The first analogy is "Scaffolding" and I like it a lot. The second analogy is a comparison to training for sports and I don't like it much. I'll copy the first one and briefly comment, then the second, and then my short comment on why I don't like the second. There is more to the article that is worth reading.
Scaffolding.Just saying - those construction workers who are building scaffolding know the purpose of it and know how it will be used during construction of the building. Students in math classes seldom know the purpose of the "scaffolding" they are supposed to be building and almost never see any point to most of it and, for MOST of those students, there is no point because they are not going to be using it (most of what is taught in an algebra class, for example) at all for the rest of their lives.When a new building is made, a skeleton of steel struts called the scaffolding is put up first. The workers walk on the scaffolding and use it to hold equipment as they begin the real task of constructing the building. The scaffolding has no use by itself. It would be absurd to just build the scaffolding and then walk away, thinking that something of value has been accomplished.
Yet this is what seems to occur in all too many mathematics classes in high schools. Students learn formulas and how to plug into them. They learn mechanical techniques for solving certain equations or taking derivatives. But all of these things are just the scaffolding. They are necessary and useful, sure, but by themselves they are useless. Doing only the superficial and then thinking that something important has happened is like building only the scaffolding.
The real "building" in the mathematics sense is the true mathematical understanding, the true ability to think, perceive, and analyze mathematically.
Ready for the big play.My problem with the sports comparison is in the conclusion - that math education produces "true mental capacity and the ability to think." First - math "education" doesn't often produce that - more likely the opposite. Math education almost always produces LESS ability to think and more mental confusion and reliance on rote memorization. Second, there are many other ways to develop mental capacity and the ability to think - math is not the only (or necessarily the best) way and certainly not even a good way for everybody. Third, and most importantly, even if learning about mathematics helps people think more logically, there is much more to "true mental capacity and the ability to think," than that. Mathematics does not, for example, help people think more empathically, more globally, more creatively, or more independently.Professional athletes spend hours in gyms working out on equipment of all sorts. Special trainers are hired to advise them on workout schedules. They spend hours running on treadmills. Why do they do that? Are they learning skills necessary for playing their sport, say basketball?
Imagine there're three seconds left in the seventh game of the NBA championship. The score is tied. Time out. The pressure is intense. The coach is huddling with his star players. He says to one, "OK Michael, this is it. You know what to do." And Michael says, "Right coach. Bring in my treadmill!"
Duh! Of course not! But then what was all that treadmill time for? If the treadmill is not seen during the actual game, was it just a waste to use it? Were all those trainers wasting their time? Of course not. It produced (if it was done right!) something of value, namely stamina and aerobic capacity. Those capacities are of enormous value even if they cannot be seen in any immediate sense. So too does mathematics education produce something of value, true mental capacity and the ability to think.