A math teacher who is teaching in Rome and writes a blog which has an impenetrable “I know a hell of a lot more about this than you” attitude to it, has written about the inferiority of “direct instruction” approaches.
In doing so, she mischaracterizes direct instruction as “The formula is provided (Area= length x width), and countless problems follow – usually, after having been modeled by the teacher. Block practice again, and not much understanding – just computation with little relevance.”
There is such a thing as scaffolded problems that have some variation so that students must think beyond the initial worked example. Using her example of finding the perimeter of a rectangle with sides of 10 and 2, one could have problems such as “A rectangle has a length of 8. If the perimeter is 24, what is the width?” This is a variation of the initial worked example. Subsequent problems can lead up to what she believes is the “holy grail” of teaching: “open-ended problems”. After a few such problems, students may then be ready for: “What possible lengths and widths can a rectangle have so that the perimeter equals 24?”
Instead, she likes to lead with that, and direct students to many different types of questions and problems. She crows with delight at the insights her students find. But in the end, what is it that the 4th graders take away? For many if not most of them, it will be like going to a shopping mall and being confronted with displays and stores all at once. Visitors to shopping malls are frequently overwhelmed and sometimes forget what they came to buy. The blog author (Christina) has the advantage of being an adult and having thought about the various things she knows about perimeter and area. She believes that her “investigations” and probing questions open their minds, and create schemas of mathematical truths. But for these young students, it is more likely to be a hodge podge of information they are left to sort through.
Lastly, she refers to a similar discussion by Robert Kaplinsky who thinks that asking students to find the relationship between perimeter and area is a good exercise. This type of problem belongs in a calculus optimization unit. Or at best an advanced high school algebra course that deals with solutions to cubics.
This is nothing more than a pointless elementary desk-work exercise that addresses no outcomes more advanced than understanding a couple of formula and doing a lot of low-level arithmetic, then sorting through options. Taken by itself, this exercise yields no interesting insights for students. And it begs one to ask, what if instead of 20 square units it was 3072 square units? Or a million? Are students asked to believe that exhausting options and picking the largest number is a reasonable way to tackle this problem? What analytics will lead them to the key insight that will help them find an appropriate general attack on it?
What is sad is that people like this blogger and others gain quite a following of believers and become thought leaders in a world where students need good, solid, scaffolded problems. Instead they are dumped in a shopping mall and told to make lists of their insights–which are soon forgotten.