Had enough time to think about this Pascal question?
Spoiler time is up!
So, which rows are in Pascal’s Triangle are we talking about here?
In theory, we are talking about “all” of the rows in the infinite Pascal’s Triangle, which makes this a bit tricky to think about for kids (and adults as well!).  But Jim shared with me slides which show the proportion of evens in increasing numbers of rows of the triangle.  You will notice that as the number of rows grows, the proportion of even entries also increases, and approaches 1.  What a neat result!  To the right is an animated gif I made using a Pascal’s coloring applet which shows the increase in the proportion of even (white space) numbers in increasing rows.

For your class, this is a fun opportunity to talk about the parallels between Pascal’s Triangle, Sierpinski’s Gasket, and fractal area.

By Michael Eiseman, "Algebra By Hand"
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Dear US teachers of algebra, I have a compliment, a question, and an answer for you. First the compliment:
I like the way you explain algebra to your students by applying equivalent arithmetic operations to both sides of an equation as depicted in Animation 1. I think this is a great approach because it explains why algebra works and leads directly to a procedure in which your students may develop fluency.

Do you know that Russia, China, India, Japan, or Malaysia use a different approach? They teach their students to treat terms and factors in an equation as "movable objects" as depicted in Animation 2. Algebra is simply introduced as a conceptually unfulfilling symbol manipulation exercise. Have you ever explained to someone from India how we teach algebra in the US? Their unanimous reaction is, "Why would you do it that way? It seems so complicated." Yes, it is more complicated, but it is better-connected to fundamentals.

Why would Eastern countries take this approach? They do this because it reduces their students' cognitive load relative to the US approach. Again, my opinion here, I think they are missing a learning opportunity.
HOWEVER, (and the question is coming) dig inside the mind of any practicing engineer, scientist, accountant, or analyst in the US who uses algebra often, and you will invariably find that they use the shorter, Eastern country method of moving terms and factors and NOT the fundamentally-based procedure you taught them.

Question:
And finally, the question: "Why do all your best students eventually find that they must cast aside the procedures you taught them?"

Answer:  
​If you can picture the look of horror on their college calculus professor's face when the student writes "-8" on each side of an equation before progressing to the next solving step of a long algebraic proof, you already know the answer. That calculus professor expects your student to have progressed past applying equivalent arithmetic operations to both sides of an equation. That calculus professor expects your student to have what I call Advanced Procedural Fluency (APF). APF is the ability to quickly solve linear equations by moving terms and factors in your head as taught in eastern countries.
As an algebra teacher, it is highly likely you possess APF. Think about WHY you developed this skill on your own. You invented it because you could see that it was faster and more accurate than the method you teach your students. You realized that success in higher math requires APF - that solving by applying equivalent arithmetic operations is too cognitively taxing, too much of a hindrance to permit advanced mathematical exploration.

But if a US student does not eventually develop APF, they are mathematically handicapped relative to their counterparts in Eastern countries who will be able to solve linear equations more quickly and accurately. AFTER students have mastered conceptual understanding, procedural fluency, and application, we need to teach APF. If we don't, they will be forced to synthesize it on their own. Not only would teaching APF more quickly advance students who wish to pursue STEM-related careers, it may make algebra attainable to students who would otherwise find it incomprehensible using today's pedagogy.

What do you think? Leave a comment and let me know.