Back to: Module: Helping Students Do Math
Orientation
A “toolkit” is a resource; it has a lot of stuff in it. This toolkit will help you learn about math, which is different form learning math itself. We hope you will take a look at it now, but also come back to the toolkit when needed. Our main hope is that you use it to learn, not more math, but more about math. Remember: you can use it to inspire students’ interest in math!
It has two compartments, following this brief orientation:
- the development of mathematics, and
- different kinds of math people can learn (ratios and proportions, algebra, geometry, trigonometry, calculus).
Every subject in school—English, Spanish, Math, Biology—has content that students learn.
- English has things like parts of speech, sentences and paragraphs, punctuation, and so on.
- Math has number-facts, lots of ideas (the idea of equality, the idea of zero, variables, and so on).
As a foundation for teaching, we should know some of the content of the subject we’re teaching or helping to teach, like math. If we’re not the main teacher, but are co-teaching or helping out, we don’t need to know a whole lot of the content. More important than knowing all of the content is to understand what makes the subject “tick”—what it’s all about. Finding out about the subject helps us see the ideas that make sense of its concepts, facts, and procedures. Learning a little about the different fields in and history of the subject (for example, math or language arts) can make a big difference.
In fact, most of us can’t explore or learn all the math content that is interesting, but we can learn a lot about the different kinds (i.e., topics or fields) of math and how they all fit together. All subjects of study have histories, and so does math. Learning about this history does not require much math content!
For helping to teach math, information about the history of mathematics and the different fields of mathematics will let you explain more about the usefulness of studying math. You don’t have to know algebra or calculus or applied math in order to know about these math fields in a general way. And if you know what topics these fields consider, you can share your knowledge with students.
Compartment #1: The Development of Mathematics-Very Briefly
(numbers, 0, fractions, arithmetic [adding, subtracting, multiplying, dividing)
Math is born: Humans thought about number and space long before they invented writing. It seems, though, that writing began its development about 6,000 years ago. And one of the first uses was to provide a record of things counted. In a sense, then, mathematics is a kind of writing. For thousands of years, the record of things counted was made in lots of different number systems. In some of these systems people also invented ways to add, subtract, multiply, and divide.
The invention of zero (0), however, was a big improvement. It made adding, subtracting, multiplying, and dividing much easier. But it also proved important as an idea.
Another milestone in math is the work on geometry done about 2,400 years ago. Euclid (YOU-klid) wrote a textbook that remains a poster-child for what mathematical thinking looks like. Euclid’s approach is still an important part of most geometry classes in U.S. schools! And there is much more. When you get a chance, explore the following resources online. There are also many other resources you can find online to help you learn about the development of math.
Resource One
Two minute video without narration; so short it leaves out the really important idea of zero.
Resource Two
Seven-minute video with narration; it gives zero the attention it needs!
Resource Three
Fifty-minute lecture on 50 centuries (5,000 years) of mathematics
(college class; challenging but worth the effort).
Resource Four
To get more detail (in writing), explore: http://www.storyofmathematics.com/
and
Wikipedia’s entry http://en.wikipedia.org/wiki/History_of_mathematics
Compartment #2: The Different Kinds of Mathematics Students Can Learn
This compartment includes an assortment of brief explanations about different kinds of math students can learn in school, plus a collection of brief You-tube videos and some short text resources to help you understand what five different kinds of math are up to: (1) fractions, decimals, percents, ratios and proportions; (2) algebra; (3) geometry; (4) trigonometry; and (5) calculus. This list covers the kinds of math that many high schools teach.
Fractions, decimals, percents, ratios, and proportions
The following table shows the idea of 1 part in 8 parts as a fraction, decimal, percent, ratio, and in a proportion. They have a lot in common:
term |
numeral |
statement (with “=” sign) |
FRACTION |
1/8 |
1/8 = 2/16 |
DECIMAL |
.125 |
1/8=.125 |
PERCENT |
12.5% |
1/8=12.5% |
RATIO |
1:8 |
1/8=1:8 |
PROPORTION |
1:8 :: ??:16 |
1/8=x/16 |
Fractions, decimals, percents, ratios, and proportions are related ideas. And they are critical to understanding math.
Students who don’t get comfortable enough to think with ratios end their learning of math right there. They can’t understand algebra, geometry, or calculus. All these types of math require students to think with ratios and proportions.
Why do people find this stuff difficult? One answer is that it requires the mind to do something unusual: to compare two things as if they were one thing (1/8 is a comparison, but it is also one number that we call one-eighth). And then, even harder, we have to do things with that comparison.
Helping kids learn to think well with ratios is worth the effort. Every realm of ordinary life uses them: gardening, cooking, driving, house-cleaning—anything that involves mixtures. Lots of jobs require ratio thinking, too: construction, nursing, teaching, engineering, and accounting.
When you have time, check out these online resources about ratios and proportions:
The first video shows how common and important ratios are in everyday life:
The second video uses ratios to play tricks on everyday life:
And once again, we can turn to Wikipedia for more detail: http://en.wikipedia.org/wiki/Ratio
Algebra
What is algebra? You may have heard that “algebra is arithmetic with letters.” That statement says what algebra looks like, but what does algebra mean? Now, we already know something important just from this toolkit: you have to think with proportions to deal with algebra.
Algebra requires people to view math not as a web of facts (e.g., 7 times 8 = 56; 56 divided by 7 = 8) but as a web of statements that include unknowns. Algebra is all about using what’s known to figure out the unknown. It is also about learning to do accurate calculations even when there’s no way to make some unknowns, known! And in this way, algebra is even more useful than arithmetic.
As we mentioned in the section above, ratio thinking is difficult because it requires you to treat a comparison (two things) as if they were one thing, and then think that way about other things combined in the same and different ways (wow): dividing fractions by fractions, for instance, is part of arithmetic. Algebra is difficult because it combines ratio thinking with webs of statements that include both known values and unknown values. The unknowns are probably the most difficult part, as explained next. What are “unknowns”?
The unknowns are all the x’s and y’s that appear in algebra. The equations in algebra are statements that combine actual numbers (for instance, 7, ¾, 3.14159) with symbols that stand for something that is a number, but one that is unknown. Here are several such statements:
73 = 2x + 3
y = 5x – 2
y = x^{2} – 4x + 2
You might be able to find the value of x in the first equation, but in the other two, something more complicated is going on. Don’t worry about that, but just observe that the first equation looks a lot like arithmetic, but not the other two!
Algebra has developed methods for finding out what the unknowns are. Here’s a simple statement with an unknown: 7x = 56 (“7x” is just a different way—the way of algebra—of writing “7 times x”). But once algebra starts working with unknowns, things get complicated pretty fast. Here are some examples: x^{2} + y^{2} = 1; y = 3x^{2} + 2x + 6. All of that looks like algebra: arithmetic with letters. But it takes a full year of hard work to feel comfortable thinking about, and with, statements that have unknowns. Algebra is really the beginning of “rocket science”—in the sense that we all mean: exact reasoning with computations.
Algebra is the foundation for many, many fields. But it’s true that only some activities and work assignments allow people to make use of it in everyday life. Whatever its immediate applicability, algebra helps people become more flexible thinkers—so long as they keep what algebra means in mind.
One explanation that contrasts algebra and arithmetic and explains why algebra is different (and harder) is given by National Public Radio’s Keith Devlin, “the math guy:”
- First, algebra involves thinking logically rather than numerically.
- In arithmetic you reason (calculate) with numbers; in algebra you reason (logically) about numbers.
- Arithmetic involves quantitative reasoning with numbers; algebra involves qualitative reasoning about numbers.
- In arithmetic, you calculate a number by working with the numbers you are given; in algebra, you introduce a term for an unknown number and reason logically to determine its value. (from http://profkeithdevlin.org/2011/11/20/what-is-algebra/)
By the way, the strange-sounding word algebra comes from an Arabic word, al-jabr, the name of the methods of dealing with unknowns described by the Muhammad al-Khwarizmi, who worked around the year 800 in one of the then-great centers of learning: Baghdad.
Here are some amusing and interesting links that can help you consider more about what algebra means and does:
What is Algebra? (4 minutes, easy)
Understanding the Vocabulary of Algebra (concentrated content, but it’s only 3 minutes long)
¿Qué es el Álgebra? (What is Algebra?) (in Spanish: could be useful if you work with a student whose first language is Spanish, or if you speak Spanish yourself! It’s 8 minutes long.)
Algebra (Wikipedia article)
http://en.wikipedia.org/wiki/Algebra
Geometry
Do you remember that math is the science of number and space? Geometry is the “space” part. Humans (most of us) see and move in space, so we have a natural sense of what geometry (as compared to algebra) might be up to. The tricky part is the logic that makes geometry a system of mathematical thinking.
In fact, geometry is one of the oldest formal systems of mathematical thinking. And it remains an excellent model for such thinking. That model is what many consider the best thing about geometry. So it’s important to describe the model to you.
That famous and excellent old system was written down by Euclid 2,000 years ago. Euclid’s system starts from a small number of things just accepted as true (e.g., parallel lines don’t meet). These simple and self-evident things are called “axioms.” The system works with these few axioms to build many, many other statements in a very logical, step-by-step process. The logic is so careful that these many, many other statements are considered proven: true for all time and space.
Actually, geometry developed further after the 1500s, but Euclid’s care for logic, for selecting starting points (axioms), and his accomplishments allcontinue to inspire generations of people who have learned to love math. The idea of “proof” is essential to mathematics. Mathematical proof is usually about deductions from a few basic ideas to a conclusion that must be true if the original ideas are true. And Euclid was the one who established this pattern firmly.
Geometry is traditionally taught in U.S. schools after algebra. The reason is probably that unlike the way algebra is usually taught, geometry teaching often tries to put the idea of proof at the center of the learning experience: much as Euclid did. After all, his famous book, The Elements, was a textbook! So geometry is supposed to teach students one method of mathematical reasoning—reasoning about space. But an appreciation of the sort of logic that geometry uses also helps make students better thinkers in general, according to many educators.
The web has lots of resources on geometry:
What’s the point of geometry? (explains the spirit of “proof”)
The geometry rap? (2 minutes)
What is Euclidean Geometry? (12 minutes)
Geometry (Wikipedia article; lots of information about geometry after Euclid)
http://en.wikipedia.org/wiki/Geometry
Trigonometry
“Trigonometry” sounds a little like “geometry,” doesn’t it? (Another “-metry.”)
The similarities are deeper, too. Trigonometry has a single focus—the study of triangles, which, of course, geometry also deals with, but differently.
So what’s to study with triangles? Triangles have been traditionally useful in a very practical field: navigation. That just sounds weird, right?
But it turns out that the relationships of the length of sides and the degree of angles in triangles open up new worlds in math that aren’t original parts of algebra or geometry. Specifically the relations of sides in right-angled triangles define a new class of mathematical relationships: trigonometric functions. These functions have names you might have heard before: sine, cosine, and tangent, for instance. If you graph them, they make beautiful curved or wavy lines. They have the interesting property of repetition or oscillation—which other families of functions lack.
These trigonometric functions have many applications in science, engineering, and, of course, navigation—where they proved their usefulness to Europeans from about the year 1500 onward. Are there everyday uses and applications? Perhaps not in cooking and cleaning and mowing the lawn, not exactly, but they are useful in just about everything that’s in your house: electric wiring and the power grid, televisions and computers, motors, the placement of satellite dishes, and so forth. Come to think of it, though, the manufacture of cooking and cleaning items—and the lawn mower—probably do involve trigonometry. Web resources follow:
What is Trigonometry? (a little bit more technical than the text above)
How to Learn Trigonometry (it’s math content, but gives an easy feel for how trig works)
http://betterexplained.com/articles/intuitive-trigonometry/
Trigonometry (Wikipedia article: it’s short and to the point, but more detailed than the others)
http://en.wikipedia.org/wiki/Trigonometry
Calculus
Calculus is usually the most “advanced” math taught in high schools. So the very word scares lots of people. Like ratios and like algebra, and like geometric proof, calculus pushes students to think differently again. It has helped stretch the minds of generations of students. Like the rest of the math taught in high school, it was invented a long time ago (about the year 1700).
Calculus deals with change. One of its major contributions is the idea of the “limit.”
Web resources:
What is Calculus? (90 seconds: good grief)
Calculus with Confidence (10 minutes: not much math, but very sensible)
Calculus (Wikipedia article; mathy, but you will understand a lot anyhow)
http://en.wikipedia.org/wiki/Calculus