What are "Basic Skills" in Math?

You know - WHY does it always have to be one or the other? Why can it never be both? Why do we always have to insist on throwing out the baby with the bath water?

Because I'll tell you something. I've got two kids - one has a naturally inclined math brain and one doesn't. And the one who doesn't has made it almost all the way through school- and I'm talking about good schools in two different developed countries- and she can't add fractions.
Why? I guess because these days it would have been looked at as 'cruel and unusual punishment' to make her learn them- along with memorizing her times tables.

You are making the assumption, without any proof, that the exercises do any good.

There is no reason to believe that these rote exercises do any good for helping people gain the critical thinking or reasoning skills that are at the heart of mathematics.

Our current education system is outdated. It was designed during the industrial revolution where we didn't need people who could reason and think. We needed great numbers of people who could work on assembly lines, meaning they could follow instructions and do rote tasks without asking questions.

Your story about you daughter doesn't support your point. It is clear that she needed attention and special help. The question is what type of help would be helpful. Fractions represent an idea. They are best taught by teaching the idea behind them, we use manipulatives to teach factions as part of a whole. Then we use real, every day problems to show fractions as numbers. Then there are all sorts of ways to address the idea between finding a common denominator.

I found a lot of time was wasted on mathematical exceptions because these were interesting to the teacher.

I found a lot of time was wasted on mathematical exceptions because these were interesting to the teacher.

ebrown p wrote:

I have never needed to know my multiplication tables. When you understand what multiplication means... you don't need to memorize tables.

Count me in those who think you need to memorize those tables. Math education is moving the way you suggest with much less emphasis on repetitive problem solving and much more on understanding "what multiplication means". The downside of that is when you are working on large, real life problems those who know the multiplication tables don't waste time on the mechanics and focus on the theory while those who don't get bogged down in pulling out their calculators. When I'm working with engineers of my generation or older, we routinely do basic math to a couple of significant digits in our heads while working out problems. I just watch the younger engineers struggle with the basic math. It's not that they don't know how to do it, just that they can't do it as second nature and it becomes a stumbling block. I remember one extrememly bright engineer who had to pause the discussion to open a spreadsheet to do a very basic math problem. I was stunned.

If you are very competent in multiplication, it makes it easier to focus on the fundamentals of algebra. If you are very competent in algebra, it makes it easier to focus on learning trig and calculus. It's hard to build up a solid math learning structure without a strong foundation. While not everyone needs this approach, I disagree that the exceptions disprove the rule.

ebrown p wrote:

Kids grow up to be Engineers in spite of grades, not because of them. I know this from personal experience... I grew up to be an Engineer. Most of us have a natural love of and aptitude in mathematics and a surprising number of us got poor grades in mathematics. Yet, we solve problems very well... and we even, at times, sit around and invent problems for ourselves to solve.

That may occasionally be true, but the majority of engineers I know got very good grades in math and science primarily because of that natural love and aptitude in mathematics and science. My experience is that those with that aptitude blow through those rote math problems even though they don't find them particularly challenging. I would hope that there is value to those without particularly strong math skills in solving a tough problem about two trains leaving cities at different times, even if it is only the reward of a tough challenge overcome. Will you ever have to do that in real life? Of course not, no more often than you will be called upon to solve a real life jigsaw puzzle. I do believe that those who can see the logic in that train system would be more likely to see the logic in more complex systems and be better candidates for harder classes. You'd be hard pressed to show that those with C's in math in high school will perform equally to those with A's in a science or engineering curriculum although I'm sure individual examples abound. Grades aren't a perfect predictor of future performance, but I think the correlation is pretty decent, whether the student got the grade due to hard work or talent.

Seriously, you find critical thinking and innovation today (in more individuals) more in evidence than during the eighteenth, nineteenth and first half of the twentieth century?

The exceptions are all the tricky little problems that are done slightly differently...example, rearranging the quadratic formula when you have already covered how to rearrange formulas.

My five year old reads. She has never had a test on the letters, what she has learned, she learned because she was focusing on her desire to read.

Strangely enough given my philosophy of education, I made flash cards for her when she started reading so-called "level 1" books (really simple three letter words). She hated them and wouldn't use them.

She learned to read by reading. Seeing the words in the story was just fine for her.

My daughter's grammar is remarkably good given that she has never had a grammar test. Although she doesn't know what a prepositional phrase is, she uses them flawlessly in normal speech as do most 5 year olds. You don't need to be quizzed on adjectives to use an adjective.

My daughter is starting to write. I don't care about her spelling. When she writes a story about a elefant who bilds a house, I am damn proud. Funny enough she is a perfectionist and is always asking how to spell each word correctly, but this is her motivation and she is motivated just fine.

She is writing and I am impressed. The fact that she has never had a spelling test is not a problem.

- Critical thinking skills. Being able to ask questions and to think about problems from a new perspective.

- Logical thinking. Being about to test your own solutions and to discuss it intelligently with your peers and teachers.

- Abstraction. Being able to model your ideas and to express ideas in different ways. And, being able to understand and discuss other people's models.

My position is in favor of ideas and understanding over rules. I don't know what an exception would be.

I like these, but I'd argue that these are advanced skills that are developed over time with layers of subtlety added as the student develops more basic skills. The love and excitement part is not so much a basic skill to me as a personality trait, certainly something to be developed and nutured, but not a skill per se.

But filling in the blanks in a preset template isn't math.

being about to use the base concept to solve problems in a form that you haven't seen before.

They want all the thrills and excitement of really numbing detail and this is at the cost of large numbers of students shutting down and turning the lights of.....keep the pace up and do it in big brush strokes.

what is the last time you have had to solve a quadratic equation?

As an engineer I use quadratic equations all the time

You are confusing pace with content.

I think the example of solving quadratic equations is a very instructive example.

Let me ask this question. You obviously learned to solve quadratic equations in high school, but what is the last time you have had to solve a quadratic equation? Actually in the past 40 years or so, human beings have very rare solved quadratic equations outside of school.

What is the purpose? What is the benefit of learning to solve quadratic equations (something that even people in math and engineering jobs never do any more).

Well, let me tell. As an engineer I use quadratic equations all the time, for example to discuss how fast one algorithm will be compared to another. But every use is different, I am not given template-form problems. No, I am proposing a design for a computer program that I am responsible for, and I am answer questions. Each situation is different.

So I rely on my ability to use the core concepts of quadratic equations every day. I never need to solve them. Each time I encounter a quadratic equation it is in a different situation and a different form to solve a different problem.