 # How to improve upon the Common Core

## Order

The Common Core does not specify an order. Khan Academy is much more useful than the Common Core, because it provides a sequence of topics.

## Puzzles and games

The Common Core doesn't mention puzzles, such as KenKen or Conway's Soldiers. No mention of games, such as Chomp.

## Insufficient detail

The Common Core lacks detail on which manipulatives and picture representations to use. For example:

3.NF.A.3.B
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.

Does visual refer to a concrete or pictorial model? Which models exactly? Two concrete models of which I'm aware are fraction slices and fraction bars.

Some standards are much too vague. For example:

5.G.B.4
Classify two-dimensional figures in a hierarchy based on properties.

Which two-dimensional figures? Which properties?

## Stories

The Common Core doesn't mention history. There's the story of Descartes and the fly on the ceiling. There's the legend of Hippasus. There's the execution of Archimedes, often cited as the catalyst for Sophie Germain's interest in math. There's the history of complex numbers. There's the history of zero. There's the history of negative numbers. There's the history of squaring the circle. There's the history of the invention of calculus. It took Napier 20 years to complete his book of logarithms. Before calculators, people would compute logarithms using tables, such as Napier's, or slide-rules. Now we can plug any number into our calculator and get an answer almost immediately. High-schoolers should be given at least a brief history of algebra, such as this one, provided by Khan Academy.

## Biographies and interviews

The Common Core doesn't mention biographies or interviews. Here are my notes on bios and interviews, all of which will eventually be woven into my curriculum.

## Example problems

Every standard in which exercises can be given should include example problems.

## Challenge problems

The Common Core does not include challenges. NRICH and Math Pickle have lots of fun challenges for K-12. TED also provides logic puzzles suitable for high-schoolers and beyond. There are also many other K-12 challenges floating around the Web. For example, the connection puzzles solvable by the wishful thinking strategy. Math competition problems can often be adapted, or given unaltered, for the purpose of a challenge. Every student deserves to be challenged. A math education without challenge is severely deficient.

## No implementation nor references to implementations

The Common Core does not specify its implementation, nor references to existing implementations. After reading the Common Core, teachers should turn immediately to Khan Academy, Eureka Math, or a textbook, highly rated by edreports.org, to understand how each standard can be implemented.

## Missing theorems and proofs

There are many interesting theorems missing from the Common Core, such as the British flag theorem.

Many proofs are missing from the Common Core. For example, the law of sines.

## Includes the what, but not the how

The Common Core includes what to teach, but not how to teach it. For example, Boaler tells us we should praise the work students do, not qualities of the student. You shouldn't say "Wow! You're really smart!" You should say "You stuck with it and figured it out. Good job!"

## Number talks

The Common Core does not mention number talks or how to conduct them. Number talks have been proven to have a positive effect on students' mental math abilities and the problem solving strategies they're able to articulate.

## Mental math

The Common Core has few references to which computations should be carried out mentally. Most problems covered by the Common Core should be carried out mentally at some point.

## Recreational math

The Common Core doesn't include recreational math. Where are the magic squares?

## Fractals

The Common Core doesn't mention fractals. The rules of fractals are simple enough to be understood by K-12 students. Fractals can be used to teach about raising fractions to powers. For example, what fraction of the Sierpinski triangle is shaded? The Mandelbrot set can enrich a student's understanding of the complex plane. Here's my list, so far, of fractal topics for K-12.

## Fibonacci numbers

Here's the only reference to Fibonacci numbers! Mensa for Kids seems to think we should teach the Fibonacci numbers to elementary school students. I agree, because all you need to generate the Fibonacci sequence is addition.

HSF.IF.A.3: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

## Visual proofs

The Common Core does not include visual proofs. Most visual proofs can be understood by high-school students. Of course, the missing square puzzle should be shown too.

## Mathematical logic for high schoolers

Mathematical logic is seldom mentioned in the high school section of the Common Core. However, to understand many geometric proofs, substantial knowledge of propositional logic is necessary. The Common Core assumes students understand not, and, or, but doesn't teach these as binary connectives. My concern is that students are using these operations without understanding them. That is, not isn't limited to Venn diagrams, it's an operation on truth values (in classical logic).

## Non-Euclidean geometries exist

Non-Euclidean geometry is briefly mentioned in the Common Core's introduction to high school geometry. However, it appears nowhere else in the standard. Students should be shown that non-Euclidean geometries exist, for the purpose of stimulating curiosity. This will also help them see that truth (proof) is dependent on your choice of assumptions and rules of inference.

## Etymology

The Common Core doesn't mention etymology, or the history of notation. If you watch this Eddie Woo video, you'll see him dissect "polynomials" into "poly," meaning "many," and "nomials" meaning "terms." In this video Eddie talks about $$\mathbb{Z},$$ and in this video, $$\mathbb{Q}$$. In Khan Academy's video on the origins of algebra, Sal tells us the word algebra comes from al-Khwarizmi, a Persian polymath.

## Fallability of inuition

The Common Core never defies the student's intuition. Bayes's theorem, the Boy or Girl paradox, the Monty Hall problem, and the missing square puzzle, show that we cannot always rely on the senses and intuition, sometimes mathematical proof is necessary. Showing a student their intuition can be faulty motivates them to understand mathematical proofs.

## Measurement is a natural science topic

I saved what I think will be the most controversial of my opinions for last! I think topics dedicated to taking measurements should be relocated to the science curriculum. Neil deGrasse Tyson seems to agree that measurement is under the purview of science, as seen here. There are several standards on measurement, such as being able to measure with unit cubes, or being able to read an analog clock, rulers, thermometers, etc. Math involving measurement is fine, I just don't think measurement alone is mathematical. For example, I think unit conversion and rates should remain in the math curriculum.