From the UW College of Education comes the very attractive publication linked from the title above. It addresses the "math wars" between "traditionalists" and advocates of "modern math education". Of course, like many political debates, they use titles such as those to pre-dispose their readers to see things their way. This is a sign right off the bat that this is not a scholarly work, but a political argument. Other signs are straw-man examples that are supposed to show how "traditional" math is misguided, such as this (p. 9):
A toy is hidden in one of two cakes. One cake is a circle, cut into fourths. The other is a rectangle, cut into sixths. Students must choose the cake that gives them the best chance of finding the toy.Of course this is a poor question for students who would answer that way, because it is varying two things at a time (cake shape -- and therefore slice shape -- and fractions of total area). Presumably, this question is trying to get at more abstract thinking; that the shape of the cake and its slices doesn't matter, all that matters is fraction of total area. But the publication doesn't say anything about this, all it does is use this as a straw man to set up the argument that we shouldn't tell students how to do things (like optimal methods for mathematical calculations). All methods are equally valid:
Some choose the rectangle. Why? Because “most toys come in square boxes.”
One student may add 28 + 34 with traditional column carryover. Another adds 2 to 28 and subtracts 2 from 34 before adding the two results. A third student adds 8 and 4 to make 12, then 12 and 30 to make 42, and 20 more to make 62. In an effective classroom, all those solutions are studied, the links between them established, and the connection made to larger mathematical concepts (such as place value, the properties of addition, and developing generalized strategies).This sounds very nice until you consider, "How did these students all arrive at different methods for addition?" The answer is that they weren't taught how to add; they were expected to "discover" it themselves. Go read a history of mathematics book sometime and consider how long humanity has worked to discover what we know about mathematics; how many geniuses have been involved. Does it make sense to systematically (not as an occasional teaching device) expect children to re-create any fraction of this? And is the only way to teach about place value, etc. to compare multiple methods?
Oh, and the implication is that "traditionalists" teach by giving out problems and just marking them right or wrong and "modernists" look at student mistakes and seek to understand why they make them. Nice false dichotomy.
I find this anecdote on page 13 especially interesting. One of the UW Education faculty has spent time observing engineers, scientists, and architects working, and here are his conclusions:
The architects, he discovered, worked problems out with visuals, not textbook algorithms. Engineers use mathematics, but much of that is embedded in their computational tools, and they too use forms of quantitative reasoning that looked very different from the activities of school math. It turned out that school math was a fairly rare species of activity outside of school.As the brochure continues, the distinction is between "school math" and the math that people actually use in the real world. Well, except for mathematicians, who are like poets, viewing the world in a different way than most people. Apparently, engineers don't need to know math; it's already in the computers (how it got there is unanswered). It's unfortunate that engineering schools and the accreditation folks require math through differential equations, multivariate calculus, etc. They must not know what engineers do as well as UW Education folks.
“If you spend a month with architects, you’ll never once see them write an equation,” says Stevens.
The story was the same when he studied roadway engineers. “All the calculations were done on the computer,” says Stevens.