Walking is one of the things I teach and I don't teach it the way the yoga teachers in this article do. I'm guessing that their material comes from Laban or Alexander Techniques, although neither gets a mention in the article.
There are many different ways to walk. The idea that there is one correct way is bogus. Just consider two possibilities.
First we want our walk to be super efficient. How would we test to see if our walking technique was up to snuff? Walk a very long distance after fasting for several days. If you aren't totally wiped out, your technique passes.
Second consider that we want our walk to create maximum circulation. The test? Walk around naked in the snow for an hour or so.
Even with out an extreme test, I think the walking method they are using is good enough for pounding the pavement in NYC.
In my experience very few people actually take the time to shift their weight forward when walking up stairs. Of course, when people become old, they have to teach themselves this if they want to keep using stairs, but it can be learned at any age, and vastly improves muscle quality and balance.
The description they give in the NYT of correct walking has five steps(pun):
1. Bones stacked. Your body should fall in a straight line — ears, shoulders, hips and ankles.
2. Buttocks released. Unless you’re going uphill, your derriere should have no active role in standing or walking.
3. Legs back. Set the muscles of your inner thighs back to allow the legs to fall directly underneath the pelvis. This will keep the pelvis level and the spine balanced.
4. Belly strong. The stomach should be strong and the middle back filled out so that you are using your entire core as you walk.
5. Head lifted. Imagine someone is pulling a string from the back of your head, allowing the chin to fall to level (most of us walk with it slightly raised) and the throat to soften.
I disagree with 3, 4 and 5. I teach: Putting the legs forward, connected as if you could pull them up to your chest at anytime. Totally relaxing the abdominal muscles (forget that core nonsense and just use your whole body!). And lift the head from the dantain, not as a separate disconnected bobble.
But whatever method a person uses, the practice of slowing down and examining ones walk has the potential to create profound positive change (as does anything you do everyday).