It was common in the late 20th century and early 21st century (AD/ACE) for high schools and colleges to teach pre algebra followed by alegebra then geometry, then trigonemtry (or Algebra 3 / Triginometry) then precalculus and finally a course equivelent to AB and BC AP calculus after that there are several calculus courses that all require a course equivelent to AB and BC calculus as prerequisites.
Pre algebra was an unnecesary course because it was a mixture of the math taught before Algebra in other math classes and Algebra that you would take a second time when you take the Algebra course. Thus if you completed all the math classes that were prerequisites for Pre Algebra you could skip it and go straight to algebra and there would be nothing you would be taught in Pre Algebra that was not either taught in Algebra or the prerequisite courses for prealgebra that you would miss.
Precalculus was a combination of Algebra 3 trigonometry and AB calculus with the exception of the possible inclusion of matrixes and probability in Alegebra 3 and or Precalculus being included in one but not the other. If matrixes and probability are included in both Algebra 3 Trigonometry and Precalculus then there is nothing someone would miss by skipping Precalculus and going straight to AB calculus from Algebra 3 Trigonometry. If Matrixes and Probability were not included in both Algebra 3 Trigonometry and Precalculus then there is nothing someone would miss by skipping Precalculus and going straight to AB calculus from Algebra 3 Trigonometry.
The reason for being able to remove Pre Algebra and Pre Calculus courses and let someone skip them is overlapping and redundant teaching material with zero new teaching material added. But there is a diffetent reason all together for the lack of necessity of the high school or college geometry course in that sequence in addition to any overlapping material.
The real life practical purpose of geometry is to measure distances or lengths of paths, distances between objects or lengths of objects, locations (coordinates) of objects, volumes of objects and cross sectional and surface areas of objects. All that can be done using material usually taught in high school algebra, trigonometry and calculus classes without ever needing to take a high school geometry course. If you understand caartesian coordinates taught in algebra class, polar coordinates taught in trigonometry and or calculus class and the meaning of the sin, cos, tan and cotangent functions and there inverses as well as what radians are and the pythagorean theoreom taught in trigonometry class you will be able to do every one of the practical purposes I have listed that you would be able to do taking a geometry class. The only one of those things you might learn in calculus that you might not learn in trigonometry is polar coordinates which really should be taught in trigonometry before AB calculus because it is more of a trigonometry related thing. Now there are some of those practical things I listed you might learn to do in calculus that you would not learn in high school trigonometry but you would not learn them in high school geometry either. Examples would be learning how to calculate the surface area or cross sectional area or volume of certain shapes by integration or the distances of certain paths that are not a single straight line segment by integration. Now there are some useful things involving geometry people might learn like names of shapes but people usually do not learn those in high school geometry but in other classes prior to high school geometry. In Geometry class you might learn complicated arcane and obscure ways to measure those practical things I listed in very specific special cases that might not be taught in trigonometry but in trigonometry you will learn general wide reaching methods to do all those things you can do in geometry which will be easier to remember because you do not have to create a special case to figure out each thing but can simply figure out the coordinates of things using trig functions then use those coordinates to get the desired measurements.
* Start side note
For further reading about how to get trigonometric measurements by means of algebra with casrtesian and polar coordinates without geometry class you can read the following article in progress
Tangent of the average of two angles and other trigonometric identities derived from a combination of it and the pythagorean theoreom
http://web.archive.org/web/*/http://teachingthenarrowway.blogspot.com/2022/01/tangent-of-average-of-two-angles-and.html
http://teachingthenarrowway.blogspot.com/2022/01/tangent-of-average-of-two-angles-and.html
* End side note
But what about the philosophy of using axioms taught in high school geometry being useful to learn how to use logic to make practical decisions. This so called philosophy taught in high school geometry is the most important reason it should be abolished except perhaps for those who wish to study how cultic thinking works. High School Geometry is full of the bad kind of dogma that is presuppositions that one is not permitted to question, as opposed to stating presuppositions you use to come to a conclusion but acknowledging the possibility your presuppositions maybe wrong. The starting point is there is a finite number of assumptions the student starts with called axioms and the claim is made that all geometric proofs in the geometry class can and will be taught either by using only those axioms or other proofs derived only through those axioms. The student is given a list of those axioms at the beginning of the course and expected to derive all proofs assigned as homework only through those axioms or through other proofs they have derived eventually tracing back to a point of only through those starting axioms. The fundamental problem with this is in reality assumptions are made to derive these proofs that are not in the list of initial axioms but the student is not permitted to admit that additional assumptions are required other than the axioms initially listed if you say another assumption is needed for the proof that is not listed in the initial list given at the beginning of the course that is classified as a thought crime and the problem is marked as wrong. The type of erroneous methodology of proofs in high school geometry class influenced most participants (who mostly were not previously educated about cult psychology to develop a resistance to the undue influence) towards pseudomathematics, pseudoscience, pseudologic and magical thinking through the act of saying and or writing things in order to agree with the consensus of an authority figure even if those things are not true.
In the Asch conformity experiment people were more likely to claim a untrue statement was true if someone else first claimed the same untrue statement was true.
The Asch conformity experiment was an experiment where volunteers were told a false answer about the length of a line segment then asked what length the line segment was. It was found that when a unanimous group of people gave a false answer first people more frequently said the same measurement as the false answer they were previously told instead of the correct measurement than under other circumstances.
Something similar to the Asch conformity experiment is replicated in geometry classes where the teachers assign what assumptions maybe used and students are not allowed to pick the assumptions they use for themselves and describe which assumptions they use and how and why they used them. Even if the teachers claims about what assumptions are needed to prove a genuinely true statement is true are false by ommission or commission the students will (or more accurately have in the past implying others will in the future) more ftequently still give the same answer as the teacher than if they never heard the teachers claim. False by ommission in this context would be where the teacher claims that such is the exact minimum list of assumptions required to validly prove a true claim when actually given the context on the list the teacher gives additional assumptions need to be added. False by commission in this context would be to claim a assumption is needed to validly prove a true statement when it is not needed in the context of the other listed assumptions, that is if all the other assumptions on the list were included but that assumption was removed it would still be a valid proof of a true statement. In addition a teacher may sometimes claim a false statement is true or a true statement is false and the students will more ftequently give the same answer as the teacher. If the students did not first hear the teacher make a false claim or say a true claim is true based on logic that was not valid the students would be much less likely to make the same error when investigating if a statement is true independently in an ungraded environment without seeing someone else's example first. Remember in geometry class people are not graded for right or wrong answers but they are graded for right or wrong reasoning.
According to the way this type of geometry class is done the reasoning should be in agreement with the reasoning predefined in the curriculum before the student even started the class in order to not be marked as wrong or less than an A or 100% grade for a set of problems. One of the fundamental problems is the assumption of minimum axioms that is the foundation for all reasoning in high school geometry classes of the type defined in this essay/article/manifesto is usually or maybe even always wrong based on a claim of being complete when more assumptions are actually needed in reality. Say you claim the teacher or textbook is missing an assumption when they show how to do a geometric proof which was an assigned homework or test problem and that assumption is not in the axiom list or derived from the initial axiom list alone and you will get your problem marked as wrong.
If a list of five assumptions were given to reach a conclusion and you were asked if these five assumptions alone with nothing added or removed were valid to show the conclusion was true or untrue and nobody told you the answer beforehand and you were not given a hint what answer someone else came to and nobody else would see or grade your answer this would not be so similar to Asch conformity. But in the case of geometry class the teacher starts by listing a certain number of assumptions and says they are sufficient. According to the principle learned in the Asch conformity experiment by giving you the answer first people are more likely to verbally say the same answer they were first given by someone else even if the answer they were given is false. Geometry classes of this sort do not start with the question of if those axioms are sufficient to prove the proofs but the statement that all the proofs you are required to prove in the class can be proved with those assumptions and no more excluding that which is derived from those assumptions alone. Nobody in their right mind would believe no more assumptions will be needed based on independent thinking without someone else first telling them no more assumptions would be needed. Those who are first told no more assumptions will be needed to prove the proofs in the book believe so (or have said they believe so) without even first reading the entire book more often than they should by which I mean more often than never. Geometry classes of this sort are like religious instiuitions such as Churches or Mosques where they far too many members say they believe every word in a certain book such as a particular Bible or Quran or book of Mormon are true without ever having read the entire book. Many self proclaimed Muslims can not read Arabic but claim every statement in the original Quran manuscript in Arabic is 100% true and many people who say they are Christian at Churches say every statement in the original Bible manuscripts of a particular Bible Canon are true without having even read an entire translation of it. It is very different to have read a translation of the Bible and say you have found zero historical claims in it that you can confirm with great certainty to be false when correctly translated than to have never even read it and to claim every statement in it is true even statements you have not read. Yet some geometry teachers will give their testimony to the axioms as being sufficient for all proofs in the assigned geometry book and far too many people will believe them without reading the entite book. I am not saying to read an entire geometry book but do not give me your testimony of the sufficiency of the axioms to prove every proof in the book if you never read the entire book.
Copyright Carl Janssen 2021, 2022
https://en.m.wikipedia.org/wiki/Asch_conformity_experiments
http://web.archive.org/web/*/https://en.m.wikipedia.org/wiki/Asch_conformity_experiments
A 2003 effort (Meikle and Fleuriot) to formalize the Grundlagen with a computer, though, found that some of Hilbert's proofs appear to rely on diagrams and geometric intuition, and as such revealed some potential ambiguities and omissions in his definitions.[11]
[11] On page 334: "By formalizing the Grundlagen in Isabelle/Isar we showed that Hilbert's work glossed over subtle points of reasoning and relied heavily, in some cases, on diagrams which allowed implicit assumptions to be made. For this reason it can be argued that Hilbert interleaved his axioms with geometric intuition in order to prove many of his theorems."
http://web.archive.org/web/20210508113205/https://en.m.wikipedia.org/wiki/Hilbert%27s_axioms
Euclid's list of axioms in the Elements was not exhaustive, but represented the principles that seemed the most important. His proofs often invoke axiomatic notions that were not originally presented in his list of axioms.[23]
[23] Heath 1956, p. 62 (vol. I)
http://web.archive.org/web/20211209025416/https://en.wikipedia.org/wiki/Foundations_of_geometry
Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates, but since these are not sufficient as a basis of Euclidean geometry, other systems, such as Hilbert's axioms without the parallel axiom, are used.[1] The term was introduced by János Bolyai in 1832.[2] It is sometimes referred to as neutral geometry,[3] as it is neutral with respect to the parallel postulate.
[1] Faber 1983, pg. 131
[2] In "Appendix exhibiting the absolute science of space: independent of the truth or falsity of Euclid's Axiom XI (by no means previously decided)" (Faber 1983, pg. 161)
[3] Greenberg cites W. Prenowitz and M. Jordan (Greenberg, p. xvi) for having used the term neutral geometry to refer to that part of Euclidean geometry that does not depend on Euclid's parallel postulate. He says that the word absolute in absolute geometry misleadingly implies that all other geometries depend on it.
http://web.archive.org/web/20211209025836/https://en.m.wikipedia.org/wiki/Absolute_geometry
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