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.
traditional math 
“She crows with delight at the insights her students find.”
I am endlessly disgusted with the use of such stories by educators. Have they no sense of logic or propriety? They speak in glowing terms of education “for all” but take 2 stories from a class of 22 kids and hold them up as representative samples of how the entire class responds.
But you know … you KNOW … that (here I am assuming that the anecdotes are real, honestly related and unembellished — themselves enormous assumptions, given that someone is trying to win a debate and knows that nobody can check the “facts” they are relating…) these “samples” are by no means representative. When you’re trying to win an argument that such-and-such has a positive effect, you don’t pick the worst case. And you don’t pick the average case. You pick the superstars — the outliers that lie far above the curve, and you try to fast-peddle these as representatives, hoping nobody sees the sleight-of-hand.
Give me access to your class, ma’am. I’d like to ask some penetrating questions, not of the two bright-eyed lassies bouncing in their seats, vibrating with eagerness to show their smarts, with their hands raised to the sky, but of the five at the table at the back, sitting on their hands with a puzzled look on their faces, waiting for someone else to speak and hoping it gives them some clue as to what they should be saying.
By all means tell me about your brilliant stars. But don’t tell me that your point must be conceded because you’re able to elevate one or two kids with your stories to the ranks of wunderkind-hood. Let us see how the average, and below-average, kids are doing in your class. Let us see your scores. Let us see how they perform in real, standard tasks over which you have less control, and which are geared to determine their actual mastery of content.
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