Consider the last few real-world problems I've solved (or attempted to solve) with math:
If you have a widget (roughly shaped like a cone with the tip chopped off) held at angle a and the widget is tapered at angle b and the tip of the widget has a diameter d, how far is it in x and y from the center of the tip of the widget to the theoretical tip were the taper continued to a point?
If two 14" pizzas feed five people, how many 16" pizzas do you need to feed eleven?
If a gallon of paint covers 400 ft^2 once, how many gallons cover a 12'x36' room with a 9' ceiling twice?
And to counter your claim that it no K-12 math can be fuzzy, if you live in a family-friendly neighborhood of x square miles, how many pieces of candy do you need to be prepared for trick-or-treating children? (This problem could easily be approached by a high school student, has real world application, but is a lot like the kind of interview problems that are often criticized as being irrelevant.)
But even if you feel these problems fit with standardized tests, the more important point is that none of these problems was presented to me the way I am presenting them to you. No one told me that angles a and b and diameter d would be both easily available and sufficient to calculate the necessary x and y. No one told me that comparing square inches would be the best way to determine how much pizza to buy. I was certainly not given multiple choices like a normal standardized test.
The real world mistakes people make with math are often more about knowing how and when to apply math, than doing the math itself. Almost everyone knows that Google can multiply and divide numbers. The real math skill is knowing what numbers to multiply and divide to see how many gallons of paint, or pizzas, or pieces of candy, you need.
Being able to pass a trigonometry test is useless if you can't figure out what to do with your widgets. Likewise for the other examples.
I'm not saying standardized tests have no correlation to real-world math ability. They obviously have some. But it's not exact. And there are lots of real-world situations that are poorly reflected in standardized tests.
I agree with all this, but consider that the type of math being tested on a standardized test is a subset of the skills you mention. You certainly can't perform complex word problems unless you can do the algebra or geometry calculations, whereas the converse is not true.
IOW, standardized testing of math is testing the bare minimum, and a successful teacher should be able to teach this bare minimum. If a teacher can do better, great!
But if "what gets measured gets done", then teachers will likely take time away from these more complex skills to spend more time going over what's on the test. This is especially true if you tie compensation and continued employment for teachers, and college admission and scholarships for students to test performance.
Functional Skills Maths and English tests were introduced nationally in the UK after concerns from employer's organisations and other stakeholders. Preparing students to take these is mostly a vocabulary exercise I have to say. I would be interested in any reactions people have to these tests. They are aimed at mature adults who work in factories, administration, medical services with non-degree qualification levels. We use them with non-academic teenagers as well as stepping stones to the 16+ standard exam called the GCSE.
There is a lot to the idea that learning needs to be situated in a context... but then assessment gets expensive!
The standardized tests I've seem often ask question similar to the ones you give, and are actually not all that bad.
The real world alternative to standardized tests is usually tests designed by the teachers themselves. There's little reason to believe that those are somehow better (or even close to the standardized ones, which are usually well thought out).
If you have a widget (roughly shaped like a cone with the tip chopped off) held at angle a and the widget is tapered at angle b and the tip of the widget has a diameter d, how far is it in x and y from the center of the tip of the widget to the theoretical tip were the taper continued to a point?
If two 14" pizzas feed five people, how many 16" pizzas do you need to feed eleven?
If a gallon of paint covers 400 ft^2 once, how many gallons cover a 12'x36' room with a 9' ceiling twice?
And to counter your claim that it no K-12 math can be fuzzy, if you live in a family-friendly neighborhood of x square miles, how many pieces of candy do you need to be prepared for trick-or-treating children? (This problem could easily be approached by a high school student, has real world application, but is a lot like the kind of interview problems that are often criticized as being irrelevant.)
But even if you feel these problems fit with standardized tests, the more important point is that none of these problems was presented to me the way I am presenting them to you. No one told me that angles a and b and diameter d would be both easily available and sufficient to calculate the necessary x and y. No one told me that comparing square inches would be the best way to determine how much pizza to buy. I was certainly not given multiple choices like a normal standardized test.
The real world mistakes people make with math are often more about knowing how and when to apply math, than doing the math itself. Almost everyone knows that Google can multiply and divide numbers. The real math skill is knowing what numbers to multiply and divide to see how many gallons of paint, or pizzas, or pieces of candy, you need.
Being able to pass a trigonometry test is useless if you can't figure out what to do with your widgets. Likewise for the other examples.
I'm not saying standardized tests have no correlation to real-world math ability. They obviously have some. But it's not exact. And there are lots of real-world situations that are poorly reflected in standardized tests.