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Are there any researchers that try to develop learning materials to teach the core mathematics knowledge without using notation (and instead just plain English -- preferably without images)?

Tom
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    Try to write the quadratic formula without notation and you'll see why this is a bad idea. Mathematicians used to do it, and old texts are hard to read for this reason. Anyway, maybe try asking on matheducators.SE? – Najib Idrissi Apr 21 '15 at 11:25
  • I think part of learning math is learning abstract reasoning skills. Using symbols and precise definitions is pretty core to such a skill set. – jack Apr 21 '15 at 11:26
  • IMO: notations is one of the most powerful tool in math. Actually, sometimes making a new notation (to write something in a simpler way) makes you discovering new results. Finding the approriate notation is also extremely important to cern the structure of the object you want to describe. – Surb Apr 21 '15 at 11:26
  • Also, do a google on Russell's Paradox. Professional mathematicians mess up when they don't use symbols. It is hard to imagine beginners being able to function without them. They are fairly different situations, but the comparison I think is pretty meaningful. – jack Apr 21 '15 at 11:27
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    @jack I don't think Russell's paradox has anything to do with notation. What do you mean by that? – Najib Idrissi Apr 21 '15 at 11:30
  • Sets were ideas people had in their heads. They were not formally defined in precise language. Thus anything could be a set, hence the paradox. Once you set out to use precise language to describe sets, it is not nearly as natural to run into that sort of a contradiction. That paradox was a big part of mathematics becoming much more technically descriptive (i.e., formal language foundations). The same principal applies in learning math. If you have to define what you mean precisely, it helps you to see if what you mean really makes sense. – jack Apr 21 '15 at 11:33
  • There are some projects which aims are to develop mathematical knowledge (for primary school students) using hands-on activity. But I don't know if you meant something like this. However, this is the link http://www.mathematicsinthemaking.eu/ – Marco Cantarini Apr 21 '15 at 11:43
  • @NajibIdrissi It's difficult to know whether one would want to have a quadratic formula if they were not using notation. Perhaps that formula is a good example of math knowledge being confounded by notation. But yes, maybe ME.SE would be better, though do they have enough people? – Tom Apr 21 '15 at 11:43
  • Formalism is not the same as notation. Here, for example, is a passage from Euclid's Elements, which is on the far end of the formalism spectrum but invokes no special notation: "If as many numbers as we please beginning from a unit are in continued proportion, then the third from the unit is square as are also those which successively leave out one, the fourth is cubic as are also all those which leave out two, and the seventh is at once cubic and square are also those which leave out five." – anomaly Feb 09 '16 at 21:51
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    I'm voting to close this question as off-topic because it would be better suited for matheducators.stackexchange . – anomaly Feb 09 '16 at 21:55

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You'll notice throughout the development of mathematics, and dependent branches such as physics, the use of mathematical notation increases more so as the field becomes more advanced. For instance, Isaac Newton's Principia is a renowned book on the early development of physics and mathematics, and yet you will find it hard to find someone who has actually read (and understood) the text itself.

The question is, why is this so? Consider this passage:

''If a body $A$ should, at its place $A$ where a force is impressed upon it, have a motion by which, when uniformly continued, it would describe the straight line $Aa$, but shall by the impressed force be deflected from this line into another one $Ab$ and, when it ought to be located at the place $a$, be found at the place $b$, then because of the body, free of the impressed force, would have occupied the place $a$ and is thrust out from this place by the force and transferred therefrom to the place $b$, the translation of the body from the place $a$ to the place $b$ will, in the meaning of this Law, be proportional to this force and directed to the same goal towards which this force is impressed. Whence, if the same body deprived of all motion and impressed by the same force with the same direction, could in the same time be transported from the place $A$ to the place $B$, the two straight lines $AB$ and $ab$ will be ...''

and so on and so forth. This is Newton's explanation of his second law, expressed conicsely today as $$\mathbf{F}=m\mathbf{a}.$$

Newton's Principia was written as to be coherent to a wider audience, but of course we can see that, as this passage is representative of the entire structure of the text, it is extremely tedious and often terribly complicated. Writing out a mathematical idea was terribly difficult to do without mathematical notation, and if it wasn't for the geometric drawings of Newton's, a whole explanation would be impossible.

It was obvious to people that mathematics needed to become symbolic so as to have more concise works, remove ambiguity, and save heaps of time writing.

  • As succinct as F=ma might seem, alone it explains little to a student. You have to continue to explain what each symbol represents, then elaborate with uses of the formula. – Tom Apr 21 '15 at 12:13
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    Newton -- a great mathematician -- explains here a law through a precise example, and although this requires more time than stating a formula, and the language (words and phrasing) is not familiar, the net result at the time (when people were more familiar with the language) might have been better communication of the ideas than a notation-driven 'explanation' today (how often do people forget(?) to explain part of a formula, or how it was derived?). – Tom Apr 21 '15 at 12:14
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    And I don't think it's so easy to say it 'saves heaps of time': the same information has to be shared... immediately I think of the time required to disect a complex formula. – Tom Apr 21 '15 at 12:14
  • Provided the student knows what the symbols mean, i.e. force, mass, and acceleration, then $F=ma$ explains a lot. If it were better to teach the student as Newton had written it out, then that would certainly be the norm, but it is not. –  Apr 21 '15 at 12:16
  • 'The norm' in teaching most things has little correspondence with what's the best approach. – Tom Apr 21 '15 at 12:18
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    That is an opinion. When you delve into complex areas of physics, such as electromagnetism, or relativity, it would be impossible to learn without the foundation of mathematics. –  Apr 21 '15 at 12:21
  • I'll avoid a flame war. But this answer doesn't answer the original question, it instead provides argument for not doing it. I'm only interested in the answer of whether there are such researchers -- preferably now, not a few hundred years ago (but older mathematicians are still welcome). – Tom Apr 21 '15 at 12:31
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    Well I doubt there are any, as, by my example, it would be a tedious, over-complicated, unnecessary human endeavour. –  Apr 21 '15 at 12:33