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slomobile

Notation study

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This thread is an attempt to quantify the methods used to communicate a mathematical concept to another person for the purpose of applying the concept.

Each response should ideally contain:

  • A single math concept, as you would present it to an inexperienced colleague in your field, who asks for help.
  • Any additions that you would include for the average person.
  • Optional comments.

Provide as many examples as you like, including historical examples, but please send each as its own post for ease of evaluating responses.

There are no restrictions on field or complexity.  The only restrictions on length and format are the practical limitations of this forum.  You may provide links to external resources.  Any document over 10 pages provided as a link or attachment should call out the most relevant page numbers in the body of the post.

We specifically request examples that include unconventional, obsolete, historical, experimental, graphical, scanned handwritten, 3 dimensional, or otherwise odd notation provided it is useful to explain a math related practical concept.  Computer code from any language is welcome.  Please note if you would use computer code exclusively, or coupled with traditional notation.  Please identify the notation type or language if it is not obvious to an English speaking C/C++ coder.

We understand there is ambiguity in the question, such as "What is a single math concept?"  Use your personal judgement.  Disambiguation is another feature of the thread.

Thank you.  I hope the results are interesting.

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I demonstrated the concept of subtraction to my children by eating their French fries. It worked quite well, and served the parallel purpose of being an education in social interaction with people more powerful,than you... that life isn’t always fair. 

Sorry... yours is a weird OP

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Yes, it is a weird question.  Stealing fries is a useful communication of the subtraction concept.  It is an example of what I am looking for.  I am trying to get back to first principles in a field with a great deal of adopted convention.  I'm trying to see if any of those conventions have become a crutch or shortcut for those in the know that slow or inhibit understanding by those unfamiliar or uncomfortable with the conventions. 

An example:  Traditional single letter variables, even with sub- and super- scripts do not convey much information and require mental symbol substitution to process.  Multi-character computer variables say what the variable is and leverage the highly practiced mental processes involved in reading and forming mental images from what is read.  Symbol substitution is fast for some and slow to others.  Perhaps this fact is unintentionally sorting people into scientific vs computer careers.  If science were done with variables composed of whole words rather than single letters would the field have access to a different set of minds?  Would existing scientists better grasp concepts because they access their emotional reading imagination, or are less burdened by cold symbol substitution.

Of course reading is also symbol substitution, but many that see single letter variables first convert to the words the variables represent then do a second substitution from the word to the meaning represented by the word.  The single letter variable convention has great space efficiency because it is unnecessary to demarcate the beginning and end of variables.  Is that better than mental efficiency?  I

Single letters allow 2 letters together to imply multiplication, which must be explicit with multi-character variables.  Do we need implicit multiplication?  Is there a different way to imply coefficients with multichar variables?  Can we settle on one representation of division?  Are there better keyboard layouts for math that just work without LaTeX?  PEMDAS applied blindly to endless practice problems (inconsistently between texts) confuses many and seems to delay understanding of how to properly form unambiguous equations.  Order of operations is not fundamental and not necessary, just convenient.  Is there value in experiencing the inconvenience?

Tackling one notation convention at a time is doomed to failure.  Tackling them all at once, is likely still doomed, but might produce some insight along the way to failure.

I wish to compile several unconventional ways to do math, present them to test subjects, and see if any of those alternate methods are more efficient or intuitive than conventional notation.  The results of that may inform development of a math software package.

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34 minutes ago, slomobile said:

Yes, it is a weird question.

So it is difficult to know how or even if to answer.

So it good that your second post has improved my understanding over your first one.

Have you considered looking at history and the development of Maths teaching in schools ?

Much of what you describe already happens so is there to be studied.

Also you should review your conception of 'first principles'.

Maths has advanced to the point where many of the most basic principles are also the most difficult.

 

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On 1/15/2021 at 11:53 AM, studiot said:

you should review your conception of 'first principles'.

That is fair given I am requesting anecdotal information.  However, the "first principles' I am interested in are those required to COMMUNICATE mathematical information in practical, not academic situations.   I am not interested in the math per se, but the form used to effectively communicate it.  In order to determine the minimum elements("as simple as possible") required to effectively communicate math concepts, I must first establish the breadth of concepts that must be communicated so that I do not leave out essential but uncommon fundamental elements("but no simpler").  I fully expect to receive examples far above my head that will require me to do research and see exactly which prerequisite concepts were implied in the communication.  This reveals a hierarchy of concepts which is a first principle.

A hierarchy of math concepts. is certainly not a new idea.  I have researched the history of notation and how math has been taught in the past as now..  It is part of what inspired the question.  I don't need to reinvent the wheel, but I do intend to prepare alternate ways to describe wheel motion and compare various measures of efficiency, depth of understanding,  and retention on various populations.

I have access to archives..  I cannot possibly read it all nor understand most of it.  I welcome pointers to specific illustrative historical examples because there is likely something Interesting I missed that you all will find.   But it cannot provide everything I am after.
Outside of internet resources, I do not have broad access to anecdotal  examples of passing math concepts verbally, or handwritten on envelopes and Post Its, or email and text message as it is actually applied.  I am interested in the incremental little bits you show someone that makes them say ":Oh, I get it now."  How do you say this in a text message?

Almost everything historical contains adopted conventions based on something that came before and was developed without the benefit of recent advances.  I am interested in what a clean sheet approach might look like today with modern knowledge and modern tools.  The nature of physics and underlying mathematical principles has not changed.  How we use math has changed a great deal.  What happens if we define integers as even multiples of PI and figure out object counting later?  What would a base 60 numeric keypad look like?  How can we use colored text, animated graphs, sound, vibration, and touch sliders as basic tools to imply  math information given that the average user of advanced math is more likely to have a smartphone than paper and pencil on them at any given time.

Matrix representations are now used far more than any time in history.  It is a very useful notation for modern purposes.  An advantage, it saves a lot of chalk by implying an awful lot .  It is efficient, for those that understand the implied information.  But chalk is mostly used for education and outdated even there.  If you want to apply matrix math you use R, or Python, or Matlab, etc... not chalk or pencils.  Maybe we should teach with those tools in second grade.  I can imagine extensions that allow you to mouse over a matrix and get a tooltip showing the expanded linear equations it is generated from while a computer voice(for auditory learners) explains the operations to be performed on that matrix and its part within the overall Jupyter notebook.

The remaining inspiration behind this thread was my own head injury and the mental adaptations I was required to make when my knowledge remained, but my processing ability diminished.  I realized the notations of higher math had unpriced externalities.  They are cheap(efficient) if you don't account for all the hidden prerequisites that allow those with ability and education to access them.  Again using https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_I_-_Mechanics_Sound_Oscillations_and_Waves_(OpenStax)/11%3A__Angular_Momentum/11.02%3A_Rolling_Motion as an example, what if you don't know how to type Greek letters, or don't even know that the squiggly w is a Greek letter.  Will you understand what to do with 

\vec{v}_{P} = -R \omega \hat{i} + v_{CM} \hat{i} \ldotp

That is equally valid from some perspective.  Are current notations making things harder than it needs to be.  A person talking directly to another person will generally communicate more efficiently.  I was hoping to gather some examples of that.  This obviously didn't work.  What is a better way to ask?

Edited by slomobile
Added code tags

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15 hours ago, slomobile said:

That is fair given I am requesting anecdotal information.  However, the "first principles' I am interested in are those required to COMMUNICATE mathematical information in practical, not academic situations.   I am not interested in the math per se, but the form used to effectively communicate it.  In order to determine the minimum elements("as simple as possible") required to effectively communicate math concepts, I must first establish the breadth of concepts that must be communicated so that I do not leave out essential but uncommon fundamental elements("but no simpler").  I fully expect to receive examples far above my head that will require me to do research and see exactly which prerequisite concepts were implied in the communication.  This reveals a hierarchy of concepts which is a first principle.

A hierarchy of math concepts. is certainly not a new idea.  I have researched the history of notation and how math has been taught in the past as now..  It is part of what inspired the question.  I don't need to reinvent the wheel, but I do intend to prepare alternate ways to describe wheel motion and compare various measures of efficiency, depth of understanding,  and retention on various populations.

 

OK so I will try to discuss communication of Mathematics, rather than principle of Mathematics.

I can't see where you have mentioned any basic Maths, computer code is hardly basic if it is indeed Maths at all.

However I beg to disagree with your outright  rejection of History.
Perhaps your experience of History at school was of the sort "History is a list of dates of battles, deaths and treaties to be learned by heart and regurgitated for the examiner".

History actually offers many lessons for those that care to peer into them.
Not the least being concerning computer code.
Coding languages have a very short lifetime; I have seen them come and go and stopped bothering to learn the new fashion decades ago now so I have little idea of the meaning of your example. The last serious program I wrote was PFortran TRIP (Trigonometric Intersection Program).

British schools went through a phase of demanding that every child learn 'programming'.
This mean resources were wasted on teaching first, different versions of BASIC, then PASCAL, then some early scripting.
None of which are current today.

History also tells us that the basic mathematical operation of counting is at least as old as writing, probably much older.
Now schools used to teach using the old fashioned balance scales. Good schools would actually get the pupils to set up pretend shops acting as customers and shoperkeeps.
They would weigh out amounts of materials, say potatoes or sand and also practice with pretend money.
This allowed a method of counting by the custemr presenting say a half crown coin and the shopkeeper saying That's one and fivepence and then making up the one and fivepence to half a crown with coins to provide the change.
Instant communication of arithmetic and fractions.
For those who were a dunce at school arithmetic there was the joke, you say you can't do maths but you can still instantly recognise that you need a treble eighteen, double top and single nineteen to finish in a darts match when you are on 113!

 

Would these be the sorts of examples you are looking for ?

 

 

 

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Having taught math successfully in Junior High and High school I though I would have a lot of great examples for you-- but on further reflection realized my best ones were not something that could be generalized.  I had the greatest success when I could connect the math lesson to the students' experiences.  For example, in my rural area the vast majority of the students have experience with guns and many also have reloaders in their families (people who make their own ammunition).  When I first tried to teach statistics I got blank looks from many students,  So that weekend I took my test equipment out to the rifle range and measured velocities of 10 rounds of ammo I had built.  On Monday I put the data up on the screen and asked the students if the load I had developed was consistent enough for hunting.  This lead to a successful lengthy discussion and the development of the idea of mean and standard deviation.

The lesson I learned and applied from then on is this:  The goal is not so much to make the students think differently, but rather to create a use for the math knowledge in a way that connects to their experiences.  I can think back to lots of examples of good teaching tricks, but realize they are were specific to a certain student or group of students.  Not much help to what you want.

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19 minutes ago, OldChemE said:

The goal is not so much to make the students think differently, but rather to create a use for the math knowledge in a way that connects to their experiences.

Yes, yes and yes.  +1

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Thank you, these answers are useful,.  They are however the low hanging fruit, as it mirrors the history of teaching basic math concepts, which I already have access to. 

I would love some examples of professional math communication between adult peers (assumed similar level of education, but disparate experience).  Those are data points which I do not have alternate access to, but which I assumed would be plentiful in this forum.

The answers thus far have exposed that the more difficult math concepts to communicate will be those that lack personal usefulness or familiar analogy.  Not everything can be made relatable.  I am acutely interested in the difficult to communicate concepts, please share those even if you feel they will not generalize.  For example, you were just hired as a very junior engineer to work with GPS satellites, but were a bit fuzzy on  time dilation, what did a colleague say that helped you "get it".

If I were pressed to identify ideal examples, I would choose the letters between Solvay Conference attendees if they had the benefit of all knowledge and tools up to now.  Modern scientists are not the ideal choice because I believe we have lost a bit of the art of communication as a society since that epic era.  I would very much like to be proven wrong on that point.  Consider it a challenge.

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