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Both of these examples are 6th grade. By discussion, finding out how and what a child thinks can help get them going in the direction you want them to go, or get them turned around if they are headed the wrong way.

A simple, progressive math test will do the same without the embarassment of showing ones' lack of knowledge (or abundance of knowledge which can be frowned upon by peers) in front of the class. I still don't see what there is to discuss. If they class can multiply and divide, then move on to bigger things.

Thinking and reasoning are very important in learning math. Math is not always about practicality. If so, use a calculator. Funtionality is based on thinking and reasoning and applying what you have learned.

I would never argue that thinking and reasoning aren't important in math. But, if a problem doesn't have a purpose or goal, it is useless. And if students find it useless, they probably won't work with it. Finding one of an infinate number of answers to a question, to me, has no goal. It sounds more like "Dr. Spock meets Pythagerus". :)

Math is changing - in some good ways and some not so good ways. My first year to teach was when "new math" started. The texts were new, and I had not had any of it in college - (like graphing inequalities on a number line). My ex tried to help our first grade son, and could not do the math - <, > were new in the texts. Now things are getting moved down further in the curriculum, and basics are getting less emphasis. Manipulatives were new 20 years ago in the US (40 in Europe), and are rarely used now. There are times when they are very beneficial, but teachers tend to teach how they were taught. Many consider them too time consuming.

Math hasn't changed a bit; the methods of teaching have. Oh granted, we do more digital now, but the quadratic equation hasn't changed and the only thing different with pi is how many decimal places it has been successfully been carried out.

I beleive that the changes in teaching methods have shown up negatively in the results on standardized testing over the last 40 years. This is particularly true on tests that haven't been dummied down since the late 1950's.

To be honest, I don't blame this entirely on teaching methods; I put most of the blame on social promotion and other feel-good claptrap that serves only to lower the standard of the class to its weakest link.

Finding certified teachers is hard, whether math or otherwise. There are too many who are ill-prepared. I blame a lot of this on universities. The profs that can do research get more time and money. Those that can teach get run off. Teachers are not highly valued in our society. Discipline is becoming non-existant in our schools. All of this leads to kids learning less.

I can understand that. Putting up with kids and their parents without the ability to flunk the ones that don't make the cut and without the ability to expel the troublemakers is probably a big reason. Why enter a profession in which you seemingly cannot do anything right?

Math has changed - not just teaching methods. I gave you two examples. Another is fractals -Benoit Mandelbrot, a mathematician at IBM, is an expert in processes with unusual statistical properties, such as those in which a random variable's average or its variance is infinite. His early work in the 1950's and 1960's suggested that the variations in stock market prices, the probabilities of words in English, and the fluctuations in turbulent fluids, might be modeled by such strange processes. Later he came to study the geometric features of these processes and realized that one unifying aspect was their self-similarity. In the mid-1970s he coined the word "fractal" as a label for the underlying objects, since they had fractional dimensions. Fractals are shapes or behaviors that have similar properties at all levels of magnification or across all times. Just as the sphere is a concept that unites raindrops, basketballs, and Mars, so fractals are a concept that unites clouds, coastlines, plants, and chaotic attractors. .

The field of statistics has also changed - box-and-whisker plots, stem and leaf plots, scatter plots, etc. Math is not static.