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 are 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 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.
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
Long gone are the days where students sat in rows and learned the same thing at the same time, usually from the authority standing in the front of the classroom. Now, fast-forward in time, and Kallick and Zmuda describe the four key attributes to personalized learning--the clear contrast to learning of the past.
Voice: Students participate in the creation of the learning, because it’s really their learning. Most people do not like to be told what to do, at least not all the time. Instead of being passive passengers heading in one direction, they are often in the driver’s seat, determining their own journey and pathway and destination.
Co-Creation: Students work with the teacher to develop the entire learning plan, from start to finish: what do they want to learn?; how will it be assessed?; how will they learn it?
Social Construction: The notion that “no man is an island” (John Donne) aptly describes student learning in the classroom; it is an social affair and construction of knowledge, according to Vygotsky. Finally, the whole is greater than the sum of the parts (Aristotle), as collaboration and cooperation amongst fellow peers can lead to much greater triumphs and accomplishments.
Self-Discovery: Creating self-aware and self-directed learners is the ultimate goal for teachers. If students can figure out their strengths and weaknesses, and determine how to improve and grow, then they will be set for life.
Differences Between Individualization, Differentiation and Personalized Learning
Students are assigned the learning tasks, and they use technology to accomplish those tasks. Khan Academy would be one such example. In blended learning environments, there may be some co-creation and social construction, but learners still have little say in the work they do.
Today’s classroom houses learners varying in skills, readiness and interest. Students can select topics (content), how to learn (process) and create the final form of learning (product). However, the teacher is still leading the design and management of the learning experience.
Kallick and Costa encourage the use of the 16 Habits of Mind, in conjunction with personalized learning, in order to fully understand their learning, and engage in higher level thinking and performing.
16 Habits of Mind:
What I notice about these habits of mind are the similarities to the core competencies of the BC curriculum: communication, critical and creative thinking, positive personal and cultural identity, personal awareness and responsibility, and social responsibility.
Source: Students at the Center, Bena Kallick & Allison Zmuda, 2017
Norma Rose Point is a beautifully designed school on the UBC campus, but it’s much more than a pretty face. It exudes innovation, collaboration, engagement and powerful learning. There are three distinctive elements that allows this unique public school to function in such a special way. All three are essential and work seamlessly together. They are the physical learning space, the overarching school philosophy, and teacher and student relationships.
PHYSICAL LEARNING SPACE
The physical space of different “communities” with rooms like the Da Vinci room and outdoor garden space, kitchen space, open spaces, hallways and much more. All spaces are communal in nature. What was interesting was the notion of instructional space at the school; everything, even little crooks and crannies, can be instructional or learning space. Rooms are as flexible as the learning, with folding tables that allow rooms to turn from a science room into a physical exercise space in a matter of minutes.
Most of the philosophy comes from the Innovative Learning Environments Project by the OECD (Centre of Educational Research and Innovation). Learning nowadays is considered socio-constructivist, meaning that in any given context, learning is actively constructed and socially negotiated. The ultimate goal is adaptive expertise--being able to use knowledge and skills in new situations. Adaptive expertise is developed through guided (teacher-led), action (student-led) and experimental (play) learning. This leads into lifelong learning. Learning is also determined by emotion and motivation, so students need to feel positive and confident yet realistic in their learning goals.
The 7 principles of learning are the following:
The building blocks for innovative learning environments are cooperative learning, service learning, technology, home-school partnerships, formative assessment and inquiry-based approaches (project, problem, design).
The school’s motto is “Learners at the Centre” and I think that pretty much sums up what happens in the school. So, anything that doesn’t contribute to learning (books or materials that sit in cardboard boxes) must be taken home. Even the shelves are considered “sacred,” so they must be essential for the teacher. The mission statement goes on to add: To meet learner needs we differentiate instruction, focus on Learner strengths, infuse technology in meaningful ways and collaborate with each other to be the best we can be.
With such a clear and powerful mission statement, and with strong buy-in by teachers, there seems to be a sense of pride, ownership, and joy in striving for success of all students.
TEACHER AND STUDENT RELATIONSHIPS
The professional office space allows for constant collaboration, beyond actual set times, and a weekly timetable helps organize that collaboration into overall teaching goals and plans. Teachers spend only about a third of their time in one learning space, and will often work with a variety of students, depending on the learning that’s happening. Students are often ability grouped, so individualized student learning is targeted.
Inquiry learning is also a key concept at this school. Inquiry learning is student-dependent, and each inquiry is different for each student. This is closely connected with Genius Hour and passion-based learning, as well. I liked how they thought of engaging ideas, such as “Ted Talks,” “Kickstarter,” “Star Wars University,” and “CSI.”
Learning is visible in more ways than one. In the most basic sense, the openness of the learning spaces--classrooms with lots of clear glass and moving doors--allows clear lines of sight of all students. Beyond that, of course, is the notion of taking what students are thinking in their brains and then showing it in a tangible way; for example, electronic and physical portfolios. Also, learning celebrations held every few months are the culmination of that learning. They sound like student-led interviews, but on a much grander and festive scale.
Another interesting relationship was even between students and classroom supervisors. These supervisors are considered staff and treated with respect. Even more so, they help give additional collaborative time to teachers, by leading learning activities like the Daily 5 in classrooms.
Finally, there was emphasis on students being able to self-regulate, using ideas of restorative justice and zones of regulation. If students are struggling with their emotions or conflict with others, then learning will suffer, so they need the tools and skills to be able to bring themselves back to the proper state of well-being.
It occurs to me that the layout and design and sharing of materials and tools are reminiscent of kindergarten and early primary. Individuality and personal space, which is keenly represented by a student's desk, does not exist. Instead, tables, floors and the outdoors now represents where learning happens--which is everywhere. What's equally interesting in my mind is that if you look at the most creative, innovative, in particular high-tech companies, you will find a similar layout and design: open space concept, communal living and working, bright lighting, and all-inclusive campus look and feel. In other words, it feels like home, not a place of work (even though you are working hard, in most cases!) Of course, individuality and a sense of uniqueness clearly exists, with their portfolio systems, differentiated learning, ability groups, and more.
Technology is a big part of the school, with a ratio of about 3:1 iPads currently, and probably lower if you include devices from home. They have short-throw projectors, a media room with some desktops and a green screen room. The Learning Resource Centre has a 3-D printer and some computers. What's most interesting is that technology was never really discussed in the two-hour tour session with principal Rosa Fazio. Maybe because it had become second nature or because it was naturally integrated into the entire learning system. It was simply another tool used for research, presentations, expression and creativity, but it never superseded other types of tools. I think the belief that "learners are at the centre" is the key, and while they "infuse technology in meaningful ways" according to their mission statement, their mindset is that learning comes from all areas of life, not simply technology.
This was an interesting session, although maybe a bit tough early in the morning to solve a potential end-of-the-Earth scenario. It's basically an escape room for the classroom. We did it as a group of about 30 teachers, and it was fun to see how people acted. There were definitely a few serious people/leaders who were working hard. Then there were some of us, just catching up with old colleagues. All in all, it was a good time and we saved the world...with 20 minutes to spare.
The educational benefits are quite clear:
The website has everything set up nicely as well with free resources for the escape rooms.
Creative Drama in the Classroom
This session started with a bang...but ended with a fizzle. I think with her drama/acting/educational background, we expected a lot. Still there were some good ideas to be had:
Artifacts Inspire Inquiry
This session was truly inspiring, engaging from start to finish, and the presenter was extremely professional, prepared and personable.
The start is key. She asked us to talk about our childhood "artifact," something from our past that was memorable, important or endearing. The initial brief moment of anxiety shifted to something quite calming: sharing something personally relevant helped create an invisible bond within that group of strangers.
The key point that was stressed was that an artifact doesn't need to be something ancient; after all, your students probably haven't seen some of the things you grew up with, given the acceleration of technology advancements combined with the nature of our disposable and consumable society.
The main activity involved having poster paper, a group of people, and a photo of an artifact. We had to brainstorm as many ideas about what the item was. It was a fruitful discussion, with plenty of varying ideas. Then we received the actual physical object, and our preconceptions or ideas from that photo changed quite dramatically. So we came up with even more refined ideas about what our item was. We actually guessed correctly: a sewing kit!
Lisa Brahms and Peter Wardrip, University of Pittsburgh researchers, have recognized learning practices in making.
1) Inquire: openness and curiosity
2) Tinker: "purposeful play, risk-taking, testing" using a variety of tools, materials and processes
3) Seek and Share Resources: sharing knowledge and expertise
4) Hack and Repurpose: reuse and combine components in new ways
5) Express Intent: find one's passion and identity
6) Develop Fluency: gain confidence in one's ability through learning and practice
7) Simplify and Complexify: gain understanding of new ways to create meaningful things
Source: Free To Make, Dale Dougherty, 2016
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.