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 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
Encouraging Creativity in Kids
Vancouver Writers Festival
Studio 1398, Granville Island
This workshop, led by Marie-Louise Gay, was informative, inspiring and entertaining. The following will be a collection of ideas shared by Ms. Gay, distilled somewhat through my perspective into four broad categories.
Richard Wagamese: Storytelling and the Face of Reconciliation
Project-Based Learning with an Indigenous Perspective
Essentially, this is an organization that deals with project-based learning, in particular, the Young Entrepreneur's Show. Of course, it's been a fixture at Surrey Centre for some time now with the grade 5s, and I'm looking forward to take part in it next year. The presentation included consideration for Aboriginal-related projects.
Paul Pantaleo: Accelerating the Progress of Readers Experiencing Difficulties
THE SIX Ts OF LEARNING:
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.