Maggie's Farm

I made a fortune as an eighth grader tutoring my brother's 5th grade friends in the New Math.
In high school I took "Illinois Math," the creation of Max Beberman. I loved it, as it had a lot of proofs. Before Illinois Math, I was indifferent to math. As a result of Illinois Math, I love math. Unfortunately, many less able students couldn't hack the proof writing that Illinois Math required. It isn't for everyone.
There were a lot of problems in taking Illinois Math/New Math down to the lower grades. A lot of elementary school teachers thought that Illinois Math/New Math meant that basic arithmetic skills-addition, subtraction, multiplication, and division - should be downgraded.

Years later I took a math course with a professor who knew Max Beberman. In a conversation after class, the professor said that Max told him that he had never intended that Illinois Math resulted in shunting basic arithmetic skills to the wayside. Why did so many elementary and junior high teachers think this was the case? Where did this misinterpretation come from?

IMHO, it came from the fact that a very high proportion of elementary school teachers are not very good at math. When Illinois Math/New Math was thrown at them, concepts which required not less math but more math, the elementary school teachers were overwhelmed, and did less math. Which meant the downgrading of basis arithmetic skills.

The presence of occasional errors is deemed less important than the overall thought process.

I tend to agree with that. You apparently do not. I have seen high school physics students write out just the numeric answer, and not the equation. This is the default result of having calculators. It is MUCH more important to write out the equation, do the process thingy, even if there is an arithmetic error, than to write down only the "correct number."

You do the process thingy, write out the equations, you will understand it a lot more in the long run. This is especially important when students have calculators, and punch in numbers, without always thinking where the numbers came from.

At the same time, I am not of the opinion that one can do well in high school math and beyond if one does not have a good grasp of basic multiplication and division.

The NEW MATH courses I had in high school back in the 60's, where I had to do a lot of work in proving various mathematical principles such as the distributive principle, were of later assistance in estimating, which is a very useful skill in later life. Due to work in the distributive principle et al, I was able to see that 370X39 is ~400X40 less 10%. But one cannot estimate well if one doesn't have a basic grasp of multiplication and division.

With Khan Academy, one can bypass one's own bad teacher and learn from a good teacher- in this case an online teacher. My 9th grade math teacher, in spite of being a Phi Beta Kappa math graduate, was a horrible teacher. I ignored her, and taught myself from the Illinois Math textbook, which was a very good text. In a sense, the textbook was my equivalent of Khan Academy, decades earlier.

Part of the "right answer versus "process" argument is on what level you are at. At the elementary level, doing basic addition, multiplication, subtraction, and division, I see getting the right answer as very important. It indicates that you have internalized the computational process the teacher has outlined. Or you have made your own process.

Before leaving elementary school a student really does need to have an automatic understanding that 8X7=56. Among other things. By whatever method.

At the same time, teachers at the elementary level should be open to students finding answers by their own methods. As the saying goes, there is more than one way to skin a cat.

One problem with math education is that educators have tried to introduce snippets of interesting mathematical information at the lower levels, such as tiling, when it is more important for more basic skills to be addressed, such as competence in fractions and decimals.

The process is more important higher up the ladder. There is a reason why science and engineering professors will take off only 10-15% if the equation is all written out, and there is a multiplication error. [Orders of magnitude errors should be taken more seriously.]

Re relevance. I have read that there are children in Brazil who work as/with small vendors who have developed excellent computational skills in their head- a necessary skill as a small vendor. When in school, they have problems. Go figure.

Another point about relevance. Word problems are an attempt to make math relevant, and many students have a big problem with them.

No simple answer.