First, I need to vent about the awful first sentence:
A common view is that students learn maths best when teachers give clear explanations of mathematical concepts, usually in isolation from other concepts, and students are then given opportunities to practise what they have been shown.
I agree that clear explanations and practice are important to learning maths. I consider myself a more traditional teacher who uses worked-examples and leading questions to develop mathematical concepts. However, my beef comes from the phrase "usually in isolation from other concepts." Why would any teacher agree that maths is best learned through isolated ideas?
For example, in my maths class for pre-service teachers, we recently discussed the number of divisors a number has. We considered two small example first: 12 and 16.
12 = 2^2 x 3 and 16 = 2^4
12 has 6 divisors (1,2,3,4,6,12) and 16 has 5 divisors (1,2,4,8,16)
Then I asked "Is there any connection to the number of divisors and the prime factorization?" A difficult question, so I allowed them to ponder a bit. We eventually connected the number of divisors to the powers of the primes in the factorization through addition of one. For example, 12 has (2+1)(1+1)=6 divisors and 16 has (4+1)=5 divisors. We discussed why this always works by considering the combinations of the powers, then made a generalization so we could find the number of divisors for larger numbers. And voilà, the concept of prime factorization has been connected to the concept of divisors. I don't believe that good mathematics teaching happens inside boxes.
So why does the author set up this idea that individuals believe that concepts are taught in isolation? Well, it is a straw man that leads into the research that he is doing on open-ended questions - the way I think you are teaching maths is incorrect, here is the way I am teaching maths. In case you missed my earlier post on poorly-designed open-ended questions, you can revisit it here. Essentially, the way he is teaching mathematics is by using open-ended questions such as
"The minute hand of a clock is on 2 and the hands make an acute angle. What might be the time?"
or
"What are some times for which the hands on a clock make a right angle?"
The main issue I have with open-ended questions is that they are often poorly thought out. For example, if we consider the first question, what time signifies a degree value of zero between the two hands (we need some kind of reference point)? Is it at 12:00? If 12:00 is zero degrees, then technically, any time after 12:15 is obtuse because we need that initial time as a reference point [here, I am assuming that we are measuring angles in a clockwise fashions, rather than the usual anti-clockwise fashion]. However, I have assumed that the hour hand doesn't move (see next paragraph)! So if we assume the hour hand moves discretely, and not continuously, then 12:15 fits the bill for 90 degrees, making any angle bigger then this obtuse. This means any time after 12:30 must be a reflex angle! So there are no acute angles if the hour hand is on 2. This question isn't mathematically sound until more information is added. Questions similar to this may develop misconceptions in your students: when given a reflex angle, they may measure the incorrect portion or fail to add the multiples of 360 degrees.
In our second question, again using 12:00 as a starting point of zero degrees, we might consider that 12:15 must be the 90 degree angle. But again, this technically isn't the case since the hour hand will have moved one-fourth of the way to 1 by the time we get to 12:15, so the hands at 12:15 are actually a little less than 90 degrees. (If interested, you can show that the time the hands will make 90 degrees is actually 12:16 and 4/11ths of a minute). Because of the argument above, this will be the only time we have a 90 degree angle! All others will be 90 + 180k degrees (k is a positive integer). Again, this open-ended question may result in the misconception that the hour hand doesn't move at all.
This article adds to my belief that open-ended questions are poor ways to teach concepts to students who are not well-versed in the content material, since poorly-designed questions may result in mathematical (or real-life) misconceptions. If these misconceptions occur during the vital younger ages and are stored as memories, it can take a significant amount of energy and time to remove these misconceptions from the long-term memory and rework in the correct concepts. The easiest solution: don't use open-ended questions, or use them sparingly.