Too often in math class, mathematically interested students are shut down in class, often by well-meaning teachers who either don't know the underlying mathematical concept, feel publicly challenged, see math as a set of rules and formulas, or perhaps is on a tight schedule to "cover" the curriculum.
Most mathematicians feel mathematics is not so much being obedient to a set of fixed rules and regulations, but rather using creativity and risk taking to solve elegant problems. Paul Lockhart, mathematician, says, "Math is not about following directions, it's about making new directions." They are most intrigued and challenged by unsolved or open problems. This requires taking risks, something you might not attribute to learning/teaching math. Mathematician James Tanton says, "Math is being able to engage in joyful intellectual play--and being willing to flail (even fail!)."
Zager talks about the joy children experience when discovering or realizing new mathematical understanding in their everyday lives. It doesn't matter to them if a mathematical law or property already exists in some math textbook.
In a grade two class, Zager observes a math lesson and notices several things. First, the teacher uses wait time, wanting deep thinking and thoughtfulness, not simple ideas and speed. Also, she keeps students focussed on mathematical thinking, not unrelated matters. The teacher makes sure they are using specific and clear mathematical language. Finally, she teaches students to challenge themselves and take risks.
PROMOTING OBEDIENCE VS. ENCOURAGING RISK
Obedience: Memorizing algorithms is necessary. "What's the rule about adding fractions?"
Risk: Different ways to solve problems exist; try to figure out which work best. "That's an interesting approach. Will it work all the time? Show your work in a way people can follow."
Obedience: Smart and easy are common words. Speed is key. "Wow, that was fast! You must be smart!"
Risk: Challenge and try are common words. "This problem is really interesting. Nice job, you really tried hard! I have a challenge for you today."
Obedience: Students speak up when they know the answer.
Risk: Students speak when they have a question, notice something, have an idea, build on a student's thinking, agree or disagree, or have an answer.
Obedience: Students are passive and do what they're supposed to.
Risk: Students are encouraged to takes risks and expect them to come up with novel ideas. "You might be inspired by Alvin's example. Is there a volunteer to try? We will help."
Source: Zager, Tracy Johnston, Becoming the math teacher you wish you'd had, 2017
Willingham's cognitive principle is children differ in intelligence, but the good news is intelligence can be improved through persistent hard work. This has been the Asian educational view for a long time, although with Dr. Dweck's growth mindset ideas, Western thought is changing in that direction. Intelligence is essentially how "people reason well and catch on to new ideas quickly." The current view of intelligence is that there is a general intelligence (g), which contributes to verbal and mathematical intelligence. Therefore, verbal scores are related to math scores, although individual verbal scores relate closer to each other. The "g" is not clearly known, but could be related to the speed or capacity of working memory.
What Makes People Intelligent?
It's the classic nature vs. nature debate; is it genetics or the environment that makes people intelligent? Through many twin studies, genes are responsible for about 50 percent of our smartness. What's interesting is that is starts off young, about 20 percent, then increases to about 60 percent in later life. The bottom line: genetic effects can make people seek out or select different environments. For example, imagine you start off life with a little better memory, more persistence, or simply more curiosity. Your parents pick up on this trait subtly, and begin to use a larger vocabulary or discuss deeper-thinking ideas. This leads you to spend more time with "smarter" kids, and grades become a natural focus. On the other hand, genetically you may not have the physical abilities, which leads you to avoid many sports and instead pick up a book and read instead.
Though genetics plays a large role, intelligence is malleable and can be improved.
Implications for the Classroom
Praise Effort, Not Ability
You want kids to understand they are in control of their intelligence. Praise effort, persistence, and taking responsibility for the work. Be careful of insincere praise, as kids are not easily fooled.
Hard Work Pays Off
Remind students that it takes hard work to be smart, just like it takes hard work and practice to be a successful athlete; natural talent can only take you so far.
Failure leads to Learning
Again, the most successful people (think entrepreneurs, inventors, athletes) take risks and fail in order to succeed. Michael Jordan talks about all his mistakes and failures on the court, which ultimately led to his greatest successes. Remind students that failure is not necessarily embarrassing or negative; it's an opportunity to learn something new.
Study Skills are Necessary
Help struggling students with techniques and methods of effective studying, memorizing, and organizing their time. They need to be self-disciplined and resourceful, as well.
Catching Up is the Goal
In order to catch up with the brighter students, they will need to work even harder than them. There is no easy solution or magic pill. They may need to revamp their entire schedule and drop activities that do not contribute to their educational goals.
Show Confidence in Them
As a teacher, set high standards and expect students to meet them. If they do an substandard job, simply state what they have done and give them feedback for improvement. Do not overpraise them for a mediocre job.
Teachers nowadays are being asked to differentiate learning by meeting students’ individual learning styles, differing cognitive abilities and multiple intelligences. Is this possible? And how effective is it? Willingham, a cognitive scientist, turns that notion on its head. He states that children are more alike than different in terms of how they think and learn.
COGNITIVE STYLES VS. ABILITIES
First, let’s differentiate between cognitive styles and cognitive abilities. Cognitive ability is the capacity for success in certain types of thought; for example, mathematical concepts. Abilities are how we deal with content and how well we think. Cognitive styles are biases or tendencies to think in a certain way, such as thinking sequentially or holistically. Styles are how we prefer to think and learn. Of course, more ability is better than less, but one style is not better than another.
COGNITIVE STYLES (a sample list)
Three characteristics of cognitive styles: 1) stable within an individual during different situations and times; 2) consequential: has implications for future actions; 3) not an ability measure
There are people who have very good visual or auditory memories. However, Willingham explains why teaching different modalities to learners with a prefered style is ineffective. He gives the example of a visual learner and an auditory learner learning vocabulary words. In theory, showing the words with pictures to the visual learner while playing a tape with words for the auditory learner should be most helpful. Yet studies show this is not the case. Why not? Because it is not the auditory or visual information that is being tested--it is the meaning of the words. Generally in schools, students need to remember what things mean, not what they look or sound like. So, if this theory is wrong, why do 90% of teachers (and students) believe it to be true? Willingham chalks it up to several reasons, the first being accepted wisdom: it must be right because everyone believes it. Another reason is because a similar fact is true: kids are different in their visual and auditory memories. Learners may have good visual and auditory memories, but this not being a “visual or auditory learner.” Lastly, the psychological phenomenon known as confirmation bias comes into play here. Once people believe something to be true, then all future ambiguous events are seen through that viewpoint. For example, people believe crazier things happen during a full moon, and, in fact, crime and births increase during a full moon. However, when there’s an uptick in crime and babies on non-full moon nights, no one bats an eyelash. In conclusion, Willingham says that all cognitive styles, not just visual-auditory-kinesthetic, suffer from the same issues; at best, the evidence is mixed.
ABILITIES AND MULTIPLE INTELLIGENCES
Over the years, studies and experiments have shown that some kids are good at math, some are musical, others athletic, but not necessarily the same kids. This must indicate there are different mental processes at work here. In the the mid-1980s, Howard Gardner, a Harvard professor, proposed his theory of multiple intelligences: linguistic, logical-mathematical, bodily-kinesthetic, interpersonal, intrapersonal, musical, naturalist, and spatial. At the time, many psychologists felt contention to Gardner’s theory. However, educators were (and are) interested in the three claims of his theory: 1) they are intelligences, not abilities or talents; 2) all eight intelligences should be taught in school; 3) many or all of these intelligences should be used to teach, matching the different intelligences of students. Gardner made the first claim, while the other two were made by others, although Gardner disagrees with them. Gardner argues that some abilities, in particular logical-mathematical and linguistic, have greater status in education than say, musical ability. He questioned why one was called “intelligence” while the other was a “talent.” Claim 2 is made on the basis of equity and fairness, that all intelligences should be acknowledged and celebrated. However, Gardner feels that curricular decisions should be made by the values of community, and his theory should only be a guide. Cognitive scientists believe Gardner has simply relabelled talents as intelligences, rather than “discovering” musical or spatial intelligence. The third claim is to use multiple intelligence modalities to introduce new knowledge. For example, when learning how to use commas, students could write a song about commas (musical), search the woods for things that look like commas (naturalist), and create sentences with their bodies (bodily-kinesthetic). So, in theory, students would come to an understanding of commas easier if taught with a particular intelligence in mind. Gardner wholeheartedly disagrees with this notion. The different abilities are not interchangeable; mathematical concepts need to be learned mathematically, and skill in music will not help. Writing a poem about your bat swing will not make you a better batter. These abilities are separate enough that one strong skill can’t compensate for a weaker one.
CONTENT VS. STUDENTS
Since catering to cognitive styles have been shown to essentially ineffective, think in terms of curricular content. For example, in socials, a country’s geography should be seen, an anthem should be heard, and a traditional meal should be made and eaten.
CHANGE PROMOTES ATTENTION
Variety is the spice of life and the surge in energy during lessons. Switch between talking and listening to something visual; go from deductive thinking to free associative thinking; quick brainstorming could lead into thoughtful, reflective responses. Give all students practice in these different mental processes.
VALUE IN EVERY CHILD
Every child is unique and valuable, regardless of their intelligence. Trying to be equitable and egalitarian and have everyone possess “multiple intelligences” may be misleading. Also, determining who is “smart” depends on which intelligences you consider and at what level; is it top 10 percent or top 50 percent? In reality, there will be many students who are not especially gifted in any of the intelligences. Telling a child they are smart or have a skill in an area they don’t rarely works. In fact, telling a child they are smart actually backfires in reality.
Source: Willingham, Daniel T., Why Don’t Students Like School? (2009)
Willingham's cognitive principle is that factual knowledge must precede skill. The current mode of thinking nowadays is that only critical thinking is necessary and the actual content, information, or knowledge is merely interchangeable; after all, one can do an Internet search and find information on any topic in seconds. However, thinking processes are intertwined with knowledge, perhaps surprisingly.
READING COMPREHENSION REQUIRES BACKGROUND KNOWLEDGE
One study shows that even poor readers with high background knowledge of the reading understood the text better than good readers with low knowledge. Background information allows chunking (grouping of information), which allows your working memory to have more space to connect ideas and thoughts, leading to better comprehension.
Four ways background knowledge aids comprehension:
The "fourth-grade slump" is a phenomenon that hits underprivileged homes. Up to grade three, most students are good decoders, but reading comprehension becomes increasing important in grade 4 and up. Because comprehension is dependent on background knowledge, privileged kids come to school with more knowledge about the world and a larger vocabulary.
COGNITIVE SKILLS REQUIRES BACKGROUND KNOWLEDGE
Thinking critically or logically often comes from what you know. To solve a problem, you first check your long-term memory to see if your solution already lies there. Think of the world's best chess players; it's not necessarily their reasoning or planning skills but rather their recall of board positions. They may have up to 50000 board game positions in their long-term memory! This goes for chefs, who can look at a kitchen pantry and whip up a delicious meal quickly, while regular folks may end up scratching their heads and end up making macaroni and cheese. In class, someone who has memorized the times tables will be able to solve a problem requiring that information faster than someone who has to figure it out by counting. This saves a lot of room in working memory to solve the rest of the problem.
Einstein said, "Imagination is more important than knowledge." Willingham hopes you realize that actually knowledge is necessary for imagination that leads to problem solving, decision making, and creativity.
Source: Why Don't Students Like School?, Willingham, Daniel T., 2009.
Language-processing problems constitute the largest proportion of learning disabilities. These include hearing sounds and words, understanding meaning, remembering verbal content, and communicating clearly.
The following are just a few examples:
Speech and Language Comprehension
Students tend to process information more slowly than usual. Sometimes teachers move on when they feel a response is not forthcoming. Often these students may be considered unmotivated or lazy. Also, language-processing disabilities affect their thinking. Language (words) are necessary to name people, places and things. Social development is influenced with this disability as they struggle with speaking, so they become fearful, shy and withdrawn; some deal in the opposite manner and become bullies. Others prefer to spend time with younger kids, using simpler language.
Word usage and comprehension is found in the left cerebral cortex. Inefficient neural "networking" can also result in processing issues. Some areas are underworked while others are handling too much. There also appears to be a genetic or heredity link with family members, as well.
The best way to intervene is with early recognition and appropriate and intense instruction. Special education is essential. They can use audio materials or simplified texts to handle the information overload. Extra time is often needed for tests and assignments. Test questions may need to be read to them. Teachers may need to speak slower and with simpler one-step instructions. Technology can assist in many ways with reading texts aloud, dictation, voice-to-text recognition, along with spelling and grammar checking.
Despite reading, writing or verbal problems, students with this language-processing disability can end up achieving amazing things, especially in professions that do not rely on advanced language skills: medical technology, architecture, finance, photography, engineering, mechanics, TV production, fine arts and computer programming, to name a few. The key thing is to maintain understanding and encouragement in order to maintain their self-confidence and enthusiasm for learning.
Source: Learning Disabilities: A to Z; Corinne Smith and Lisa Strick, 2010
Matthew Crawford, a writer and research fellow at the Institute for Advanced Studies in Culture at the University of Virginia feels that today's education needs to return to its practical, hands-on roots, not its current state of representation in the virtual world. Crawford worries that since attention is a stimulus-driven, goal-directed and a limited resource, children, in particular, are subjected to and bombarded with continuous stimulus-driven attention of ads and manipulative messages. Social media is designed solely to have users engaged constantly and returning to their platforms. "Distractibility," says Crawford, "might be regarded as the mental equivalent of obesity." He worries that all this clutter of digital noise may dampen imagination, as well as the clear sense of self. Who are we as people or individuals, when so much of our self-image is now being shaped by marketers, friends and followers? Crawford also feels the philosophical movement of individualization and autonomy has gone too far. "I think, therefore I am, " stated Descartes, in the Age of Enlightenment. However, so much of reality, argues Crawford, now resides in our minds as representations, and the physical reality of the world has lost its meaning and value. Crawford wants genuine individuality and agency, which comes from skilled practice and experience affecting objects in the real world.
Professor Guy Claxton of Winchester University feels that attentional habits are a part of good learning habits, such as collaboration or listening. He believes this disposition of attention can be shaped over time, but not explicitly in the form of a workshop or lecture. He suggests approaching it from the point of losing mindfulness. The goal is when students are distracted, how quickly can they return to attention? Some classes work on a scale of 1 to 10, to see how distracted they have been in a week. Other classes will use a show of five fingers: 1 = not distracted; 2= vaguely distracted; 3= minor distraction; 4 = major distraction; 5 = I was the distraction! The goal is to get kids interested in their own distractibility and to gain greater control and assume responsibility. Another method is for students to keep track of their own distractions, marking a tick every time they are off task on a line scale of completely distracted and completely focused.
Source: Attention: Beyond Mindfulness, Gay Watson, 2017
Most of us have heard the analogy that our brain is a thinking machine. But, according to Willingham, our brains are not really designed for thinking, because it is slow and unreliable, and requires much effort. In fact, your brain uses most of its processing power to see things and to move around physically. Nonetheless, the good news is that people are curious, as long as the problem is not too easy or too difficult--the Goldilocks special.
So how do we manage to get through life if we don't think well? Essentially, we rely on our memories. Once we've figured out how to do something once (or twice), then we rely on our memory system to recall that piece of information, so that our brains don't have to work hard and figure it out again. For example, when driving a car, you don't have to relearn how to press the accelerator, apply the right amount of pressure on the brakes for stopping, shifting gears, checking for cars on the side, and much more. All those discrete steps are memorized and now recalled perfectly and efficiently. That explains why travelling to a country with a different language and culture is so tiring: you have to relearn all of the simple rules and customs of that particular place.
How does thinking work in basic terms? There are four factors: information from the environment, facts in long-term memory, procedures in long-term memory, and the amount of space in working memory. If any of these is lacking, then thinking will likely fail.
Therefore, one of the reasons why students don't like school is because the tasks and problems they face are either too easy or too difficult, or the thinking required to solve them breaks down in one of the four key areas. So what can be done to alleviate this conundrum?
Have solvable problems: Make sure students have a variety of cognitive work during the day that pose moderate challenge. Are there cognitive breaks? Consider their suitability.
Respect Students' Cognitive Limits: Do students have the necessary background information to solve the mental challenge? If not, prepare them accordingly. Also, don't overload their working memory. Slow the pace and use memory aids, such as writing on the board.
Clarify the Problems: It's difficult for any problem to be "relevant" to an entire group of diverse learners with unique interests. When planning a lesson, start with the information you want students to learn. Then prepare key questions at the right level of difficulty to engage your students and respect their cognitive limitations.
When to Puzzle Students: Do we start with a thought-provoking question, or conduct an interesting demonstration or present a fact? Which is more effective? Sometimes a startling experiment can capture students' attention, but without the proper background information, the temporary thrill will be akin to a magic trick.
Student variance and differentiation: Because students come to class with varying levels of preparedness, understanding, motivation, it is best to assign work that best suits their current level of readiness.
Change the Pace: If you feel you're losing the attention or interest of the learners, then switch gears, change topics, start a new activity or find out what they are having difficulty with, or if it is too easy.
Keep a Diary: As a teacher to improve professionally, it's important to keep track to successes and failures, in order to build up a library of best practices. What worked best for the students? What failed miserably?
Source: Why don't Students like school? Daniel T. Willingham, 2009
How did such creative and novel ideas such as the iPhone, the Apollo 13 rescue mission, or Picasso's Les Demoiselles come about? Did they merely appear out of thin air, or is there a more logical and replicable explanation? According to Brandt and Eagleman in The Runaway Species, there are three categories (cognitive operations or strategies) that all innovations can fall into: bending, breaking and blending.
Bending takes the original item and then changes one or more aspects of it, such as shape, size, colour, or viewpoint. One basic example is Monet's many views of Rouen Cathedral in the 1890s. A more practical example of bending would be the invention of the polarized windshield. In order to not be blinded by headlights in the past, the idea of glare-resistant windshields were suggested. The problem: a calcite crystal was six inches thick! However, Edwin Land used "orthogonal thinking" and made sheets of glass with thousands of tiny embedded crystals. Miniaturization solved the dilemma. Other examples of bending include umbrellas, cars, jazz, language, architecture and television signals.
Breaking involves taking something whole, taking it apart, and reassembling something new out of the fragments. The first cell phone systems followed in the footsteps of radio and TV broadcasting, a single cell tower transmitting signals everywhere, but only a few people could make calls at one time. Bell Lab engineers then divided the single coverage area into small cells, each with its own tower, thus solving the problem of too many users. Another example of breaking comes from motion pictures. In the early cinemas, scenes in movies were told in real-time. Soon, filmmakers began to cut the beginning and endings of scenes. Then in Citizen Kane, we see time moving rapidly in years, and soon montages of long scenes can be done in seconds. Other examples of breaking include the following: computers, carbon copy, LCD screens, acronyms, and MP3 files.
Blending combines two or more sources in novel fashion. A classic example would be the Egyptian Sphinx, part human, part lion. With advances in genetics, professor Randy Lewis was able to splice the DNA of a spider to a goat to create Freckles the spider-goat; she's a goat but her "superpower" is she can secrete spider silk in her milk. This spider silk will be used to weave ultra-light bulletproof vests in the future. We also have fish and pigs that glow thanks to the jellyfish gene. Creoles are the blending of languages. Children in a remote village in Australia used their parents' baby talk (which combined 3 languages, Warlpiri, Kriol, and English) and then created their own syntax, known as Light Warlpiri. Blending is nearly limitless: a kingfisher + train = Japanese bullet train; soccer + volleyball = futevolei in Brazil; copper + tin = alloy bronze.
Source: The Runaway Species, Anthony Brandt & David Eagleman, 2017
Attention-Deficit/Hyperactivity Disorder (ADHD)
Children with ADHD make up 3 to 5 percent of the school population. There are three forms: predominately inattentive; predominantly hyperactive/impulsive; and the combined type. Inattention includes the inability to sustain attention in tasks or play, great difficulty getting organized or does not follow through on instructions or finish schoolwork or chores. Hyperactivity includes fidgeting and squirming, leaving one's seat in the classroom or talking excessively. Impulsivity includes blurting out answers, trouble taking turns or intruding on others.
Boys and girls are both equally likely to be affected, but girls tend to be less aggressive and disruptive, so are often less diagnosed than boys. Students struggle with attention and disruptive behavior, which leads to academic struggles. When they can't manage their emotions with peers, their social lives begin to suffer, often being left out of playdates and parties. Even at home, children with ADHD can strain the relationship between parent and child, because of their stubbornness or embarrassing behavior. If left unaddressed, these children often end up becoming teens that indulge in more thrill-seeking behavior, more drug and alcohol abuse, and are at risk for mental health issues, such as anxiety and depression. Ultimately, unchecked, their chances for completing school, pursuing higher education and finding satisfying jobs are reduced.
First, determine whether it is indeed ADHD and other factors, such as allergies, hearing or vision problems, stress, diet, inappropriate placement, and maturity level. Generally, students with ADHD require more time and guidance to master information. They may need reinforcing lessons and methods to monitor their own attention, to bring them back on task. Assistive technology can also be a benefit. Parent support and education, as well as family counselling, may be helpful. By their teens, many children show improvement though they remain energetic. Medication may be necessary for some with an extreme condition of ADHD.
Source: Learning Disabilities: A to Z; Corinne Smith and Lisa Strick, 2010
This chapter definitely gets into some practical classroom suggestions related to mindfulness: mindful memory, field of vision, mindful seeing, and taking a pause.
But it starts off with the first thing of the day: attendance, and its value. The authors stress the importance of not only knowing who’s present, but acknowledging their value and their state of mind. It’s important to establish that connection, first thing. I realize that with online attendance, I’m not making that eye-to-eye contact that I normally had in the past (partly because of the way my computer is facing), so I plan on going back to a paper and pen method (along with the online, as well.)
The root of mindfulness is attention, focussing on what’s important. Mindfulness also reduces mind-wandering. Being able to prioritize between relevant and irrelevant information is critical for mindfulness. Learning mindfulness is a useful skill for both teachers and students alike.
Mindful Memory is a technique that stresses process over product. In other words, although the goal of memorizing is important, the process and experience of paying attention while memorizing is equally as important. One activity is the Memory Game. Phase 1: Put a dozen objects on a table and cover them. Then students look for a minute, cover them up, and then recall the items on a piece of paper. Phase 2: Place new objects, but use Take 1 to calm their minds to focus attention prior to seeing the objects. Variation #2: While teaching a regular class, suddenly pull then over to a table and do the Memory Game again. Variation #3: Play music or read a story aloud while the students memorize new objects. This is testing the effects of distraction and multitasking on concentration. Finally, reflect and find out how students felt after each of these three variations of the memory game.
The Field of Vision activity shows where you look affects what you see--a kind of “mental shortsightedness.” What you expect to see affects what you notice. This activity entails going outside, and covering the ground with many little twigs. Pick a short twig about 7cm, and say you will place in amongst the twigs and the students need to find it. They face away and then you pretend to place it on the ground but actually put in behind your ear but with a little bit showing. See how many students actually see the twig in your ear, the “last” place they expect. Finally, ask the students what the point of the activity was.
Mindful Seeing involves simply seeing and looking--and nothing else: no judgements, no opinions, no assumptions. Then apply mindfulness to look for something--”thinking, and deciding or attributing meaning.” The difference is looking at something vs. looking for something. This exercise leads into the notion of first impressions, and how mindful seeing can help you from getting caught up in making quick judgements, which later might be found to be false.
Taking a beat or a Pause aids in mindfulness, even if it’s just one breath in and out. This momentary pause can help you determine four things: 1) the simple facts with no emotion or interpretation; 2) how you feel about it; 3) what you think about it; 4) what might be the best to do. Start with neutral objects, before moving onto objects with more emotion attached. This will give you the best chance to come up with a good plan or idea in any difficult situation or circumstance.
Source: Mindful Teaching and Teaching Mindfulness, Deborah Schoeberlein David, 2009
Daniel H. Lee
This blog will be dedicated to sharing in three areas: happenings in my classroom and school; analysis and distillation of other educators' wealth of knowledge in various texts; insights from other disciplines and areas of expertise that relate and connect with educational practices.