World-Play as Self-Apprenticeship

...worldplay can be studied as a kind of self-apprenticeship in creative practice, rather than prodigious application in discipline or craft. The childhood inventor of imaginary lands often elaborates his or her world in multiple ways at once. He or she may write stories and compose music, draw maps and build models, design games, and possibly construct a secret language—all within the context of play. It is likely, therefore, that childhood woridplay confers benefits that differ substantia...

 The Double Multiplicative Nature of Fraction or Ratio Equ...

Most real-world numbers aren’t always so nice and neat, with wholenumber multiples. If, say, Plant A grew from 2 to 3 feet, and Plant B grew from 6 to 8 feet, then we would say that Plant A grew 1/2 of its original height, whereas Plant B only grew 1/3 of its original height. Such reasoning exemplifies multiplicative thinking and necessarily involves rational numbers. Consider a final example. If you ask a rising 6th grader to compare 13/15 and 14/ 16, chances are that the student will say...

 Mathematics is Hard Work, Not Genius

What I fight against most in some sense, [when talking to the public,] is the kind of message, for example as put out by the film Good Will Hunting, that there is something you're born with and either you have it or you don't. That's really not the experience of mathematicians. We all find it difficult, it's not that we're any different from someone who struggles with maths problems in third grade. It's really the same process. We're just prepared to handle that struggle on a much larger scal...

 Effects of Good Teachers on Student Outcomes

These findings would suggest that the difference in achievement gains between having a 25th percentile teacher (a not so effective teacher) and a 75th percentile teacher (an effective teacher) is over one-third of a standard deviation (0.35) in reading and almost half a standard deviation (0.48) in mathematics. Similarly, the difference in achievement gains between having a 50th percentile teacher (an average teacher) and a 90th percentile teacher (a very effective teacher) is about one-third...

 Intellectuals Must Engage the Public, Not Hide From It

A final point, something I've written about elsewhere (e.g., in a discussion in Z papers, and the last chapter of "Year 501"). There has been a striking change in the behavior of the intellectual class in recent years. The left intellectuals who 60 years ago would have been teaching in working class schools, writing books like "mathematics for the millions" (which made mathematics intelligible to millions of people), participating in and speaking for popular organizations, etc., are now large...

 Mathematics is one of humanity's great achievements

Mathematics is one of humanity’s great achievements. By enhancing the capabilities of the human mind, mathematics has facilitated the development of science, technology, engineering, business, and government. Mathematics is also an intellectual achievement of great sophistication and beauty that epitomizes the power of deductive reasoning. For people to participate fully in society, they must know basic mathematics. Citizens who cannot reason mathematically are cut off from whole realms of ...

 Five Strands of Mathematical Literacy

Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to capture what we think it means for anyone to learn mathematicssuccessfully. Mathematical proficiency, as we see it, has five strands: • conceptual understanding—comprehension of mathematical concepts, operations, and relations • procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and a...

 Number

Number is a rich, many-sided domain whose simplest forms are compre- hended by very young children and whose far reaches are still being explored by mathematicians. Proficiency with numbers and numerical operations is an important foundation for further education in mathematics and in fields that use mathematics. Because much of this report attends to the learning and teaching of number, it is important to emphasize that our perspective is considerably broader than just computation. First, nu...

 The Nintendo Effect

Technology can play an important role, but we haven’t had the time or wherewithal to explore it fully. I’m waiting for a breakthrough process, which I think may happen in mathematics. The people at Nintendo figured out billions of dollars ago that you pull kids in, you get them engaged, and that’s the model: engagement of intensive focused effort. The result is rapid incremental development of new skills and capabilities. These kids operate at a speed and accuracy level unheard of outsi...

 Humans Are Abstraction Masters

What distinguishes us from cavemen is the level of abstraction we can reach. Abstraction enabled humans to move from barter to money, and from gold coins to plastic cards. These days, what's left of "money" is often just an account record we read on a computer screen, and soon it could just be a line of code in a bitcoin ledger. Today, abstraction is all around us — and math is the language of abstraction. In the words of the great mathematician Henri Poincare, mathematics is valuable beca...