The question in the title mostly covers the question I want to ask. After seeing a number of questions here on ME and having taught/TA'd a number of introductory math classes, I wonder what people think the essential skills high school should teach students in terms of math. It might be interesting to see both what math major type people think is necessary and what just general population think is necessary. To limit the question though, let's say that we care what people who teach STEM based math (for non US, that's essentially the non-humanities based academics disciplines) think. What would you like your incoming students to have learnt in high school?
This is entirely opinion based, but I don't really care what content is "covered" at all. I care that students are engaged with thinking about problems which involve quantitative and spatial reasoning, and that they develop progressively more sophisticate logical reasoning abilities. They should be able to communicate their ideas effectively, produce logically coherent arguments, critique the arguments of others, produce examples and counter examples of claims generated by themselves and their peers, etc. In other words, they should engage with the real process of mathematics. Too often prescribing content leads to mimicry of these fundamental skills. We need the real deal.
It would be nice if the topics they are thinking about "build up" to something. Another important feature of mathematics is how mastery of one body of knowledge can lay the groundwork for beginning on another. What was at first insurmountable becomes routine. I would like students to have this experience so that they know what it feels like, and that it is possible.
The nominal entry level into university-level mathematics is first-semester freshman calculus. The way that class is customarily taught, it makes only extremely modest demands on students' high-level reasoning skills, such as reading comprehension, creativity, and sense-making. There are almost no "word problems," and the class consists almost entirely of differential calculus, at which students can succeed simply by mastering rules.
Students need skills like the following in order to succeed in such a class:
- ability to manipulate fractions
- order of operations
- ability to solve an equation for an unknown
- knowledge of some very basic trigonometry, such as being able to tell what is the sine of 90 degrees and explain why without recourse to rote memorization
- knowledge of exponents and logarithms and basic properties such as $\log(ab)=\log a+\log b$
Many students at this level have only been exposed to solving multiple equations in multiple unknowns in the case where the equations are linear. That is probably sufficient in most cases for success in such a class.
The classes that will more stringently test their preparation are second-semester calculus (because integration is not algorithmic) and first-semester physics (because it's all word problems and interpretation, and one often has to solve multiple nonlinear equations in multiple unknowns).
TL;DR: Most students need to already have a basic understanding of common concepts to cope with doing them again more quickly and abstractly at university.
My perspective: I am involved in teaching physics to students of physics and some other disciplines (e.g., math) in Germany. I studied physics and maths in Germany myself.
University mathematics education does almost everything from scratch. Moreover, before the first semester, every physics department in Germany offers a mathematical repetition course of about three weeks, covering all the mathematics you require for the first semesters. This will cover mathematics that the physics lectures require and that the parallel mathematics lectures haven’t got to (e.g., integrals). If a student has missed a particular topic in school, they will see it here. So, from a naïve perspective, a student doesn’t need anything except the ability to engage in critical mathematical thinking of any kind.
However, all these courses only really work if you have some familiarity with the general topics. For example, it simply takes some time (and repetition) until a student has wrapped their mind around derivatives and automatised the basic concepts to some extent (also see this answer of mine). If a student hasn’t already done this in school, it will be very difficult for them to follow the much more quick-paced and abstract introduction to calculus at university and the physics lectures that simply use it. Some exceptional student may be able to grasp university calculus to the required extent even though this is their first encounter with calculus, but they are, well, the exception.
The good news is that the details don’t matter that much. For example, I care that students have an understanding of what a derivative is, but if I don’t care very much whether they know certain differentiation methods. Those can be learnt quickly and are not the basics for more advanced layers of mathematics. (Of course working with different differentiation methods is a good way to fortify the understanding of derivatives.)
With all that being said, here are things that we just assume as given in first-semester physics education and for which it is already very helpful if you have already seen them for the mathematics courses:
- solving equation systems (linear or easy non-linear)
- solving quadratic equations
- basic trigonometry
- vectors (geometrical interpretation and arithmetics)
- exponents and logarithms
- probability (not required in the first semester, but later)
With all these topics, I mostly want students to have a robust understanding of the basics. There is no use if a student can quickly solve all sorts of integrals but has no idea how to apply integrals to real problems or is lost once the variable of integration is not $x$ anymore or if presented a straightforward double integral.
- Checking answers
- Showing and checking work. Algebraic work can benefit from the two-column proof approach: one column for the algebra work and another column explaining what happened in each step. This is a lot to ask of students though since they will grow impatient at how much longer an algebra problem takes.
Processing information attentively for meaning;
thinking systematically (structure, connections, etc.);
being empowered to unpack stuff and continually make revisions and sense;
skills of inquiry.
In short, active thinking. Quoting my recent comments (Why do some students struggle so much with fractions):
Or how about just cultivating confidence (by sufficiently exposing learners to reasoning that involves more than one step, being methodical, and the process of intellectual discovery)? Technical topics naturally seem formidable when it continues to feel alien to be applying attention span to them.
I was riffing generally that, especially when it comes to hierarchical and technical subjects, waving the white flag at the first opportunity is a vicious cycle. Like reflexively dismissing alien cultures and never getting to broaden horizons.
Edit to clarify that the above illustration is an inherently didactic concept map, not an inspirational poster.
For most STEMs, they will have a standard calculus course, freshman year. Strong working ability, B level, of most of Frank Ayres First Year College Math is a good enough foundation. Of course it would be nice to have more. And some will have less. But that's a decent expectation.
Note that, in the US, college algebra is a misnomer since the 50s. Calculus freshman year is the normal track. If you aren't ready for it, you're on a remedial track. If you place out, you are on an accelerated track.
Also, I'm not saying they need to study the Ayres book. Duh. Just that it is a convenient synthesis of high school work, pre calculus. If you can work the Ayres problems, or the equivalent, you have enough to move into a normal calc class.