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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…
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…
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…
The familiar, hierarchical sequence of math instruction starts with counting, followed by addition and subtraction, then multiplication and division. The computational set expands to include bigger and bigger numbers, and at some point, fractions enter the picture, too. Then in early adolescence, students are…
Imagine you had to take an art class in which you were taught how to paint a fence or a wall, but you were never shown the paintings of the great masters, and you weren't even told that such paintings existed. Pretty soon you'd be asking, why study art? That's absurd, of course, but it's surprisingly close to the way…
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…
Geometry enlightens the intellect and sets one's mind right. All its proofs are very clear and orderly. It is hardly possible for errors to enter into geometrical reasoning, because it is well arranged and orderly. Thus, the mind that constantly applies itself to geometry is not likely to fall into error. In this…
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…
If we possessed a thorough knowledge of all the parts of the seed of any animal (e.g. man), we could from that alone, by reasons entirely mathematical and certain, deduce the whole conformation and figure of each of its members, and, conversely if we knew several peculiarities of this conformation, we would from those…
For the things of this world cannot be made known without a knowledge of mathematics. For this is an assured fact in regard to celestial things, siDce two important sciences of mathematics treat of them, namely theoretical astrology and practical astrology. The first. . . gives us definite information as to the number…
Rounded numbers are cultural attractors: They are easier to remember and provide better symbols for magnitudes. So we celebrate twentieth wedding anniversaries, hundredth issues of journals, the millionth copy sold of a record, and so on. This, in turn, creates a special cultural attractor for prices, just below…
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…
Now these numbers can be appreciated in many different ways. From the standpoint of calculation, they're as easy to understand as one plus one, which is two. Then one plus two is three, two plus three is five, three plus five is eight, and so on. Indeed, the person we call Fibonacci was actually named Leonardo of…
So why do we learn mathematics? Essentially, for three reasons: calculation, application, and last, and unfortunately least in terms of the time we give it, inspiration. Mathematics is the science of patterns, and we study it to learn how to think logically, critically and creatively, but too much of the mathematics…
So what was Fourier’s discovery, and why is it useful? Imagine playing a note on a piano. When you press the piano key, a hammer strikes a string that vibrates to and fro at a certain fixed rate (440 times a second for the A note). As the string vibrates, the air molecules around it bounce to and fro, creating a wave…
A good card magician knows many tricks that depend on luck—they don’t always work, or even often work. There are some effects—they can hardly be called tricks—that might work only once in a thousand times! Here is what you do: You start by telling the audience you are going to perform a trick, and without telling them…
Before there was an accurate seafaring clock, the sailor seeking his bearings had to be a trained mathematician. The accepted way to find longitude at sea was by precise observations of the moon, which required refined instruments and subtle calculations. An error as small as 5' in observing the moon meant an error of…
I request a last indulgence from the reader. The introductory material, thus far, has been written in the friendly and confiding first person singular voice. Starting in the next paragraph, I will inhabit the first person plural for the duration of the mathematical expositions. This should not be construed as a “royal…
Combinatory analysis refers to a group of techniques that can be used to determine the number of elements in a particular setwith- out having to count them one-by-one. Theelements in question could be the results froma scientific experiment or the different potential outcomes of a random event. [...] Combinatory…