## Powerful ideas in math

There are some ideas in mathematics which are powerful tools for thinking. I’ve enumerated some of them below.

1. Equivalence

This is the idea that some things are the same as each other, and that some things are not. We see equivalence taught through early arithmetic and algebra. We also see this idea taught through trigonometric identities and proofs.

I watched a colleague of mine working with his class to introduce trigonometric identities (such as sin2x + cos2x = 1) and over and over again the idea of equivalence between different expressions came up. At one point a student tried to cancel the 1s in the following expression: $\frac{cos^2x-1}{cosx+1}$. My colleague then asked students to look at $\frac{2+1}{3+1}$ and students noticed that $\frac{3}{4}$ is not the same as $\frac{2}{3}$ which means that it must not be possible to do the same thing with the trigonometric expression. This idea is only possible to understand if you understand the bigger idea of equivalence.

2. Sequencing

We see this idea comes up in patterns, ordering numbers, but it also comes up in limits. In the earliest years, the idea of sequencing is introduced as children learn to count by ones, twos, and so on, and in the later years, students look at lists of numbers and try and find a pattern to predict further numbers in the sequence.

In the final years of high school, if they learn calculus, students learn about how sequences apply to understanding the limit of a series. For example with $S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + …$, what should the value of S be to make this a true statement, assuming that the sequence of numbers continues forever? How does this idea relate to the idea of differentials and integrals?

More generally, any time we arrange anything in any kind of order, we necessarily have to use our ability to sequence ideas.

3. Representation

This idea is where much of mathematical notation comes from as any mathematical notation is in fact a representation of another concept. Even number symbols are in fact representations of the idea of a number, rather than actually being the number themselves, eg. the numeral 5 is a representation of the idea of fiveness.

Another place representation comes up in a standard curriculum is when students explore functions, and they learn that functions can be graphed, given as ordered pairs, listed in a table, and represented by an equation. Each of these representations means the same thing, but written in a different form for our convenience.

4. Precision

Early in school students learn the difference between an estimate and a calculation, and they learn that the way we use words (especially in mathematics) can sometimes lead to some interesting issues. For example, a common definition of a prime number is that it is any number which is divisible by one and itself. Unfortunately, many children read this definition and think that one is a prime number. Understanding that one has been chosen not to be included in the definition of the prime numbers so that we can more easily identify and use the properties of primes later on is something that many children either do not understand, or are never told about.

5. Procedures

This is one powerful idea that most children spend a lot of time learning about. They learn that someone can create a procedure, or a list of steps to follow in order to achieve a particular goal, and that when that procedure is followed accurately, the result is predictable. This idea itself is seen in almost all areas of mathematics, but notably, it often fails when applied to mathematical problem solving because so many rich mathematical problems defy a simple procedure to solve.

6. Subroutines

Learning how to decompose and reduce the complexity of problems with which we are faced is one of the most critical thinking tools we can learn. Over and over again, this issue comes up in mathematics. In fact, in order to support students ability to find and create subroutines is one of the reasons mathematics curriculums often follow similar sequences.

Here’s an activity: find a problem from secondary school mathematics, and work out how many different subroutines are necessary in order to solve that problem. One benefit of seeing this issue of subroutines is that it gives you a lot more empathy about what children have to go through in order to learn mathematics.

7. Classification

This is the ability to take a list of traits, identify them in an object, and use those traits to understand the differences between one object and another. For example, if you look at the properties of a rectangle, you may recognize that opposite sides of the rectangle should be the same length, and that every angle in the rectangle should be $\frac{1}{4}$ of a full turn. One thing that many people struggle with is understanding that this definition of a rectangle also necessarily includes a sequence, and that in fact a square is a specific example of a form of rectangle.

Classification is one of the earliest thinking tools we learn. It is likely in fact that our ability to classify objects may be preprogrammed into our brains, given how early in life you can see children using it.

8. Relationships

Mathematics is also about finding relationships between different objects. Look at the video below.

Is it surprising that through paper folding with a point and a circle that we can create the shape of a hyperbola?

Finding relationships between objects and finding how one area of mathematics is connected to another area of mathematics are critical skills to learn when solving mathematical problems. Some of the most challenging problems in mathematics have been solved by looking for parallels in other areas of mathematics that are easier to work with.

These powerful ideas also apply to non-mathematical problems. If students can learn these thinking skills, and learn how to apply them in a variety of different contexts, then they will hopefully be able to apply them to the problems they will face in life. Learning these skills empowers childrens to think.

What other powerful ideas do you think children should learn?

• Equivalence. Amen. To add to your examples, I offer congruence and similarity. Equivalence is about sameness, and we need to carefully state what we mean by the same

Someone (not me) is gonna pounce on you for the sequence/series language. It may be worth editing that carefully through the eyes of a fussily precise mathematician.

I dig your list. And I am pretty sure that I can find examples of Talking Math with Your Kids where we address each of these. In fact, I may challenge myself to do this very thing.

Thanks for the post.

• David Wees wrote:

I wonder if there is a way to bring equivalence, congruence, and similarity under one umbrella term, since they are all obviously closely related to the idea of comparison?

Maybe the big idea is comparison itself?

• David Wees wrote:

Discussed this on Twitter. Equivalence is the big ideal, similarity and congruence are examples.

• David Wees wrote:

Eddi Vulic, via Twitter, suggested that Proof and Generalization are distinct powerful ideas which should be included in this list. Generalization seems to me to be the opposite of Subroutines (as I have defined it above), and Proof seems to require many of these powerful ideas as pre-requisities, suggesting that it is easier to think of proof as being formed by a few of these powerful ideas together, but I could be wrong on this.