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 am reminded of a talk I attended in college about (elementary-level) math education. One of the first intro problems to estimation was something like 27+75, and the presenter showed us a (real) dialogue where the student gave the exact answer, and the teacher had to "correct" them to the estimate, and then proceeded to talk about how this wasn't exactly correct. The teacher's guiding the student through the "proper" reasoning was convoluted, and not effective teaching. The presenter concluded that if you teach estimation, you should use an example like 209385324579+394875293745, where estimation is actually faster and easier than exact arithmetic.
Now, what real-life example demands systems of equations? I suspect none, until you get into industrial applications. Therefore, maybe you should ask what the kid wants to be when he grows up, and create an example in that setting.
My wife painted our porch floor in sort of an argyle pattern with a border. That takes two equations and two variables to work out a pleasingly proportioned pattern that also meets the border in a uniform way.
Being able to brute-force mentally these kind of vans/cars problems is a useful skill in life, but it's not going to be learned by solving systems of equations.
Actually, if you fudged the numbers a bit, you'd end up with equations that yield non-integer solutions, which you'd then have to play with in order to get an acceptable solution in real life.
As soon as I read that problem my brain went on instinct and quickly calculated you need 3 vans and 2 cars. And somehow I'd skipped over the part that implies you need the number of vehicles to total five.
But I guess I'm just wired like that. And one thing I'm getting out of this how-do-we-teach-math kerfuffle is, people who are "just wired like that" think differently from the people who most need to learn math. Paul Lockhart would advocate dropping the pretense of real-world relevance altogether: "Suppose I were thinking of groups of five and seven things (as I very often am). How would I divide 31 things into groups of five and seven with none left over? It's the beauty of solutions to problems like this, which exist purely of and in the imagination, that we are denying today's children!"
But, 3 vans and 2 cars isn't the only valid solution, unless you assume every vehicle has to be full. Four vans and 1 car works, and a previous poster gave the example of 5 vans, which also works. The fact that there are multiple valid solutions when you remove the constraint that each vehicle should be as full as possible makes me think this is a lousy example for illustrating systems of equations. It is, however, a great problem for teaching combinatorics, since, for each solution, you can count the number of ways to fill the vehicles, and thus the total number of distinct ways to transport the 31 people.
Might this be titled as: "Why Students Hate Word Problems?"
The OP's argument is that common word problems in high-school textbooks are very contrived and that it problems made to be less contrived and more like what a student might expect to experience they might have more success in keeping their interest.
I'm not so sure of this conclusion. Making examples that aren't contrived but still illustrative and simple enough to retain their didactic value is very difficult, especially if you are attempting to make it relatable to teenagers.
However, making contrived examples, even if illustrative, does make students think math is pointless and detestable. Many high school students, in my experience, really do develop a belief that they dislike math because they are led to believe that math is only good for things as boring, useless, and contrived as solving a linear system to figure out how many cars and vans to bring.
In this case there are even many other simpler and faster solutions compared to the proposed one. The obvious solution if the problem actually occurred in reality, barring other unmentioned limitations, is to take five vans. In many math courses, if a student given that problem during the lesson on linear systems were to suggest taking five vans, the teacher would mark the answer wrong. Seeking the simplest solution involving the fewest and simplest steps(and thus fewest opportunities to make a mistake) is a virtue in mathematics. Is not a shorter and simpler proof of the same theorem usually the most well-respected amongst mathematicians? Likewise should our mathematics education teach us to seek the shortest and simplest route to a correct solution.
I hated math. Absolutely loathed it. I had a terrible algebra teacher (8th grade) that left me a gap I never truly closed until college.
However, the 'lights' finally came on in my senior year of high school. I took an experimental trigonometry course, instead of calculus. This course focused on the application of trigonometry to real life problems.
It was by far the best class I've ever had, and it opened me up to understanding the application of ALL of the math that I had been taking for 12 years. It was truly seminal to me. That I had gone 11 years without ever being taught how to actually apply those principles is, in retrospect, mind boggling.
I'm still in education, doing my A levels - and I experience questions of a fixed form all the time.
Now this seems quite good in principle. In lessons you learn about the same ladder against the same wall or whatever, with very slightly changed values each time. Then you do the past papers - and it has the same problems - so you end up with a nice qualification at the end.
The problem with this is what I experienced today. The examiners changed the exam - not massively - just enough so that the standard solutions people had learned became worthless.
This is actually a useful word problem, they only left out one part that would make it more practical: cars cost $50 to rent, vans cost $100 to rent. Now you've got a reason to prefer cars over vans.
I (and in my experience most students feel the same) always liked learning better when it was about concrete problems and not just an abstract operation. If I'd had to learn programming without real-world applications, I wouldn't have bothered.
I explain this by pointing out that physics has theoreticians and experimentalists. Seems to me that's about mental style. Also, it's easier to communicate an idea when you've got a physical metaphor.
It's funny, algebra is when I first started liking math. I hated arithmetic and got poor grades because for some reason I could not memorize addition and multiplication facts very well so I was slow on the tests.
With algebra I started to see that I could actually do something useful with math. It became more about thinking than memorization. Yeah a lot of the problems were contrived but at least it was more interesting than long division.
I always liked math for the sake of math. Math that has "real-world" applications is boring.
(Of course, Calculus, number theory, and differential equations have lots of real world applications, but the "real world" that these apply to is not "daily life", and is interesting as a result.)
Why on earth would we want to "waste" the time of teachers and students on such un-"realisitic" problems? Its not like they would have to solve the usary equations in the fine print in a bank contract in the future ... and if we ever need people who have encountered systems of two (yikes!) equations, I'm sure we can outsource it to India and China and elsewhere.
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.