The previous post (Articles I wish I never finished reading, Dept) stimulated several insightful and useful comments which I am reproducing here. The first, from Greg Ashman, a PhD candidate in education who resides in Australia, and is outspoken on the issue of the ineffectiveness of constructivist teaching strategies, commented at the article itself. He writes:

A few points to note:

Finland is declining in its PISA maths performance and has been doing so since 2006. If we therefore wish to look to Finland as a model then we probably need to examine what it was doing prior to 2006. If we do so then we see it was pretty traditional: Reference.Teaching for conceptual understanding is actually quite a strong theme in American maths education and not something that has been ignored. A comparative study of the attitudes of mathematics teachers in different countries found that American teachers prioritised the idea of understanding-first more than teachers in East Asia. East Asian teachers still thought understanding was important but were relaxed about whether it came before or after a knowledge of procedures (East Asian countries do quite well in international tests): Reference.When teachers prioritise understanding-first then they are often drawn to ‘constructivist’ pedagogies where students are asked to design their own strategies for solving problems rather than learn canonical ones. These pedagogies are ineffective: Reference.Canonical procedures have advantages over student invented procedures, particularly for more complex work: ReferenceJon Star has done a great deal of work on the fact that what we often call ‘procedural knowledge’ actually contains a great deal of conceptual understanding. Procedural and conceptual knowledge are bidirectional and iterative: Reference

Far from a challenge to the prevailing orthodoxy, this opinion piece is a good expression of it. It is thus part of the problem.

And one of our readers, Chester Draws, has this to say about the belief (or poster child of the progressivists/math reformers) that the distributive rule leads students to a lack of understanding in solving certain algebra problems. Specifically, he is referring to this part of the previous post: “When they see an equation such as 3(x+5)=30, they will distribute rather than divide both sides by 3 to get the simpler equation of x + 5 = 10. The author claims ‘but a child who learned the distribution method might have great difficulty recognizing the alternate method – or even that both procedures are equally correct.’ ”

Chester comments:

But both procedures are not equally useful.

If you have a standard 14 year old and you give them 7(x + 5) = 30, then they are going to stuff it up if they don’t distribute first. x + 5 = 30/7 is not something you want them to be attempting. Whereas 7x + 35 = 30 produces no such problems, despite yielding the same fractional negative.

The divide first method is usually more difficult and much more prone to error. Why would you even want them to know about it? I have nothing but derision for teachers that show students methods that aren’t universally applicable so they have “choice”. They don’t need or want choice, they need and want to get the answers correct with a reliable method. The time lost teaching a trivially useful technique would be much better spent getting the ones that they do need properly organised in their heads.

I teach all my students that normally the first thing you do in any algebra solving problem is get rid of fractions and brackets. Then you can see what you have. They then have their minds freed of what to do first — remove brackets and fractions — and that leaves more brain power for the hard bits.

Finally, we have SteveH, addressing this same issue:

In a traditional approach to algebra, you learn that there is no one way to solve anything, even though pedagogues really push ideas of order of operations. Learning this is not an understanding issue. It’s a practice issue, where mastery of problem sets give you plenty of chances to solve problems in different ways. Practice for SAT also teaches you to look for tricks and short cuts, but that is neither necessary or sufficient. Practice, practice, practice on problem sets is the solution. That level of understanding is only driven by individual practice on problem variations, not transference of a few in-class group projects covering general ideas. Words are not understanding.

There are ways to talk and provide proper and more abstract algebraic understandings, but most of these rote pedagogues don’t have a clue. In the end, the only way to create proper understandings is via lots of individual practice on problem sets. Practice is not just about speed.