We have all at one point or another learned some variation of a mathematical formula involving trains and their timetables. For example: if a train leaves Boston for New York at 7am and travels at 60mph, will it beat a train leaving Providence at 6am traveling 45mph? The idea behind this kind of "story" problem is to engage a student with a real-world example to which they can relate. The thinking follows that that engagement will solidify the mathematical concept. It's one of those conceits that has hung around for seemingly as long as math has been taught. And it may very well be completely wrong.
Researchers at Ohio State University have set out to determine whether or not these kinds of examples reinforce the math behind them. They conducted a study in which college students were taught a simple but unfamiliar system. Some learned it through abstract symbols and some learned it through concrete examples. They were then tested on what they were told was a children's game, the rules of which used the same math as they were just taught.
Those who had learned the system through symbols did well in figuring out the game; those who had learned through examples did no better than if they were completely guessing. The experiment proved the researcher's hypothesis that real-world examples tend to distract from the math and that students are likely to have difficulty transferring their knowledge to new problems without a purely abstract foundation. They are now turning their attention to younger learners to see just how pervasive the effect can be.
Via NY Times
A train leaves the station at 5:00 due east going 55mph.
Another train leaves a station 600 miles away heading due west at 4:30 going 40 mph.
How does this make you feel?
I actually beg to differ with the article. By giving the students specific concrete examples instead of general equations; though of course they will grasp the concept less often and less strongly, it forces the students to go through a process of abstraction and generalization to induce the general formula. To be able to see several instances of something and then infer a pattern, common thread, or generally applicable method of solution is an invaluable skill, and of certainly more value than the specific material itself.
This is my favorite thing to do - abstraction - I remember there were once a lot of clock problems we had to do in geometry, determining what the angles between the hands were - it seemed to me kind of boring that we had so many - and i worked really hard to find a formula that based simply on the time would find the angle, and it was awesome. Did the same thing for all the different basic kinematics problems involving angles initial or final values of velocity, etc. This abstraction is (for me) the most terrific thing about learning, and necessary to infer things about life as well.
By giving the students specific examples and then testing them on their ability to apply inferred methods of solving them to new specific situations rewards the students with great abilities, and also teaches and reinforces those induction abilities. Unfortunately, as it would seem obvious, fewer students do as well when they are expected to infer the process on their own.
The best method of teaching is probably a hybrid, where one starts by showing several examples and allow the students to try to solve slightly altered examples based on their knowledge and possible inferences. Then, eventually, show them the general formulae.
Another valuable skill is simply the ability to relate math to real life. Numerous times, I've heard students ask teachers 'what is this good for?' in math, and these problems help to answer that. A connection to real life is important to those who don't appreciate math on its own.
But this finding is very interesting. Maybe there should be a bigger focus on abstraction to begin with? It does make sense to me learn the concepts in as efficient a form as possible, and then learn to apply them. We should probably make sure the underlying math is understood before complicating it (and thus making it harder to understand) by relating it to real life.
I hated this example on every standardized test I ever took. I only learned it by eliminating the train metaphor, which was - get this - distracting to solving the problem.
Real world metaphor examples make the assumption that abstract concepts need to be concretized in corporeal form to be obvious. I think we vastly underestimate student's ability to grasp abstract concepts.
Turns out that knowing the approach to how to solve a problem is more important than knowing how to solve a particular problem. Here's what I ran into in school: busy work. Smart kids want to learn how to solve problems, learn the approach, and move on, which is how they'll function in the real world - learn what matters, and what they're interested in with passion - ignore the rest. The system ignores that, and instead focuses on preparing "students for the real world," which is complete and utter crap, because I can honestly say after 15 years in the real world I have used - at most - 10% of the math and science skills I learned in school. And I work in technology. EVERYTHING I had to learn I learned on the job, and it was a lot. School was an epic failure as a preparation for the real world, with the exception of writing skills.
The above comment arguing for why these real world examples are needed (and I beg to ask: are you a teacher?) simply demonstrates the reason we're still dealing with these outdated relic problems: they've been on the books a long time. Hybridizing the problem simply ignores the point of this research - that an abstract foundation provides an understanding for solving real world problems.
Why not just focus on teaching kids how to solve problems, rather than making them solve worksheets full of examples nobody cares about? The research shows that this is the best approach.
It's time we started to re-examine the antiqued agriculturally-founded education system that plagues Western countries and stunts our youth.
If I read the article correctly, I am aghast. The 'test' that was performed was
- teach group A the math behind the game
- teach group B by showing several examples of game
I'll bet group B was confused!
This does not test the concept that practical problems reinforce math concepts. It tests a concept that would have you learn math from example real world problems.
So, to learn about addition, subtraction, multiplication, and division -- you would be given a bunch of train/airplane/apple picking problems-- apparently with minimal explanation of the fundamental arithmetic concepts. Then you would be shown the solutions of these specific problems, and be expected to generalize to the math behind them. What!!???
Practical problems show that there might be some reason for all of the goofy math stuff. They also teach you how to set up
the calculations. In real life, people will seldom ask you ( at work or home) --"What is (12x64)/(4^12)?" BUT, you might be called upon to figure out how much wallpaper you need, or how long some new equipment would take to pay for itself.
--It's good to have some experience in setting up problems.
I hope I'm wrong, but I read the Times article, and I fear that I am not. Of course math is taught through abstract symbols.
I'd never heard that there was any debate about that. Practical problems reinforce the math, and demonstrate that someone needs to translate the real world into dry equations.
I give the folks who confucted this study a resounding F-
It amazes me when people look at research and dismiss it because of their own opinion. I would suggest actually getting a copy of the report and reading it instead of an article about the report.
If the research is correct, it says two somewhat conflicting things.
1st. Students learn things better with real world type connections.
2nd Those real world connections interfere with the student transferring their learning to a situation that appears different but actually uses the same math concept.
So which is more important? As a high school math teacher, I go with the second.
yonogro -- I tried to qualify my comments " if I'm right", etc.
I'm commenting on an article -- and I did find as much source as is readily accessible --i.e. the NY Times piece.
These folks are not asking me to review their research. They are presenting a conclusion based on the descriptions that they choose to give. I am challenging that conclusion. Nowhere in the PS stuff or the NY Times stuff did I see any link to the actual report.
I did see comments by some who chose to agree, but who also apparently had not seen the detailed infomation.
The description of the test appears to have been given by those who conducted the test. I disagree with the validity of the test as I understand it.
when it comes to a train beating another, but they both leave from different locations-
what if the student in question doesn't know their geography?
I know where boston and new york are, but where is providence? What one is closer to the destination?
Those are the questions I'd expect to hear if i were the instructor in that class.
The truth is lots of people have lots of problems and you want kids to care about them. Who cares about kid's problems? School is more or less a prison for kids if you dare to investigate with the kids. Yet they are forced to attend.