning environment and the videos follow the same procedure.

3.2.5 Faded worked example

The digital learning environment with the faded WE follow the same structure as the version with the non-faded WE. However, in this version, the information in the WE videos is successively reduced. In the first section, all three videos are identical to the version with the non-faded WE. In the second section, faded content was presented as follows: the tip at the end was omitted in all three videos. In the third section, the tip and the practical example were omitted. In the fourth and fifth sections, no more videos were presented, only the work instructions.

3.3 Procedure

The data collection took place on four continuous days on the university campus, with a maximum group size of 15 participants on each day. The students were randomly assigned to one of the three conditions (no WE vs. faded WE vs. non-faded WE). After a short introduction to the procedure, the participants were handed the paper-based FA worksheet and one tablet per person. Students scanned the QR code on the first page of the worksheet that opened the pretest questionnaire, which took about 20 min to complete. After completing the questionnaire, the group walked for about 15 min to the nearby small river that was to be mapped. Upon arrival, there was first a short introduction to the digital learning environment and a check that the login (via university account on Moodle) worked. During the next 4 h, the participants individually mapped five segments of the river using the cartography worksheet. They were guided through the steps of scientific observation using the digital learning environment on the tablet. The results of their scientific observation were logged within the digital learning environment. At the end of the digital learning environment, participants were directed to the posttest via a link. After completing the test, the tablets and mapping sheets were returned. Overall, the study took about 5 h per group each day.

3.4 Instruments

In the pretest, sociodemographic data (age and gender), the study domain and the number of study semesters were collected. Additionally, the previous scientific observation experience and the estimation of one’s own ability in this regard were assessed. For example, it was asked whether scientific observation had already been conducted and, if so, how the abilities were rated on a 5-point scale from very low to very high. Preparation for the FA on the basis of the learning material was assessed: Participants were asked whether they had studied all six videos and all four PDF documents, with the response options not at all, partially, and completely. Furthermore, a factual knowledge test about scientific observation and questions about self-determination theory was administered. The posttest used the same knowledge test, and additional questions on basic needs, situational interest, measures of CL and questions about the usefulness of the WE. All scales were presented online, and participants reached the questionnaire via QR code.

3.4.1 Scientific observation competence acquisition

For the factual knowledge (quantitative assessment of the scientific observation competence), a single-choice knowledge test with 12 questions was developed and used as pre- and posttest with a maximum score of 12 points. It assesses the learners’ knowledge of the scientific observation method regarding the steps of scientific observation, e.g., formulating research questions and hypotheses or developing a research design. The questions are based on Wahser (2008 , adapted by Koenen, 2014 ) and adapted to scientific observation: “Although you are sure that you have conducted the scientific observation correctly, an unexpected result turns up. What conclusion can you draw?” Each question has four answer options (one of which is correct) and, in addition, one “I do not know” option.

For the applied knowledge (quantified qualitative assessment of the scientific observation competence), students’ scientific observations written in the digital learning environment were analyzed. A coding scheme was used with the following codes: 0 = insufficient (text field is empty or includes only insufficient key points), 1 = sufficient (a research question and no hypotheses or research question and inappropriate hypotheses are stated), 2 = comprehensive (research question and appropriate hypothesis or research question and hypotheses are stated, but, e.g., incorrect null hypothesis), 3 = very comprehensive (correct research question, hypothesis and null hypothesis are stated). One example of a very comprehensive answer regarding the research question and hypothesis is: To what extent does the lack of riparian vegetation have an impact on water body structure? Hypothesis: The lack of shore vegetation has a negative influence on the water body structure. Null hypothesis: The lack of shore vegetation has no influence on the water body structure. Afterward, a sum score was calculated for each participant. Five times, a research question and hypotheses (steps 1 and 2 in the observation process) had to be formulated (5 × max. 3 points = 15 points), and five times, the research questions and hypotheses had to be answered (steps 5 and 6 in the observation process: evaluation and conclusion) (5 × max. 3 points = 15 points). Overall, participants could reach up to 30 points. Since the observation and evaluation criteria in data collection and analysis were strongly predetermined by the scoresheet, steps 3 and 4 of the observation process (planning and conducting) were not included in the analysis.

All 600 cases (60 participants, each 10 responses to code) were coded by the first author. For verification, 240 cases (24 randomly selected participants, eight from each course) were cross-coded by an external coder. In 206 of the coded cases, the raters agreed. The cases in which the raters did not agree were discussed together, and a solution was found. This results in Cohen’s κ = 0.858, indicating a high to very high level of agreement. This indicates that the category system is clearly formulated and that the individual units of analysis could be correctly assigned.

3.4.2 Self-determination index

For the calculation of the self-determination index (SDI-index), Thomas and Müller (2011) scale for self-determination was used in the pretest. The scale consists of four subscales: intrinsic motivation (five items; e.g., I engage with the workshop content because I enjoy it; reliability of alpha = 0.87), identified motivation (four items; e.g., I engage with the workshop content because it gives me more options when choosing a career; alpha = 0.84), introjected motivation (five items; e.g., I engage with the workshop content because otherwise I would have a guilty feeling; alpha = 0.79), and external motivation (three items, e.g., I engage with the workshop content because I simply have to learn it; alpha = 0.74). Participants could indicate their answers on a 5-point Likert scale ranging from 1 = completely disagree to 5 = completely agree. To calculate the SDI-index, the sum of the self-determined regulation styles (intrinsic and identified) is subtracted from the sum of the external regulation styles (introjected and external), where intrinsic and external regulation are scored two times ( Thomas and Müller, 2011 ).

3.4.3 Motivation

Basic needs were measured in the posttest with the scale by Willems and Lewalter (2011) . The scale consists of three subscales: perceived competence (four items; e.g., during the workshop, I felt that I could meet the requirements; alpha = 0.90), perceived autonomy (five items; e.g., during the workshop, I felt that I had a lot of freedom; alpha = 0.75), and perceived autonomy regarding personal wishes and goals (APWG) (four items; e.g., during the workshop, I felt that the workshop was how I wish it would be; alpha = 0.93). We added all three subscales to one overall basic needs scale (alpha = 0.90). Participants could indicate their answers on a 5-point Likert scale ranging from 1 = completely disagree to 5 = completely agree.

Situational interest was measured in the posttest with the 12-item scale by Lewalter and Knogler (2014 ; Knogler et al., 2015 ; Lewalter, 2020 ; alpha = 0.84). The scale consists of two subscales: catch (six items; e.g., I found the workshop exciting; alpha = 0.81) and hold (six items; e.g., I would like to learn more about parts of the workshop; alpha = 0.80). Participants could indicate their answers on a 5-point Likert scale ranging from 1 = completely disagree to 5 = completely agree.

3.4.4 Cognitive load

In the posttest, CL was used to examine the mental load during the learning process. The intrinsic CL (three items; e.g., this task was very complex; alpha = 0.70) and extraneous CL (three items; e.g., in this task, it is difficult to identify the most important information; alpha = 0.61) are measured with the scales from Klepsch et al. (2017) . The germane CL (two items; e.g., the learning session contained elements that supported me to better understand the learning material; alpha = 0.72) is measured with the scale from Leppink et al. (2013) . Participants could indicate their answers on a 5-point Likert scale ranging from 1 = completely disagree to 5 = completely agree.

3.4.5 Attitudes toward worked examples

To measure how effective participants rated the WE, we used two scales related to the WE videos as instructional support. The first scale from Renkl (2001) relates to the usefulness of WE. The scale consists of four items (e.g., the explanations were helpful; alpha = 0.71). Two items were recoded because they were formulated negatively. The second scale is from Wachsmuth (2020) and relates to the participant’s evaluation of the WE. The scale consists of nine items (e.g., I always did what was explained in the learning videos; alpha = 0.76). Four items were recoded because they were formulated negatively. Participants could indicate their answers on a 5-point Likert scale ranging from 1 = completely disagree to 5 = completely agree.

3.5 Data analysis

An ANOVA was used to calculate if the variable’s prior knowledge and SDI index differed between the three groups. However, as no significant differences between the conditions were found [prior factual knowledge: F (2, 59) = 0.15, p = 0.865, η 2 = 0.00 self-determination index: F (2, 59) = 0.19, p = 0.829, η 2 = 0.00], they were not included as covariates in subsequent analyses.

Furthermore, a repeated measure, one-way analysis of variance (ANOVA), was conducted to compare the three treatment groups (no WE vs. faded WE vs. non-faded WE) regarding the increase in factual knowledge about the scientific observation method from pretest to posttest.

A MANOVA (multivariate analysis) was calculated with the three groups (no WE vs. non-faded WE vs. faded WE) as a fixed factor and the dependent variables being the practical application of the scientific observation method (first research question), situational interest, basic needs (second research question), and CL (third research question).

Additionally, to determine differences in applied knowledge even among the three groups, Bonferroni-adjusted post-hoc analyses were conducted.

The descriptive statistics between the three groups in terms of prior factual knowledge about the scientific observation method and the self-determination index are shown in Table 1 . The descriptive statistics revealed only small, non-significant differences between the three groups in terms of factual knowledge.

Table 1 . Means (standard deviations) of factual knowledge tests (pre- and posttest) and self-determination index for the three different groups.

The results of the ANOVA revealed that the overall increase in factual knowledge from pre- to posttest just misses significance [ F (1, 57) = 3.68, p = 0.060, η 2 = 0 0.06]. Furthermore, no significant differences between the groups were found regarding the acquisition of factual knowledge from pre- to posttest [ F (2, 57) = 2.93, p = 0.062, η 2 = 0.09].

An analysis of the descriptive statistics showed that the largest differences between the groups were found in applied knowledge (qualitative evaluation) and extraneous load (see Table 2 ).

Table 2 . Means (standard deviations) of dependent variables with the three different groups.

Results of the MANOVA revealed significant overall differences between the three groups [ F (12, 106) = 2.59, p = 0.005, η 2 = 0.23]. Significant effects were found for the application of knowledge [ F (2, 57) = 13.26, p = <0.001, η 2 = 0.32]. Extraneous CL just missed significance [ F (2, 57) = 2.68, p = 0.065, η 2 = 0.09]. There were no significant effects for situational interest [ F (2, 57) = 0.44, p = 0.644, η 2 = 0.02], basic needs [ F (2, 57) = 1.22, p = 0.302, η 2 = 0.04], germane CL [ F (2, 57) = 2.68, p = 0.077, η 2 = 0.09], and intrinsic CL [ F (2, 57) = 0.28, p = 0.757, η 2 = 0.01].

Bonferroni-adjusted post hoc analysis revealed that the group without WE had significantly lower scores in the evaluation of the applied knowledge than the group with non-faded WE ( p = <0.001, M diff = −8.90, 95% CI [−13.47, −4.33]) and then the group with faded WE ( p = <0.001, M diff = −7.40, 95% CI [−11.97, −2.83]). No difference was found between the groups with faded and non-faded WE ( p = 1.00, M diff = −1.50, 95% CI [−6.07, 3.07]).

The descriptive statistics regarding the perceived usefulness of WE and participants’ evaluation of the WE revealed that the group with the faded WE rated usefulness slightly higher than the participants with non-faded WE and also reported a more positive evaluation. However, the results of a MANOVA revealed no significant overall differences [ F (2, 37) = 0.32, p = 0.732, η 2 = 0 0.02] (see Table 3 ).

Table 3 . Means (standard deviations) of dependent variables with the three different groups.

5 Discussion

This study investigated the use of WE to support students’ acquisition of science observation. Below, the research questions are answered, and the implications and limitations of the study are discussed.

5.1 Results on factual and applied knowledge

In terms of knowledge gain (RQ1), our findings revealed no significant differences in participants’ results of the factual knowledge test both across all three groups and specifically between the two experimental groups. These results are in contradiction with related literature where WE had a positive impact on knowledge acquisition ( Renkl, 2014 ) and faded WE are considered to be more effective in knowledge acquisition and transfer, in contrast to non-faded WE ( Renkl et al., 2000 ; Renkl, 2014 ). A limitation of the study is the fact that the participants already scored very high on the pretest, so participation in the intervention would likely not yield significant knowledge gains due to ceiling effects ( Staus et al., 2021 ). Yet, nearly half of the students reported being novices in the field prior to the study, suggesting that the difficulty of some test items might have been too low. Here, it would be important to revise the factual knowledge test, e.g., the difficulty of the distractors in further study.

Nevertheless, with regard to application knowledge, the results revealed large significant differences: Participants of the two experimental groups performed better in conducting scientific observation steps than participants of the control group. In the experimental groups, the non-faded WE group performed better than the faded WE group. However, the absence of significant differences between the two experimental groups suggests that faded and non-faded WE used as double-content WE are suitable to teach applied knowledge about scientific observation in the learning domain ( Koenen, 2014 ). Furthermore, our results differ from the findings of Renkl et al. (2000) , in which the faded version led to the highest knowledge transfer. Despite the fact that the non-faded WE performed best in our study, the faded version of the WE was also appropriate to improve learning, confirming the findings of Renkl (2014) and Hesser and Gregory (2015) .

5.2 Results on learners’ motivation

Regarding participants’ motivation (RQ2; situational interest and basic needs), no significant differences were found across all three groups or between the two experimental groups. However, descriptive results reveal slightly higher motivation in the two experimental groups than in the control group. In this regard, our results confirm existing literature on a descriptive level showing that WE lead to higher learning-relevant motivation ( Paas et al., 2005 ; Van Harsel et al., 2019 ). Additionally, both experimental groups rated the usefulness of the WE as high and reported a positive evaluation of the WE. Therefore, we assume that even non-faded WE do not lead to over-instruction. Regarding the descriptive tendency, a larger sample might yield significant results and detect even small effects in future investigations. However, because this study also focused on comprehensive qualitative data analysis, it was not possible to evaluate a larger sample in this study.

5.3 Results on cognitive load

Finally, CL did not vary significantly across all three groups (RQ3). However, differences in extraneous CL just slightly missed significance. In descriptive values, the control group reported the highest extrinsic and lowest germane CL. The faded WE group showed the lowest extrinsic CL and a similar germane CL as the non-faded WE group. These results are consistent with Paas et al. (2003) and Renkl (2014) , reporting that WE can help to reduce the extraneous CL and, in return, lead to an increase in germane CL. Again, these differences were just above the significance level, and it would be advantageous to retest with a larger sample to detect even small effects.

Taken together, our results only partially confirm H1: the integration of WE (both faded and non-faded WE) led to a higher acquisition of application knowledge than the control group without WE, but higher factual knowledge was not found. Furthermore, higher motivation or different CL was found on a descriptive level only. The control group provided the basis for comparison with the treatment in order to investigate if there is an effect at all and, if so, how large the effect is. This is an important point to assess whether the effort of implementing WE is justified. Additionally, regarding H2, our results reveal no significant differences between the two WE conditions. We assume that the high complexity of the FA could play a role in this regard, which might be hard to handle, especially for beginners, so learners could benefit from support throughout (i.e., non-faded WE).

In addition to the limitations already mentioned, it must be noted that only one exemplary topic was investigated, and the sample only consisted of students. Since only the learning domain of the double-content WE was investigated, the exemplifying domain could also be analyzed, or further variables like motivation could be included in further studies. Furthermore, the influence of learners’ prior knowledge on learning with WE could be investigated, as studies have found that WE are particularly beneficial in the initial acquisition of cognitive skills ( Kalyuga et al., 2001 ).

6 Conclusion

Overall, the results of the current study suggest a beneficial role for WE in supporting the application of scientific observation steps. A major implication of these findings is that both faded and non-faded WE should be considered, as no general advantage of faded WE over non-faded WE was found. This information can be used to develop targeted interventions aimed at the support of scientific observation skills.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics statement

Ethical approval was not required for the study involving human participants in accordance with the local legislation and institutional requirements. Written informed consent to participate in this study was not required from the participants in accordance with the national legislation and the institutional requirements.

Author contributions

ML: Writing – original draft. SM: Writing – review & editing. JP: Writing – review & editing. JG: Writing – review & editing. DL: Writing – review & editing.

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/feduc.2024.1293516/full#supplementary-material

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Keywords: digital media, worked examples, scientific observation, motivation, cognitive load

Citation: Lechner M, Moser S, Pander J, Geist J and Lewalter D (2024) Learning scientific observation with worked examples in a digital learning environment. Front. Educ . 9:1293516. doi: 10.3389/feduc.2024.1293516

Received: 13 September 2023; Accepted: 29 February 2024; Published: 18 March 2024.

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Copyright © 2024 Lechner, Moser, Pander, Geist and Lewalter. 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) and the copyright owner(s) 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: Miriam Lechner, [email protected]

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Published: 24 March 2024

A survey on fractal fractional nonlinear Kawahara equation theoretical and computational analysis

Laila A. Al-Essa 1 &
Mati ur Rahman 2 , 3

Scientific Reports volume 14 , Article number: 6990 ( 2024 ) Cite this article

Metrics details

Applied mathematics
Computational science

With the use of the Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu-Caputo (ABC) fractal fractional differential operators, this study offers a theoretical and computational approach to solving the Kawahara problem by merging Laplace transform and Adomian decomposition approaches. We show the solution’s existence and uniqueness through generalized and advanced version of fixed point theorem. We present a precise and efficient method for solving nonlinear partial differential equations (PDEs), in particular the Kawahara problem. Through careful error analysis and comparison with precise solutions, the suggested method is validated, demonstrating its applicability in solving the nonlinear PDEs. Moreover, the comparative analysis is studied for the considered equation under the aforementioned operators.

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Introduction

Partial derivatives of an unknown function with respect to many variables are part of a partial differential equation (PDE), a particular kind of mathematical equation. In the realms of physics, engineering, and other disciplines, PDE’s are used to model a broad variety of events. They are frequently used to represent intricate systems that change over time and space, such as wave propagation, heat transfer, and fluid flow 1 , 2 . PDE’s can be categorised according to their order, linearity, and coefficients. The highest derivative in an equation determines the order of a PDE. For instance, the heat equation, a second-order PDE, defines how heat diffuses across a medium. A PDE’s linearity decides whether it is a linear or nonlinear equation 3 . Superposition techniques can be used to solve linear PDE’s, however numerical approaches are frequently needed to solve nonlinear PDE’s. PDE’s can be difficult to solve, hence numerous analytical and numerical techniques have been created to help. The separation of variables, the characteristic method, and Green’s functions are examples of analytical techniques. Finite element, boundary element, spectral, and finite difference approaches are examples of numerical techniques. In multiple branches of science and engineering, including heat transport, fluid mechanics, electromagnetic, and quantum physics, PDE’s are used. They are also utilised in biomedical engineering, finance, and image processing 4 , 5 .

In the 17th century, when the concept of fractional calculus (FC) first evolved, the mathematician Leibniz first wondered what would happen if the order of differentiation or integration was not a whole number. However, it wasn’t until the 20th century that FC began to develop as an independent subject 6 , 7 , 8 . One of the core notions of FC is the concept of fractional differential equations, fractional derivatives, and fractional integrals. We are able to define fractional derivatives by using fractional order operators, which are commonly denoted by the symbol D, where is a non-classical order 9 , 10 , 11 , 12 . Several issues, including the modeling of viscoelastic materials, the research of non-local mechanics, and the study of fractional diffusion processes, have been tackled via FC 13 , 14 , 15 , 16 . It is an extension of the integer order integrals and derivatives used in classical calculus. FC is used in many fields, including as physics, engineering, economics, and biology 17 , 18 , 19 , 20 .

The ideas of fractals and fractional calculus are joint to propose fractal fractional (FF) differential equations (DEs). Fractals are complex patterns produced at distinct sizes by self-similar geometric objects, while FC deals with derivatives and integrals of non-integer orders. FFDEs have recently attracted significant attention due to their ability to demonstrate complex phenomena that exhibit self-similarity at different scales, and their potential to give new insights into fundamental problems in physics and other fields. FF DEs are used in a variety of applied science subjects, including physics and mathematics 21 , 22 , 23 , 24 . FF DEs can be solved using analytical and numerical techniques like the spectral approach, integral transform techniques, and the finite difference method because they are often nonlinear in nature. The fractional Laplace equation, which extends the Laplace equation to non-integer order derivatives, is one of the most significant FFDEs. It is used to simulate diffusion in fractal media 25 . Another important FFDE is the fractional diffusion-wave equation, which is a generalization of the diffusion and wave equations to non-integer order derivatives. This equation is utilized to model wave propagation in fractal media and has uses in different subjects such as seismology, cosmology and mathematical physics 26 .

The KE is a nonlinear PDE that explain wave propagation in shallow water. It was initially given by Toshiaki Kawahara in 1978 as a model for wave propagation in a channel with slowly varying width 27 . The equation has since found uses in other areas, such as mathematical physics and fluid mechanics 28 . We consider the FF non-linear KE as

Where, $ \omega $ is the dependent variable, t is temporal, and x is spatial variables. and also $\sigma $ and $\vartheta $ denote the fractional and fractal orders, respectively. The first term on the left-hand side demonstrates the time evolution of $ \omega $ , while the other terms provide nonlinear and dispersive effects. Initial condition (IC) is given by

The KE has been extensively analyzed in the literature, and many numerical and analytical techniques have been proposed to study its features. One interesting characteristic of the KE is the existence of solitary wave solutions 29 . Another interesting feature is the presence of instability regions in the parameter space, which can lead to the formation of chaotic patterns. The KE has been analyzed via various numerical and analytical methods, such as the kernel particle method 30 , the homotopy perturbation transform method 31 , 32 , 33 , the Galerkin procedure 34 , and the inverse scattering transform. These methods have been utilized to study the behavior of solitary waves, the stability features of the equation, and the formation of chaotic solutions.

In this work, we study the use of these nonlocal operators in solving the FF KE using the Laplace Adomain decomposition method (LADM). This method involves decomposing the equation into a set of simpler equations that can be handled analytically, followed by the inversion of the Laplace transform to figured out the solution in the original time domain. Additionlay, some qualitative features of FF KE are presented via fixed point theory. Our results illuminate the effectiveness of the Caputo, Caputo Fabrizio (CF), and Atangana-Baleanu (AB) fractional operators in accurately capturing the evolution of the system, and we show how our method can be used to analyze various physical phenomena, such as the propagation of waves in different domains.

Basic definitions

Here, we elucidate some basic notions related to fractal-fractional (FF) calculus.

Definition 1

Suppose $\omega \in H^1(p,~q)$ , then the Caputo fractional operator (CFO) sense is

Suppose $\omega (t)$ is FF- differentiable in ( p , q ), with fractal order $\vartheta $ , then FF operator with power law kernel is

Definition 2

The FF integral with CFO is 35

Definition 3

Let $\omega (t)\in H(p,~q)$ , $b > a$ and $\sigma \in [0,1]$ , then the Caputo-Fabrizio fractional operator (CFFO) sense is

Definition 4

Let $\omega (t)$ is FF- differentiable. So the FF operator with CF operator having order $(\sigma ,\,\vartheta )$ of $\omega (t)$ is

Definition 5

Suppose that $\omega (t)\in H(p,~q)$ , then the FF operator having ABC operator is

Definition 6

Let $\omega (t)$ is a fractal fractional differentiable, then the FF operator with ABC kernel is

Definition 7

Laplace transform ${\textbf{L}}$ , of a function $\omega (t)$ , with $t>0$ as:

Definition 8

If ${\textbf{L}}^{-1}\omega (\varsigma )=\omega (t)$ , then ${\textbf{L}}^{-1}$ is:

Definition 9

The Laplace transform (LT) of CFO as:

Definition 10

The LT of CFFO is:

Definition 11

The LT of ABC sense is:

where $r=[\sigma ]+1$ .

Existence of the initial value problems

The existence and uniqueness of the initial value problems are studied in this section by using $\sigma $ -type ${\mathfrak {F}}$ -contraction 36 . For this purposes, Suppose $({\textbf{Z}}, d)$ be a complete metric space and $\varsigma $ be the family of strictly increasing functions ${\mathfrak {F}}:\mathfrak {R_{+}}\rightarrow {\mathfrak {R}}$ having the following properties:

$\lim \limits _{n\rightarrow \infty }{\mathfrak {F}}(a_{n}) = -\infty $ if and only if, for each $\{a_{n}\}$ , $\lim \limits _{n\rightarrow \infty }(a_{n})=0;$

there exist $\upsilon \in (0,1)$ such that $\lim \limits _{a\rightarrow 0^{+}}a^{\upsilon }{\mathfrak {F}}(a)=0.$

Definition 12

Let $\text{T}:{\textbf{Z}}\rightarrow {\textbf{Z}}$ be self mapping and $\sigma :{\textbf{Z}}\times {\textbf{Z}}\rightarrow [0,\infty )$ , if

for all $\mathcal {Y},\mathcal {V}\in {\textbf{Z}}$ , then $\text{T}$ is called $\sigma $ -admissible.

Definition 13

Let $({\textbf{Z}},d)$ be a complete metric space (CMS), $\text{T}:{\textbf{Z}}\rightarrow {\textbf{Z}}$ and $\sigma :{\textbf{Z}}\times {\textbf{Z}}\rightarrow \{-\infty \}\cup [0,\infty )$ , there exist $\omega >0$ such that

for each $\text{Y},\mathcal {U}\in {\textbf{Z}}$ with $d(\text{T}\text{Y},\text{T}\mathcal {U})>0$ . then $\text{T}$ is called $\sigma $ -type ${\mathfrak {F}}$ -contraction.

Let $({\textbf{Z}},d)$ be a CMS and $\text{T}:{\textbf{Z}}\rightarrow {\textbf{Z}}$ be an $\sigma $ -type ${\mathfrak {F}}$ -contraction such that

there exist $\text{Y}_{\circ }\in {\textbf{Z}}$ such that $\sigma (\text{Y}_{\circ },\text{T}\text{Y}_{\circ })\ge 1;$

if there exist $\{\text{Y}_{n}\}\subseteq {\textbf{Z}}$ with $\sigma (\text{Y}_{n},\text{Y}_{n+1})\ge 1$ and $\text{Y}_{n}\rightarrow \text{Y}$ , then $\sigma (\text{Y}_{n},\text{Y})\ge 1$ for all $n\in \mathcal {P};$

${\mathfrak {F}}$ is continuous.

Then $\text{T}$ has a fixed point $\text{Y}^{*}\in {\textbf{Z}}$ also for $\text{Y}_{\circ }\in {\textbf{Z}}$ the sequence $\{\text{T}^{n}\text{Y}_{\circ }\}_{n\in \mathcal {P}}$ is convergent to $\text{Y}_{\circ }$

Let ${\textbf{Z}}=\mathcal {E}([0,1]^{2},{\mathfrak {R}})$ where $\mathcal {E}$ the space of all continuous functions $\text{Y}: [0,1]\times [0,1]\rightarrow {\mathfrak {R}}$ and $d(\text{Y}(x,t),\mathcal {U}(x,t))=\sup _{x,t\in [0,1]}[|\text{Y}(x,t)-\mathcal {U}(x,t)|]$ , then we can write the IVP ( 1 ) in CF fractional derivative sense as:

where ${\mathfrak {F}}(x,t,\text{Y}(x,t))=\omega _{xxxxx}-\omega _{xxx}-\omega \omega _{x}.$

The theorem provided below gives the existence of solution of the problem ( 3 )

There exist ${\mathfrak {G}}:{\mathfrak {R}}^{2}\rightarrow {\mathfrak {R}}$ such that:

$|{\mathfrak {F}}(x,t,\text{Y})-{\mathfrak {F}}(x,t,\mathcal {U})|\le \frac{(2-\sigma )A(\sigma )}{2} e^{b}|\text{Y}(x,t)-\mathcal {U}(x,t)|$ for $(x,t)\in [0,1]^{2}$ and $\text{Y},\mathcal {U}\in {\mathfrak {R}};$

there exist $\text{Y}_{1}\in {\textbf{Z}}$ such that ${\mathfrak {G}}(\mathcal {Y_{1}},\text{T}\text{Y}_{1})\ge 0$ , where $\text{T}:{\textbf{Z}}\rightarrow {\textbf{Z}}$ defined by

for $\text{Y},\mathcal {U}\in {\textbf{Z}}, {\mathfrak {G}}(\text{Y},\mathcal {U})\ge 0$ implies that ${\mathfrak {G}}(\text{T}\text{Y},\text{T}\mathcal {U})\ge 0$ ;

$\{\text{Y}_{n}\}\subseteq {\textbf{Z}}, \lim \limits _{n\rightarrow \infty }\text{Y}_{n}=\text{Y}$ , where $\text{Y}\in {\textbf{Z}}$ and ${\mathfrak {G}}(\text{Y}_{n},\text{Y}_{n+1})\ge 0$ implies that ${\mathfrak {G}}(\text{Y}_{n},\text{Y})\ge 0,$ for all $n\in \mathcal {P}$

Then there exist at least one fixed point of $\text{T}$ which is the solution of the problem ( 3 ).

To prove that $\text{T}$ has a fixed point, therefore

Thus for $\text{Y},\mathcal {U}\in {\textbf{Z}}$ with ${\mathfrak {G}}(\text{Y},\mathcal {U})\ge 0,$ we obtain

Taking $\ln $ on both sides, we have:

if ${\mathfrak {F}}:[0,\infty )\rightarrow {\mathfrak {R}}$ defined by ${\mathfrak {F}}({\mathfrak {u}})=(\vartheta T^{\vartheta -1})\ln [{\mathfrak {u}}^{2}+{\mathfrak {u}}],~~{\mathfrak {u}}>0$ , then ${\mathfrak {F}}\in \delta $ .

Now define $\sigma :{\textbf{Z}}\times {\textbf{Z}}\rightarrow \{-\infty \}\cup [0,\infty )$ as

for $\text{Y},\mathcal {U}\in {\textbf{Z}}$ with $d(\text{T}\text{Y},\text{T}\mathcal {U})>0$ . Now by ${\mathfrak {G}}3,$

for all $\text{Y},\mathcal {U}\in {\textbf{Z}}$ . From ${\mathfrak {G}}2$ there exist $\text{Y}_{\circ }\in {\textbf{Z}}$ such that $\sigma (\text{Y}_{\circ },\text{T}\text{Y}_{\circ })\ge 1.$ therefore by ${\mathfrak {G}}4$ and Theorem 1 , there exist $\text{Y}^{*}\in {\textbf{Z}}$ such that $\text{Y}^{*}=\text{T}\text{Y}^{*}$ . Hence $\mathcal {Y}^{*}$ is the solution of the problem ( 3 ) $\square $

Similarly we can write the IVP ( 1 ) in ABC sense as

The following theorem show the existence of solution of the problem ( 4 )

There exist ${\mathfrak {G}}:{\mathfrak {R}}^{2}\rightarrow {\mathfrak {R}}$ such that

$|{\mathfrak {F}}(x,t,\text{Y})-{\mathfrak {F}}(x,t,\mathcal {U})|\le \frac{(\Gamma \sigma ) \sigma }{(1-\sigma )\Gamma (\sigma +1)} e^{\frac{-b}{2}}|\text{Y}(x,t)-\mathcal {U}(x,t)|$ for $(x,t)\in [01]^{2}$ and $\text{Y},\mathcal {U}\in {\mathfrak {R}};$

Consequently

Applying $``\ln ''$ which implies that

Let ${\mathfrak {F}}:[0,\infty )\rightarrow {\mathfrak {R}}$ define by ${\mathfrak {F}}(\lambda )=\ln \lambda ,$ where $\lambda >0$ , then it is easy to show that ${\mathfrak {F}}\in \varsigma $ .

Now define $\sigma :{\textbf{Z}}\times {\textbf{Z}}\rightarrow \{-\infty \}\cup [0,\infty )$ by

Thus $b+\sigma (\mathcal {Y},\mathcal {V}){\mathfrak {F}}(d(\text{T}\mathcal {Y},\text{T}\mathcal {V}))\le {\mathfrak {F}}(d(\mathcal {Y},\mathcal {V}))$ for $\mathcal {Y},\mathcal {V}\in {\textbf{Z}}$ with $d(\text{T}\mathcal {Y},\text{T}\mathcal {V})\ge 0$ . therefore $\text{T}$ is an $\sigma $ -type ${\mathfrak {F}}$ -contraction. From $({\mathfrak {G}}3)$ we have

for all $x,t\in [0,1]$ . Thus $\text{T}$ is an $\sigma $ -admissible. From $({\mathfrak {G}}2)$ there exist $\mathcal {Y}_{\circ }\in {\textbf{Z}}$ with $\sigma (\mathcal {Y}_{\circ },\text{T}\mathcal {Y}_{\circ })\ge 1.$ From $({\mathfrak {G}}4)$ , there exist $\mathcal {Y}^{*}\in {\textbf{Z}}$ such that $\text{T}\mathcal {Y}^{*}$ . Hence $\mathcal {Y}^{*}$ is the solution of the initial value problem ( 4 ). $\square $

Proposed method

Here, we develop the Laplace transform of the FF operators with different kernels. Next, we’ll use LADM to roughly solve the system under consideration.

Scheme for the proposed model with CFO

Equation ( 1 ) in terms of the Caputo operator, which is provided by.

Equivalent form of Eq. ( 5 ) is:

Applying LT to Eq. ( 7 ), we get:

In “ Basic definitions ” section on the power law kernel, we discussed the definition of the LT.

The series solution can be expressed as:

The decomposed non-linear terms are as follows:

where $E_n$ denotes Adomian polynomials $\omega _0,\omega _1,\omega _2, \dots $ ,

Applying ${\textbf{L}}^{-1}$ to Eq. ( 9 ) together with Eq. ( 10 ), we get:

The series solution is obtained by comparing the terms on both sides of Eq. ( 11 ).

The series can written as: $\omega (x,t)=\sum _{n=0}^\infty \omega _n(x,t)$ .

Scheme for the proposed model with CFFO

Equivalently Eq. ( 12 ) gets the form:

Applying LT to Eq. ( 14 ), we get:

In “ Basic definitions ” section on the exponential decay kernel, we addressed the definition of the LT.

The whole series solution can be scripted as,

the non-linear terms are decomposed with Adomian-polynomial discussed above.

Applying ${\textbf{L}}^{-1}$ to Eq. ( 16 ), we get

Equating terms on both sides in Eq. ( 17 ), we get:

Scheme for the proposed model with ABC operator

Equivalently Eq. ( 18 ) can written as:

Applying LT to Eq. ( 20 ), we obtain:

Using the definition of LT discussed in “ Basic definitions ” section on Mittag-Leffler kernel, we get:

The whole series solution can be scripted as:

the non-linear terms are decomposed with Adomian-polynomial discussed above. Applying ${\textbf{L}}^{-1}$ to Eq. ( 22 ), we get

Equating terms on both sides in Eq. ( 23 ), we get:

The whole series solution can be scripted as $\omega (x,t)=\sum _{n=0}^\infty \omega _n(x,t)$ .

Validation of the proposed method

Here in this section we solve some of the examples by the proposed method discussed in “ Existence of the initial value problems ” section.

Consider Eq. ( 1 ) in Caputo sense which is given by

with initial condition and exact solution is,

We find $\omega _0$ , $\omega _1$ , $\omega _2$ and so on in order to solve Eq. ( 24 ) .

Since we know that:

after some calculation we get:

Thus, the solution is:

The above table shows the error in approximate vs Exact solution of the considered model with CFO for parameters taken as $\sigma =1,~\vartheta =1,~\nu =\frac{1}{2\sqrt{13}},~\rho =\frac{-72}{169},~ \eta =\frac{105}{169},~and~c=\frac{36}{169}$ . From the Table 1 , it is observable that the absolute error decreases as space variable x increases, at small time t .

The surface plot of Error analysis for exact versus approximate with CFO.

Here, in Fig. 1 the parameters taken is same as above for the Table 1 . Figure 1 shows the 3-dimensional graph of Error in Approximate solution of considered model with CFO. Anyone can get an Idea at a glance that how much the proposed method with CFO is efficient by giving such a negligible error vs Exact solution of considered model.

The surface plot of approximate solution Eq. ( 27 ).

Figure 2 is the 3d behavior of approximate solution (Eq. 27 ) for the parameters taken as same as above.

Comparison of Eqs. ( 26 ) vs ( 27 ) for different values of $\sigma $ and $\vartheta $ respectively.

In Fig. 3 , we set the parameters as follows: $\nu =\frac{1}{2\sqrt{13}},~\rho =\frac{-72}{169},~ \eta =\frac{105}{169},~and~c=\frac{36}{169}$ . The left plots depict a comparison between the Approximate solution Eq. ( 27 ) and the Exact solution Eq. ( 26 ) for different values of the fractional order variable, i.e., $\sigma =0.9$ and $\sigma =0.7$ , while keeping the fractal variable $\vartheta $ fixed at 1, with a time variable $t=6.5$ .

In the right plot, we illustrate the 2D behavior of the solution Eq. ( 27 ) of the model considered, in a fractal-fractional sense, comparing it with the exact solution Eq. ( 26 ) with integer order, for a fixed fractional variable $\sigma =1$ and varying fractal parameter $\vartheta =0.6,0.2$ .

This analysis pertains to the fractal fractional Kawahara equation, which describes the evolution of wave phenomena in complex media with fractal and fractional characteristics.

We observe that for small values of t , the waves exhibit a close proximity to each other, indicating a subtle interplay between the fractional and fractal effects. Moreover, as time ( t ) increases, the system’s behavior becomes more pronounced, revealing intricate patterns and dynamics. Notably, we note a convergence of the approximate soliton solution towards the exact solution of the considered model, underscoring the robustness of the theoretical framework in capturing the underlying physics of the system.

Time behavioral plots of considered model with CFO.

In Fig. 4 , the left plot is the illustration of Eq. ( 27 ) vs time t for $x=6.5$ with varying fractional parameter $\sigma =0.9,~\sigma =0.8,~\sigma =0.6$ with fixed fractal variable $\vartheta =1$ , moreover, the remaining parameters are taken same as above, and the right plot illustrate Eq. ( 27 ) for different value of time in order with fixed fractal and fractional variables $i-e$ $\sigma =1=\vartheta $ .

Consider Eq. ( 1 ) with CFFO

Equations ( 25 ) and ( 26 ) are the initial condition and exact solution, respectively. The following formula can be used to calculate Eq. ( 28 ) approximate solution:

after simplification we get:

Same like above, we can also calculate $\omega _2$ and so on. The solution is describes as

The above table shows the error in approximate vs Exact solution of the considered model with CFO for parameters taken as $\sigma =1,~\vartheta =1,~\nu =\frac{1}{2\sqrt{13}},~\rho =\frac{-72}{169},~ \eta =\frac{105}{169},~and~c=\frac{36}{169}$ . From the Table 2 , it is observable that the absolute error decreases as space variable x increases, at small time t .

The surface plot of Error analysis for exact versus approximate with CFFO.

Here, in Fig. 1 the parameters taken is same as above for the Table 2 . Figure 5 shows the 3-dimensional graph of Error in Approximate solution of considered model with CFO. Anyone can get an Idea at a glance that how much the proposed method with CFO is efficient by giving such a negligible error vs Exact solution of considered model.

The surface plot of approximate solution Eq. ( 29 ).

Figure 6 is the 3d behavior of approximate solution (Eq. 29 ) for the parameters taken as same as above.

Comparison of Eqs. ( 29 ) vs ( 26 ) for different values of $\sigma $ and $\vartheta $ respectively.

We explore the dynamics of the Caputo-Fabrizio fractal fractional operator in the framework of the Kawahara equation in Fig. 7 . $\nu =\frac{1}{2\sqrt{13}},\rho =\frac{-72}{169},\eta =\frac{105}{169}, andc=\frac{36}{169}$ are the parameters that have been specified.

The left plots provide a comparison of the different fractional order variables ( $\sigma =0.9$ and $\sigma =0.7$ ) between the approximate solution Eq. ( 29 ) and the exact solution Eq. ( 26 ). The time variable stays at $t=6.5$ , while the fractal variable is held constant at $\vartheta =1$ .

As for the right figure, it explores the 2D behavior of the precise solution Eq. ( 26 ) under integer order in comparison to the solution Eq. ( 29 ) under the Caputo-Fabrizio fractal fractional operator. In this case, the fractal parameter ( $\vartheta =0.6,0.2$ ) is varied while the fractional variable is fixed at $\sigma =1$ . These studies illuminate wave processes in complicated media, capturing fractional and fractal properties in the context of the Kawahara equation. After investigation, we find that the waves show an impressive coherence at tiny time steps, suggesting complex interactions between fractional and fractal constituents. The system’s behavior becomes increasingly apparent with time, exposing complex dynamics and changing patterns. We see that the approximation soliton solution converges noticeably to the precise solution, demonstrating the ability of the theoretical framework to accurately describe the physics underlying the system.

Time behavioral plots of considered model with CFFO.

In Fig. 8 , the left plot is the illustration of Eq. ( 29 ) vs time t for $x=6.5$ with varying fractional parameter $\sigma =1.0,~\sigma =0.9,~\sigma =0.8$ with fixed fractal variable $\vartheta =1$ , moreover, the remaining parameters are taken same as above, and the right plot illustrate Eq. ( 29 ) for different value of time in order with fixed fractal and fractional variables $i-e$ $\sigma =1=\vartheta $ .

Consider Eq. ( 1 ) with Mittag Leffler kernel

Equations ( 25 ) and ( 26 ) are the initial condition and exact solution, respectively. The following formula can be used to calculate Eq. ( 26 ) approximate solution:

Same like above we can find other terms. The solution is describes as:

The above table shows the error in approximate vs Exact solution of the considered model with ABC fractional operator for parameters taken as $\sigma =1,~\vartheta =1,~\nu =\frac{1}{2\sqrt{13}},~\rho =\frac{-72}{169},~ \eta =\frac{105}{169},~and~c=\frac{36}{169}$ . From the Table 3 , it is observable that the absolute error decreases as space variable x increases, at small time t .

The surface plot of Error analysis between exact and approximate with ABC fractional operator.

Here, in Fig. 9 the parameters taken is same as above for the Table 3 . Figure 9 shows the 3-dimensional graph of Error in Approximate solution of considered model with ABC. Anyone can get an Idea at a glance that how much the proposed method with ABC fractional operator is efficient by giving such a negligible error vs Exact solution of considered model.

The surface plot of approximate solution Eq. ( 31 ).

Figure 10 is the 3d behavior of approximate solution (Eq. 31 ) for the parameters taken as same as above.

Comparison of Eqs. ( 26 ) and ( 31 ) for different values of $\sigma $ and $\vartheta $ respectively.

We investigate the fractal fractional Kawahara equation dynamics using the ABC fractal fractional operator, as shown in Fig. 11 . The parameters that were selected are as follows: $\nu =\frac{1}{2\sqrt{13}}$ , $\rho =\frac{-72}{169}$ , $\eta =\frac{105}{169}$ , and $c=\frac{36}{169}$ . The figure’s left plots provide a comparison of the equation’s approximate Eq. ( 31 ) and exact Eq. ( 26 ) solutions. We examine how the fractional order variable $\sigma $ behaves at various values, namely $\sigma =0.7$ and $\sigma =0.3$ , while maintaining the constant value of the fractal variable $\vartheta $ at $\vartheta =1$ . The temporal evolution is fixed at $t=5$ in this case. However, when compared to the integer-order exact solution Eq. ( 26 ), the right plot reveals the complex 2D trajectory of the solution presented in Eq. ( 31 ). The fractal parameter $\vartheta $ assumes different values: $\vartheta =0.7$ and $\vartheta =0.3$ , while the fractional variable $\sigma $ stays constant at $\sigma =1$ . During our investigation, we find remarkable physical insights. First, at shorter time intervals, the waves show a remarkable closeness, highlighting the interaction of the system’s aspects. Furthermore, the system exhibits a measurable progression throughout time, with each instant enhancing its dynamic diversity. Especially, we see a strong convergence: over time, the approximation soliton solution smoothly approaches the precise solution, demonstrating the stability of our model and its ability to describe real-world processes.

Time behavioral plots of considered model with ABC fractional operator.

In Fig. 12 , the left plot is the illustration of Eq. ( 31 ) vs time t for $x=5$ with varying fractional parameter $\sigma =1.0,~\sigma =0.9,~\sigma =0.8$ with fixed fractal variable $\vartheta =1$ , moreover, the remaining parameters are taken same as above, and the right plot illustrate Eq. ( 31 ) for different value of time in order with fixed fractal and fractional variables $i-e$ $\sigma =1=\vartheta $ .

Comparative analysis

Here in this section we shows the results obtained on all of the above mentioned operator.

Comparison analysis of different fractional order.

While fractal fractional operators such as the Caputo, Caputo Fabrizio, and ABC operators are scrutinized by means of Laplace Adomian decomposition techniques, a comparison of the outcomes displays in Table 4 and graphically represented in Fig. 13 , we observe that the ABC operator accomplishes better than the others. Even though all operators are convenient for simulating sophisticated systems with fractional derivatives, the ABC operator is more accurate and efficient in capturing the complex behavior of fractal events. The ABC operator’s supremacy arises from its exceptional capacity to provide a more adaptable framework for explaining anomalies and non-local behaviors seen in fractal systems. In contrast to the Caputo and Caputo Fabrizio operators, the ABC operator adds a new parameter that allows for more precise modifications to the fractional order, improving its flexibility to adapt to a wider range of fractal concepts. As a result, the ABC operator is the go-to option for both researchers and practitioners in various domains, including engineering and the natural sciences, since it produces more accurate findings and makes it easier to comprehend the underlying dynamics of fractal systems.

Ethical declaration

In this study, human data has not been used for modeling.

This research study has given a way for solving the Kawahara equation using the Laplace transform Adomian decomposition method (LADM) under three different fractal fractional differential operators: Caputo, Caputo-Fabrizio, and ABC. We have proved the solution’s existence and uniqueness of solution via advanced fixed point theorems. Our results show that the suggested approach offers an effective and precise solution for nonlinear partial differential equations like the Kawahara problem which has been observed in absolute error in the tables. A comparison between results under three different have been presented via tables and graphs to figure out that ABC operator provide good results due to nonsingular and nonlocal kernel. Contributions of this paper include a new approach to partial differential equations solution based on fractal fractional differential operators and the LADM. The LADM can be used to other nonlinear issues in physics, engineering, and other disciplines. It also has the benefit of offering effective numerical solutions that work for a variety of issues. We hope that our results will stimulate additional investigation into the application of fractal fractional differential operators and the LTADM for resolving complex issues and promote the creation of more effective and precise numerical methods for solving partial differential equations. By means of our investigation, we provide a valuable contribution to the rapidly developing domain of fractional calculus and establish a foundation for forthcoming research paths that aim to fully use fractal fractional approaches in the modeling of intricate physical processes.

Now a days delay differential equations, neural network approach and fractional calculus have many applications in the different fields of sciences such as bifurcation and BAM neural nework 37 , 38 , predator-prey and Lotka-Volterra system 39 , 40 , plankton-oxygen model 41 , and others 42 , 43 . Using these approaches, one can find soliton solutions for the considered system.

Data availability

The datasets generated during the current study are available from the corresponding author (Mati ur Rahman) on reasonable request.

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Acknowledgements

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R443), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

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Laila A. Al-Essa

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Mati ur Rahman

Department of computer science and mathematics, Lebanese American university, Beirut , Lebanon

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March 12, 2024

The Simplest Math Problem Could Be Unsolvable

The Collatz conjecture has plagued mathematicians for decades—so much so that professors warn their students away from it

By Manon Bischoff

Close up of lightbulb sparkling with teal color outline on black background

Mathematicians have been hoping for a flash of insight to solve the Collatz conjecture.

James Brey/Getty Images

At first glance, the problem seems ridiculously simple. And yet experts have been searching for a solution in vain for decades. According to mathematician Jeffrey Lagarias, number theorist Shizuo Kakutani told him that during the cold war, “for about a month everybody at Yale [University] worked on it, with no result. A similar phenomenon happened when I mentioned it at the University of Chicago. A joke was made that this problem was part of a conspiracy to slow down mathematical research in the U.S.”

The Collatz conjecture—the vexing puzzle Kakutani described—is one of those supposedly simple problems that people tend to get lost in. For this reason, experienced professors often warn their ambitious students not to get bogged down in it and lose sight of their actual research.

The conjecture itself can be formulated so simply that even primary school students understand it. Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x : if x is odd, you calculate 3 x + 1; otherwise calculate x / 2. Repeat these instructions as many times as possible, and, according to the conjecture, you will always end up with the number 1.

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For example: If you start with 5, you have to calculate 5 x 3 + 1, which results in 16. Because 16 is an even number, you have to halve it, which gives you 8. Then 8 / 2 = 4, which, when divided by 2, is 2—and 2 / 2 = 1. The process of iterative calculation brings you to the end after five steps.

Of course, you can also continue calculating with 1, which gives you 4, then 2 and then 1 again. The calculation rule leads you into an inescapable loop. Therefore 1 is seen as the end point of the procedure.

Bubbles with numbers and arrows show Collatz conjecture sequences

Following iterative calculations, you can begin with any of the numbers above and will ultimately reach 1.

Credit: Keenan Pepper/Public domain via Wikimedia Commons

It’s really fun to go through the iterative calculation rule for different numbers and look at the resulting sequences. If you start with 6: 6 → 3 → 10 → 5 → 16 → 8 → 4 → 2 → 1. Or 42: 42 → 21 → 64 → 32 → 16 → 8 → 4 → 2 → 1. No matter which number you start with, you always seem to end up with 1. There are some numbers, such as 27, where it takes quite a long time (27 → 82 → 41 → 124 → 62 → 31 → 94 → 47 → 142 → 71 → 214 → 107 → 322 → 161 → 484 → 242 → 121 → 364 → 182 → 91 → 274 → ...), but so far the result has always been 1. (Admittedly, you have to be patient with the starting number 27, which requires 111 steps.)

But strangely there is still no mathematical proof that the Collatz conjecture is true. And that absence has mystified mathematicians for years.

The origin of the Collatz conjecture is uncertain, which is why this hypothesis is known by many different names. Experts speak of the Syracuse problem, the Ulam problem, the 3 n + 1 conjecture, the Hasse algorithm or the Kakutani problem.

German mathematician Lothar Collatz became interested in iterative functions during his mathematics studies and investigated them. In the early 1930s he also published specialist articles on the subject , but the explicit calculation rule for the problem named after him was not among them. In the 1950s and 1960s the Collatz conjecture finally gained notoriety when mathematicians Helmut Hasse and Shizuo Kakutani, among others, disseminated it to various universities, including Syracuse University.

Like a siren song, this seemingly simple conjecture captivated the experts. For decades they have been looking for proof that after repeating the Collatz procedure a finite number of times, you end up with 1. The reason for this persistence is not just the simplicity of the problem: the Collatz conjecture is related to other important questions in mathematics. For example, such iterative functions appear in dynamic systems, such as models that describe the orbits of planets. The conjecture is also related to the Riemann conjecture, one of the oldest problems in number theory.

Empirical Evidence for the Collatz Conjecture

In 2019 and 2020 researchers checked all numbers below 2 68 , or about 3 x 10 20 numbers in the sequence, in a collaborative computer science project . All numbers in that set fulfill the Collatz conjecture as initial values. But that doesn’t mean that there isn’t an outlier somewhere. There could be a starting value that, after repeated Collatz procedures, yields ever larger values that eventually rise to infinity. This scenario seems unlikely, however, if the problem is examined statistically.

An odd number n is increased to 3 n + 1 after the first step of the iteration, but the result is inevitably even and is therefore halved in the following step. In half of all cases, the halving produces an odd number, which must therefore be increased to 3 n + 1 again, whereupon an even result is obtained again. If the result of the second step is even again, however, you have to divide the new number by 2 twice in every fourth case. In every eighth case, you must divide it by 2 three times, and so on.

In order to evaluate the long-term behavior of this sequence of numbers , Lagarias calculated the geometric mean from these considerations in 1985 and obtained the following result: ( 3 / 2 ) 1/2 x ( 3 ⁄ 4 ) 1/4 x ( 3 ⁄ 8 ) 1/8 · ... = 3 ⁄ 4 . This shows that the sequence elements shrink by an average factor of 3 ⁄ 4 at each step of the iterative calculation rule. It is therefore extremely unlikely that there is a starting value that grows to infinity as a result of the procedure.

There could be a starting value, however, that ends in a loop that is not 4 → 2 → 1. That loop could include significantly more numbers, such that 1 would never be reached.

Such “nontrivial” loops can be found, for example, if you also allow negative integers for the Collatz conjecture: in this case, the iterative calculation rule can end not only at –2 → –1 → –2 → ... but also at –5 → –14 → –7 → –20 → –10 → –5 → ... or –17 → –50 → ... → –17 →.... If we restrict ourselves to natural numbers, no nontrivial loops are known to date—which does not mean that they do not exist. Experts have now been able to show that such a loop in the Collatz problem, however, would have to consist of at least 186 billion numbers .

A plot lays out the starting number of the Collatz sequence on the x-axis with the total length of the completed sequence on the y-axis

The length of the Collatz sequences for all numbers from 1 to 9,999 varies greatly.

Credit: Cirne/Public domain via Wikimedia Commons

Even if that sounds unlikely, it doesn’t have to be. In mathematics there are many examples where certain laws only break down after many iterations are considered. For instance,the prime number theorem overestimates the number of primes for only about 10 316 numbers. After that point, the prime number set underestimates the actual number of primes.

Something similar could occur with the Collatz conjecture: perhaps there is a huge number hidden deep in the number line that breaks the pattern observed so far.

A Proof for Almost All Numbers

Mathematicians have been searching for a conclusive proof for decades. The greatest progress was made in 2019 by Fields Medalist Terence Tao of the University of California, Los Angeles, when he proved that almost all starting values of natural numbers eventually end up at a value close to 1.

“Almost all” has a precise mathematical meaning: if you randomly select a natural number as a starting value, it has a 100 percent probability of ending up at 1. ( A zero-probability event, however, is not necessarily an impossible one .) That’s “about as close as one can get to the Collatz conjecture without actually solving it,” Tao said in a talk he gave in 2020 . Unfortunately, Tao’s method cannot generalize to all figures because it is based on statistical considerations.

All other approaches have led to a dead end as well. Perhaps that means the Collatz conjecture is wrong. “Maybe we should be spending more energy looking for counterexamples than we’re currently spending,” said mathematician Alex Kontorovich of Rutgers University in a video on the Veritasium YouTube channel .

Perhaps the Collatz conjecture will be determined true or false in the coming years. But there is another possibility: perhaps it truly is a problem that cannot be proven with available mathematical tools. In fact, in 1987 the late mathematician John Horton Conway investigated a generalization of the Collatz conjecture and found that iterative functions have properties that are unprovable. Perhaps this also applies to the Collatz conjecture. As simple as it may seem, it could be doomed to remain unsolved forever.

This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.

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The Scientific Method Of Problem Solving. The Basic Steps: State the Problem - A problem can't be solved if it isn't understood.; Form a Hypothesis - This is a possible solution to the problem formed after gathering information about the problem.The term "research" is properly applied here. Test the Hypothesis - An experiment is performed to determine if the hypothesis solves the problem or not.
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The first step in the scientific method is to identify and observe a phenomenon that requires explanation. This can involve asking open-ended questions, making detailed observations using our senses or tools, or exploring natural patterns, which are sources to develop hypotheses. 2. Formulation of a Hypothesis.
Scientific method
The scientific method is an empirical method for acquiring knowledge that has characterized the development of science since at least the 17th century. (For notable practitioners in previous centuries, see history of scientific method.). The scientific method involves careful observation coupled with rigorous scepticism, because cognitive assumptions can distort the interpretation of the ...
Solving sports problems with science
The scientific method is a systematic way to solve problems and answer questions in science and engineering. List the steps of the scientific method. Use an external resource if necessary. Make ...
Sample Problems #1-10: Scientific Method
Mark only those parts of the scientific method that are explicit in the problem description. Parts of the Scientific Method. (A) Goal. (B) Model. (C) Data. (D) Evaluation. (E) Revision. 6. The U.S. Forest Service is given the task of managing Forest Service lands for multiple uses (e.g., lumber, recreation, conservation).
Frontiers
Science education often aims to increase learners' acquisition of fundamental principles, such as learning the basic steps of scientific methods. Worked examples (WE) have proven particularly useful for supporting the development of such cognitive schemas and successive actions in order to avoid using up more cognitive resources than are necessary. Therefore, we investigated the extent to ...
A survey on fractal fractional nonlinear Kawahara equation ...
Here in this section we solve some of the examples by the proposed method discussed in "Existence of the initial value problems" section. Example 1 Consider Eq.
The Simplest Math Problem Could Be Unsolvable
Take a natural number. If it is odd, multiply it by 3 and add 1; if it is even, divide it by 2. Proceed in the same way with the result x: if x is odd, you calculate 3 x + 1; otherwise calculate x ...

Using the Scientific Method to Solve Problems

The Scientific Method

Applying the Scientific Method to Problem-Solving

Inductive and Deductive Reasoning

Social Sciences and the Scientific Method

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What Is the Scientific Method?

The History of the Scientific Method

The Ancient World

The Middle Ages

Scientific Revolution

The Scientific Method Steps

#1: Make Observations

#2: Ask a Question

#3: Make a Hypothesis

#4: Experiment

#5: Analyze Data

#6: Communicate Your Results

A Scientific Method Example: Falling Toast

#1: Make Observations

Key Scientific Method Tips

Don’t Worry About Proving Your Hypothesis

Be Prepared to Try Again

What’s Next?

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A Guide to Using the Scientific Method in Everyday Life

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1.3: The Scientific Method - How Chemists Think

Learning Objectives

Step 1: Make observations

Step 2: Formulate a hypothesis

Step 3: Design and perform experiments

Step 4: Accept or modify the hypothesis

Step 5: Development into a law and/or theory

Example \(\PageIndex{1}\)

Exercise \(\PageIndex{1}\) ﻿

Contributions & Attributions

1.1: The Scientific Method

Proposing a Hypothesis

Testing a Hypothesis

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What is the Scientific Method?

The Flashlight Mystery...

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Using the Scientific Method to Solve Mysteries

15 Scientific Method Examples

Definition of Scientific Method

15 Examples of Scientific Method

Origins of the Scientific Method

Steps in the Scientific Method

1. Observation

2. Formulation of a Hypothesis

3. Testing of the Hypothesis

4. Data Analysis

5. Drawing Conclusions

Exercise \(\PageIndex{1}\)