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When dealing with a sequence of length N, the elements of which we wish to distinguish by subscript, the next vexing question is what subscript value to assign to its starting element. Adhering to convention a) yields, when starting with subscript 1, the subscript range 1 ≤ i < N 1; starting with 0, however, gives the…
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…
For example, if the problem was to fi nd the answer to 8 × 6, students may suggest three options: memorizing the multiplication table for 6, knowing that 8 × 5 = 40 and adding another 8 to equal 48, or adding a column of six 8s. Allowing students to personally choose among approaches all confi rmed as correct and to…
Periodically remind students that their mental effort is relevant to pleasure in the near future. The younger the children, the less tolerant their brains are to activities that are not pleasurable now or expected to be so in the very near future. Fortunately, the dopamine-reward network releases motivating dopamine…
I recognize Map Readers because they often like to work independently, but they are most comfortable when they have specifi c instructions or procedures to follow. Th ey often take more time and work deliberately, showing all their steps on homework or taking detailed notes in class or during group work. Explorers are…
It’s a common misconception that someone who’s good at math is someone who can compute quickly and accurately. But mathematics is a broad discipline, and there are many ways to be smart in math. Some students are good at seeing relationships among numbers, quantities, or objects. Others may be creative problem…
The bombshell is that the Flynn Effect is almost certainly environmental. Natural selection has a speed limit measured in generations, but the Flynn Effect is measurable on the scale of decades and years. Flynn was also able to rule out increases in nutrition, overall health, and outbreeding (marrying outside one’s…
“Mathematics is not just a sequence of computations to be carried out by rote until your patience or stamina runs out—although it might seem that way from what you’ve been taught in courses called mathematics. Those integrals are to mathematics as weight training and calisthenics are to soccer. If you want to play…
no matter what the odds are at one point in spacetime probability materialize
The way it was described to me when I was in high school was in terms of 'levels'. Sometimes, in your mathematics career, you find that your slow progress, and careful accumulation of tools and ideas, has suddenly allowed you to do a bunch of new things that you couldn't possibly do before. Even though you were…
I am miles away from Eric or Linus, but the "curse of the gifted" is very real. Thankfully I wasn't smart or gifted enough that I could ride it for long, but when it comes to math and problem-solving I rode it well into my high school years. I never learned to do algebra "by the book," because I didn't need to. Or…
I think that there is a moral to this story, namely that it is more important to have beauty in one's equations than to have them fit experiment. If Schroedinger had been more confident of his work, he could have published it some months earlier, and he could have published a more accurate equation .... It seems that…
There are some subtle facts about scale analysis that make it more powerful than simply comparing orders of magnitude. A most remarkable example is that scale analysis can be applied, through a systematic use of dimensions, even when the precise equations governing the dynamics of a system are not known. The great…
Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful…
Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution…
Occasionally he made a comment similar to Archimedes' "Eureka!" about a discovery, as he did when he wrong T T T=N. This is part of number theory, the field that Gauss called the Queen of Mathematics. A triangular number T is a number that you can get by adding up a string of consecutive whole numbers beginning with…
The simplest definition I can give for Big-O notation is this: Big-O notation is a relative representation of the complexity of an algorithm. There are some important and deliberately chosen words in that sentence: relative: you can only compare apples to apples. You can't compare an algorithm to do arithmetic…
As the DM, the tools of your trade are dice — platonic solid-shaped or just about any other sort. The random numbers you generate by rblling dice determine the results based on the probabilities determined herein or those you have set forth on your own. In case you are not familiar with probability curves, there are…
Hieron asked Archimedes to discover, without damaging it, whether a certain crown or wreath was made of pure gold, or if the goldsmith had fraudulently alloyed it with some baser metal. While Archimedes was turning the problem over in his mind, he chanced to be in the bath house. There, as he was sitting in the bath…
Perhaps we see equations as simple because they are easily expressed in terms of mathematical notation already invented at an earlier stage of development of the science, and thus what appears to us as elegance of description really reflects the interconnectedness of Nature's laws at different levels.