I taught high school level algebra for two years, so I have a moderate familiarity with this subject.
I, too, dislike these types of problems, and I refer to them as "math book" problems, because that's the only place you'll ever see them. I mean, if Luigi can paint 1/2 of a room in 3 hours and Mario can paint 1/4 of a room in 2 hours, the way you find out in the real world how fast they can paint the room together is to throw them in the freaking room with two sets of painting equipment and let them go to it. If it's not going fast enough for you, then you go find Mario's friend Bowser and hire him to help them.
But, I don't think lack of realism is the worst thing about these types of problems. I think lack of fun is. I have a graduate degree in mathematics, and I frequently work on problems that have little connection to anything (so far as I know) in the real world, just because it's fun for me to do so. To me, the real problem with high school algebra textbooks is they're so goddamn boring -- and I'm saying this as someone who really, really likes math.
If I had my way, there wouldn't be such a thing as "algebra 1", "algebra 2," "trigonometry," etc. as high school math courses. I'd teach "Year 1 math," "year 2 math," and so on, and put the focus on problem solving rather than any one particular corner of math. If the problems are interesting, the students will do the work and love it. I've seen it happen. I can't imagine that such a problem-solving based course would prepare students for either college or the real world any worse than what we're currently doing.
I agree with your basic concept. I dislike this type of problem as being obvious contrived. This is especially galling in basic Algebra where it is so easy to take genuine and real world problems from things like managing money, basic Newtonian physics, etc that are real and that will see in other contexts, even if only other classes.
I really think you are right that most students would be better served with emphasis on problem solving that is at least more like real world problems and less obviously contrived.
I disagree with your renaming suggestion. For one thing, you will need to split students by ability long before high school, so what would be "Year 1 Math" for one student might be more appropriate as "Year 2 Math" for another. That makes a lot more sense if you have names like "Pre-algebra" and "Algebra". Also having meaningful names will make it easier when they go to college (for those that go).
I don't think lack of realism is the worst thing about these types of problems. I think lack of fun is.
I can recommend an algebra textbook with fun. And it was written by a mathematician with rather better credentials (theorems that bear his name in higher mathematics) than most algebra textbook authors. The book is Algebra by Israel Gelfand and Alexander Shen
Askey's review is actually a review of Ma Liping's book Knowing and Teaching Elementary Mathematics that quotes a passage of Gelfand's book in a sidebar. I later saw a glowing description of Gelfand's books in the same series in a bibliography by a U of Chicago mathematics student.
I use the Gelfand textbook to teach supplemental math lessons for gifted elementary-age students. They LOVE the Gelfand problems. They haven't even gotten to the really funny parts of the book yet, which such sections as "How to Confuse Students on an Exam." Gelfand's book(s) exemplify what you are looking for in math textbooks. Of course they are not used in very many public schools in the United States, nor in very many remedial college classes.
Thank you so much for introducing me to this textbook. I've only read a little sample (what Amazon will let me see when I "look inside"), but I can see it's an improvement over at least 95% of the books I've seen.
The latter proposal is just punting on the problem of what math to teach when. You're going to teach some subset of math first. You might disagree with the particular ordering that is mainstream now, but calling it 'year 1' and 'year 2' just begs the question. Yes, focus on problem solving is good. Yes, these problems are dumb. But people still are going to learn what a cosine is sooner or later, and you have to decide when, and you're probably going to put in the course description that yeah, you learn what a cosine is this year.
I, too, dislike these types of problems, and I refer to them as "math book" problems, because that's the only place you'll ever see them. I mean, if Luigi can paint 1/2 of a room in 3 hours and Mario can paint 1/4 of a room in 2 hours, the way you find out in the real world how fast they can paint the room together is to throw them in the freaking room with two sets of painting equipment and let them go to it. If it's not going fast enough for you, then you go find Mario's friend Bowser and hire him to help them.
But, I don't think lack of realism is the worst thing about these types of problems. I think lack of fun is. I have a graduate degree in mathematics, and I frequently work on problems that have little connection to anything (so far as I know) in the real world, just because it's fun for me to do so. To me, the real problem with high school algebra textbooks is they're so goddamn boring -- and I'm saying this as someone who really, really likes math.
If I had my way, there wouldn't be such a thing as "algebra 1", "algebra 2," "trigonometry," etc. as high school math courses. I'd teach "Year 1 math," "year 2 math," and so on, and put the focus on problem solving rather than any one particular corner of math. If the problems are interesting, the students will do the work and love it. I've seen it happen. I can't imagine that such a problem-solving based course would prepare students for either college or the real world any worse than what we're currently doing.