The Post-Truth Era

Multidimensional Approach in Math Education

 

 

Carol Dweck herself has written that the information on the value of changing mindsets needs to be accompanied by a different approach to teaching, one that enables students to learn differently. One of the things that she says keeps her up at night is when students are told to put in effort and that success is all about hard work, without their being given the tools by teachers to learn more effectively. As she says, “Effort is key for students’ achievement, but it is not the only thing. Students need to try new strategies and seek input from others when they are stuck.”

**

Fluid and flexible brains, neuroscientists conclude, come from the synchrony that occurs when multiple brain areas are working together. Communication between brain areas comes about when we approach knowledge through multiple avenues, encountering ideas in different forms and representations.

A multidimensional approach can be used in the teaching of all subjects to bring about higher engagement and achievement. Many subject areas, particularly in the humanities, already value treating the subject in multiple ways by asking students to give their own interpretations of texts they read and employing such forms as group discussions, debates, and plays. In most cases they could still become more multidimensional, but they are rarely as narrowly taught as some other subjects. In my experience the subjects that seem most in need of change are mathematics, science, and language teaching. Coming at the subject matter from multiple angles is an ideal learning approach for all of these disciplines.

**

Working in these multiple ways encourages brain communication while also bringing the content to life. The vast majority of students think about math as a set of numbers and methods and about English as books and words. When we approach math, English, science, or other subjects as opportunities for creativity and seeing things in multiple ways, it changes everything, stimulating vital brain growth and neural connections. Additionally, as teachers diversify the curriculum, moving from a simple list of numerical answers, pages of text, or scientific equations to visuals, models, words, videos, music, data, and drawings, the classroom changes from a place where all the work looks the same to one where the variety is enticing and creativity can be celebrated.

**

This new openness to challenges and uncertainty seems to be a common reaction to becoming unlocked—people realize that it is good to struggle, that it is not a sign of a brain weakness, but of brain growth. This leads to more confidence in times of struggle and a willingness to share ideas that they are unsure of. One of the saddest, most central characteristics of fixed-brain thinking is the fear of being wrong. People’s minds are literally locked, immobilized, by their fear, which is why an approach to life that values multidimensionality, growth, and struggle is so liberating. Holly said: “I have so many more ideas because I let myself have ideas.”

**

Another core benefit of working and living with a multidimensional approach is that when roadblocks appear, you know there are alternate routes. Many of the adults I interviewed for this book said they would no longer stop when they met challenges or roadblocks; they simply would find another strategy, another approach. A multidimensional approach to knowledge reveals that there isn’t only one way to do anything; there are always multiple ways forward.

**

In many classrooms students are given problems they do not know how to access—which causes them to think negatively about themselves and their learning. When problems are changed to become “low floor and high ceiling”—problems that are accessible by all but lead to more challenging work—everybody can access them and take them to different places.

You do not have to live your life as an “expert”

 

Part of the process of change and of becoming limitless involves letting go of the idea that your past failures came about because there was something wrong with you. A similarly important change is realizing that you do not have to live your life as an “expert,” that you can go into situations and proudly share uncertainty. Jesse Melgares told me about these two aspects of the change he went through as he became unlocked. Jesse is an assistant principal in east LA, but in earlier years he taught mathematics and was, as he said, “extremely self-conscious,” thinking he did not know enough and nothing could change. When Jesse became an assistant principal, he needed to coach math teachers, but he was fearful that others would find out he was a fraud:

To be honest I would get a lot of paralyzing stress when someone asked me a math-related question. . . . It was terrible. It was like a boot on my chest. It’s what I woke up with in the morning, wondering, “Am I gonna be asked something that I don’t know the answer to? And will I be discovered as some sort of fraud?”

The feeling of paralyzing stress Jesse described, the fear of being asked something he could not answer, is a feeling shared by millions of people in different situations and jobs, and it is a feeling that I hope this book can change. For Jesse the change began when he took one of my online courses and realized: “Everything that I had been taught as a student of math when I was in the K–12 system and as a math educator was wrong.”

For Jesse, the first step in becoming unlocked was realizing that any trouble he had had learning in the past was not due to some deficit in him, but to the faulty system in place. This is a shift I have seen others make, and it is vital for those who have had bad learning experiences.

Jesse not only started feeling better about mathematics; he started a new “journey” discovering that mathematics was his passion. He shifted from feeling defeatist about math to seeing it as an exciting challenge. Jesse is now the director of mathematics for twenty-five schools—quite a change for a person who used to feel paralyzing stress when he thought about math. New knowledge about the brain allowed him to shift his perspective, his mindset, and his belief in himself. Jesse still meets questions he cannot answer, but instead of being afraid, he thinks: “Well, I don’t know what the answers are but, you know, we’ll figure it out. This is a challenge.” This shift in perspective is typical for people who have become unlocked. When people change their mindset and become aware of the positive benefits of struggle, they take a new and much more positive approach to challenge and uncertainty. They let go of the need to be the expert and replace it with curiosity and the desire to collaborate.

The Power of Yet

 

When students become disillusioned because others are ahead of them or complain that they don’t understand something, a word that Carol Dweck champions using with them is “yet.” When I ask adults to visually represent an idea, I often hear them say, “I am terrible at drawing.” I tell them, “You mean you have not learned to draw well yet.” This may seem like a small linguistic change, but it is an important one. It moves the focus from the perceived personal lack to the process of learning.

Beliefs vs Brains

 

In order to study the impact of our beliefs on our health, Stanford researchers Alia Crum and Octavia Zahrt collected data from 61,141 people over an extensive time span, twenty-one years. The researchers found that those people who thought they were doing more exercise were actually healthier than those who thought they were doing less, even when the amount of exercise they were doing was the same.

**

**

If you enter a challenging situation believing in yourself, but then mess up, your brain will react more positively than if you go into a situation thinking, “I don’t think I can do this.” If we have a difficult job or a problematic situation at home, this result should prompt us to go into those situations believing in ourselves. If we enter difficult situations with positive beliefs, our brains will become more resilient and adaptative when we make errors than if we are doubting ourselves. This change in belief alters the physical structures of the brain and creates avenues for higher-level thinking and creative problem solving. Just as those who believed they were engaging in healthy exercise became healthier, those who believe they are learning more productively actually learn more.

**

One of Dweck’s studies revealed the immediate impact of the word “smart.” Two groups of students were given a challenging task. On completion, one group was praised for being “really smart,” and the other was praised for working hard. Both groups were then offered a choice between two follow-up tasks, one that was described as easy and one that was described as challenging. Ninety percent of the students praised for working hard chose the harder task, whereas the majority of the students praised for being “smart” chose the easy task. When students are praised for being smart, they want to keep the label; they choose an easy follow-up task, so they can continue to look “smart.”

**

It is just as important to take on ideas about social comparison with students as it is to make them aware of the value of struggle. I have had many conversations with learners of all ages who argue that brains must be fixed, because some people appear to get ideas faster and to be naturally “gifted” at certain subject areas. What they do not realize is that brains are growing and changing every day. Every moment is an opportunity for brain growth and development. Some have simply developed stronger pathways on a different time line. It is critical that students understand that they too can develop those pathways at any time—they can catch up with other students if they take the right approach to learning.

Making mistakes are the best times for brain growth

Most of us have grown up with the idea that mistakes are bad, especially if we attended test-driven schools, where we were frequently marked down for making mistakes, or our parents punished mistakes with harsh words and actions. This is unfortunate, and this is why. 

**

... [In a workshop, Carol Dweck] announced that every time we make mistakes, synapses fire in the brain, indicating brain growth. All the teachers in the room were shocked, as they had all been working under the premise that mistakes are to be avoided. Carol was drawing from work that has researched the brain’s response when we make mistakes, particularly investigating the different ways brains respond when people have a growth or a fixed mindset.

Jason Moser and his colleagues extended Carol’s work investigating the brain’s response when we make mistakes. Moser and his team found something stunning. They had asked participants to take tests while they monitored the participants’ brains with MRI technology. They looked at the scans when people got questions correct and when they got them incorrect. The researchers found that when people made mistakes, brains were more active, producing strengthening and growth, than when people got work correct. Neuroscientists now agree that mistakes positively contribute to the strengthening of neural pathways.

This learning key is particularly significant because most teachers design classes so that everyone is successful. Curricula and textbooks are designed with trivial, unchallenging questions, so that students will get a high percentage of answers correct. The common belief is that getting most answers correct will motivate students toward greater 

success. Here’s the problem, though. Getting questions right is not a good brain exercise.

For students to experience growth, they need to be working on questions that challenge them, questions that are at the edge of their understanding. And they need to be working on them in an environment that encourages mistakes and makes students aware of the benefits of mistakes. This point is critical. Not only should the work be challenging to foster mistakes; the environment must also be encouraging, so that the students do not experience challenge or struggle as a deterrent. Both components need to work together. 

**

One of the significant characteristics of the highly effective learning described is the presence of mistakes and the role of struggle and error in transforming people from beginners into experts. This is consistent with the brain research showing increased brain activity when people struggle and make mistakes and decreased activity when they get work correct. Unfortunately, most learners think they should always be getting work correct, and many feel that if they make mistakes or struggle, they are not good learners—when this is the very best thing they can be doing.

Practice is important for the development of any knowledge or skill. Anders Ericsson helped the world understand the nature of expert performance and found that most world-class experts—pianists, chess players, novelists, athletes—practiced for around ten thousand hours over twenty years. He also found that their success was not related to tests of intelligence but to the amount of “deliberate practice” they undertook. Importantly, although people succeed because they are trying hard, the people who become experts are trying hard in the right way. A range of different researchers describe effective practice in the same way—people pushing at the edge of their understanding, making mistakes, correcting them, and making more.

** 

Japan has always scored well in mathematics—it has always finished in one of the top-five TIMSS positions—and was one of the countries visited in the study. The researchers found that Japanese students spent 44 percent of their time “inventing, thinking, and struggling with underlying concepts,” whereas students in the US engaged in this kind of behavior less than 1 percent of the time.

Jim Stigler, one of the authors of the study, writes that the Japanese teachers want the students to struggle—and recalls the times when they would purposely give the wrong answer so that students would go back and work with foundational concepts. In my thousands of observations of classrooms over many years in the US and the UK, I have never seen this kind of practice; more typically I have seen teachers who seem to want to save students from struggle. Many times I have observed students asking for help and teachers structuring the work for students, breaking down questions and converting them into small easy steps. In doing so they empty the work of challenge and opportunities for struggle. Students complete the work and feel good, but often learn little.

I saw a very similar teaching approach, focused on struggle, in a visit to classrooms in China, another country that scores highly in mathematics. I had been asked to visit China to give a talk at a conference and managed, as I like to do, to sneak away and visit some classrooms. In a number of high-school math classrooms, lessons were approximately one hour long, but at no time did I see students working on more than three questions in one hour. This contrasts strongly with a typical US high-school math classroom, where students chug through about thirty questions in an hour—about ten times more. The questions worked on in Chinese classrooms were deeper and more involved than the ones in US classrooms. Teachers would ask provocative questions, deliberately making incorrect statements that students would be challenged to argue against.

One of the lessons I watched was on a topic that is often uninspiring in US classrooms—complementary and supplementary angles. The teacher in China asked the students to define a complementary angle, and the students gave their own ideas for a definition. Often the teacher would push the students’ definition to a place that made it incorrect and playfully ask, “Is this right, then?” The students would groan and try to make the definition more correct. The teacher bantered with the students, playfully extending and sometimes twisting their ideas to push the students to deeper thinking. The students probed, extended, clarified, and justified for a long time, reaching depths that were impressive.

**

**

"I am going to figure this out if it kills me."

When I talk with teachers, they often say this sort of persistence is missing in the students they teach. One of the most common complaints I hear from teachers is that students don’t want to struggle; they want to be told what to do. To the teachers it seems as though students just can’t be bothered with struggling, which is probably what it looks like. The truth is, however, that when students don’t want to struggle, it is because they have a fixed mindset; at some point in their lives they have been given the idea that they cannot be successful and that struggle is an indication that they are not doing well.

**

As an academic, I experience a lot of failure. To keep our youcubed center at Stanford running, supporting staff salaries and providing free materials for teachers and parents, we have to apply for lots of grants—most of which are rejected. I also have to submit our papers to journals, where rejection is part of the process. If they are not rejected, they are subject to reviewers’ comments. I have had reviewers dismiss my work entirely, saying that it is “not research, just a story.” It is nearly impossible to keep going as an academic without viewing “failure” as an opportunity to improve. A wise professor named Paul Black, my PhD advisor, once said to me: “Whenever you send a paper to a journal, have in mind the next journal you will send it to when the paper is rejected.” I have used his advice a number of times.

Taking a limitless approach—particularly when embracing challenge and struggle—also helps when we encounter difficult people. In today’s world of social media, it seems impossible to make a statement about anything without getting pushback, some of it aggressive. I have experienced extreme and aggressive pushback many times, and I now know that it is important to stay strong in those moments and to look for something positive. Instead of dismissing a challenge or beating yourself up, think, “I will take something from this situation and use it to improve.”

Your brain is constantly reorganizing, growing, and changing

The first step in living a limitless, unlocked life is to know brains are constantly reorganizing, growing, and changing. Remembering that every day of our lives, we wake up with a changed brain. In every moment of our lives our brains have opportunities to make connections, to strengthen pathways, and to form new pathways. When we face a challenging situation, rather than turn away because of fear of not being good enough, we should dive in, knowing that the situation presents opportunities for brain growth. As we start to recognize the huge implications of the adaptability of our brains, we will start to open our minds, and live differently.

Giftedness??

 

 

The idea of giftedness is not only inaccurate and damaging; it is gender and racially biased. We have many different forms of evidence showing that those who believe in fixed brains and giftedness also believe that boys, men, and certain racial groups are gifted and girls, women, and other racial groups are not.

One of the forms of evidence that shows this clearly was collected by Seth Stephens-Davidowitz, who focused his attention on google searches. His study revealed something very interesting and disturbing. He found that the most commonly googled word following “Is my two-year-old son . . .” is “gifted.” He also found that parents search the words “Is my son gifted?” two and a half times more than the words “Is my daughter gifted?” This is despite the fact that young children of different genders have equal potential.

Sadly, the problem is not limited to parents. Daniel Storage and his colleagues conducted analyses of anonymous reviews on RateMyProfessors.com, and they found that students were twice as likely to call male rather than female professors “brilliant” and three times as likely to call male rather than female professors “geniuses.”20 These and other studies show that ideas of giftedness and genius are intertwined with racist and sexist assumptions.

I am convinced that the majority of people who have gender or racial biases do not think about them consciously or perhaps even realize they have them. I also contend that if we were to dispel the idea that some people are “naturally” gifted and instead recognize that everyone is on a growth journey and can achieve amazing things, some of the most insidious biases against women and people of color would disappear. This is needed in the STEM fields more than anywhere else; it is no coincidence that STEM subjects evidence the strongest fixed thinking and the starkest inequities in participation. 

Part of the reason so many students are dissuaded from thinking they are capable of learning math is the attitudes of the teachers and professors who teach them.

Brain Plasticity

 

Unfortunately our schools, colleges, businesses, and culture have, for hundreds of years, been built around the idea that some people can and some people can’t. This is why putting young students into different groups and teaching them differently made perfect sense. If individuals within a school or company weren’t reaching their potential, it was not due to teaching methods or environmental factors, but to their limited brains. But now, with decades of knowledge about brain plasticity, it is time that we eradicate this damaging myth about learning and potential.

Energized by the new evidence showing brain plasticity in animals, researchers began to look at the potential of human brains to change. One of the most compelling studies of the time came from London, the city where I had my first teaching and university job. London is one of the most vibrant cities in the world—and it is always filled with millions of residents and visitors. On any day in London you will see “black cabs” zipping around the thousands of major thruways, streets, and lanes. The drivers of these iconic taxicabs hold themselves to very high professional standards. Londoners know that if they get in a black cab and tell the driver a road to find, and the driver does not know it, the driver should be reported to black-cab authorities.

Knowing all the roads in London is quite a feat—and drivers go to huge lengths to learn them. In order to become a black-cab driver, you need to study for at least four years. The most recent cab driver I traveled with told me he had studied for seven years. During this time drivers must memorize every one of the twenty-five thousand streets and twenty thousand landmarks within a six-mile radius of the centrally located Charing Cross station—and every connection between them. This is not a task that can be accomplished through blind memorization—the drivers drive the roads, experiencing the streets, landmarks, and connections, so they can remember them. At the end of the training period, the drivers take a test that is aptly named “The Knowledge.” On average, people have to take the test twelve times in order to pass it.

The extent and focus of the deep training needed by black-cab drivers caught the attention of brain scientists, who decided to study the brains of the black-cab drivers before and after the training. Their research found that, after the intense spatial training, the hippocampus of the cab drivers’ brains had grown significantly.4 This study was significant for many reasons. First, the study was conducted with adults of a range of ages, all of whom showed significant brain growth and change. Second, the area of the brain that grew—the hippocampus—is important for all forms of spatial and mathematical thinking. 

Researchers also found that when black-cab drivers retired from cab driving, the hippocampus shrank back down again—not from age, but from lack of use. This degree of plasticity of the brain, the amount of change, shocked the scientific world. Brains were literally growing new connections and pathways as the adults studied and learned, and when the pathways were no longer needed, they faded away.

Math Anxiety and Brain

 

Researchers now know that when people with math anxiety encounter numbers, a fear center in the brain is activated—the same fear center that lights up when people see snakes or spiders. As the fear center of the brain becomes activated, activity in the problem-solving centers of the brain is diminished. It is no wonder that so many people underachieve in mathematics—as soon as people become anxious about it, their brains are compromised. Anxiety in any subject area has a negative impact on the functioning of the brain. It is critical that we change the messages that are given to learners about their ability and rid education and homes of anxiety-inducing teaching practices. 

Working on mathematics has four stages

 

In a TED talk watched by over a million people, Wolfram (2010) proposes that working on mathematics has four stages:

1. Posing a question

2. Going from the real world to a mathematical model

3. Performing a calculation

4. Going from the model back to the real world, to see if the original question was answered

The first stage involves asking a good question of some data or a situation—the first mathematical act that is needed in the workplace. The fastest-growing job in the United States is that of data analyst—someone who looks at the “big data” that all companies now have and asks important questions of the data. The second stage Wolfram describes is setting up a model to answer the  question; the third is performing a calculation, and the fourth is turning the model back to the world to see whether the question is answered. Wolfram points out that 80% of school mathematics is spent on stage 3—performing a calculation by hand—when that is the one stage that employers do not need workers to be able to do, as it is performed by a calculator or computer. Instead, Wolfram proposes that we have students working on stages 1, 2, and 4 for much more of their time in mathematics classes.

What employers need, he argues, is people who can ask good questions, set up models, analyze results, and interpret mathematical answers. It used to be that employers needed people to calculate; they no longer need this. What they need is people to think and reason.

Kişi Bilmediğine Düşmandır

  

Bu yazı Alin Özinian'ın Kronos'taki Önyargılarımız ve Profiterolün İncisi yazısından alındı. Başlıktaki söz Bediüzzaman'a ait.

 Annie Spratt on Unsplash

Hemen hemen hepimizin önyargıları, ön koşulları ve tabuları var.

En çok karşılaşan önyargılardan biri ise kişideki “kendinde ön yargı bulunmadığı zannı”.

Her insanın çocukluğu, geçmişi, hayatı, içinde bulunduğu ortam, aldığı eğitim, dünyaya bakışı farklı ve özel; tam da bu yüzden her zihin kendine özel filtreler içeriyor. Bu filitrelere algılıyoruz dünyayı.

Ne yazık ki çoğu insan dünyayı herkesin kendisi gibi algıladığını, olayları kendi filtresinden geçirdiğini zannediyor, başka filitrelerden süzülen algılar, onlara sadece farklı değil bazen de düşmanca geliyor. İletişimsizliğin temel sebebi çoğu zaman bu bakış açısı lakin, “gelişmiş” insan bu filitreleri biliyor. Bu sebeble farklılık ve yenilik onun için korkutucu değil, farklılığı tehdit olarak algılamıyor. Yeniyi tanımaya, anlamaya, denemeye hatta zevk almaya ve sevmeye meyilli.

Bunu yaparken kendinden vazgeçmek zorunda olmadığını da biliyor, kendi özgünlüğünü de koruyor.

Ezelden beri genellemeler ışığında bilinmeyen hakkında fikir yürütmek gibi bir yönelim, bir tarz ve bir gerçek var bu topraklarda. Bilgi sahibi olmadan fikir sahibi olanların ahkam kesmesi artık normal bir durum haline geldi. Biz böyleyiz, diyoruz, geçiyoruz.