A while back when our Relief Society was in high gear doing these extremely high frequency enrichment activities, I casually mentioned to my wife how it would be cool to have a session on how to teach math to your kids. Word got back to the person scheduling these sessions, and somehow I was signed up to teach a session. Personal time constraints and a fundamental lack of experience in the area prevented me from pulling anything together, and the thing was ultimately shelved.
But one of the things I did do was go to the library to see what kind of books I could muster up on the subject. In the search I came across the book, Young Children Reinvent Arithmetic, Implications of Piaget's Theory. Its a pretty dense book really meant for classroom teachers and leans heavily on Paiget's theories on how children learn. It has an interesting premise, if I remember correctly, Piaget wanted to discover how human beings discovered knowledge in the first place, and to do so, he studied how children acquired knowledge as they aged.
But what hooked me in to this idea was chapter 4 entitled: "Autonomy: The Aim of Education for Piaget" claiming that obtaining knowledge really is about a child's development toward autonomy.
Here's a quote:
"An unusual example of moral autonomy is the struggle of Martin Luther King, Jr. for civil rights. King was autonomous enough to take relevant factors into account and to conclude that the laws discriminating against African Americans were unjust and immoral. Convinced of the need to make justice a reality he worked to end the discriminatory laws, in spite of the police, jails, dogs, water hoses, and threats of assassination used to stop to stop him. Morally autonomous people cannot be manipulated with reward and punishment."
The chapter continues with a discussion on how schools are geared to largely manipulate especially in mathematics. Math is pretty unique because there are truly right and wrong answers usually, but there are not necessarily wrong or right ways of coming up with the answer. The point of the book is that its not necessarily as important to get the answer right as it is in the struggle to get to the answer autonomously. To come up with internal methods to find it, and to learn to be sure enough in the answer to defend it to others. If in the process of defending your answer, you discover our methods are wrong, room is provided to reassess and fix the mistakes until the right answer is found.
The danger in our current teaching approach in mathematics is when the student views the teacher as the ultimate source knowledge, rather than learning to search from within. In reality, mathematics is intrinsically true. Numbers and equations that model nature around us existed before humans did. We came up with symbols and ways to represent them, but addition, subtraction, numbers were not invented they were discovered. The process of learning math, in Piaget's view, is to allow a child to go through the process of mathematical discovery in their own way working collaboratively with other children in the classroom using the teacher more as a guide and less as source for truth.
So, obviously this hooked me in. Any book that can use Martin Luther King Jr. to explain mathematical teaching is going to hook me in.
Well, with that book in my head, recently, I read about why programmers (like me) should learn math.
Here are some points:
"In fact, I don't think you need to know anything, as long as you can stay alive somehow."
On math education:
"They teach math all wrong in school. Way, WAY wrong. If you teach yourself math the right way, you'll learn faster, remember it longer, and it'll be much more valuable to you as a programmer."
"Math is... ummm, please don't tell anyone I said this; I'll never get invited to another party as long as I live. But math, well... I'd better whisper this, so listen up: (it's actually kinda fun.)"
How you should learn math:
"The right way to learn math is to ignore the actual algorithms and proofs, for the most part, and to start by learning a little bit about all the techniques: their names, what they're useful for, approximately how they're computed, how long they've been around, (sometimes) who invented them, what their limitations are, and what they're related to. Think of it as a Liberal Arts degree in mathematics.
Why? Because the first step to applying mathematics is problem identification. If you have a problem to solve, and you have no idea where to start, it could take you a long time to figure it out. But if you know it's a differentiation problem, or a convex optimization problem, or a boolean logic problem, then you at least know where to start looking for the solution."
I think this is more true in high school than in early grades because you need to have the foundations of numbers, addition, subtraction, multiplication, etc. down solid. You can't understand algebra or geometry if you don't know how numbers relate to one another (the network of numbers as the Piaget book describes it). And you really need to know algebra and geometry.
Finally, I read Outliers and this quote:
"We sometimes think of being good at mathematics as an innate ability. You either have 'it' or you don't. But to Schoenfeld, it's not so much ability as attitude. You master mathematics if you are willing to try. That's what Schoenfeld attempts to teach his students. Success is a function of persistence and doggedness and the willingness to work hard for twenty-two minutes to make sense of something that most people would give up on after thirty seconds."
The point is that people who don't get math, don't get it because they give up too early. Math, with its heavy emphasis on confusing-at-first symbols and equations is on the surface one of the most intimidating subjects around. But if you approach it systematically with focus and persistence, it is really not that hard. But it takes a certain kind of mindset, the willingness not to give up, and patient practice. The kind of attributes that translate really nicely in all kinds of other areas.
Summing all of this information together, I want to see my children really getting the basics principles deeply embedded. They need to not just know their math and subtraction tables, they need to understand deeply how numbers relate to each other.
Its why I think doing the repetitiveness of a worksheet is important, but also doing the math games and word problems Piaget would suggest, are even more important. I want to de-emphasize having the correct answers in favor for helping my children learn how to derive answers on their own and be able to defend their results to me, answers they came up with after a struggle. If their answer is wrong, I want them to be able to work through it until they get it right.
As they get older, I like the idea of gaining an overall awareness and an ability to identify problems that have mathematical solutions. I would probably emphasize statistics and probability and linear algebra over calculus. I just think probability and statistics is so important, so relevant in day to day life. Calculus is also cool and important, but maybe not as much so unless a science or engineering degree is pursued (basic calculus should be known, though. Basic calculus is really not that hard and can be learned by most anyone).
We'll see, but my personal goal is that every single one of my kids both enjoy and be good at math.