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I am a young postdoc at a large crowded university in the USA. Also, I am not from the USA, so my school experience has been very different from the high school experience of my students.

Since I am teaching mathematics classes, I wonder whether there is a good and comprehensible summary the American students have seen in their high school mathematics classes.

Given the fractured education system in the USA, good and comprehensible may be mutually exclusive, but the question is worth a try.

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    I think this would be better on MathEducators – Alexander Woo Oct 12 '17 at 3:59
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    There is a well-known phenomenon: If you ask them "Have you seen topic X?", they will almost always answer "No." That can happen even if topic X was covered just last week. – GEdgar Oct 12 '17 at 11:40
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    Do you perhaps mean comprehensive and not comprehensible? – J... Oct 12 '17 at 12:25
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    Your institution probably has a math placement exam; you could ask to take a look at that, to see what kinds of things your students are expected to have covered to achieve various scores/placements. – 1006a Oct 12 '17 at 16:58
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Every state has their own standards of what is covered in high school math classes. Many states (but not all) modeled their standards on the Common Core standards found in this link. http://www.corestandards.org/Math/

The topics covered are sorted into the categories of 1. Numbers 2. Algebra 3. Functions 4. Geometry and 5. Statistics and Probability. Each category gives more specific standards students are expected to master.

If you would like to see how topics covered by standards are presented good resources are EngageNY (https://www.engageny.org) which provides sample lessons/worksheets for the entire Curriculum. One note is most most teachers I knew who used EngageNY would modify/scaffold the material extensively for their students. Also to see sample exams JMAP has archives of New York's standardized tests by year/subject http://www.jmap.org/JMAP_REGENTS_EXAMS.htm and also with questions sorted by standard http://www.jmap.org/JMAP_REGENTS_BOOKS.htm

An important note is that Common Core standards go up through Algebra II but it is possible that some or even most of your students also took Pre-calculus and maybe even Calculus in High School as well. If you want to see what might be covered in a high school Calculus class one option is you can look at the course overviews for Advanced Placement calculus https://apcentral.collegeboard.org/pdf/ap-calculus-ab-course-overview.pdf?course=ap-calculus-ab and http://media.collegeboard.com/digitalServices/pdf/ap/ap-course-overviews/ap-calculus-bc-course-overview.pdf

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I write as someone who has taught calculus/linear algebra to first year engineering/science students in the US and Spain. In a typical European system (and probably most of the rest of the world) a student studying a technical area (science, engineering, medicine) in the university will have seen in the equivalent of high school a full year of calculus and a full year of some sort of matrix calculus course that involves using Gaussian elimination to solve systems of linear equations. In the US even a student studying a technical area may have seen in high school neither calculus nor matrices. Probably more students will have seen calculus than not, but very few will have seen a systematic treatment (meaning using elimination, involving formalizing parameter counts in terms of rank and dimension, etc.) of the solution of systems of linear equations.

For example, in Spain one can reasonably assume that first-year engineering students know that the solutions of a system of $m$ linear equations in $n$ variables depend on $n-r$ parameters, where $r$ is the rank of the system, and, moreover, that they can compute a parametric description of these solutions via row reduction. This material appears on the exams used to determine placement in the university, and is covered in the last year of high school. One does have to review this material in a first semester linear algebra class, because they probably do not know it as well as they should, nor have they seen it presented with much sophistication (their understanding is purely operational). In the US, teaching engineering students (the situation might be different teaching math majors) one cannot suppose familiarity even with the matricial representation of a system of linear equations, nor even that students have seen systems of linear equations involving more than two variables (which they were taught to solve "by hand"). Also linear algebra would usually be a second or third semester course, not a first semester course.

The conservative assumption is that US students entering the university have seen no linear algebra (to the point of not even knowing what a matrix is) and have not learned much more than very mechanical manipulation of derivatives and integrals. With non-technical students the conservative assumption is that they do not know basic trigonometry and with polynomials can do little more than factor two variable polynomials. It is also important to remember that their backgrounds are far more heterogeneous than they would be in many other systems. The US "system" is not a system at all, and it is a mistake to assume uniformity of preparation. Moreover, even for nominally well prepared students the expected level is less than what it is in many other countries. On the other hand, the flexibility underlying this heterogeneity also means that an occasional talented student will have studied outside the standard curriculum, and will have learned linear algebra, vector calculus and more, but often that student gets channeled into level appropriate courses in the university too.

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    will have seen in the equivalent of high school a full year of calculus and a full year of some sort of matrix calculus course that involves using Gaussian elimination to solve systems of linear equations. Don't want to argue against but I have been teaching last year in a Northern European university (Informatics) and a good percentage of students did not know what Pythagorean theorem is, not to say about Calculus, Algebra etc. One even asked me what is that trick that puts two fractions on a common denominator. Most of them didn't know basic induction. – PsySp Oct 12 '17 at 18:32
  • @PsySp: Good point. Having spent two semesters in Northern Europe, I believe that their education system is really washed down. My general impression is that Dan's answer gives a good description of Spain, France, Germany, Italy, and Russia. With the smaller countries in Continental Europe, it really depends. – Ambicion Oct 12 '17 at 20:31
  • @Ambicion from my limited experience, it seems that within EU there can be some very large variance in mathematical education in high schools, depending on the country of origin. Some countries have notoriously difficult high school system, which is reflected on the selectivity and admission process in universities. – PsySp Oct 12 '17 at 21:08
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Some high school students take the equivalent of a semester of calculus. For some, the highest course is Algebra 2, which does trigonometry and transformations, mainly. There is practice graphing functions, finding the inverse of a function, that sort of thing.

If the student is interested in math, science or engineering, it's likely s/he will have taken "Precalculus" which does some analytical geometry to get students ready for the first semester of calculus.

Sadly, high school students in the U.S. almost never learn any linear algebra.

You could take a look at New York's Regents tests and EngageNY materials.

Generally, some statistics are taught in each year of school.

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  • So, e.g., systems of linear equations are generally not part of the school curriculum? Thanks for your reply. – Ambicion Oct 12 '17 at 3:39
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    Solving a system of 2 linear equations in 2 variables is usually – user81191 Oct 12 '17 at 3:45
  • My 14yo is currently solving 2x2 and 3x3 systems, but they don't do row reduction. – aparente001 Oct 16 '17 at 4:46

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