Mathematics Curriculum.
Establishing a rigourous and effective mathematical curriculum is difficult, to say the least. As stated in the last section, I think students are rushed through calculus. I do see the reason for this: Calculus is the basis of most modern science and engineering techniques; providing calculus at an early age should give them an edge. The problem is that it doesn't.
It is unanimously agreed that Calculus 2 is the hardest calculus course. Why? It isn't because integrating and differentiating trig functions is difficult; it isn't because integration by parts is difficult. So what is it? Trig identities, infinite series, partial fractions. These all have something in common: they are algebra, not calculus. Perhaps if students were given a thorough understanding of algebra before taking calculus, calculus professors could spend their time explaining calculus and proving the theorems, instead of teaching algebra. This way, calculus classes would prove to be the joke they are; hell, you could even get rid of calc 3: it's just calc 1 and 2 with extra variables. If you understand algebra, this is mind-numbingly tedious.
Imagine if history teachers talked about WWII in high school, but didn't mention the Treaty of Versailles, the Weimar republic, or the rise of Totalitarian States until college! It's absurd, isn't it? Well, that's what math instructors do.
I think my tangent is over, let me outline my curriculum. It contains a wide range of useful (to engineering) mathematical topics, as well as some absolutely necessary mathematical knowledge.
Kindergarten/1st grade: Introduction to numbers. Adding and subtracting. Basic shapes. Interesting properties.
2nd Grade: Adding/subtracting continued. Basic multiplication. introduction to simple division. Intro to decimal system. Intro to fractions.
3rd Grade: Multiplication and division. Remainders, decimal expansions. Basic fraction arithmetic.
4th Grade: More on arithmetic. More on shapes. Fractions.
5th Grade: Intro to generalizations and abstraction. Exponents. Order of operations. Intro to shapes. Intro to counting.
6th Grade: Cartesian Plane. More on abstraction. pre-algebra.
7th Grade: Cartesian Plane, graphing. Pre-algebra, algebra.
8th Grade: Mathematical logic. Sets. Basic Probability. Basic proofs. Algebra.
9th Grade: Geometry. Intro to trigonometry. Proofs.
10th Grade: Functions, algebra, variables. Sums. Sets. Probability.
11th Grade: Trigonometry. Stats. Intro to graph theory. Intro to game theory. Modelling.
12th Grade: Infinite series and sequences. Limits. Fundamentals of number theory and combinatorics.
Note that 10th, 11th, and 12th grade represent a foundation for all higher level math.
Next: Brief "lectures"
Showing posts with label The "New" School. Show all posts
Showing posts with label The "New" School. Show all posts
Wednesday, November 21, 2007
Sunday, November 18, 2007
If I Ran a School... Part 1 Section 0
Introduction to Mathematic Education.
It's hard to know where to start when considering the problems I have with math education. So, let's start with the difficulties:
*Math is built upon math. The point of math is to start with something simple and abstract increasingly complex ideas from that. Because of this, falling behind in math means you will be behind in math for the rest of your education, unless either it stops, you work hard to get back on track, or a miracle occurs.
*Math works for certain people better than others. Certain people can just see math. Others need it put in their face and wiggled around for a while. The problem is, very few people can see it, grab it and wiggle it around in front of someone else. This leads to two types of math teachers (there are exceptions):
**The brilliant mathematician: This is the math teacher that clearly knows his math and teaches you useful tricks. The problem is, math just works for him, he doesn't need to think about, so he can't explain it very well.
**The formula teacher: This teacher can do math fairly well, but doesn't understand the churning of the gears well enough to teach how to think mathematically. Instead, they try to teach formulas to their students. Those who can see math understand why the formulas work, the rest are flipping the magic switch.
Most of the problems I have with math education stem from these 2 problems. There is one more large problem: Schools have a tendency to try and rush students through calculus. This is quite possibly the worst idea ever. Calculus is terribly easy so long as you have a thorough grasp of algebra. If you do not have a thorough grasp of algebra, calculus is impossible, absolutely impossible. There are other subjects in math that require a less thorough grasp of algebra, and help to expand your grasp of algebra. Further, because students only learn geometry, elementary algebra, and calculus, they have a very false impression of what math is. Mostly, what students learn is arithmetic.
This part will have multiple sections: first a brief explanation of how I would do math in our current school system, then a brief "lecture" on each subject, and finally a hierarchy for use in "my system"
It's hard to know where to start when considering the problems I have with math education. So, let's start with the difficulties:
*Math is built upon math. The point of math is to start with something simple and abstract increasingly complex ideas from that. Because of this, falling behind in math means you will be behind in math for the rest of your education, unless either it stops, you work hard to get back on track, or a miracle occurs.
*Math works for certain people better than others. Certain people can just see math. Others need it put in their face and wiggled around for a while. The problem is, very few people can see it, grab it and wiggle it around in front of someone else. This leads to two types of math teachers (there are exceptions):
**The brilliant mathematician: This is the math teacher that clearly knows his math and teaches you useful tricks. The problem is, math just works for him, he doesn't need to think about, so he can't explain it very well.
**The formula teacher: This teacher can do math fairly well, but doesn't understand the churning of the gears well enough to teach how to think mathematically. Instead, they try to teach formulas to their students. Those who can see math understand why the formulas work, the rest are flipping the magic switch.
Most of the problems I have with math education stem from these 2 problems. There is one more large problem: Schools have a tendency to try and rush students through calculus. This is quite possibly the worst idea ever. Calculus is terribly easy so long as you have a thorough grasp of algebra. If you do not have a thorough grasp of algebra, calculus is impossible, absolutely impossible. There are other subjects in math that require a less thorough grasp of algebra, and help to expand your grasp of algebra. Further, because students only learn geometry, elementary algebra, and calculus, they have a very false impression of what math is. Mostly, what students learn is arithmetic.
This part will have multiple sections: first a brief explanation of how I would do math in our current school system, then a brief "lecture" on each subject, and finally a hierarchy for use in "my system"
If I Ran a School... Part 0: Introduction
This is going to be a multipart series, which will be updated periodically. Our school system is horrible. Everything revolves around the philosophy that "college means success", which is absolutely false. I propose a system, K-12 which provides students with a rounded education and options, rather than forgetting about those that would do better in a less "academic" career.
Firstly, at grade 8, the school is broken into an 8 class, alternating block schedule; That is, 4 classes on 1 day, the 4 the next. I would assume each class session would be 1:.5 - 1.75 hours. (I will address problems with the extended class/block system in a later section.) Perhaps a 1 hour homeroom or study hall would also be appropriate.
Along with the introduction of the block schedule, the classes will be split into 3 "tracks": Arts and Trades, Humanitites, Math and Science. (This will also be addressed in a later section.) Each student must select a track. years 8 and 9, each student must take 3 classes from their track, and 2 from each other; the remaining class is an elective. The remaining 3 years, 3, 2, 1, with 2 elective classes.
That is a quick and dirty run through the basics. I know there are problems, which I will address, and I know I will change some of the specifics, but I figure it's best to get something down.
Part 1 will be many-sectioned, and focus on everything I think is wrong with grade school math education.
Firstly, at grade 8, the school is broken into an 8 class, alternating block schedule; That is, 4 classes on 1 day, the 4 the next. I would assume each class session would be 1:.5 - 1.75 hours. (I will address problems with the extended class/block system in a later section.) Perhaps a 1 hour homeroom or study hall would also be appropriate.
Along with the introduction of the block schedule, the classes will be split into 3 "tracks": Arts and Trades, Humanitites, Math and Science. (This will also be addressed in a later section.) Each student must select a track. years 8 and 9, each student must take 3 classes from their track, and 2 from each other; the remaining class is an elective. The remaining 3 years, 3, 2, 1, with 2 elective classes.
That is a quick and dirty run through the basics. I know there are problems, which I will address, and I know I will change some of the specifics, but I figure it's best to get something down.
Part 1 will be many-sectioned, and focus on everything I think is wrong with grade school math education.
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Cory Knapp.