In her book titled Limitless Mind, Professor Jo Boaler discusses six keys of learning to create opportunities for students, adults, and workers to excel in areas they want to. This article will summarize one of the points in the first key: The Problems of Giftedness. She has conducted sixty-two interviews in six different countries with people from different walks of life. Dr. Boaler developed her “The Limitless Approach” to learning and education. The core principle of this approach is that anybody can learn any subject as long as they think positively about their talents and abilities, and put in a lot of effort and practice.

In her interviews, Dr. Boaler met people that had a wide array of myths and psychological pitfalls that prevented them from accomplishing their goals. The interviewees came from all walks of life and indicated that they gave up studying subjects that they loved including math, science, and English. They believed that they lacked the right mind for these fields, or that they were incapable of learning them, because they struggled in learning them. They also gave up on all math related subjects, such as engineering, science, and technology.

To address these problems, Dr. Boaler had collaborated with brain scientists to learn more about the brain and how we learn in order to help teachers, students, and parents in their subject learning. There is a common misconception among subject fields, especially math, where many children grow up believing that they either are a math person or are not. This is partially due to math anxiety that is widespread in the US and the world. This has become a major hinderance in millions’ mathematics learning. For example, according to one of Dr. Boaler’s studies 48% of young adults working in a work-apprentice program and 50% of students taking introductory math courses (cited in Boaler 2019) have math anxiety. Boaler claims that this might be affecting half of the population.

The idea that people are born with fixed learning abilities is a false misconception and is similar to the misconception that some of society’s highest achievers are successful simply because of their genetics. The idea that our brains are “fixed”, and that we are naturally predisposed to be good or bad at different subjects or activities, is a myth. Research in the last decade has revealed that our brains are incredibly adaptable. Our brains change and reorganize each time we learn something new. These discoveries are all thanks to the research on brain plasticity—neuroplasticity.

Neuroplasticity

Neuroplasticitiy was first discovered in the early 20^th century by Anders Ericsson, a Swedish-born psychologist, and is one of the pioneers who became aware of the brain’s amazing ability to grow and change.

He designed a study to investigate the limits of people’s ability to memorize a random string of digits. He showed that people could improve their ability to memorize with a study that he published in 1929. He chose an average person—Steve—who happened to be an athlete. He started working with the researchers to memorize digits. His performance was average and he memorized seven numbers in the first day. Steve spent four more days to memorize only nine numbers. Although Ericsson and his research team thought he reached his limit, something remarkable happened: Steve continued to improve and then memorized ten numbers. He did not stop and regularly improved until he had successfully memorized eighty-two random digits. He was just an average college student that had unlocked his learning potential to accomplish a very unique challenge.

Later, Ericsson tried the same study with another average college student, Renee. She started better and learned twenty digits without any trainings. Then, she received fifty hours of training but did not improve at all. She eventually quit the study. This intrigued Ericsson and inspired him to find out what caused Steve to succeed and Renee to fail. Ericsson realized that Steve’s love for running had made him more competitive and motivated to succeed. Steve also developed a new strategy each time he struggled memorizing new digits such as grouping numbers into four four-digit strings.

One of the main takeaways from Steve’s example is that it is smart to develop a new strategy or approach when you encounter a roadblock. Dr. Boaler does not say this is easy and explains it by reminding us why so many people fail to make changes in the time of struggles or running into barriers.

Ericsson repeated his experiment one more time to support his claims from the previous studies. He chose another runner named Dario. Dario was more successful than Steve in memorizing numbers. He remembered more than one hundred numbers. This was not because of Dario’s genetics, but instead because he put in lots of effort and hard work. The idea of genetic ability is both incorrect and damaging as we see today in our education systems where fixed-ability thinking controls and designs our children’s education.

Growth and fixed mindsets

Carol Dweck is a professor at Stanford and her research revealed that how we think about our talents has a huge impact on our potential (Dweck 2006). To Dweck, there are two groups of mindsets. The first one is called “growth mindset.” People with this mindset believe that they can learn anything as long as they put in enough effort. The second group has a fixed mindset. People with this mindset believe that their intelligence is more or less fixed and they cannot learn everything. For example, they might believe that they are not a math person so they cannot learn math.

In one of the studies conducted at Columbia University, Dr. Dweck and her colleagues found that stereotyping is still alive and affecting student’s lives. Young female students were given the message that they did not belong in math discipline. Later, they found that this message stuck only with those with a fixed mindset. These people heard the message that math was not for women and they dropped out. However, students with a growth mindset, who believed that anyone can learn anything, remained firm and completed their programs.

We learn a lot of crucial information about the importance of self-beliefs and the role of teachers and parents in influencing students’ lives. But we still receive widespread message containing fixed mindsets and giftedness.

Dr. Boaler underlines the damage of incorrect usage of praises such as “you are smart” or “you are a genius,” coming from parents or others who regularly praise their children by telling them how smart they are in order to build up their self-confidence. Children that only rely on their genius and are not truly taught the value of hard work may face severe obstacles in life later on if they start struggling. Instead, Boaler recommends us to use some alternatives such as, “You can divide fractions? That is great that you have learned how to do that, you must have worked really hard.”

The downside of gifted & talented program in schools

Fixed-brain mindsets have consequences on the vast majority of the population but can also negatively harm society’s most intelligent children, most of which are often labeled as “gifted”. Dr. Boaler explored its negative effects with a film she produced with her youcubed.org team including Sophie Constantinou from Citizen Film. She recruited twelve Stanford students who had the experiences of being labeled as “gifted.” The students were asked to reflect upon their experiences of the labeling. They all gave the same message—they received some advantages but at some costs. They felt the pressure of the label all the time such that they could not ask any questions when they struggled understanding a topic. The students indicated that they had to hide any challenges in order people not to think they do not have a gift. One of the students summarized the burden that came with their gifted labeling as, “If I grew up in a world where no one was labeled as gifted then I would have asked a lot more questions.”

The purpose of classifying some students as gifted is to ensure that high-ability kids are challenged with rigorous and accelerated programs in regular classroom settings. But the problem with this idea is that believing some students are worthy of this program because they have a fixed gift-like present that they had been given as Dr. Boaler points out. Although these students might need special challenging programs, ignoring the rest is not a solution as Dr. Boaler puts it, “the message is that some people are born with something that others cannot achieve” (p. 40).

Another downside of this labeling for the students in Dr. Boaler’s study is that they are not expected to struggle, and when they do it becomes devastating for them. Dr. Boaler shares one of her student’s experiences when she was teaching about research on brain growth and the damage of fixed labels. Her student, Susannah, started talking about her experience of being a gifted student. She had been told frequently that she had a math brain and was very smart. This led her to enroll a math program at UCLA, but things did not go as they were expected to. Susannah took a challenging course in the second year of the program and struggled. She felt that she was not a math person or did not have a math brain after all and ended up dropping out of the program. Unfortunately, Susannah did not know that struggle is a very basic and necessary process for brain growth which could grow the neural pathways she needed to learn more mathematics. If she had known that, Susannah might not had quit and could have graduated with a math major. This is a very real “fixed-ability” scenario that happens every day.

Dr. Boaler does not claim that everyone is born the same. Although everyone has a unique brain at birth, and there are differences between people’s brains, people can change their brain in many ways in terms of achieving things. The proportion of people born with brains so exceptional is literally tiny—less than 0.001 percent of the population. To the contrary of the common belief, Dr. Boaler says, “there is not such thing as a math brain, writing brain, artistic brain, or musical brain” (p 42). That is, we all have to develop the brain pathways to be successful because we all have the potential to learn and achieve at the highest levels as long as we believe that we can succeed and that we are willing to work hard.

Anders Ericsson has studied IQ and hard work for decades and found that people like Einstein, Mozart, Newton and many others are made to be, not born, genius, and their achievements come from extraordinary hard work. This reflects a reality of human condition that “human has only that for which he or she labors” (Qur’an 53:39). Therefore, we need to communicate to all students that they are in a world where growing or evolving is part of everyday life and nothing is fixed about them including gift and/or disability.

In conclusion, the first step towards having a limitless and unlocked life is to know that our brains are given the capacity to constantly reorganize, grow, and change. We need to remember that we are waking up with a changed brain every morning. Our brains are created to make new connections all the time and strengthen older pathways. Once we understand the adaptability of our brains we will start to open our minds and live differently. Then we will not worry about any categorizations we go through at school or work because we know that we have a great growth mindset with which we are enabled to achieve anything we put our minds toward.

References

• Boaler, J. (2019). Limitless mind: Learn, lead, and live without barriers. Harper Collings Publishers.
• Dweck, D. S. (2006). Mindset: The new psychology of success. Penguin Random House LLC, New York.

New developments in scientific thought and hypotheses testing are changing the ways we think about truth and certainty.

Not only do scientific decisions rely heavily on tools and procedures of quantitative analysis, but also on social life and daily decisions. I want to start with a story from the judiciary system about the misuse of an important quantitative tool of the scientific community, Hypothesis Testing. It’s about the overturned Amanda Knox case in Italy. In 2007, she was accused of killing her roommate and sent to jail. The odds of DNA matching in the case of accidental death were reported as incredibly rare by a forensic data analyst, who examined data with regard to incidents of crime, concluding that the accident is not statistically an accident. The judges then made their decision based on this rareness, a single value, named p-value. Later, this was overturned (New York Times, March 27, 2013), because of a misinterpretation by the judges and lawyers, of the rare probability. What factors forced the judiciary system to make this decision based on a single value alone? Honestly, this is the story of the century and a story of Hypothesis Testing. This article introduces the logic behind their decision, draws attention to the misuse of data, and mentions alternative logics to this traditional approach on the matter (in March 2013, Italy’s Supreme Court ordered a retrial and found the two suspects guilty on January 30, 2014).

What is Hypothesis Testing?

We live in a data-driven world. Statistical inference is the process of drawing conclusions or decisions from data. The Hypothesis Testing procedure in the Neyman-Pearson paradigm is one of the procedures of this process, widely used in the scientific community for over a century. In this mindset, two complementary hypotheses are first defined: one is the conventional thesis (the null hypothesis) accepted to be true by default; the other is the alternative thesis (the alternative hypothesis) that needs evidence from data to falsify the null hypothesis. It results in a single value, called p-value, which is calculated from the data using theoretical model assumptions. A p-value is a measure – in probability sense, ranging from 0% to 100% – of how much evidence you have against the null hypothesis. The smaller the p-value, the more evidence you have against the null hypothesis. One may combine the p-value with the significance level to make decisions on a given hypothesis. This is also called significance testing. It has an accept-reject mindset (dichotomous decision or binary logic); in such a case, if the p-value is less than some threshold (usually .05) then the null hypothesis is rejected. Basically, it is a black or white decision, without considering the contrasts between them. This interpretation has been widely accepted in the twentieth century of scientific research, and many scientific journals routinely publish papers using this interpretation for the results of hypothesis tests, even though there are current tendencies not to use this. Let’s take a look at the history of this black-white logic.

History

The history of the process of hypothesis testing starts with Fisher (1890-1962), around the early twentieth century. Later, the contributions of Pearson (1857-1936) and Neyman (1894-1981), who were early fathers of statistics, about the interpretation of the hypothesis tests were integrated into present-day applications. Fisher’s approach focuses on inductive inference about a single hypothesis, whereas the Neyman-Pearson approach informs future behavior based on a test using two complementary hypotheses (Newman 2007).

These approaches are strongly influenced by Popper’s logic of falsification. Falsification can be defined as the act of disproving a proposition, hypothesis, or theory. This logic asserts that sufficiently improbable events can be considered impossible. A p-value suggests whether a null hypothesis is sufficiently improbable to be considered practically falsified in the sense of a logical refutation (Newman 2007). When we come back to the Amanda Knox case, the decision the judges made and justified is followed from this falsification logic in hypothesis testing. Another criticism with this mindset is: how fair it is to rely on a single value that yields a rare result? Are we going to generalize one lucky drawing of raffle tickets to believe all tickets are winners?

Logics and proofs

We use logical flaws and biases in our daily life. Finding the logical flaw in scientific papers is something most readers aren’t interested in. Instead, the results are believed as a fact. One of the widely used flaws/biases is in seeking or interpreting evidence in ways that are partial to existing beliefs or a hypothesis in hand (Mercie and Sperber, 2011). We tend to prove or show the accessibility or superiority of arguments by bringing evidence. Failure to prove that a treatment – say, a low fat diet – is effective is not the same as proving it is ineffective.

Let me give another example to clarify this: The New York Times editorial news reported on February 9, 2006, “Millions of Americans have tried to reduce the fat in their diets, and the food industry has obligingly served up low-fat products. Yet now comes strong evidence that the war against all fats was mostly in vain.” Actually, in the hypothesis testing mindset, the evidence collected from research should either reject the null hypothesis which is “diet is ineffective,” or fail to reject it. The mindset in testing is not about finding evidence to support the null statement. It is also not about proving the alternative hypothesis, “diet is effective.” Rather, it is basically looking for evidences to falsify the null statement. In the report, the “evidence” the reporter meant should be against the null hypothesis, not against the alternative hypothesis. Data is collected to falsify the null hypothesis, not to prove its truthiness, because the null is already accepted to be true unless convincing evidence is collected against it.

Let me give another example of this paradigm. As Newfoundland (2013) described, a person is innocent until proven guilty by bringing convincing evidence. The jury can’t say “he is innocent,” instead, the jury uses evidence that produces reasonable doubt to reject his innocence.

Developments in inferential paradigms

The accept-reject paradigm in inference represents a conventional wisdom. There are other, or currently developing, paradigms in inference, too. One of the dominating paradigms is the Bayesian-likelihood approach. After enjoying much wider acceptance in social and natural sciences, the Bayesian method suggests different views of hypothesis testing. The Bayesian approach is a method of data analysis in which subjectivity, conditionality, or past information is used to update the hypothesis as additional evidence is acquired. This approach in hypothesis testing offers a dynamic structure in probability calculation so hypotheses, and decisions, are not seen as a static truth; instead, they are updated with current data and the decision is stated in the sense of likelihood.

In our court room example, the Bayesian inference is applied to all evidence presented, with the past information being combined with the current evidence. The benefit of a Bayesian approach is that it gives all historical information so the decision is unbiased all along as the past data (prior knowledge) is used correctly. This paradigm changes the way statistics are calculated and how the result and inference are interpreted. Currently it is widely appreciated in quantitative data analysis, especially after convenient software exists for its implementation. However, the criticism to this approach, made by many, is found in its subjectivity. Objectivity and handling prior knowledge is a concern here so that different people, having different opinions, may arrive at different results.

Another developing paradigm: Fuzzy Logic

Fuzzy logic is another method in quantitative decision making. In contrast to binary logic (yes-no, or accept-reject), fuzzy logic can be thought of as gray logic, which allows a way to express in-between data values. It emerged in the development of the theory of fuzzy sets, by Lotfi Zadeh (1965). Fuzzy logic is mostly seen as a branch of artificial intelligence that deals with reasoning algorithms used to emulate human thinking and decision making. It handles the concept of partial truth using linguistic variables, where the truth value may range between completely true and completely false. The concept of partial truth would be subjective and would depend on the observer. For example, for the temperature of the weather, we want to determine when to say cold, warm, and hot. The meaning of each of these concepts can be represented by a certain membership (called a fuzzy set). The concept of each would be subjective. One might consider cold for all values up to 40 0F. In Figure 1, the meanings of the expressions cold, warm, and hot are represented by functions mapping a temperature scale to “truth values” ranging from 0 to 1. A point on that scale has three truth degrees, one for each of the three functions (expressions). The vertical line in the figure represents a particular temperature that the truth values (see three arrows) are measured. Since the red arrow points to zero, this temperature may be interpreted as “not hot.” The orange arrow (pointing at 0.2) may describe it as “slightly warm,” and the blue arrow (pointing at 0.8) “fairly cold” (Fuzzy Logic, 2013, para. 6). A rule could then be adopted, like for example, if “slightly warm” then stop the fan.

Picture was obtained from

Fuzzy logic uses truth degrees as a mathematical model of the vagueness phenomenon, and it summarizes data analysis in facts with truth degrees, and leaves the decision to the observer. Its advantage is its ability to deal with vague situations with respect to linguistic variables. While the significance testing for a hypothesis declares one accept or reject, fuzzy logic allows for degrees of acceptance or rejection. However, in fuzzy logic, the notion of truth doesn’t fall by the wayside, but it is expressed in degrees and offers possibilities for different situations. Regarding inference, fuzzy logic uses the mindset ‘everything is a matter of degree and open to interpretation,’ and this mindset is also adopted to machine learning, which is considered a more suitable mindset with human reasoning instead of binary logic. The constraints in fuzzy logic are found in its tools and interpretations when complex inputs are considered.

The developments of paradigms in statistical inference have similar fates as in the developments of mathematics, geometry, and physics. According to the Euclidean parallel postulate, in space, there exist no two parallel lines that intersect each other. However, after Non-Euclidean geometry was developed in the nineteenth century, a wider geometrical and mathematical reasoning stemmed from it; accordingly, the geometry of the physical universe and particles came to be understood better. The development of Riemannian geometry, offering that distance properties might vary, resulted in the synthesis of diverse results concerning the geometry of higher dimensional surfaces and the behavior of geodesics on them. It also made Einstein’s theory of general relativity justifiable. Likewise, as time passes, we witness wider paradigms or logics that abandon or correct former approaches in scientific decision making tools. This is a good reason why teachers and professors should update their current teachings as to be consistent with convincing trends, as well as to train students to be ready for wider paradigms in the future.

Conclusion

In today’s scientific community, the way decisions are made and fallacies proved, are changing as new paradigms emerge. In order to make better decisions or to validate claims, many aspects and methodologies should be considered. One way to avoid mistakes as much as we can would be to expect fallacy points to exist in human mental processing during decision making, and improving and seeking better alternatives with well-established wisdoms.

(Thanks Ugur Sahin for reviewing the preliminary copy of this article.)

References

• Fuzzy Logic. In Wikipedia. Retrieved August 1, 2013, from http://en.wikipedia.org/wiki/Fuzzy_logic
• Kass, Robert E. 2011. Statistical Inference: The Big Picture 1. Statistical Science, 2011, Vol. 26, No. 1, 1-9, DOI: 10.1214/10-STS337, Institute of Mathematical Statistics.
• Mercier, Hugo, Dan Sperber. 2011. “Why do humans reason? Arguments for an argumentative theory.” Behavioral and Brain Sciences. 34, 57-111.
• Newfoundland, Jason. 2013. “The Cell Phone-Brain Cancer Controversy.” The Fountain Magazine, Jan-Feb, Issue 91.
• Newman, Michael C. 2008. “‘What exactly are you inferring?’ A closer look at hypothesis testing.” Environmental Toxicology and Chemistry, Vol. 27, No. 5, pp. 1013-1019, 2008.