Some considerations for true beginners.
The equal sign “=” is a mathematical symbol (coined by Welsh mathematician Robert Recorde in 1557) of two parallel horizontal lines used to show that two expressions have the same value or are identical.
It is taught very early on in elementary school, but I don’t think the power of its deep meaning is explained well. At least, that’s what we can see from the difficulties experienced by slightly older students.
Equal means exactly equal. What is on the left of the “=” sign has the same value as what is on the right. So if there is a symbol on the left and a number on the right, in that context that symbol is worth exactly the amount indicated. And if two symbols are equal, they can be used interchangeably. Trivial, but a source of great doubt for those who have not fully grasped the basic concept.
From a formal point of view, the equal sign satisfies the following conditions:
– Reflexive Property: any value is equal to itself, a = a
– Symmetric property: if a = b then b = a (the order of the expressions can be swapped without changing the truth)
– Transitive Property: if a = b and b = c then a = c
– Substitution Property: informally, this just means that if a = b , then a can replace b in any mathematical expression or formula without changing its meaning; formally, for every a and b , and any formula ϕ(x) with a free variable x , if a = b , then ϕ(a) implies ϕ(b) ; we can call this a function application.
Even without going into further detail, these simple properties allow us, for example, to solve first-degree equations directly. They allow us to invert formulas simply by adding or multiplying the same quantity on both sides of the equation. OK, taking care not to divide by zero.
If the unknown quantities (the so-called variables) are of a higher degree, things get more complicated, but the effectiveness of “=” remains the same.
A powerful little trick based on this is to use new variables to replace more complicated expressions or those with higher powers.
molto ben fatto , i miei complimenti. Bisognerebbe riuscire a farlo transitare nelle scuole
grazie!
temo che non avrebbe gran riscontro… se non ci sono gli insegnanti a far venir voglia di porsi domande e cercare di capire, il mondo oggi va avanti solo a risposte, tutte pronte sulle AI dove i giovani sono certi di trovarle (se c’è campo e le batterie sono cariche). nel senso di risposte a quello che viene chiesto loro (tipo compiti per casa), non risposte ai loro dubbi, che ormai hanno disimparato ad avere.
vedo una decadenza esiziale che trascinerà la civiltà in un oblio dal quale forse riemergerà, come storicamente è già successo. non credo in tempi brevi. prima sarà (già è) un oblio culturale, poi cominceranno a mancare i soggetti pensanti, chi possa scoprire nuove leggi. ci si illuderà che basti la tecnologia, mentre la scienza dei fondamenti della fisica già è inefficace da 50 anni e ben pochi gridano allarmi…
vista l’efficienza impositiva del ‘pensiero unico’, magari non servirà nemmeno una nuova inquisizione… 🙁