[kscaldef]

Step 1 is figuring out why someone would want, or need, to solve that equation. He's by no means saying that students don't have to learn how to do algebraic manipulations. He's saying that without motivating the process and letting students understand the process that leads to the algorithm, they aren't really going to learn how to use math in a way that will be helpful to them.

I suspect that he'd also argue that an hour spent slowly and carefully exploring 1 problem like this is better than having them do 30 examples with no context and no motivation.

[yequalsx]

Do they really learn such motivation from solving word problems? Especially when many word problems are not practical or even correct?

How about this for motivation, we solve such equations because we can. Solving equations is useful and the more equations we can solve the better.

It's not really motivating when we give a word problem whose model is a radical equation and then say solve the equation. Most equations can't be solved by algebraic methods and anyone who really needs to solve an equation and trust the answer is better off having a computer do the computation.

In solving the 30 problems one might get to a point in understanding why sqrt(2x+1) = sqrt(x) + 1 is slightly harder than solving sqrt(x+1) = sqrt(x) + 1. In solving 30 problems one might get to a point to discover why sqrt equations can be solved but why we don't solve cube root equations. Or why sqrt(2x+3) = x is fundamentally easier than sqrt(2x+3) = x + 1. You won't get this from solving word problems.

[kscaldef]

Your second paragraph contradicts itself. You say solving equations is useful. So, why not show your students the use?

[yequalsx]

Having the ability to solve equations is useful. Randomly write down an equation. That particular equation isn't likely to have practical applications. Is it worthwhile to explore whether or not it can be solved by algebraic methods?

Why is it that first, second, third, and fourth degree polynomials can be solved by algebraic methods but not arbitrary polynomials of higher degree? Is this worth studying only if it has practical applications? It does have applications today but it didn't for the first thousand years this problem was tackled. The requirement that something be 'useful' before it is worthy to be studied is too great a burden. It's a detriment to intellectualism.