Mathematical Mindset Quotes

A lot of scientific evidence suggests that the difference between those who succeed and those who don't is not the brains they were born with, but their approach to life, the messages they receive about their potential, and the opportunities they have to learn.

Every time a student makes a mistake in math, they grow a synapse.” There

Many parents have asked me: What is the point of my child explaining their work if they can get the answer right? My answer is always the same: Explaining your work is what, in mathematics, we call reasoning, and reasoning is central to the discipline of mathematics.

It is really this “mathematical mindset” that seems to be most useful to those who are not trained to think as mathematicians.

Always give help when needed, always ask for help when you need it

STOP APPLYING PATCHES. RECODE YOURSELF NOW. 24 Dec National Mathematics Day

five suggestions that can work to open mathematics tasks and increase their potential for learning: Open up the task so that there are multiple methods, pathways, and representations. Include inquiry opportunities. Ask the problem before teaching the method. Add a visual component and ask students how they see the mathematics. Extend the task to make it lower floor and higher ceiling. Ask students to convince and reason; be skeptical.

Performance cannot be based on one assessment. You cannot determine the slope of a line given only one point, as there is no line to begin with. A single point in time does not show trends, improvement, lack of effort, or mathematical ability.… Sincerely,

It turns out that even believing you are smart—one of the fixed mindset messages—is damaging, as students with this fixed mindset are less willing to try more challenging work or subjects because they are afraid of slipping up and no longer being seen as smart.

Another misconception about mathematics that is pervasive and damaging—and wrong—is the idea that people who can do math are the smartest or cleverest people. This makes math failure particularly crushing for students, as they interpret it as meaning that they are not smart.

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.

Darwin didn’t consider himself a quick or highly analytical thinker. His memory was poor, and he couldn’t follow long mathematical arguments. Nevertheless, Darwin felt that he made up for those shortcomings with a crucial strength: his urge to figure out how reality worked. Ever since he could remember, he had been driven to make sense of the world around him. He followed what he called a “golden rule” to fight against motivated reasoning: . . . whenever a published fact, a new observation or thought came across me, which was opposed to my general results, to make a memorandum of it without fail and at once; for I had found by experience that such facts and thoughts were far more apt to escape from the memory than favourable ones. Therefore, even though the peacock’s tail made him anxious, Darwin couldn’t stop puzzling over it. How could it possibly be consistent with natural selection? Within a few years, he had figured out the beginnings of a compelling answer.

The researchers found that when students were given problems to solve, and they did not know methods to solve them, but they were given opportunity to explore the problems, they became curious, and their brains were primed to learn new methods, so that when teachers taught the methods, students paid greater attention to them and were more motivated to learn them. The researchers published their results with the title “A Time for Telling,” and they argued that the question is not “Should we tell or explain methods?” but “When is the best time do this?

I'm still just as slow… At the end of the eleventh grade, I took the measure of the situation, and came to the conclusion that rapidity doesn't have a precise relation to intelligence. What is important is to deeply understand things and their relations to each other. This is where intelligence lies. The fact of being quick or slow isn't really relevant. (Schwartz, 2001)

Such results should prompt educators to abandon the traditional fixed ideas of the brain and learning that currently fill schools—ideas that children are smart or dumb, quick or slow. If brains can change in three weeks, imagine what can happen in a year of math class if students are given the right math materials and they receive positive messages about their potential and ability.

Money is mathematical figures on a computer screen. It's not the definition of your self-worth or the value of your success.

This chart contrasts predictive and prospective thinking: Predictive Thinking Prospective Thinking Mindset Forecasting, “We expect …” Preparing, “But what if …” Goal Reduce or even discard uncertainty, fight ambiguity Live with uncertainty, embrace ambiguity, plan for set of contingencies Level of uncertainty Average High Method Extrapolating from present and past Open, imaginative Approach Categorical, assumes continuity Global, systemic, anticipates disruptive events Information inputs Quantitative, objective, known Qualitative (whether quantifiable or not), subjective, known or unknown Relationships Static, stable structures Dynamic, evolving structures Technique Established quantitative models (economics, mathematics, data) Developing scenarios using qualitative approaches (often building on megatrends) Evaluation method Numbers Criteria

This chart contrasts predictive and prospective thinking: Predictive Thinking Prospective Thinking Mindset Forecasting, “We expect …” Preparing, “But what if …” Goal Reduce or even discard uncertainty, fight ambiguity Live with uncertainty, embrace ambiguity, plan for set of contingencies Level of uncertainty Average High Method Extrapolating from present and past Open, imaginative Approach Categorical, assumes continuity Global, systemic, anticipates disruptive events Information inputs Quantitative, objective, known Qualitative (whether quantifiable or not), subjective, known or unknown Relationships Static, stable structures Dynamic, evolving structures Technique Established quantitative models (economics, mathematics, data) Developing scenarios using qualitative approaches (often building on megatrends) Evaluation method Numbers Criteria Attitude toward the future Passive or reactive (the future will be) Proactive and creative (we create or shape the future) Way of thinking Generally deduction Greater use of induction

Mathematics is a very broad and multidimensional subject that requires reasoning, creativity, connection making, and interpretation of methods; it is a set of ideas that helps illuminate the world; and it is constantly changing.

Every time a student makes a mistake in math, they grow a synapse.

The powerful thinkers are those who make connections, think logically, and use space, data, and numbers creatively.

Teachers greatly influence how students perceive and approach struggle in the mathematics classroom. Even young students can learn to value struggle as an expected and natural part of learning, as demonstrated by the class motto of one first-grade math class: If you are not struggling, you are not learning. Teachers must accept that struggle is important to students' learning of mathematics, convey this message to students, and provide time for them to try to work through their uncertainties. Unfortunately, this may not be enough, since some students will still simply shut down in the face of frustration, proclaim, 'I don't know,' and give up. Dweck (2006) has shown that students with a fixed mindset--that is, those who believe that intelligence (especially math ability) is an innate trait--are more likely to give up when they encounter difficulties because they believe that learning mathematics should come naturally. By contrast, students with a growth mindset--that is, those who believe that intelligence can be developed through effort--are likely to persevere through a struggle because they see challenging work as an opportunity to learn and grow.

Darwin didn’t consider himself a quick or highly analytical thinker. His memory was poor, and he couldn’t follow long mathematical arguments. Nevertheless, Darwin felt that he made up for those shortcomings with a crucial strength: his urge to figure out how reality worked. Ever since he could remember, he had been driven to make sense of the world around him. He followed what he called a “golden rule” to fight against motivated reasoning: . . . whenever a published fact, a new observation or thought came across me, which was opposed to my general results, to make a memorandum of it without fail and at once; for I had found by experience that such facts and thoughts were far more apt to escape from the memory than favourable ones.

The discussion is designed to get students to engage with one another's sorts and deduce the attribute that defines each group.

Performance cannot be based on one assessment. You cannot determine the slope of a line given only one point, as there is no line to begin with. A single point in time does not show trends, improvement, lack of effort, or mathematical ability….

Imperfection is a part of any creative process and of life, yet for some reason we live in a culture that has a paralyzing fear of failure, which prevents action and hardens a rigid perfectionism. It's the single most disempowering state of mind you can have if you'd like to be more creative, inventive, or entrepreneurial.

Many of the mathematical models for how a trait will spread in a population have failed—they don’t tell you this. No, I don’t talk about miracles, whatever words you put them under. And the “design” is there, but it is by no means benevolent or intelligent, nor comprehensible. You see in the spider’s web a creature of rudimentary nervous system and little intelligence “design” something beautiful and complex, and this is key to understanding also all of nature. There is an inherent “intelligence” inside things, uncanny, silent and demonic. Its workings and aims are obscure to us. Our own intelligence is only a crude deviation of it, an approximation. There is an “intelligence” in all things, and inborn in our bodies before anything to do with the brain or the nervous system. And all “adaptations,” no matter how much natural or unnatural selection may have gone to spreading them within a population, occur not by random but by a spontaneous correspondence of some kind between the organism and the environment.

You cannot determine the slope of a line given only one point, as there is no line to begin with. A single point in time does not show trends, improvement, lack of effort, or mathematical ability….

You need to take positions big enough that they make you feel legitimately at risk if things go wrong. You need to be able to drive the car fast enough to win the Grand Prix… But not so fast you slam into the wall.

Ellis was intrigued by the possibility of security created by just one party to the secret and thought there must be a way to create a similar technology for transferring data. One summer's evening he went to sleep and, as he said later, "It was done in my head overnight." Being a good spy, he didn't write it down at home. He just hoped he would remember it. And he did. In July 1969 Ellis's report hit the desk of GCHQ's chief mathematician Shaun Wiley. Wiley's response gives an insight into the 'glass half empty' mindset of an intelligence chief. "Unfortunately," he said. "I can't see anything wrong with this.

The game is played by partners. Each child has a blank 100 grid. The first partner rolls two number dice. The numbers that come up are the numbers the child uses to make an array on their 100 grid. They can put the array anywhere on the grid, but the goal is to fill up the grid to get it as full as possible. After the player draws the array on their grid, she writes in the number sentence that describes the grid. The game ends when both players have rolled the dice and cannot put any more arrays on the grid

In a TED talk watched by over a million people, Wolfram (2010) proposes that working on mathematics has four stages: Posing a question Going from the real world to a mathematical model Performing a calculation 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.

Numerous research studies (Silver, 1994) have shown that when students are given opportunities to pose mathematics problems, to consider a situation and think of a mathematics question to ask of it—which is the essence of real mathematics—they become more deeply engaged and perform at higher levels.

You cannot determine the slope of a line given only one point, as there is no line to begin with. A single point in time does not show trends, improvement, lack of effort, or mathematical ability. . . .

long time, step by step, to work through the same process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics. (Thurston, 1990)

The brain researchers concluded that automaticity should be reached through understanding of numerical relations, achieved through thinking about number strategies (Delazer et al., 2005).

the researchers found that the students who memorized more easily were not higher achieving; they did not have what the researchers described as more “math ability,” nor did they have higher IQ scores (Supekar et al., 2013). The

In one study, researchers at the National Institute for Mental Health gave people a 10‐minute exercise to work on each day for three weeks. The researchers compared the brains of those receiving the training with those who did not. The results showed that the people who worked on an exercise for a few minutes each day experienced structural brain changes. The participants’ brains “rewired” and grew in response to a 10‐minute mental task performed daily over 15 weekdays (Karni et al., 1998).

Trips like this forced me to confront the terror that snuck in when I couldn’t sleep. If I was a cannonball, I had already landed, but was calculating the mathematical probabilities after I had already missed. When I envisioned my paths, I had to face my truth. At the time, I wasn’t prepared to allow the rough winds of therapy to repair my broken mindset. I had come to terms that no one could save me. Trips like this were how I sorted things out.