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Thursday, January 28, 2016

Mathematical Mindsets: Myths and Facts

Jo Boaler's book will have a great impact on my teaching in the days to come. As I read the first half of the book today, I was struck by idea after idea and fact after fact about how to improve my work with regard to teaching math well.

As I go forward I first want to dispel the myths, and teach the facts about learning math and learning in general. Below I wrote down a number of myths and facts highlighted in her book. The words in quotes, unless noted to another individual, are Boaler's words. This list is a beginning list to help me teach math better. In no way is it a complete look at what Boaler is saying in her book. You'll have to read Mathematical Mindsets: Unleashing Students' Potential Through Creative Math, Inspiring Messages and Innovative Teaching to discover that.

Myth: Only some can learn math.
Fact: Everyone can learn math  You have to believe in yourself.

Myth: Learning is fixed.
Fact: A growth mindset is necessary for optimal learning.

Myth: Repetition of same practice sets will build skill.
Fact: Too much of the same practice does not build skill, but worthy problem solving activities, projects, discussions, games, and puzzles do build skill as well as engagement.

Myth: Math is mostly about procedures, calculation, and rules.
Fact: Math is reasoning, creativity, connection making, interpretation, and different pathways.

Myth: You have to overhaul an entire system to improve math education.
Fact: "Relatively small changes in teaching and parenting can change students' mathematical pathways."

Myth: Telling students that they are smart is the best way to encourage deeper math learning.
Fact: Rather than statements such as "you are smart," parents and teachers need to encourage the action of learning math: tenacity, creativity, collaboration, making, and thinking.

Myth: Mistakes demonstrate problems in students' learning.
Fact: "Every time a student makes a mistake in math, they grow a synapse." - Dweck. "Mistakes cause your brain to spark and grow." "Imperfection, in fact, is great for creativity and entrepreneurship. When mistakes are encouraged, "incredible things happen."

Myth: Avoid confusion
Fact: Disequilibrium, as Piaget claimed, leads to greater learning by creating problems to solve as well as prompting creative, challenging work.

Myth: Math is a performance subject.
Fact: Instead math is a subject where one can "appreciate the beauty of mathematics. . .ask deep questions. . .explore the rich set of connections that make up the subject. . . learn about the applicability of the subject."

Myth: The study of math results in right or wrong.
Fact: Math is a subject "full of uncertainty" and multiple paths. "When you learn a new idea in mathematics, it is helpful to reinforce that idea, and the best way to do this is by using it in different ways."

Myth: Math is mostly memorizing.
Fact: Math solves important problems. ". . ..those that learned through strategies achieved "superior performance over those who memorized." "The more we emphasize memorization to students, the less willing they become to think about numbers and their relations and to use and develop number sense."

Myth: Non-examples will confuse students.
Fact: It's often best to match an example with a non-example for deep learning.

Myth: Only humans employ math.
Fact: Animals display sophisticated math such as the way a dolphin communicates or how a spider spins its web.

Myth: It's useless to explain my thinking in math.
Fact: "Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics." "The powerful thinkers are those who make connections, think logically, and use space, data, and numbers creatively."

Myth: Math is a solitary subject.
Fact: "Math is a very social subject, a proof comes about when mathematicians can convince other mathematicians of logical connections." "A lot of mathematics is produced through collaborations. . "

Myth: You don't need to learn or use much math after high school or college.
Fact: "Almost all new jobs in today's technological world involve working with massive data sets, asking questions of the data and reasoning about pathways."

Myth: The best mathematicians are fast with calculation and problem solving.
Fact: ". . .mathematicians, whom we could think of as the most capable math people, are often slow with math."  Math "can be taught in a way that values depth and not speed, that enhances brain connections and that engages many more students."

Myth: Even though many parents "hated mathematics in school. . .they still argue for traditional teaching because they think it has to be that way."
Fact: School math is often very distant from what real math is--real math is "a creative, visual, connected, and living subject." "Mathematics is at the center of thinking about how to spend the day, how many events and jobs can fit into the day, what size of space can be used to fit equipment or turn a car around, how likely events are to happen, knowing how tweets are amplified and how many people they reach."

Myth: It's best to start teaching what we think of as traditional math as early as possible.
Fact: "In Finland, one of the highest-scoring countries in the world on PISA tests, students do not learn formal mathematics methods until they are seven."

Myth: Puzzles are just for fun.
Fact: "When students see math as a broad landscape of unexplored puzzles in which they can wander around, asking questions and thinking about relationships, they understand that their role is thinking, sense making, and growing. When students see mathematics as a set of ideas and relationships and their roles as one of thinking about the ideas, and making sense of them, they have a mathematical mindset." "Successful math users search for patterns and relationships and think about connections."

Myth: There is one good way to calculate and solve problems.
Fact: "The low achievers did not know less, they just did not use numbers flexibly--probably because they had been set on the wrong pathway, from an early age, of trying to memorize methods and number facts instead of interacting with numbers flexibly."

Myth: Math is a passive subject.
Fact: "A mathematical mindset reflects an active approach to mathematics knowledge, in which students see their role as understanding and sense making."

Myth: Challenging math that produces anxiety teaches students to work harder.
Fact: Anxiety is associated with the blocking of working memory.

Myth: Scripted math programs are best.
Fact: "When teachers are designers, creating and adapting tasks, they are the most powerful teachers they can be. Any teacher can do this; it does not require special training. It involved knowing about the qualities of positive math tasks and approaching tasks with the mindset to improve them."

Myth: There's no room for intuition in math learning and thinking.
Fact: "When students are asked to think intuitively, many good things happen. First, they stop thinking narrowly about single methods and consider mathematics more broadly. Second, they realize they have to  use their own minds--thinking, sense making, and reasoning. They stop thinking their task is just to repeat methods, and they realize their task is to think about the appropriateness of different methods. And third. . .their brains become primed to learn new methods."