I had quoted what the mathematician Peter Hilton had said about skills and education versus training. And Peter Hilton makes a valid point because what skills are necessary are constantly changing, and what skills are necessary will also depend on what students will do later in their lives; thus, any mathematics classroom consists of students whose skills they will need for the future we cannot predict except those skills that we call "soft skills." Soft skills are the ability to think critically and creativity, the ability to think mathematically, the ability to read and write and communicate effectively, the ability to work and learn independently, the ability to work with others, and so on. So-called "hard skills" cannot be predicted. Hard skills are the ability to use particular technology and other highly specific skills that work only for a narrow range of tasks. Hard skills, especially those related to technology, tend to go out of date quickly.
Education is not career training but preparation for life and for society. Education is about helping us learn to think for themselves, to learn for ourselves, to learn to be creative, to learn about the world, to learn about our society--in short, education is about giving us all the tools we will need to succeed in our lives and careers but those tools that we all need, regardless of what our futures will be. None of our teachers can predict all the career skills we will need though some pretend that they can do so. In fact, none of us can say with certainty what our own futures will be. For instance, can I say for myself that I will continue teaching college mathematics the rest of my life? I cannot.
What about mathematics education? What should that give us? For one thing, a good mathematics education should give us the ability to think mathematically and the ability to make sense of numerical data all around us. Good number sense goes along with this. These abilities will serve us regardless of how mathematics plays a role in our lives later on. It does no good to have all these so-called "skills" in mathematics if one does not have good number sense and if one has no clue as to the meaning of what they are doing beyond merely scratching symbols onto a piece of paper.
Good mathematics education should help us to learn to mature mathematically so that we can learn whatever mathematics later on we are interested in learning or need to learn. Low mathematical maturity accounts for many of the struggles with mathematics that I see among students, and low mathematical maturity causes crippling math anxiety in students, which in turn makes them falsely believe that they are no good at mathematics. College mathematics will make no sense to anybody with low mathematical maturity. But focusing on teaching all these so-called "skills" and procedures and formulas does not contribute to students learning to develop mathematical maturity. What contributes to them learning mathematical maturity is being encouraged to think about the mathematics they are learning and working on by asking themselves questions that mathematicians often ask themselves: Why is this result true? Why does this procedure work? Can I find an alternate way to do this problem? Can I find another algorithm for carrying out such-and-such a computation? Why is this result important? Where does this definition come from? And so on. In short, for students to learn mathematical maturity, they must actively engage with the mathematics itself by seeking meaning behind the mathematics. Learning mathematical maturity will serve them more in their mathematics education than anything else mathematics education can offer them.
And a good mathematics education should give us a view of what mathematics is really like and why mathematicians find it fascinating. And a good mathematics education should help us learn to see where mathematics has come from, how it has impacted culture, how culture has impacted mathematics (these last two ideas are ideas I wish I knew a lot more about)--in short, to learn to see the human side to mathematics. Without seeing the human side to mathematics, students end up falsely believing that mathematics is purely mechanical, uncreative, and disconnected from people's lives (except from the lives of some "weirdos" or "nerds" or "geeks" called mathematicians) and end up seeing math as simply one of those things we have to learn to get through school. What a waste! No wonder that mathematics suffers from a low status in our culture: Many schools and teachers so badly distort mathematics in the classroom that it ends up looking like nothing like the mathematics we mathematicians have fallen in love with. In short, school mathematics is often boring and an extreme drudgery to students.
And it irritates me that so many of these mathematicians who support MC/HOLD see no problem with their beloved subject so badly maligned in the classrooms. How can those fools refuse to see that most of the problems with mathematics education are simply because mathematics is taught so that it becomes forbidding to students--even when you mention that to them?
What I had learned in mathematics that serves me best is the ability to think mathematically, the ability to learn mathematics for myself, and the reasons that mathematicians find mathematics fascinating. These things have served me much more than any specific content knowledge I have learned because any specific knowledge would not help me in the least bit without these other skills (in other words, without these skills, all I could do is quote mathematical facts but not be able to use them in any meaningful way).