What Is Wrong with How We Are Teaching Math?

Although quite a number of people would be quick to respond, the above question is in fact complex and difficult to answer. There is a tendency to dislike one specific algorithm or way to solve a problem, yet some people unknowingly subscribe to one specific way of teaching children how to do math. Some even go as far as teaching so many ways to do math that not subscribing to this diverse set is now viewed as wrong. Rote learning is frowned upon, but now students need to go through mindless and seemingly endless examples of various ways that learning by drill during my time as a grade school student seems like a walk in the park. For instance, here are five ways to add 47 and 35:

Above image copied from Five Ways to Add Multi-digit Whole Numbers

When I was in grade school, we were drilled only on the first method. One could only imagine the pain and frustration a child experiences if drilled on all five different ways. Practicing several times is certainly not wrong especially on the fundamental skills or concepts. Neil Aggarwal writes the following thoughtful note as an answer to the question, "How are we teaching math wrong?":
I was a seventh- and eighth-grade math teacher.  From my experience, it seems that the basic problem with math ed is the lack of focus on the fundamentals.
Let me give you an example: Most teachers taught 2(x+y) using dolphins. They would draw two dolphins traveling from the 2 to the x and the y. This was meant to indicate that we distribute the 2 to get 2x + 2y.
Teaching this way allows kids to answer that specific construction of problem, integer(variable + variable) = integer * variable + integer * variable, but not much else. What happens when the kid sees (x+y)2 or (x+2)(y+2)? The dolphins can only allow a child to guess at what to do in these new circumstances.
Rather, if the teacher had taught that 2(x+y) = (x+y) + (x+y), therefore giving the students some insight as to how and why the distributive property works, then kids might be more able to approach new circumstances and prevail by force of logic rather than speculation.
A student facing a new construct—say (x+2)(y+2)—has a shot at realizing that the (x+2) can be seen as its own number, thus (x+2)y + (x+2)2. The kid didn't have a shot with the dolphins.
Moreover, this drilling approach that is commonplace in schools is very inefficient. Think about it: The above two examples take up the better part of two quarters over the course of two years in most schools using the drilling method. Is that really necessary? And imagine you are the student. How boring! Three to four months of what?
The drilling approach requires that the teacher drill almost every new circumstance with the same ferocity as the first circumstance. Rather, when teachers focus on fundamentals, new circumstances still need to be taught, but usually not drilled to the same extent.
Perhaps the worst consequence of the drilling paradigm is that students and adults have no ability to use the math they learned in school outside of the boxes within which they were drilled. The value of math is in its predictive powers. You combine math with economics, or math with biology, or math with physics, even with law (see Coase) etc., and suddenly you can predict the future. But these uses of math require extremely strong fundamentals, which most people were never taught.
The National Council of Teachers in Mathematics (NCTM) in the United States recently summarized what is wrong with math education:
Too much focus is on learning procedures without any connection to meaning, understanding, or the applications that require these procedures.
◆ Too many students are limited by the lower expectations and narrower curricula of remedial tracks from which few ever emerge.
◆ Too many teachers have limited access to the instructional materials, tools, and technology that they need. 
◆ Too much weight is placed on results from assessments—particularly large-scale, high-stakes assessments—that emphasize skills and fact recall and fail to give sufficient attention to problem solving and reasoning.
◆ Too many teachers of mathematics remain professionally isolated, without the benefits of collaborative structures and coaching, and with inadequate opportunities for professional development related to mathematics teaching and learning.
As a result, too few students—especially those from traditionally underrepresented groups—are attaining high levels of mathematics learning.
NCTM undoubtedly sees teachers as major factors behind quality in math education. NCTM sees that not equipping teachers with the resources and support they need makes it impossible for children to receive good education in math. The problem, however, goes far beyond resources. Even with all the resources, teachers could still fail if the goals are either unclear or misunderstood. The Common Core provides guidelines on what math education should be. Sadly, teachers, textbooks and tests seem totally unprepared to realize these guidelines.
The teaching channel provides a video showcasing four teachers who have started implementing the Common Core. Their examples, in my opinion, provide the necessary first step. Teaching is responsive - It is after all a relationship between two individuals, the teacher and the pupil. The five different ways to add 47 and 35 shown near the top of this post are surely correct methods of addition. It is up to the teacher and student to choose what works best....
To watch this video, visit Teaching Math to the Core