My "modest proposal" for reforming K-12 math education
July 2010 (perspective of a Ph.D. student)
I propose a hypothetical curriculum for K-12 math education where only one essential concept (or set of related concepts) is taught per school year. This curriculum can lead to deep understanding and retention of knowledge.
My modest proposal
My modest proposal for reforming K-12 math education is to teach only one essential concept per year. The only goal of my proposed curriculum is to enable every high school graduate to have a basic level of literacy regarding numerical intuition and quantitative thinking.
We currently teach students hundreds of disparate mathematical concepts throughout their 13 years of elementary, middle, and high school. It's nearly impossible for students to distinguish the useful ones (e.g., ratios) from useless antiquated relics (e.g., trigonometry), so I'm not at all surprised that most American students develop a deep disdain for and phobia of everything math-related.
Even math-loving people like myself and my friends in science and engineering admit that most of what we learned throughout our 13 years of pre-college math classes was useless and forgettable—remember sinh functions? Gauss-Jordan elimination? integration by parts? Didn't think so.
Instead, I think that if we only teach students 13 concepts (or sets of closely-related concepts)—one per school year—then we will avoid overwhelming students with information overload. Hopefully they will be less likely to exhibit a knee-jerk rejection of math-related concepts. It's much better to have students deeply understand and retain 13 crucial concepts than to have them learn and then forget hundreds of disparate ones.
Elementary school curriculum
The goal of elementary school math is to establish the basics of arithmetic.
Kindergarten: Non-negative integers
Emphasize how numbers correspond to real-world objects; counting apples, bananas, oranges, etc.
1st grade: Addition of non-negative integers
Emphasize the advantages of addition over simple repeated counting
2nd grade: Subtraction of integers
Introduce negative integers for the first time
3rd grade: Multiplication of integers
Emphasize the advantages of multiplication over simply doing repeated additions; can also sneak in some basic geometry, using multiplication to calculate areas of simple 2-D polygons
4th grade: Division
Introduce rational numbers for the first time
5th grade: Ratios and proportions
Applications of division to comparing real-world objects
Middle school curriculum
The goal of middle school math is to start introducing abstractions via variables, but still ground lessons with applications to tangible, real-world objects rather than teaching algebra.
6th grade: Measurement of real-world objects
Measuring lengths, areas, volumes, weights; comparing measurements using ratios and proportions; converting between units of measurement
7th grade: Logical reasoning
Basics of propositional and first-order logic, deductive reasoning, introducing variables for the first time (in the context of logic, not algebra), teaching students to recognize common logical fallacies
8th grade - Percentages, and a review of ratios and proportions
Using basic algebra to solve problems related to percentages, ratios, and proportions
High school curriculum
The goal of high school math is to give teenagers a basic sense of numerical literacy, which will be useful for managing personal finances, assessing risk, and understanding the implications of numbers that the media presents to them.
9th grade: Visually interpreting data
Introduce different types of data visualizations (e.g., scatter plots, bar graphs); teach intuitions about how to interpret and draw conclusions from visually-presented data; could motivate these skills by critically studying data visualizations in newspapers and news magazines
10th grade: Mean, median, standard deviation, and shapes of data distributions
Basic statistics without all the jargon and formalism
11th grade: Basic and conditional probabilities
An understanding of probabilities, especially conditional probabilities, is essential for properly assessing risk
12th grade: Exponential growth
Teach students about the unintuitive nature of exponential growth, and how they can get screwed by it if they're not careful; the most pragmatic lesson will be on compound interest rates, since that is the form of exponential growth that young adults will most directly encounter
Isn't a year too much time to devote to one concept?
One common objection to my proposal is that one year seems like too long of a time to reinforce just one core concept. What if kids "get it" after only a month? Should you just move on to another concept?
I'm going to argue that NO, you shouldn't move on, since there's a big difference between kids nodding their heads with a superficial understanding versus having a true understanding to the point where it becomes second nature. In our current public school math curriculum, we teach hundreds of different math concepts, spending at most a week on each one. I suspect that, in the best case scenario, the most perceptive kids "get it" at the time they're learning it but quickly forget after moving onto the next concept. I want kids to understand these core concepts so deeply and intuitively that it becomes second nature, just like how being able to read and write is second nature.
For example, native English speakers know when an English sentence "feels" grammatically incorrect, even though they might not be able to pinpoint exactly which grammar rule is being violated. This occurs because they've been continuously immersed in a language for their entire lives. I want something similar to happen with math: that is, for kids to intuitively "feel" that a certain mathematical claim is likely to be correct or incorrect. I don't think we'll get as far as we do with language, since the human brain is hard-wired for primary language acquisition, but there's a lot of room for improvement from the status quo.
How to implement this proposal
In reality, I know there is no way that this proposal could ever be adopted in the mainstream American school system. There is way too much inertia-generating infrastructure around the current curriculum, most notably accreditation requirements and standardized tests. If kids were taught using this minimal curriculum, then there is no possible way that they could pass their annual state-mandated standardized tests, since they simply won't know most of the required material.
If I ran a school district, I would have teachers use my minimal curriculum for 7 months out of the school year, and then force-feed students with standardized test prep material for the 2 months leading up to year-end exams. That is: 7 months of minimal math, followed by 2 months of cramming. Two months of cramming is nothing compared to schools in Asia, where teachers spend all 12 months out of the year force-feeding students with standardized test prep material! With this hybrid proposal, kids would get to learn their one important concept each year and still satisfy testing requirements.
How to REALLY implement this proposal
Obviously most of us aren't in control of our local K-12 curriculum. However, there is a super-simple way to implement this proposal at home: As a parent, teach your kid these concepts in your spare time. They still need to go to school and learn the mainstream math curriculum, so that they can get decent grades, pass their SATs, and get admitted into colleges. But at home, you don't have to worry about accreditation requirements or "teaching to the tests".
The most pragmatic aspect of this plan is that you only need to teach one concept per year. For instance, when your kid is in 6th grade (the year I propose to teach "Measurement of real-world objects"), find every opportunity you can to sneak in some lessons about measuring objects (e.g., when shopping at the supermarket).
You don't need to brutally force your kid to do draconian math exercises every night after school. Even two hours a week of informal tutoring (one on Saturday and one on Sunday), coupled with a keen eye on finding informal opportunities to teach, can go quite far in developing your child's numerical literacy skills!