Date: Nov 30, 2012 10:57 PM
Author: Jonathan J. Crabtree
Subject: Re: Some important demonstrations on negative numbers
> > OK thanks Bob. That was a great answer! Please keep
> playing along with the next question. You might even
> get your son to answer with just a single word...
> > A number added to itself once, is said to have been
> > A number added to itself once, is said to have
> > ==========
> > Thank you!
> Bob Hansen
Correct! A number added to itself once is said to have been doubled.
This all gets back to an old example of the mismatch between Euclid's algorithmic definition of multiplication summarised as mk = m added to itself k times.
1 x 1 = one added to itself once = 2
Old news as I have written about this on many occasions.
Now with 'i' it is often said to be the square root of take away one, or the square root of subtract one.
How can you have a square root of a verb? You can't!
Marcus du Sautoy and other professors keep talking about the square root of MINUS one when they mean the square root of NEGATIVE one. People reading this probably have as well.
People state the plus and minus sign when they mean the positive and negative sign.
Complex numbers are actually lots of fun and easier than you think! Maybe they could be called simplex numbers and we could get kids to play variants of the game battleships with them. So why hide how simple complex number arithmetic can be, because of using the WRONG word in the definition of an imaginary number? ie minus instead of negative?
Synonyms are interchangeable as are math terms. All the way through elementary school children are exposed to the verbs of minus, subtract and take away. Negative as an adjective makes an appearance after seven years of math education only BECAUSE the mk = m added to itself k times is false. If it was accepted that mk = m added to zero k times, we could introduce integers years earlier. Same with the rationals.
I bought a ticket to travel by: plane | aircraft | airplane | aeroplane etc.
Minus means 'take away' or 'subtract' therefore the definition of i in which the incorrect word minus is used, becomes meaningless.
One path to better understanding of math is to simply SAY WHAT YOU MEAN and use the correct word rather than a shorter word with different meaning for convenience.
The minus temperature below zero is often read as 'minus three' yet that is usually by attractive weather presenters, not mathematicians. We know they mean three below and that zero (freezing point) minus three is the temperature on that vertical number line called a thermometer.
So the square root of take away one is mush, just as the square root of minus one is mush. We either talk, speak and write with precision or we have marks taken off by those who follow the rules of math.
That is why I talk about the necessity of explaining the nature of multiplicand and multiplier.
Unless that is made abundantly clear, then there can be no demonstrations of a negative number repeatedly taken away from zero on the number line. (Note I did NOT say negative times negative or adjective times adjective!)
Positively bending the rules of negative multiplicands with either subtractive multipliers or additive multipliers must not be attempted until people know EXACTLY what the real multiplication process involves.
It doesn't matter if you agree with my approach or you don't. You explanation will simply continue to be sub-optimal and continue to confuse children.
The simple fact is Euclid NEVER included the words 'to itself' in his algorithmic definition of multiplication.
I have checked and rechecked over many centuries of history in Latin, French, and of course, the Greek. I use the Vatican edition, not the one the haberdasher Billingsley used in 1570 that people continue to cite.
So who keeps the 'added to itself' myth alive?
Steven Stogatz writes on page 4 of The Joy of X released Oct 2012.
Does "seven times three" mean "seven added to itself three times?" Or "three added to itself seven times?"
There we go again, the mindless repetition by yet another professor of the haberdasher Henry Billingsley from 1570 who WRONGLY included the words "to itself" in Euclid's definition of multiplication.
Henry had every right to be confused as he wrote before the invention of the zero based number line in 1685.
Yet for professors like Wu and Stogatz and many others to keep talking non-sense about something as simple as multiplication of discrete quantities is truly bizarre.
Do we worship the cult of celebrity that much that we have no choice but to slavishly follow what people write just because everybody says the same old (wrong) thing?
Plagiarising stupidity and passing it off without citing the 'faulty' source just keeps getting rewarded. **Sigh**
Yes I have Children Doing Mathematics and Adding it Up.
Yes I have also bought, the Development of Multiplicative Reasoning. Once again, the false and untrue definition wrongly attributed to Euclid this time appears in a 400+ page book on multiplication. Maybe if a correct definition of multi-plication of multi-tudes was taught this book could be 40+ pages rather than 400+. Then another 40+ page book could handle proportions, scalars and magnitudes.
Yes we can all prove -2 x -3 =6 algebraically or with lines on a graph yet that does NOT translate to an understanding of the PROCESS involved in doing multiplication on discrete objects.
Whenever the simple application of correct words is available to explain something, it should be embraced.
I have provided a concrete example of how to explain a negative multiplicand paired with a 'minus' multiplier via bumps and holes as proxies for positive and negative with the multiplier providing the repeater instruction.
I have also provided a precise verbal mapping of the instructions that reveal the logic of why the products of both -ve x - and - x -ve are positive.
The commutative law works with precision when the unit swaps sides with the multiplicand. Three apples added 2 times is the same as twice three apples were added. Two apples added three times is a DIFFERENT story and process which may happen to lead to the same result.
Again there is no 'debt times debt = wealth'. There is only debt taken away repeatedly which becomes wealth once we get to zero. Another 'concrete' example from India misquoted for 1500 years.
Sometimes I wonder if it is the minds of mathematicians that are set in concrete that prevent them from revealing math to be as simple as it truly is!
I suspect Jeffrey Kluger has nailed it best in his book 'Simplexity - Why Simple Things Become Complex and How Complex Things Can Be Made Simple' He writes...
"We're suckers for scale. Things that last for a long time impress us..." p.11
Euclid's Elements became infected by the addition of the words 'to itself' and from 1570 nearly every English language edition has multiplied this 'additional' mistake.
As a result people waste time debating whether or not the repeated addition model works for multiplication while ignoring the FACT that for discrete multitudes, multiplication is defined by addition. For magnitudes, multiplication is defined by scalars and the proportional logic of the 'rule of three'. Euclid gave us that insight 2300 years ago. Willam Oughtred gave as the symbol :: as well as the equals sign yet :: was pretty much ignored and the x had to perform double-duty on multitude as well as magnitude, which is a compromise situation.
Because of the 'infection' contained within the definition of multiplication involving discrete objects (as opposed to proportional and scalar magnitudes) children have been taught garbage math for centuries now in which the precise terms are absent. Is it any surprise we wonder why children so often drop out of math before they fail it altogether?
Children deserve better math insights from mathematicians not better pedagogy from educators.
This thread originated by Peter is very important as it identifies the need for a proper and rigorous investigation of the 'mush' that passes for basic math.
It seems every elementary math book now skips from addition to arrays without any attempt to differentiate between objects and actions, nouns, adjectives and verbs.
A lot of word stories in which children write the equation would be beneficial BEFORE we jump to the arrays and area models that bury the critical differences between multiplicands and multipliers.
In the classroom next door Ms Smith is teaching about nouns, verbs and adjectives in English class. So why doesn't the SAME Ms Smith use the same linguistic logic with her children? Because mathematicians as a group are not that smart! They refuse to do what used to work centuries ago so they can write more and more books on the SAME ARITHMETIC! What else could it be?
["Here we go again. Crabtree is being pedantic. Perhaps just treat him like the autistic child that gets upset every time the rules are broken by US."]
Children are not allowed to be sloppy and lazy, so why should mathematicians be allowed to be sloppy and lazy? When, where and why did mathematicians stop caring about obeying the laws of linguistic and symbolic logic developed by those who cared enough to create arithmetic in the first place?
Do we 'really' want students to keep doing remedial arithmetic as they enter colleges and universities?
Without the rich and precise 'original' vocabulary of arithmetic, the meaningful becomes meaningless.
Arithmetic and both the verbal logic and symbolic logic that underpin it, provides the foundation for nearly all subsequent mathematics.
The stronger and deeper we make those foundations, the higher we can build upon them and marvel at the beauty of mathematics and come to love it!
Thank you for reading!
Jonathan Crabtree :)