Why Measuring Can Be So Difficult To Learn.

Learning to measure is a fundamental skill in science, yet it often proves surprisingly challenging for students. This seemingly straightforward concept involves a complex interplay of factors, making accurate measurement a hurdle for many young learners.

One obstacle lies in the abstract nature of measurement units. Concepts like centimeters or milliliters don't have a tangible meaning for young students. Relying solely on memorizing conversions between units proves ineffective. Students need to develop a spatial understanding of these units through hands-on activities.

Another challenge is the inherent variability present in measurement. No two students will hold a ruler or pour a liquid with identical precision. Factors like hand-eye coordination and fine motor skills can significantly impact results.Acknowledging this variability and incorporating estimations into the learning process helps students understand the inherent uncertainty in measurement.

Furthermore, the choice of measuring tool itself can be confusing. Students need to understand the appropriate tool for the job – a ruler for length, a graduated cylinder for liquids, and a balance for mass. Learning to read the scale on each instrument adds another layer of complexity.

So there are ways to bridge these gaps and help students acquire the skills needed to accurately measure. First, look at including hands on activities that encourage students to visualize the unit they are measuring. For instance, asking them to imagine the width of their thumb as a centimeter can solidify the concept. Similarly, incorporating estimation exercises helps students develop a sense of scale and recognize when a measurement seems unreasonable.

Consider having students use different measuring tools and repeat using these tools.Expose students to various measuring tools and have them repeat measurements with different tools or even classmates. This not only helps them learn about different instruments but also reinforces the concept of variability and the importance of multiple trials.

Focus on the process. Don't prioritize achieving a specific numerical answer initially. Instead, emphasize the steps involved in the measurement process – selecting the right tool, reading the scale accurately, and recording the data.

By incorporating these strategies, educators can transform measurement from a rote task to a dynamic learning experience. Through hands-on activities, visualization techniques, and a focus on the process, students can develop a deeper understanding of measurement concepts and the skills necessary to measure accurately in the real world.  Let me know what you think, I'd love to hear.  Have a great weekend.

Guinness Beer And The Statistical Method They Pioneered.

The world of statistics is filled with complex formulas and abstract concepts. However, one of the most fundamental and widely used methods, the Student's t-test, has a surprisingly frothy origin story – it was born in the vats of the Guinness brewery.

In the early 1900s, William Sealy Gosset, a young chemist and statistician, joined Arthur Guinness Son & Co. Unlike most brewers who guarded their secrets closely, Guinness, recognizing the value of science, actively recruited scientists to optimize their brewing process. Gosset, with his keen eye for detail, quickly realized the importance of statistics in analyzing data from countless experiments with barley, yeast, and brewing techniques.

However, Gosset faced a challenge. Most statistical methods of the time were designed for large datasets. In a brewery,where experiments were often limited by factors like cost and time, analyzing small datasets was crucial. Traditional methods produced unreliable results when applied to these smaller sample sizes.

Determined to find a solution, Gosset embarked on a groundbreaking exploration. He developed a new statistical test that took into account the inherent variability present in small datasets. This new method, christened the "Student's t-test" due to Gosset's publication under the pseudonym "Student" to protect trade secrets, allowed brewers to analyze the significance of differences observed in small-scale experiments.

The Student's t-test revolutionized the field of statistics. At its core, the t-test helps determine if the observed difference between two averages (means) is likely due to random chance or reflects a genuine underlying difference. This seemingly simple question has profound implications across numerous disciplines.

Beyond the world of brewing, the t-test has become an indispensable tool in scientific research. From analyzing the effectiveness of new drugs to comparing psychological phenomena, the t-test allows researchers to draw statistically sound conclusions from smaller studies. It's used in social science research, economics, and even education to evaluate the impact of different teaching methods.

The story of the Student's t-test is a delightful reminder that innovation can come from unexpected places. It highlights the transformative power of applying scientific rigor to everyday processes. And next time you raise a glass of Guinness, remember – you're not just enjoying a delicious beverage, you're celebrating a statistical marvel born in the heart of a brewery.  Let me know what you think, I'd love to hear.  Have a great day.

The Secret Society Who Discovered Irrational Numbers

The discovery of irrational numbers stands as a pivotal moment in the history of mathematics. These numbers, unlike rational numbers, cannot be expressed as a simple fraction. Their existence challenged the very foundation of the Pythagorean belief system, leading to both intellectual triumph and a touch of mystery.

The story is often attributed to the Pythagoreans, a secret society in ancient Greece obsessed with the harmony of numbers and their connection to the cosmos. While details are fuzzy, credit for the discovery is usually given to Hippasus of Metapontum, a Pythagorean philosopher.

Hippasus' revelation likely stemmed from the study of perfect squares – integers that can be obtained by squaring an integer (e.g., 4 is a perfect square because 2 x 2 = 4). The Pythagoreans believed that all lengths could be expressed as ratios of whole numbers. However, Hippasus, while investigating the diagonal of a square relative to its side,demonstrated that this wasn't always true.

Imagine a square. The length of a side is a simple, whole number (let's say 1). According to the Pythagorean theorem, the diagonal of this square would be equal to the square root of 2 (√2). However, try as they might, the Pythagoreans couldn't express √2 as a fraction with a whole number numerator and denominator. They reached a logical contradiction: if the side length is rational and the Pythagorean theorem holds true, the diagonal must also be rational. But their attempts to express √2 as a fraction kept failing.

This discovery of irrational numbers, numbers that were "incommensurable" with rational numbers, was a major blow to the Pythagorean worldview. Their belief system was built on the idea that everything could be neatly explained by whole numbers and their ratios. The existence of irrational numbers threw a wrench into their philosophy, highlighting the limitations of their understanding of mathematics.

The legend surrounding Hippasus' discovery adds a dramatic flair. Some accounts claim he was ostracized or even drowned at sea by his fellow Pythagoreans for revealing this unwelcome truth. While the details may be embellished, the story underscores the profound impact of this discovery.

The revelation of irrational numbers forced mathematicians to re-evaluate their understanding of numbers themselves. It opened doors to new areas of mathematical exploration and paved the way for advancements in geometry, algebra, and beyond. Although the exact details surrounding the discovery remain shrouded in the mists of time, the story of irrational numbers serves as a reminder of the power of challenging established beliefs and the ongoing quest for mathematical knowledge.  Let me know what you think, I'd love to hear.  Have a good day.

The Silk Dress Mystery.

 The unassuming Victorian-era silk dress, purchased at a Maine antique mall, held a hidden treasure – a coded message tucked away in a secret pocket. Discovered in 2013 by archaeologist Sara Rivers-Cofield, the "Silk Dress Cryptogram" baffled experts for over a decade. Two short, cryptic messages seemed to defy conventional code-breaking techniques.But in 2023, the veil of secrecy was finally lifted, revealing the fascinating story behind the codes.

The initial attempts at deciphering the messages focused on established methods. Cryptologists explored the possibility of military codes, given the dress's approximate dating to the 1880s. However, this theory was quickly dismissed due to inconsistencies with known wartime communication codes of the era. Another possibility involved telegraphic codes, a more common communication method in the late 19th century. This track held more promise, aligning with the historical context.

The breakthrough came from Nick Pelling, a code-breaking enthusiast who analyzed the symbols and patterns within the messages. He noticed a consistency in the number of symbols used and a potential repetition structure. This led him to suspect a "bigram substitution cipher," where pairs of letters were replaced with single symbols.

Meanwhile, another crucial piece of the puzzle emerged. Historian Christopher Stuart stumbled upon a reference to a specific weather code used by the U.S. Army Signal Service (later the U.S. Weather Bureau) around the same period. This code, designed for efficient telegraphic transmission of weather data, employed a system of two-letter codes representing locations, weather conditions, and other meteorological observations.

Combining Pelling's observations with Stuart's historical discovery, cryptologists were finally able to crack the code. The seemingly random symbols revealed themselves to be weather observations, meticulously recorded by an unknown individual – perhaps a military officer stationed at a remote outpost. The messages detailed weather conditions like wind direction, barometric pressure, and cloud cover, offering a glimpse into a bygone era's meteorological practices.

The Silk Dress Cryptogram serves as a testament to the enduring power of human ingenuity. A seemingly mundane object held a secret message for over a century, waiting to be deciphered. The collaborative effort of cryptologists, code enthusiasts, and historians ultimately unlocked the code, revealing a fascinating historical tidbit and showcasing the evolution of weather data collection. More importantly, it highlights the potential for hidden stories to lie dormant within everyday objects, waiting to be discovered by curious minds.

The Changing View Of Equal Signs.

 

The equals sign (=), a seemingly innocuous symbol, has long been the cornerstone of mathematical expression. It signifies a fundamental truth: the expressions on either side represent the same quantity or relationship. However, a growing number of mathematicians are rethinking this seemingly absolute concept. This shift in perspective is leading to a new understanding of the equals sign, one that embraces the concept of equivalence rather than strict equality.

The traditional view of the equals sign implies a one-to-one correspondence between the left and right sides of the equation. 2 + 3 = 5 is a classic example – the sum on the left is demonstrably equal to the number on the right. However,mathematics ventures beyond simple arithmetic. As we delve into abstract concepts like sets, categories, and transformations, the question of "what exactly are we equating?" becomes more nuanced.

Category theory, a powerful branch of mathematics that focuses on relationships between mathematical structures, offers an alternative perspective. Here, the equals sign is reinterpreted as an isomorphism, a concept signifying a deeper structural equivalence, not necessarily a numerical match. Imagine two different maps of the same city. While the maps themselves might look different, they both represent the same underlying structure (the city) and allow you to navigate it.

This shift from equality to equivalence unlocks new ways of understanding and manipulating mathematical objects. For instance, two seemingly different equations might, under closer scrutiny, be revealed as equivalent representations of the same underlying concept. This can lead to deeper insights and potentially simpler solutions to complex problems.

The changing interpretation of the equals sign isn't a rejection of its traditional use. In basic arithmetic and algebra, the old understanding remains perfectly valid. However, as mathematics grapples with more complex structures and relationships, the language of equivalence offers a more nuanced and flexible way to express these connections.

This rethinking of the equals sign is not merely an academic exercise. It has potential applications in various fields, from computer science, where it can streamline data processing and algorithm design, to physics, where it can help unify seemingly disparate theoretical frameworks.

The equals sign, once a symbol of absolute certainty, is evolving. By embracing the concept of equivalence,mathematicians are opening doors to new avenues of exploration and discovery. This shift in perspective reflects the dynamic nature of mathematics itself – a vast and ever-evolving landscape where even the most fundamental symbols can take on new and unexpected meanings.  Let me now what you think, I'd love to hear.  Have a great day.

Pyramids And Degrees

The other day, at band, the director mentioned  something called the "Pyramid Concept" and then I went with a family member to their first physical therapy appointment and the physical therapist had a cool implement that I've never seen before.  So today, I'll be talking about both today as they are a different application of mathematical concepts.

First, we'll start with the musical "Pyramid Concept". In music production, the "Pyramid Concept" is a framework for structuring a song. It visualizes the song as a pyramid, with the foundation playing a crucial role in supporting the elements that build on top.

The pyramid's base represents the foundation of the song – harmony and rhythm. Chords provide the harmonic framework, creating the overall tonal structure. The rhythm section, consisting of drums, bass, and sometimes percussion,lays down the groove that drives the song forward. Both these elements are crucial for creating a strong foundation that listeners can connect with.

Moving up the pyramid, we encounter the melody and lyrics. The melody, often carried by the vocal line or a lead instrument, provides the catchy hook that listeners remember. Lyrics add another layer of meaning and emotion, weaving a story or expressing feelings. These elements build upon the foundation laid by harmony and rhythm.

The peak of the pyramid represents production and effects. This is where additional instruments, vocals, and sonic enhancements are layered on top. While these elements can add depth and excitement, they should not overshadow the core foundation.

The Pyramid Concept emphasizes that strong songwriting starts from the bottom up. A well-defined harmonic and rhythmic foundation allows melodies and lyrics to shine. Production and effects then add the finishing touches, creating a cohesive and impactful song.

As for the other topic, the tool used in physical therapy to measure how much someone can move a joint, like the arm, is called a goniometer. It resembles a large compass with two arms connected by a hinge. In the middle is a round circle that goes from 0 to 180 in each half, much like the markings in a protractor.  

The therapist positions the goniometer on specific landmarks around the joint. One arm is aligned along the bone before the joint, and the other arm aligns with the bone after the joint. The hinge of the goniometer acts as a fulcrum, mimicking the joint's movement. As the arm moves, the therapist observes the degrees marked on the hinge, indicating the range of motion.

Goniometers are not just for measuring arm movement. They can be used on any joint in the body, like the knee, ankle, or spine. This information helps physical therapists assess a patient's progress, set rehabilitation goals, and track improvement over time. 

Two new ways to use mathematical concepts.  Let me know what you think, I'd love to hear.  Have a great day.

What Mathematical Concepts Should Be Taught In Pre-School?

 

Math is often viewed as a complex subject, but the seeds of mathematical understanding are planted much earlier than many might think. Preschool provides a crucial window to introduce young children to fundamental concepts that will pave the way for future success in math.

First, number sense since it forms the bedrock of math. Preschoolers should be exposed to activities that develop their ability to recognize, count, and compare numbers. Singing counting songs, playing games with numbered objects, and sorting toys by quantity are all engaging ways to solidify this foundation.

Next, spatial reasoning is another vital area. Activities involving shapes like blocks, puzzles, and shape sorters help children recognize basic shapes like circles, squares, and triangles. Additionally, exploring concepts like "up" and "down," "big" and "small," and "in" and "out" lay the groundwork for understanding spatial relationships.

Identifying patterns is a key mathematical skill. Preschoolers can be introduced to patterning through activities like sorting objects by color or size, completing simple sequenced patterns (ABAB), or singing songs with repetitive structures. These activities nurture their logical thinking and problem-solving abilities.

Early exposure to measurement helps children understand concepts like "long" and "short," "heavy" and "light," and "full" and "empty." Preschoolers can compare objects using their senses or simple tools like measuring cups. Activities like pouring water from different sized containers or sorting objects by length can introduce basic measurement concepts.

The key to fostering a love of math in preschoolers is to make learning fun and engaging. Sing songs that involve counting, play games that reinforce number recognition, and provide opportunities for exploration and discovery.Learning math shouldn't feel like a chore; it should be an exciting adventure filled with exploration and play.

By focusing on these core concepts, preschool teachers can equip young children with the foundational skills they need to thrive in math throughout their academic journey. A strong foundation in math empowers students to tackle more complex concepts with confidence, setting them on the path to success in this fascinating and ever-evolving subject.

Remember, early math education isn't about rote memorization or advanced calculations. It's about nurturing a love of exploration, fostering curiosity, and providing a solid foundation for future learning. With the right approach, preschool can be the launchpad for a lifelong journey of mathematical exploration and discovery.  Let me know what you think, I'd love to hear.  Have a great weekend.

Decision Theory And The Classic Secretary Problem.

 

The classical secretary problem, a thought experiment in decision theory, poses a dilemma: you have a single position to fill and interview a set number of candidates in random order, ranking them as you go. The challenge? You can only accept or reject a candidate after the interview, and there are no second chances.

This seemingly simple scenario becomes surprisingly complex. But what if we could peek into the future, not literally, but statistically? Enter the realm of decision-making analysis, a variation of the classical secretary problem that injects a strategic element.

Decision-making analysis introduces the concept of a "stopping rule." Imagine having access to historical data or statistical models that can predict the quality of future candidates based on the pool you've already interviewed. This allows you to make informed decisions rather than relying solely on the candidate in front of you.

There are two main approaches to decision-making analysis in the secretary problem: the "reservation strategy" and the "threshold strategy."

The reservation strategy involves setting a minimum acceptable standard for a candidate (the reservation level). You reject all candidates below this threshold, waiting for one who exceeds it. This strategy is safe but might lead you to miss out on the very best candidate, who could appear later in the sequence.

The threshold strategy, on the other hand, involves calculating a "stopping probability" – the chance of encountering a better candidate later on. Based on this probability and the quality of the current candidate, you can decide to accept or reject. This strategy is riskier but potentially leads to a higher chance of selecting the best candidate.

Decision-making analysis isn't just theoretical. It has applications in various fields where ranking and selection are crucial. For example, imagine a hiring manager interviewing candidates for a sales position. Historical data on past hires might predict the quality of future applicants. By using a threshold strategy, the manager can decide whether to accept a promising candidate or hold out for someone with a potentially higher sales potential.

While the idea of predicting future candidates seems attractive, real-world applications have limitations. Historical data might not always be reliable, and creating accurate statistical models can be challenging. However, decision-making analysis offers a glimpse into a more strategic approach to the secretary problem, highlighting the potential benefits of incorporating statistical analysis into selection processes.

The classical secretary problem, with its decision-making analysis variation, demonstrates the complexities of selection in a world of limited information. While the perfect prediction might be elusive, understanding the role of statistics and calculated risk-taking can help us make better decisions in a world full of uncertainties.  Let me know what you think, I'd love to hear.  Have a great day.

The Everyday Power Of Exponents.

Exponents, those tiny superscript numbers, might seem like relics of high school math. But their applications extend far beyond textbook problems. These seemingly abstract rules govern a surprising number of real-world phenomena, from the growth of bacteria colonies to the mind-boggling calculations within your computer. Let's see how the magic of exponents shapes our world.

One of the most fundamental applications of exponents lies in understanding exponential growth and decay. Imagine a bacteria colony doubling every hour (represented by 2). After two hours, you'll have 2 x 2 bacteria, or 4. This concept,where the base (2) is multiplied by itself according to the exponent (the number of times it doubles), allows scientists to model population growth, predict the spread of viruses, or even calculate the decay of radioactive materials (with a decaying base like 0.5).

In addition, exponents are the hidden language within your computer. Every piece of information, from images to text, is stored as a series of binary digits (0s and 1s). Exponents come into play when dealing with large amounts of data. For instance, 2^10 represents 1024, a common unit of data storage. Understanding exponents allows us to efficiently compress and transmit vast amounts of information in the digital world.

Furthermore, fractals, those self-similar geometric patterns found in nature (like snowflakes or coastlines), are often described using exponents. These complex shapes exhibit repeating patterns at different scales. By understanding how these patterns change with magnification (often involving exponents), mathematicians can model natural phenomena and even create stunning computer-generated fractals.

Also, understanding exponents is crucial in finance. Compound interest, where interest earned is added to the principal amount and then itself earns interest, is based on the concept of exponential growth. Even small interest rates, compounded over time (represented by the exponent), can significantly grow your savings, making exponents a valuable tool for financial planning and understanding the power of long-term investments.

Exponents may seem like a mathematical abstraction, but their reach extends far beyond textbooks. From understanding the growth of populations to the intricate workings of computers and the magic of compound interest, exponents are the hidden language that governs many real-world phenomena. So, the next time you encounter these tiny superscript numbers, remember – they hold the key to unlocking a deeper understanding of the world around us. Let me know what you think, I'd love to hear. Have a great day.

Is Pre-Algebra A Bridge Or A Barrier to Algebra Success?

Algebra, the cornerstone of mathematics, unlocks powerful problem-solving skills. But before diving into the world of variables and equations, some students benefit from a preparatory course: pre-algebra. Let's explore who needs this bridge and why it can be a valuable stepping stone, not a barrier, to success in algebra.

Not all students require pre-algebra. Those with a strong foundation in arithmetic – understanding operations with whole numbers, fractions, decimals, and percentages – might be well-equipped to tackle algebra directly. However, some students might benefit from pre-algebra.

If a student has trouble with certain basic math skills such as calculations,  conversions between fractions, decimals, and percentages, or a lack of fluency in applying arithmetic operations.  If they have weak problem-solving skills like identifying key information, struggling to break down problems, or are unable to develop a logical approach to solving them. Or if they have an attitude towards math that hinders learning or evening engaging with new concepts, they might need pre-algebra. 

Students who demonstrate that they have a strong foundation in arithmetic where they have a confidence and are accurate in performing calculations and an ability to manipulate numbers.  They have the ability to analyze problems, identify relevant information, and they can develop strategies to find solutions.  They have a genuine interest in learning new math concepts and a willingness to tackle challenges.  These are the ones who are ready to move straight into algebra.

Pre-algebra shouldn't be seen as a roadblock but as a bridge. It provides a targeted environment to solidify foundational skills that are crucial for success in algebra. A pre-algebra class can help fill in gaps. It allows teachers to identify and address specific areas of weakness in arithmetic, ensuring students have a strong foundation before moving on. Pre-algebra courses often emphasize problem-solving techniques, teaching students how to break down problems, develop logical approaches, and build confidence in their ability to tackle challenges.  It helps build mathematical confidence by helping students master foundational skills that fosters a sense of accomplishment and reduces math anxiety, creating a more positive foundation for learning algebra.

Think of pre-algebra as a bridge not a barrier. Ultimately, the decision of whether a student needs pre-algebra should be made on an individual basis, considering their current skillset and learning style. Pre-algebra serves a vital role, not by holding students back, but by equipping them with the tools they need to thrive in algebra and beyond. By building a strong foundation and fostering a positive learning experience, pre-algebra paves the way for a successful journey into the fascinating world of mathematics.  Let me know what you think, I'd love to hear.  Have a great weekend.

What Essential Concepts Are Needed For Success In Algebra?

 

Algebra is often perceived as a mysterious collection of variables and equations but it can open up a powerful world of problem solving. However, what concepts do students need in their foundational knowledge to navigate and master the topic.  Time to look at the key building blocks students need to succeed in algebra. 

First, students need to understand the power of variables. Variables, represented by letters like x and y, are the cornerstone of algebra. They act as placeholders for unknown values,allowing us to express general relationships and solve for unknowns. Understanding how to manipulate variables – performing operations like addition, subtraction, multiplication, and division – is crucial for success.

Next they need to see that equations are balanced. Equations are statements that declare two expressions equal. They are like scales – for an equation to hold true, both sides must be balanced. Understanding how to solve equations – finding the value of the variable that makes the equation true – is a core skill in algebra. This involves using addition, subtraction, multiplication, and division to isolate the variable on one side of the equation.

In addition, they should understand that expressions are the building blocks of equations. Expressions are combinations of numbers, variables, and mathematical operations. They can be simple, like 2x + 3, or more complex, involving brackets and various operations. Simplifying expressions – combining like terms and using the order of operations (PEMDAS) – is essential for manipulating expressions and solving equations.

Furthermore, they should know how the order of operations actually works rather than being able to recite the rules. PEMDAS, an acronym for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right), dictates the order in which we perform operations within an expression. Following PEMDAS ensures we obtain the correct result when simplifying expressions and solving equations.

It is important that students know how to factor or expand equations. Factoring involves breaking down a polynomial (an expression with multiple terms) into its component parts. Expanding is the opposite – combining factored terms back into a single expression. These skills are vital for solving various types of equations and simplifying complex expressions.

Then they should understand how functions work. Functions are relationships between variables. They describe how one variable (the input) affects another variable (the output). Understanding basic function notation (f(x) for the output of function f evaluated at input x) and graphing functions is essential for many areas of algebra and beyond.

These core concepts form the foundation for success in algebra. By mastering these building blocks, students can progress to solving linear equations, tackling inequalities, understanding systems of equations, and venturing into more advanced topics like quadratic equations and exponential functions.

The journey through algebra may seem daunting at first, but with a solid grasp of these fundamental concepts,perseverance, and a willingness to practice, students can unlock the secrets of this mathematical realm and unlock a world of problem-solving abilities.  Let me know what you think, I'd love to hear. Have a great day.

Fluid Dynamics And Tissue Movement

 Imagine watching a video showing two streams of droplets of ink in an ink jet printer and realizing that one of the streams was not a video, it was a simulation.  A young researcher realized that it was possible to predict everything about ink droplets.

The simulation was based on the laws of fluid dynamics which explains how gases and liquids behave. This one event, lead to a young researcher and his team to publishing a paper concluding that sheets of epithelial tissues, those tissues that make up skin and sheathe internal organs act as liquid crystals.

Liquid crystals are materials whose structure is ordered like a crystal but flows like a liquid.  It was discovered there are two different symmetries within the epithelial tissues.  These symmetries determine how liquid crystals react to physical forces but each appears at different scales.

This discovery could make it much easier to apply fluid dynamic based simulations to living tissues. it is hoped people can use this information to predict how tissues react in various situations ranging from healing to cancer metastasizing. 

We know that liquid crystals flow like liquids but they have a certain structure but their reaction to stimuli  tends to be based on symmetry and orientation.  Most people see this when they watch a liquid crystal display.  The crystals refract light based on their orientation.  In addition to being found in televisions, liquid crystals are found in cells and cell membranes.  Scientists hit a wall because they found that small groups of crystals had a six fold hexatic symmetry while tissues had a two fold nematic symmetry which created a contradiction.

Eventually one scientists said this was possible if both symmetries existed in the same time based on simulations. With the help of a biologist, they were able to see how both symmetries existed but they needed a real situation to distinguish the two symmetries in the tissue.  They created a shape tensor which captured information about cell shapes and their and directions of cells in a tissue sample. In very small clusters, they saw that the cells had a sloped hexagonal shape but in larger clusters of cells the two fold symmetry emerged confirming the earlier model.

Although they were able to prove the simulation, they still aren't sure how the change from six fold to two fold symmetry happens or what signals the change. it is theorized that there is either a cell that causes the change or due to the way the cells adhere to each other.  However, scientists have also suggested that this occurrence helps cells be more than their whole.

With this confirmation, scientists have a direction to move in their research.  Let me now what you think, I'd love to hear.  Have a great day.