# Math for Science and Science for Math

When you study math with us, you will learn how scientists use math. In particular,
you will learn how scientists *model* the real world using the abstractions of math.
That makes our approach different from what your child gets in
a conventional math class, which take what we call a "cookbook approach."
Our approach is also different from that of many other math enrichment
programs. These programs claim to take a "problem solving" approach, but the
problems they focus on are more the sorts of puzzles that are used in math
competitions, rather than scientific problems.

Our approach is not just different, but for many students, better. Students will leave our program better prepared to study physics, engineering, economics, or any other subject where math is used to model reality. They will be better prepared to study these subjects at the AP level or in college.

They will also be better prepared to succeed on math tests. The new perspective that students gain from seeing how math is actually used makes it easier for them to recall and use individual topics when they need them. They can raise their grades in math class, and their scores on high school entrance exams, PSAT/NMSQT, SAT, ACT, AP exams, and the many other tests that will be coming their way.

Your teachers at *Annandale Academics*, Dr. Rebecca and Mr. Dan, both have
backgrounds in the physical and social sciences. Rebecca has a PhD. in Economics
and an AB in Chemistry, both from Harvard University. Dan studied physics and
math, before a career as an economist.

## What Is Math, Why Is It Hard, and How Does Science Help?

As students go through school, what the word "math" means to them changes.
In elementary school, "math" means "doing calculations." Students learn
step-by-step recipes, or *algorithms*, to perform various calculations. For the
most part, elementary age students are not asked to think deeply about what they
are doing. If anything, the opposite. Students are rewarded for performing
calculations flawlessly, quickly, and without having to think about them.
They are drilled to that end with homework, and tested to see that they can
precisely repeat the steps of each recipe that they have been taught.

One thing elementary math students are never asked to do is a calculation that
they have not seen before. There isn't much *thinking* or *understanding*
involved at this level of math, nor much *initiative* or *creativity*. Which is
all as it should be. Most elementary school students are not yet capable of the
kind of abstraction that would allow them to engage with math creatively,
and it is reasonable that they exert their energies towards memorizing the basic
algorithms to the point where they are able to do them automatically.

But things change when the students get to algebra. Because to learn solve problems that one has never seen before is the main point of algebra.

Making that shift requires students to reach a new understanding of what "math" means. The foundation of truth in math can no longer be "I executed the steps of the recipe exactly the way the teacher told me to." Students solving problems they haven't seen before don't have recipes. So they need to learn to rely on a different, more fundamental foundation for mathematical truth.

Students in algebra need to learn to approach math as an *axiomatic system of
deductive reasoning*.

It's an *axiomatic* system, because at the foundation of math are a set of basic
laws, or *axioms*. It is a system of *deductive reasoning* because it has rules
by which, given some mathematical statements that are true (or false), it is
possible to *deduce* the truth or falsehood of other statements.

If that sounds very dry and esoteric, that's because it is. Which is why math is hard. We think that students have an easier time if they learn how math is used in science at the same time that they use math.

The scientist always has one foot in the abstract world of math, and the other in the concrete world of reality. Science connects reality to abstraction by "models." A model is a statement that certain physical phenomena are related by certain mathematical constraints. The image at the top of this article depicts Newton's law of gravitation, a fundamental constraint relating the motions of all objects in the universe.

The behavior of the model is entirely determined deductively, according to the
abstract rules of mathematics. The truth of the model, on the other hand, is
determined *inductively*, by conducting experiments to determine how
closely the relationships predicted by the model actually occur in physical
systems.

Physical scientists, social scientists, and engineers are people who use math
every day. And most of what they do with math is to build, manipulate, and test
mathematical models of reality. Yet this is not an activity that students
practice very much in math class. At *Annandale Academics*, we think this
is a grave oversight.

We identify three ways that learning to use models helps students succeed:

Students gain practice doing what they will need to do when they study science, engineering, and the social sciences.

Learning how mathematical laws represent physical behaviors with which the students are already familiar helps them to build their intuition about those mathematical laws.

Many real scientific models are simple enough that middle school and high school students can start using them right away to design things, make things, and understand things they care about. So math, instead of something that they spend years learning before they actually get to use it, becomes something that they start using on their own right away.

## How Is This Different from the "Real World Examples" in my Math Book?

At *Annandale Academics*, we draw on the "real world" to help students
undestand math. Most school math programs claim to do the same thing. But we
think what they actually do is very different.

When children learn fractions, they are told to imagine a slice of a pizza pie. Ever after that, they are given "real world examples" to illustrate the mathematical concepts that they learn. These examples have their place as aids to visualization and to memory, but they are very different from scientific models.

Actually, in a way, they are the opposite of scientific models. A slice of pizza is a physical model of an abstract mathematical entity, whereas Newton's law of gravitation, depicted at the top of this article, is a mathematical model of physical reality.

Relying too heavily on pizza-pie type models causes problems.
The visualization can become a substitute for understanding the underlying
abstraction. Indeed, to a disturbing degree, modern secondary school math books
seem committed to avoiding as much as possible any discussion of the abstract
structure of math. At *Annandale Academics*, we teach scientific modeling
not to avoid teaching the abstractions, but as an aid to help students
become more familiar with the abstractions.

That problem is compounded when the physical example fails. And the physical example always fails. Because mathematical abstractions can be applied to mathematical entities that have no correspondence to the particular physical system used to visualize them.

How do pizza slices help me understand how to multiply fractions? What is five eighths of a pizza times three eighths of a pizza?

The answer is fifteen sixty-fourths of a square pizza, of course, but what does that even mean?

Do pizza slices help me visualize $\frac{180}{\pi}$ or $\frac{{x}^{2}-1}{x+1}$? What about complex numbers like $\frac{1}{2+i}$?

At the end of the day, classes that over-rely on pizza pies and similar visualization aids end up teaching their students bad habits. I have encountered many students who have run into trouble in math when the visualizations that they had been using as a crutch suddenly fell apart.

## What about the Problem Solvers? No Pizza Here.

You have probably encountered programs that are geared to prepare students to compete in the International Mathematical Olympiad and other math competitions. Make no mistake, we like these programs much better than the pizza pie approach. However, they do have certain limitations that you should consider when comparing them to our programs.

First, they are relentlessly abstract. Your child will not learn bad habits in such a class. You do not have to worry about the teachers shying away from explaining math's abstract structure. But your child may well find the approach to be so abstract as to be inaccessible.

Second, they don't emphasize the sort of math that scientists use, or the sort of mathematical reasoning that scientists do. A disproportionately large emphasis is placed on such topics as number theory (i.e., the math of the positive integers), and various topics in discrete math, such as graphs and trees. These topics are important in solving the sorts of problems that show up on competition tests.

But they are not topics that are hugely important in the physical sciences, social sciences or engineering, which are mostly concerned with continuous functions, the math of the real numbers, and in more advanced classes, the math of complex numbers. They use the ordinary algebra of linear equations and polynomials, trigonometry, exponents and logarithms, and calculus.

So if your child's goal is to succeed in math competitions, and has already determined that the particular flavor of math that shows up in those competitions is one that they are comfortable with, one of the problem-solving programs may be a very good fit. But if your child is more interested in science and engineering, and how math can be used to understand the world, and to design and make things, then we think our program would be a better fit.